Computation of the quark-line disconnected part of the hadronic vacuum polarization of the muon

The anomalous magnetic moment of the muon, au, is one of the most precisely measured quantities in Nature. It can also be calculated to a very high precision theoretically in the framework of the Standard Model. Interestingly, there is a persistent deviation of the theoretical value from the experimentally measured value at the level of of about three standard deviations. This discrepancy is a possible indication of the incompleteness of the Standard Model and sign of undiscovered, fundamental physical processes. Thus, it is of paramount importance to sharpen or resolve this uncertainty. On the experimental side, a new experiment is underway at the Fermi National Accelerator Laboratory with the aim of reducing the experimental uncertainty by a factor of four. A comparable reduction of the theoretical uncertainty is much desired. The anomalous magnetic moment receives contributions from all sectors of the Standard Model, including the electromagnetic, weak, and strong interactions. They have all been calculated with varying degrees of uncertainty. The largest theoretical uncertainty comes from the contribution from the strong-interaction sector, as described by quantum chromodynamics. In our study, we contribute to the improvement of the theoretical prediction using a numerical method based on lattice quantum chromodynamics (LQCD). In LQCD, this contribution can be expressed as a sum of two terms: a quark-line-connected part and a quark-line-disconnected part. We focus on the latter. We rst motivate our study, provide the theoretical background, review the theoretical calculation, brie y discuss the experiments, describe the algorithm and parameter tuning, and give the final result.

COMPUTATION OF THE QUARK-LINE DISCONNECTED PART OF THE HADRONIC VACUUM POLARIZATION OF THE MUON by Shuhei Yamamoto A dissertation submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Department of Physics and Astronomy The University of Utah December 2019 Copyright c Shuhei Yamamoto 2019 All Rights Reserved The University of Utah Graduate School STATEMENT OF DISSERTATION APPROVAL The dissertation of Shuhei Yamamoto has been approved by the following supervisory committee members: Carleton DeTar , Chair(s) 06 June 2019 Date Approved Pearl Sandick , Member 06 June 2019 Date Approved Dmytro Pesin , Member Date Approved Douglas Bergman , Member 06 June 2019 Date Approved Michael Kirby , Member 06 June 2019 Date Approved by Peter Trapa , Chair/Dean of the Department/College/School of Physics and Astronomy and by David B. Kieda , Dean of The Graduate School. ABSTRACT The anomalous magnetic moment of the muon, aµ , is one of the most precisely measured quantities in Nature. It can also be calculated to a very high precision theoretically in the framework of the Standard Model. Interestingly, there is a persistent deviation of the theoretical value from the experimentally measured value at the level of of about three standard deviations. This discrepancy is a possible indication of the incompleteness of the Standard Model and sign of undiscovered, fundamental physical processes. Thus, it is of paramount importance to sharpen or resolve this uncertainty. On the experimental side, a new experiment is underway at the Fermi National Accelerator Laboratory with the aim of reducing the experimental uncertainty by a factor of four. A comparable reduction of the theoretical uncertainty is much desired. The anomalous magnetic moment receives contributions from all sectors of the Standard Model, including the electromagnetic, weak, and strong interactions. They have all been calculated with varying degrees of uncertainty. The largest theoretical uncertainty comes from the contribution from the strong-interaction sector, as described by quantum chromodynamics. In our study, we contribute to the improvement of the theoretical prediction using a numerical method based on lattice quantum chromodynamics (LQCD). In LQCD, this contribution can be expressed as a sum of two terms: a quark-line-connected part and a quark-line-disconnected part. We focus on the latter. We first motivate our study, provide the theoretical background, review the theoretical calculation, briefly discuss the experiments, describe the algorithm and parameter tuning, and give the final result. To my late grandfather, Kikuo Yamamoto CONTENTS ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv CHAPTERS 1. MOTIVATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1 Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Anomalous Magnetic Moment of the Muon . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.1 Magnetic moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.2 Anomalous magnetic moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.3 New-physics effects in aµ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.4 Derivation of aµ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3. REVIEW OF STANDARD MODEL COMPUTATION OF aµ . . . . . . . . 14 3.1 QED Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 One-loop order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Two-loop order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Three-loop order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Four-loop order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Five-loop order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.6 Total aQED ............................................... µ 3.2 EW Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 One-loop order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Two-loop order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Total aEW ................................................ µ 3.3 Hadronic Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 LOHVP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 NLOHVP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 LOHLbL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 NNLOHVP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 NLOHLbL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. 15 16 16 18 20 24 25 26 27 28 31 31 31 44 44 46 46 47 EXPERIMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.1 Basic Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.2 Decay Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.3 E821 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.4 E989 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5. LATTICE GAUGE THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.1 Lattice Gauge Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Symanzik improvement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1.1 Perturbative approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1.2 Tadpole improvement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Fermion Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Naive fermion action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Doubling problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Staggered fermion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Taste exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 HISQ action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Monte Carlo integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1.1 Fermionic determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1.2 Hamiltonian Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1.3 Metropolis method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Molecular dynamics simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Rational Hybrid Monte Carlo method . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3.1 Hybrid molecular dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3.2 Hybrid Monte Carlo method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3.3 Rational hybrid Monte Carlo method . . . . . . . . . . . . . . . . . . . . . . . 6. 69 74 74 77 79 79 80 81 82 84 89 89 90 92 93 94 96 97 98 98 LATTICE COMPUTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.1 Hadronic Vacuum Polarization (HVP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 HVP Disconnected Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Calculation of the current density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Deflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Truncated solver method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Dilution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 Calculating the current correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Parameter Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Determination of εgoal corr and εfine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Determination of εslp and R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 Determination of Nslp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.5 Determination of Nev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.6 Determination of εev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 108 110 110 115 118 122 123 126 126 130 132 135 137 140 146 152 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 vi LIST OF FIGURES 2.1 The diagram for the lowest-order radiative correction, called Schwinger term, that corrects Dirac’s tree-level value of ge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 The Feynman diagrams for the contribution to aµ from possible leading-order new physics effects. Diagram (a) shows the diagram for the contribution from new states of mass MN P . Diagram (b) shows the diagram for the contribution from the new fermion of mass MN P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.1 The Feynman diagrams for the leading-order term from each category of contributions listed in Eq. (3.1) [15]. On the upper row are the leading QED diagram, the Schwinger term, the diagram for the leading-order hadronic contribution, and the diagram for the leading-order light-by-light contribution. On the bottom row are the leading contributions from EW sector. . . . . . . . . . 15 3.2 The second-order diagrams for the QED contributions [15]. The diagrams from (1) to (7) correspond to the universal part. The diagrams (8) and (9) represent the mass-dependent part. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature The Anomalous Magnetic Moment of the Muon by Friedrich Jegerlehner Copyright by Springer International Publishing AG (2008) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 The Feynman diagrams at the three-loop order from the QED sector [15]. The diagrams for universal part can be obtained by assuming all the lines to be muon lines. The nonuniversal part can be obtained by replacing at least one closed muon loop by the lepton loop of another flavor. The diagrams from (1) to (6) are classified as QED light-by-light graphs, and the ones from (7) to (22) contain the photon vacuum polarization. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature The Anomalous Magnetic Moment of the Muon by Friedrich Jegerlehner Copyright by Springer International Publishing AG (2008) . . . . . . . . . . . . . . . . . . . . . . . . 19 3.4 Some of the typical QED Feynman diagrams at the fourth-order in α [15]. The number in the bracket indicates how many diagrams of the given type there are. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature The Anomalous Magnetic Moment of the Muon by Friedrich Jegerlehner Copyright by Springer International Publishing AG (2008) 20 3.5 Some of typical diagrams in Group I [15]. li indicates the lepton inserted to form the vacuum polarizing closed loop into the internal photon line. The subgroup Ia contains 7 diagrams, Ib 18, Ic 9, and Id 15 [27]. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature The Anomalous Magnetic Moment of the Muon by Friedrich Jegerlehner Copyright by Springer International Publishing AG (2008) . . . . . . . . . . . . . . . 22 3.6 Some of typical diagrams belonging to Group II [15]. The number in the circle represents the order in e of the inserted closed lepton loop [27]. So 2 indicates the one-loop subdiagram, and 4 two-loop. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature The Anomalous Magnetic Moment of the Muon by Friedrich Jegerlehner Copyright by Springer International Publishing AG (2008) . . . . . . . . . . . . . . . . . . . . . . . . 22 3.7 Some of typical diagrams belonging to Group III [15]. The number in the circle represents the order in e of the inserted closed lepton loop as in Fig. 3.6 [27]. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature The Anomalous Magnetic Moment of the Muon by Friedrich Jegerlehner Copyright by Springer International Publishing AG (2008) 23 3.8 Some typical diagrams belonging to Group IV [15]. The number in the circle represents the order in e of the inserted closed lepton loop as in Fig. 3.6 [27]. The subgroup IVa contains 54 diagrams, IVb 60 diagrams, IVc 48 diagrams, and IVd 18 diagrams. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature The Anomalous Magnetic Moment of the Muon by Friedrich Jegerlehner Copyright by Springer International Publishing AG (2008) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.9 The leading-order Feynman diagrams from the EW sector. . . . . . . . . . . . . . . . 27 3.10 Sample diagrams for EW contributions at two-loop order. . . . . . . . . . . . . . . . . 28 3.11 Two major contribution from the hadronic sector. . . . . . . . . . . . . . . . . . . . . . . 31 3.12 Hadronic part of iΠ2 , which is the sum of all possible QCD intermediate states. It can be written in terms of the quark-line current-current correlation function as is indicated in the figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.13 The diagram corresponding to σ0 (e+ e− → γ ∗ → hadrons). The virtual photon from the vertex to hadronic blob is the tree-order photon propagator. q represents the quark, and arrows on the right of the hadronic blob indicate the various hadronic products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.14 The compiled data for R-ratio with E from 0 to 5 GeV [132]. “The compilation of R(s) data utilized” by F. Jegerlehner published in Acta Physica Polônia B is licensed under CC BY 4.0 / Desaturated from original. . . . . . . . . . . . . . . . . 37 3.15 The compiled data for R-ratio with E from 5 to 13 GeV [132]. “The compilation of R(s)-data utilized” by F. Jegerlehner published in Acta Physica Polônia B is licensed under CC BY 4.0 / Desaturated from original. . . . . . . . . 37 3.16 The compilation of the experimental data for the modulus of the pion form factor, |Fπ |2 , near the π + π − → ρ resonance [132]. This figure by F. Jegerlehner published in Acta Physica Polônia B is licensed under CC BY 4.0 / Desaturated from original. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 viii 3.17 A summary of the R-ratio results based on other data sets: DHMZ10 [170], JS11 [167], HLMNT11 [171], BDDJ15 [172], DHMZ16 [173], FJ17 [15, 168], DHea09 [166], BDDJ12 [174], KNT18 [150] [132]. The shaded box in blue emphasizes the value cited in the above. The experimental value, aexp is µ indicated by the vertical band. The grey vertical band shows the current experimental error, and the narrower band inside the goal uncertainty for the E989 experiment [132]. Two of the estimates are computed without Initial State Radiation (ISR) method [175, 176]. The listed values depend on what value to use for HLbL contribution. JS11 and BDDJ13 uses 116 (39) × 10−11 [125]. Others use 105 (26) × 10−11 [177]. FJ17 uses τ spectral data [167] and ππ scattering phase-shift data [178]. This figure by F. Jegerlehner published in Acta Physica Polônia B is licensed under CC BY 4.0. . . . . . . . . . . . . . . . . . 42 3.18 The next-to-leading order contributions from the hadronic sector [205]. The shaded blob represents the HVP, and loops with no correction, lepton loops. The figures by Alexander Kurz, Tao Liu, Peter Marquard, and Matthias Steinhauser are licensed under CC BY 3.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.19 The diagrams for the hadronic light-by-light contribution at the next-to leading order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.20 The diagrams contributing to aµ at the next-to-next-to leading order from hadronic sector [128]. As before, the shaded blob represents the HVP, and unshaded blob a lepton loop. The figures by Alexander Kurz, Tao Liu, Peter Marquard, and Matthias Steinhauser are licensed under CC BY 3.0. . . . . . . . . 46 3.21 An example of the NLO HLbL diagram that is sufficiently suppressed at the present desired theoretical uncertainty goal. The loop correcting an internal photon line is formed by a lepton. Other possibilities for a similar correction are not shown here. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.1 A diagram for decay modes of the pion in the pion’s rest frame. C represents the charge conjugation operator, and P, the parity operator. The black one-sided arrow indicates the momentum, and the blue arrow indicates the direction of the spin. The crossed-out decay mode is forbidden. . . . . . . . . . . . . 53 4.2 Illustration of muon decay in muon’s rest frame [15]. White arrows indicate the direction of the spin. Arrows of other colors indicate the direction of the momentum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.3 The distribution of the relative number and asymmetry number as a function of y. (a) shows the plots of the functions in the Center-of-Mass frame. (b) shows those in the laboratory frame [235]. Reprinted figure with permission from G. W. Bennett et al. (Muon g-2 Collaboration), Phys. Rev. D 73, 072003, 2006. Copyright (2006) by the American Physical Society. https://doi.org/ 10.1103/PhysRevD.73.072003. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ix 4.4 The schematic diagram of the top view of the experimental facility at BNL for measuring muon g − 2 [235]. Here, Q indicates quadrupoles, K collimators, and D magnetic dipoles [235]. Reprinted figure with permission from G. W. Bennett et al. (Muon g-2 Collaboration), Phys. Rev. D 73, 072003, 2006. Copyright (2006) by the American Physical Society. https: //doi.org/10.1103/PhysRevD.73.072003. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.5 The layout of the muon storage ring. The numbers from 1 to 24 label the twenty-four calorimeters, Q indicates the electric quadrupole section, K, the kicker plates, and C, full-aperture and 12 C the half-aperture calorimeters [235]. The kicker pushes the muons to a stable orbit concentric to the storage ring by a fast, nonferric, pulsed magnetic kicker [237]. Reprinted figure with permission from G. W. Bennett et al. (Muon g-2 Collaboration), Phys. Rev. D 73, 072003, 2006. Copyright (2006) by the American Physical Society. https: //doi.org/10.1103/PhysRevD.73.072003. . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.6 The schematic diagram for the measurement of the decayed positrons [15]. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature The Anomalous Magnetic Moment of the Muon by Friedrich Jegerlehner Copyright by Springer International Publishing AG (2008). . . . . . 59 4.7 A plot of the number of decayed electrons as a function of the time [235]. These data are an accumulation of 3.6 × 109 muon decays in the R01µ− data-taking period [235]. These data are wrapped around, modulo 100 µ s. Reprinted figure with permission from G. W. Bennett et al. (Muon g-2 Collaboration), Phys. Rev. D 73, 072003, 2006. Copyright (2006) by the American Physical Society. https://doi.org/10.1103/PhysRevD.73.072003. . . . . . . . . . . . . . . . 59 5.1 Three loops of perimeter six composed of gauge links. . . . . . . . . . . . . . . . . . . . 76 5.2 A typical tadpole diagram. This contributes to a quark self-energy. . . . . . . . . . 78 5.3 The leading tree-level taste exchange interaction involving highly virtual gluon of the momentum ζπ/a [261]. q indicates a quark, q̄ antiquark, and g gluon. The momentum of each quark is indicated at the tip of the line. Reprinted figure with permission from E. Follana, Q. Mason, C. Davies, K. Hornbostel, G. P. Lepage, J. Shigemitsu, H. Trottier, and K. Wong, Phys. Rev. D 75, 054502, 2007. Copyright (2007) by the American Physical Society. https: //doi.org/10.1103/PhysRevD.75.054502. . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.4 q q̄ → q q̄ taste-exchange diagrams at one-loop order [261]. They are of the order O(αs2 (pa)2 ) [261]. Other diagrams with additional external quarks are suppressed by higher powers of pa [261]. Reprinted figure with permission from E. Follana, Q. Mason, C. Davies, K. Hornbostel, G. P. Lepage, J. Shigemitsu, H. Trottier, and K. Wong, Phys. Rev. D 75, 054502, 2007. Copyright (2007) by the American Physical Society. https://doi.org/10.1103/PhysRevD. 75.054502. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.1 The Feynmann diagram for the leading-order HVP contribution to aµ . The bubble labeled by Hadronic contains all the QCD radiative corrections but not corrections from other sectors of the SM. . . . . . . . . . . . . . . . . . . . . . . . . . 103 x 6.2 The Feynman diagrams for the contribution to aµ from the leading-order HVP. The quark loop in (a) is radiatively corrected by virtual gluons and sea quarks (not shown). In the same way, the quark loops of flavor f and f 0 in (b) are connected by virtual gluons and sea quarks (not shown). . . . . . . . . . . . . . . . . . 106 6.3 The value of the time-slice correlation function Cdisc (t) on one gauge configuration at t = 1 as a function of the inverse random source block size. Also shown is the extrapolation to infinite block size. The extrapolated value is plotted in blue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.4 The point-to-point correlation function C(r) over a single gauge configuration with Nev = 350, εev = 10−9 , εslp = 2.70 × 10−2 , εfine = 10−5 , Nslp = 1408, and Nfine = 72. The values for C(r) are plotted against the distance |r|. Also note that beyond |r| > 5, the data are binned. Uncertainties arise from the stochastic estimation of the current density and from gauge fluctuations in the context of a single configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.5 Average change ∆C(r) in the point-to-point correlation function C(r) as defined in Eq. (6.22) as a function of the choice of the solver residual εfine ranging from 10−1 to 5 × 10−7 . The change is measured relative to the fiducial values of C(r) defined at εfine = 10−7 . Each point corresponds to a displacement. The cross indicates the average absolute value. The blue region indicates a less-than-1% error. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.6 Same as Fig. 6.5 but for Nev = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 6.7 The average r̄ of the correlation in the current densities between fine and sloppy solves as a function of the sloppy residual εslp defined in Eq. (6.17) for various degrees of deflation. Curves are fits based on Eq. (6.23). . . . . . . . . . . . 133 6.8 The number of CG iterations ninv required to reach a given residual for selected Nev and quark flavors. The deflation has very little effect on the charm and strange CG iteration number. So these values are small and almost constant. The curve fit was made for the range (10−1 , εfine = 10−5 ) . . . . . . . . . . . . . . . . . 134 6.9 The dependence of the average square of the statistical errors in the correlation C(r), ε2corr , as a function of the number of sloppy stochastic sources, Nslp . The number of eigenvectors used was Nev = 350. . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.10 The dependence of the cost of inversion Cinv and the cost of deflation Cproj in minutes as a function of Nslp . The legend in each plot shows a fit of the data to a linear function going through the origin, ax. . . . . . . . . . . . . . . . . . . . . . . . 137 6.11 The dependence of the cost of generation of eigenpairs Cev and the estimated est as a function of N . The total estimated computer time cost of inversion Cinv ev est = C est + C Ctot is plotted with green symbols. Also shown is the measured ev inv total computer time Ctot as a function of Nev in the identified range, plotted with cross symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.12 Shifts of estimated eigenvectors vs. eigenvalue index for various eigensolver residuals εev . Shifts are measured relative to the fiducial values from εev = 10−15 .141 6.13 Shifts of estimated eigenvalues vs. index for various eigensolver residuals εev . Shifts are measured relative to the fiducial values from εev = 10−15 . . . . . . . . . 142 xi 6.14 The number of CG iterations required to reach a given residual of inversion for light quark for various values of εev . The number of deflated eigenvectors are fixed at 350. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.15 The dependence of log10 εcut on log10 εev . As deflation has negligible impact on the performance of inversion for εev < 10−3 , a linear fit was taken using only the values for εev ≥ 10−4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 6.16 The cost of inversion in minutes as a function of εev . . . . . . . . . . . . . . . . . . . . . 145 6.17 The computer time for Cinv and Cev as a function of the eigenresidual εev . The blue line shows Ctot at the preferred value of εev = 10−9 . The total computer time is plotted with green symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 6.18 The computer time Cev in minutes as a function of Nev for different values for εev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6.19 Time-slice disconnected current density correlator vs. the temporal separation in lattice units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 6.20 10 as a function of the cut-off time T in Eq. (6.24). A constant fit is adisc µ × 10 taken from T = 12 to 20. The shaded region indicates the error. . . . . . . . . . . . 149 6.21 The plot of −adisc vs. the lattice spacing a2 provided by Kotaroh Miura in µ private communication. Different fit lines are obtained using different cuts on a150 xii LIST OF TABLES (4) 3.1 The preliminary values of A1 reported in various proceedings and books [15, 25]. The list shows the progress of the numerical computation. . . . . . . . . . . . . 21 3.2 The summary of the numerical values for various subcontributions from each subgroup of mass-dependent part of aQED at fourth-loop order [23]. Note that µ these values are not based on the latest values of the mass ratios. Reprinted table with permission from Tatsumi Aoyama, Masashi Hayakawa, Toichiro Kinoshita, and Makiko Nio, Phys. Rev. Lett., 109, 111808, 2012. Copyright (2012) by the American Physical Society. https://doi.org/10.1103/ PhysRevLett.109.111808 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 The values of contributions to aµ from QED sector at each loop order and of aQED based on the two different determinations of α from Ref. [34]. These µ values are in units of 10−11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.4 SM The various leading contributions, acont µ , to aµ where LO means leading order, NLO next leading order, and LbL light-by-light in units of 1010 . . . . . . . . . . . . 48 4.1 The list of major sources of the systematic errors in measuring ωa . CBO stands for Coherent Betatron Oscillation, REFC, for radial electric field correction, and VPC for vertical pitch correction [238]. This table by Antoine Chapelain was licensed under CC BY 4.0/ Desaturated from original. . . . . . . . . . . . . . . . 61 5.1 Parameter values used in the simulation for generating lattice ensembles used in our study. The first column gives the approximate lattice spacing in fm. The second gives the size of each lattice. Ns indicate the spatial size in units of a, and Nt , the temporal size. 10/g 2 gives the value for the coupling constant. The masses are given in lattice units. ams and amc are equal or close to its physical value. u0 is a tadpole coefficient, which is explained in Section 5.1. It was sufficient to set it equal to 1 for the purpose of our study. N is the Naik term for the charm quark, the mass-dependent correction to the tree-level improvement of the charm quark dispersion relation, discussed in Section 5.2. 101 6.1 The values of the constants of the fits for ln(1 − r̄) = a ln εslp + b and ln εinv = cninv + d and εslp for various Nev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 ACKNOWLEDGMENTS Computation for this work was done using resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under Contract No. DE-AC02-05CH11231. This study was done with constant assistance from my Ph.D. advisor, Dr. Carleton DeTar. Also, it is a collaborative work of Fermilab, HPQCD, UKQCD, and MILC and conducted with the help of many collaborators, in particular, Ruth Van DeWater, Aida X. El-Khadra, Craig McNeile, Alejandro Vaquero, and Christine H. Davies. This research was in part supported by the U.S. National Science Foundation under a grant PHY17-19626 and by Teaching Assistantship and Summer Research Assistantship from The University of Utah. CHAPTER 1 MOTIVATION The Standard Model (SM) of particle physics describes the dynamics and the fundamental structure of Nature at the subatomic scale in the absence of gravity [1]. The model is formulated in the language of quantum field theory (QFT). At the subatomic scale, the fundamental degrees of freedom are matter fields and gauge fields, and they interact with each other via the gauge mechanism, i.e., by exchanging gauge bosons. There are two kinds of spin 1/2 matter particles, quarks and leptons, and three kinds of gauge forces, electromagnetic (EM), strong, and weak, relevant to particle phenomenology. Here, we consider the simplest SM with minimal couplings, a single elementary Higgs scalar, and no neutrino masses. The SM has achieved tremendous success and is well-established [2, 3]. It has successfully explained most phenomena on all experimentally accessible scales from the subatomic scale of elementary particles to the scale of the Universe. The Large Hadron Collider (LHC) in Switzerland explores the physics at the 13 TeV frontier [4], and all the precision measurements so far have not deviated from their SM predictions by more than 5σ [3, 5]. The SM has also been applied to cosmology and has helped us to explain the observed cosmological data such as the temperature of the cosmic microwave background based on the Big Bang theory [6]. It is the most comprehensive theory of Nature we have right now. Although the theory is highly successful, it is still incomplete. As it stands, the SM requires nineteen input parameters to be specified from experiment if we ignore neutrino masses [2] and twenty-six free parameters if we take into account nonzero neutrino masses and their mixing angles [7]. These parameters are not explained by the SM. Specifically, it cannot derive the masses of the elementary particles from the first principle nor explain the huge spread of the masses from the less-than 2 eV masses of neutrinos to 173 GeV mass of the top quark as well as the strange pattern of magnitudes of the mixing angles, 2 called “flavor puzzles” [8]. The mechanism of neutrino mass generation is also not part of the simplest SM and is not completely resolved. Gravity is not part of the SM either. Furthermore, the theory does not answer some questions such as the hierarchy problem, matter-antimatter asymmetry, and dark matter [3, 9–11]. So its extension to a more complete theory is much desired. These insufficiencies clearly show that the current form of the SM is conceptually unsatisfactory and moreover fails to capture some of the observed phenomena. Hints to go beyond the SM are strongly needed, and wide-ranging searches are underway. In this regard, researchers have found possible deviations of experimental measurements from the theoretical predictions such as the anomalies in B physics [12]. They are indications that may point toward physics beyond Standard Model. Another is the 3σ deviation of theoretical prediction for the anomalous magnetic moment of muon from its measured value [13]. This quantity may also give us a hint toward a new theory and is the focus of this dissertation. In this dissertation, the anomalous magnetic moment of the muon is studied. In particular, we compute the quark-line disconnected part of its leading-order hadronic contribution using the techniques from lattice quantum chromodynamics (LQCD). LQCD is a cut-off regularized QCD and provides a way to compute correlators in quantum field theory from first principles. The evaluation of the quantities is performed numerically, and with sufficient computer power, it is possible to estimate the values as precisely as desired. With this method in hand, we attempt to calculate the hadronic contribution to leading order in the electromagnetic interaction – a contribution that has the most significant uncertainties among various contributions to the anomalous magnetic moment. Our goal is to reduce this uncertainty. For this purpose, we use the highly-improved staggered quark (HISQ) formulation for the current density with gauge configurations generated with four flavors of HISQ sea quarks. The computation is performed by stochastic estimation of the current density using the truncated solver method combined with deflation of low-modes and dilution. The parameters are tuned to minimize the computational cost for a given target uncertainty in the current-current correlation function. After estimating the current density, the time-slice correlation function is constructed for various gauge configurations on a lattice of the given lattice spacing. The correlation function is used to evaluate the anomalous magnetic 3 moment at the lattice spacing. The simulation was run on lattices of different lattice spacing, and extrapolation to zero lattice spacing is taken to find the value at the continuum. This dissertation is organized as follows. In Chapter 2, we provide theoretical background necessary to understand how the anomalous magnetic moment of the muon, aµ , is theoretically computed. Then, in Chapter 3, we review how each contribution to aµ has been computed so far. Chapter 4 gives an overview of the experimental measurement. Chapter 5 explains the theoretical framework employed in this project, namely LQCD. Finally, in Chapter 6, we provide details of the methods we have employed to compute aµ with the future outlook and conclusion in Section 6.6. CHAPTER 2 THEORY In this chapter, we first briefly describe the SM in Section 2.1. Then, we define the anomalous magnetic moment along with its history and show how it can be computed using the QFT in Section 2.2.4. 2.1 Standard Model Mathematically, the SM is an SU (3) × SU (2)L × U (1) gauge theory [1, 14]. As is mentioned in Chapter 1, there are three forces treated in the SM, and they are related to their respective gauge symmetries. The gauge symmetries are mathematically described by gauge groups. These gauge groups arise from the respective global symmetry groups. Here, SU (3) describes a global internal color symmetry, and SU (2)L × U (1), the symmetries in the electroweak sector related to the EM and weak forces. Often, the part of the SM related to the strong force is called quantum chromodyamics (QCD) sector, and the part related to the EM force, the quantum electrodynamics (QED) sector. The gauge fields associated with these gauge groups are denoted, respectively, by G, W, and B [14]. The G-field couples to the color charge, the W -field to the weak isospin, and the B-field to the weak hypercharge [14]. After the spontaneous breaking of symmetry, the remaining symmetries and corresponding gauge groups are associated with the electric charge and weak isospin [14]. The matter fields transform under these gauge groups, forming a representation space at each point in the space-time. The subscript, L, in SU (2)L indicates that the elements in this group act only on the left-handed part of the matter fields. The right-handed part transforms as a singlet. As there are known to be six flavors of quarks and leptons and these six flavors are grouped into three generations consisting of two flavors each, we can collectively write the matter fields as follows: 5 νei  ei i L =  u di L  ψLi = i lL qLi  i νe,R  i  i   , ψR =  eR    ui  R diR   where i = 1, 2, 3 labels the generation of fermions. The left-handed lepton and quark tuples emphasize that their components mix under SU (2)L . Finally, the Higgs fields give rise to mass via Higgs mechanism. They are given by + φ Φ= , Φ̃ = iτ 2 Φ∗ φ0 where τ i is ith Pauli matrix. Thus, the SM Lagrangian that contains its dynamics and remains invariant under SU (3) × SU (2)L × U (1) gauge transformations is 1 1 1 LSM = − trGµν Gµν − trWµν Wµν − Bµν B µν 2 2 4 f f f f / B ψR / BW ψL + ψ̄R iD + ψ̄L iD / Gqf + q̄ f iD (2.1) µ Φ − V (Φ) + (DµBW Φ)∗ DBW f f f ΦefR + h.c − ge ¯lL Φ̃νe,R − gu q̄Lf Φ̃ufR − gd q̄Lf ΦdfR − gν ¯lL where the g’s represent coupling constants. The covariant derivative operators in Eq. (2.1) are defined by DB,µ = ∂µ − igB YW Bµ i DBW,µ = ∂µ − igB YW Bµ + igW TW Wµi DG,µ = ∂µ − igG λa a G 2 µ i = τ i /2, the weak with YW , the weak hypercharge generator associated with U (1) and TW isospin generator associated with SU (2)L . They are related by [1] 3 YW = Q − TW where Q is the electric charge generator. Predictions are made via Green’s functions in QFT. In this dissertation, we adopt the functional-integral formulation of QFT. So we start from the classical fields and the classical Lagrangian and then move to the quantum theory via path-integration over classical fields. 6 In this approach, fermionic fields are represented by Grassmann variables in order to take into account the anticommuting nature of the fermionic fields. Details on path-integral formalism and Grassman calculus can be found in Ref. [1]. The Green’s functions can be generated from the generating functional, Z[J] = lim lim hΩ; t | Ω; −tiJ →0 t→∞(1−i ) RQ Q Q DψDψ̄ DA DΦDΦ̄eiS[ψ,A,Φ,J] Q Q = RQ . DψDψ̄ DA DΦDΦ̄eiS[ψ,A,Φ] (2.2) (2.3) where ψ collectively denotes the fermionic fields, A, the gauge fields, Φ, the Higgs fields, J, the external sources for all the fields, and |Ω, tiJ the interacting physical vacuum in the presence of external currents J. S[ψ, A, Φ, J] can be decomposed into S[ψ, A, Φ] and the sum of products of the external source and its associated field as follows: Z S[ψ, A, Φ, J] = d4 xL[ψ, A, Φ, J] = Z d4 xLSM + Z d4 LJ = S[ψ, A, Φ] + Z d4 LJ where LJ = J¯f ψ + ψ̄Jf + AJA + ΦJΦ . Here, Jf collectively represents the external sources for fermions, JA , for gauge bosons, and JΦ , for Higgs fields. Since ψ’s are anticommuting Grassmann variables, their associated currents are also Grassmann variables. Now, let us form an observable corresponding to O(x1 , · · · , xn ) by putting together field operators in such a way that it is a function of φi ’s that is gauge-invariant where φi is now a generic field operator. The Green’s function for O is then G(n) (x1 , · · · , xn ) ≡ hT Oi = 1 δZ[J] n i δJ1 (x1 ) · · · δJn (xn ) J=0 where T is a time-ordering operator, Ji represents the external current associated with a field φi , hOi the expectation value of the observable O, and δ/δJi (xi ) is a functional derivative with respect to Ji (xi ). Note that if the ith external current Ji is associated with a fermionic field, ψ, then the corresponding derivative becomes δ δ =− ¯ δJi δ Jψ where Jψ is the external current associated with ψ. 7 Green’s functions can be related to the scattering amplitudes via the LSZ reduction formula. The scattering amplitudes are the elements of the scattering matrix S. In momentum space, it is defined as [1] in h{p}i i | |{k}i iout ≡ h{p}i | S |{k}i i where {k}i is a set of incoming momenta, {p}i a set of outgoing momenta, |{k}i iin a state at the infinite past with the momenta ki ’s, |{p}i iout a state of the momenta pi ’s at the infinite future, and |{k}i i and |{p}i i are the states at some common time. We generally write S as S = 1 + iT (2.4) where T is called the transition matrix. The square of the scattering amplitude gives the probability of an initial state scattering into a final state. We can use this to compute the scattering cross section, an important observable in particle physics. The details can be found in Ref. [1]. 2.2 Anomalous Magnetic Moment of the Muon This section loosely follows Ref. [15]. The muon (µ− ) is a lepton with the same electric charge and spin as the electron (e− ), but comes with a mass of 105.658 MeV, whereas the mass of the electron is 0.511 MeV. This difference in mass leads to a difference in lifetime. As the lightest lepton, the electron is stable on a time scale of the age of the Universe, while the muon is unstable and decays. Its lifetime is τµ = 2.197019(21) × 10−6 s. Due to its stability, the properties of the electron can be probed very precisely by experiment, but the muon properties can also be studied with extreme precision. The anomalous magnetic moment of the muon is one of such properties that are investigated both theoretically and experimentally with great precision. In what follows, we introduce how it is defined, explain why we study it, and show how we derive it using the QFT. We begin with the magnetic moment. 2.2.1 Magnetic moment For a circulating particle of charge q and mass m, its orbital magnetic dipole moment is given classically by q q L µL = gL r × v = gL 2 2m 8 where L = mr × v, and gL is an orbital g-factor. For an electron, q = e = −|e| with e being the charge of the electron, and gL = 1. The particle with intrinsic spin S also comes with a spin magnetic dipole moment and is given analogously by µS = gS q S 2m where gS is the spin g-factor. The magnetic dipole interacts with magnetic fields and gives rise to a contribution to the energy of the muon Hm = −µS · B. (2.5) We also define gl to be the spin g-factor for leptons l = e, µ, τ . In 1922, Stern and Gerlach conducted an experiment in which a beam of silver atoms was passed through the region of non-uniform magnetic field [16]. Such non-uniform magnetic fields create a force of F = −∇ (Hm ) = ∇(µS · B). Since the atom is electrically neutral, the spin effect dominates. In addition, the silver atom has a single 5s electron below which are fully filled shells so that the effect of magnetic moment of the 5s electron is exposed. The beam of silver atoms splits into two. This indicates that the spin is quantized, and the length of the split is a measure of the g-factor of the electron. The Stern-Gerlach experiment measured it to be ge = 2.0(2) in 1924. Later in 1928, Dirac showed theoretically that ge is exactly 2 based on relativistic quantum mechanics. His prediction was consistent with the experimental measurement within the experimental uncertainty, albeit large. Only twenty years later did experiment indicate that there was a slight deviation of the value for ge from Dirac’s prediction of 2. This is the topic of Section 2.2.2. 2.2.2 Anomalous magnetic moment The first precision measurement of ge reporting the deviation of ge from Dirac’s prediction of 2 was conducted in 1948 by Kusch and Foley via the study of the Zeeman spectrum of gallium by measuring the Lande g-factor, gJ [17]. The Zeeman spectrum is a manifestation of the Zeeman effect, which is induced by the interaction term in Eq. (2.5): D E q ∆E = − (L + gl S) · B = −gJ µ0 mj B 2mc where µ0 = q~/2mc is the Bohr magneton and mj is the quantum number for Jz = Lz + Sz . Set up the coordinate axes so that B = B ẑ. Then, we can relate gJ to gl as follows [15]: 9 h(L + gl S) · Bi = j(j + 1) − l(l − 1) + s(s + 1) mj ~B. 1 + (gl − 1) 2j(j + 1) So the measurement of gJ for the electron leads to the value for ge , which they determined to be ge = 2.00229 (8). This value is slightly different from Dirac’s value of ge , namely 2. In the same year, Schwinger calculated the lowest-order radiative correction to ge using quantum field theory and found [18] ge = 2 + α/π = 2.00232 · · · . This correction is the simplest radiative correction and is called the Schwinger term. The associated Feynman diagram is shown in Fig. 2.1. So the g-factor was found to deviate from Dirac’s prediction of 2, and the deviation is entirely due to the vertex radiative corrections, the simplest of which Schwinger computed. That is, the deviation is a purely quantum field theoretic effect. The difference is called the anomalous magnetic moment and quantified by al = 2.2.3 gl − 2 2 (l = e, µ, τ ). New-physics effects in aµ As is pointed out in Ref. [19], the sensitivity to new physics (NP) showing up at the scale ΛN P increases as m2l /Λ2N P where ΛN P here is a UV cut-off characterizing the scale of new physics. To see this, suppose that there are new states of mass MN P ∼ ΛN P mµ not part Figure 2.1: The diagram for the lowest-order radiative correction, called Schwinger term, that corrects Dirac’s tree-level value of ge . 10 of the SM. The lowest-order potential contributions from these supposed new states have Feynman diagrams shown in Fig. 2.2. Note that the diagram in Fig. 2.2a looks identical to the lowest-order weak contribution. So the general form of the contribution from the supposed new bosonic states of this type can be written in a manner similar to the W -boson contribution as [15] P aN = Cb l m2l 2 MN P where Cb is a constant. If we assume the simplest bosonic NP interaction suggested in Fig. 2.2a, Cb would be of the order O(α/π) where α is the fine-structure constant. New physics interactions could be more intricate and of higher order in α. If there is a new fermionic state of mass MN P , then its potential leading-order contribution has a diagram shown in Fig. 2.2b, which is analogous to the diagram for the τ -lepton loop contribution, and so the contribution from the new fermionic state can be written as a l = Cf m2l 2 MN P where we would expect Cf to be of the order O((α/π)2 ) or higher in α. So if there are contributions from NP of those forms not accounted for by the SM, the difference between the experimental and theoretical value for al is given by ∝ al − aSM l m2l . Λ2N P Thus, this quantity provides an opportunity for a precision test of the SM. Here, note that due to m2l in the expression, heavier particles are more sensitive to physics at a higher energy scale. This means the heavier leptons are more sensitive to the effect of new physics. (a) W -boson-like NP Contribution (b) NP Fermion Contribution Figure 2.2: The Feynman diagrams for the contribution to aµ from possible leading-order new physics effects. Diagram (a) shows the diagram for the contribution from new states of mass MN P . Diagram (b) shows the diagram for the contribution from the new fermion of mass MN P . 11 With this regard, τ lepton, which has the same charge as the electron but is much heavier with a mass of about 1777 MeV, is the best choice as a probe of NP. However, at present, due to its short life-time, it is challenging to make measurement of the anomalous magnetic moment of the τ lepton. So we turn to the second best choice, the muon. As for the electron, its anomalous magnetic moment is mostly accounted for by QED effects. In the case of the electron, contributions from other sectors account for only 2.26 parts per billion of the total theoretical value for ae [20], meaning that it is not very sensitive to QCD and weak effects, not to mention the effects of new physics. However, since (mµ /me )2 ∼ 4 × 104 , the possibility of detecting a deviation from SM prediction is higher for the muon. Furthermore, we can measure the anomalous magnetic moment of the muon to a high precision. This enables us to explore the effects of physics beyond the SM by probing the anomalous magnetic moment of the muon. In fact, we currently observe a deviation of about 3σ to 4σ between the theoretical prediction and experimental measurement, depending on which theoretical calculation is selected. The goal is to resolve or sharpen the discrepancy by increasing the precision of both theoretical and experimental value of the quantity beyond 5σ. 2.2.4 Derivation of aµ In this subsection, we derive the QFT expression for aµ . For this purpose, we first consider the scattering amplitude of the muon by a classical background EM field. Then, we take its nonrelativistic limit to reveal the connection of the amplitude to the anomalous magnetic moment. Finally, we review the quantum-field-theoretic computation of aµ . For this purpose, consider the scattering of the muon by classical background EM field in the limit of q 2 = (pf − pi )2 → 0 so that we replace Aµ by Aµ + Aext µ . This induces an R ext additional term in the Hamiltonian HIext = d3 xeψ̄γ µ ψAext µ . If Aµ is weak enough, we consider only the leading-order term in the expansion in Aext µ , and higher-order terms can be neglected. In addition, to make a connection between the form factors and magnetic moment, we take Aµext (x) = (0, Aext (x)) so that ∂t Aµext (x) = 0. Then, Aµext (pf − pi ) = 2πδ(Ef − Ei )Aµext (pf − pi ) = 2πδ(Ef − Ei )Aµext (q). Now, according to Ref. [1], we can write µ− (pf ) in iT µ− (pi ) out = −iM2πδ(Ef − Ei ) 12 where T is the transition matrix introduced in Eq. (2.4) and iM ≡ i µ− (pf ) µ− (pi ) Aext µ (q) µ 2 2 = −ieū(pf ) γ FE (q ) + i FM (q ) u(pi )Aext µ (q) 2mµ σ iν qν = ieū(pf ) γ i FE (q 2 ) + i FM (q 2 ) u(pi )Aiext (q). 2mµ j µ (0) in em out σ µν qν (2.6) For interpretation of the expression, we take nonrelativistic limit. Keeping terms through the first order in momenta, we obtain [1] √ √ p · σξ (1 − p · σ/2mµ )ξ √ ' mµ u(p) = (1 + p · σ/2mµ )ξ p · σ̄ξ where ξ is a 2-spinor. Here, we have adopted Weyl basis. Then, i σ̄ 0 i † ū(pf )γ u(pi ) = u (pf ) u(pi ) 0 σi = ξf† (pf · σσ i + σ i pi · σ)ξi (2.7) = ξf† (pjf σ j σ i + σ i pji σ j )ξi = ξf† [pi − pf − i ijk (pjf − pji )σ k ]ξi = −ξf† (q + i ijk q j σ k )ξi where we have used σ i σ j = δ ij + i ijk σ k . The first term in the last line of Eq. (2.7) corresponds to p · A + A · p in (p − ec A)2 of the nonrelativistic Hamiltonian in the electromagnetic field associated with the potential (φ, A) 1 e 2 HNR = p − A + eφ. 2m2 c The second term in the last line of Eq. (2.7) comes from interaction of the spin with the magnetic field. We focus on that term in the rest. The second part of Eq. (2.6) is already linear in q, and so we keep only the leading-order term in the expansion of the spinors to obtain i iν i † ū(pf ) σ qν u(pi ) = u (pf )γ 0 σ i0 q0 + σ ij qj u(pi ) 2mµ 2mµ 1 † 0 σ̄ i 0 u (pf ) q0 + = σi 0 σiσj 2mµ ' −ξf† (pf ) i ijk q j σ k ξi σiσj 0 where we have used the fact i 6= j due to σ ii = 2i [σ i , σ i ] = 0 in the last step. qj u(pi ) 13 Combining the two terms, by keeping only the relevant contributions, we obtain h i iM −−−→ − ieξf† i ijk q j σ k (FE (0) + FM (0)) ξi Aiext (q) q→0 = − ieξf† [FE (0) + FM (0)]σ k ξi i ijk q j Aiext (q) k = − ieξf† [FE (0) + FM (0)]σ k ξi Bext (q) σk k ξi Bext (q) 2 D E e k S k Bext (q) = − i2mµ 2[FE (0) + FM (0)] 2mµ = − i2[FE (0) + FM (0)]eξf† where S is a spin operator, and in the last line we take out the relativistic normalization factor of 2mµ arising from ūu = 2mξ † ξ. Recall that hp | p0 i = 2Ep (2π)3 δ (3) (p − p0 ) in QFT, while in QM hp | p0 i = (2π)3 δ (3) (p − p0 ) so that there is extra factor of 2Ep ' 2m in the QFT expression. Taking this difference in the normalization convention into account, we have in the first Born’s approximation [21], µ− (pf ) in iT µ− (pi ) out = −2πiδ(Ef − Ef ) hpf | V |pi i Z = −2πiδ(Ef − Ef ) d3 xeiq·x V (x) = −2πiδ(Ef − Ef )V (q). Hence, V (q) = M = −2[FE (0) + FM (0)] e † σk k ξ ξi Bext (q) = − hµi · B(q) 2mµ f 2 where hµi = 2[FE (0) + FM (0)] e hSi . 2mµ Thus, since FE (0) = 1, g = 2[FE (0) + FM (0)] = 2[1 + FM (0)]. Therefore, FM (0) = g−2 = aµ . 2 CHAPTER 3 REVIEW OF STANDARD MODEL COMPUTATION OF aµ In Section 2.2.4, we have concluded that aµ is equal to the magnetic form factor FM at zero momentum transfer. So the value of FM (0), and hence the prediction of aµ , comes from the evaluation of µ− (pf ) j µ (0) in em µ− (pi ) out ≡ −ieū(pf )Γµren (pf , pi )u(pi ) at q = pf − pi → 0 where the renormalized vertex function is defined as Γµren (pf , pi ) = γµ FE (q 2 ) + iσ µν qν FM (q 2 ). 2m As we pointed out in Section 2.2.2, aµ contains only the effect of radiative corrections. This implies that, experimentally speaking, aµ potentially contains the effects of all the particles in Nature that interact with the muon either directly or indirectly [22]. So it captures the physics within the SM as well as what is beyond it. The standard model prediction of aµ can be obtained by computing the Feynman diagrams that contribute to Γµren (pf , pi ). These diagrams can be grouped into several leading contributions to aSM µ plus all the other higher order terms. The leading-order diagrams from each group are shown in Fig. 3.1. Accordingly, we write aSM µ as QED Had aSM + aEW + aLOHVP + aHLbL + aHO + (Higher Order) µ = aµ µ µ µ µ (3.1) where aQED denotes the purely QED contribution, aEW µ µ , the electroweak (EW) contribution, aLOHVP , the leading-order hadronic vacuum polarization (LOHVP) contribution, µ Had , some additional hadronic aHLbL , hadronic light-by-light (HLbL) contribution, and aHO µ µ contributions still higher order in α such as the next-to-leading order or the next-tonext-to-leading order HVP or the next-to-leading order HLbL contributions. The EW contribution includes terms involving the electromagnetic interaction, weak interaction, 15 Figure 3.1: The Feynman diagrams for the leading-order term from each category of contributions listed in Eq. (3.1) [15]. On the upper row are the leading QED diagram, the Schwinger term, the diagram for the leading-order hadronic contribution, and the diagram for the leading-order light-by-light contribution. On the bottom row are the leading contributions from EW sector. and Higgs particles. We grouped any other diagrams that have not been included in the listed contributions in (Higher Order). This group contains diagrams such as the Had or diagrams with mixing hadronic contributions higher order in α than contained in aHO µ electroweak and hadronic effects. They are not considered as they are suppressed by higher 2 /4π, where g powers of α and αW = gW W is the dimensionless weak coupling constant, and thus do not make significant contribution to aSM or improvement of precision of the µ theoretical prediction. 3.1 QED Contribution Generically, aQED , the purely QED contribution to aµ , can be written as [23] µ aQED = µ ∞ X α n QED (n) aµ 2π n=1 where (n) (n) (n) (n) (n) aQED = A1 + A2 (mµ /me ) + A2 (mµ /mτ ) + A3 (mµ /me , mµ /mτ ). µ (3.2) (n) Here, A1 is called the universal part, and the rest, the mass-dependent part. The universal part is a sum of terms involving only a single flavor of lepton, whose mass is the only physical scale of the problem. Since aµ is a dimensionless number, this implies that its 16 value is common to all leptons l [15]. On the other hand, the mass-dependent part contains terms involving more than one lepton and thus depends on mass ratios of the leptons [15]. The QED part of aSM µ has been computed up to 5-loop order using perturbation theory [23–34]. The universal part is given as [15, 33–36] α 2 α − 0.328 478 965 579 193 78 · · · 2π π α 3 + 1.181 241 456 587 · · · π α 4 − 1.912 245 764 926 · · · π α 5 + 663 (20) π where dots indicate that the value is evaluated analytically to significant digits beyond what auniv. l QED = is displayed. For the level of precision we are aiming at, we do not need to use all the digits, and so the sequence of the digits is truncated. The mass-dependent part is evaluated using mµ /me = 206.768 2826 (46) and mµ /mτ = 5.946 49 (54) × 10−2 [37, 38] except for some contributions. Details of the calculation are provided in the following. The combined QED contributions up to 5-loops is then aQED = µ α 2π + 0.765 857 425(17) α 2 π α 3 + 24.050 509 82(28) π α 4 + 130.878 2(6 0) π α 5 + 750.93(89) . π 3.1.1 One-loop order The calculation of the QED contribution has a long history. The lowest-order correction, one-loop QED contribution, was computed by Schwinger in 1948 [18]: Z α 1 α QED (2) dx(1 − x) = ' 116 140 973.364 (72) × 10−11 . aµ = π 0 2π Its associated Feynman diagram was presented in Fig. 2.1. 3.1.2 Two-loop order At second-order in α, there are nine terms contributing to aQED . They contribute to µ (2) A1 (2) and A2 (2) in Eq. (3.2). There are no diagrams contributing to A3 at this order. The 17 Feynman diagrams for those can be found in Fig. 3.2. Of these, seven diagrams from (1) to (7) in Fig. 3.2 are universal. This part was first evaluated by Petermann [39] and then by Sommerfield [35] in 1957: (2) A1 univ = 197 π 2 π 2 3 + − ln 2 + ζ(3) = −0.328 478 965 579 193 78 · · · 144 12 2 4 where ζ(n) is a Riemann zeta function of argument n [15]. The mass-dependent part at this order corresponds to the last two diagrams in Fig. 3.2. If we set xl = ml /mµ , where ml with l = e, τ is the mass of the virtual lepton in the loop in (8) and (9) of Fig. 3.2, we can write the associated terms as a double integral [15, 40] Z 1 Z 1 u2 (1 − u)v 2 (1 − v 2 /3) (2) dv 2 du . A2 (1/xl ) = u (1 − v 2 ) + 4x2l (1 − u) 0 0 (2) The mass-dependent term A2 (1/xl ) is then evaluated by expanding in 1/xe for the electron and in xτ for the τ [32, 41, 42] to give (2) A2 (mµ /me ) = 1.094 258 3092 (72) (2) A2 (mµ /mτ ) = 0.000 078 079 (14) Figure 3.2: The second-order diagrams for the QED contributions [15]. The diagrams from (1) to (7) correspond to the universal part. The diagrams (8) and (9) represent the mass-dependent part. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature The Anomalous Magnetic Moment of the Muon by Friedrich Jegerlehner Copyright by Springer International Publishing AG (2008) 18 where the error arises from the experimental uncertainties of the mass ratios [20]. Hence, (2) aQED = 0.765 857 423(16) µ 3.1.3 α 2 π ' 413 217.6280 (86) × 10−11 . Three-loop order At three-loop order, there are 72 diagrams that contribute to aQED , which are shown in µ Fig. 3.3. The analytic calculation of the universal part was mainly performed by Remiddi and his collaborators [43–50]. Later, Remiddi and Laporta invented a trick to calculate the nonplanar triple-cross topology diagram, which is Fig. 3.3(25), and they completed the calculation in 1996 [51]. Kinoshita has also performed numerical evaluation of these terms using an adaptive Monte Carlo integration routine VEGAS [36] and confirmed the analytic (3) values in Ref. [52]. The analytical value for the universal part at three-loop order A1 evaluated by Remiddi and his collaborators is (3) A1 = 1.181 241 456 · · · . The nonuniversal part of aQED at this order can be found by replacing a muon in a closed µ loop in the diagrams containing a loop by another lepton. At this order, there is at most one (3) closed fermion loop in each diagram. The computation of A2 was done in Ref. [40, 53–56], (3) and that of A3 in Ref. [42, 57–59]. The calculation was conducted via series expansion first, and later it was cross-checked against the exact results [32, 60]. These values are reevaluated using the updated value for the mass ratios, mµ /me = 206.768 282 6 (46) and mµ /mτ = 5.946 49 (54) × 10−2 [37, 38, 61–65] in Ref. [20]. The results are summarized as follows: (3) A2 (mµ /me ) = 22.868 380 00 (17), (3) A2 (mµ /mτ ) = 0.000 360 63 (12), (3) A3 (mµ /me , mµ /mτ ) = 0.000 527 76(10). Here, the error arises from uncertainties associated with mass ratios. QED (3) Combining the universal and mass-dependent part, aµ (3) aQED = 24.050 509 82 (28) µ α 3 π is computed to give [20] ' 30 141.902 34 (36) × 10−11 . 19 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 21) 22) 23) 24) 19) 20) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46) 47) 48) 49) 50) 51) 52) 53) 54) 55) 56) 57) 58) 59) 60) 61) 62) 63) 64) 65) 66) 67) 68) 69) 70) 71) 72) Figure Fig. 3.3: 4.3. TheThe Feynman diagrams at the three-looptoorder from the QED [15]. The universal third order contribution aµ . All fermion loops sector here are diagrams for universal part can be obtained by assuming all the lines to be muon lines. The muon–loops. Graphs 1) to 6) are the light–by–light scattering diagrams. Graphs nonuniversal can bephoton obtained by replacing at least one closed muon loop byconthe lepton 7) to part 22) include vacuum polarization insertions. All non–universal follow by at least a closed loopas byQED some light-by-light other loop of tributions another flavor. Thereplacing diagrams from one (1) muon to (6)inare classified graphs, fermion and the ones from (7) to (22) contain the photon vacuum polarization. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature The Anomalous Magnetic Moment of the Muon by Friedrich Jegerlehner Copyright by Springer International Publishing AG (2008) 20 (3) When computing the error, the errors for two terms from A2 (3) are added in quadrature, (3) and the errors for A2 and A3 are added linearly as these errors both originate from the uncertainties of the same mass ratios [15]. 3.1.4 Four-loop order At the fourth-order in α, there are more than a thousand diagrams that contribute to aQED [15]. The universal part includes 891 diagrams [26], and the nonuniversal diagrams µ can be obtained from the universal diagrams by the replacement of muon closed loops by the loops of other flavors. Figure 3.4 shows some of the typical QED Feynman diagrams at this order. Among them, only a few terms are known analytically [66–70]. The evaluation of the diagrams at this order has been mainly done numerically by Kinoshita and his collaborators over the past 38 years [71–81] with revisions and improvements [23, 25, 27, 82–86]. The QED (4) computation of the dominant universal part of aµ took a long time due to the large number of integrals and renormalization terms to be computed for obtaining the value (4) for A1 , as well as the difficulty in the algebraic construction of the integrands and their numerical evaluation [25]. The estimate of the time scale for the computation at that time was up to 10 years using the high-performance computers available at the time [25]. So the progress of the numerical evaluation was reported in various conferences and meetings to meet numerous requests on the status of the computation [25]. Table 3.1 lists the estimate of (4) A1 over time printed in the proceedings and books [25]. For the computation, they have Figure 3.4: Some of the typical QED Feynman diagrams at the fourth-order in α [15]. The number in the bracket indicates how many diagrams of the given type there are. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature The Anomalous Magnetic Moment of the Muon by Friedrich Jegerlehner Copyright by Springer International Publishing AG (2008) 21 (4) Table 3.1: The preliminary values of A1 reported in various proceedings and books [15, 25]. The list shows the progress of the numerical computation. (4) A1 −1.434(138) −1.557(70) −1.409 2(384) −1.509 8(384) −1.736 6(60) −1.726 0(50) −1.728 3(35) −1.914 4(35) year 1983-1990 1995 1997 1999 1999 2004 2005 2007 Ref. [71–79] [87] [88] [81, 89] [90] [29, 64, 91] [25] [84] developed an automated calculation scheme for Feynman diagrams [84]. In the scheme, the topological structure as well as the UV and IR structure of each Feynman diagram is analyzed, and the corresponding FORTRAN code is generated, which is then processed by a numerical integration system such as VEGAS [84]. They found [86] (4) A1 = −1.912 98 (84) where the error comes from the Monte Carlo integration. Recently, Laporta has obtained a highly precise value in a semianalytical manner after twenty years of research effort. The first digits of his result are [33] (4) A1 = −1.912 245 764 926 · · · . (3.3) (4) The value for A1 was evaluated up to 1100 digits in Ref. [33]. To obtain this highly precise (4) value, he first reduced each diagram contributing to A1 to a linear combination of 334 master integrals where the coefficients are polynomials in the number of dimension D in the dimensional regularization [92, 93]. The master integrals are expanded in = (4−D)/2, and the coefficients are evaluated numerically to very high precision [92–94]. These coefficients are then replaced by analytical expressions involving harmonic polylogarithms, a family of one-dimensional integrals of products of elliptic integrals, and the finite terms of the -expansions of some master integrals [92–94] by fitting analytic expressions to the highprecision numerical values of the master integrals [95, 96]. The mass-dependent part is evaluated numerically first in Ref. [25, 26, 97] and then updated in Ref. [23]. There are 469 diagrams that contribute, and they are classified 22 into four gauge-invariant groups [27]. They are grouped according to the type of proper vertex diagrams from which the diagrams in a particular group originate and of closed loops inserted [27]. Group I contains 49 diagrams, and some of the typical diagrams are shown in Fig. 3.5. These diagrams are obtained by inserting vacuum polarizing lepton loops into the internal photon line of the Schwinger diagram. Group II contains 90 diagrams. Some of typical diagrams are shown in Fig. 3.6. The diagrams of this class are obtained from the second-order proper vertex diagram, i.e., a proper vertex diagram with two internal photon lines, found by inserting lepton loops at one- and two-loop order [15, 27]. Group III consists of 150 diagrams. Some typical diagrams are shown in Fig. 3.7. The diagrams are Figure 3.5: Some of typical diagrams in Group I [15]. li indicates the lepton inserted to form the vacuum polarizing closed loop into the internal photon line. The subgroup Ia contains 7 diagrams, Ib 18, Ic 9, and Id 15 [27]. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature The Anomalous Magnetic Moment of the Muon by Friedrich Jegerlehner Copyright by Springer International Publishing AG (2008) Figure 3.6: Some of typical diagrams belonging to Group II [15]. The number in the circle represents the order in e of the inserted closed lepton loop [27]. So 2 indicates the one-loop subdiagram, and 4 two-loop. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature The Anomalous Magnetic Moment of the Muon by Friedrich Jegerlehner Copyright by Springer International Publishing AG (2008) 23 Figure 3.7: Some of typical diagrams belonging to Group III [15]. The number in the circle represents the order in e of the inserted closed lepton loop as in Fig. 3.6 [27]. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature The Anomalous Magnetic Moment of the Muon by Friedrich Jegerlehner Copyright by Springer International Publishing AG (2008) obtained from proper vertex diagrams with three internal photon lines by inserting lepton closed loops [27]. Group IV consists of 180 diagrams. Some typical diagrams are shown in Fig. 3.8. The diagrams are obtained from proper vertex diagrams with a light-by-light subgraph by inserting lepton closed loops [27]. Reevaluation in Ref. [23] used the mass ratios mµ /me = 206.768 284 3 (52) and mµ /mτ = 5.946 49 (54) × 10−2 [38]. The values for mass ratios they have used are not the most up-to-date values cited in the introductory paragraph of Section 3.1. However, the numerical values of the mass-dependent part of QED (4) aµ have not been updated yet using the up-to-date values of mass ratios found in Ref. [37, 38]. The latest available results are summarized in Table 3.2. at this order There is an independent calculation of the mass-dependent part of aQED µ by Kurts et al. [30], which gives Figure 3.8: Some typical diagrams belonging to Group IV [15]. The number in the circle represents the order in e of the inserted closed lepton loop as in Fig. 3.6 [27]. The subgroup IVa contains 54 diagrams, IVb 60 diagrams, IVc 48 diagrams, and IVd 18 diagrams. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature The Anomalous Magnetic Moment of the Muon by Friedrich Jegerlehner Copyright by Springer International Publishing AG (2008) 24 Table 3.2: The summary of the numerical values for various subcontributions from each subgroup of mass-dependent part of aQED at fourth-loop order [23]. Note that µ these values are not based on the latest values of the mass ratios. Reprinted table with permission from Tatsumi Aoyama, Masashi Hayakawa, Toichiro Kinoshita, and Makiko Nio, Phys. Rev. Lett., 109, 111808, 2012. Copyright (2012) by the American Physical Society. https://doi.org/10.1103/PhysRevLett.109.111808 Group Ia Ib Ic Id IIa IIb IIc III IVa IVb IVc IVd (4) A2 (mµ /me ) 7.745 47 (42) 7.582 01 (71) 1.624 307 (40) −0.229 82 (37) −2.778 88 (38) 4.552 77 (30) −9.341 80 (83) 10.7934 (27) 123.785 51 (44) −0.4170 (37) 2.90 72 (44) −4.432 43 (58) (4) A2 (mµ /mτ ) 0.000 032 (0) 0.000 252 (0) 0.000 737 (0) 0.000 368 (0) −0.007 329 (1) −0.002 036 (0) −0.005 246 (1) 0.045 04 (14) 0.038 513 (11) 0.006 106 (31) −0.018 23 (11) −0.015 868 (37) (4) A3 0.003 209 (0) 0.002 611 (0) 0.001 807 (0) 0 0 −0.009 008 (1) −0.019 642 (2) 0 0.083 739 (36) 0 0 0 (4) A2 (mµ /me ) = 132.86 (48), (4) A2 (mµ /mτ ) = 0.042 494 1 (53), (4) A3 (mµ /me , mµ /mτ ) = 0.062 722 (10). In total, using the most precise evaluations of each component, we obtain (4) aQED = 130.878 2 (60) µ 3.1.5 α 4 π ' 381.004 (17) × 10−11 . Five-loop order At fifth-order, there are 9080 diagrams that contribute to aQED [15]. Originally, this µ class of diagrams was evaluated using the renormalization group argument in Ref. [74, 98]. However, due to the overwhelming number of diagrams and their complexity, a numerical approach via Monte Carlo integration was taken [23, 24, 29, 34, 83, 86]. The initial numerical estimate was made by Kinoshita and Nio [29, 83] to give (5) A2 (mµ /me ) = 663 (20), which was later cross-checked by Kataev using the renormalization group argument [31, 99]. The universal part was evaluated numerically in Ref. [23, 24, 34, 86]. The updated value is 25 [34] (5) A1 = 6.675 (192). The mass-dependent part is also evaluated numerically via the adaptive Monte Carlo integration by Aoyama and her collaborators [23]: (5) A2 (mµ /me ) = 742.18 (87), (5) A2 (mµ /mτ ) = −0.068 (5), (5) A3 (mµ /me , mµ /mτ ) = 2.011 (10). So using the value for universal part from Ref. [34] and nonuniversal part from Ref. [23], we obtain (5) aQED = 750.80 (89) µ 3.1.6 α 5 π = 5.077 8 (60) × 10−11 . Total aQED µ To compute aQED , we need the value for α. The best non-QED evaluation of the value µ comes from the measurement involving the Rb atom [37, 100–102]. It is given as α−1 (Rb) = 137.035 998 995 (85). The error is 0.62 ppb where ppb means parts per billion, i.e., 10−9 . This is the value of α we have consistently used above and will use below. The most precisely known value for α, however, comes from the measurement of electron g − 2 [103, 104]: α−1 (ae ) = 137.035 999 149 1 (15)(14)(330) (3.4) where the first uncertainty is from the uncertainty of tenth-order QED contribution to ae , the second, from the hadronic contribution, and the third, the experimental uncertainty in measuring ae [34]. When the errors in Eq. (3.4) are added in quadrature, it amounts to 0.25ppb. This value of α, however, is obtained by assuming that the SM gives the complete prediction of ae and there is no new physics contributing to it. So it is not appropriate to use α−1 (ae ) in probing new-physics effects in aµ . 26 The QED contribution to aµ based on these two estimates of α is summarized and compared in Table 3.3. In short, [34] aQED α−1 (Rb) = 116 584 718.971 (75) µ aQED α−1 (ae ) = 116 584 718.841 (58). µ The difference between aQED (α−1 (Rb)) and aQED (α−1 (ae )) gives an upper bound on the µ µ NP effects. 3.2 EW Contribution The electroweak contributions, or weak contribution in short, to aµ is also computed in perturbation theory up to two-loop order [105–113]. The diagrams in this category involve W and Z bosons as well as Higgs bosons, also interacting electromagnetically. Their contributions to aµ are collectively labeled by EW [13]. We group diagrams in this category according to the number of loops formed by fermions and bosons interacting electroweakly or in powers of ~. Then, we write the contribution as [113] EW (2) (1) aEW = aEW + aµ,b µ µ EW (1) where aµ EW (2) + aµ,f H.O. + aEW µ EW (2) is the leading one-loop order term proportional to GF m2µ , aµ,b EW (2) two-loop order term, aµ,f the bosonic H.O. is for the the fermionic two-loop order term, and aEW µ higher-order EW terms where GF is the Fermi constant. These contributions are suppressed at least by a factor of 2 α mµ π m2W ' 4 × 10−9 [13]. More specifically, the suppression factors are [20] Gµ m2µ √ ' 111.613 × 10−11 8 2π 2 Table 3.3: The values of contributions to aµ from QED sector at each loop order and of aQED based on the two different determinations of α from Ref. [34]. These values are in µ units of 10−11 . loop order 1 2 3 4 5 aQED µ with α−1 (Rb) 116 140 973.364 (72) 413 217.6280 (87) 30 141.902 34 (36) 381.004 (17) 5.077 8 (60) 116 584 718.971 (75) with α−1 (ae ) 116 140 973.233 (29) 413 217.6271 (86) 30 141.902 24 (35) 381.004 (17) 5.077 8 (60) 116 584 718.841 (58) 27 for one-loop order terms and Gµ m2µ α √ ' 2.70868 × 10−12 8 2π 2 π for two-loop order terms. In general, n-loop order terms are suppressed by GF αn−1 [113]. So their contributions are small [13]. 3.2.1 One-loop order At one-loop order, there are three diagrams to compute, which are shown in Fig. 3.9. The weak contribution at one-loop order is given by [105–108] " ! m2µ GF m2µ 5 1 2 2 one-loop EW + 1 − 4sW + O +O aµ = √ m2W 2 8π 2 3 3 m2µ m2H !# = 194.80(1) × 10−11 . The error is due to the uncertainties of the parameters, mainly of the W -boson mass [113]. Here and in the following, the values of the parameters used are taken from Ref. [114]: mµ = 105.658 371 5 (35) MeV mZ = 91.187 6 (21) GeV mt = 173.5(10) GeV GF = 1.166 378 7 (6) × 10−5 GeV−2 α = 1/137.035 999 where mµ , mZ , and mt are masses of the muon, Z-boson, and top quark, respectively. From these parameter values, the mass of the W -boson can be computed in the SM [115]: mW = 80.363 (13)GeV (a) W -boson contribution (b) Z-boson contribution (c) Higgs boson contribution Figure 3.9: The leading-order Feynman diagrams from the EW sector. 28 Then, the square of the weak mixing angle in the on-shell renormalization scheme is [13, 113] 2 s2W = 1 − MW /m2Z ' 0.233. As a reference, the parametric form of the Higgs-dependent term at one-loop order is [116] m2µ m2H Z 1 2x2 (2 − x) dx 0 1−x+ m2µ 2 x m2H . With mH = 125 GeV, it is 2.2 × 10−14 [116]. The computation of aEW at two-loop order µ requires the value for the Higgs mass. Although the latest value of the Higgs mass is mH = 124.98 (28) GeV [117], a conservative error is taken, and the value for mH used in Ref. [113] is mH = 125.6 (1.5) GeV, which is the average central value with 2σ error band of mH = 125.5 (0.2) (+0.5 −0.6 ) GeV by ATLAS [118] and mH = 125.7 (0.3) (0.3) GeV [119] where the first error is statistical and the second systematic. 3.2.2 Two-loop order Sample diagrams at two-loop order are shown in Fig. 3.10. Figure 3.10a shows a typical EW (2) bosonic EW contribution to aµ,b (a) Higgs dependent bosonic contribution to aµ,b . at two-loop order. The diagrams of this class are the (b) A diagram with a fermion loop and no Higgs particles conEW (2) tributing aµ,f . (c) A typical diagram of the rest of fermionic contributions containing a Higgs particle. It contributes EW (2) to aµ,rest-f,H . (d) A typical diagram of the rest of fermionic contributions with no Higgs particles. It contributes to EW (2) aµ,rest-f,no H . Figure 3.10: Sample diagrams for EW contributions at two-loop order. 29 EW (2) two-loop diagrams without fermionic loops. The first full computation of aµ,b was done in Ref. [120] and improved in Ref. [121, 122] [113]. It was reevaluated in Ref. [113] to give EW (2) aµ,b EW (2) The fermionic part of aµ = −19.97 (3) × 10−11 . EW (2) , aµ,f , consists of the diagrams with a closed fermion loop. For the diagrams of type Fig. 3.10b, a VVA triangle appears, and due to the U (1) anomaly cancellation between lepton and quark loops, the hadronic and leptonic effects need to be treated together. In addition, if the circulating fermion in the loop is light, the evaluation of the diagram using perturbation theory may not be appropriate [113]. For the rest of the diagrams, quark loops and lepton loops can be treated separately [116]. For these reasons, the contribution is split, and each new term is computed separately [113]: EW (2) aµ,f EW (2) = aµ,f EW (2) (e, µ, u, d, c, s) + aµ,f EW (2) EW (2) (τ, t, b) + aµ,rest-f,H + aµ,rest-f,no H The first two terms correspond to the diagrams of the type Fig. 3.10b. We split these diagrams into two groups depending on the mass of the circulating fermion. The fermions in the respective argument show up in the closed fermion loop. Also they contain no Higgs particles. The rest of the diagrams are divided into two groups. The first of the EW (2) two is associated with aµ,rest-f,H and consists of the diagrams with the Higgs particle such EW (2) as the one in Fig. 3.10c. The second group corresponds to aµ,rest-f,no H and collects all the remaining diagrams at two-loop order such as the W -boson exchange diagram or the diagram in Fig. 3.10d [113]. EW (2) The first full computation of aµ,f (e, µ, u, d, c, s) involving the first and second gen- erations was done in Ref. [110] via a naive perturbative calculation with constituent-like quark masses which is later improved by taking into account nonperturbative information in Ref. [109, 112]. They found [111] EW (2) aµ,f (e, µ, u, d, c, s) = 6.91 (20) (30) × 10−11 where the first error comes from the first generation and the second from the second EW (2) generation [113]. The third-generation contribution aµ,f (τ, t, b) can be computed using perturbation theory [113]. The result reads [109–111] EW (2) aµ,f (τ, t, b) = −8.21 (10) × 10−11 . 30 EW (2) The remaining contributions with a Higgs particle, aµ,rest-f,H , can be computed without approximation using two methods [113]. One method is to use an asymptotic expansion and integral reduction techniques, which were originally developed in Ref. [121, 123], and the other is to integrate the inner loop first and insert it into the outer loop, a method developed by Barr and Zee [124]. The result is [113] EW (2) aµ,rest-f,H = −1.50 (1) × 10−11 . EW (2) The rest of the electroweak contributions at this order, aµ,rest-f,no H , are given as [113] EW (2) aµ,rest-f,no H G2F m2µ α 1 5 m2t m2t 7 + ln 2 + = −√ 3 mW 2 8π 2 π 2s2W 8 m2W 2 2 c mt + W2 (1 − 4s2w ) 2 2s m W W 8 mZ 4 mZ + ln + ln (1 − 4s2W )2 9 mµ 9 mτ 3 2 + × 6.88(1 − 4sW ) 4 (3.5) where c2W = 1 − s2W . The first line was computed in Ref. [110] and rewritten in the above form in Ref. [125, 126]. The terms in the second and third lines were added in Ref. [111], and the explicit formula in Eq. (3.5) was presented in Ref. [113]. The last term comes from the diagrams with γ-Z mixing of type Fig. 3.10d [113] computed using a renormalization-group technique [111, 127]. The number 6.88 was obtained in Ref. [111] by replacing the perturbative result by a nonperturbative numerical value [113]. Then, EW (2) aµ,rest-f,no H = −4.64 (1) × 10−11 . The error is mainly due to the diagrams of this class that have not been evaluated [113]. By combining the contributions at two-loop order, we obtain EW atwo-loop = −41.2 (1.0) × 10−11 . µ The leading-logarithmic terms at the three-loop order are of the order O(10−12 ) and are thus negligible [13, 127]. 31 3.2.3 Total aEW µ Adding up the results from the previous sections, we get the total contribution from the electroweak sector, namely [113, 128] aEW = 153.6(1.0) × 10−11 . µ 3.3 Hadronic Contributions There are two major contributions from the hadronic sector: hadronic vacuum polarization (HVP) and hadronic light-by-light (HLbL) contribution. The Feynman diagrams for these can be found in Fig. 3.11. The HVP contribution is the leading order (LO) hadronic contribution in α and is much larger than HLbL, but its uncertainty is comparable to the one from HLbL. Other hadronic contributions that are higher order in α have smaller errors associated with them, as they are suppressed by higher powers of α. 3.3.1 LOHVP Currently, the most precise value of the HVP contribution is obtained from a dispersion relation and the R-ratio [128] R(s) = σ (0)) (e+ e− → γ ∗ → hadons) 4πα2 /3s by integrating the R-ratio over a kernel function K(s) to obtain Z R(s)K(s) 1 α 2 ∞ ds ahad = µ 2 3 π s 4m 0 π (a) The leading-order HVP contribution (b) HLbL contribution Figure 3.11: Two major contribution from the hadronic sector. (3.6) 32 where γ ∗ is the intermediate virtual photon, σ (0) (e+ e− → hadons) is the bare total cross section without radiative correction to the initial state, and Z K(s) = 1 dx 0 x2 (1 − x) . x2 + s(1 − x)/m2µ (3.7) We now derive this equation starting from causality and unitarity of the SM. Causality implies the dispersion relation, and the conservation of probability, unitarity. We begin with photon vacuum polarization tensor Πµν , where iΠµν is a sum of all the 1PI insertions to the free photon propagator. According to the dispersion relation, the trace part of the tensor Πµν (q) = (q 2 g µν − q µ q ν )Π(q 2 ), where q is the momentum transfer, satisfies [1, 15] q2 Π̂(q ) ≡ Π(q ) − Π(0) = π 2 2 Z ∞ ds 4m2π Im Π(s) . s(s − q 2 − i ) (3.8) Now, consider the Schwinger diagram in Fig. 2.1. We first replace the virtual photon in the diagram by the hadronic part of the full photon propagator, Dµν , at leading order in α but to all orders in QCD: e20 Dµν had (q 2 ) = −ie20 g µν q 2 (1 − Π(q 2 )) had ' O(α) −ie2 g µν had 2 1 + Π̂ (q ) 2 q2 (3.9) where iΠhad represents the hadronic vacuum polarization of a photon propagator in first 2 order in α but all orders in QCD, which is diagrammatically represented in Fig. 3.12. The coupling e20 comes from the vertexes connecting virtual photons to incoming and outgoing muon in Fig. 3.11a. The first term gives rise to the Schwinger diagram, and the second produces the LOHVP contribution. To see this, first apply the dispersion relation Eq. (3.8) Figure 3.12: Hadronic part of iΠ2 , which is the sum of all possible QCD intermediate states. It can be written in terms of the quark-line current-current correlation function as is indicated in the figure. 33 to iΠhad 2 . Then, − 2 Π̂had 2 (q ) = q2 ∞ Z 4m2π ds 1 1 Im Πhad . 2 (s) 2 s π q −s (3.10) Note that the only q 2 dependence of the term on the right-hand side in Eq. (3.10) comes from q 2 − s in the denominator. So, dropping the subscript 2 from now on, if we replace 2 Π̂had 2 (q ) in the second term in Eq. (3.9) by the right-hand side of Eq. (3.10), the replaced √ term generates a diagram for a fictitious photon of mass s, which is at one-loop order higher than the Schwinger diagram as Π̂had (q 2 ) contains a factor of α [15]. This term corresponds to the diagram in Fig. 3.11a. To obtain the contribution from this term to aµ , consider again the Schwinger diagram √ where a virtual photon is now replaced by a fictitious, massive virtual photon of mass s. If the mass of the fictitious photon is set to zero, it reduces to the familiar Schwinger diagram. Its magnetic form factor at zero net momentum transfer is given by [19, 129] massive γ a(1) µ α = π 1 Z dx 0 x2 (1 − x) . x2 + (s/m2µ )(1 − x) Thus, the LOHVP contribution to aµ can be written as aLOHVP µ α = π Z ∞ 4m2π ds s 1 Z dx 0 x2 (1 − x) −Im Πhad (s) . x2 + (s/m2µ )(1 − x) π (3.11) The quantity of interest, Πhad (s), contains all QCD effects and thus is not feasible to compute using perturbation theory. Fortunately, however, we can use the optical theorem to relate this quantity to the total scattering cross section of e+ e− -pair into all hadronic products, which we can measure experimentally. To begin with, unitarity of the S-matrix, S † S = 1, together with analyticity leads to the optical theorem, which states [1, 15] that the imaginary part of the forward-scattering amplitude is proportional to the total cross section, namely 2Im Tii = X Y Z d3 pi 1 |Tni |2 3 2E (2π) i n i∈n where T is the transition matrix, S = I + iT , n labels an intermediate state, and |ii is an initial state. The sum is taken over all the intermediate states |ni. Since Πhad (s) can be thought of as the forward-scattering amplitude of the photon at one-loop order in α, and since the hadronic blob, or the hadronic vacuum polarization, in Fig. 3.11a is connected with the virtual photon, the cross section related to Πhad (s) via the optical theorem is e− e+ 34 annihilation into hadronic states without iterative insertions of the hadronic blobs into the virtual photon. We denote it as σ0had ≡ σ0 (e+ e− → γ ∗ → hadrons). Figure 3.13 shows the diagram. The optical theorem relates the total cross section, σ0had , to the imaginary part of Πhad as (with kinematic factor included) [1, 15] σ0had (s) 4π α =− q 2 s e 1 − 4m s 2m2e Im Πhad (s), 1+ s where s here denotes the Mandelstam variable equal to the total center-of-mass energy squared. Since s ≥ 4m2π m2e so as to produce hadronic intermediate states at all, we can approximate σ0had (s) ' − 4πα Im Πhad (s) s (3.12) so that aLOHVP = µ α π Z 0 ∞ ds s Z 1 dx 0 x2 (1 − x) sσ0had . x2 + (2/m2µ )(1 − x) 4π 2 α We conventionally express the total cross section in terms of the normalized ratio [15] Rhad (s) = σ had (s) σ had (s) = 4πα2 (s)/3s σµµ (s) where σ had (s) represents the cross section going to hadronic states where the free photon propagator in Fig. 3.13 is replaced by the full photon propagator in QED and σµµ (s) ≡ Figure 3.13: The diagram corresponding to σ0 (e+ e− → γ ∗ → hadrons). The virtual photon from the vertex to hadronic blob is the tree-order photon propagator. q represents the quark, and arrows on the right of the hadronic blob indicate the various hadronic products. 35 σ(e− e− → γ ∗ → µ− µ+ ) is the high-energy asymptotic form of Bhabha scattering into a muon pair [15]. The cross section σ had (s) is related to σ0had (s) approximately as [15] α σ0 (e e → γ → hadrons) = σ(e e → γ → hadrons) α(s) + − ∗ + − 2 ∗ where α(s) is a running coupling constant evaluated at the center-of-mass energy s. Then, Rhad (s) = α2 σ0had (s) , 4π/3s which also means 3 Im Πhad (s), α Rhad (s) = so that aLOHVP µ α = π Z ∞ 0 ds s Z 1 dx 0 x2 (1 − x) α had R (s). 2 2 x + (2/mµ )(1 − x) 3π (3.13) Using K(s) defined in Eq. (3.7), we can write it as aLOHVP µ 1 α 2 = 3 π Z 0 ∞ ds K(s)Rhad (s). s (3.14) Experimentally, however, σ had (s) is not what is actually measured. The cross section σ had (s) is in a sense a pseudo-observable, as it does not contains bremsstrahlung effects such as initial and final state radiation and initial-final state interference [15]. Fortunately, at energies up to about 40 GeV where the Z-boson exchange effect comes in, the one-photon exchange approximation σ(e+ e− → hadrons)(s) ≈ σ(e+ e− → γ ∗ → hadrons)(s) is an excellent approximation [15]. The contribution to aLOHVP from the higher energy µ region above 11.5 GeV, beyond which the R-ratio is computed completely in the above approximation theoretically using pQCD without recourse to the experimental data, is only about 0.28% of the total aLOHVP [20]. So the approximation causes negligible error µ compared with the errors from lower energy region. Below 11.5 GeV, the experimental data from the cross section measurements are used to compute the R-ratio except where pQCD is applicable. At high enough energy, greater than about 2.1 GeV in practice [20], the running QCD coupling constant αc becomes small 36 enough, and perturbative QCD becomes valid except at the resonances. There, we can assume that hadron-quark duality [130, 131] holds to a good approximation: σ(e+ e− → hadrons)(s) = X σ(e+ e− → Xh )(s) ' X Xh σ(e+ e− → q q̄)(s), (3.15) q where Xh is a hadronic state consistent with all conservation laws, and the sum over all quarks q is taken for which 4m2q s, so as to ensure the production of the quark pair [15]. Also, experimentally, hadronic final states are produced in jets. So it is easy to separate the hadronic events from the leptonic events, making inclusive measurements possible [15]. At lower energy below 2.1 GeV, however, there are no characteristics such as jets that can be used to separate the hadronic events from the nonhadronic final states in the background [15]. So it is necessary to identify each channel individually [15]. The reasons are as follows. Beyond 1.2 GeV [132], the cross section of multihadron channels is significantly larger than that of two-body channels [133], and the nonhadronic events of multiple leptons are suppressed at least by a factor of α [20]. So we can identify the hadronic events by counting the number of final products [133]. However, it is a difficult task to distinguish muon-pairs from charged pion pairs, neutral pions from photons, π + π − π 0 from π + π − γ, and so on [20]. Also, channels with only neutral final products can be invisible in some experiments [133]. Further, detector difficulties such as finite resolution and cuts due to unavoidable blind spots of the detector need to be taken into account [15]. These lead to difficulty in estimating the acceptance and reconstruction effciency in a reliable manner by simulations [134]. Thus, in this energy range, namely from ππ threshold to 2.1 GeV, exclusive measurements are performed to obtain the R-ratio, and due to the difficulties, about 81% of the total error comes from this low-energy region. had (s) is mainly used up to some The strategy is then as follows. The measured Rdata energy Ecut , beyond which pQCD is a good approximation. After this cut, we compute had (s) using pQCD, which is precise enough due to asymptotic freedom of QCD except RpQCD had (s) below E at several resonance energies. So we use Rdata cut and near resonance energies had (s) beyond E and RpQCD cut except at the resonances [15]. Figure 3.14 and Fig. 3.15 show the compiled data for the R-ratio over the energy range used for the computation [132]. had in the energy region below about 1 In the graph, Rdata GeV is dominated by the ρ(770) resonance at 770 MeV of the π + π − channel [135]. Due 37 Figure 3.14: The compiled data for R-ratio with E from 0 to 5 GeV [132]. “The compilation of R(s) data utilized” by F. Jegerlehner published in Acta Physica Polônia B is licensed under CC BY 4.0 / Desaturated from original. Figure 3.15: The compiled data for R-ratio with E from 5 to 13 GeV [132]. “The compilation of R(s)-data utilized” by F. Jegerlehner published in Acta Physica Polônia B is licensed under CC BY 4.0 / Desaturated from original. to the 1/s-factor in Eq. (3.14), the contribution to aLOHVP from the lower energy region µ is enhanced, and the contribution from the π + π − channel in particular accounts for about 75% of aLOHVP [132]. Also in this low energy range, sQED, in which the pion is treated as µ a scalar particle, is valid [15]. So since hadronic production is dominated by ππ-production when 4m2π < s < 9m2π , using sQED, we can write Rhad (s) in terms of the nonperturbative amplitude Fπ (s) [15]: 38 R (0) had 1 (s) = 4 4m2π 1− s 32 Fπ(0) (s) 2 2 where Fπ (s) (s) = |Fπ (s)|2 |α/α(s)|2 . Figure 3.16 shows |Fπ |2 as a function of energy [132]. The recent improvement in the measurements of e+ e− → π + π − channel (CMD2 [136–139], SND/Novosibirsk [140], KLOE/Frascati [141–145], BaBar/SLAC [146, 147], BES-III/Beijing [148], CLEOc/Cornell [149]) reduced the error from this region considerably [132]. Thanks to the improvement, the error from this region is comparable to the error from the energy region from 1.2 GeV to 2.0 GeV. In the energy range from 1.2 GeV to 2.0 GeV, there are more than 30 exclusive channels that need to be measured [132]. Due to the aforementioned difficulty in identifying each had in this region is exclusive channel and disagreement in the measurements, the error of Rdata large as can be seen from Fig. 3.14. Consequently, while the contribution from this region is only about 14% of the total, it accounts for about 42% of the uncertainty of ahad to ahad µ µ [132]. Perturbative QCD is applied from 5.2 GeV to 9.46 GeV and above 11.5 GeV [132] in order to avoid the region with resonances associated with hadronic states, namely upsilon mesons Υ(1S), Υ(2S), Υ(3S), Υ(4S), Υ(10860), and Υ(11020), where pQCD fails. The Figure 3.16: The compilation of the experimental data for the modulus of the pion form factor, |Fπ |2 , near the π + π − → ρ resonance [132]. This figure by F. Jegerlehner published in Acta Physica Polônia B is licensed under CC BY 4.0 / Desaturated from original. 39 upsilon mesons are composed of the bottom and antibottom quark, and different excitations of this mesonic state are labeled either in the spectroscopic notation or by their mass in MeV. Combing the data and pertubative calculation, the final result using this approach is obtained to be [132] aLOHVP = 688.07 (4.14) × 10−10 . µ As is mentioned before, at energy beyond about 40 GeV, the Z-boson exchange contribution, e+ e− → Z ∗ → hadrons, becomes non-negligible [15]. However, K(s) becomes smaller at higher energy, and such a contribution is highly suppressed [15]. In the late 90s, the electron-positron annihilation data were of low statistics and suffered from systematic uncertainties, e.g., from normalization (luminosity determination), radiative correction due to initial and final state radiation as well as a running coupling constant [150, 151], and the difficulty in distinguishing π + π − pairs from µ+ µ− pairs [20]. So in 1997, Alemany, Davier, and Hocker proposed an alternative method for determining the I = 1 isovector part of e+ e− → hadrons process such as ππ channel using the precisely known τ spectral functions from τ → ντ + hadrons decays to improve the precision by relating the two quantities using isospin rotation in Ref. [152]. The tau-spectra data collected in the ALEPH collaboration were more precise than the electron-positron annihilation data at that time [152]. The use of τ spectral function led to a reduction in the uncertainty of aLOHVP by a factor of about 2 [153]. The method assumes isospin symmetry and conserved µ vector current (CVC) hypothesis [154, 155]. Simply put, under the CVC hypothesis, we regard the isovector part of the quark electromagnetic current and the weak vector currents as members of an isotriplet of current operators, and these currents are all conserved [156, 157]. To be concrete, let us introduce ψ= u d . Then, the electromagnetic current can be written as 1¯ µ 2 µ d = Q+ ψ̄γ µ ψ + Q− ψ̄γ µ τ3 ψ Jem = ūγ µ u − dγ 3 3 40 where Q+ = (2/3 − 1/3)/2 = 1/6, Q− = (2/3 + 1/3)/2 = 1/2, and τ3 is a Pauli matrix acting on the isospin space. The hadronic parts of the charged weak currents are 1 − γ5 d 2 ¯ µ 1 − γ5 u. = dγ 2 +µ Jch = ūγ µ −µ Jch The vector parts of the above operators are proportional to spherical components of the isovector operators [158] conventionally defined as [14] Jiµ = ψ̄γµ τi ψ. 2 (3.16) Now, we apply this technique to the computation of the cross section of the neutral channel e+ e− → π + π − as an example. For the channel, we can write the cross section for the 2π channel at lowest order in α as σ (0) ≡ σ0 (e+ e− → π + π − ) = 4πα2 v0 (s) s where v0 (s) is a spectral function for the neutral π + π − channel. From the relevant τ decay at the lowest order in the weak coupling, namely τ − → π − π 0 ντ , we can measure the spectral function for the charged π 0 π − channel [20, 155]. For these two channels, the relevant spectral functions are given in terms of the respective pion form factors by [154] β03 0 2 F (s) 12 π β3 2 v− (s) = − Fπ− (s) 12 v0 (s) = where β(s) = β0 (s) = β− (s) is a pion velocity and Fπi (s) is a pion form factor. Isospin symmetry implies Fπ0 (s) = Fπ− (s), and we have [20, 155] v− (s) = v0 (s). (3.17) Thus, we can obtain σ (0) out of τ spectral data [159–164]. In reality, isospin symmetry is only approximate. So a correction must be made to account for isospin violation such as different final state radiation effects, mass-splitting of pions and ρ mesons, the difference in width, and appearance of ρ − ω mixing [20, 154]. After all these corrections, there has been long-standing discrepancy in the value for aHVP , obtained based on e+ e− annihilation data and on τ -spectrum data [159, 165, 166]. µ 41 Generally, the determination based on e+ e− → hadrons data is preferred as it is more directly related to the HVP showing up in Eq. (3.10) [20]. This puzzle has been resolved after taking into account γ − ρ mixing present in the radiative photon correction, namely virtual and real soft- as well as hard-photon radiation, to the charged channel but absent in the neutral channel [167]. What needed to be done was to model correctly the vector meson dominance (VMD) mechanism with ρ, ω, and φ included, which we can use to parameterize the low energy data, also using the results from the hidden local symmetry model, up to and including the φ resonance [20]. With this correction made, the result is [168] aLOHVP (e+ e− + τ ) = 689.46 (3.25) × 10−10 . µ This value agrees within the uncertainty to the one presented earlier using σ(e+ e− → hadrons) data. For more detail, refer to Ref. [14, 20, 152]. Depending on the experimental data sets used and the method of their combination had , different values for ahad have been obtained. Figure 3.17 shows the for constructing Rdata µ various estimates of aµ based on different data sets [169]. There are also other factors had and so ahad as well as their uncertainties. First of all, in affecting the value of Rdata µ some energy range, there are only old data sets available for use [20]. Usually, these data sets have low statistics, which presents a problem when combining data sets from different experiments [20]. The situation, however, has been improving significantly [132] thanks to the aforementioned experiments on the dominant e+ e− → π + π − channel as well as the data in the important energy range from 1.2 GeV to 2.4 GeV by the BaBar exclusive channel measurements in the ISR mode [173, 179–192], CMD-3 on the processes, e+ e− → 3(π + π − ), e+ e− → p̄p, and e+ e− → KS0 KL0 , K + K − [193–195], and SND on the processes e+ e− → n̄n, e+ e− → ηπ + π − , e+ e− → π 0 γ, e+ e− → ωηπ 0 , e+ e− → ωη,e+ e− → K + K − , and e+ e− → ωπ 0 → π 0 π 0 γ [196–202]. There are still problems in combining the data sets from different experiments. For instance, as the quality of the data is different from one experiment to another, if we simply combine all data sets into one data set, precise data might be buried in imprecise data [20]. We need to give more weight to the data that are sparse but more precise than dense data with large errors when averaging the points in a single bin. To take more information from precise data rather than the data with a large error, we perform “clustering” [150–152, 203]. 42 Figure 3.17: A summary of the R-ratio results based on other data sets: DHMZ10 [170], JS11 [167], HLMNT11 [171], BDDJ15 [172], DHMZ16 [173], FJ17 [15, 168], DHea09 [166], BDDJ12 [174], KNT18 [150] [132]. The shaded box in blue emphasizes the value cited in the above. The experimental value, aexp is indicated by the vertical band. The grey µ vertical band shows the current experimental error, and the narrower band inside the goal uncertainty for the E989 experiment [132]. Two of the estimates are computed without Initial State Radiation (ISR) method [175, 176]. The listed values depend on what value to use for HLbL contribution. JS11 and BDDJ13 uses 116 (39) × 10−11 [125]. Others use 105 (26) × 10−11 [177]. FJ17 uses τ spectral data [167] and ππ scattering phase-shift data [178]. This figure by F. Jegerlehner published in Acta Physica Polônia B is licensed under CC BY 4.0. 43 A different method is proposed in Ref. [166, 173]. Another question is, if there are different data sets from different experiments in the same energy range, whether it is better to combine the data sets to a single data set to obtain a had and then integrate, or integrate first using Rhad from each experiment and combined R̄data data take the weighted sum of the integrals. The former approach is advocated in Ref. [133]. Also, obtaining a combined systematic error from various experiments is problematic [20]. Combining the systematic errors from different experiments linearly overestimates the error as it does not take into account the fact that the experiments are conducted independently [20]. At the same time, these experiments use some common techniques and methods for estimating the error, such as the one from initial and final state radiation, so that there is some correlation among the experiments [20]. Even after we manage these issues, we still encounter the problem of an appropriate interpolation. If we integrate the data using the trapezoid rule by connecting the points had , the possibility is that such a procedure might fail to capture the of the combined Rdata correct form of Rhad in some regions where Rhad changes rapidly. In this case, the trapezoid rule is not reliable. So the use of some model is necessary, and it introduces another source of systematic error [20]. The issue here is that fitting data to a model by minimizing the χ2 -value could produce a misleading result [204]. As was mentioned above, at higher energy, pQCD is a good had by Rhad had approximation, and we replace Rdata pQCD away from resonances. However, RpQCD might miss some nonperturbative effect, and the use of hadron-quark duality expressed as Eq. (3.15) may not be valid, although the experimental uncertainty in the region where pQCD holds is not precise enough to test the validity of the assumption [20]. So the selection of the energy region in which pQCD should be preferred over the experimental data is not based on clear criteria. Also, adjustment of the normalization of the experimental data to conform with pQCD is usually done, but it has no solid foundation [20]. Thus, the integration of experimental data to obtain aLOHVP is not a clear-cut procedure with a large space for qualitative µ and subjective judgment, and consequently there are many estimates of the quantity as is displayed in Fig. 3.17. For more detail, refer to Ref. [20]. Given this ambiguity, it is desirable to have a first-principles approach which gives 44 an unequivocal result. Lattice QCD provides such an approach. It is a nonperturbative first-principles approach to QCD, free from ambiguity. The basics of the lattice approach to QCD are described in Chapter 5, and its application to the evaluation of aLOHVP is detailed µ in Chapter 6. 3.3.2 NLOHVP The kernel function, K(s), receives radiative corrections from perturbation theory in α [128]. Figure 3.18 shows the diagrams contributing to aµ at the next-to-leading order, O(α3 ), hadronic contributions except for the HLbL contribution. Their values are [168, 205, 206] had. aNLO µ (2a) = −206.13 (1.30) × 10−11 had. aNLO µ (2b) = 103.49 (63) × 10−11 had. aNLO µ (2c) = 3.37 (5) × 10−11 , and so had. aNLO = −99.27 (67) × 10−11 . µ 3.3.3 LOHLbL The HLbL contribution showing up at the next-to-leading order is very difficult to evaluate. A glance at Fig. 3.19 suggests that the evaluation of the diagram requires the computation of the four-point Green’s function [20], h0| T {Aµ (x1 )Aν (x2 )Aρ (x3 )Aσ (x4 )} |0i (a) 2a (b) 2b (c) 2c Figure 3.18: The next-to-leading order contributions from the hadronic sector [205]. The shaded blob represents the HVP, and loops with no correction, lepton loops. The figures by Alexander Kurz, Tao Liu, Peter Marquard, and Matthias Steinhauser are licensed under CC BY 3.0. 45 Figure 3.19: The diagrams for the hadronic light-by-light contribution at the next-to leading order. where T is a time-ordering operator. As only one photon is external in Fig. 3.19, the momenta of the rest of the photons can be off-shell, and so we do not have the direct had [20]. A short , namely Rdata experimental input that was available for computing aLOHVP µ distance tail of the contribution to aHLbL can be evaluated using pQCD. However, the µ dominant part of the contribution to aHLbL comes from the long-distance part where pQCD µ fails [132]. At long-distance, effective field theory can be used, but only for the very short long-distance tail [20]. Fortunately, the contribution from pseudoscalar exchanges π 0 , η, and η 0 , described by the effective Wess–Zumino Lagrangian, contributes a substantial portion of aHLbL . So for µ , we can resort to a low-energy effective field model containing the evaluation of aHLbL µ pseudoscalar and vector mesons, although this makes the estimate model dependent [13, 132]. The examples include VMD-type models, such as hidden local symmetry (HLS), extended Nambu–Jona-Lasinio (ENJL) models, and examples of the resonance Lagrangian approach (RLA) [132]. At present, aHLbL cannot be evaluated from first principles with enough precision [13], µ although the lattice approach to provide the ab initio numerical calculation is underway, and some promising results are reported [207–217]. There is also a very different alternative first-principles approach in which aHLbL is obtained by numerically solving the trunµ cated tower of the Dyson-Shwinger equations together with the Bethe-Salpeter equations of QCD [218–221]. A model-independent approach via dispersion relations taking inputs from experiments was recently initiated for general light-by-light scattering by Colangero, Hoferichter, Passera, and Stoffer in Ref. [222–224] and mainly by Pauk and Vanderhaeghen 46 in Ref. [225–227]. A recent improvement with this approach can be found in Ref. [228, 229]. The estimate based on low-energy effective models is [169] aHLbL = 10.0 (2.9) × 10−10 . µ 3.3.4 NNLOHVP The HVP contributions at next-to-next-to leading (NNLO), O(α4 ), have been computed in Ref. [205, 230, 231]. Some sample Feynman diagrams at this order are shown in Fig. 3.20. Their values are [205] had. aNNLO µ (3a) = 0.80 × 10−10 had. aNNLO µ (3b) = −0.41 × 10−10 had. aNNLO µ (3LbL) had. aNNLO µ (3c) = −0.06 × 10−10 had. aNNLO µ (3d) = 0.0005 × 10−10 . 3.3.5 = −0.91 × 10−10 NLOHLbL The HLbL type diagram where a lepton loop corrects the internal photon propagator, shown in Fig. 3.21, is sufficiently suppressed and can be neglected for the comparison of the theoretical aSM µ with the new experimental data [232]. The estimate made in Ref. [232] is Figure 3.20: The diagrams contributing to aµ at the next-to-next-to leading order from hadronic sector [128]. As before, the shaded blob represents the HVP, and unshaded blob a lepton loop. The figures by Alexander Kurz, Tao Liu, Peter Marquard, and Matthias Steinhauser are licensed under CC BY 3.0. 47 Figure 3.21: An example of the NLO HLbL diagram that is sufficiently suppressed at the present desired theoretical uncertainty goal. The loop correcting an internal photon line is formed by a lepton. Other possibilities for a similar correction are not shown here. aNLOHLbL = 0.3 (0.2) × 10−10 . µ Thus, had. aNNLO = −1.24 (1) × 10−10 . µ 3.4 Summary Table 3.4 summarizes the various contributions to aµ [13, 132]. As is indicated in the table, the FNAL E989 [233] experiment, which recently started its first run, aims to reduce the experimental uncertainty by a factor of four. The goal of the planned J-PARC E34 [234] experiment is to provide a completely independent measurement, also with improved precision compared with the Brookhaven experiment. The table also shows that among all the contributions, the QED contribution is dominant, but it comes with the smallest uncertainty. The other leading contributions are smaller than QED contributions but have relatively larger uncertainty. In particular, contributions from the hadronic sector have the largest uncertainty. Of these, the LOHVP has the highest uncertainty, and the LOHLbL comes next and has smaller but comparable uncertainty. The other higher-order hadronic diagrams are much smaller than these leading contributions compared to these two contributions. So in order to improve the precision of the theoretical value of aµ , we need to improve the precision of the hadronic effects. Specifically, we need to work on the LOHVP and 48 SM Table 3.4: The various leading contributions, acont µ , to aµ where LO means leading order, NLO next leading order, and LbL light-by-light in units of 1010 . Contributions QED (up to 5 loops) EW (up to 2 loops) LOHVP HLbL NLO HVP NNLO HVP aSM µ aexp (BNL E821) µ aexp (FNAL E989/J-PARC E34 - goal) µ SM aµ − aexp µ acont × 1010 µ 11 658 471.8971 (75) 15.36 689.46 10.34 −9.927 1.24 11 659 178.3 11 659 209.1 −30.6 δacont × 1010 µ 0.0075 0.10 3.25 2.88 0.067 0.01 5.1 6.3 ∼ 1.6 7.6 Ref. [34] [30] [168] [132, 169] [168] [205] [34, 132] [235] [233]/[234] [132] HLbL. We focus on LOHVP in this dissertation. The goal is to achieve a reduction of the theoretical uncertainty in aSM µ comparable to experiment. For this purpose, a systematically improvable calculation is desirable, given the limitations of the phenomenological approach based on the R-ratio discussed above. This is what we try to achieve with a first-principles calculation of the HVP based on lattice QCD. As a last note, a new, more precise value for α has been measured using atomic interferometry of Cs133 at the University of California, Berkeley [236]. The value is α−1 (Cs133 ) = 137.035999046 (27) With this new value, the anomalous magnetic moment of the electron, aSM e , is reevaluated to give 0.00115965218157 (23). so that SM aexp = −84 (36) × 10−14 , e − ae which is a 2.3σ deviation [132]. The previous value was aSM = 0.00115965218165 (77) with e aexp − aSM = −92 (82) × 10−14 , which is a 1.1σ deviation. Thus, the discrepancy has e µ widened and may suggest the existence of missing new-physics effects in analogy with the muon anomaly. CHAPTER 4 EXPERIMENT In this chapter, the experimental scheme for measuring aµ is reviewed. We primarily focus on the E821 experiment at Brookhaven National Laboratory (BNL) [235]. The newly proposed experiment, E989 at Fermi National Accelerator Laboratory (FNAL) [233] aims at a reduction of the experimental uncertainty by a factor of four. This experiment follows a similar experimental design with some minor changes to suppress statistical as well as systematic uncertainties. As a matter of fact, the main experimental apparatus, the muon storage ring, was transported from BNL to FNAL. So this subsection mainly describes the E821 experiment at BNL and then discusses some changes in the E989 experiment at FNAL for better precision. 4.1 Basic Principles When the muon is in the region of uniform magnetic field, B, perpendicular to the direction of its motion, it completes the cyclotron motion. The magnetic field also causes a precession of the spin of the muon about the magnetic field. As the gyromagnetic ratio −2eg/2m contains the g-factor, g, by measuring the difference in the cyclotron frequency and precession frequency, we can measure the anomalous magnetic moment. This is the basic idea of the experiment. Mathematically, assuming that β · B = 0 where β is the velocity of the muon, the cyclotron frequency and the Larmor precession frequency are given by [15] qB mµ cγ qB qB ωs = + aµ mµ cγ mµ cγ ωc = where γ = 1/ p 1 − β 2 and q = ±e. The difference is proportional to aµ and given by ωa = ωs − ωc = aµ qB . mµ c (4.1) 50 If aµ = 0, ignoring all other corrections, the cyclotron frequency and precession frequency are the same, i.e., when the particle completes a circle, its spin also completes full rotation. So the difference measures the anomalous magnetic moment. In reality, in order to keep the muons on the plane perpendicular to the applied magnetic field, a quadrupole electric field, E, is applied for vertical focusing [235, 237]. Assuming that β ⊥ E and β ⊥ B, this as well as relativistic effect adds a term to ωa given as [235] E×v 1 aµ − 2 (4.2) γ −1 c2 due to corrections to ωc and ωs . This electric field for vertical focusing also induces oscillatory motion called betatron oscillation, and its effect on oscillation frequencies, namely ωc and ωs , needs to be taken into account. However, if we choose to work with a muon whose velocity is equal to the so-called magic value, at which aµ (γ 2 − 1) = 1, this term vanishes [235]. On the other hand, at such a value, the energy takes the value Emagic = γmµ c2 ' 3.098GeV 105.7MeV = mµ c2 and the momentum, pmagic = 3.094 GeV so that βmagic = 0.99942 [15]. Such muons are highly relativistic, and so relativistic effects are not negligible. For one thing, this implies an elongated lifetime of the muon in the laboratory frame. Its proper lifetime is τproper ' 2.197 × 10−6 s, while in the laboratory frame it is observed to be τlab ' 6.454 × 10−5 s [15]. A measurement period is thus chosen typically 700 µs after each injection [235]. For the other, Thomas precession becomes significant, the effect of which is pointed out and has been included in Eq. (4.2) [15]. Since aµ ≈ 1.66 × 10−3 , the deviation of the spin direction of the muon from the direction of its motion grows very slowly. This slow development of the deviation, combined with the longer lifetime, is what makes precise determination of the muon anomalous magnetic moment possible. This means that what limits the precision of experimental muon g − 2 is the difficulty in producing a constant homogeneous magnetic field and its precise determination [15]. 4.2 Decay Kinematics This section loosely follows Ref. [15]. The production of the muons starts with smashing of the protons from the proton storage ring onto a target material in a way that pions are the most predominant secondary particles. This is achieved via resonant decays. 51 The process proceeds as p + (N, Z) → ∆∗ + X → (N + 1, Z + 1 ∓ 1) + π ± where N is the number of neutrons and Z, the atomic number. The basic processes are p + p → p + n + π+ p + n → p + p + π−. For pion production, the intermediate state is dominated by the ∆33 isobar. The ratio σ(π + )/σ(π − ) approaches 1 at high enough Z. In the BNL experiment, the Alternating Gradient Synchrotron (AGS) produces 60×1012 protons (mass about 1 GeV) at 24 GeV per AGS cycle of 2.5 s, which are then projected onto a Nickel target. As a result of the collision, a large number of low-energy pions of mass about 140 MeV are produced. These pions are momentum-selected, and then about one-third of pions decay into muons as follows: π + → µ+ + νµ π − → µ− + ν̄µ If we approximate the mass of the neutrinos to be zero, we know in the rest frame of the pion m2π = (pµ + pν )2 = m2µ + 2Eµ Eν − 2~ pµ · p~ν = m2µ + 2(Eπ − Eν )Eν − 2|~ pµ |2 = m2µ + 2mπ |~ p|µ so that |~ p|µ = q m2π − m2µ m2π + m2µ , Eµ = m2µ + |~ pµ |2 = . 2mπ 2mπ From now on, we focus on π − for concreteness. The π − consists of a d-quark and a u-antiquark. It decays mainly via weak interaction. The relevant part of the effective weak Lagrangian is Gµ Lweak, eff = − √ Vud [µ̄γ α (1 − γ5 )νµ ][ūγα (1 − γ5 )d] + h.c 2 52 where Gµ is the Fermi constant and Vud is a CKM matrix element. Since pions are pseudoscalar, we introduce a constant capturing the structure of the pion bounded state, called the pion decay constant Fπ , as follows ¯ µ γ5 u |π(p)i ≡ iFπ pµ . h0| dγ Then, we can write the transition matrix element at tree level as T = out µ, ν̄µ π − in Gµ = −i √ Vud Fπ [ū(µ) γ α (1 − γ5 )v (νµ ) ]pα . 2 Note that the matrix element contains the left-handed projection operator 1 − γ5 . So in the massless limit of the leptons, the resulting particles must consist of left-handed particles and right-handed antiparticles. In this case, this means that the resulting muon must be left-handed, and the muon antineutrino must be right-handed. However, this goes against the conversation of angular momentum for the following reasons. In the pion’s rest frame, the muon and the muon antineutrino emerge with equal but opposite spatial momentum. If we define the direction of the motion as the z-axis, conservation of angular momentum requires that their combined angular momentum be Jz = 0, but since the muon is left-handed, and the antineutrino is right-handed, the total angular momentum has the value Jz = 1. Hence, this process is not possible in the massless limit. This is called helicity suppression. At nonzero mass, the mass term effects a helicity flip, which permits the decay to proceed. The experimental evidence indicates that the neutrinos are almost always left-handed and antineutrinos right-handed, i.e., the neutrinos are nearly massless [14]. Hence, the emergent muons must be in a right-handed helicity state, and the emergent antimuons must be in a left-handed helicity state to conserve angular momentum. Figure 4.1 summarizes the situation. The decay rate of π − to µ− ν̄µ is [14] Γπ− →µ− ν̄µ G2µ = |Vud |2 Fπ2 mπ m2µ 8π m2µ 1− 2 mπ !2 × (1 + δQED ) where δQED is QED radiative correction. The decay rate for the π + is obtained by applying CP. The effect of helicity suppression is larger for the charged lepton with smaller mass. So 53 Figure 4.1: A diagram for decay modes of the pion in the pion’s rest frame. C represents the charge conjugation operator, and P, the parity operator. The black one-sided arrow indicates the momentum, and the blue arrow indicates the direction of the spin. The crossed-out decay mode is forbidden. this means that the decay of π − to e− ν̄µ is heavily suppressed compared with that of π − to µ− ν̄µ , theoretically after radiative correction, by a factor of [14] Rsupp. = me mµ 2 m2π − m2e m2π − m2µ 2 α mµ 1 − 3 ln + ··· π me = (1.2353 ± 0.0001) × 10−4 . As a matter of fact, experimentally, the pions predominantly decay to muons with a branching ratio of (99.98770 ± 0.00004)% [61]. To obtain highly polarized muons resulting from pion decay, then, we note that when the pions decay in flight, forward muons decaying in the direction of pion motion in the pion rest frame are boosted, and backward muons decaying in the opposite direction to the pion motion have smaller momentum [116]. We need to select higher and lower momentum muons to get highly polarized muons. Muons then decay into electrons and neutrinos as follows µ− → e− + ν̄e + νµ µ+ → e+ + νe + ν̄µ . The relevant part of the effective Lagrangian in charge-exchange form is Gµ Leff,weak = − √ [ēγ α (1 − γ5 )νe ][ν̄µ γα (1 − γ5 )µ] + h.c 2 54 Then, again at the tree-level. T = out e− ν̄e νµ µ− in Gµ = √ [ū(e) γ α (1 − γ5 )v (νe ) ][ū(νµ ) γα (1 − γ5 )u(µ) ] 2 Note again that, because of 1−γ5 , e− is in the left-handed helicity state and the positron in the right-handed helicity state in the limit of a massless electron. Figure 4.2 illustrates this process. Also, due to the parity-violating nature of the weak interaction, there is a correlation between the direction of the muon spin and the direction of the emitted electron [235]. In the muon’s rest frame, the electron (positron) is preferentially emitted in the direction antiparallel (parallel) to the direction of the (anti-)muon’s spin. After integrating out the momenta of the two nonobservable neutrinos in the decay process and neglecting radiative corrections, we obtain an expression for the differential decay width in the muon’s rest frame [13, 14, 235] G2µ m5µ 2 d2 Γ± = x [3 − 2y ± Pµ (2y − 1) cos θ∗ ] dy d cos θ∗ 192π 3 G2µ m5µ ∗ = n (y)[1 ± Pµ α∗ (y) cos θ∗ ] 192π 3 1 = n∗ (y)[1 ± Pµ α∗ (y) cos θ∗ ] τµ (4.3) where we indicate by * the quantities in the muon’s rest frame. Here, we denote a normalized energy for the electron to emerge by y = Ee∗ /Emax with Emax = (m2µ + m2e )/2mµ = 52.8 MeV at an angle θ∗ with respect to the muon spin, τµ , the proper lifetime of the muon due to the decay µ → eνe νµ , given by [14] τµ ≡ τµ→eνe νµ = 192π 3 , G2µ m5µ and Pµ , the longitudinal muon polarization, which is essentially one. In fact, Pµ & 0.95% in the E821 experiment [235]. This is due to the helicity suppression discussed in the above from 1 − γ 5 in the V − A charged weak current coupling to the charged weak boson. If the neutrino masses are zero, we have Pµ = 1 [14]. Figure 4.2: Illustration of muon decay in muon’s rest frame [15]. White arrows indicate the direction of the spin. Arrows of other colors indicate the direction of the momentum. 55 Also, we have defined n∗ (y) = y 2 (3 − 2y) 2y − 1 . 3 − 2y R1 Here, 2n∗ (y) is a “normalization spectrum” ( 0 dy2n∗ (y) = 1), and α∗ (y) represents the α∗ (y) = asymmetry due to parity violation and reflects the correlation between the muon spin direction and the emitted electron momentum [20]. In writing this, we have neglected y0 = me ∼ 9.67 × 10−3 . ∗ Emax We boost back to the laboratory frame. Then, n∗ → N α∗ → A. where N and A are of the same form as n∗ and α∗ , but now they are a function of a ∗ . transformed reduced energy y = E/Emax (Emax ≈ 3.1 GeV [235]) instead of E ∗ /Emax Also, since the muons are highly relativistic, the decay angle in the lab frame, θ, is highly compressed into the direction of the muon momentum [235], and electrons decay more or less into the direction of the muon momentum in the laboratory frame. Figure 4.3 shows the plots of α∗ , n∗ , N , and A as a function of y [235]. The energy of the emitted electron or positron in the laboratory frame is [235] Elab = γv (E ∗ + βp∗ cos θ∗ ) ≈ γv E ∗ (1 + cos θ∗ ). (4.4) So electrons or positrons emitted in the direction of the muon spin have higher energy. This means that when the emitted electrons or positrons are close to their maximum energy, they are more likely to have been emitted in the direction of the muon spin. Hence, if we count only the electrons or positrons with energy higher than some threshold energy, Eth , we can select the electrons emitted in a certain range of angles [235]. Due to the presence of cos θ∗ in Eq. (4.3), the number of electrons with their energy greater than Eth emitted after time t since injection of the muons into the storage ring modulates at frequency ωa . The modulation comes from parity violation and is characterized by a threshold-dependent asymmetry. The number of surviving muons after time t can be 56 (a) Center-of-Mass frame (b) Lab frame Figure 4.3: The distribution of the relative number and asymmetry number as a function of y. (a) shows the plots of the functions in the Center-of-Mass frame. (b) shows those in the laboratory frame [235]. Reprinted figure with permission from G. W. Bennett et al. (Muon g-2 Collaboration), Phys. Rev. D 73, 072003, 2006. Copyright (2006) by the American Physical Society. https://doi.org/10.1103/PhysRevD.73.072003. obtained from Eq. (4.3) using Pµ ≈ 1 after integrating over E and θ∗ beyond the cut-off, [235] N (t) = N0 (Eth ) exp −t γv τµ [1 + A(Eth ) sin(ωa t + ψ(Eth ))] where N0 (Eth ) is a normalization factor, and A is an asymmetry factor as a function of Eth . This correlation (anticorrelation) between the spin direction of the µ+ (µ− ) and the momentum of highly energetic positrons (electrons) is the key to determining the muon spin direction at the moment of its decay. The plot of the observed oscillation can be found in Section 4.3. The statistical variance of the measurement of ωa is inversely proportional to the product N A2 , the figure-of-merit (FOM) averaged over measurement ensemble. Thus, the statistical error is given as follows [235]: 1 δωa ∝ γv τµ r 2 . N A2 As N and A both depend implicitly on Eth , Eth is determined so as to minimize δωa [235]. In the E821 experiment, Eth = 1.8 GeV [235]. 4.3 E821 Experiment The E821 experiment at the BNL is based on the basic principles and decay kinematics discussed in Section 4.1 and Section 4.2, respectively. The schematic diagram of the 57 experimental facility is shown in Fig. 4.4 [235]. In the BNL experiment, first of all, protons are extracted as a bunch of 7 × 1012 protons at 24 GeV/c from the AGS and smashed onto the 1-interaction-ength target. As a result of the collision, a large flux of pions is produced along with other debris particles such as e± and protons. The pions from the primary target are steered through magnets and collimators to produce a beam with momentum of approximately 3.2 GeV/c and desired charge, while reducing other particles [237]. The pions are then directed into an 80 m long pion decay channel. The pions are unstable and quickly decay to muons of the corresponding charge in the channel. The produced muons are passed through magnets and slits, resulting in a beam with the storage ring momentum [235]. Approximately 95% are polarized in the longitudinal direction [237]. Also, the slits reject most of the pions as well as momentum-selecting some of them. Finally, muons injected into the muon storage ring circulate around the ring while decaying into electrons and positrons, depending on the charge of the injected muons, plus neutrinos. These decayed electrons and positrons are detected by the twenty-four electromagnetic calorimeters placed Figure 4.4: The schematic diagram of the top view of the experimental facility at BNL for measuring muon g − 2 [235]. Here, Q indicates quadrupoles, K collimators, and D magnetic dipoles [235]. Reprinted figure with permission from G. W. Bennett et al. (Muon g-2 Collaboration), Phys. Rev. D 73, 072003, 2006. Copyright (2006) by the American Physical Society. https://doi.org/10.1103/PhysRevD.73.072003. 58 symmetrically around the inside of the storage ring. Figure 4.5 shows the layout of the muon storage ring [235]. The calorimeters are attached next to the vacuum chamber of the storage ring and measure the energy and time of arrival of the decayed electrons [235]. From their momentum, the approximate spin direction of muons at the moment of their decay can be inferred. Figure 4.6 illustrates this procedure. An example plot of the number of decayed electrons as a function of the time is shown in Fig. 4.7 [235]. Now that we have a measurement of ωa , we can determine aµ experimentally. According to Eq. (4.1), assuming that the various corrections to it such as the one due to deviations from β · E = β · B = 0 have been taken into account, Figure 4.5: The layout of the muon storage ring. The numbers from 1 to 24 label the twenty-four calorimeters, Q indicates the electric quadrupole section, K, the kicker plates, and C, full-aperture and 21 C the half-aperture calorimeters [235]. The kicker pushes the muons to a stable orbit concentric to the storage ring by a fast, nonferric, pulsed magnetic kicker [237]. Reprinted figure with permission from G. W. Bennett et al. (Muon g-2 Collaboration), Phys. Rev. D 73, 072003, 2006. Copyright (2006) by the American Physical Society. https://doi.org/10.1103/PhysRevD.73.072003. 59 Figure 4.6: The schematic diagram for the measurement of the decayed positrons [15]. Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature The Anomalous Magnetic Moment of the Muon by Friedrich Jegerlehner Copyright by Springer International Publishing AG (2008). Figure 4.7: A plot of the number of decayed electrons as a function of the time [235]. These data are an accumulation of 3.6 × 109 muon decays in the R01µ− data-taking period [235]. These data are wrapped around, modulo 100 µ s. Reprinted figure with permission from G. W. Bennett et al. (Muon g-2 Collaboration), Phys. Rev. D 73, 072003, 2006. Copyright (2006) by the American Physical Society. https://doi.org/10.1103/ PhysRevD.73.072003. 60 ωa = 2µB aµ B where µB = e~/2mµ c is the Bohr magneton of the muon. The magnetic field B is measured via an NMR procedure applied to protons in water to give B= ωp 2µp where ωp is the Larmor precession frequency of protons in the uniform magnetic field inside of the storage ring and µp is the magnetic dipole moment of the protons. In the BNL experiment, this measurement is done at many points around the ring to map and monitor the value of B, and the averaged value ω̃p is used in the subsequent calculation. Then, µB = Rµp aµ where R = ωa /ω̃p . Now, the magnetic moment of the muon is by definition µµ = (1 + aµ )µB so that by denoting λ = µµ /µp , λaµ = (1 + aµ )R ∴ aµ = R λ−R In the BNL experiment, R is measured using both µ− and µ+ to be [235] Rµ− = 0.0037072083 (26) Rµ+ = 0.0037072047 (26) where the error is the combined error of statistical and various systematic errors in ωa and ω̃p . The major sources of systematic uncertainties are identified in Section 4.4 and are summarized in Table 4.1. They are intrafill gain instability, pileup, lost muons, coherent betatron oscillation (CBO), and vertical pitch correction (VPC). Intrafill gain instability is due to the change of the scale of the energy of the electrondetection between two consecutive muon injections [235, 238]. This accounts for 120 ppb of the total systematic error. Pileup occurs when two electrons below threshold energy hit the calorimeter close enough in time so that they are counted as a single particle of higher energy [235]. This effect is about 80 ppb. The error due to lost muons occurs because the 61 Table 4.1: The list of major sources of the systematic errors in measuring ωa . CBO stands for Coherent Betatron Oscillation, REFC, for radial electric field correction, and VPC for vertical pitch correction [238]. This table by Antoine Chapelain was licensed under CC BY 4.0/ Desaturated from original. Intra-fill gain change E821 [ppb] 120 Pileup 80 Lost muons CBO REFC and VPC Total 90 70 50 180 E989 Improvement Plans Better laser calibration Low-energy threshold Low-energy samples recorded Calorimeter segmentation Better collimation in ring Change CBO frequency Precise storage ring simulations Quadrature sum Goal [ppb] 20 40 20 < 30 30 70 injected muons did not have proper momentum at the time of entry into the storage ring or were perturbed by electric and magnetic field so that circulating muons become unstable and deviate from the stable orbit [235, 237, 238]. Coherent betatron oscillation is caused by the mismatch between the inflector and storage ring apertures [235]. The diameter of the former is 18 mm, and the diameter of the latter is 90 mm [235]. This results in a non-uniform distribution of the muons in the storage ring aperture and also an oscillation of the shape of the distribution in the plane perpendicular to the azimuth direction of the ring due to the magnetic field and focusing electric field. Since the acceptance of the emitted electrons depends on the radial coordinate of the muon at the moment of the decay, CBO affects both the spin motion of the muon and the decay positron acceptance [239]. Radial electric field correction and vertical pitch correction arise from deviations from the conditions β · E and β · B [235]. The systematic error on measuring ω˜p was estimated to be 170 ppb [237]. The error comes from the calibration and the uncertainty of the average muon position when estimating the averaged effect of the magnetic field on muons [237]. Please refer to [235, 237, 239] for detailed discussion on the error analysis. Combined with λ = 3.183345439 (10) by an independent measurement [240, 241] aµ− = 11659214 (8)(3) × 10−10 aµ+ = 11659204 (7)(5) × 10−10 . Assuming CPT invariance, we can combine the two value for R to obtain the average R̄ [235] 62 R̄ = 0.003 707 206 4 (2 0). Using this value of R, they obtained [235] aµ = 11659208.0 (5.4)(3.3) × 10−10 (4.5) where the first error is statistical and the second systematic. The relative total uncertainty is 0.54 ppm [235]. 4.4 E989 Experiment The E989 experiment at FNAL follows the same experimental principles as the E821 experiment at the BNL [233]. The factor of four improvement is achieved by meeting the following three challenges [238]: 1. increasing the statistics to achieve a four-fold reduction in the statistical uncertainty 2. reducing the systematic error in measuring ωa by a factor of about 3 3. reducing the systematic error in measuring ωp by a factor of about 2.5 from 170 ppb to 70 ppb [242] In order to increase the statistics of the experiment so as to achieve the desired statistical uncertainty requires a factor of 84 improvement in the integrated luminosity, because the Fermilab proton beam is 4 times less intense than that of the BNL experiment and because we need 21 times more statistics than that of the BNL experiment for the desired statistical precision [238]. For this purpose, the muon injection efficiency is significantly improved, and the repetition rate is increased from 0.37 Hz for the E821 experiment to 12 Hz on average for the E989 experiment. In the E989 experiment, 4 out of 16 batches of approximately 4×1012 protons of energy 8 GeV, compared with 24 GeV at the BNL [233], are produced from the Booster synchrotron and injected into the Recycler ring at a 15 Hz rate [238]. Due to the lower energy of protons, the pion production yield is 2.5 times higher than at the BNL [233]. At this energy, the production cross section of π + is 2.5 times larger than that of π − so that this experiment uses π + for its measurements [116]. In the Recycler ring, these protons are reorganized into four bunches with 10 ms separation and then sent to the target hall after momentum selection [238]. The collision 63 produces pions along with the debris particles just as in the E821 experiment, and the resulting particles are directed into the Delivery ring. Pions then make three turns in the Delivery ring, after which they are sent to the storage ring relocated from BNL [116]. The total muon path is then about 1900 m, which is ≈ 800 m longer than at BNL, so that, given that the pions of momentum 3.1 GeV/c have a decay length of 173 m, most of the pions in the beam will have decayed when they arrive at the storage ring corresponding to a factor of 100 reduction in the number of undecayed pions [116, 233]. Also, as the speed of the protons in the Recycler ring is about 5% smaller than the speed of the muon, when exiting the Recycler ring after making three turns, the arrival time to the exit differs by about 250 ns, which can be used to exclude protons from the beam [116]. The combined effect of the experimental design is to significantly reduce the hadronic “flash” at the beginning of the muon decay measurement, e.g., the burst of background hadrons in the storage ring hitting the counters [116, 233, 238]. This was one of the key limitations to increasing the statistics of the E821 experiment [116, 233, 238]. This arrangement also lets E989 use a momentum much closer to the magic momentum, pmagic [116]. The reduction of the flash enables the use of the data at an earlier time [238]. These measures increase the statistics of the E989 experiment. All the aspects of the E821 detector systems were scrutinized and updated to decrease the systematic uncertainty in measuring ωa. The list of the main sources of systematic uncertainty and proposed improvement plans is given in Table 4.1. CHAPTER 5 LATTICE GAUGE THEORY In Section 2.1, we briefly described the SM written in the language of the QFT, which in turn was formulated using a path-integral. We also explained how we make a prediction using this machinery by computing the Green’s functions. We also presented all the sectors of the SM. In this dissertation, we are primarily interested in the contribution to aSM µ from the hadronic sector, i.e., from QCD. Thus, here, we focus on the QCD part of the SM. As is suggested in the previous chapters, we carry this program in the lattice framework, which is a path-integral formulation of QFT on a discretized space-time lattice. The discretization regulates the theory, removing ultraviolet divergences [243–245]. The discretized pathintegral then is performed numerically using a computer after restricting to a finite spacetime. In this chapter, we explain what we mean by the lattice formulation of QCD and introduce the lattice action used in our work. In the next chapter, we discuss how we use . this formalism to compute part of aLOHVP µ We start from the QCD part of the generating functional presented in Eq. (2.3) without the normalization factor in the denominator. In the lattice approach, the integration is done numerically. In this case, it is much more convenient if the integrand is a real number rather than a complex number, because we can avoid cancellation of complex phases for which we need precise numerical integration. Also, we can interpret the exponential as the weight factor and use a Monte Carlo method in numerical simulation. So we perform Wick rotation on the the Minkovski generating functional to obtain the Euclidean generating functional as follows: Z ZQCD = t→−iτ E DAµ DψDψ̄eiSQCD −−−−→ ZQCD = Z E DAµ DψDψ̄e−SQCD where E SQCD Z = 4 d x 1 µν f /E tr FE + mf )ψ f µν FE + ψ̄ (D 2 (5.1) 65 E a / E = ∂µ + ig λ2 AEa with D γµ for µ = 1, · · · , 4. Here, g is the strong coupling constant, µ a f and FE µν now represents the gluon field-strength tensor and Aµ , the gluon field. The ψ represents the quark fields, which interact strongly, and the flavor f runs over u, d, s, c, t, b. Since FE µν is defined as usual as in the Minkovski metric and satisfies E E /µ, D / ν ] = igFE [D µν , we have E E E E FE µν = ∂µ Aν − ∂ν Aµ + ig[Aµ , Aν ] Ea abc Eb Ec λa = ∂µ AEa − ∂ A − gf A A ν µ ν µ ν (5.2) 2 aE λa = Fµν 2 where the λa ’s are infinitesimal generators of SU (3), called Gell-Mann matrices in the fundamental representation of the group and f abc is a structure constant. It is defined as [λa , λb ] = 2ifabc λc . The gamma matrix γµE is defined to be Hermitian as follows: γ4E = γ 0 , γiE = −iγ i . i Also, Aa4 = iAa0 and AE i = −Ai = A . Now that we have switched to the Euclidean metric and laid out our conventions, henceforth, the superscript E will be omitted for notational simplicity. Also, as we focus on the QCD sector from now on, the subscript QCD will not be explicitly written unless it is necessary to avoid confusion, and the gluon field strength tensor and gauge fields will not be bold-faced. As is explained in Ref. [1], Dµ ψ is gauge-covariant. This simply means that Dµ ψ transforms as a vector, i.e., contravariantly under the gauge transformation, a characteristic of a physical quantity demanded by the general invariance principle. Under the gauge transformation G(x) ∈ SU (3) [1], ψ(x) → G(x)ψ(x) λa λa i a a Aµ (x) → G(x) Aµ (x) + ∂µ G† (x), 2 2 g and so λa † i a † Dµ ψ(x) → ∂µ + igG(x) Aµ (x) G (x) + ∂µ G (x) Gψ(x) 2 g = G(x)Dµ ψ(x). 66 / is a gauge invariant scalar, as it does not change its value Then, the product of ψ † and D under the gauge transformation. In fact, / / / ψ † (x)Dψ(x) → ψ † (x)G† (x)G(x)Dψ(x) = ψ † (x)Dψ(x). If we turn the continuous space-time into a discrete lattice of space-time points, whose unit cell has dimension a = (a1 , a2 , a3 , a4 ), the continuous Euclidean generating functional, which is also called the partition function, can be written as Z Y Y Y (lat) Z = dAµ (x) dψ̄(x) dψ̄(x)e−Slat x x (5.3) x where x now denotes a lattice site. Here, Slat is the discretized version of S. We divide it into two parts: Slat = SG + Sq (5.4) where SG is a lattice gauge action and Sq is a lattice fermion action. They can be in any form so long as sthey converge to their continuum counterparts in the continuum limit. We will use this fact to our advantage later in developing the lattice action that leads to better computational efficiency. These actions as well as their Symanzik-improved versions used in this work will be discussed in what follows. With the discretization of the space-time, the momentum is confined within a finite Brillouin zone. So with this discretization, we can avoid ultraviolet divergences in loop integrals, which in essence arise from an integration over the infinite momentum space. This technique is called cut-off regularization. Also, we can label each lattice site by four integers n = (n1 , n2 , n3 , n4 ), which we call the site index, as x = n · a = (n1 a1 , n2 a2 , n3 a3 , n4 a4 ). We denote by µ̂ the unit vector with its site index nν = δµν and also for simplicity take aµ = a for all µ = 1, · · · , 4 where a is some constant. From now on, the subscript lat will be omitted. Observables are calculated via Z Y Y Y 1 dAµ (x) dψ̄(x) dψ̄(x)Oe−S hOi = Z x x x (5.5) where O is a gauge-invariant function of Aµ , ψ, and ψ̄. In this path-integral formulation, classical fields are integrated over. The classical fermionic fields are represented by a field of anticomuting numbers, called Grassmann numbers, as required by Fermi-Dirac statistics. 67 We summarize here some of their properties. Two Grassmann variables, θ and η satisfy [1] θη = −ηθ. This implies θ2 = 0, which in turn means for a general function f its Taylor expansion terminates after two terms: f (θ) = A + Bθ where A and B are commuting numbers. The integral of f (θ) is defined to be [1] Z dθf (θ) = B. We adopt the following convention: Z Z dθ dηηθ = +1. Similarly for differentiation, ∂θ η = −η∂θ , ∂θ ∂η = −∂η ∂θ , ∂θ f (θ) = B. The complex Grassmann variables are defined as θ= θ1 + iθ2 √ , 2 θ∗ = θ1 − iθ2 √ , 2 where ∗ indicates complex conjugation, and θ and θ∗ are treated as independent [1]. The sign convention for complex conjugation is fixed by (θη)∗ ≡ η ∗ θ∗ = −θ∗ η ∗ . These properties are enough to let us perform explicit path-integration over the fermionic degrees of freedom [246]. We could estimate the integral over fermions statistically. However, that would introduce statistical errors and incur computational cost. So using these 68 properties of the Grassmann variables, we integrate out the fermionic degrees of freedom explicitly without any statistical errors. For this, we use the following two formulae [243]:   Z Y N N X dηi dη̄i exp  η̄i Mij ηj  = det M (5.6) i=1 Z N Y i,j=1  ! dηk dη̄k ηi η̄j exp  N X  η̄m Mmn ηn  = −(M −1 )i,j det M (5.7) m,n=1 k=1 where η’s are Grassman variables and M is a matrix of complex numbers. These two relations easily follow from the change of basis ηi0 = Mi,k ηk where we have used the Einstein summation notation. Then, due to the anticommuting nature of Grassmann variables, N Y dηi dη̄i = det M i=1 N Y dηi0 dη̄i . i=1 and, since the Grassmann variables are nilpotent, i.e., ηi2 = 0, Z Y dηk dη̄k ηi η̄j eη̄M η = det M k Z Y 0 −1 0 ηn η̄j eη̄η = −(M −1 )i,j det M. dηk0 dη̄k Mi,n k In case of lattice QCD, the fermionic part of the integrand in the partition function, Z, can be written in terms of the Grassmann variables, ψ, as exp(Sq [Aµ ]) = exp ψ̄i Mi,j [Aµ ]ψj (5.8) where i = (x, s, a) runs over the lattice sites, x, the spin of the fermion, s, and the collection of internal quantum numbers, a. The matrix M [Aµ ] is called the Dirac matrix and encodes the kinematics of the fermions as well as their interaction with the gauge bosons [244]. Its inverse is the fermion propagator [1, 243] 0 ,b SF (y, s0 , b; x, s, a) = (M −1 )y,s x,s,a Using Eq. (5.7, 5.4,5.5,5.8), we obtain [244] Z Y 1 hOi = dAµ (x)O SF [Aµ det M [Aµ ] e−SG Z x (5.9) where O[SF ] is the sum of the terms obtained after all possible contractions of the quark fields in O and thus depends on the fermion propagators SF and Aµ . 69 5.1 Lattice Gauge Action In lattice QCD, the spacing between the lattice sites is finite, and the derivative is approximated by the finite difference. In doing so, we need to compare the value of the fermionic field ψ, which belongs to the fundamental representation of SU (3), at two different sites. In the continuum theory of the local gauge symmetry, the bases in the color space formed by ψ’s at each site differ continuously by a phase, and the gauge field of the Lie algebra of SU (3), su(3), describes the infinitesimal change in the phase at infinitesimally separated points [1]. In lattice QCD, the separation is finite. So we use an element of SU (3) to describe the finite change in the phase as the representation of the gauge degrees of the freedom. In fact, this leads to a natural gauge covariance of the finite difference approximation to the derivative term in the fermion action. In the continuum, the derivative of ψ(x) in the direction nµ for the free theory is defined as [1] 1 nµ ∂µ ψ(x) = lim [ψ(x + εn) − ψ(x)]. ε→0 ε This is invariant under a global SU (3) transformation, but it is not gauge covariant as ψ(x + n) and ψ(x) transform differently. Under gauge transformation by G(x) ∈ SU (3), ψ transforms as [1] ψ(x) → G(x)ψ(x) ψ̄(x) → ψ̄(x)G(x)† . (5.10) Then, 1 nµ ∂µ ψ(x) → lim [G(x + ε)ψ(x + εn) − G(x)ψ(x)] 6= G(x)nµ ∂µ ψ(x). ε→0 ε To make it gauge-covariant and thus make the action gauge-invariant, we need to paralleltransport ψ(x) from x to y using the gauge transporter [1] Z UP (y, x) = P exp i dxµ Aµ (x) P where P is a path-ordering operator, Aµ (x) ∈ su(3), and P , a path from x to y, with the following transformation property with a caution that we are using the Euclidean metric [1]: UP (y, x) → G(y)UP (y, x)G† (x). (5.11) 70 Note that we rescaled gAµ → Aµ so that the coupling constant g does not show up in the formula. The parallel transported ψ(x) to y is given as ψ(y) = UP (y, x)ψ(x). Then, the derivative looks like [1] 1 nµ Dµ ψ(x) ≡ lim [UP (x, x + εn)ψ(x + εn) − ψ(x)]. ε→0 ε This is evidently gauge-covariant. In fact, 1 nµ Dµ ψ(x) → lim [G(x)UP (x, x + εn)G† (x + εn)G(x + εn)ψ(x + εn) − G(x)ψ(x)] ε→0 ε = G(x)nµ Dµ ψ(x) Here, Dµ denotes a covariant derivative, which involves parallel transport. It reduces to the familiar form if we expand UP (x + εn, x) in ε as [1] UP (x, x + εn) = UP† (x + εn, x) = 1 − iεnµ Aµ (x) + O(ε2 ) so that Dµ ψ(x) = [∂µ − iAµ (x)]ψ(x). (5.12) On the discrete lattice, the derivative is approximated by finite difference. To make it gauge covariant, we introduce lattice version of the gauge transporter, Uµ (n) ∈ SU (3), called the “link variable” connecting ψ at n and at n+ µ̂. In analogy to the gauge transporter UP (x) in the continuum, Uµ (n) has the following gauge transformation property [243] Uµ (n) → G(n)Uµ (n)G† (n + µ̂). We also define U−µ (n) ≡ Uµ† (n − µ̂). It transforms as U−µ (n) → G(n)U−µ (n)G† (n − µ̂). In the light of Eq. (5.11), these transformation properties suggest that Uµ (n) takes ψ from n + µ̂ to n and U−µ (n) from n − µ̂ to n. Using this object, we can form the gauge covariant finite difference Uµ (n)ψ(n + µ̂) − Uµ† (n − µ̂)ψ(n − µ̂) . 2a 71 This also makes intuitive sense, as it compares ψ at n + µ̂ and n − µ̂ after bringing them to n by Uµ (n) and U−µ (n), respectively. Now, ψ̄(n)Uµ (n)ψ(n + µ̂) shows up in the action, and it transforms as ψ̄(n)Uµ (n)ψ(n + µ̂) →ψ̄(n)G† (n)G(n)Uµ (n)G(n + µ̂)† G(n + µ̂)ψ(n + µ̂) = ψ̄(n)Uµ (n)ψ(n + µ̂) so that the action is now gauge invariant. Thus, the gauge degree of freedom is represented by the field of elements of SU (3), Uµ (n). For any element G ∈ SU (3), we can parameterize it as [1] G = exp iα a aλ 2 where αa ’s are real numbers and λa ’s are Gell-Mann matrices. In particular, [245] Uµ (x) = exp (iaAµ (x)) where Aµ (x) = Aaµ (x) λa 2 and we require Aaµ (x) to converge to the continuum gauge field in the continuum limit. This form of Uµ (x) is used in lattice perturbation theory [244]. Alternatively, we could have defined Uµ (x) = exp (iaAµ (x + aµ̂/2), which is adopted in Ref. [245]. To first order in a, Uµ (x) = 1 + iaAµ (x) U−µ (x) = 1 − iaAµ (x − aµ̂). Then, if we imagine that the discrete lattice is embed in a continuum space-time, keeping everything to the leading order in a, since ψ(x + aµ̂) = ψ(x) + a∂µ ψ(x) + O(a2 ) ψ(x − aµ̂) = ψ(x) − a∂µ ψ(x) + O(a2 ), we have, to first order in a, Uµ (x)ψ(x + aµ̂) − Uµ† (x − aµ̂)ψ(x − aµ̂) = [∂µ + iaAµ (x)]ψ(x) 2a 72 where we have used Aµ (x − aµ̂) = Aµ (x) − a∂µ Aµ (x) + O(a2 ). We now want to construct the gauge action out of Uµ (x). We follow Ref. [243]. Since the action is a gauge-invariant scalar, it consists of gauge invariant products of link variables. We start from a slight generalization of Uµ (n). Let P be a path connecting n to m consisting of k links identified by the ordered pairs (ni , µi ) with n0 = n and consider the product of these links: P [U ] = Y Uµ (n) = Uµ0 (n)Uµ1 (n1 + µˆ0 ) · · · Uµk−1 (m − µ̂k−1 ). (n,µ)∈P This transforms as P [U ] → G(n)P [U ]G† (m). (5.13) So if n = m, tr[P [U ]] will be gauge invariant. The simplest nontrivial gauge-loop is called a plaquette and defined as Uµν (n) ≡ Uµ (n)Uν (n + µ̂)U−µ (n + µ̂ + ν̂)U−ν (n + ν̂) = Uµ (n)Uν (n + µ̂)Uµ (n + ν̂)† Uν† (n). (5.14) This plaquette takes ψ around a loop defined by µ̂ and ν̂ going from n in the ν-direction, then to the µ-direction and finally coming back to n by moving in the negative ν direction and then in the negative µ direction. It is the lattice analogue of the Wilson loop [1]. We can build the gauge action by summing the trace of plaquettes. This action was proposed by Wilson and properly called the Wilson gauge action. Thus, at its simplest level, the gauge action takes the form SG = 2 XX Re tr[1 − Uµν (x)]. g 2 x µ<ν (5.15) The factor of 2/g 2 is set so that its continuum limit matches with the continuous counterpart. In the limit a → 0, this action reduces to the continuum action. To show this, recall 1 exp(A) exp(B) = exp A + B + [A, B] + · · · , 2 73 and note a2 Uµν (x) = exp iaAµ (x) + iaAν (x + aµ̂) − [Aµ (x), Aν (x + aµ̂)] 2 a2 − iaAµ (x + aν̂) − iaAν (x) − [Aµ (x + aν̂), Aν (x)] 2 2 a2 a + [Aν (x + aµ̂), Aµ (x + aν̂)] + [Aµ (x), Aν (x)] 2 2 2 2 a a + [Aµ (x), Aµ (x + aν̂)] + [Aν (x + aµ̂), Aν (x)] + O(a3 ) . 2 2 The term Aν (x + aµ̂) also needs to be expanded in a. Aν (x + aµ̂) = Aν (x) + a∂µ Aν (x) + O(a2 ). Combining these, we obtain 2 (∂ Uµν (x) = eia 3 µ Aν (x)−∂ν A(x)+i[Aµ (x),Aν (x)])+O(a ) 2F 3 µν (x)+O(a ) = eia . Thus, a4 X X 2 XX Re tr[1 − Uµν (n)] = 2 tr[Fµν (an)Fµν (an)] + O(a6 ). 2 g n µ<ν 2g n µ,ν h a bi R P Since lim a4 n = d4 x and tr λ2 λ2 = δ2ab , a→0 Z Z 1 1 a→0 4 a a SG −−−→ 2 d x tr[Fµν (x)Fµν (x)] = 2 d4 xFµν (x)Fµν (x). 2g 4g SG = (5.16) Now, if we restore the coupling constant g by replacing Aaµ → gAaµ , Fµν = ∂µ Aν (x) − ∂ν A(x) + i[Aµ (x), Aν (x)] → g (∂µ Aν (x) − ∂ν A(x) + ig[Aµ (x), Aν (x)]) , which agrees with the definition of Fµν in the continuum shown in Eq. (5.2). Therefore, 1 a a 1 1 a a Fµν Fµν → Fµν Fµν = tr Fµν Fµν , 2 4g 4 2 which agrees with the gauge part of the integrand in the continuum action presented in Eq. (5.1). Thus, it has a correct limit. For historic reasons [246], we often write SG in terms of β ≡ 6/g 2 instead of g 2 as [247] β XX Re tr[1 − Uµν (n)] 3 n µ<ν i XX 1 h † =β 1 − tr Uµν (n) + Uµν (n) . 6 n µ<ν SG = 74 5.1.1 Symanzik improvement Equation (5.16) shows that if we embed the lattice in a continuum space-time and expand the lattice gauge action SG in a, it can be written as the continuum action plus correction terms proportional to a [243]. In our case of the Wilson action, the correction terms are of the order O(a2 ). This quantifies the deviation of the discrete approximation from the continuum limit, which we call the discretization error. We can reduce the discretization error in the Wilson action, SG , using a technique called Symanzik improvement [248–250]. In actuality, Symanzik improvement is a technique to reduce discretization errors not only from the lattice operator but also from the prediction: hOi = 1 Z Z DUµ O[SF ] det M e−SG [Uµ ] . The goal is to remove higher powers of a from the expansion of hOi in a by adding terms of higher mass dimension to the definition of each factor of the integrand where the mass dimension is the power of the mass dimension of a quantity under consideration when expressed in natural units [1]. This leads to faster convergence to its continuum limit when extrapolating to a = 0 as well as higher precision at the same computational cost. In lattice QCD, we have two expansion parameters, a and g. Perturbative improvement discussed below aims at removing O(an ) terms from the perturbative expansion of hOi in g up to some order g 2m for the desired values of n, m ∈ Z. This gives us mth -order in g 2 O(an ) improvement. Nonperturbative improvement aims at removing O(an ) terms from hOi to all orders in g using some nonperturbative relation. 5.1.1.1 Perturbative approach Firstly, we discuss the perturbative improvement program. For concreteness, we focus specifically on SG . We loosely follow Ref. [244]. We start from the Wilson action in Eq. (5.15). As we pointed out, when expanded, it leads to correction terms of the order O(a2 ). To remove the effect of these terms from SG , we need to add extra lattice terms to cancel the lattice artifacts up to the desired accuracy. We proceed as follows: 1. Expand the original action in a up to the desired order of accuracy an in the continuum expression. 75 2. Count the number of independent continuum terms of the mass dimension up to and including 4 + n, consistent with the gauge symmetry of the original action. 3. Add as many independent lattice terms as there are continuum terms found in the previous step. 4. Determine the improvement criteria, i.e., an observable O, to be used. 5. Evaluate hOi using lattice perturbation theory up to some order in g m . 6. Adjust the coefficients of these newly added terms to cancel the lattice artifacts at an in hOi from the terms at O(g m ). For an introductory explanation of lattice perturbation theory, refer to Ref. [244, 245]. To apply this program to the Wilson action, we need to expand SG in a first. For that purpose, we note there is a single dimension-four gauge-invariant operator: O4 = X tr Fµν Fµν , µ,ν which is proportional to the continuum gauge action. Beyond the lowest-order term in a, we have no dimension-five operator and three dimension-six operators O6a = X tr Dµ Fµν Dµ Fµν µν O6b = X tr Dµ Fνρ Dµ Fνρ µνρ O6c = X tr Dµ Fµρ Dν Fνρ . µνρ So there are three correction terms at O(a2 ), and the general form of tr Uµν expanded in a is X 1 ri O6i + · · · tr Uµν = const. − a4 O4 + a6 2 for i = a, b, c and ri ∈ R. This means that we need at least three lattice terms to cancel O(a2 ) terms in SG which is a sum of tr Uµν . The simplest choice of gauge-loops to be added to SG after the plaquette (pl), Upl ≡ Uµν , would be a rectangular loop of perimeter six (rt), a closed parallelogram consisting of six unit lines (pg), and chair (ch) [244]. Each of these gauge-link loops are shown in Fig. 5.1. The shape of the terms suggests its name. Then, the action can be 76 (a) Rectangle Loop (b) Parallelogram Loop (c) Chair Loop Figure 5.1: Three loops of perimeter six composed of gauge links. written as SG = β X ci Oi i where now i = pl, rt, pg, ch, and Oi = 1X Re tr[1 − Ui (l)] 3 l∈Li where Li is a set of all loops of type i with rotation and reflection. Once we determine the general form of the improved action, we need to decide on the values for the coefficients. For that, we need some improvement criteria. For example, we can use a correlation function of Polyakov loops, a closed loop going around the lattice due to the periodicity of the lattice, and demand the absence of O(a2 ) terms in the correlation function at tree-level. Lüscher and Weisz studied this correlation function and found that by adjusting the values for ci ’s, this improvement criterion could be satisfied [251]. In particular, they determined that O(a2 ) improvement can be achieved by setting the coefficients to cpg = 0 cch = 0 crt = − 1 12 where cpl + 8crt + 8cpg + 16cch = 1 [251]. This choice leads to what is called “tree-level Lüscher-Weisz” action [251]. To obtain higher-order in g O(a2 ) improvement, we need to expand the coefficients up to the desired order, which are then written as (0) (1) ci = ci + g 2 ci + · · · . 77 One-loop improvement using the quenched approximation, i.e., treating the fermionic determinant constant, is done in Ref. [251, 252] and with tadpole improvement (see below) in Ref. [253]. The coefficients are reevaluated using dynamical fermions up to one loop order including both one-gluon loop and one-quark loop in Ref. [254, 255]. In the simulation, they used the HISQ action, which will be explained in Section 5.2.5. An on-shell improvement [251, 252] approach was taken in performing the Symanzik improvement program on the Wilson gauge action. In the original Symanzik improvement program [248–250], the goal was to remove lattice artifacts from the lattice correlation functions [255]. This led to many complications and difficulties as the correlation functions are not gauge invariant [251, 255]. Lüscher and Weisz avoided these difficulties by focusing on the on-shell physical observables such as the gauge invariant spectral quantities in Ref. [251], and the approach is called on-shell improvement. Then, they showed that we are free to set one of the three coefficients, cpg , crt , cch , to zero without any loss to all orders in perturbation theory in g [251, 252, 256]. The conventional choice is to let cch (g 2 ) = 0 [244, 256]. This choice was made in obtaining the one-loop on-shell improved Lüscher-Weisz action with tadpole improvement. This is the gauge action employed in this study. The details can be found in Ref. [254, 255]. The summary can be found in Ref. [257, 258] 5.1.1.2 Tadpole improvement We now turn our attention to nonperturbative improvement. We focus on a type of nonperturbative improvement called tadpole improvement, which we have adopted in our study. To start, we recall that in lattice perturbation theory, we expand the link variable Uµ in powers of a as [244] 1 Uµ (x) = exp (igaAµ (x)) = 1 + igaAµ (x) + g 2 a2 Aµ (x)2 + · · · . 2 Since ψ̄(n)Uµ (n)ψ(n + µ̂) is contained in the fermion action as is briefly discussed in Section 5.1 and will be discussed in more length in Section 5.2, the third term in the expansion produces an interaction term of the form ψ̄A2µ ψ, and this in turn gives rise to the so-called tadpole diagrams [244]. An example showing up in the quark self-energy is presented in Fig. 5.2. We denote this contribution by Σtad . It can be expressed in momentum space as 78 Figure 5.2: A typical tadpole diagram. This contributes to a quark self-energy. [244, 245] Σtad g2 = 2 2 Z ≈g a Z a dk 2 λa λb , 2 2 π/a kdk ∼ X [cos(pρ a) − iγρ sin(pρ a)] ρ 1 k̂ 2 (4 − ζ) g2 a where k̂µ = 2 sin(kµ a/2) and ζ is a gauge-fixing term. In lattice calculation, we express quantities in dimensionless form by multiplying them by the appropriate power of a [244]. As Σtad gives rise to a mass shift, we multiply it by a to obtain aΣtad ≈ g 2 . This correction is suppressed only by g 2 (no factor of a) and produces a rather large discretization error. Higher-order terms in the expansion of Uµ (x) in a also generate similarly large lattice artifacts. The good news is that these contributions are process-independent [259], and so it is possible to measure the effect in one quantity and then remove it from all other quantities [259]. The simplest way to implement this idea is to cancel it out by dividing Uµ (x) by the parameter accumulating the tadpole contributions [259]. This approach was proposed by Lepage in Ref. [260]. He suggested measuring the mean value u0 defined as u0 = 1 htr Uµ i . 3 Since this consists of only tadpoles [259], we can cancel the tadpole effect by transforming Uµ (x) → Uµ (x) . u0 79 Another common choice uses the expectation value of the plaquette: u40 = 1 htr Uµν i . 3 Thus, by replacing every link operator Uµ in the action including the fermion part by Uµ /u0 , we can largely eliminate the tadpole contributions from our lattice simulations [259, 260]. This is called tadpole improvement. 5.2 Fermion Action There are several lattice fermion actions, all converging to the same continuum action. They differ in terms involving powers of a when expanded in a. The action used in our study is called the highly improved staggered quark (HISQ) action [261]. It is based on the staggered or Kogut-Susskind action [262] and is Symanzik improved. The staggered action is a modification of the naive discretization of the continuum fermion action. So we start with the native fermion action. 5.2.1 Naive fermion action The continuum free action for a single flavor reads [1] Sqfree Z = d4 xψ̄(x) ∂/µ + m ψ(x). Its discrete version replaces the integral by a sum over lattice sites and the derivative by a finite difference. Here, we choose a symmetric finite difference as the lattice approximation to the derivative because it suffers from smaller discretization errors compared with other alternatives such as left and right derivatives taking the difference of ψ at two adjacent sites [247]. This choice gives [247] Sqfree naive 4 =a X n ψ(n + µ̂) − ψ(n − µ̂) + mψ(n) . ψ̄(n) γµ 2a We rescale the variables as follows to obtain a dimensionless expression [247] m/a → m ψ/a3/2 → ψ ψ̄/a3/2 → ψ̄, 80 and then Sqfree naive → X n ψ(n + µ̂) − ψ(n − µ̂) ψ̄(n) γµ + mψ(n) . 2 As is explained in Section 5.1, we can make this action gauge invariant by introducing gauge links, Uµ (x), and thus include the interaction with the gauge field as follows [244, 247] ! X Uµ (n)ψ(n + µ̂) − Uµ† (n − µ̂)ψ(n − µ̂) naive Sq = ψ̄(n) γµ + mψ(n) . (5.17) 2 n 5.2.2 Doubling problem The position and momentum wave functions are related as usual: ψ(k) = X e−ik·x ψ(x) x 1 X ik·x e ψ(k) ψ(x) = V k where V = Lx Ly Lz Lτ is the volume of the lattice with its dimensions Li in lattice units and −π/a < ki = nπ/Li a < π/a with n ∈ Z. Then, the free naive action can be written as X ψ(n + µ̂) − ψ(n − µ̂) free naive 4 + mψ(n) Sq =a ψ̄(n) γµ 2 n 1 X i(q−p)·x e−aqµ − e−iaqµ = 2 4 e ψ̄(p)γµ + m ψ̄(q) V a x,p,q 2a X i sin(qµ a) 1 X = + m ψ̄(q) δ(p − q) ψ̄(p) γµ V p,q 2a p,q i sin(pµ a) 1 X = + m ψ̄(p) ψ̄(p) γµ V p 2a so that the free propagator is given by [244] −1 i sin(pµ a) S(p) = γµ + ma a 2 −iγµ sin(pµ a) + ma =P . 2 2 2 µ sin (pµ a) + m a Since pµ ∈ (−π/a, π/a), this propagator receives a large contribution from the region pµ ≈ 0 as well as the region around the fifteen corners of the Brillouin zone. So the naive theory contains sixteen duplicate modes of the fermion for each fermion type found in the continuous theory, and we call the additional fermions near the corners of the Brillouin 81 zone, except for the one near the origin, as “doublers” [243, 244, 247]. These doubler modes may be interpreted as equivalent fermions called “tastes” [261]. 5.2.3 Staggered fermion This subsection follows loosely Ref. [261]. The formulation of staggered action begins with the naive fermion action Sqnaive = a4 X x ! Uµ (x)ψ(x + aµ̂) − Uµ† (x − aµ̂)ψ(x − aµ̂) ψ̄(x) γµ + mψ(x) . 2a We observe that by performing spacetime-dependent transformation Ω(x) = γx = Q4 µ=1 γ xµ /a to ψ, ψ(x) → Ω(x)χ(x) ψ̄(x) → χ̄(x)Ω† (x), we can diagonalize the fermion action in spinor indices to Sqnaive = a4 X x ! Uµ (x)χ(x + aµ̂) − Uµ† (x − aµ̂)χ(x − aµ̂) χ̄(x) αµ (x) + mχ(x) 2a where Pν<µ αµ (x) = Ω† (x)γν Ω(x ± µ) = (−) ν=1 xν /a 1 with 1 being the unit matrix. Also, we have used Ω† (x)Ω(x) = 1. Note that each component of χ(x) obeys the same equation of motion and thus is redundant. So we keep only one component, say µ = 1, and discard the rest. Then, we make following replacement:   1  0   χ(x) → χ1 (x)   0  = χ1 (x)1̂. 0 This gives us the staggered fermion action in the “one-component basis” [244] Sqstag. 4 =a X x ! Uµ (x)χ1 (x + aµ̂) − Uµ† (x − aµ̂)χ1 (x − aµ̂) χ̄1 (x) αµ (x) + mχ1 (x) . 2a Now, define (t) ψB (xB ) = 1 X χ1 (xB + δx)Ω(δx)t̂. 16 δxµ ∈aZ 82 where    0 1      0 , 1 t̂ ∈   0   0    0 0 In short, t̂a = δt,a .   0 0   0   0 , ,   1   0 0 1        .    Different values for t are associated with different tastes. This representation is called the staggered fermion in the spin-taste basis [244], and this process is called blocking. Then, in the free case, we can rewrite the Lagrangian in terms of ψB [244, 247], SFfree stag. = a4 XX t h i (st) ψ̄B (xB ) (γµ ⊗ 1) M(st) µ +a(γ5 ⊗ γµ γ5 ) µ + 8m(1 ⊗ 1) ψB (xB ) xB where the derivative is now taken on the blocked lattice as (t) (t) ψB (xB + 2aµ̂) − ψB (xB − 2aµ̂) 4a (t) (t) (t) ψB (xB + 2aµ̂) + ψB (xB − 2aµ̂) − 2ψB (xB ) (st) (t) µ ψB (xB ) = . 4a2 (t) M(st) µ ψB (xB ) = The first Dirac gamma matrix acts on the spinor components of ψB and the second (t) gamma matrix acts on the taste indices (t). This indicates that we can interpret ψB as different species of spin-1/2 fermion. Hence, the original sixteen redundant excitations produce four tastes with four spinor components. Note that the second term in the square bracket in Eq. (5.2.3) is proportional to a. So in the limit of a → 0, tastes decouple and becomes degenerate. There is a numerical evidence for this [261]. 5.2.4 Taste exchange The most striking feature of the lattice fermion is that when the fermion in the lowenergy mode near pµ = 0 absorbs momentum close to ζπ/a where ζµ ∈ Z2 , it turns into another low-energy fermion with another taste, instead of acquiring high momentum and moving far away from the origin on the energy shell [261]. So the process involving exchange of high momentum close to ζπ/a leads to taste exchange [261]. The simplest such tasteexchange process is the interaction of the quarks via a highly virtual gluon of momentum ζπ/a [261]. The diagram is presented in Fig. 5.3. As the taste-exchange interaction involves highly virtual gluons, the diagram in Fig. 5.3 is approximately equivalent to a four-point vertex diagram, which arises from a local four-quark operator of dimension six, and so its effect is suppressed by (pa)2 where p is a typical external momentum [261]. Also, the 83 Figure 5.3: The leading tree-level taste exchange interaction involving highly virtual gluon of the momentum ζπ/a [261]. q indicates a quark, q̄ antiquark, and g gluon. The momentum of each quark is indicated at the tip of the line. Reprinted figure with permission from E. Follana, Q. Mason, C. Davies, K. Hornbostel, G. P. Lepage, J. Shigemitsu, H. Trottier, and K. Wong, Phys. Rev. D 75, 054502, 2007. Copyright (2007) by the American Physical Society. https://doi.org/10.1103/PhysRevD.75.054502. one-gluon exchange process with gluon-momentum p ≈ ζπ/a depicted in Fig. 5.3 is the most dominant taste-changing interaction as it is lowest-order in αs ζπ/a and involves only four external particles [261]; processes with more quark legs are suppressed by higher power of pa than (pa)2 [261]. Another implication is that even if we start with initial states of pµ ≈ 0, fermions with different tastes show up in the loops, and the effect of taste exchange is unavoidable [244]. As we saw in Section 5.2.3, after blocking, we have only four tastes. In the absence of a taste-exchange interaction and neglecting the term in Eq. (5.2.3) proportional to a, the Dirac matrix in spin-component basis diagonalizes into M ∼ diag(M1 , M1 , M1 , M1 ) where M1 is a Dirac matrix indexed by xB for a single taste. Then, det M 1/4 = det M14 1/4 = det M1 . This is the basis for removing taste degeneracy by taking the root of the determinant, which will be discussed in Section 5.3.1.1. In general, the taste-exchange effect is present in the simulation, and the numerical results suggest that the discretization error induced by this type of taste exchange is of the order of O(αa2 ) [261]. The action, which will be introduced later in Section 5.2.5, aims at suppressing these errors. In the continuum limit, 84 fortunately, there is strong numerical evidence that these additional taste degrees of freedom decouple from each other and become degenerate [257, 258, 261, 263]. 5.2.5 HISQ action This subsection follows closely Ref. [261]. Here, since the staggered fermion in the one-component basis is equivalent to a naive quark, we introduce the highly improved staggered quark (HISQ) action in terms of naive fermions, in which the presentation becomes simpler. The staggered fermion action introduced in Section 5.2.3 contains O(a2 ) errors. There are two major sources of error: One is the discretization error due to the finite difference approximation to the derivative in the continuum fermion action. The other is more subtle and has the origin in the taste-exchange interaction discussed in Section 5.2.4. We first eliminate O(a2 ) error due to the finite difference approximation to the derivative by adding the so-called Naik term [264] − a2 3 M . 6 µ To see that the Naik term indeed removes O(a2 ) error from the derivative term, consider the action of Mµ on some function f (x) expanded in a [243], Mµ f (x) = f 0 (x) + f 000 (x) 2 f (5) (x) 4 a + a + O(a6 ) 6 120 where we have again used f (x ± a) = f (x) ± af 0 (x) + a3 a2 00 f (x) ± f 000 (x) + O(a4 ). 2 6 Then, since M3µ f (x) = f (x + 3aµ̂) − f (x − 3aµ̂) − 6a Mµ f (x) = f 000 (x) + O(a), 8a3 by adding −a2 M3µ /6 to Mµ , we have M3µ Mµ −a 6 2 ! f (x) = f 0 (x) + O(a3 ). So we have successfully removed the O(a2 ) error arsing from the derivative term in the staggered action. Other O(a2 ) errors comes from the taste-exchange interactions. The goal is to remove the taste-exchange effects up to one-loop order O(αs2 a). As is pointed out in Section 5.2.4, 85 taste exchange arises when quarks exchange high momentum close to ζπ/a, and so the tasteexchange interaction diagrams involve highly virtual gluons. Then, its effect is perturbative for most lattice spacings [261]. At tree-level, they are the Feynman diagrams with a single gluon exchange with the exchange momentum close to ζπ/a that is most dominant where ζ 6= ~0, as shown in Fig. 5.3. The tree-level taste-exchange effect can be removed if we replace ψ̄γµ Uµ ψ by ψ̄(p)fµ (q)γµ Uµ ψ(p0 ) such that ( 1, if q → 0 fµ (q) → . 0, if q → ζπ/a where ζ 2 6= 0 and ζµ = 0 (5.18) We can effect such a form factor by smearing the gauge link via [265] (2) Fµ = Y ρ6=µ a2 δ ρ 1+ 4 ! symm. where the subscript “symm.” indicates that the product is symmetrized over all possible orderings of the operators [265] and δρ(2) Uµ (x) = 1 (Uρ (x)Uµ (x + aρ̂)Uρ† (x + aµ̂) − 2Uµ (x) a2 (5.19) +Uρ (x − ρ̂)Uµ (x − aρ̂)Uρ† (x − aρ̂ + µ̂) approximates a covariant second derivative on the link variables. This is called smearing because we replace a single link by an average over short paths of gauge links connecting (2) the endpoints of the original link variable [243]. When we apply δρ on a link variable with (2) its momentum qρ = π/a, we find that δρ ≈ −4/a2 and thus Fµ ≈ 0 so that the condition Eq. (5.18) is satisfied [261]. However, this smearing introduces a new O(a2 ) discretization error, which can be removed if we replace Fµ with [265] F asqtad = Fµ − X a2 (δρ )2 ρ6=µ 4 where δρ Uµ (x) = 1 [Uρ (x)Uµ (x + aρ̂)Uρ† (x + aµ̂) 2a − Uρ† (x − aρ̂)Uµ (x − aρ̂)Uρ (x − aρ̂ + aµ̂)]. (5.20) 86 This term approximates the covariant first derivative on the link variables. Replacing the gauge link operator Uµ in ψ̄γµ Uµ ψ by Vµ (x) = F asqtad Uµ and performing tadpole improvement of the gauge links in the fermion action, we obtain the asqtad action [265]: SFasqtad = X x ψ̄(x) γµ a2 3 4µ − 4µ (U ) + m ψ(x). 6 (5.21) This action produces Feynman diagrams free of O(a2 ) errors at the tree-level. Usually, Vµ (x) is also tadpole-improved. However, this is not necessary when links are smeared and reunitarized [259, 261, 266]. This is what we do next. To remove O(a2 ) errors at the one-loop order in g 2 , we first note that there are five taste-exchange diagrams which are important, shown in Fig. 5.4 [261]. They are of the order O(αs2 (pa)2 ) [261]. Other diagrams with additional external quarks are suppressed by higher powers of pa and not significant at the order αs2 [261]. As is suggested by Fig. 5.4, these diagrams are equivalent to a contact term with four legs to a good approximation [261]. So it is possible to remove the taste-exchange effect at one-loop order by adding contact terms [261]. However, this is not easy to implement [261]. It is much simpler to implement smearing, and Ref. [267] suggests that repeated smearing Figure 5.4: q q̄ → q q̄ taste-exchange diagrams at one-loop order [261]. They are of the order O(αs2 (pa)2 ) [261]. Other diagrams with additional external quarks are suppressed by higher powers of pa [261]. Reprinted figure with permission from E. Follana, Q. Mason, C. Davies, K. Hornbostel, G. P. Lepage, J. Shigemitsu, H. Trottier, and K. Wong, Phys. Rev. D 75, 054502, 2007. Copyright (2007) by the American Physical Society. https://doi.org/10. 1103/PhysRevD.75.054502. 87 reduces the taste-exchange effect. This is expected given the perturbative origin of taste exchange [261]. Since the asqtad smearing introduces a form factor that suppresses the high-momentum part of the taste-exchange interaction leaving the low-momentum behavior intact [261], it should suppress these one-loop taste-exchange effect too. The problem is that while smearing the link variable in µ-direction, to preserve gauge invariance (see Eq. (5.19)), Fµasqtad introduces additional links in the orthogonal directions that are not smeared [261]. However, if we smear the links multiple times, we can smear these newly introduced orthogonal links too, thereby reducing lattice artifacts. Unfortunately, this procedure comes with two problems. One is that multiple smearings induce new O(a2 ) errors, just as Fµ did. We can fix this problem by using O(a2 ) improved smearing such as F asqtad as F asqtad → F asqtad F asqtad The second problem is that multiple smearing replaces a single link variable with a sum of a large number of products of link variables. This proliferation increases the size of two-gluon √ vertices by N where N is the number of terms in the sum [261]. To circumvent this problem, we can combine smearing with unitary projection. That is, we unitarize the smeared result by a unitarization operator U and then smear it again, i.e., F asqtad → Fµasqtad UFµasqtad . Unitarization of the link variable bounds its size by unity [261], and reunitarization does not √ affect the single-gluon vertex [261]. So we can avoid this N -enhancement of the one-loop diagrams with two-gluon vertices without introducing new O(a2 ) errors by reunitarization between the double smearing [261]. We can simplify the doubly smeared link with unitarization by rearranging as follows:  F HISQ = Fµ − X a2 (δρ )2 ρ6=µ 4   UFµ . Note that here the corrections to O(a2 ) errors are moved in the outermost smearing. The HISQ action is thus given as X x ψ̄(x)(γµ DµHISQ + m)ψ(x) 88 where DµHISQ =Mµ (W ) − a2 (1 + ) M3µ (X). 6 In the finite difference operator, the smeared links are used, which are defined as Wµ (x) = FµHISQ Uµ (x) Xµ (x) = UFµ Uµ (x). The constant in the definition of HISQ action is there to further improve the action. In order to completely eliminate the discretization errors at order O(αs2 a2 ), we need to add more taste-preserving operators to the action [261]. These operators are not necessary for light quarks, however, because the effects are at the few-percent level and not relevant [261]. On the other hand, for heavier quarks, they have noticeable impact on simulation results [261]. While for b and t quarks, high-quality simulation can be performed using rigorously defined effective field theories such as nonrelativistic QCD, such a method does not work for c quark, as it is much lighter [261]. The analysis shows that the Naik term causes the largest error of the order O(αs2 (amc )2 ) [261], and factor multiplying the Naik term parameterizes radiative corrections to it and removes all the remaining O(αs (amc )2 ) errors [261]. We can find the appropriate value to achieve the desired improvement nonperturbatively by adjusting the value of to ensure the correct dispersion relation [261] c2 (p) ≡ E 2 (p) − m2r =1 p2 where mr is a renormalized mass. Or we can compute it perturbatively. At the tree-level taste-exchange interaction, the dispersion relation can be expanded in a as [261] E 2 (p) − m2r . p→0 p2 c(0) = lim To remove the leading error, we adjust the value for , which has the expansion at the tree-level [261] =− 27 327 5843 15367 (amr )2 + (amr )4 − (amr )6 + (amr )8 + · · · 40 1120 53760 3942400 where mr is related to the bare mass m in the action via [261] 3 23 4 6 m = mr 1 + (amr ) − (amr ) + · · · . 80 2240 89 To one-loop order in perturbation theory in g 2 , can be determined by requiring the correct dispersion relation for a c-quark at one-loop order. The calculation gives [261]. = 1 αs2 − 27 (amc )2 + O(αs3 , (amc )4 ). 40 The coefficient 1 depends on (amc )2 where mc is the mass of the charm quark [261]. In our work, the correction is made only at the tree-level, and so 1 is neglected and set to 0. 5.3 Numerical Simulation In lattice QCD, the path-integral is taken numerically in a stochastic manner using Monte Carlo methods or in terms of classical statistical mechanics using molecular dynamics. The two methods differ in how the representative ensemble of the domain of integration is obtained. The simulation is performed on a computer so that the lattice under consideration in a practical calculation is finite, and the extrapolation to infinite volume as well as to vanishing lattice spacing needs to be taken in the end to extract the value of the observable in the continuum, i.e., zero lattice spacing. 5.3.1 Monte Carlo integration In the Monte Carlo method, we sample points randomly from the domain of integration, evaluate the integrand at each point, and then average the values of the operator O to get the approximation to the integral. Naively, we sample points uniformly from the integration domain and take the expectation value of the integrand multiplied by the volume of the domain [243]. If we sample enough points from the domain of integration, the Law of Large Numbers says that we obtain a good estimate to the integral [243, 268]. One problem is that if we want to make our lattice reasonably large, the necessary number of sample points increases rapidly, and the approximation by naive Monte Carlo sampling methods quickly becomes impractical [247]. Also, in doing naive sampling, we might collect points from regions where the integrand is small and does not contribute much to the integral, making the algorithm inefficient. To resolve these issues, we use a statistical technique called “Importance Sampling.” In the importance sampling method applied to the Monte Carlo integration, we sample points from the integration domain according to some weight so that the points are taken from the region where the integrand is large and important, thereby reducing the variance [268]. 90 In this way, we can avoid sampling points from the region where the integrand is small and does not contribute much to the integral. In lattice QCD, the expression of hOi in Eq. (5.9) contains the factor e−SG . So we can regard e−SG as a weight function in the space of gauge configurations and sample configurations from the space according to this weight. 5.3.1.1 Fermionic determinant Note that in Eq. (5.9), there is another factor det M . In the “quenched approximation,” det M is treated as a constant and taken out of the integrand. This factor is then canceled by the same factor in Z. In the “dynamic” simulation, we take into account the dependence of det Mf on Aµ , which we need to integrate over, for some or all flavors f . In this method, det M remains in the integrand. As the determinant is nonlocal, the computation of its change as a function of Aµ is very expensive [244] and should be avoided for a tractable simulation. So we try to put it in an exponential form and include it in the weight factor. There are several ways to do this. One way is to introduce a color-triplet, scalar field, called “pseudofermion” φ and write det M as [244] Z det M = Dφ∗ Dφ exp −φ∗ M −1 φ . This is called the φ-algorithm. The problem with this approach is that the eigenvalues of the Dirac matrix M in lattice QCD are in general complex, and their real parts may not be positive definite [244]. Then, since we can write the exponent in terms of the eigenvectors, vi , and their eigenvalues, λi , as −φ∗ M −1 φ = − X j φ∗ vj 1 ∗ v φ, λj j unless φ is chosen carefully, regardless of the global properties of the matrix, each term in the sum could be complex, or the entire exponent is positive so that the exponential cannot be interpreted as a probability weight [244]. To circumvent this complexity, we perform explicit doubling, i.e., we work with det M 2 = det M † M where we have used the γ5 -hermiticity of the lattice Dirac matrix, M † = γ5 M γ5 [244]. This requires taking a square root of the determinant for types of lattice fermions such as Wilson and overlap fermions [244]. For the type of lattice fermions used in this dissertation, staggered fermions, this is not necessary as M † M decouples even and odd 91 sites, so that, by restricting the pseudofermion field to even or odd sites, we can remove the degeneracy. Taste-degeneracy still remains, and we need to take a fourth-root to remove it, as we discussed in Section 5.2.4. If we switch off the correction by not taking the root of the square, we have Z 2 det M = Dφ∗ Dφ exp −φ∗ (M † M )−1 φ . We can generate the pseudofermion field, φ, distributed according to exp −φ(M † M )−1 φ , out of a complex Gaussian random field R distributed as exp −R† R by setting φ = M † R [244]. Then, Z Z= DAµ det M 2 e−SG = Z DAµ DR† DRe−Seff where Seff = SG + R† R. For our study, we use staggered fermion, which comes with four tastes, and we remove this taste-degeneracy by taking the fourth root of the fermionic determinant. So in using the φ-algorithm, we need to deal with the fractional power of the determinant. To make the numerical computation more amenable, we approximate the root of M † M by a rational function [269, 270] (M † M )−m/n ≈ r(M † M ) = a0 + N X l=1 al M † M + bl where m and n are integers and al and bl are some constants. These constants as well as N are set so that the sum of poles is the best rational approximation by a sum of poles in the range of eigenvalues of M † M [271]. The thorems and algorithms can be found in Ref. [271]. Alternatively, we can use the identity ln det A = tr ln A for a matrix A to turn the determinant into an exponential form and write Z Z −SG Z = DAµ det M e = DAµ e−Seff where Seff = SG − tr ln M † M while still using the squared Dirac matrix. This is called R-algorithm. In this algorithm, when M † M carries an exponent as (M † M )m/n , we simply have Seff = SG − m tr ln M † M n 92 Lastly, removing the degeneracy by taking a square or fourth root might make lattice predictions unreliable [244, 261, 263]. The procedure is not rigorously justified and controversial [244]. There is no proof that the rooted determinant leads to the correct continuum limit and the approximation is valid [244]. However, there is ample numerical evidence that the continuum limit is smooth [261]. 5.3.1.2 Hamiltonian Monte Carlo Now that we have a factor that can be used to do importance sampling, we sample points from integration domain based on the weight P = exp(−Seff ) . Z (5.22) In Eq. (5.9), we integrate over all classical gauge configurations. So in lattice QCD, the integration domain is a space of all gauge configurations. Denoting a single configuration by C, if the collection of samples Ci reaches an equilibrium distribution exp(−Seff )/Z after sampling a further N times, we have [247] hOi ≈ N 1 X O(Ci ). N i=1 If the measurements O(Ci ) are statistically independent, the error of the estimate reduces √ as 1/ N in accordance with the law of large numbers [247]. To generate such a representative ensemble, we use a Markov process to obtain a sequence of configurations called a Markov chain {Ci } [247]. In general, the index i could be either discrete or continuous. In the continuous case, i is usually called simulation time or computer time [247]. A Markov process generates a Markov chain out of the state space, which is a space of gauge configurations in this case, and the transition probability T (Cf |Ci ), which gives the probability of going from a state Ci to another state Cf [272]. If the sample distribution, Ps , reached the equilibrium distribution P , then we expect T to satisfy the so-called balance equation [243] X C T (C 0 |C)P (C) = X T (C|C 0 )P (C 0 ) = P (C 0 ). (5.23) C This simply says that the probability of hopping into a state C 0 at some Markov step is equal to the probability of leaving the state at this step, which is just the probability of 93 finding a state at this step to be C 0 , due to the normalization condition P C0 T (C 0 |C) = 1 [243]. A sufficient condition on T for this property to hold is to require the detailed balance condition [243] ∀C∀C 0 T (C 0 |C)P (C) = T (C|C 0 )P (C 0 ). Another important requirement on T is that Markov process with T must have access to all the states in the state space, i.e., a Markov chain reaches any state in a finite number of steps, in order to ensure that we get the correct result [247]. This property is called strong ergodicity [247]. 5.3.1.3 Metropolis method To actually generate the sequence of configurations, we need an updating method to suggest a new configuration Cf given the present configuration Ci . One frequently used method to achieve this goal is the Metropolis method [273]. It works as follows. First of all, we need another probability distribution called selection probability T0 (Cf |Ci ), which is used to select a candidate for the next state Cf given the present state Ci [243]. Then, it proceeds as [243]: 1. Choose some candidate configuration Cf based on the present configuration Ci according to the selection probability T0 (Cf |Ci ). 2. Accept the candidate Cf as the next configuration with the probability T0 (Ci |Cf ) exp(−Seff [Cf ]) . TA (Cf |Ci ) = min 1, T0 (Cf |Ci ) exp(−Seff [Ci ]) 3. If accepted, Cf is the new state, and set it as the present state. Otherwise, the present state remains to be Ci . 4. Repeat. In practice, there is a correlation between consecutive configurations in the sequence [247]. So in order to ensure statistical independence of configurations in the sequence, it is generally necessary to select a subsequence where the entries are far enough in the original sequence to decrease the correlation [244, 247]. Also, there are techniques used to reduce this correlation such as overrelaxation and heat bath method. More on these updating 94 methods can be found in Ref. [243, 244, 247]. For algorithms implementing the Metropolis method, please refer to Ref. [243, 244, 247]. Other Monte Carlo methods such as Langevin algorithm, please also see Ref. [247]. 5.3.2 Molecular dynamics simulation Another way of performing numerical integration is by Molecular Dynamics. In this approach, we first transform the quantum problem into a classical one by rewriting the discrete partition function for lattice QCD into one for classical statistical mechanics and apply the ergodic hypothesis to evaluate hOi [247]. The goal is to evaluate 1 hOi = Z Z Y dUµ (x)O[Uµ ]e−Seff . x For notational simplicity, we consider a scalar field and replace Uµ (x) with φi , where the index i runs over lattice sites as well as any other quantum numbers such as spin, and remove the subscript “eff” in Seff . The method, applied to the SU (3) gauge theory, is discussed in Ref. [244]. Then, we have 1 hOi = Z Z Y dφi O[φ]e−S(φ) (5.24) i where φ collectively represents φi ’s. If we introduce a fictitious canonical momenta πi conjugate to φi and consider the classical Hamiltonian for a system of many particles H(π, φ) = X π2 i i 2 + S(φ), we can rewrite Eq. (5.24) as the canonical ensemble average of O 1 hOi = Z Z Y dφi O[φ]e −S(φ) = i 1 Zcan Z Y dφi dπi O[φ]e−H(π,φ) i where Zcan = Z Y dφi dπi e−H(π,φ) . i Now, 1 Zcan Z Y i dφi dπi O[φ]e −H(π,φ) RQ R dφi dπi O[φ] dEδ(H(π, φ) − E)e−E i R = RQ . −E i dφi dπi dEδ(H(π, φ) − E)e 95 The integral for hOi is reduced to the integral over a constant surface H(π, φ) = Ē in the phase space divided by its volume where Ē is given implicitly as [247] ∂s(E) =1 ∂E E=Ē with the “entropy,” s(E), defined as [247] Z Y s(E) = ln dφi dπi δ(H(π, φ) − E). (5.25) i The ergodic hypothesis [274] now says that this integral is equal to the long time average of O over the integral curve of the Hamiltonian vector field, which is a solution to the Hamilton’s equations of motion [275] ∂H(π, φ) = πi ∂S(π, φ) ∂πi ∂S(π, φ) ⇒ φ̈i = − ∂φi ∂H(π, φ) =− π̇ = − ∂φi ∂φi φ̇ = (5.26) That is, Z hOimic E=Ē T = lim T →∞ 0 dτ O(φ(τ )). (5.27) where τ is the simulation time. Note since O does not depend on π, we can average only over the trajectory φ(τ ). The integration along with the trajectory φ(τ ) is done numerically, for example, by the leap-frog method [247]. In doing so, we discretize Eq. (5.26) in the time with the time step . Then, [247] 2 φ̈i (τ ) + O( 3 ) 2 2 πi (τ + ) = πi (τ ) + π̇i (τ ) + π̈i (τ ) + O( 3 ). 2 φi (τ + ) = φi (τ ) + φ̇i (τ ) + Since φ̇i = πi (τ ) and φ̈i (τ ) = πi (τ ) = −∂S/∂φi (τ ) according to the Hamilton’s equations, X ∂2S 1 ∂S ∂S π̈(τ ) = − πj (τ ) = − + O( 3 ). ∂φi (τ )∂φj (τ ) ∂φi (τ + ) ∂φi (τ ) j Then, up to O( 3 ), [247] ∂S φi (τ + ) = φi (τ ) + πi (τ ) − 2 ∂φi (τ ) ∂S ∂S ∂S πi (τ + ) − = πi (τ ) − − . 2 ∂φi (τ + ) 2 ∂φi (τ ) ∂φi (τ + ) 96 Since up to O( 3 ), [247] πi (τ ) − ∂S 2 ∂φi (τ ) = πi (τ + /2), we have φi (τ + ) = φi (τ ) + πi (τ + /2) 3 ∂S πi τ + = πi τ + − . 2 2 ∂φi (τ ) (5.28a) (5.28b) In order to use this updating algorithm starting from the initial φ(0) and π(0), we need to first update π(0) by a half time-step /2 as πi (τ + /2) = πi (τ ) − ∂S + O( 2 ). 2 ∂φ(τ ) This initial updating step has a systematic error of order O( 2 ). A single updating step in Eq (5.28) has the error of order O( 3 ). However, after 1/ steps, the accumulated error from the updating steps is of the order O( 2 ). So the method suffers from a systematic error of O( 2 ) [247]. We can at least make the coefficient of 2 smaller in the systematic error by introducing a tunable parameter λ ∈ [0, 0.5] and updating π and ψ multiple times in a single leap-frog step [276–278]. The parameter λ is tuned so that the coefficient is minimized [276–278]. This is called the Omelyan algorithm and given as [276–278]: ∂S 1. πi (τ + λ ) = πi (τ ) − λ ∂φ(τ ) 2. φi (τ + /2) = φi (τ ) + 2 πi (τ + λ ) ∂S 3. πi (τ + (1 − λ) ) = πi (τ + λ ) − (1 − 2λ) ∂φ(τ ) 4. φi (τ ) = φi (τ + /2) + 2 πi (τ + (1 − λ) ) ∂S 5. πi (τ ) = πi (τ + (1 − λ) ) − λ ∂φ(τ ). In our numerical simulation, this integrator has been chosen. 5.3.3 Rational Hybrid Monte Carlo method In this subsection, we describes an algorithm by which we generated gauge configurations used in our study. The algorithm is called rational hybrid Monte Carlo method, a variant of hybrid Monte Carlo method, which in turn is a combination of molecular dynamics and Monte Carlo algorithms. 97 5.3.3.1 Hybrid molecular dynamics Equation (5.27) is correct only in the limit of an infinite number of degrees of freedom [247]. However, in a practical calculation, the simulation is performed in a finite lattice. So equality is only approximately valid. Also, since the ergodicity is not explicitly built into the algorithm, the ergodic hypothesis could fail [279]. The trajectory could fail to cover the energy shell uniformly due to some hidden conservation laws [279]. Even if it does, it could be very slow and inefficient [244]. We can measure this efficiency using the rate of decay of the autocorrelation coefficient for an observable O of arbitrary choice [244]. The autocorrelation, C, of O is defined as [244] Z ∞ C(τ ) = N dτ 0 δO(τ 0 + τ )δO(τ 0 ) 0 where N is a normalization constant chosen to ensure C(0) = 1 and δO = O[φ(τ )] − hOi [244]. The slower the decay of this autocorrelation, the longer we need to integrate O[φ(τ )] along with the trajectory φ(τ ) to make the variance in the expected value of O suitably small [244]. So it is suitable to adopt a trick to avoid this possible inefficiency. These considerations suggest that it is better to incorporate some stochastic processes into this algorithm. Duane and Kogut proposed to combine the Langevin approach with Molecular Dynamics, a hybrid molecular dynamics method [279]. Ref. [247] presents the hybrid molecular dynamics (HMD) algorithm incorporated into the leap-frog scheme. According to it, 1. Pick some initial configuration φ consistent with the energy of the simulated system. 2. Choose π according to the probability distribution ! " # X1 Y 1 2 √ exp − π P (π) = 2 i 2π i i 3. Perform the numerical integration using, for example, the leap-frog method according to Eq. (5.28), for several time steps and collect the configurations. 4. Go back to (1). Step (2) introduces randomness into the algorithm, and the resulting path is more ergodic than the one generated using molecular-dynamics algorithm. However, it still suffers from a systematic error due to discretization in the Monte Carlo time step. For more detail, please see Ref. [280]. For details of improvements to the leap-frog method, please see Ref. [244]. 98 5.3.3.2 Hybrid Monte Carlo method To overcome the discretization error, we can introduce a Metropolis step in the hybrid method to ensure that the detailed balance condition is met [247]. The idea was originally proposed by Scalettar, Scalapino, and Sugar in Ref. [281] as a modification to Langevin approach [247]. This method was applied to full QCD later by Gottlieb et al. in Ref. [282]. Duane, Kennedy, Pendleton, and Roweth also proposed a similar modification to the hybrid method in Ref. [283]. They changed Step (3) of HMD slightly and inserted a Metropolis test after the step. Namely, [247] 1. Pick some initial configuration φ consistent with the energy of the simulated system. 2. Choose π according to the same distribution used in Step (2) of the HMD algorithm. 3. Perform the leap-frog integration according to Eq. (5.28) for several time steps. 4. To the last configuration (φf , πf ) obtained in the previous step, apply the Metropolis test. Namely, accept it as a new configuration with the probability p = min{1, e−H[πf ,φf ] /e−H[π,φ] } where H is the Hamiltonian used for Molecular Dynamics update. 5. If accepted, update the configuration to φf ; otherwise keep the configuration as it is. 6. Go back to (2). If Hamilton’s equations are integrated exactly, the updated configuration will always be accepted because there will be no difference in the value of H evaluated at the initial and final states. In practice, we solve the equations numerically, and there will be a discretization error. So the Metropolis algorithm is nontrivial and eliminates the systematic error [247]. As a matter of fact, the resulting transition probability satisfies the detailed-balance condition [247]. For its proof, see Ref. [247]. 5.3.3.3 Rational hybrid Monte Carlo method The rational hybrid Monte Carlo (RHMC) method is the hybrid Monte Carlo method where Seff is obtained using the φ-algorithm in Section 5.3.1.1 with the fractional power of the fermion matrix approximated by a rational function [269, 270]. This is the algorithm 99 used to generate most of the configurations used in this study. For the ensembles with smaller lattice spacing, in particular a ≤ 0.09 fm, the rational hybrid molecular dynamics (RHMD) method is used, which is HMD method with an analogous replacement of the fractional power of the fermion matrix by a rational function and reduces computer time significantly compared with RHMC [257]. Details of the simulation can be found in Ref. [257, 258]. Here, we briefly summarize it. The action used to generate the lattices consists of a one-loop Symanzik improved gauge action and the HISQ action. The gauge action is discussed in Section 5.1, and the fermionic action in Section 5.2. The action contains four quark flavors, namely u, d, s, and c. It also contains two other parameters: lattice spacing a and the color coupling constant g. The ensembles of lattices are generated at four different approximate values for a: a ' 0.06, 0.09, 0.12, and 0.15 fm [257]. The masses of the strange and charm quarks, ms and mc , respectively, are fixed at their physical values [257]. The light-quark masses are approximated to be degenerate and set to be equal to each other, i.e., mu = md = ml [257]. Configurations with three different values of ml are generated, namely ml = ms /5, ms /10, and the value such that the Goldstone pion mass is as close as possible to the physical pion mass, approximately given as ms /27 [257]. For a more detailed account of the parameter setting and determination of the parameter values, see Ref. [257, 258]. In generating configurations using RHMC, different step sizes are used for the gauge and fermion parts of the action [257]. Three gauge steps are taken for each fermion step [257]. To see what it means, first recall that the numerical integration was done using the Omelyan algorithm [276–278]. This algorithm is employed both for the gauge and fermion part of the action [257]. In deriving the fermion part from the determinant, the φ-algorithm from Section 5.3.1.1 is used. Here, there are four determinants for each flavor of sea quarks used in our simulation, namely up, down, strange, and charm, and the fractional power of each fermion determinant is approximated with a rational function approximation [257]. So we evaluate hOi = 1 Z Z Y h i1/4 e−SG dUµ (x)O[Uµ ] det M † M x 1/4 while stochastically estimating the fermionic determinant det M † M using the pseud −1/4 ofermion after replacing M † M by a rational function r M † M as 100 h i1/4 Z det M † M = Dϕ∗ Dϕ exp −ϕ∗ r M † M ϕ In the φ-algorithm proposed in Ref. [284], the variable ϕ is distributed according to the Gaussian distribution. Ideally, we refresh it at each updating step as r M † M depends on Uµ . However, this is very expensive. So r M † M is fixed for several steps for updating the gauge links, and we refresh r M † M only after those steps. Then, 1 hOi = Z Z Y dUµ (x)dϕ∗ (x)dϕ(x)O[Uµ ]e−Seff x where Seff = SG + SF = SG + ϕ∗ r M † M ϕ. The resulting action is used to evolve the trajectory in molecular dynamics time. In our case, we have φi = Uµ (x) where i = (x, µ), and we need to update the link variables according to the algorithm in Section 5.3.3.2. This would require evaluation of ∂Seff /∂Uµ (x) and ∂Seff /∂ϕ(x). The discussion on how they are done can be found in Ref. [244]. At the end of the trajectory, the resulting configuration is tested for acceptance at the Metropolis step. There, the exact Hamiltonian is used to make the RHMC algorithm exact [257]. RHMD is used when the lattice is very large or lattice spacing is very small, both requiring a tiny step size to ensure that the change in the action is small over the trajectory for a good acceptance rate [257]. RHMD simply omits the Metropolis test from RHMC [257]. The lattices with a ≤ 0.09 fm are generated with RHMD, although those with a ' 0.09 fm and ml = ms /5 and ms /10 are generated with RHMC [257]. The parameter values of the lattices used in our study are summarized in Table 5.1 [257]. 101 Table 5.1: Parameter values used in the simulation for generating lattice ensembles used in our study. The first column gives the approximate lattice spacing in fm. The second gives the size of each lattice. Ns indicate the spatial size in units of a, and Nt , the temporal size. 10/g 2 gives the value for the coupling constant. The masses are given in lattice units. ams and amc are equal or close to its physical value. u0 is a tadpole coefficient, which is explained in Section 5.1. It was sufficient to set it equal to 1 for the purpose of our study. N is the Naik term for the charm quark, the mass-dependent correction to the tree-level improvement of the charm quark dispersion relation, discussed in Section 5.2. ≈ a (fm) 0.15 0.12 Ns3 × Nt 323 × 48 483 × 64 10/g 2 5.80 6.00 aml 0.002426 0.001907 ams 0.06730 0.05252 amc 0.8447 0.6382 N −0.358920 −0.2309 CHAPTER 6 LATTICE COMPUTATION As is explained in Chapter 3, the precise theoretical determination of the anomalous magnetic moment of the muon requires a reduction in the uncertainties from hadronic contributions to the quantity. Since the leading hadronic contribution is from hadronic vacuum polarization, we need a precise calculation of this quantity. In this chapter, the lattice QCD computation of hadronic vacuum polarization is detailed. In particular, the focus is on a disconnected component of such contribution. The computation is performed by using the highly-improved staggered quark (HISQ) formulation for the current density with gauge configurations generated with four flavors of HISQ sea quarks. 6.1 Hadronic Vacuum Polarization (HVP) From Section 2.2.4, we know that aµ = FM (q = 0), and so we need to compute µ− (pf ) j µ (0) in em µ− (pi ) out ≡ −ieū(pf )Γµren (pf , pi )u(pi ) at q = pf − pi → 0. This means that we need to evaluate the various contributions to Γµren (pf , pi ) = γµ FE (q 2 ) + iσ µν qν FM (q 2 ). 2m The Feynman diagram for the leading-order hadronic contribution is shown in Fig. 6.1. The leading-order hadronic contribution to Γµ and thus to the anomalous magnetic moment arises from the QCD radiative corrections to the virtual photon in the Schwinger term. As is indicated in the figure, the computation requires the calculation of the HVP of 103 Figure 6.1: The Feynmann diagram for the leading-order HVP contribution to aµ . The bubble labeled by Hadronic contains all the QCD radiative corrections but not corrections from other sectors of the SM. the photon propagator: µν 2 µν 2 µ ν 2 Z Π (q ) = (δ q − q q )Π(q ) = d4 xeiqx hJ µ (x)J ν (0)i , (6.1) and the leading-order HVP contribution to aµ can be written in terms of Πµν (q 2 ) as [285] aLOHVP µ = 4α 2 Z ∞ dq 2 f (q 2 )Π̂(q 2 ). (6.2) 0 Here, f (q 2 ) is the kernel function, f (q 2 ) = m2µ q 2 A3 (1 − q 2 A) , 1 + m2µ q 2 A2 where mµ is the mass of the muon, α is the fine-structure constant, q A= q 4 + 4m2µ q 2 − q 2 2m2µ q 2 , and Π̂(q 2 ) = Π(q 2 ) − Π(0) is the renormalized photon vacuum polarization. We denote the current-current correlation tensor as Cµν (x, y) = hJµ (x)Jν (y)i . Then, 2 Πµν (q ) = Z d4 xeiqx Cµν (x, 0). 104 When ~q = 0, we can write Π̂(q 2 ) in terms of the time-slice correlation function of the QCD electromagnetic current, C(t), which is defined as the average over spatial components of the diagonal entries of Cµν projected onto zero momentum component, namely, 1X C(t) = 3 Z d~xCii (x, 0). i For this, first note that when ~q = 0, Eq. (6.1) becomes for i = 1, 2, 3 Πii (ω, ~q = 0) = ω 2 Π(ω 2 ) where we have written q4 = ω. Since this is independent of i, ω 2 Π(ω 2 ) = 1X 1X Πii (q) = 3 3 i Z i ∞ dteiωt Z Z ∞ dteiωt C(t), d~xCii (x, 0) = −∞ −∞ and so Z 1 Π(ω ) = 2 ω 2 ∞ dteiωt C(t). −∞ When ω is small, we can expand Π(ω 2 ) as 2 Z ∞ Π(ω ) ∼ dt −∞ t2 C(t) it + C(t) − C(t) + · · · ω2 ω 2 Now, using the translation invariance of Cµν (x, y), Z 1X d~xCii (−t, ~x, 0) 3 i Z 1X = d~xCii (−x, 0) 3 i Z 1X = d~xCii (0, −x) 3 i Z 1X = d~xCii (x, 0) 3 C(−t) = i = C(t). Hence, C(t) is an even function of t. Thus, 2 Z Π(ω ) ∼ ∞ dt 0 2C(t) − t2 C(t) + · · · ω2 . (6.3) 105 so that Π̂(ω 2 ) = Π(ω 2 ) − Π(0) iωt Z ∞ e − 1 t2 dtC(t) = + ω2 2 −∞ Z ∞ cos(ωt) − 1 t2 + =2 dtC(t) ω2 2 0 2 Z ∞ 1 − cos(ωt) t dtC(t) =2 − 2 ω2 0 Z ∞ 2 4 sin (ωt)/2 2 dtC(t) t − . = ω2 0 Therefore, in the continuum, Π̂(q 2 ) where ~q = 0 can be computed via [286] ∞ Z 2 Π̂(q ) = 0 4 sin2 (qt/2) dt t − q2 2 C(t). On the lattice, we have X 4 sin2 (qt/2) 2 C(t). Π̂(q ) = t − q2 t 2 (6.4) In the above equation, C(t) is defined analogously to the continuum counterpart, i.e., C(t) = 1X hJk (~x, t)Jk (0)i 3 (6.5) ~ x,k with k = 1, 2, 3 and Jk a QCD electromagnetic current. Here, hi indicates the average over gauge as well as fermion configurations. The lattice correlation function C(t) can be divided into two parts: quark-line connected and disconnected. Figure 6.2 shows the schematic diagrams associated with these two pieces. To illustrate how these two terms arise, we consider the nonconserved local lattice current density in naive fermion theory for simplicity. It is given as Jµ = i X Qf ψ̄f γµ ψf f where f = u, d, s, c and Qf is the charge of the quark of flavor f in units of the electron charge e. We start with the correlation tensor: Cµν (x, y) = hJµ (x)Jν (y)i . (6.6) 106 (a) Connected Contribution (b) Disconnected Contribution Figure 6.2: The Feynman diagrams for the contribution to aµ from the leading-order HVP. The quark loop in (a) is radiatively corrected by virtual gluons and sea quarks (not shown). In the same way, the quark loops of flavor f and f 0 in (b) are connected by virtual gluons and sea quarks (not shown). We can take the fermionic path integral explicitly as follows. hJµ (x)Jν (y)iF = − X D E Qf1 Qf2 ψ̄ f1 (x)γµ ψ f1 (x)ψ̄ f2 (y)γµ ψ f2 (y) f1 ,f2 = X Q2f tr [γµ Mf−1 (x, y)γµ Mf−1 (y, x)] f − X Qf1 Qf2 tr [γµ Mf−1 (x, x)]tr [γν Mf−1 (y, y)]. 1 2 f1 ,f2 The first contribution in the above expression is called connected part corresponding to Fig. 6.2a, and the second disconnected part to Fig. 6.2b. To compute the current-current correlation function needed for computation of aLOHVP , µ we first need to find an expression for the electromagnetic current density on the lattice with a given action. For this, we need to perform global phase transformation of ψ(x) → (1 + iω)ψ(x) where ω is a small real number. The Noether current associated with it will be the conserved electromagnetic current on the lattice. However, we know that the electromagnetic current couples to the electromagnetic gauge field Aem µ . Also, any definition of currents is acceptable on the lattice as long as it converges to the correct continuum limit. So we compute the correlation function in Eq. (6.4) by first restoring the electromag netic link variable Uµem (x), which we write as Uµem (x) = exp −ieAem µ (x) , and taking the functional derivative of Z in Eq. (5.3) from Chapter 5. For example, for the simplest naive 107 action, we would have SFnaive =a 4 X ψ̄(x) x ! em (x)ψ(x − aµ̂) Uµ (x)Uµem (x) − U−µ (x)U−µ γµ + mψ(x) . 2a X µ In evaluating the time-slice correlation function in Eq. (6.5), we need only the spatial diagonal entries of Cµν , i.e., Cii for i = 1, 2, 3. So we consider only the spatial components em em 2 of the current. Also, we rescale Aem µ as eAµ → Aµ . The factor of e is explicitly taken out from Π̂(q 2 ) in Eq. (6.2) and brought to the front, resulting in 4α2 in the equation. In the end, we set Aem µ = 0. As a first step toward the current-current correlation function Cµν (x, y), we compute Jµstg (x) = δ ln Z δAem µ (x) ≡ D Jµstg Aem µ =0 F (x) E G where F indicates the path-integration over fermionic degrees of freedom and G, the pathintegration over gauge degrees of freedom. Since the part in the action that depends on Aem µ is the determinant i h det M 1/4 = exp tr ln M 1/4 , we have Jµstg F δ det M 1/4 1 (x) = det M 1/4 δAem µ (x) . Aem µ =0 Inside of the trace, we have for general matrix X eX δX = δXeX so that Jµstg F δ det M 1/4 1 = det M 1/4 δAem µ (x) Aem µ =0 1 δM = tr M −1 em . 4 δAµ (x) Aem =0 µ Then, 108 Cµν (x, y) = hJµ (x)Jν (y)i|Aem =0 µ δ2Z 1 em Z δAem µ (x)δAν (y) Aem =0 µ Z h i 1 δ † stg 1/4 −SG = DU DU J (y) det M e ρ ρ ν F Z δAem µ (x) Aem µ =0 + * δ 1 Jνstg (y) F + Jµstg (x) F Jνstg (y) F = (x) det M 1/4 δAem µ = . G Aem µ =0 From now on, we make the setting of Aem µ = 0 implicit for notational simplicity. The first term corresponds to the connected part and the second, the disconnected part. In this / HISQ + mf . We are interested in work, we have adopted the HISQ action, and so Mf = D disconnected part. 6.2 HVP Disconnected Part The disconnected part of Cµν (x, y) is disc Cµν (x, y) = D Jµstg (x) F Jνstg (y) E F G where Jµstg F 1 δM −1 = tr M . 4 δAem µ (x) To reduce the computational complexity, instead of taking the functional derivative of the / HISQ Dirac matrix Mf = D HISQ + mf for each flavor f , we use the simpler staggered Dirac matrix with no improvement: Mfstg (x, y) = X em (x) αµ (x) Uµ (x)e−iQf Aµ µ em (y) δx+aµ̂,y − Uµ† (y)δx−aµ̂,y eiQf Aµ 2 + mf δx,y . Then, for the connected part, we have δ δAem µ (x) Jνstg (y) F δM stg −1 = tr M δAem δAem µ (x) ν (y) δ 2 M stg δM stg δM −1 −1 = tr M + em em δAem δAem µ (x)δAν (y) ν (y) δAµ (x) stg δM stg −1 δM = tr M M −1 em (x) δAem (y) δA ν µ " # stg stg δM δM f = tr M −1 em M −1 em δAν (y) δAµ (x) δ 109 where we have noted δ 2 Mfstg " em (x0 ) Uν (x0 )e−iQf Aν − iQ α (x ) (x , y ) = ν f em δAem δAem 2 µ (x)δAν (y) µ (x) 0 δ 0 0 δx0 +aν̂,y0 δx0 ,y Uν† (y 0 )δx0 −aν̂,y0 eiQf Aν + 2 !# em (y 0 ) " = −Q2f αν (x0 ) em (x0 ) Uµ (y)e−Qf Aµ 2 δx0 +aν̂,y0 δy,y0 δx0 ,x em (y 0 ) Uν† (y 0 )δx0 −aµ̂,y0 eiQf Aµ − 2 # δy0 ,y δx,y δµν stg 0 0 = −δx,y δµν Q2f Mf,µ (x , y ), i h stg Mf−1 vanishes when averaged over gauge configuration due to rotational and tr Mf,µ symmetry of the vacuum. The remaining term is a sum of four terms: tr αµ (x)Uµ (x)M −1 (x + aµ̂, y)αν (y)Uν (y)M −1 (y + aν̂, x) h i tr −αµ (x)Uµ (x)M −1 (x + aµ̂, y + aν̂)αν (y + aν̂)Uν† (y)M −1 (y, x) h i tr −αµ (x + aν̂)Uµ† (x)M −1 (x, y)αν (y)Uν (y)M −1 (y + aν̂, x + aµ̂) h i tr αµ (x + aµ̂)Uµ† (x)M −1 (x, y + aν̂)αν (y + aν̂)Uν† (y)M −1 (y, x + aµ̂) . For the disconnected part, δMfstg " tr δAem µ (x) # Mf−1 = − i iQf αµ (x) h tr Uµ (x)Mf−1 (x + aµ̂, x) + U † (x)Mf−1 (x, x + aµ̂) . 2 Since M is antihermitian, Mf−1 (x, x + aµ̂) = − Mf−1 (x + aµ̂, x) † so that ∗ tr Uµ† (x)Mf−1 (x, x + µ̂) = − tr Uµ (x)Mf−1 (x + µ̂, x) . This implies " tr δMfstg (x, y) δAem µ (x) # Mf−1 † iQf αµ (x) −1 −1 † =− tr Uµ (x)Mf (x + aµ̂, x) − U (x) Mf (x + aµ̂, x) 2 h i = Qf αµ (x)Im tr Uµ (x)Mf−1 (x + aµ̂, x) . 110 Thus, Jµstg F 1 δM −1 (x) = tr M 4 δAem µ (x) h i X αµ (x) = Qf Im tr Uµ (x)Mf−1 (x + aµ̂, x) 4 (6.7) f = 1X Qf jf,µ (x) 4 f where we have defined h i jf,µ (x) ≡ Im tr αµ (x)Uµ (x)Mf−1 (x + aµ̂, x) . (6.8) As we discussed when deriving the expression for the current to be used in our study, we have made some simplifications, and the resulting current is different from the conserved current which can be derived based on Noether’s thoerem. This means that the current in Eq. (6.7) is not conserved, and we need a vector-current renormalization constant ZV . Keeping this in mind, we can write the correlation function in Eq. (6.5) as 3 Cdisc (t) = = 1 XX 2 ZV hhJistg (t, ~x)iF hJistg (0)iF iG 3 i=1 ~ x 3 XX 1 48 i=1 ZV2 X Qf Qf 0 jf,µ (~x, t)jf 0 ,µ (0, 0) G . (6.9) f,f 0 ~ x The disconnected part of the HVP can then be obtained from Πdisc HVP (q 2 ) = a4 X eiqt Cdisc (t). t 6.3 Algorithm In this section, we will describe the algorithm used in computing the disconnected part of the correlation function. 6.3.1 Calculation of the current density To compute Cdisc (t), we need to compute disc Cµν (x, y) = D Jµstg (x) F Jνstg (y) E F G . D E D E The product Jµstg (x) Jνstg (y) is averaged over gauge configurations. Since the curF D EF rent density Jµstg (x) has dependence on gauge configurations, we need to compute F 111 D Jµstg E F (x) in Eq. (6.7) for each configuration. We focus on its calculation on a single gauge configuration. For this, we need to invert each Mf where f runs over flavors. In the MILC convention, Mf is defined as twice the conventional Dirac matrix. As the space-time is now discrete, Mf can be treated as a matrix indexed by lattice sites as well as other quantum number such as spin and color. In doing so, it is convenient to classify lattice sites into even sites and odd sites and reorder site indicies so that even sites come before the odd sites. The matrix written in this manner is said to be in the even-odd decomposed form. We write Mf in the even-odd decomposed form. We call a lattice site x even if X xi i a ≡ 0 mod. 2. Odd sites are similarly defined. Then, we can write Mf as [244] / + 2mf = MfMILC = 2Mf = D 2mf Doe Deo 2mf / = 2D / HISQ . Also, for staggered fermion, due to antihermiticity of Mf [244], where D † Deo = −Doe . Thus, MfMILC / + 2mf = = 2Mf = D 2mf Deo † −Doe 2mf . We will drop the superscript MILC. Unless specified, Mf means the Dirac matrix in the MILC convention. In this form, its inverse can be written as Mf−1 = 2mf A−1 −A−1 f f Deo † −1 Deo Af 2mf Bf−1 where † Af = 4m2f + Deo Deo † Bf = 4m2f + Doe Doe . ! (6.10) 112 Note, by definition, Af acts only on even sites, and Bf on odd sites. Also, they are hermitian. † To see that Mf−1 in Eq. (6.10) is indeed the inverse of Mf , note that since Deo = −Doe , −1 † † −1 Deo A−1 = A (D ) f eo f † −1 † −1 = 4m2f (Deo ) + (Deo ) Deo Deo † −1 † −1 † = 4m2f (Deo ) + (Deo ) Doe Doe † −1 = (Deo ) Bf † = Bf−1 Deo . Similarly, −1 A−1 f Deo = Deo Bf . Using these two properties, we can show that Mf−1 in Eq. (6.10) is the inverse of Mf . As the equation for the current density in Eq. (6.7) contains the off-diagonal elements of Mf−1 , † −1 namely Mf (x + aµ̂, x), we need to know either −A−1 f Deo or Deo Af . This in turn requires A−1 f . We need to invert Af . In particular, we need to know a particular entry of Mf−1 , namely Mf−1 (x + aµ̂, x) for each x and µ for the computation of jf,µ (x). One way to do this is to solve a linear system with a color point source sx0 ,a (y, b) = δx0 ,y δa,b for site x0 and color a Mf gx0 ,a = sx0 ,a where gx0 ,a is a color-vector field and a, b are color indicies. If we multiply gx0 ,a by sx1 ,b , then we obtain Mf−1 (x1 , x0 )a,b = s†x1 ,b gx0 ,a . So we can compute the current density at each lattice site using the point sources sx,a by setting x0 = x and x1 = x + aµ̂ and repeating this for every combination of color indicies. The problem with this approach is that we need to perform an inversion as many times as there are lattice sites multiplied by the number of components of the current density we need to know, which in our case is three, and the number of entries of the color matrix, which is nine. The smallest lattice we have employed in this study contains 324 ×48 = 1 572 864 ∼ 1.5 million sites. So we need to perform about 40.5 million inversions, and this requires massive computer power and a long time, which is impractical. 113 Hence, in this work, we estimate the entries of Mf−1 stochastically using random Z(4) color-vector fields with support on the entire four-volume Ω [287–299]. In particular, we have used the field such that each complex-valued color vector at each site takes the value randomly from the set of four elements: 1 1 1 1 √ (1 + i), √ (1 − i), √ (−1 + i), √ (−1 − i) . 2 2 2 2 The hope is that we can obtain a good enough estimate with much a smaller number of inversions. A stochastic color-vector field ξk (x) with k ranging from 1 to Nr satisfies [300] 1 X ξk (x)ξk† (y) = δx,y 1c . Nr →∞ Nr lim k where 1c is a unit color matrix. Each entry is normalized as ∗ (x)ξk,a (x) = 1. ξk,a (6.11) where a denotes a color index. Using these properties, we can estimate the color matrix Mf−1 (x, y) as 1 X 1 X −1 ζk (x)ξk† (y) = lim Mf (x, z)ξk (z)ξk† (y) Nr →∞ Nr Nr →∞ Nr Mf−1 (x, y) = lim k k where ζk = Mf−1 ξk . If we write (e) ξk = ξk (o) ξk ζk = ζk (o) ζk and (e) ! (6.12) ! (6.13) we have ζk = Mf−1 ξk (e) = (o) −1 2mf A−1 f ξk − Af Deo ξk ! † −1 Deo A−1 f ξk + 2mf Bf ξk   (e) (o) A−1 2m ξ − D ξ eo k f k f  =  −1 † (e) (o) Bf Deo ξk − 2mf ξk ! † (e) A−1 [M ξ ] k f f = . Bf−1 [Mf† ξk ](o) (e) (o) 114 Now, for x even 1 X h −1 i Mf ξk (x + aµ̂)ξk† (x) Nr →∞ Nr k i 1 X h † −1 (e) (o) (e)† Deo Af ξk + 2mf Bf−1 ξk (x + aµ̂)ξk (x) = lim Nr →∞ Nr k 1 X h † −1 (e) i (e)†(x) Deo Af ξk (x + aµ̂)ξk = lim , (6.14) Nr →∞ Nr Mf−1 (x + aµ̂, x) = lim k and for x odd 1 X h −1 i Mf ξk (x + aµ̂)ξk† (x) Nr →∞ Nr k i 1 Xh (e) (o) (o)† −1 = lim 2mf A−1 ξ − A D ξ (x + aµ̂)ξk (x) eo f k f k Nr →∞ Nr k i 1 Xh (o) (o)† D ξ (x + aµ̂)ξk (x). (6.15) −A−1 = lim eo k f Nr →∞ Nr Mf−1 (x + aµ̂, x) = lim k (p) (p0 )† Terms involving ξk (x)ξk (y) with p 6= p0 average out because x cannot be equal to y. We can make this exactly 0 by starting with ξk = (e) ! 0 ! ξk 0 or ξk = (o) ξk . Substituting these expressions into Eq. (6.8), we obtain a stochastic estimation of jf,µ : jf,µ (x) = Im tr αµ (x)Uµ (x)M −1 (x + aµ̂, x) " # 1 X h † −1 (e) i (e)† = Im tr αµ (x)Uµ (x) Deo Af ξk (x + aµ̂)ξk (x) Nr k h h i i 1 X (e)† (e) † = Im ξk (x)αµ (x)Uµ (x) Deo A−1 ξ (x + aµ̂) f k Nr k for x even, and jf,µ (x) = Im tr αµ (x)Uµ (x)M −1 (x + aµ̂, x) " # i 1 Xh (o) (o)† = Im tr αµ (x)Uµ (x) −A−1 (x + aµ̂)ξk (x) f Deo ξk Nr k h h i i 1 X (o)† (o) =− Im ξk (x)αµ (x)Uµ (x) A−1 D ξ (x + aµ̂) eo k f Nr k for x odd. 115 6.3.2 Deflation Stochastically estimating jf,µ requires a lot of inversions (solving a linear system Af g = (e) (o) ξk or Deo ξk ) to be performed, and this can be expensive, especially when the ratio of the largest eigenvalue to the smallest eigenvalue of Af , called the condition number, κ(Af ) = λmax /λmin , (the ratio of the largest to smallest eigenvalue), is large. The condition (e) number quantifies the potential sensitivity of the solution to the change in either ξk or Af and puts an upper bound on the related error in the solution g [301]: ||δAf || |δg| ≤ κ(Af ) |g + δg| ||Af + δAf || and (e) δξk |δg| ≤ κ(Af ) . (e) (e) |g + δg| ξk + δξk We can obtain a solution with the same accuracy but less computational cost if we can increase the condition number. Since κ(Af ) depends on the eigenvalues, we can achieve this by using a better-conditioned matrix with higher λmin . This technique of removing the low-eigenvalue part from Af is called deflation [295, 302–314]. In our work, we apply † , instead of Af , which deflation to Af in terms of the eigenvectors and eigenvalues of Deo Deo † is related to Deo Deo by a constant shift of 4m2f . / in Mf in the even-and-odd decomposition as For this, first write D /= D 0 Deo . † 0 −Deo Then, (e) vn / D (o) vn ! (e) vn (o) vn = iλn ! , / and vn(e) and vn(o) are even and odd parts of the nth Here, λn is an nth eigenvalue of D, / The eigenvalues are ordered in an ascending manner, and eigenvectors eigenvector of D. (e) (o) are normalized to 1. The relative phase of vn and vn are fixed by the above definition, and even and odd nth eigenvectors are related to each other as follows: (o) vn(e) = Deo vn iλn † Deo vn . iλn (e) vn(o) = − 116 † This also means that vn is the nth eigenvector of Deo Deo with its eigenvalue λ2n , and (e) † vn is the nth eigenvector of Doe Doe with its eigenvalue λ2n . In fact, (o) † (e) Deo Deo vn = −iλn Deo vn(o) = λ2n vn(e) , and † (o) † † (e) Doe Doe vn = Deo Deo vn(o) = iλn Deo vn = λ2n vn(o) . −1 Then, we can write A−1 f and Bf without making stochastic estimation using the eigenvalues and eigenvectors as follows: A−1 f (x, y) = X vn(e) (x)vn(e)† (y) λ2n + 4m2f n Bf−1 (x, y) = X vn(o) (x)vn(o)† (y) λ2n + 4m2f n . † , we can obtain A−1 If we compute all eigenpairs of Deo Deo f exactly with no statistical error. However, in reality, each eigenvector has as many components as there are lattice sites and colors, and there are as many eigenvectors as there are lattice sites and colors. So it costs too much in memory and computation time. We need to strike a balance. For deflation −1 of A−1 f , we remove the low-mode part of Af by computing the lowest Nev eigenpairs and write: A−1 f = Nev (e) (e)† X vn (x)vn (y) n=1 λ2n + 4m2f + A−1 f − Nev (e) (e)† X vn (x)vn (y) n=1 λ2n + 4m2f ! . If we define the projection operator onto the space spanned by the low-lying Nev eigenvectors as (e) PNev = Nev X vn(e) vn(e)† . n=1 we can write A−1 f as (e) −1 (e) −1 A−1 = A P + A 1 − P Nev Nev . f f f (e) By computing A−1 1 − P , which now has a lower condition number, via stochastic N f ev estimation with Nr random sources, we now obtain for even x h i (e)† † (e) Nev Deo vn (x + aµ̂)vn (x) X 1 X (e) (e)† + ζk (x + aµ̂)ξk (x) Mf−1 (x + aµ̂, x) = 2 2 Nr λn + 4mf n=1 k where (e) (e) (e) † ζk = Deo A−1 1 − P Nev ξk . f 117 Similarly for odd x, Mf−1 (x + aµ̂, x) = − h i (e) (e)† Nev vn (x + aµ̂) vn Deo (x) X λ2n + n=1 4m2f + 1 X (o) (o)† ζk (x + aµ̂)ξk (x) Nr k where (o) (o) (o) ζk = −A−1 f (1 − PNev Deo ξk . −1 −1 Here, since A−1 f Deo = Deo Bf , we can write the low-mode part of Mf (x + aµ̂, x) as h i h i (e) (e)† (o) (o)† N N ev ev v (x + aµ̂) v D (x) D v (x + aµ̂)vn (x) X n X n eo eo n − =− . λ2n + 4m2f λ2n + 4m2f n=1 n=1 Then, we can write jf,µ (x) as a sum of the low-mode part constructed from the eigenvectors and the stochastically estimated high-mode part: high low (x). (x) + jf,µ jf,µ (x) = jf,µ Denoting p = (e, o), the parity of the given site x, and p̃ = (o, e), its conjugate, and using † low (x) as Deo = −Doe , we can write jf,µ h i  (p)† (p) vn (x)αµ (x)Uµ (x) Dp̃p vn (x + aµ̂) low  (x) = − Im  jf,µ 2 + 4m2 λ n f n=1 Nev X  high (x) as and jf,µ high jf,µ (x) = Nr X Im (p)† ξk (x)αµ (x)Uµ (x) Mf−1 p̃p k=1 (1 − (p) (p) PNev )ξk (x + aµ̂) . The even eigenpairs are generated using a software package called PRIMME [315]. (e) The numerically estimated nth eigenvector ṽn and eigenvalue λ̃n are determined with the residual εev = Deo Deo ṽn(e) − λ̃2n ṽn(e) . The odd eigenvectors are reconstructed from the even eigenvectors as follows: † Deo ṽn (e) ṽn(o) = − † Deo ṽn (e) (e) . (e) The estimated even eigenvector ṽn can be written in terms of true eigenvectors vn : ṽn(e) = X m (e) cm vm 118 (e) where cm ∈ R. For a precise estimate of vn , the coefficients cm ’s must be small except for † m = n. However, when we multiply ṽ (e) by −Deo , we have † (e) −Deo ṽ = −i X (e) λm cm vm , m and the coefficients for larger m are relatively enhanced by the eigenvalues λm ’s. Those with smaller λm are suppressed. So the reconstructed estimate for odd eigenvectors become more inaccurate. For this reason, when constructing the low-mode part of the current density at odd sites, we work with even eigenvectors, namely, jf,µ (x) = − h i (e) (e)† Nev vn (x + aµ̂) vn Deo (x) X λ2n + 4m2f n=1 . † . This requires a smaller number of multiplications by Deo or Deo 6.3.3 Truncated solver method high requires Nr inversions to be performed. Since Af is a large Stochastic estimation of jf,µ (e) sparse matrix, we can solve the system Af g = ξk numerically using an iterative algorithm such as conjugate gradient (CG) method [244]. If we perform more iterations, the residual of the solution via the CG method (e) εinv = ξk − Af gn , where gn is the estimate at nth step, decreases with increasing n. Also, the variance high associated with the stochastic estimation of jf,µ goes as O(1/Nr ). So to get an accurate estimate with good precision, we need to invert Af with a large ninv for a lot of random (e) sources ξk . This can be very expensive. Fortunately, people have found out that there is a correlation between the solutions with a large iteration number nfine and a fewer iteration number nslp if both nfine and nslp are large enough. Here, nfine is the number of iterations required to achieve a desired residual εfine , and nslp , the number of iterations to reach a larger residual εslp than εfine . This observation led to the development of the method to compute the stochastic estimate accurately with good precision at a cheaper cost than solving the equation at the residual εfine for all random sources. It is called the truncated solver method (TSM) [316–318]. This is the method we have adopted. 119 In the TSM algorithm, the iteration of solvers is truncated after nslp steps, which is smaller than the number of iterations required to achieve a desired accuracy with its associated residual εfine . The truncation of the iterative procedure introduces a bias in the solution and thus in the estimate. This bias is estimated separately by taking the difference of a sloppy truncated solve and a fine solve with nfine > nslp iterations. Due to the correlation, the bias can be estimated using a smaller number of random sources, nfine . Then, we can achieve a smaller variance with the same computer time if we solve the system sloppily many times to reduce the variance and compute the bias correction to improve the accuracy with a few solves. high Applying the TSM to our case, we write jf,µ (x) as the sum of the sloppy estimate and bias correction: (slp) (diff) high (x) = jf,µ (x) + jf,µ (x) jf,µ (slp) where jf,µ (x) is stochastically estimated using Nslp random sources with nslp iteration, while (diff) (fine) (slp) jf,µ (x) = jf,µ (x) − jf,µ (x) is estimated more precisely with nfine iterations using Nfine random sources, a much smaller number than Nslp . (slp) (diff) We use different sets of random sources to estimate jf,µ (x) and jf,µ (x). Then, the high variance of jf,µ (x), σ 2 , is given by [317, 318] 2 2 + σdif σ 2 = σslp f = (slp) fslp ffine + Nslp Nfine (6.16) (diff) 2 and σ 2 where σslp dif f are the variance of jf,µ (x) and jf,µ (x), respectively. The factor fslp depends on nslp , and ffine , nfine and nslp in general. To find the optimal value for nslp while constraining the computer cost, we first define a measure of the computer cost of stochastic estimation of the current density jf,µ , which is approximately given as C = nslp Nslp + nfine Nfine . Here, we have ignored the overhead such as generating a random source and data manage† † −1 † ment when shifting Deo A−1 f ξ in the µ̂ direction and taking the product of Deo Af ξ and ξ to get jf,µ . This is constant and negligible compared with the cost of CG iterations. 120 For finding the optimal values, we use the method of Lagrange’s multipliers. In doing so, we note that nfine is fixed, i.e., not optimized, and assume fslp is approximately independent of nslp to obtain an analytical expression. This is usually acceptable in practice, as fslp reaches to a plateau quickly well below nslp [319]. The result is [316–318] 1 fslp ffine 0 )2 nfine (ffine s fslp nfine = , fslp nopt slp nopt slp = Ropt which gives the optimal number of sloppy iterations nopt slp and the optimal ratio of sloppy inversions to fine inversions Nslp /Nfine , Ropt . Here, 0 ≡ ffine ∂ffine . ∂nslp 0 . So we need fslp , ffine , and ffine To proceed, following the discussion in Ref. [320], we define the correlation E slp fine (x) (x)∆jf,µ ∆jf,µ r(x) = rD E ED slp fine (x)2 (x)2 ∆jf,µ ∆jf,µ D (6.17) slp fine (x) is (x) is the sample deviation of the sloppy estimate from its mean and ∆jf,µ where ∆jf,µ the corresponding deviation of the fine estimate. Then, the various quantities are averaged over samples of size Nfine indicated by hi. We can write the variances as 2 E σslp slp ∆jf,µ (x)2 = Nfine D E σ2 f ine fine ∆jf,µ (x)2 = Nfine D slp 2 and σ 2 fine th where σslp f ine are standard deviation of jf,µ,k (x) and jf,µ,k (x) obtained with k random source ξk , respectively. Assume 2 σslp = σf2ine = σ02 . slp diff (x) are statistically independent, Then, since jf,µ (x) and jf,µ 121 D E D E slp diff σ 2 = ∆jf,µ (x)2 + ∆jf,µ (x)2 D E slp slp = ∆jf,µ (x)2 + ∆ jf,µ (x)2 Nfine D E slp fine + ∆jf,µ (x)2 − 2Cov jf,µ (x) σ02 σ02 Nfine fine , jf,µ (x) σ02 +2 − 2r(x) √ Nslp Nfine Nfine Nfine 1 2 2r(x) 2 = σ0 + − Nslp Nfine Nfine = = where Nfine σ02 [1 + 2R (1 − r(x))] . Nslp indicates that the quantity is estimated using Nfine random sources. Comparing this with Eq. (6.16), we obtain fslp = σ02 ffine = 2σ02 (1 − r(x)) . Thus, nopt slp = Ropt = 2 (1 − r(x)) nfine r0 (x)2 s nfine 2 (1 − r(x)) nopt slp where r0 (x) = ∂r(x) . ∂nslp As we will see in the next section, r can be modeled as ln(1 − r) = a ln(εslp − εfine ) + b where a, b ∈ R and εslp and εfine are the residuals of the solutions g in Af g = ξk corresponding to nslp and nfine , respectively. Numerical evidence suggests that after a few initial steps it decreases exponentially as εslp = exp (cnslp + d) . That this is indeed so in our simulation is shown in Section 6.4.3. Therefore, exp −acnopt − ad − b slp nopt , slp = 2 2a c2 nfine 122 which we need to solve for nopt slp and Ropt = −acnfine . In general, nopt slp depends on lattice sites as r does, but we need to use the same stopping condition nslp for estimating the values of the current densities at all sites. So in our study, we have averaged r(x) over all sites r̄ = 1 X r(n) |Ω| (6.18) n∈Ω to obtain a single number capturing an overall behavior of the correlation r. 6.3.4 Dilution Another technique to reduce the uncertainty of the stochastic estimate is called dilution [300]. Recall that when we stochastically estimate jf,µ (x), we compute X ξk† (y)N (x, z)ξk (z) = tr N (x, y) + z X ∗ (y)Na,b (x, z)ξk,b (z) ξk,a z6=y,a6=b = tr N (x, y) + δN (x, y) for each random source. The idea was that after averaging over Nr random sources, the second term is averaged out due to the properties of ξk in Eq. (6.11). This term gives rise to the noise of the estimate. If we use point sources, sx,a and sy,a instead of ξk and sum over color indicies a, we can completely eliminate δN (x, y). But we pointed out that this would require as many inversions as there are lattice sites. In between, we can set some of the components of ξk to zero. Suppose that we partition the lattice into Np disjoint sublattices as Ω= Np [ Ωj . j=1 We define subfields of ξk each of which has support only on one of Ωj ’s so that ξk = Np X (j) ξk . j=1 Then, δ Ñ (x, y) = X X z6=y,z∈Ωj a6=b (j)∗ (j) ξk,a (y)Na,b (x, z)ξk,b (z) < δN (x, y). 123 This is dilution. We can achieve the same uncertainty using a smaller number of random sources. As we can see, to obtain the current density at all sites, we need to repeat Nk inversions Np times as a particular site is contained in only one Ωp . So for this method to work, the cost of additional inversions needs to be offset by the improvement in the variance. This indeed works because in our case, N (x, z) decreases with increasing distance |x − z|, which enhances the reduction in the variance [318]. To (j)∗ (j) see this, first note that for our choice of Z(4) source, ξk,a (x)ξk,b (z) is a complex number (±1 ± i)(±1 ± i)/2. So for a given x, z, a, b, j, the product is a random variable taking the values from {±1, ±i}. The magnitude of the expectation value of the sum of the random variables with their outcomes being {±1, ±i} is 1. So on the average, δN (x, z) ∼ 1 Nr X Na,b (x, z) = z6=y,a6=b α . Nr If we assume that we have achieved the same variance using Nr0 random sources with dilution, δN (x, z) = δ Ñ (x, z) ∴ Nr = α 0 N α̃ r where α̃ is defined similarly to α. The computational cost is smaller with dilution if Nr < Np Nr0 ⇒ α < Np α̃ If N (x, z) is constant, there is no gain. However, N in our case is a propagator and shows exponential suppression. This in most cases leads to decrease in the computer time to reach the same uncertainty [318]. In our work, dilution with stride 2 is also applied to the random sources to reduce the variance [321]. That is, on each lattice, we use 16 random sources, each with support on a subset Ωη = {r ∈ Ω | η ≡ r (mod 2)} where ηµ = 0 or 1. Since N (x, y) = α(y)U (y)Mf−1 (x, y) and the propagator Mf−1 (x, y) is smaller as the distance |x − y| becomes longer, this method suppresses the larger effects of unnecessary short range propagation from Eq. (6.14) and Eq. (6.15). 6.3.5 Calculating the current correlation To summarize, we compute jf,µ (x) by combining deflation and stochastic estimation as high low jf,µ (x) = jf,µ (x) + jf,µ (x) (6.19) 124 low (r) is the low-mode part of the current constructed from the N where jf,µ ev eigenpairs, and the rest is the stochastically estimated high-mode part. The even eigenvectors are numerically computed using PRIMME, and the odd eigenvectors are reconstructed by high / The high-mode part jf,µ multiplying them by D. (x) is stochastically estimated using TSM, with Nslp sloppy inversions and Nfine sloppy and fine inversions to compute the correction term. In the process, dilution was used to reduce the variance. Now that we have computed jf,µ , we can proceed to estimate the time-slice correlation function in Eq. (6.5). For that, we have a product of the current densities Jµstg (x)Jµstg (y). Ideally, we need to use statistically independent sets of random sources to estimate two current densities in the product. However, in our work, we have saved stochastic current densities for each random source, color and Lorentz component µ, and flavor. So employing two different sets of random sources requires a large storage, which is impractical. So we have used the current densities estimated using the same set of random sources. This introduces a bias. To see this, for notional simplicity, neglect the low-mode part and write Jµstg (x) = 1 X † ξk (x)N (x̃, z)ξk (z) Nr k where Einstein summation convention is used for site index z and x̃ = x + aµ̂. Then, Jµstg (x)Jµstg (y) = 1 X † ξk (x)N (x̃, z)ξk (z)ξl† (y)N (ỹ, z 0 )ξl (z 0 ) Nr2 k,l 1 X † = 2 ξk (x)N (x̃, z)ξk (z)ξl† (y)N (ỹ, z 0 )ξl (z 0 ) Nr k6=l + 1 X † ξk (x)N (x̃, z)ξk (z)ξk† (y)N (ỹ, z 0 )ξk (z 0 ). Nr2 (6.20) k=l In the first term with k 6= l, averaging over random sources forces z = x and z 0 = y and gives the result we want, except for a factor of (Nr − 1)/Nr . The second term includes an unwanted contribution when z = y and z 0 = x. To eliminate this contribution from the second term, we note that these terms for which z = y or z 0 = x in the second term in Eq. (6.20) add up so that 1 X † 1 ξk (x)Nx̃,z ξk (z)ξk† (y)Nỹ,z 0 ξk (z 0 ) ∼ . 2 Nr Nr k=l 125 So we can eliminate the bias by taking the limit of Nr → ∞. We have performed this elimination as follows: 1. Divide Nslp and Nfine random sources into Nb blocks of random sources so that after partitioning there are Nb sets of NBS ≡ Nslp /Nb random sources for sloppy inversion diff . and Nfine /Nb fine random sources for estimating the correction jf,µ high 2. Compute jf,µ using random sources from each block separately and take the average and standard deviation of the mean. 3. Repeat this several times by changing the block number. 4. Finally, we extrapolate to infinite number of random sources by taking the limit 1/NBS → 0. When we compute the time-slice correlation function in Eq. (6.5), we apply this extrapolation procedure to the values of Cdisc (t) at each t. Figure 6.3 shows an example of this extrapolation for the time-slice correlation function at t = 1 for a single gauge configuration. Figure 6.3: The value of the time-slice correlation function Cdisc (t) on one gauge configuration at t = 1 as a function of the inverse random source block size. Also shown is the extrapolation to infinite block size. The extrapolated value is plotted in blue. 126 6.4 Parameter Tuning The calculation presented in this section is carried out on a single gauge-field ensemble of size 323 × 48 with an approximate lattice spacing of 0.15 fm. As is discussed, the simulation employs stochastic estimation using TSM and deflation along with dilution with stride 2 for computing current densities. Accordingly, there are several parameters in the simulation that need to be tuned to achieve the target statistical uncertainty at minimum computational cost. The total statistical uncertainty comes from the stochastic estimation of the current-current correlation function and from gauge fluctuations. We first minimize the statistical error from the stochastic estimation at a given computational cost by tuning the parameter values. The total statistical error coming from gauge fluctuations is then reduced by increasing the number of gauge configurations analyzed in the tuned parameter setting. 6.4.1 Overview The tuning was done by referring to a point-to-point correlation function Cdisc (x, y) = XD Jµstg F (x) Jµstg F µ E 1 X X stg Jµ (y) = NG g µ G where g is a gauge configuration in which P D µ Jµstg E F F (x) Jµstg F (y) g E D (x) Jµstg (y) is evaluated, and F an average over NG gauge configurations is taken. For notational simplicity, we will omit the subscript “disc” on Cdisc from now on. So the total stochastic error of C(r) comes from the averaging over gauge configurations as well as the stochastic estimation of the current densities. By translation invariance of C(x, y), C depends only on the displacement r = y − x, and so C(r) = 1 X C(x, x + r) V x where V = |Ω| is the volume of the lattice. We introduce Cg (r) on a single gauge configuration as C(r) = 1 X Cg (r) NG g where Cg (r) = 1 X Cg (x + r, x). V x 127 For purposes of tuning, we fix a gauge configuration and so drop the subscript g from now on. When we present data, we average C(r) over rotational-symmetry-related displacements. Also, when r is greater than 5, there are a number of displacements, and the values of C(r) are close to 0. So we binned these values according to the distance. An example of the point-to-point correlation function is shown in Fig. 6.4. We need to decide on the precision of the correlation function before averaging over gauge configurations. For that, we first define what we mean by the precision of the correlation function on a single gauge configuration. We have adopted the definition: Figure 6.4: The point-to-point correlation function C(r) over a single gauge configuration with Nev = 350, εev = 10−9 , εslp = 2.70 × 10−2 , εfine = 10−5 , Nslp = 1408, and Nfine = 72. The values for C(r) are plotted against the distance |r|. Also note that beyond |r| > 5, the data are binned. Uncertainties arise from the stochastic estimation of the current density and from gauge fluctuations in the context of a single configuration. 128 εcorr v u u ≡t 1 Ncorr X σ (C(r))2 |r|<5 where Ncorr is the number of displacements whose distance is less than 5 and σ (C(r)) is the standard deviation of C(r) at such a displacement r. Our ultimate goal is to achieve a 10% error of adisc using the lattice method. There µ are several sources of this uncertainty such as a finite volume error coming from the approximation of the infinite space-time by a finite lattice, isospin breaking effects due to the nonzero mass difference between up and down quarks, and the error arising when taking extrapolation to zero lattice spacing. We do not know how much they will be beforehand. Also, only in the end we will know how the error in C(r) propagates to aµ . Furthermore, the total stochastic error of C(r), is a combination of the error from stochastic estimation of current densities and that from gauge fluctuation. However, before we average over gauge configurations, we do not know how much uncertainty that average will incur. So we have set a conservative goal that εcorr should be 1% of the average value of the correlation function on a single gauge configuration at displacements with their distance less than 5, which we call εgoal corr , namely εcorr ≤ εgoal corr ≡ 1 1 X |C(r)|. 100 Ncorr (6.21) |r|<5 The average of the absolute values is taken in order to obtain a scale of the problem. We will tune the parameters to meet this goal at minimal computational cost. The parameters to be tuned are 1. The residual of the fine inversion, εfine = |ξk − Af gnfine |/|ξk | where gnfine is the solution of Af g = ξk at the nfine step of the CG method and nfine is the iteration number necessary to reach the desired precision. 2. The residual of the sloppy inversion, εslp = ξk − Af gnslp /|ξk | where gnslp is the solution of Af g = ξk at the nslp step of the CG method and nslp is the iteration number necessary to reach to the desired precision. 3. The ratio of the number of stochastic sources for use in fine inversions to the number of stochastic sources for use in sloppy inversions, R = Nslp /Nfine . 129 4. The number of stochastic sources for use in sloppy inversions, Nslp . 5. The number of eigenvectors used for deflation, Nev . (e) (e) (e) (e) 6. The residual of the even-site eigenvectors, εev = Deo Deo ṽn − λ̃2n ṽn /ṽn where ṽn and λ̃n are the estimated nth eigenpair. We give an overview of our tuning procedure here and then give details in subsections below. After setting εgoal corr , we proceed to fix the parameters in the order listed above. (1) First, we determined the precision of the fine solve required to ensure that the systematic uncertainty due to the finite solver precision is less than the 1% accuracy. The systematic shift of the values of the point-to-point correlation function due to the use of nonzero εfine needs to be smaller than εgoal corr . Here, as we change the precision of the solve εfine , the central values of C(r) shifted, but the stochastic uncertainty of the correlation function C(r) showed negligible change. So we focused only on the systematic error due to finite εfine . (2,3) Next, the residual of the sloppy solve, εslp , and the ratio of the number of sloppy solves to fine solves, R, are determined to minimize the uncertainty at a given computational cost at a fixed number of deflating eigenvectors Nev [322]. The analysis was presented in Section 6.3.3. Since R = Nfine /Nslp , at a fixed Nev , Nslp is the only free parameter at this point. So we compute C(r) and also the corresponding εcorr by gradually increasing nslp to find the value for nslp such that εcorr ≤ εgoal corr . This step is repeated for several different numbers of deflating eigenvectors Nev but not for different residuals εev with the goal of minimizing the total cost of Cev and Cinv . The residual of the eigenpairs εev was set to 10−9 so that the first few smallest estimated eigenvalues were close to their true eigenvalues. What we mean by this is described in Section 6.4.6. While going through this exercise, we assumed that the correlation between the fine and sloppy current densities is independent of the eigenresidual to a good approximation. We did not attempt to optimize this choice at this point. (4) Next, we obtain the number of sloppy inversions Nslp required to achieve the goal uncertainty εgoal corr as a function of Nev . Now, except for the fixed parameters, εgoal corr and εfine , all the parameters are dependent on Nev . This also means that the computer cost for the stochastic estimation of C(r) with deflation, Cinv , is a function only of Nev . (5) We next determined the optimal values for 130 the free parameters by minimizing the total computer cost of the numerical estimation of C(r), which is a sum Ctotal = Cev + Cinv where Cev is the computer cost for generation of eigenpairs. The value for Nev at which the minimum is attained and the corresponding value for Nslp is their optimal values. (6) Finally, we considered the effect of varying εev on the variance and cost. As a summary, the optimum parameter values after the tuning procedure are determined to be • εcorr = 3 × 10−9 • The number of eigenpairs for deflation: Nev = 350 • The residual of fine and sloppy solve: εslp = 2.70 × 10−2 and εrmf ine = 1 × 10−5 , respectively • The number of fine and sloppy solves per configuration: Nslp = 1408 and Nfine = 72, respectively. • εev = 10−9 We provide the details of each step of the tuning procedure next. 6.4.2 Determination of εgoal corr and εfine To determine the value for a small enough εfine such that the systematic shift of the −1 central values of C(r) is smaller than εgoal corr , we computed C(r) while varying εfine from 10 to 10−7 and observing the shift in C(r). This shift was defined as ∆C(εf id , εfine ) = 1 Ncorr X Cεf id (r) − Cεinv (r) . (6.22) |r|<5 We found that when we changed εfine from 5 × 10−7 to 10−7 , the averaged shift of correlation values was about 10−13 , which is less than 0.01% of εgoal corr determined with εfine = 10−7 . Thus, we concluded that the correlation values have converged for our purpose at the residual of 10−7 . Figure 6.5 shows ∆C(εf id , εinv ) as a function of εfine in the range [10−1 , 5 × 10−7 ]. Based on this observation, from now on, the values of the correlation function determined with the residual of 10−7 are used as the fiducial correlation values. 131 Figure 6.5: Average change ∆C(r) in the point-to-point correlation function C(r) as defined in Eq. (6.22) as a function of the choice of the solver residual εfine ranging from 10−1 to 5 × 10−7 . The change is measured relative to the fiducial values of C(r) defined at εfine = 10−7 . Each point corresponds to a displacement. The cross indicates the average absolute value. The blue region indicates a less-than-1% error. Using the values of the correlation function C(r) determined with the residual of 10−7 , the −9 value for εgoal corr in Eq. (6.21) was determined to be 3 × 10 . For the determination, we choose values for other parameters that are much more conservative than our final choice, namely, Nr = 1024, Nev = 1000, εev = 10−10 . Now that we have the fiducial values for the correlation function C(r), for the determination of εfine , we compared the values of C(r) determined with various CG residuals ranging from 10−1 to 5 × 10−7 against the values with a residual of 10−7 . This was repeated for various numbers of deflated eigenvectors from 0 to 1000 separated by 50. Figure 6.6 shows ∆C(r) when Nev = 100. As it can be seen, the absolute differences and the average of the differences are well within the less-than-1%-error region. This was true for various numbers of deflated eigenvectors as long as the residual is smaller than 10−5 . 132 Figure 6.6: Same as Fig. 6.5 but for Nev = 100. Another observation we have made is that reducing the residual of the inversion εfine below 10−5 does not affect the statistical uncertainty of the correlation function. So we focus only on the systematic error due to the shift of jf,µ , and we conclude that a residual of 10−5 is sufficiently small to achieve a 1% systematic error. That is the value we used. 6.4.3 Determination of εslp and R To determine εslp and R, we adapted the optimization procedure in Ref. [322] to our case, which was described in Section 6.3.3. As we recall from Section 6.3.3, for the stochastic estimates of high mode part of the current density with the fixed εfine , the dependence on εslp of the correlation value r̄ between sloppy and fine inversions can be modeled as ln(1 − r̄) = a ln εslp + b. (6.23) To find out the fitting parameters a and b, we first stochastically estimated the current 133 density using a range of values for εslp from 10−1 to εfine = 10−5 with 256 random sources and compared it with the stochastic estimation of the current density with the residual εfine using the same set of random sources. The comparison was made at each lattice site for each component. This provides the value for the correlation, r(x), at each lattice site x, and the average r̄ and standard deviation of the mean σ(r̄) as a function of εslp were evaluated to obtain an indicator of the correlation as is discussed in Section 6.3. This process was repeated for each number of deflated eigenvectors from 0 to 1000, separated by 50. The data were fitted to the model equation Eq. (6.23) to yield the values for a and b. Figure 6.7 shows the fit for each number of deflating eigenvectors. Also, the residual of CG inversion εCG depends on the number of CG iterations nCG . We can model this dependence as [322] ln εinv = cninv + d Figure 6.7: The average r̄ of the correlation in the current densities between fine and sloppy solves as a function of the sloppy residual εslp defined in Eq. (6.17) for various degrees of deflation. Curves are fits based on Eq. (6.23). 134 † To find the values for c and d, we inverted 4m2f +Deo Deo for each flavour f = l, s, c using the CG method for a single random source and observed how the residual changes as we do more CG iterations for a given number of deflated eigenvectors ranging from 0 to 1000 separated by 50. Figure 6.8 shows the number of CG iteration required to achieve a given residual. This figure also shows that deflation is most effective in reducing the cost of inverting the light quarks, up and down. Deflation does not have observable effect in inverting the strange and charm quarks. When deflation ceases to be effective, the slope of εinv as a function of ninv changes. As we increase Nev , the slope becomes smaller. Also, the value at which deflation becomes ineffective becomes smaller as Nev becomes larger. Using the obtained values of the constants a, b, c, d for each Nev , we get the optimal values for nslp and R using the formulae presented in Section 6.3.3. They are tabulated in Figure 6.8: The number of CG iterations ninv required to reach a given residual for selected Nev and quark flavors. The deflation has very little effect on the charm and strange CG iteration number. So these values are small and almost constant. The curve fit was made for the range (10−1 , εfine = 10−5 ) 135 Table 6.1. In modeling the dependence of the residual of the CG inversion on the number of CG iterations, we have neglected a constant overhead associated with CG inversion because the overhead was smaller compared with the cost of a single CG inversion iteration. So we have the optimal values for εfine , εslp , and the ratio R. The values for εslp and the ratio R are a function of Nev . The remaining parameters are Nslp and Nev . We will tune Nslp next. 6.4.4 Determination of Nslp Now that we know the optimal values for εslp and R for each Nev , the number of deflating eigenvectors, we can finally set values for Nev and Nslp . The goal is to compute the current density to the desired precision at a minimum computational cost. The total computational cost Ctot is a sum of the cost of inversion Cinv and the cost of generating eigenvectors Cev . Here, the cost of generating eigenvectors Cev is measured in units of time. So to find out Ctot in units of time, we need to know the cost of inversion Cinv in units of time to achieve the Table 6.1: The values of the constants of the fits for ln(1 − r̄) = a ln εslp + b and ln εinv = cninv + d and εslp for various Nev . Nev 0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 a 2.2 2.3 2.1 2.1 2.1 2.1 2.1 2 1.9 1.9 1.8 1.8 1.7 1.7 1.6 1.6 1.6 1.6 1.5 1.5 1.5 b 2.8 3.4 3.1 2.9 2.7 2.7 2.4 2.1 1.8 1.6 1.3 1.1 0.89 0.7 0.53 0.36 0.21 0.056 −0.1 −0.21 −0.31 c −0.0015 −0.0024 −0.004 −0.0056 −0.0072 −0.0087 −0.01 −0.011 −0.012 −0.014 −0.015 −0.016 −0.017 −0.018 −0.018 −0.019 −0.02 −0.021 −0.022 −0.023 −0.023 d −2 −2 −1.8 −1.8 −1.7 −1.6 −1.6 −1.6 −1.5 −1.5 −1.5 −1.5 −1.5 −1.5 −1.5 −1.4 −1.4 −1.4 −1.4 −1.4 −1.4 εslp −0.0014 −0.0014 −0.0014 −0.0014 −0.0014 −0.0015 −0.0015 −0.0015 −0.0016 −0.0015 −0.0015 −0.0015 −0.0015 −0.0015 −0.0016 −0.0015 −0.0016 −0.0016 −0.0015 −0.0016 −0.0016 136 target precision εcorr of the correlation function for a given number of deflating eigenvectors. From the discussion in Section 6.3.3, we know that σ2 = σ02 1 (1 + 2(1 − r̄)R) ∝ . Nslp Nslp If we assume for simplicity ε2corr = βσ 2 , where β is a constant of proportionality, we also have ε2corr ∝ 1 . Nslp This suggests that there is a relation between ε2corr and Nslp . So we fit the data of the computer time and εcorr at each Nslp . From the fit of εcorr at each Nslp , we determined the value for Nslp needed to achieve εgoal corr of the correlation function. Then, we found our the value for the computer time necessary to achieve εgoal corr from the fit of Cinv measured in units of time as a function of Nslp , by evaluating the fit function at the value of Nslp just found. This is how we determined the cost of inversion Cinv in units of time necessary to achieve εgoal corr for a given number of deflating eigenvectors. An example of this fit procedure for illustration is as follows. Figure 6.9 shows how ε2corr −8 depends on Nslp . For Cdisc (t), we identified εgoal corr to be 4 × 10 . The fit to the power law Axn gives A = 1.424 × 10−6 and n = −2.838. So we need Nslp = εgoal corr A !−1/n = 1408. Then, the fit of Cinv as a function of Nslp in Fig. 6.10 tells us that the cost of inversion in units of time is Cinv = 0.1173Nslp ' 165 (min). We have also presented the cost of deflation as a function of Nslp , which we need later when we estimate the time for stochastic estimation of current densities, which includes the time for inversion Cinv , the time for deflation Cproj , and the time for saving the solutions to inversion Csave . 137 Figure 6.9: The dependence of the average square of the statistical errors in the correlation C(r), ε2corr , as a function of the number of sloppy stochastic sources, Nslp . The number of eigenvectors used was Nev = 350. Figure 6.10: The dependence of the cost of inversion Cinv and the cost of deflation Cproj in minutes as a function of Nslp . The legend in each plot shows a fit of the data to a linear function going through the origin, ax. 6.4.5 Determination of Nev In the previous subsection, we discussed how we determined the number of stochastic sources in use for sloppy inversion Nslp to achieve εgoal corr and the corresponding cost of inversion Cinv measured in units of time for a given number of the eigenvectors Nev . So Cinv now depends only on Nev . Then, the total cost Ctot = Cinv + Cev is a function of Nev . There is an optimal value for Nev that minimizes Ctot because there is a trade-off here. The number of itrations nslp for sloppy inversion depends on the effectiveness of deflation – the 138 more deflation, the fewer iterations. But the more deflation, the more eigenvectors needed. To find the optimal value for Nev , we need to compute Ctot in the range of the values for Nev and finding the values of Nev at which the minimum is attained. The optimal value for Nslp is then the number of stochastic sources in use for sloppy inversion to achieve εgoal corr . The discussion in the previous subsection on how the cost of inversion Cinv is measured in units of time to achieve εgoal corr suggests that determination of Cinv as a function of Nev requires actually doing the simulation with varying values of Nslp to do the fit for each Nev . This would be expensive if we did it for all the values for Nev in the range from 0 to 1000. So we first make an estimate of Cinv and identify the range of Nev where the minimum occurs. Since r R= nfine nfine = √ 2(1 − r̄)nslp α where we have introduced α = 2(1 − r̄)nslp nfine , we can write nfine Cinv = Nslp nslp + R √ = Nslp nslp + α . Also, 2 σ = σ02 1 2(1 − r̄) + Nslp Nfine σ02 (1 + 2(1 − r̄)R) Nslp √ σ02 α = 1+ . Nslp nslp = So since we have assumed ε2corr = βσ 2 , √ Cinv = Nslp nslp + α √ √ σ02 β α = goal 1 + nslp + α nslp εcorr 2 √ 2 σ0 β = nslp + α . goal nslp εcorr This is an expression of Cinv (Nev ) in terms of the number of solver iterations. 139 To obtain the estimates for Cinv (Nev ) in units of time for various Nev , we first measure the total cost of inversion with a specific number of deflating eigenvectors in units of time, following the method presented in Section 6.4.4, while actually performing the simulations. We have chosen Nev = 750 as the number of eigenvectors with which the cost of inversion is measured in units of time for normalization. Then, we normalized the cost of inversion with Nev deflating eigenvectors by taking the ratio Rinv (Nev ) = Cinv (Nev )/Cinv (Nev = 750). To obtain an estimate of the total cost of inversion with a given number of deflating eigenvectors Nev , we multiply Rinv (Nev ) by Cinv (750) to obtain Cinv (Nev ) measured in units est depends on of time. Figure 6.11 shows how Cev and the estimated cost of inversion Cinv Nev . One note is that this provides only an estimate as we have neglected the dependence of σ02 on Nev so that Cinv (Nev ) required to achieve the goal uncertainty of the correlation Figure 6.11: The dependence of the cost of generation of eigenpairs Cev and the estimated est as a function of N . The total estimated computer time C est = cost of inversion Cinv ev tot est + C Cinv ev is plotted with green symbols. Also shown is the measured total computer time Ctot as a function of Nev in the identified range, plotted with cross symbols. 140 function drops more sharply with increasing Nev than is predicated using this estimation. This is because σ02 becomes smaller if we use more eigenvectors for deflation. Based on Fig. 6.11, we identified the range of Nev where the total cost of Cinv + Cev is minimum. The identified range was from Nev = 250 to 600. For each number of deflated eigenvectors in the identified range, the simulation was conducted to figure out the value for Nslp to achieve εgoal corr . This was done by computing the correlation function and εcorr at the optimal R and Nslp while varying the number of sloppy inversion Nslp , as was discussed in Section 6.4.4. This gives us the value of εcorr as a function of Nslp . By fitting the data and interpolating the curve, we obtain the value of Nslp sufficient to achieve εgoal corr . We then interpolate the data points of Cinv vs Nslp to find the cost of inversion to attain εcorr = εgoal corr . A fit to Cproj as a function of Nslp is also taken. The obtained simulation time Cinv at each value of Nev was added to the computer time for generating the corresponding number of eigenvectors. The value for Nev and associated value for nslp at which the minimum of the sum is attained are the optimal values. Figure 6.11 shows the total computer time for stochastic estimation of current densities, indicated by cross symbols. Here, Cinv includes not only the time for inversion but also deflation and saving solutions of inversion. The plot suggests that the total cost starts to flatten past Nev = 350. After the point, the cost fluctuates due to the difference in computer time depending on how computer nodes are allocated for the calculation. When the nodes are close to each other, the communication among nodes takes a shorter time. In any case, in order to save memory, we have chosen to work with the smallest number of eigenvectors beyond which the effect of deflation in reducing the total computer time becomes less significant. 6.4.6 Determination of εev In Section 6.4.1, we have stated that εev was temporarily set so that the first few smallest estimated eigenvalues are close to their true eigenvalues. What we meant by this is that we require the distance of the ith estimated eigenvector ṽi to its true eigenvector vi to be smaller than the distance to any other eigenvector vj with j 6= i. Similarly, we require the difference between ith estimated eigenvalue λ̃i and its true eigenvalue λi be smaller than λ̃i − λj with j 6= i. To find the value for εev that ensures these conditions, we 141 computed the first 50 smallest eigenvalues and their associated eigenvectors determined with the residual of 10−7 , reduced the residual by a factor of 10, and observed how they converge. To test convergence, we plotted the difference in eigenvalues in Fig. 6.12 and the norms of difference of eigenvectors against those determined with 10−15 in Fig. 6.13. Here, the eigenpairs determined at εev = 10−15 are used as fiducial values. Based on these figures, it is clear that εev = 10−9 is sufficient to ensure convergence. We now take a closer look at the case where Nev = 350, which was determined to be the optimal value in Section 6.4.5. We can put the criterion on sufficient precision of the estimate of the eigenpairs on a more quantitative foundation. As a matter of fact, through the analysis, we will find that we can loosen the above condition and use eigenestimates with a larger residual. First, we note that the deflation ceases to be effective if the residual of the sparse-matrix solution is set smaller than a cutoff value that depends on the eigenresidual εev [323]. This is indicated by the change in the slope of ninv as a function of log10 εinv in Fig. 6.14. Let the cut be denoted as εcut . We can see that εcut is a function of εev from Fig. 6.14. In the figure, we plotted the dependence only for the light quark. This is because deflation has Figure 6.12: Shifts of estimated eigenvectors vs. eigenvalue index for various eigensolver residuals εev . Shifts are measured relative to the fiducial values from εev = 10−15 . 142 Figure 6.13: Shifts of estimated eigenvalues vs. index for various eigensolver residuals εev . Shifts are measured relative to the fiducial values from εev = 10−15 . the strongest impact on inverting light quarks as can be seen from Fig. 6.8. This dependence means that in order for deflation to be effective when aiming for some inversion redisual εinv , the residual of the eigensolutions should be small enough to make sure that εcut < εinv [323]. The data showed that the eigenresidual can be increased by a factor of 10 without affecting the performance of the inversion. This adjustment of the eigenresiduals was not done when we tuned the parameter values for this work. Since this decreases the cost for generating eigenpairs, had we done the adjustment, we could have obtained a higher number of deflating eigenvectors as the optimal parameter setting and decreased the total computational cost. We pursue this possibility in the following. To see the effect of the change in εev on Ctot , first recall that in our case, εfine = 10−5 . Figure 6.14 indicates that when εev becomes larger than 10−7 , the change in the slope occurs at around εinv = 10−5 , indicating that had we used eigenpairs with εev > 10−7 , deflation would not have reduced the simulation time. For each value of εev , we determined the value εcut as the value of εinv at which the change in the slope occurs. Figure 6.15 shows how log10 εcut changes as we change log10 εev . Using the fit, we found 143 Figure 6.14: The number of CG iterations required to reach a given residual of inversion for light quark for various values of εev . The number of deflated eigenvectors are fixed at 350. εev (εcut = 10−5 ) = 1.39 × 10−7 . Figure 6.14 suggests that the cost of inversion would have remained constant even if we had increased the eigenresidual from 10−9 to 1.39 × 10−7 . There is one thing one needs to be careful about here. The plots of iteration number as a function of εinv are obtained by averaging the numbers of iterations to reach to a certain value for εinv for all diluted random sources. Solving for random sources with support on odd sites takes more iterations to reach the same εinv than the ones with even sites. This is because odd eigenvectors are / The obtained from the even eigenvectors by multiplying even eigenvectors by the matrix D. multiplication inflates the eigenresidual of the reconstructed odd eigenvectors as is pointed out in Section 6.3. This implies that deflation with the reconstructed odd eigenvectors becomes less effective at a larger solver residual than with the even eigenvectors. Ideally, since the slope of the iteration number as a function of log10 εinv changes only after εinv becomes smaller than εev , the cost of inversion should remain constant at fixed εfine as long as εcut (εev ) < εfine . However, due to the imprecision of the reconstructed odd eigenvectors, the computational cost increases slightly when we increase εev from 10−9 to 144 Figure 6.15: The dependence of log10 εcut on log10 εev . As deflation has negligible impact on the performance of inversion for εev < 10−3 , a linear fit was taken using only the values for εev ≥ 10−4 . 10−7 and significantly when we increase εev beyond 10−7 . We can see this in Fig. 6.16. This shows that deflation does not help in reducing the total computer time when εev & 10−7 . At the optimal parameter setting to achieve the desired precision, the computational cost of inversion increased by about 157 min from 747 min when we increase εev from 10−7 to 10−6 , while the computer time for generating eigenpairs decreased by about 100 min from 176 min if we made the same change in the precision of the eigenpairs. Also, if we go from εev = 10−9 to 10−7 , the decrease in Cev is about 71 min, while the increase in Cinv is about 70 min. The decrease in the cost of generating eigenvectors was completely offset by the increase in the cost of inversion if we increased εev from 10−9 to 10−7 . Figure 6.17 illustrates the results. However, theoretically we expect that the cost of inversion remains constant as we increase εev as long as εcut (εev ) < εfine . This is indeed true when going from εev = 10−9 to 10−8 , shown in Fig. 6.17. The conclusion is that at least for Nev = 350, we can reduce the total computational cost by going to larger eigenresidual than 10−9 but smaller than 10−7 , beyond which reduction in the computational cost to generate eigenpairs by increasing εev is completely offset by the 145 Figure 6.16: The cost of inversion in minutes as a function of εev . increased cost of inversion. In addition, as we recall from Section 6.4.3, εcut also depends on Nev , indicated by Fig. 6.8, and its identification is easier for larger values for Nev . Also, reduction in Cev due to increase in εev is greater for larger Nev as is indicated in Fig. 6.18. This dependence also means that εev can be larger than the convergence test requires if we use more eigenvectors but at a fixed εev . So the effect of deflation is grater for a larger Nev , although this effect was insignificant. These observations lead to two implications for simulations with Nev 6= 350: 1. If we use more eigenvectors for deflation, relaxing εev has a greater impact on the computer time for generation of Nev eigenpairs. 2. Figure 6.8 suggests that εcut = εcut (εev = 10−9 ) is smaller for larger Nev . So εev = εev (εcut = 10−5 ) is larger for larger Nev . A consequence is that at least for Nev ≥ 350, we could have reduced the total computer time by using a larger eigenresidual. This would have shifted the optimal value for Nev 146 Figure 6.17: The computer time for Cinv and Cev as a function of the eigenresidual εev . The blue line shows Ctot at the preferred value of εev = 10−9 . The total computer time is plotted with green symbols. higher. 6.5 Results Now that we have determined the optimal parameter values to stochastically estimate the current densities and compute the time-slice correlation function with the target uncertainty of εgoal corr at the minimal computer time on a single gauge configuration, we go on to compute the correlation function on a number of gauge configurations and take the average to obtain Cdisc (t) in Eq. (6.9). In this study, we have employed 1326 gauge configurations with HISQ sea-quarks with masses set to their physical values. The sea quarks included in the simulation are up, down, strange, and charm quarks. In the calculation, isopsin-symmetric limit is taken so that the up and down quarks have been approximated to be of the same mass. Also, the QED effects 147 Figure 6.18: The computer time Cev in minutes as a function of Nev for different values for εev . have been omitted. The computational resource we relied on is the Cray XC30 computer, named Edison, at the DOE facility NERSC [324]. The supercomputer comes with 5586 computer nodes, and each node has two sockets, each of which is populated with a 12-core Intel “Ivy Bridge” processor [324]. So in total, there are 24 cores per node. We have used 8 nodes with 24 cores for estimation of current densities on each gauge configuration and 1 node with 16 cores for computation of the time-slice correlation function. The stochastic estimation of the current densities took about 11 hours. The computation of the time-slice correlation function took about half an hour. Generation of 350 eigenvectors using 8 nodes with 24 cores took about 4 hours for each gauge configuration. The preliminary report of this work was first presented in Ref. [325]. Figure 6.19 shows our result for Cdisc (t) at a lattice spacing of ∼ 0.15 fm. The size of the lattice is L3s × Lt = 323 × 48 where Ls is the spatial dimension and Lt is the temporal dimension in units of lattice spacing. In the plot, an oscillation around the curve is clearly visible. This is caused by the opposite parity states excited by the single-time-slice operator in the correlation function expected with staggered fermions [244]. This happens because in 148 Figure 6.19: Time-slice disconnected current density correlator vs. the temporal separation in lattice units. the staggered formulation, a single component fermion behaves as a fermion of four spinor components with taste degeneracy as discussed in Section 5.2.3. When written in the spin-taste basis, the single-time-slice mesonic operator is expressed as a linear combination of bilinears of the fermions in the spin-taste basis, which comes with opposite parity. What we want is J P C = 1−+ component as we are concerned with the correlation function formed by two vector currents. The unwanted effect of the lattice artifact is suppressed when we sum Cdisc (t) with the coefficients wt . To compute the disconnected contribution to aHVP , we use the time-moment representaµ tion (TMR) [326]. The expression can be obtained by substituting Eq. (6.4) into Eq. (6.2): HVP adisc. µ = 4α 2 Z ∞ dq 2 f (q 2 )Π̂(q 2 ) 0 = 4α 2 Z 0 = TX max ∞ T X 4 sin2 (qt/2) 2 dq f (q ) t − Cdisc (t) q2 2 2 t=0 wt Cdisc (t) t=0 where Tmax is the size of the lattice in temporal direction and 149 wt = 4α 2 ∞ Z 4 sin2 (qt/2) 2 dq f (q ) t − . q2 2 0 2 These values are evaluated with Mathematica. With the value of ZV−1 = 0.837 (3) [327], the values of the partial sum adisc µ (T ) as a function of T in lattice units is shown in Fig. 6.20 where adisc µ (T ) is defined as: adisc. µ HVP (T ) = T X wt Cdisc (t). (6.24) t=0 As Cdisc (t) gets smaller and smaller as t increases, the sum should become flat and converges as T < Tmax increases. At the same time, the signal-to-noise ratio for Cdisc (t) gets smaller as t increases so that adisc. µ value and uncertainty of adisc. µ HVP (T ) consistent with the values of adisc. µ HVP (T ) gets noisier at larger T . If we choose the at T = 12 as the value for adisc. µ HVP (T ) HVP , it is mostly at T > 12 except for the values at T > 20 where the effect of noise becomes significant and the values cannot be trusted. So we say that adisc. µ HVP (T ) has reasonably converged at T = 12 and take its value and uncertainty at T = 12 as our values for adisc and its uncertainty. µ Figure 6.20: adisc × 1010 as a function of the cut-off time T in Eq. (6.24). A constant fit µ is taken from T = 12 to 20. The shaded region indicates the error. 150 adisc = −2.6 (4) × 10−10 . µ (6.25) This value is less negative than the value cited in Section 3.3.1, which is the value in the continuum limit. As is observed by the BMW group [328], adisc has a strong lattice-spacing µ dependence; the smaller the lattice spacing is, the more negative adisc becomes. Figure 6.21 µ shows how adisc becomes larger as the lattice spacing a gets smaller. The values for adisc µ µ are computed by the BMW group using staggered fermions in the isospin limit using up, down, and strange sea quarks. The red squares indicate the simulation results at various lattice spacings, and the blue diamond, the continuum limit. The solid line indicates that the value of adisc would be about −2.5 when a2 = 0.020. Since the square of our lattice µ spacing is a2 = 0.025, our result is plausibly consistent with the result obtained by the BMW group. Recently, significant progress has been made in the numerical lattice QCD computation of the HVP contribution by the RBC/UKQCD group [329]. They combine the lattice result and R-ratio result, which they called the window method [329]. The motivation comes from the observation that wt C(t) calculated from the R-ratio data is more precise in the smaller and larger t regions, and wt C(t) computed in the lattice method is more precise in the middle interval. So to obtain the values of wt C(t) with smaller uncertainty at each t, they wrote aLOHVP as a sum of three contributions [329]: µ SD W LD aLOHVP = aLOHVP + aLOHVP + aLOHVP µ µ µ µ where Figure 6.21: The plot of −adisc vs. the lattice spacing a2 provided by Kotaroh Miura in µ private communication. Different fit lines are obtained using different cuts on a. 151 SD aLOHVP = µ X wt C disc (t) [1 − Θ(t, t0 , ∆)] t W aLOHVP µ = X = X wt C disc (t) [Θ(t, t0 , ∆) − Θ(t, t1 , ∆)] t LD aLOHVP µ wt C disc (t) [1 − Θ(t, t1 , ∆)] . t Here, they have used a smeared step function [329] Θ(t, t0 , ∆) = 1 + tanh t−t0 ∆ 2 . The smearing parameter ∆ was tuned to control the effect of discretization errors [329]. SD and Based on the relative magnitude of uncertainties, they have evaluated adisc µ LD using the R-ratio data and adisc W based on lattice computation. Using ∆ = 0.15 adisc µ µ fm in their calculation, they have obtained for the disconnected part including sea quarks of flavor u, d, s in the isospin-symmetric limit without QED using the window method at the lattice spacing of about 0.11 fm [329] adisc = −1.0 (0.1) × 10−10 . µ Using the pure lattice QCD computation, they have obtained [329] adisc = −11.2 (4.0) × 10−10 . µ (6.26) The Fermilab/MILC/HPQCD collaboration made an estimate of the disconnected contriin the continuum limit [330] bution to aLOHVP µ adisc = −13 (5) × 10−10 . µ (6.27) There is yet another independent lattice calculation of the disconnected part by the BMW collaboration, which reads [328, 330] adisc = −12.8 (1.9) × 10−10 . µ The BMW collabotion obtained the continuum value through the extrapolation to the continuum limit using the values of adisc at six different lattice spacings. 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