Gauge and gravity scattering amplitudes from chy formalism

In this dissertation, we first review some recent progress on exploring the nature of scattering amplitudes. Then we present our recent work on direct evaluation of tree level maximally helicity violating (MHV) amplitudes by Cachazo-He-Yuan (CHY) formula, which naturally reproduce the Parke-Taylor and Hodges formula, respectively, for gauge and gravity. We also verify that they are supported only by one single solution to the scattering equation. In addition, we derive a new compact formula for tree level single trace MHV amplitudes for Einstein-Yang-Mills theory, which is equivalent to, but much simpler than the known Selivanov-Bern-De Freitas-Wong (SBDW) formula. It can be shown that other solutions do not contribute to the MHV amplitudes of Yang-Mills, gravity and Einstein-Yang-Mills theory. We further propose a method to characterize the solutions to the scattering equations using the rank of two discriminant matrices. In four dimensions, such a characterization can be used to understand the correspondence between the helicity configurations of external scattering particles and the solutions to the scattering equation.

GAUGE AND GRAVITY SCATTERING AMPLITUDES FROM CHY FORMALISM by Fei Teng 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 May 2017 c Fei Teng 2017 Copyright All Rights Reserved The University of Utah Graduate School STATEMENT OF DISSERTATION APPROVAL The dissertation of Fei Teng has been approved by the following supervisory committee members: Yong-Shi Wu , Chair(s) 14 Dec 2016 Date Approved Pearl Elizabeth Sandick , Member 14 Dec 2016 Date Approved Zheng Zheng , Member 14 Dec 2016 Date Approved Oleg Starykh , Member 14 Dec 2016 Date Approved Graeme W. Milton , Member 30 Dec 2016 Date Approved by Benjamin C. Bromley , Chair/Dean of the Department/College/School of Physics and Astronomy and by David B. Kieda , Dean of The Graduate School. ABSTRACT In this dissertation, we first review some recent progress on exploring the nature of scattering amplitudes. Then we present our recent work on direct evaluation of tree level maximally helicity violating (MHV) amplitudes by Cachazo-He-Yuan (CHY) formula, which naturally reproduce the Parke-Taylor and Hodges formula, respectively, for gauge and gravity. We also verify that they are supported only by one single solution to the scattering equation. In addition, we derive a new compact formula for tree level single trace MHV amplitudes for Einstein-Yang-Mills theory, which is equivalent to, but much simpler than the known Selivanov-Bern-De Freitas-Wong (SBDW) formula. It can be shown that other solutions do not contribute to the MHV amplitudes of Yang-Mills, gravity and Einstein-Yang-Mills theory. We further propose a method to characterize the solutions to the scattering equations using the rank of two discriminant matrices. In four dimensions, such a characterization can be used to understand the correspondence between the helicity configurations of external scattering particles and the solutions to the scattering equation. For my family. CONTENTS ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii NOTATION AND SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix CHAPTERS 1. 2. 3. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Modern development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Outline of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 7 GAUGE AND GRAVITY AT TREE LEVEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1 Spinor helicity formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Spinor identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Polarization vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Yang-Mills amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Color-ordered amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Helicity classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2.1 Three-point amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2.2 MHV amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Gauge amplitude relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Gravity amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Perturbative gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Kawai-Lewellen-Tye (KLT) relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2.1 Pure gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2.2 Einstein-Yang-Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Hodges formula for MHV gravity amplitudes . . . . . . . . . . . . . . . . . . . . . 2.4 Color kinematic duality and double copy relation . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Formal construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Double copy examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Witten-RSV formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 9 11 12 13 17 18 20 20 22 24 27 28 29 32 33 33 39 43 CACHAZO-HE-YUAN FORMALISM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1 The scattering equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Properties of the scattering equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Polynomial form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The integrated CHY formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The CHY integrands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Yang-Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 49 51 52 55 55 3.3.2 Pure gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.3.3 Einstein-Yang-Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4. SPECIAL RATIONAL SOLUTION AND MHV AMPLITUDES . . . . . . . . . . . . . . 59 4.1 4.2 4.3 4.4 5. Calculation of the reduced determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculation of the reduced Pfaffian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yang-Mills and gravity MHV from CHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Remarks and summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SINGLE TRACE EINSTEIN-YANG-MILLS MHV AMPLITUDES . . . . . . . . . . . 68 5.1 ( g− g− ) and ( g− h− ) MHV amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The vanishing of (h− h− ) MHV amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Spanning forests and MHV amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Generating the spanning forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 A seven-point example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 General relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. 69 71 73 75 76 81 SOLUTIONS AND HELICITY CONFIGURATIONS . . . . . . . . . . . . . . . . . . . . . . 85 6.1 6.2 6.3 6.4 6.5 7. 60 61 66 67 Discriminant matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rank characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rank and Eulerian sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relating solution sectors to helicity sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . General single trace Einstein-Yang-Mills amplitudes . . . . . . . . . . . . . . . . . . . . 85 88 90 94 97 DISCUSSION AND CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 APPENDICES A. PERMUTATIONS IN CHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 B. THE PARKE-TAYLOR FACTOR AND AMPLITUDE RELATIONS . . . . . . . . . . . 106 C. EULERIAN SECTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 vi LIST OF FIGURES 2.1 The color-ordered Feynman diagrams that contribute to A4 (1234). . . . . . . . . . . 15 2.2 en,r . . . . . . . . . . . . . . . . . . 31 Feynman rules for partially color ordered amplitude A 2.3 Three diagrams whose color factors satisfy the Jacobi identity. . . . . . . . . . . . . . . 35 4.1 The shape of matrix Ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.2 e nb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 The shape of matrix Ψ nb 4.3 1,s The shape of [ψm ]nn− −1,s with s > 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.1 Some spanning forest examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.2 The 16 spanning trees of K4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.3 A classification of spanning forests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 NOTATION AND SYMBOLS ηµν gµν pµ , kµ , qµ , etc eµ (σµ )αα̇ (σµ )α̇α eαβ , eαβ , eα̇ β̇ , eα̇ β̇ flat space-time metric with the most-negative signature general space-time metric with the most-negative signature space-time momentum polarization vector four-vector composed of the 2 × 2 Pauli matrices (1, σi ) four-vector composed of the 2 × 2 Pauli matrices (1, −σi ) 2 × 2 antisymmetric matrices: e12 = −e21 = 1, e12 = −e21 = −1 λα e λα̇ s ab ...ir Mij11 ...j r left-handed Weyl two-spinor right-handed Weyl two-spinor Mandelstam variable, s ab = ( p a + pb )2 a submatrix of M with rows {i1 . . . ir } and columns { j1 . . . jr } deleted MI J a submatrix of M with row and column index ranging within I and J ACKNOWLEDGEMENTS I would like to thank my advisor, Prof. Yong-Shi Wu, for guiding me into the field of theoretical physics. I have benefited a lot from his patient instruction on many interesting topics, which is invaluable to my future career. I also would like to thank Prof. Pearl Sandick and her postdoc Kuver Sinha, for teaching me the dark matter phenomenology. It is really a wonderful experience to discuss and work with Pearl and Kuver. It is now a good time to thank all my collaborators: Yi-Jian Du from Wuhan University, Jason Kumar from University of Hawaii, and my fellow graduate student, Takahiro Yamamoto. I appreciate the wonderful experience of working with all of you. I am indebted to my beloved wife, Siyuan Feng, who has offered tremendous encouragement and care of my wellness during my long journey towards graduation, despite the long distance and time difference. Finally, I am grateful to all my fellow students, friend, and my entire family. You are the people who make my life in this city brilliant. CHAPTER 1 INTRODUCTION Relativistic quantum field theory is undoubtedly a big triumph in the last century. It not only sets the playground for the high-energy particle physics, but also finds numerous applications in many other branches of modern physics. The nonabelian gauge theory (or Yang-Mills theory) is the cherry on the cake, due to its mathematical beauty and pivotal role in describing how fundamental particles interact in the Standard Model. On the experimental side, studying how particles scatter with each other at colliders plays a very crucial role in test existing theories and probing new physics. One of the most important physical observables in a scattering experiment is the differential cross section, which describes the angular dependence of the scattering probability. This quantity provides a link between theory and experiment. Thus, calculating the cross section accurately is of paramount importance since the correct interpretation of experimental results at colliders hinges on it. In the perturbative regime of field theory, the cross section can be obtained through a phase space integral over the norm-square of the scattering amplitude. Because of its importance, the scattering amplitude has been a main subject of research ever since the birth of quantum field theory. In the 1970s, this effort seemed to conclude with the Lagrangian formalism of scattering amplitudes. The scheme is as follows: 1. Fix the gauge freedom in the Lagrangian using the Fadeev-Popov trick; 2. Read off the Feynman rules of each interaction vertex and propagator; 3. At the leading order, sum over all tree level Feynman diagrams; 4. At higer orders, sum over all loop diagrams with the loop momentum integration properly regularized; 5. Renormalize the theory by absorbing the divergence into the counterterms. Nowadays, most quantum field theory textbooks, for example [1], follow this scheme. 2 However, people soon realized that the computational complexity grew factorially with the number of external particles, especially for gauge interactions. This is not so alarming since many-particles scattering are expected to be complicated by then. In the mid-1980s, people started to discuss the concept of the hadron collider. Subsequently, understanding the gluon1 scattering became very important since it dominated the hadron jet production at the collider. Parke and Taylor first calculated the gluon two-to-four scattering amplitude at the tree level. In their eleven-page paper [2], the result spanned about eight pages. With such a messy outcome, they concluded that this result is "suitable for fast numerical calculations." However, at the end of the paper, they still hoped to "obtain a simple analytic form for the answer, making our result not only an experimentalist's, but also a theorist's delight." The effort indeed paid off. They soon realized that the norm-square of the amplitude, which is a gauge invariant quantity, was much simpler. In a famous paper [3], they proposed the following formulas for arbitrary n-point color ordered gluon tree level amplitude (to be discussed in Section 2.2): A n (1+ 2+ 3+ . . . n + ) 2 = A n (1− 2+ 3+ . . . n + ) 2 = 0 ( p1 · p2 )4 A n (1− 2− 3+ . . . n + ) 2 = , ( p1 · p2 )( p2 · p3 ) . . . ( pn · p1 ) (1.1) where all particles are assumed to be outgoing. When we gradually flip the helicity configuration away from the uniform, An (1− 2− 3+ . . . n+ ) is the first nonzero amplitude, which is called the maximally helicity violating (MHV) amplitude. Parke and Taylor's confidence in this simple result comes from the correct collinear limit behavior. Meanwhile, Xu, Zhang, and Chang developed the spinor helicity formalism for massless momentum and polarization vector [4]. Using this later called "Chinese magic" variable (to be discussed in Section 2.1), Mangano, Parke, and Xu put the gluon MHV amplitude itself into an explicit gauge invariant form [5]: A n (1− 2− 3+ . . . n + ) = h12i4 . h12ih23i . . . hn1i (1.2) This form, which is effectively the "square-root" of Eq. (1.1), is nowadays referred as the Parke-Taylor formula. This discovery opens the field of modern scattering amplitude. 1 The gluon here stands for general SU ( N ) gauge boson, not just the phenomenologically important SU (3) gauge boson that mediates the strong interaction. 3 The surprising simplicity in the Parke-Taylor formula strongly suggests that the complexity in the traditional Feynman diagram approach largely comes from the fact that it fails to capture the most essential structures of scattering amplitudes. In hindsight, the reason is twofold. First, the Feynman diagram approach insists on off-shell gauge invariance and locality, while physical amplitudes are all on-shell. As a result, the off-shell calculation involves a lot of unphysical baggage only to be cancelled when all the external particles are taken on-shell. If we can develop an algorithm that involves only on-shell quantities, the overly complicated Feynman rules and the massive summation over Feynman diagrams can be avoided. The second reason is that tree level gluon amplitudes can be embedded into N = 4 superamplitudes. Then it turns out that there are some hidden symmetries (dual conformal and Yangian) that constrain the possible forms of amplitudes. These symmetries are not reflected in the Lagrangian. The unnecessarily complicated calculation following the Feynman diagrams does not only plague the gauge theory, but it also plagues more severely in gravity. Following the traditional quantization prescription, the graviton three-point and four-point vertex involve 171 and 2850 terms, respectively [6]. Moreover, as a nonrenormalizable theory, there are an infinite number of higher-point contact terms. Consequently, extracting a meaningful result for graviton amplitudes using Feynman diagrams is extremely difficult. String theory provides a solution: to construct a closed string, one can just glue two open string together. Following this intuition, Kawai, Lewellen, and Tye found that a closed string amplitude can also be obtained by two open string amplitudes [7]. Subsequently, in the field theory limit, this relation stated that a gravity amplitude can be obtained by two color ordered gluon amplitudes with different particle orders. This discovery, later called the KLT relation (to be discussed in Section 2.3.2), indicated that although the perturbative gravity and gauge theory shared no similarity in the Lagrangian, they were secretly related through a "double copy." The Lagrangian again failed to make such a connection. The above discussion hints that the traditional Lagrangian formalism might not be suitable to describe the structure of gauge and gravity scattering amplitudes. In Section 1.1, we give a brief introduction to the modern ways of calculating amplitudes, with no reference to the Lagrangian. Then in Section 1.2, we give an outline on our work in this field, which will be discussed in detail in the following chapters. 4 1.1 Modern development The discovery of Parke-Taylor formula and KLT relation motivated people to break the shackle of the Lagrangian formalism of scattering amplitudes. A more promising approach would be to study the mathematical structure of the amplitudes directly to gain a deeper understanding on the nature of field theory. In the following section, we provide a quick survey on important developments of modern scattering amplitude in the past three decades. We emphasize new and relevant development that led to the Cachazo-He-Yuan formalism for tree level amplitudes. We refer the readers to the textbooks [8, 9] for a comprehensive treatment of other interesting subjects. After the Parke-Taylor formual was proposed, Berends and Giele (BG) abandoned the Feynman diagrams and developed a recursive algorithm [10] to construct the n-point tree amplitude with only one particle off-shell. Although from today's point of view, BG recursive relation does not embrace the full on-shell simplicity, it does provide an efficient way to attack higher-point amplitudes and is still of use today. With the help of BG recursive relation, Kleiss and Kuijf found the so-called KK relation among the color ordered gluon tree amplitudes [11]. This discovery reduces the number of independent color ordered amplitudes to from (n − 1)! to (n − 2)!. In the 1990s, the unitarity cut method for loop integrand was developed [12, 13]. The basic idea is that when some internal loop momenta are on-shell, unitarity guarantees that the loop integrand reduces to the product of tree amplitudes. Then, by using the knowledge of tree amplitudes, we can reproduce certain coefficients of the loop integrand. In this way, we can avoid calculating the integrand from Feynman rules, which involves even more complicated calculation than the trees. The previous mentioned development was motivated by the pursuit of fast numerical calculation of hadron jet production at the Large Hadron Collider (LHC). Along a different line of thought, string theorists also found interesting structures contained in the Parke-Taylor formula. Nair embedded the gluons into the N = 4 super Yang-Mills theory and noticed that the Parke-Taylor formula could be reproduced by a string correlation function [14]. 5 Interestingly, the target space of this string theory is the space of supertwistor.2 Although Nair's approach cannot reproduce the amplitudes beyond MHV, the insight on supertwistor string inspired a series of important breakthroughs fifteen years later. In 2003, Nair's expectation was realized by Witten, who found that general N = 4 super Yang-Mill tree amplitude can be calculated from the correlation function of a topological string theory in the supertwistor space [17]. This work, together with a later paper by Roiban, Spradlin, and Volovich [18], proposed the Witten-RSV formalism for N = 4 super Yang-Mills amplitude in four dimensions (all the terminologies will be defined later in this dissertation): 1. General Nk MHV sector superamplitude is supported by a set of degree d = k + 1 polynomial curves in the supertwistor space; 2. Those polynomial curves are determined by solving a set of polynomial equations with external momentum data as input; 3. The entire permutation information is encoded in a Parke-Taylor-like factor: PT (12 . . . n) = 1 . (σ1 − σ2 )(σ2 − σ3 ) . . . (σn − σ1 ) The set {σ} is a solution to the above-mentioned polynomial equation. Quite remarkably, the KK relation is realized just as a complex number identity of PT. This work shifts our understanding on the nature of scattering amplitude. It soon motivates the CSW [19] and BCFW [20] on-shell recursive relation for amplitudes. On one hand, these new recursive relations only use on-shell gauge invariant quantities, with no extra unphysical baggage. Therefore, the computational complexity is significantly reduced. The BCFW recursion is indeed intensively used at LHC to calculate jet production. On the other hand, the on-shell recursive relations put the amplitudes into new forms. Using the dual momentum variables, the BCFW construction naturally put the N = 4 superamplitudes into an explicit dual superconformal invariant form [21, 22]. However, from the Lagrangian, one can only tell the superconformal symmetry. Indeed, failing to realize a symmetry in the problem usually causes unnecessary complications. Moreover, superconformal and dual superconformal algebra generate an infinite dimensional Yan2 The twistor is first introduced by Penrose [15], while the supersymmetrization is first written down by Ferber [16]. 6 gian algebra [23], which has some implication on the integrability of the N = 4 super Yang-Mills. Another feature of the amplitudes obtained from BCFW is that spurious poles exist and cancel each other, such that the amplitude is still a local quantity. Hodges first realized a geometric picture behind this cancellation [24]: the amplitude corresponds to the volume of a polytope, while BCFW produces a summation over the constituent simplices of this polytope. Then those spurious poles are just the internal boundaries in the triangulation of the polytope. This geometric interpretation was further generalized into the amplituhedron [25]. The BCFW can be also used to construct the loop integrand for N = 4 super Yang-Mills. This effort finally culminated in the Grassmannian and on-shell diagram formalism, which is claimed to be valid at all loops [26]. We note that the Witten-RSV style twistor based formula for supergravity have been established in [27, 28]. Alternatively, Bern, Carrasco and Johansson [29, 30] proposed an very impressive construction for gravity amplitudes, which is now referred as the BCJ double copy relation. This pursuit starts with an implausible requirement at the first glance: to find a set of kinematic numerators of gauge amplitudes that satisfy the same Jacobi identity as the color factor (color kinematic duality). Surprisingly, such numerators can always be constructed. Once it is completed, by simply replacing the color factor by another copy of the kinematic numerator, we can obtain gravity from gauge amplitudes! The reason why those color kinematic duality satisfying numerators exist is interesting. The gauge amplitudes actually satisfy a new set of amplitude relations: the BCJ relation, which reduces the number of independent color ordered amplitudes from (n − 2)! (after using KK relation) to (n − 3)!. This equivalence will be explored in Section 2.4. As we previously mentioned, in the Witten-RSV formula, the only quantity that knows the amplitude relations is the Parke-Taylor factor PT. It turns out that on the polynomial curves that support amplitudes, the BCJ relation is indeed satisfied. On the other hand, we can impose PT to satisfied the BCJ relation. Then surprisingly, this condition alone can reproduce all the polynomial curves. Cachazo first indicated this point in [31], and later with He and Yuan, proposed the scattering equation [32, 33], which is equivalent to the condition that PT should satisfy BCJ. The benefit is that the scattering equation takes Mandelstam variables as input such that it should hold in arbitrary dimensions. Indeed, both KK and BCJ relation can be inferred from Feynman rules (although cumber- 7 some), they should be generally true in any dimensions. Starting form 2013, Cachazo, He and Yuan (CHY) published a series of papers on the integrands for various field theory amplitudes [34-37]. The importance of the CHY formalism lies in that it extends the modern on-shell method to a spectrum of field theories beyond the gauge and gravity, for example, nonlinear sigma model, Born-Infeld, etc. Second, CHY is independent of the spinor helicity formalism, which provides the first on-shell formalism for amplitudes in arbitrary dimensions. The following is an outline on the layout of this dissertation, and a brief summary on our contribution to the CHY formalism. 1.2 Outline of this work This thesis is organized as follows. In Chapter 2, we review some introductory knowledge on the structure of gauge and gravity amplitudes. This review is not comprehensive, but emphasizes results that are useful in the subsequent chapters. Then we formally introduce the CHY formalism on tree level gauge and gravity in Chapter 3. The first two chapters are devoted to prepare the readers for understanding more technical studies presented next:3 • Chapter 4 is based on our paper [39]. It explores how to reproduce the known MHV Yang-Mills and gravity amplitudes (Parke-Taylor formula and Hodges formula). • Chapter 5 is based on our paper [40], where we derived a new compact formula for single trace tree level MHV amplitudes for Einstein-Yang-Mills theory, using the CHY formalism. • Chapter 6 is based on our paper [41], where we studied the correspondence between helicity configurations with the solutions to the scattering equation in four dimensions. We then provide some possible future directions before we conclude in Chapter 7. Finally, in the three appendices, we give some important technical details that are used in our main text. 3A concise treatment can be found in [38]. CHAPTER 2 GAUGE AND GRAVITY AT TREE LEVEL This chapter reviews and summarizes several well-known properties of tree level gauge and gravity amplitudes. The emphasis is on the gauge amplitude relations and the double copy between gauge and gravity. In Section 2.1, we introduce the spinor helicity formalism. In terms of spinor variables, gauge amplitudes are surprisingly simple. This is totally unexpected from what Feynman diagrams suggest. It was the invention of the spinor helicity formalism that gave birth to the field of modern scattering amplitude. In Section 2.2, we briefly review the traditional Feynman diagram approach to calculate the Yang-Mills amplitudes. The simplicity of the amplitudes and the existence of various amplitude relations clearly indicate that the gauge amplitudes have a very elegant mathematical structure. A lot of effort has been put on understanding this structure, and the pursuit, in turn, provides novel and more efficient ways to evaluate amplitudes. In principle, amplitudes of the perturbative gravity can also be obtained from the Feynman diagram approach, but as we show in Section 2.3, the complexity soon becomes unwieldy. String theory tells us that gravity amplitude is secretly an inner product of color ordered gauge amplitudes, and their expressions are well organized. Similarly, we show in Section 2.4 that the color kinematic duality is the key to ensure that the double copy of gauge amplitudes can lead to a meaningful gravity amplitude, even with certain class of matter interactions. 2.1 Spinor helicity formalism First proposed by Xu, Zhang, and Chang [4] in 1984,1 the spinor helicity formalism significantly simplifies the result of multigluon amplitudes, which earned its name "Chinese magic." In 4d space-time, given a null vector pµ , we have the identity: 1 The preprint came in 1984 while the paper was not published until 1987. 9 p2 = ηµν pµ pν = det pµ σµ = 0 . Consequently, the 2 × 2 matrix pαα̇ ≡ pµ σµ αα̇ (2.1) must have co-rank one, such that we can decompose it into the direct product of two spinors: λα̇ ≡ | p]h p| . pαα̇ = λα e (2.2) The left-handed spinor λα transforms as the (1/2, 0) representation of SL(2, C) while the right-handed one e λα̇ transforms as (0, 1/2). Here | p] and h p| are the two short-hand notations for spinors. We can raise the index α and α̇ in the following way to form the dual spinors: eαβ λ β = λα ≡ [ p| , e α̇ ≡ | pi . eα̇ β̇ e λ β̇ = λ (2.3) If pµ is a real momentum, then pαα̇ is an Hermitian matrix, which imposes that e λα̇ = (λα )∗ . Once we complexify pµ , which is a very useful technique in this field, then λα and e λα̇ become totally independent. If we manage to find one set of {λα , e λα̇ } that makes Eq. (2.2) hold, the spinors rescaled by a complex number t: λ α → t −1 λ α , e λα̇ λα̇ → te (2.4) will also do the job. For real momentum, t is restricted to be a phase. We call this operation little group rescaling.2 This point is important to make the degrees of freedom balanced in Eq. (2.2): now both sides have 3 real (complex) degrees of freedom, depending on whether pµ is real or complex. By construction, pαα̇ is invariant under the little group rescaling. 2.1.1 Spinor identities If we have two null momenta pµ and qµ , their inner product can be written in terms of spinor inner products as: 2p · q ≡ 2pµ qµ = pα β̇ q β̇α = −h pqi[ pq] , (2.5) where the angular and square brackets are the abbreviation of the spinor inner products: h pqi ≡ (e λ p )α̇ (e λq )α̇ , 2 This [ pq] ≡ (λ p )α (λq )α , redundancy is the little group associated with massless particles. (2.6) 10 agreeing with the short-hand notations introduced in Eq. (2.2) and (2.3). The inner products are antisymmetric: h pqi = −hqpi , [ pq] = −[qp] . (2.7) In particular, we have the identity: h ppi = [ pp] = 0 , (2.8) both due to the antisymmetric nature of eαβ . Similarly, we have the product [ p|σµ |qi, which essentially states that the (1/2, 1/2) representation of SL(2, C) is the complex fourmomentum representation of the Lorentz group. We can easily derive that: [ p|σµ |qi = hq|σµ | p] , (2.9) eαβ eα̇ β̇ (σµ ) β β̇ = (σµ )α̇α . (2.10) which follows directly from: If we further simplify our notation by using |i ] ≡ | pi ], etc, as we will do repeatedly in this dissertation, we can write down two important identities, which hold for arbitrary spinors, in a very nice form: Fierz identity: h1|σµ |2]h3|σµ |4] = −2h13i[24] , [1|σµ |2i[3|σµ |4i = −2[13]h24i . (2.11) This identity directly comes from the fact that: (σµ )α̇α (σµ ) β̇β = 2eα̇ β̇ eαβ , (σµ )αα̇ (σµ ) β β̇ = 2eαβ eα̇ β̇ , which is sometimes also called Fierz identity. Schouten identity: |1ih23i + |2ih31i + |3ih12i = 0 , |1][23] + |2][31] + |3][12] = 0 . (2.12) It follows from the fact that the spinor space is two dimensional, such that the symmetrization of any three indices vanishes identically: eα( β eγδ) = 0 , eα̇( β̇ eγ̇δ̇) = 0 . 11 If we have n scattering massless particles, there total momentum must be conserved:3 n ∑ ( pi )αα̇ = 0 . (2.13) i =1 Once we contract it with two arbitrary spinors h j| and |k], we get another identity, imposed by physics: n ∑ h jii[ik] = 0 . (2.14) i =1 This enables one to eliminate certain momentum spinors in a physical scattering process. Occasionally, one needs to "cross" a particle from initial state to final state. This corresponds to the analytic continuation p → − p. In this disseration, we use the convention: | − pi = −| pi , | − p] = +| p] , (2.15) namely, we flip the sign of angular spinor. Henceforth, we will use the angular and square spinor notation consistently, for the sake of simplicity. The index α and α̇ will be kept implicit whenever possible. 2.1.2 Polarization vector The polarization eµ is a complex null vector that describes the helicity state of external gluon and graviton. In four dimensions, we only have two helicities for each particle with spin: positive and negative, the polarization of which is denoted as e± . For a momentum pi , the polarization ei± must satisfy the transverse condition: pi · ei± = 0 . (2.16) According to Eq. (2.5), the transverse condition means that ei± either shares |i ] or hi | with pi = |i ]hi |. Therefore, we can write them as ei+ (η ) ∝ |i ]hη | , ei− (ξ ) ∝ |ξ ]hi | , where hη | and |ξ ] are two arbitrary reference spinors. They encode exactly the gauge freedom in the polarization vectors: ei → ei + Cpi , ∀C ∈ C . (2.17) We can use the little group rescaling to fix the form of ei up to an overall constant depending on conventions. Gauge invariance of physical amplitudes requires hη | and |ξ ] be 3 It is customary to choose all particles be outgoing. 12 cancelled in final results, so that e± should be invariant under a rescaling of them. This prompts us to express: ei+ (η ) = where the coefficient √ √ 2× |i ]hη | , hηi i ei− (ξ ) = √ 2× |ξ ]hi | , [ξi ] (2.18) 2 is just a convention. The vector form of e± is: 1 [i |σµ |η i (ei+ )µ = √ , 2 hηi i 1 hi | σ µ | ξ ] (ei− )µ = √ , 2 [ξi ] (2.19) in which we simply use e± , with the reference spinor dependence suppressed. Using the Fierz identity (2.11), we have the following results for inner products of polarization vectors: [ij]hηi η j i , hiηi ih jη j i [iξ j ]h jηi i ei+ · e− j = hiη i[ jξ ] , i j ei+ · e+ j = − hiji[ξ i ξ j ] , [iξ i ][ jξ j ] hiη j i[ jξ i ] ei− · e+ j = [iξ ]h jη i . i j ei− · e− j = − (2.20) Under the litte group rescaling (2.4), ei± transforms as: ei+ → ti−2 ei+ , ei− → ti+2 ei− . (2.21) Gluons are spin one massless particles, and their helicity states are described by a single polarization vector (e± )µ . Gravitons are spin two massless particles, and their helicity states are described by a polarization tensor (e± )µ (ẽ± )ν . The tilde over the second vector indicates that we can choose a different gauge from the first one. 2.2 Yang-Mills amplitudes The Yang-Mills theory with the gauge group U ( N ) is given by the Lagrangian: 1 LYM = − Tr F µν F µν . 4 (2.22) The field strength tensor F µν equals: ig F µν = ∂µ A ν − ∂ν A µ − √ A µ , A ν , 2 (2.23) where the gluon field A µ = Aµa T a is a Lorentz vector taking value in the Lie algebra u( N ). The gluons are thus in the adjoint representation, with a runs from 1 to N 2 − 1. As an 13 U ( N ) gauge theory, we have one more particle, the "photon," associated with the identity matrix 1. The generators close the Lie algebra: h i T a , T b = i f˜abc T c , a, b, c ∈ {1, 2, . . . , N 2 − 1} . (2.24) 2 Since all the generators commute with the photon generator T N = 1, the photon does not interact with any gluon. Our structure constant f˜abc is related to the usual one through √ f˜abc = 2 f abc , such that the generators are normalized as Tr( T a T b ) = δ ab . This also √ accounts for the 2 in F µν . Similar to the spinor Fierz identity (2.11), we have the U ( N ) Fierz identity for the N × N matrix ( T a ): 4 N2 ∑ (T a ) j i (T a )l k = δl i δj k . (2.25) a =1 Instead of a comprehensive introduction, we only focus on the amplitudes of Yang-Mills. Unless otherwise stated, all the amplitudes considered in this dissertation are at the tree-level. 2.2.1 Color-ordered amplitudes Color and kinematic factors are entangled in gluon n-point amplitudes calculated from traditional Feynman rules. As the first step towards revealing the unsung beauty in the amplitudes, we are going to strip off color factors and only study color-ordered (partial) amplitudes in the following. Quantization of a gauge theory always start with the gauge fixing. We choose possibly the most amplitude-friendly gauge, Gervais-Neveu gauge [42]. First, we introduce the Lie algebra valued tensor: ig H µν = ∂µ A ν − √ A µ A ν . 2 (2.26) µ The gauge choice G that leads to Gervais-Neveu gauge is G = H µ . The field component of G can be extracted by a trace: ig G a ≡ Tr(G T a ) = ∂µ Aµa − √ Tr(A µ A µ T a ) . 2 The path integral including this gauge fixing is Z Z Z i 4 a a a a 4 A ) F (A ) ∏ δ( G − f ) exp i d x LYM . Z = exp − d xf f ( DA 2 x,a (2.27) (2.28) (σµ )αα̇ means that four vector is the bi-fundamental representation (1/2, 1/2) of SL(2, C), the adjoint representation ( T a ) j i is the bi-fundamental representation ( N, N ) of U ( N ). 4 While 14 The Fadeev-Popov determinant F (A ) leads to a ghost C , described by the Lagrangian: ig ig µ µ √ C C C C H , Dµ = ∂ µ − √ A µ , Dµ D + Lgh = Tr µ 2 2 which can be ignored from now on since at tree-level ghosts do not appear. Actually, Gervais-Neveu gauge trades an ugly ghost Lagrangian for a very nice-looking gauge fixed Lagrangian: √ g2 1 L = Tr − (∂µ A ν )(∂µ A ν ) − i 2 g(∂µ A ν )A ν A µ + A µ A ν A µ A ν 2 4 , (2.29) from which the Feynman rules are derived as: 1 √ µ µ µ 2 = g 2 Tr( T a1 T a2 T a3 )(η µ1 µ2 p1 3 + η µ2 µ3 p2 1 + η µ3 µ1 p3 2 ) + (2 ↔ 3) , (2.30) 3 2 1 = g2 Tr( T a1 T a2 T a3 T a4 ) × η µ1 µ3 η µ2 µ4 + (234) permutation . (2.31) 3 4 Notably, the color factors are encoded in a color trace. When two lower-point diagrams amalgamate into one higher-point diagram, we need to sum over the internal color index at the point they are glued. The U ( N ) Fierz identity then makes the two color traces into one, for example, 3 2 N2 a a = ∑ Tr(T a T a T a ) Tr(T a T a T a ) = Tr(T a T a T a T a ) . 1 2 3 4 1 2 3 4 a =1 1 4 Therefore, the total n-particle tree-level amplitude An must be of the following form: An = gn−2 Tr( T a1 T a2 . . . T an ) An (12 . . . n) + ( noncyclic perm. ) , (2.32) where the summation is over the noncyclic permutation of {1, 2, . . . , n}. The amplitude An associated to each color trace is called color-ordered (partial) amplitude. As a very commonly used shorthand notation, we use i to stands for the data { pi , ei } of the particle i. For simplicity, we will use solid line in the following to represent gluons in the Feynman diagrams, instead of wavy lines. 15 The calculation of An is notorously difficult using the traditional Feynman diagram approach. One of the reasons is that even for moderate n, the number of Feynman diagrams involved quickly increases to one million [43], as shown in Table 2.1. As the first step towards simplification, Eq. (2.32) enables us to divide and conquer the total amplitude An : we divide the Feynman diagrams into subsets, each of which contributes to only one partial amplitude An . For example, only the three diagrams in Figure 2.1 contribute to the partial amplitude A4 (1234). The summation over permutation is not needed since other orders do not contribute to this partial amplitude. Such a division makes physical sense only if each partial amplitude is gauge invariant by itself. Namely, when replacing ei by pi , not only the total amplitude An , but also each partial amplitude An , satisfy the Ward identity: A n ( ei → p i ) = 0 . (2.33) This point is guaranteed by the partial orthogonality of the color traces: N2 ∑ Tr(T a1 a2 an T . . . T ) [Tr( T a σ (1) T a σ (2) ...T aσ(n) ∗ )] = N n δσI + O a i =1 1 N2 , (2.34) where σ ∈ Sn /Zn is a noncyclic permutation of {1, 2, . . . , n}, and δσI = 1 if and only if σ is the identity element (up to a cyclic ordering). Although this is not a strict orthogonality, it is sufficient to ensure the gauge invariance of each color-ordered amplitude since those amplitudes An do not contain N while the gauge invariance must hold at each order of O( N12 ) expansion. Table 2.1. Number of Feynman diagrams in gluon n-point amplitudes. n 4 5 6 7 8 9 10 # of diagrams 4 25 220 2485 34300 559405 10525900 2 2 3 1 4 1 3 2 3 1 4 4 Figure 2.1. The color-ordered Feynman diagrams that contribute to A4 (1234). 16 Before closing this subsection, we give a brief proof of Eq. (2.34). First, we notice that when the two T a 's are separated, Eq. (2.25) just rearranges the traces. If they are multiplied together, we gain a factor of N: N2 ∑ (T a ) j i (T a )i k = Nδj k . (2.35) a =1 This fact leads to the following proof: proof of partial orthogonality: Since all the Lie algebra generators are Hermitian matrices, the complex conjugate on the trace just reverses its order: [Tr( T aσ(1) T aσ(2) . . . T aσ(n) )]∗ = Tr( T aσ(n) . . . T aσ(2) T aσ(1) ) . Now we further simplify the notation by using ( a1 a2 . . . an ) ≡ Tr( T a1 T a2 . . . T an ) . First, if we have σ = I, Eq. (2.34) becomes ∑ (a1 a2 . . . an−1 an )(an an−1 . . . a2 a1 ) = ∑ (a1 a2 . . . an−1 an−1 . . . a2 a1 ) ai ai = N ∑ ( a 1 a 2 . . . a n −2 a n −2 . . . a 2 a 1 ) ai =N n −2 ∑ ( a1 a1 ) = N n . a1 This accounts for the part N n when σ = I. For σ 6= I, let us assume that after the first step, the second an−1 is at another position: ∑ (a1 a2 . . . an−1 an )(an X . . . Yan−1 Z . . .) = ∑(a1 a2 . . . an−1 X . . . Yan−1 Z . . .) ai ai = ∑( Z . . . a1 a2 . . . an−2 )( X . . . Y ) , ai namely, non-neighboring Fierz contraction does not contribute the factor N. Now the best hope is that ( Z . . . a1 a2 . . . an−2 )( X . . . Y ) = ( a1 a2 . . . an−2 )( an−2 . . . a2 a1 ) such that we get N n−2 after the contraction. In this way, we have proved that the right hand side of Eq. (2.34) is N which is the partial orthogonality. n δσI + O 1 N2 , 17 2.2.2 Helicity classification Before delving into calculations, we can get some valuable information from a simple dimensional analysis of the gauge amplitudes. The conclusion is that if there are only less than two particles having different helicities from the others, the amplitude must vanish. It also motivates us to classify amplitudes by the helicity configurations, which also represent the level of complexity in the amplitudes. Our first task is to find out the mass dimension of An (also An ) in four dimensions. The cross section for the 2 to n − 2 scattering process g A gB → (n − 2) g can be expressed formally as: −1 Z dσ |v A − vB | = ( E A EB ) | {z } | | {z } relative velocity initial energy d3 p 1 (2π )3 2Ep {z phase space n −2 δ4 ( p1 + . . . pn−2 − p A − p B ) |An |2 . (2.36) }| {z } momentum conservation The full expression can be found in any standard quantum field theory textbook, for example [1]. After writing down the mass dimension of each component, we can solve that of An : [m]−2 = [m]−2 [m]2n−4 [m]−4 [An ]2 =⇒ [An ] = [ An ] = [m]4−n . (2.37) An n-point tree-level Feynman diagram with only 3-point vertices has exactly n − 2 vertices and n − 3 internal propagators. Since the 3-point vertex has mass dimension one, we work out correctly that: [ An ] = [ m ] n −2 = [ m ] 4− n . [m]2n−6 (2.38) For each presence of 4-point vertices, we eliminate two adjacent 3-point vertices and the propagator in between so that the above counting still works. Now here comes the punchline: if we dot in n polarization vectors, there must be at least two of them get multiplied together, since there are at most n − 2 momentum factor upstairs: An ∼ ∑ (ei · e j )(ek · pl )n−2 ( ei · e j ) 2 ( e k · p l ) n − 4 ( p m · p s ) + +··· . ∑ ( PI2 )n−3 ( PI2 )n−3 (2.39) Now let us look at the case when all gluons have the same helicity. According to Eq. (2.20), if we choose all the reference spinors be the same: | η1 i = | η2 i = · · · = | η n i = | η i , |ξ 1 ] = |ξ 2 ] = · · · = |ξ n ] = |ξ ] , 18 their inner products must vanish: ei± · e± j = 0. Then the amplitudes must also vanish, according to Eq. (2.39): A n (1− 2− . . . n − ) = 0 . A n (1+ 2+ . . . n + ) = 0 , (2.40) Next, we flip one helicity, say particle 1, and choose the following reference spinors: | η2 i = · · · = | η n i = | 1 i , | ξ 2 ] = · · · = | ξ n ] = |1] , ∓ ± we can achieve ei± · e± j = 0 and e1 · e j = 0, such that the following amplitudes also vanish: A n (1− 2+ 3+ . . . n + ) = 0 , A n (1+ 2− 3− · · · n − ) = 0 . (2.41) This luck ends if we flip one more particle. Suppose now we have particle 1 and 2 of negative helicity and all the others positive. With the reference spinors: | η1 i = | η2 i = η , | η3 i = | η4 i = · · · = | η n i = | 1 i , the only nonzero inner products are e2− · e+ j , so that the amplitude looks like: − − + + A n (1 2 3 . . . n ) ∼ ∑ n −2 (e2− · e+ j )( ek · pl ) ( PI2 )n−3 . (2.42) This amplitude is called maximally helicity violating (MHV) amplitude. Similarly, we can define Nk MHV amplitudes, which have k + 2 negative helicity particles. The case k = n − 4 is called anti-MHV, which has only two positive helicites. The anti-MHV amplitude can be obtained by a complex conjugation of the MHV. 2.2.2.1 Three-point amplitudes We now show that all the 3-point Yang-Mills amplitudes can be determined without consulting Feynman diagrams. The 3-point special kinematics has one peculiarity that is not shared by any other higher points: p1 + p2 + p3 = 0 =⇒ ( p1 + p2 )2 = − p23 = 0 , where p1,2,3 are complex null momenta. The above two relations together lead to h12i[23] = 0 , h12i[13] = 0 , h12i[12] = 0 , namely, if h12i 6= 0, all the square brackets vanish: [12] = [23] = [13] = 0. If all momenta are real, then angular brackets also vanish since square and angular spinors are conjugate. 19 The conclusion is that if we allow complex momenta, then under 3-point special kinematics, we can only have either nonzero angular brackets or square ones, not both. Now let us look at amplitudes. First, we notice that when evaluating Feynman diagrams, polarization vector of each particle only appears in the numerator once per diagram. Thus under the rescaling (2.4), the amplitude must transform in the same way as the polarization vector: An (. . . {t|i i, t−1 |i ], hi } . . .) = t−2hi An (. . . {|i i, |i ], hi } . . .) . (2.43) Using this property, we can determine the possible forms of 3-point gauge amplitudes. Suppose angular brackets are nonzero, on which the amplitude can only depends, then we must have: A3 (1h1 2h2 3h3 ) ∝ h12ih3 −h1 −h2 h13ih2 −h1 −h3 h23ih1 −h2 −h3 . (2.44) Next, we need to investigate the coefficient at each helicity configuration, modulo permutations: h12i3 , h23ih31i h23ih31i , A 3 (1+ 2+ 3− ) = c 1 h12i3 1 A 3 (1+ 2+ 3+ ) = c 2 , h12ih23ih31i A3 (1− 2− 3− ) = c3 h12ih23ih31i . A 3 (1− 2− 3+ ) = g (2.45a) (2.45b) (2.45c) (2.45d) Since A3 has dimension [m] in four dimensions, the coupling g in Eq. (2.45a) is thus dimensionless, which is exactly the one in Yang-Mills. In the real momentum limit, all angular brackets also vanish such that A3 (1− 2− 3+ ) = 0, which is physical since real 3-point kinematics forbids this scattering. On the other hand, Eq. (2.45b) and (2.45c) blow up in the real limit, and the couplings have positive mass dimension: [c1 ] = [m]2 , [c2 ] = [m]4 . This is the typical behavior of nonlocal interactions, which is absent in Yang-Mills. Therefore, we require c1 = c2 = 0. Finally, in Eq. (2.45d) the coupling c3 has negative mass dimension: [c3 ] = [m]−1 , which corresponds to the nonrenormalizable µ ρ F3 interaction Tr(F ν F ν ρ F µ ). Therefore, when angular brackets are nonzero, the only nonvanishing Yang-Mill 3-point amplitude in four dimensions is the MHV one (2.45a). The same analysis can be performed for the case of nonzero square bracket. 20 Our final result is that the only possibly nonzero 3-point amplitudes of four dimensional Yang-Mills are MHV and anti-MHV:5 A 3 (1− 2− 3+ ) = h12i3 , h23ih31i A 3 (1+ 2+ 3− ) = [12]3 . [23][31] (2.46) However, they cannot be nonzero at the same time. From the prospective of real kinematics, which one being nonzero depends on how we complexify the momenta: whether we deviate |i i or |i ] from their real kinematic values while momentum conservation still holds. To wrap up, we emphasize that to reach Eq. (2.46), we have only used: (1) rescaling property of massless spin one particles, (2) 3-point special kinematics, (3) locality and renormalizability. It is well known that in four dimensions the only local renormalizable quantum field theory for a massless spin one particle is Yang-Mills, and we do get uniquely the amplitude of what we want. 2.2.2.2 MHV amplitudes The behavior of Eq. (2.46) actually persists to arbitrary n-point MHV and anti-MHV amplitude. They can be described by the famous Parke-Taylor formula [3]: h12i4 , h12ih23ih34i . . . hn1i [12]4 . A n (1+ 2+ 3− . . . n − ) = [12][23][34] . . . [n1] A n (1− 2− 3+ . . . n + ) = (2.47) This behavior has been proved by Berends-Giele [10] and BCFW [44] recursive relation. Therefore, despite the sky-rocketing complexity in Feynman diagrams, the MHV amplitudes always turn out to be a single term at arbitrary n. 2.2.3 Gauge amplitude relations The calculation of the color-ordered amplitude An is much simpler than the total amplitude An , not only because it corresponds to fewer Feynman diagrams, but it also satisfies a number of nice relations, some are inherited from the color trace: Cyclic symmetry: An (123 . . . n) = An (23 . . . n1) = An (3 . . . n12) = · · · . 5 The common factor g is neglected in all the following equations. (2.48) 21 This property comes directly from the cyclic symmetry of the color traces, which ensures that in An (123 . . . n) all the cyclic rotations of particles have been summed over. We can use symmetry to fix the position of particle 1, so that the total number of independent color-ordered amplitudes is reduced to (n − 1)!. Reflection: An (123 . . . n) = (−1)n An (n . . . 321) . (2.49) For example, we can check that from Eq. (2.30), A3 (123) = (e1 · e2 )( p1 · e3 ) + (e2 · e3 )( p2 · e1 ) + (e3 · e1 )( p3 · e2 ) , A3 (321) = (e3 · e2 )( p3 · e1 ) + (e2 · e1 )( p2 · e3 ) + (e1 · e3 )( p1 · e2 ) = −(e1 · e2 )( p1 · e3 ) − (e2 · e3 )( p2 · e1 ) − (e3 · e1 )( p3 · e2 ) . The general proof comes from dividing the total amplitude in the usual Feynman gauge, whose color factor is a chain of structre constants, into each color-ordered amplitude. U (1) decoupling identity: An (123 . . . n) + An (213 . . . n) + An (231 . . . n) + . . . + An (23 . . . 1n) = 0 . (2.50) This identity follows from taking T a1 = 1, namely, make the particle 1 be a photon. Since the photon does not interact with the other gluons, the total amplitude must vanish. Then all the amplitudes in Eq. (2.50) carry the color trace Tr( T a2 T a3 . . . T an ) so that they must add to zero due to the partial orthogonality. All the above relations come with a careful analysis of the color trace structure. On the other hand, there exist other unexpected relations from the Feynman diagram perspective: Kleiss-Kuijf (KK) relation [11]: An (1, α , n, β ) = (−1)|β | ∑ An (1, σ , n) , (2.51) βT ) σ ∈OP(α ,β where α and β are two collections of particles, β T is the reverse ordering of β , and OP(α , β T ) are those permutations of α ∪ β T that preserve the ordering within α and β T . We can thus fix the positions of two particles, say, 1 and n, by the KK relation, so that the number of independent color-ordered amplitudes is reduced to (n − 2)!. This relation can be derived from the properties of color trace. 22 Bern-Carrasco-Johansson (BCJ) relation [29]: ! n i i =3 j =3 ∑ ∑ p2 · p j An (1, . . . , i, 2, i + 1, . . . , n) = 0 . (2.52) After we apply the U (1) decoupling identity to An (134 . . . n2), all the amplitudes are of the KK independent form An (1, σ , n), so that Eq. (2.52) indeed is a new set of relations among those KK independent amplitudes. The number of independent color-ordered amplitudes is further reduced to (n − 3)!. The existence of this relation is tied with the color kinematic duality, which will be discussed in Section 2.4 We now provide some examples to these highly nontrivial relations. In Eq. (2.51), if |β | = 1, the KK relation reduces to the U (1) decoupling identity (2.50). As a nontrivial 5-point example, we choose α = {2} and β = {4, 3}, such that the KK relation leads to: A5 (12543) = A5 (12345) + A5 (13245) + A5 (13425) . (2.53) Also at 5-point, the BCJ relation gives: s23 A5 (13245) + (s23 + s24 ) A5 (13425) − s12 A5 (13452) = 0 , (2.54) where s ab = 2p a · pb , and the momentum conservation has been used in the last term. After applying the U (1) decoupling identity (2.50) to the last amplitude, we obtain a relation between the three KK independent amplitudes: s12 A5 (12345) + (s12 + s23 ) A5 (13245) − s25 A5 (13425) = 0 . (2.55) The existence of these relations and the simplicity of the Parke-Taylor formula strongly suggest that some elegant structures in the color-ordered amplitudes are obscured by the unnecessarily complicated Feynman diagram evaluation. The culprit is actually easy to spot: our insistence on off-shell gauge invariance. It introduces too many extra baggages that on-shell amplitudes do not depend on. Therefore, modern approach to amplitudes works with on-shell particles from the beginning. 2.3 Gravity amplitudes The classical theory for gravity is the general relativity. If the matter field is given by the Lagrangian Lmatter , then the following Einstein-Hilbert action solves how the spacetime curves and how the matter propagates in it at the same time: SEH = − 1 16πGN Z d4 x p − g ( R + 2Λ) + Z d4 x p − g Lmatter . (2.56) 23 The definition of each symbol in SEH is given by: GN R Λ g Newton's constant, we have κ 2 ≡ 8πGN = 1.69 × 10−37 GeV−2 at D = 4 Ricci scalar, see below for its definition cosmological constant, Λ ∼ (10−3 eV)4 in our universe the determinant of the metric gµν Quantum field theory and general relativity has long been plagued by various conventions in the literature, which constantly leads to a sign difference in expressions. In this dissertation, we use the most-negative signature for gµν , and the following expressions for the geometric quantities involved: Christoffel connection Γλµν : Riemann curvature tensor Γλµν = 1 λρ g ∂µ gνρ + ∂ν gρµ − ∂ρ gµν 2 Rλσµν = ∂µ Γλνσ − ∂ν Γλµσ + Γλµρ Γνσ − Γλνρ Γµσ ρ Rλσµν : Ricci tensor Rµν : Rµν = Rλµλν Ricci scalar R : R = gµν Rµν . ρ (2.57) Our convention agrees with [45]. The equation of motion for gµν , the Einstein equation, is obtained by δSEH = 0. δgµν First, by varying the metric, the matter action produces the energy-momentum tensor: Tµν = √ 2 δSmatter , − g δgµν (2.58) which satisfies the conservation law ∇µ T µν = 0. With all the preparation, we are now ready to write down the Einstein equation: δSEH =0 δgµν =⇒ 1 Rµν − Rgµν − Λgµν = κ 2 Tµν . 2 (2.59) The left hand side is usually called the Einstein tensor: Gµν ≡ Rµν − (1/2) Rgµν − Λgµν . First, we assume that the matter distribution is fixed, namely, Tµν is given and Eq. (2.59) is used to solve gµν . Both Gµν and Tµν are symmetric tensors, so that there are 10 equations in Eq. (2.59) in four dimensions. However, Gµν must satisfy 4 constraints ∇µ G µν = 0, so that there are only 6 independent equations in Eq. (2.59). At the first glance, the metric gµν should also have 10 degrees of freedom in four dimensions such that Eq. (2.59) could not 24 solve them all. However, since two metrics related by a diffeomorphism are considered equivalent: geµν ( xe) = ∂x ρ ∂x λ gρλ ( x ) , ∂e x µ ∂e xν (2.60) and the diffeomorphism invariance is a gauge symmetry. Therefore, by a gauge fixing, we can eliminate 4 degrees of freedom in gµν , so that the 6 independent equations in Eq. (2.59) is enough to solve gµν . The harmonic gauge (or de Donder gauge): ∂µ ( gµν p − g) = 0 (2.61) is very commonly used for perturbative gravity. If we allow the matter field to interact with gravity, we need to associate the Einstein equation with the Euler-Lagrange equation of the matter field. For example, if we have a scalar field Lmatter = gµν (∂µ φ)(∂ν φ) − V (φ), we need to add in the equation of motion: gµν ∇µ ∇ν φ − dV (φ) = 0. dφ (2.62) The physical meaning of these two equations are nicely summarized by John Wheeler: Eq. (2.59): Eq. (2.62): matter tells spacetime how to curve spacetime tells matter how to move In this dissertation, we will study one such interaction, the Einstein-Yang-Mills theory (EYM), where Lmatter = LYM . According to Eq. (2.58), the energy momentum tensor of Yang-Mills reads: ρ Tµν = − Tr(F µρ F ν ) + 1 gµν Tr(F ρσ F ρσ ) , 4 (2.63) through which gluons interact with gravitons. The definition of graviton in the weak field limit is the subject of the next subsection. 2.3.1 Perturbative gravity In the weak field limit, we can expand the metric around the flat spacetime as gµν = ηµν + hµν , (2.64) where hµν is the graviton field. Now we can treat hµν as a usual field on the flat background spacetime, just as a matter field. Therefore, the contraction of indices is made through the √ flat metric ηµν . Both the expansion of gµν (the inverse metric) and − g do not terminate: 25 gµν = η µν − hµν + h ρ hρν + O(h3 ) , p 1 1 1 − g = 1 + h + hµν hµν + h2 + O(h3 ) , 2 2 8 µ (2.65) where h ≡ η µν hµν . Consequently, the expansion generates an infinite series of interaction vertices that are cubic and higher in hµν . As a local quantum field theory, the perturbative gravity is clearly nonrenormalizable, since we need to introduce an infinite number of counterterms. However, we can still use it as an effective field theory, valid at energy scale much lower than the Planck scale. First, let us look at the pure gravity, with Lmatter = 0. It describes how gravitons propagate and scatter in an otherwise empty spacetime. Under the expansion (2.65), the first and second order of the Einstein-Hilbert Lagrangian are (with proper integral by parts and discarding boundary terms): p − g R = ∂µ ∂ν hµν − ∂2 h 1 1 1 ρ µν µν µ µν ρ + ∂ h ∂µ hρν − ∂µ h ∂ν h + ∂µ h∂ h − ∂ρ h ∂ hµν 2 2 2 · · · O(h) · · · O(h2 ) (2.66) where ∂2 ≡ ∂µ ∂µ . The O(h) term integrates to zero in the action since it is a total derivative. The O(h2 ) terms are actually the free part of the weak field Lagrangian: 1 p −g R 2 2κ 1 1 1 µν ρ µν µν ρ µ = 2 ∂µ h ∂ν h − ∂ h ∂µ hρν + ∂ρ h ∂ hµν − ∂µ h∂ h + O(h3 ) . 4κ 2 2 LEH = − (2.67) Clearly, there is no mass term hµν hµν in the Lagrangian, so that graviton is a massless particle. Varying the Lagrangian with respect to δhµν gives the equation of motion: 1 ρ ∂ ∂µ hρν + ∂ρ ∂ν hρµ − ∂2 hµν − ∂µ ∂ν h − ηµν ∂ρ ∂λ hρλ + ηµν ∂2 h = 0 , (2.68) 2 where the left hand side is just the O(h) order of the Einstein tensor Gµν . In Eq. (2.60), if we take gµν = ηµν + hµν , an infinitesimal coordinate transformation xeµ = x µ − ξ µ will lead to a gauge transformation in hµν : hµν → e hµν = hµν + ∂µ ξ ν + ∂ν ξ µ . (2.69) Indeed, we can check that LEH is invariant at O(h2 ). Actually, it is the only possible combination quadratic in hµν that is gauge invariant [46]. The weak field expansion of the harmonic gauge is: ∂µ gµν p 1 − g = −∂µ hµν + ∂ν h + O(h3 ) . 2 (2.70) 26 After this gauge fixing, the quadratic Lagrangian contains only two terms: 1 LEH + 2 4κ ∂µ h µν 1 − ∂ν h 2 2 1 = 2 8κ 1 ∂ρ h ∂ hµν − ∂µ h∂µ h 2 µν ρ , (2.71) which leads to the graviton propagator: Dµ1 ν1 µ2 ν2 ( p) = i ηµ1 µ2 ην1 ν2 + ηµ1 ν2 ην1 µ2 − ηµ1 ν1 ηµ2 ν2 . 2 2p (2.72) Like the Lorentz gauge ∂µ Aµ = 0 in electrodynamics, there is left-over gauge freedom under the harmonic gauge. Actually, on-shell graviton only has two internal states: helicity ±2 state described by the polarization tensor (e± )µν ≡ (e± )µ (e± )ν . The interaction between graviton and matter can be obtained by expanding √ − gLmatter with respect to hµν : Lh-matter = p 1 1 − gLmatter = T − hµν Tµν + O(h2 ) , 2 2 (2.73) where T ≡ Tµν η µν is the trace of the energy momentum tensor Tµν . Together with LEH , we can reproduce the full Einstein equation at the weak field limit: ∂ρ ∂µ hρν + ∂ρ ∂ν hρµ − ∂2 hµν − ∂µ ∂ν h − ηµν ∂ρ ∂λ hρλ + ηµν ∂2 h = κ 2 Tµν . (2.74) For EYM, the leading order graviton-gluon interaction is 3-point, involving one graviton and two gluons. The self-interaction between gravitons are described by O(h3 ) and higher order terms in LEH . Just from the index structure, we can tell that these terms are very complicated. For example, O(h3 ) is given by all possible contractions between three hµν and two ∂µ , with indices properly symmetrized. In all we have 171 separate terms at O(h3 ), while this number grows to 2850 at O(h4 ) [6]. Therefore, calculation of graviton amplitudes using traditional Feynman rule approach is very involved. For example, in calculating 4-point graviton amplitudes, the expressions of s, t and u amplitudes span one and a half page each, while the final result boils down to only one single term, like the Yang-Mills case [47]. Clearly, a new algorithm with complexity matching final results may better reveal the nature of perturbative gravity. 27 Despite the complexity in Feynman rules, the graviton 3-point amplitude can also be fixed through little group rescaling and locality. We can redefine the graviton field as hµν → κhµν , and write the action schematically as: SEH = Z d4 x h∂2 h + κh2 ∂2 h + κ 2 h3 ∂2 h + . . . . (2.75) The coupling constant of the 3-point vertex is κ, with mass dimension [m]−1 . Plugging hi = ±2 into Eq. (2.44), we get: h12i6 , h23i2 h31i2 1 M3 (1+ 2+ 3+ ) = c2 , 2 h12i h23i2 h31i2 M3 (1− 2− 3+ ) = κ M3 (1+ 2+ 3− ) = c1 h23i2 h31i2 , h12i6 M3 (1− 2− 3− ) = c3 h12i2 h23i2 h31i2 . (2.76) Like the Yang-Mills case, the κ one is the correct graviton 3-point MHV amplitude, while [c1 ] = [m]3 and [c2 ] = [m]7 are two nonlocal interactions, absent in the gravity theory. However, the [c3 ] = [m]−5 one corresponds to the R3 interaction, in which 3-point vertex has the form (∂2 h)(∂2 h)(∂2 h). It appears in some modified gravity theory, but not in our Einstein gravity. Therefore, in four dimensions, the only nonzero 3-point graviton amplitudes are again MHV and anti-MHV: M3 (1− 2− 3+ ) = h12i6 , h23i2 h31i2 M3 (1+ 2+ 3− ) = [12]6 . [23]2 [31]2 (2.77) Remarkably, comparing Eq. (2.76) and Eq. (2.44), we get M3 (123) = [ A3 (123)]2 , which holds not only between the Einstein gravity and Yang-Mills, but also between those nonminimal gravity and gauge theories with higher dimensional and nonlocal terms. Since the little group rescaling essentially means that for each Feynman diagram, the external wavefunction of each particle can only appear once in the numerator, it actually sorts out all possible field theoretical interactions. At three point, we have just derived the relation gravity = (gauge)2 . For higher point amplitudes, since we do not have any new coupling constants both in gravity and gauge, we hope this nice pattern to persist. Indeed it does in some less trivial form than a simple square. This will be explored our next two subsections. 2.3.2 Kawai-Lewellen-Tye (KLT) relation Nonrenormalizable in nature with infinite interaction terms, the perturbative gravity resembles little with the much more organized Yang-Mills. However, gravity is secretly 28 a double copy of gauge theory in some sense. The clue first comes in the study of string amplitudes. In 1986, Kawai, Lewellen, and Tye (KLT) showed that any n-point tree level closed string amplitude can be written as a sum over two n-point tree level open string amplitudes [7]. In the field theory limit (or technically "α0 → 0 limit"), closed and open string amplitudes reduce to gravity and gauge amplitudes, respectively. Therefore, the simplest double-copy behavior shown in Eq. (2.77) does have a generalization to higher points. 2.3.2.1 Pure gravity Using the modern amplitude language, we can write the generic KLT relation between gravity and Yang-Mills amplitudes in the form of an inner product over the (n − 3)! permutations of {2, 3, . . . , n − 2}, denoted as Sn−3 : Mn (12 . . . n) = − ∑ An (1, α , n, n − 1)S[α |β ] An (1, β , n − 1, n) , (2.78) β ∈ S n −3 α ,β where the KLT kernel S[α |β ] is: n −2 S[α |β ] = ∏ i =2 " i −1 s α i 1 + ∑ θ ( α j , α i )β s α j α i # (2.79) j =2 We define the function θ to be: 1 α j is also before αi in β . 0 otherwise θ ( α j , αi ) = (2.80) Note that α j always comes before αi in α . This formula is first written down and proved in [48], while we follow the notation of [33]. With this definition, S[α |β ] is symmetric:6 S[α |β ] = S[β |α ] , (2.81) since the criterion in θ is symmetric in α and β : suppose we have αi = β k , then i −1 k −1 j =2 j =2 ∑ θ ( α j , α i )β s α j α i = ∑ θ ( β j , βk )α sβ β j k , and Eq. (2.81) follows. At n = 3, we can immediately verify Eq. (2.77) from Eq. (2.78); at n = 4, we have S[2|2] = s12 such that: M4 (1234) = − A4 (1234)s12 A4 (1243) . 6 Using (2.82) the notation of the original paper [48], we have to reverse both α and β : Sthere [α |β ] = Sthere [β T |α T ]. 29 Now using the 4-point Parke-Taylor formula, we can get the 4-point graviton MHV amplitude: M4 (1− 2− 3+ 4+ ) = − A4 (1− 2− 3+ 4+ )s12 A4 (1− 2− 4+ 3+ ) h12i7 [12] h13ih14ih23ih24ih34i2 h12i4 [34]4 , =− stu =− (2.83) where the symbol s, t, and u are the usual Mandelstam variables: s = ( p1 + p2 )2 , t = ( p1 + p4 )2 , u = ( p1 + p3 )2 . As a less trivial example, the n = 5 KLT kernel consists of: S[23|23] = s12 (s13 + s23 ) S[23|32] = s12 s13 , S[32|23] = s12 s13 S[32|32] = s13 (s12 + s32 ) , such that the KLT relation reads: M5 (12345) = A5 (12345)s12 s34 A5 (14352) + A5 (13245)s13 s24 A5 (14253) , (2.84) in which the BCJ relation (2.52) has been used to simplify the result. The KLT relation (2.78) can be understood in the following way. The Yang-Mills amplitude An is a summation over Feynman diagrams, each of which is multilinear in the polarizations ei . Thus we need exactly two An to match the polarization degrees of freedom of gravitons. The summation over different color ordering is essential since graviton amplitudes are symmetric under permutation while An is not. Finally, the coefficients sij come to cancel the double poles introduced by multiplying two gauge amplitudes. For example, in Eq. (2.82) the product A4 (1234) A4 (1243) introduces a double pole at ( p1 + p2 )2 = 0, which is exactly canceled by the coefficient s12 . This is the reason why the KLT kernel S[α |β ] soon becomes very complicated at higher points. 2.3.2.2 Einstein-Yang-Mills Besides pure gravity, the total EYM amplitudes can also be constructed by the KLT product of amplitudes from Yang-Mills and a special double-colored scalar theory [49]: L = Tr g2 iκ ∇µΦ a (∇µΦ a ) − f abc Tr [Φ a , Φ b ]Φ c + Tr [Φ a , Φ b ][Φ a , Φ b ] 3! 4 (2.85) 30 0 0 0 where Φ a ≡ φ a a T a . The scalar field φ a a is in the adjoint representation of U ( N ) × U ( N ). However, only the primed color charges couple to our gluons. The n-point total EYM amplitudes with s gravitons and r = n − s gluons can be expressed as: Mn,r = − ∑ en,r (α )S[α |β ] An (β ) , A (2.86) β ∈ S n −3 α ,β where An (β ) is the n-point gluon color ordered amplitude complied to the permutation β : A n ( β ) ≡ A n [ g1 , β ( g2 , . . . , g n − 1 ) , g n − 1 , g n ] . en,r is the n-point partially color ordered As the other component of the KLT inner product, A amplitude with r scalars and n − r gluons, where the order of external legs are fixed according to (1, α , n, n − 1): en,r (α ) ≡ A en,r [s a1 , α (s a2 , . . . , srar , gr+1 , . . . , gn−2 ), gn , gn−1 ] . A 2 1 They are given by the Feynman rules derived from Eq. (2.85). The pure gluon part is the same as Eq. (2.30) and (2.31), while those involving scalars are shown in Figure 2.2 on the en,r is color ordered in a0 , the color charge that couples following page. We emphasize that A to the gluons, while the uncouple color charge a is NOT color ordered. This is why we call en,r partially color ordered. If we require the resultant amplitudes to have only one single A trace, then the external scalar legs should be connected. We can understand Eq. (2.86) like en,r , it gets "squared" and promoted into a graviton. For this: if a gluon appears both in An and A the other gluons, they obtain color factors from the scalars and get summed over all possible color orderings. Now we give a few simple examples. First, we compute the 4-point pure scalar amplie4,4 (s a1 s a2 s a4 s a3 ), given by the following Feynman diagrams: tude A 1 2 4 3 2 2 4 4 + 1 3 =− 1 3 Then using i f˜a1 a2 b = Tr([ T a1 , T a2 ] T b ), we have: f˜a1 a2 b f˜ba4 a3 f˜a2 a4 b f˜ba3 a1 − . s12 s24 (2.87) 31 2 2 3 i = − √ δ a1 a2 ( p 1 − p 2 ) µ 2 = i f˜a1 a2 a3 1 1 3 2 2 3 i = δ a1 a2 η µ3 µ4 2 1 4 2 3 = −iδ a2 a4 η µ1 µ3 1 4 i = −iδ a1 a3 δ a2 a4 + (δ a1 a2 δ a3 a4 + δ a1 a4 δ a2 a3 ) 2 1 4 en,r . Figure 2.2. Feynman rules for partially color ordered amplitude A e4,4 (s a1 s a2 s a4 s a3 )s12 A4 ( g1 g2 g3 g4 ) −A 1 2 4 3 s23 Tr( T a1 T a2 T a4 T a3 ) = A4 ( g1 g2 g3 g4 ) Tr( T a1 T a2 T a3 T a4 ) + Tr( T a1 T a4 T a3 T a2 ) + s24 s12 s12 s23 a1 a3 a4 a2 a1 a3 a2 a4 a1 a4 a2 a3 Tr( T T T T ) + Tr( T T T T ) + Tr( T T T T ) + s24 s24 s24 = Tr( T a1 T a2 T a3 T a4 ) A4 ( g1 g2 g3 g4 ) + (234) permutations , (2.88) namely, the final result is nothing but the total 4-point gluon amplitude A4 . The last identity can be verified by expressing all the color ordered 4-point amplitudes in terms of A4 ( g1 g2 g3 g4 ), using KK and BCJ relation. We do not include the scalar 4-point vertex here since it contains a double trace structure. However, this vertex is crucial for constructing correct multitrace amplitudes through KLT. We will study such an example in Section 2.4.2. To obtain a single trace EYM amplitude with a graviton, we can replace leg 4 by a e4,3 (s a1 s a2 g4 s a3 ): gluon, for example, such that we now have A 3 1 2 2 2 4 4 √ p 3 · e4 p 2 · e4 a1 a2 a3 ˜ = if 2 − . s12 s24 + 1 3 1 (2.89) 3 Suppose we choose gluon 4 be of negative helicity, with reference spinor |2], we have: e4,3 (s a1 s a2 g− s a3 ) = [Tr( T a1 T a2 T a3 ) − Tr( T a1 T a3 T a2 )] h34i[23] , A 1 2 4 3 s23 [24] 32 such that the amplitude M4,3 ( g1+ g2+ g3− h4− ) can be calculated by: e4,3 (s a1 s a2 g− s a3 )s12 A4 ( g+ g+ g− g− ) M4,3 ( g1+ g2+ g3− h4− ) = − A 1 2 3 4 1 2 4 3 = [Tr( T a1 T a2 T a3 ) − Tr( T a1 T a3 T a2 )] × h34i4 . h12ih23ih31i (2.90) As a general pattern, the single trace MHV amplitudes of EYM carry a Parke-Taylor denominator ranging within the gluons. The existence of such a KLT relation for EYM can be inferred from the heterotic string amplitudes. However, the list of functioning KLT pairs does not stop, and we require a better understanding on under what condition does this double copy trick work. A beautiful answer is given in Section 2.4 below. 2.3.3 Hodges formula for MHV gravity amplitudes According to the KLT construction Eq. (2.78) and (2.86), we can see clearly that the helicity classification of pure gravity and EYM amplitudes is identical to that of the gauge theory, since they both contain one gauge amplitude in the KLT inner product. The most compact formula for n-point tree level gravity amplitude is given by Hodges in 2012 [50, 51]: Mn (− − + . . . +) = h12i8 M̄ (12 . . . n) , (2.91) where M̄ is called the reduced gravity amplitude, which does not contain any information on helicity configurations. This quantity can be expressed by: ijk M̄(12 . . . n) = (−1)n+1 (−1)i+ j+k+ p+q+r cijk c pqr det(φ pqr ) , (2.92) where the c symbols are c abc = c abc = 1 . h abihbcihcai (2.93) The n × n Hodges matrix φ is defined as: φab = [ ab] h abi n ( a 6= b) , [ al ]hlξ ihlη i . h al ih aξ ih aη i l =1 φaa = − ∑ (2.94) l 6= a In the diagonal elements φaa , we have two reference spinors ξ and η. They represent a gauge freedom and the value of φaa does not depend on them: if we choose another spinor η̃, then we have: 33 [ al ]hlξ ihl η̃ i [ al ]hlξ ihl η̃ ih aη i ∑ hal ihaξ ihaη̃ i = ∑ hal ihaξ ihaη̃ ihaη i l 6= a l 6= a = l 6= a l 6= a = [ al ]hlξ ihlη i [ al ]hlξ ihη η̃ i ∑ haξ ihaη̃ ihaη i + ∑ hal ihaξ ihaη i [ al ]hlξ ihlη i ∑ hal ihaξ ihaη i , (2.95) l 6= a where the first term in the second line vanishes due to the momentum conservation. Fiijk nally, φ pqr is an (n − 3) × (n − 3) submatrix of φ with row {i, j, k } and column { p, q, r } deleted. It has been proved in [51] that φ has co-rank three and M̄ is independent of the choice of {i, j, k } and { p, q, r }. However, we are going to gain a better understanding on this point from the calculation of Cachazo-He-Yuan (CHY) formalism, to be discussed in detail in Chapter 4. 2.4 Color kinematic duality and double copy relation In hindsight, the existence of KLT gives the first evidence on that gravity is secretly the square of a gauge theory. However, the KLT relation is still not transparent and powerful enough to fully reveal such a connection. Immediately, one can ask whether there is a way to "diagonalize" the KLT kernel such that gravity amplitudes are really some kind of square of gauge amplitudes. A deeper question one can ask is: what properties must the two component amplitudes satisfy such that their KLT inner product can produce a meaningful gravity theory? The best answer up to now is the color-kinematic duality [29], which is the main subject of this section. 2.4.1 Formal construction First, it is instructive to study the total gluon amplitude An instead of the color ordered one, since it respects the Bose symmetry, just as graviton amplitudes. Without losing generality, we can formally write the total amplitude as: An = ∑ i ∈ Γ3 ci ni . Pi2 (2.96) The summation is over all trivalent diagrams Γ3 and Pi2 is the product of all internal propagators in the diagram i: Pi2 ≡ ∏ p2αi , αi αi ∈ internal edges of diagram i . 34 In the numerator, we can always factor out the color factor ci from the kinematic part ni . Here, ci is a polynomial of the structure constant f˜abc , while ni is a Lorentz invariant quantity, composed of external momenta and polarizations.7 Note that Γ3 is not the set of Feynman diagrams. However, it is easy to reorganize Feynman diagrams into Γ3 . First, all trivalent Feynman diagrams belong to Γ3 . In the other diagrams, we can break all the 4-point vertices into two trivalent ones by inserting 1 = s/s = t/t = u/u: = 1 s = 1 t = 1 u . (2.97) Which choice do we use depends on the color structure. In addition, the numerators ni are not uniquely defined. This is not surprising since we allow a deformation of ni by the gauge transformation ei → ei + αpi , which does not change the total amplitude An . The numerators actually have a larger class of transformations that keep the total amplitude invariant. For example, the color factors in Figure 2.3 satisfy the Jacobi identity: ci + c j + ck ∼ f˜ABe f˜eCD + f˜ADe f˜eBC + f˜ACe f˜eDB = 0 . (2.98) In the denominator, the only propagator that is different in those three diagrams is si , s j , and sk highlighted by the thick gray lines. It is easy to check that with an arbitrary function ∆ of kinematic variables, the following numerator transformation does not change the total amplitude: ni → ni + si ∆ nj → nj + sj ∆ nk → nk + sk ∆ , (2.99) due to the color Jacobi identity: δ A n ∼ ∆ ( ci + c j + c k ) = 0 . Eq. (2.99) is called the dual gauge transformation. A natural question to ask is whether we can find a color basis in which the component (partial) amplitudes are invariant under the dual gauge transformation. Such a basis and the associated partial amplitudes can be constructed by the following way. First, we can 7 In this context, it is more convenient to stick with the usual Feynman rules written in the Feynman gauge, where the color factor is given in terms of f˜abc . They can be found in, for example, [1]. Without causing misunderstandings, we will simply call ni numerator later. 35 cj ci ck sj si sk Figure 2.3. Three diagrams whose color factors satisfy the Jacobi identity. pick out two external legs, say, 1 and n, and then land all the other external legs onto the edge connecting 1 and n by the color Jacobi identity. For example: 2 3 2 4 3 3 4 =− 5 1 2 4 − 5 1 . 5 1 Then by this manipulation, we can transform each trivalent diagram into the following multiperipheral form: σ2 σn−1 σ3 ··· n , 1 (2.100) whose color factor is f˜a1 σ2 b2 f˜b2 σ3 b3 f˜b3 σ4 b4 . . . f˜bn−2 σn−1 n . Since any three multiperipheral diagrams cannot be related through a color Jacobi identity, we can use them as a basis, called Del Duca-Dixon-Maltoni (DDM) basis [52], to expand the total amplitude: A n = ( i ) n −2 ∑ f˜a1 σ2 b2 f˜b2 σ3 b3 f˜b3 σ4 b4 . . . f˜bn−2 σn−1 n An (1, σ , n) , (2.101) σ ∈ S n −2 where An (1, σ , n) are the usual color ordered amplitudes. This pursuit is in parallel with that in Section 2.2.1, where we define the color trace basis in which the color ordered amplitudes are invariant under the usual gauge transformation. The DDM basis demonstrates that there are (n − 2)! independent color factors under the Jacobi identity. This number agrees exactly with the (n − 2)! independent color ordered amplitudes after the particle 1 and n are fixed by cyclic and KK relations. 36 Now let us return to generic trivalent diagrams and discuss the properties of their numerators. We first make a rather bold claim and justify its legitimacy later: for each diagram, it is alway possible to find a numerator ni such that it satisfies the same algebraic identities as the corresponding color factors: ci = − c j ⇔ ni = − n j , ci + c j + c k = 0 ⇔ ni + n j + n k = 0 . (2.102) This relation is the color-kinematic (C-K) duality [29]. First, we need to verify that the condition (2.102) is gauge invariant. Since each numerator is multilinear in polarizations, µ we can write ni = e1 1 (ni )µ1 . Since ni is by definition an on-shell quantity, it does not µ contain any term X that makes e1 1 Xµ1 = 0. As a result, the component (ni )µ1 must also satisfy Eq. (2.102). Under a gauge transformation e1 → e1 + αp1 , the numerators change as ni → ni + ∆i , where ∆i = αp1 1 (ni )µ1 . Then it is easy to verify that: µ ni = − n j ⇒ ∆i = − ∆ j , ni + n j + n k = 0 ⇒ ∆i + ∆ j + ∆ k = 0 , namely, the C-K duality is gauge invariant. For a set of ni satisfying Eq. (2.102), we can perform Jacobi moves on the numerators and transform them into dual DDM basis, as what we have done to the color factors. According to Eq. (2.101), there are in all (n − 2)! C-K duality satisfying independent numerators n̂i under Jacobi identities. We know that (i ) there are also (n − 2)! independent color ordered amplitudes An under the cyclic and KK relations, such that there must be a linear relation in between: (i ) An ( n −2) ! = ∑ Θij n̂ j i = 1, 2, . . . , (n − 2)! . (2.103) j =1 Just from dimension analysis, we can tell that the entries of Θij are solely composed of massless propagators. If the matrix Θ is invertible, then we would obtain a unique C-K duality satisfying basis n̂i from any color ordered amplitudes, from which we can generate a unique set of C-K duality satisfying numerators by Jacobi identities. Actually, such numerators cannot be uniquely determined. It is always possible to add a set of higher dimensional operators to the Lagrangian in such a way that they add up to zero due to the Jacobi identity, but modify the numerators with the C-K duality preserved [53]. Therefore, if the C-K duality is true, there must exist one more kinematic linear relation between color 37 ordered amplitudes such that Θ is degenerate. This relation is nothing but the famous BCJ relation (2.52): n i i =3 j =3 ∑ ∑ p2 · p j ! An (1, . . . , i, 2, i + 1, . . . , n) = 0 . (2.104) More generally, the BCJ relation and the C-K duality are equivalent to each other: if the amplitudes of a gauge theory (possibly interacting with matter fields) satisfy the BCJ relation, it signals the existence of the C-K duality. Finally, we note that for Yang-Mills, the C-K duality satisfying numerators can indeed be constructed systematically at any n [54-58]. However, a closed and compact formula is still missing. Now we arrive at the punchline of this story: if we have two gauge theories (possibly interacting with matter fields) whose color ordered amplitudes satisfy the BCJ relation, then we can construct a gravity amplitude (possibly interacting with matter fields) through the double copy relation: Mn = ∑ i ∈ Γ3 ei ni n . Pi2 (2.105) ei come from the gauge amplitudes: The numerator ni and n (1) An = ∑ i ∈ Γ3 ci ni , Pi2 (2) An = ∑ i ∈ Γ3 ei ci n , Pi2 (2.106) in which one numerator, say ni , should explicitly satisfy the C-K duality.8 Again, it is instructive to prove that Mn constructed this way is gauge invariant. Proof. Under e → e + αp, the numerator changes as ni → ni + ∆i . Since the gauge amplitudes are invariant, we must have: ∑ i ∈ Γ3 ci ∆i =0 Pi2 ∑ i ∈ Γ3 ei ci ∆ = 0. Pi2 Throughout our discussion, we have never specified the gauge group, such that the above two equations hold merely due to the Jacobi identity of the color factor. Therefore, we can 8 The second numerator n ei does not need to satisfy the C-K duality explicitly, but it can be written in such a form due to the BCJ relation. 38 replace ci by any other quantities that satisfy the same Jacobi identity, in particular, the C-K duality satisfying numerators: ∑ i ∈ Γ3 ei ni ∆ =0 Pi2 ∑ i ∈ Γ3 ei ∆i ∆ = 0. Pi2 This immediately leads to: ∑ i ∈ Γ3 ei) e i + ∆i n ei ei + ∆ e + ni ∆ ei + ∆i ∆ e +∆ n e (ni + ∆i )(n nn nn = ∑ i i = ∑ i i 2 i i. 2 2 Pi Pi Pi i ∈ Γ3 i ∈ Γ3 (2.107) ei does not explicitly satisfy the C-K duality, there must exist another numerator Next, if n (2) ei n ei + Ωi that does so. Since both numerators give the same gauge amplitude, we = n must have the identity: ∑ i ∈ Γ3 ci Ωi =0 ⇒ Pi2 ∑ ∆i Ωi = 0, Pi2 ∑ ei − ∆i Ωi ∆i n ei ∆i n = ∑ = 0, 2 Pi Pi2 i ∈ Γ3 i ∈ Γ3 ∑ i ∈ Γ3 ci ∆i =0 ⇒ Pi2 ∑ i ∈ Γ3 (2) ei ∆i n = 0. Pi2 Therefore, we get: i ∈ Γ3 (2) (2.108) and then the gauge invariance of Mn follows immediately. Comparing Eq. (2.105) with Eq. (2.96), we see that to get gravity from gauge amplitudes, we only need to replace the color factors ci by the C-K duality satisfying ni . Then following the same steps leading to Eq. (2.101), we can write the gravity amplitude in the dual DDM basis [53]: Mn = ∑ σ ∈ S n −2 n1|σ2 σ3 ...σn−1 |n An (1, σ , n) , (2.109) where each numerator factor n1|σ2 σ3 ...σn−1 |n corresponds to one multiperipheral form in (2.100), and An is the gauge color ordered amplitude. The KLT relation (2.78) suggests a very natural arrangement: n1|α |n−1,n = − ∑ S[α |β ] An (1, β , n, n − 1) , β ∈ S n −3 and n1|σ |n = 0 if σn−1 6= n − 1. For example, at n = 4, we have: n1|23|4 = −s12 A4 (1243) n1|32|4 = 0 . At n = 5 the only two nonzero factors are: n1|234|5 = s12 s34 A5 (14352) n1|324|5 = s13 s24 A5 (14253) . (2.110) 39 Then using Jacobi identities, we can reproduce a set of C-K duality satisfying numerators {ni }. In this sense, the existence of C-K duality is equivalent to the KLT relation. Less obviously, we can even transform Eq. (2.105) into the dual trace basis form [59]: Mn = ∑ σ ∈ S n −1 τ(1σ2 σ3 ...σn ) An (1, σ1 , σ2 , . . . , σn ) , (2.111) where the dual trace τ(12...n) satisfies both the cyclic and KK relations, just as the usual color trace. The constructions of τ(12...n) can be found in [60-63]. To summarize, in this section we have shown that the scheme: gravity = (gauge)2 is valid if and only if the gauge amplitudes satisfy the BCJ relation, although there is no rigorous proof yet in the mathematical sense. Furthermore, we have shown the equivalence: BCJ relation ⇔ C-K duality ⇔ double copy relation ⇔ KLT relation (2.112) constructively at the tree level. However, the KLT relation is only valid at tree level, while the C-K duality and double copy relation can also be used to construct gravity loop integrands from the gauge theories ones [30]. 2.4.2 Double copy examples After a rather formal discussion, we provide several explicit 4-point calculations to manifest the relation (2.112). First, let us consider the total 4-point Yang-Mills amplitude, written in the color trace basis as: A4 = Tr( T a1 T a2 T a3 T a4 ) A4 (1234) + (234 permutations) (2.113) Using the KK relation, we can express all the partial amplitudes in terms of A4 (1234) and A4 (1324), and transform A4 into the DDM basis: A4 = − f˜a1 a2 b f˜ba3 a4 A4 (1234) − f˜a1 a3 b f˜ba2 a4 A4 (1324) , where we have used the identities: (2.114) 40 cs ≡ f˜a1 a2 b f˜ba3 a4 = − Tr( T a1 T a2 T a3 T a4 ) + Tr( T a1 T a2 T a4 T a3 ) + Tr( T a1 T a3 T a4 T a2 ) − Tr( T a1 T a4 T a3 T a2 ) cu ≡ f˜a1 a3 b f˜ba4 a2 = Tr( T a1 T a3 T a2 T a4 ) − Tr( T a1 T a2 T a4 T a3 ) − Tr( T a1 T a3 T a4 T a2 ) + Tr( T a1 T a4 T a2 T a3 ) ct ≡ f˜a1 a4 b f˜ba2 a3 = Tr( T a1 T a4 T a3 T a2 ) − Tr( T a1 T a4 T a2 T a3 ) − Tr( T a1 T a3 T a2 T a4 ) + Tr( T a1 T a2 T a3 T a4 ) . (2.115) We can verify that indeed cs + ct + cu = 0. In the representation of (2.96), we have A4 = + + = cs ns ct nt cu nu + + . s t u (2.116) Then using Eq. (2.115), we can express A4 (1234) and A4 (1324) in terms of the numerators: A4 (1234) = − ns nt + s t A4 (1324) = − nt nu + . t u (2.117) Now if we impose the C-K duality ns + nt + nu = 0 and choose the dual DDM basis as (n̂1 , n̂2 ) ≡ (ns , nt ), we get Θ= − 1s − u1 ! 1 t − u1 − 1 t , (2.118) according to Eq. (2.103). It is easy to verify that indeed det(Θ) = 0, which indicates a linear relation within the two color ordered amplitudes: sA4 (1234) = uA4 (1324) = −ns + snt . t (2.119) Following the double copy relation (2.105), the 4-point pure gravity amplitude should be: M4 (1234) = n2s n2 n2 n2 n2 ( n s + n t )2 + t + u = s + t + . s t u s t u (2.120) In this equation, if we replace ns in terms of A4 (1234) and nt , according to Eq. (2.119), we find that in M4 , the nt dependence is cancelled. What left is exactly: M4 (1234) = − st [ A4 (1234)]2 = −tA4 (1234) A4 (1324) , u (2.121) which is nothing but the KLT relation. Therefore, whether the double copy and KLT relation work hinges on the BCJ relation (2.119). At 4-point, the only nonzero amplitude is MHV, so it suffices to check it for the Parke-Taylor factor: s −[12]h24i −[13]h13i u = = = . h12ih23ih34ih41i h23ih34ih24ih41i h13ih32ih24ih41i h13ih32ih24ih41i 41 Indeed, the Yang-Mills amplitudes satisfy the BCJ relation, such that their double copy correctly reproduces a pure gravity amplitude. As a second example, we consider the pure scalar 4-point amplitudes of the Lagrangian (2.85). The total 4-point amplitude has both the single trace and double trace contribution: ed-t Ae4 = Aes-t 4 + A4 . (2.122) ed-t The single trace amplitude Aes-t 4 and the double trace amplitude A4 have exactly the same color structure as the Yang-Mills ones: ets-t eus-t ess-t c0s n c0u n c0s n + + Aes-t = 4 s t u etd-t eud-t esd-t c0s n c0u n c0s n Aed-t + + , = 4 s t u (2.123) where c0s , c0t , and c0u are the same as those in Eq. (2.115), but with color index a0 . Consequently, we have the color ordered amplitudes: s-t s-t e es n es-t (1234) = − n A + t 4 s t d-t d-t e e n n ed-t (1234) = − s + t A 4 s t s-t s-t et e n es-t (1324) = − n A + u 4 t u d-t d-t e e n n ed-t (1324) = − t + u . A 4 t u (2.124) Then we can impose the C-K duality on the numerators such that using the double copy relation (2.105) with Yang-Mills, we can derive the single and double trace pure gluon 4-point EYM amplitudes in the KLT form: es-t As-t 4 = − tA4 (1234) A4 (1324) ed-t Ad-t 4 = − tA4 (1234) A4 (1324) . (2.125) Next, we verify whether the pure scalar amplitudes satisfy the BCJ relation. Otherwise the double copy would not give a meaningful gravity theory. The color ordered amplitude of the single trace part has already been calculated in Eq. (2.87): ˜a1 a2 b f˜ba3 a4 f˜a1 a4 b f˜ba2 a3 es-t (1234) = − f A + 4 s t ˜f a1 a3 b f˜ba4 a2 ˜f a1 a4 b f˜ba2 a3 es-t (1324) = A − . 4 u t We can immediately see that they satisfy the BCJ relation: es-t (1234) = u A es-t (1324) . sA 4 4 (2.126) 42 Actually, we have already checked in Eq. (2.88) that the KLT product correctly reproduces the total 4-point Yang-Mills amplitude as expected: As-t 4 = A4 . The color ordered amplitude of the double trace part can be calculated from the Feynman rules in Figure 2.2: 2 ed-t (1234) A 4 2 4 4 + = 3 1 = −δ a1 a3 a2 a4 δ −δ 3 1 4 + 3 1 a1 a2 a3 a4 δ 2 u s −δ a1 a4 a2 a3 δ u t . (2.127) By exchanging the leg 2 and 3, we get: ed-t (1324) = −δ a1 a2 δ a3 a4 − δ a1 a3 δ a2 a4 A 4 s u − δ a1 a4 δ a2 a3 s t . (2.128) Again, it is easy to see that the BCJ relation is satisfied: ed-t (1234) = u A ed-t (1324) . sA 4 4 We note that if we forget the 4-point contact term in Figure 2.2, this BCJ relation will not hold.9 The consequence is that all the multi-trace EYM amplitudes constructed from KLT would be wrong. The KLT product (2.125) gives: a1 a2 a3 a4 Ad-t δ [tA4 (1234)] + δ a1 a3 δ a2 a4 [tA4 (1324)] + δ a1 a4 δ a2 a3 [sA4 (1234)] . 4 = δ (2.129) The 4-point partial amplitude with the double trace structure ( a1 a2 )( a3 a4 ) has the uniform expression: − − Ad-t 4 [( a1 a2 )( a3 a4 )|i j ] = − s a1 a2 hiji4 , h a1 a2 ih a2 a1 ih a3 a4 ih a4 a3 i (2.130) where the gluon i and j carry negative helicity. The generic n-point double trace pure gluon EYM amplitude is first written down in [36] using the Cachazo-He-Yuan (CHY) formalism, which is the main topic of Chapter 3. 9 This happens in [49]. Using the Feynman rules therein, we can only derive the correct single trace EYM amplitudes. 43 2.5 Witten-RSV formalism According to the work of Witten [17] and Roiban, Spardlin, Volovich [18], the superamplitude of the four dimensional N = 4 super Yang-Mills can be written as: An = n −3 Z ∑ d =1 h i n dMn,d ∏ δ2 (λi )α − ti λα (σi ) i =1 " d × n ∏ δ2 ∑ ti σim eλα m =0 # δ4 i =1 n ∑ ti σim ηiA ! . (2.131) i =1 The symbols in this formula have the following definitions: • The two-spinor λi and e λi are given by the null four momentum k i through: (k i )αα̇ = (λi )α (e λi )α̇ . (2.132) The fermionic part of the momentum is given by ηiA , with A = 1, 2, 3, 4. • The first delta function confines λi projectively to a degree d curve: d λα (z) = ∑ (ρm )α zm . (2.133) m =0 • The degree d is connected to the Nk MHV sector through d = n − k − 3. • The integral measure dMn,d is defined as: dMn,d = (d2d+2 ρ) dn σdn t Vol[ GL(2, C)] n 1 ∏ ti (σi − σi+1 ) ( n + 1 ≡ 1) . (2.134) i =1 In the last equation, We need to divide the volume of GL(2, C) since the integrand is invariant under this world sheet symmetry. The first two delta functions in Eq. (2.131) give the support of the superamplitude A n . The third fermionic delta function gives an expansion with respect to ηi , from which we can obtain all the component amplitudes, for example, pure gluon amplitudes An ( gg . . . g). The support of the Witten-RSV delta function is: d ( λi )α − ti ∑ (ρm )α σim (i = 1, 2 . . . n) m =0 n ∑ ti σim (eλi )α̇ = 0 (m = 0, 1 . . . d) . (2.135) i =1 In all, there are 2n + 2d + 2 equations. However, four of them are rather constraints interpretted as the momentum conservation: n ∑ (λi )α (eλi )α̇ = i =1 d n m =0 i =1 ∑ (ρm )α ∑ ti σim (eλi )α̇ = 0 . (2.136) 44 Therefore, there are only 2n + 2d − 2 independent equations.10 The unknown variables involved in Eq. (2.135) are {σi , ti , (ρm )α }, whose number is 2n + 2d + 2. Using the world sheet GL(2, C) freedom, we can fix three σ's and one t, such that the total number of unknown variables is also 2n + 2d − 2. Consequently, the solutions of Eq. (2.135) consist of only discrete points. Interestingly, in the Witten-RSV integrand, only the Parke-Taylor factor, defined as: PT (I ) = 1 σ12 σ23 . . . σn1 (σij ≡ σi − σj ) , (2.137) where I stands for the identity permutation of {1, 2, . . . , n}, depends on the particle permutations. Thus, it should manifestly satisfy the KK and BCJ relation. Indeed, according to the calculation in Appendix B, the KK relation is satisfied by PT trivially as a complex number identity. However, PT satisfies the BCJ relation if and only if {σ} solves the equation: n s ab =0 σ b=1 ab ∑ a = 1, 2 . . . n . (2.138) b6= a This is exactly the scattering equation later called by Cachazo, He, and Yuan [33-37]. In the following, we prove that the {σ} solved from Eq. (2.135) must satisfy the scattering equation. To start with, we propose a useful identity [18]:  m 6 n−2  0 n σim 1 m = n−1 . pm = ∑ = (σ − σj )  m − n +1 i =1 ∏ j 6 = i i O(σi ) m>n (2.139) Namely, pm is a polynomial with leading terms: n ∑ σim−n+1 i =1 when m > n, and it is constant otherwise. This identity follows directly from the global Cauchy theorem of multivariate complex analysis. Next, we define a degree de = n − d − 2 polynomial: de e λα̇ (z) = ∑ (ρel )α̇ zl , l =0 and another projection eti such that eti ti = ∏ j6=i σij . 10 The conservation of supermomentum ∑i (λi )α ηiA can be similarly derived. (2.140) 45 In hindsight, we come to evaluate the following integral: I= h i n e d2d+2 ρe ∏ δ2 (e λi )α̇ − eti e λα̇ (σi ) , Z (2.141) i =1 which will eventually get related to the Witten-RSV integrand. We first perform a linear transformation to the variables of the delta function according to the n × n matrix Mmi = ti σim (m = 0, 1 . . . n − 1 and i = 1, 2 . . . n) . (2.142) The result is: I= Z d 2de+2 " ρe [det( M )]2 #2 Z n = V ∏ ti d 2de+2 " n −1 n n i =1 i =1 ∏ δ2 ∑ Mmi (eλi )α̇ − ∑ Mmieti eλα̇ (σi ) m =0 n −1 ρe ∏ δ2 " m =1 i =1 n de i =1 l =0 ∑ ti σim (eλi )α̇ − ∑ (ρel )α̇ pl+m # # , (2.143) where V is the Vandermonde determinant: ∏ V= (σi − σj ) . 16i < j6n e the second term is identically Inside the delta functions, since the largest value of l is d, zero when 0 6 m 6 n − de− 2 = d, such that " #2 n I = V ∏ ti × d 2de+2 n m =0 i =1 ∏ δ2 ∑ ti σim (eλi )α̇ i =1 Z d " n −1 ρe ∏ δ 2 n ∑ ! ti σim (e λi )α̇ i =1 m = d +1 # de − ∑ (ρel )α̇ pl +m . (2.144) l =0 The integral in the second line is trivial: Z d 2de+2 " n −1 ρe ∏ δ n 2 ∑ ti σim (e λi )α̇ i =1 m = d +1 de # − ∑ (ρel )α̇ pl +m = 1 , l =0 since we have det( pl +m ) = 1. The reason is that the d × d matrix pl +m is upper triangular, with all diagonal elements being one. This derivation establishes the relation: d ∏ m =0 δ2 n ∑ ti σim (eλi )α̇ i =1 ! " n = V ∏ ti # −2 Z i =1 h i n e λi )α̇ − eti e λα̇ (σi ) , d2d+2 ρe ∏ δ2 (e (2.145) i =1 such that the second equation of Eq. (2.135) is equivalent to (e λi )α̇ − eti e λα̇ (σi ) = 0 (i = 1, 2 . . . n) . (2.146) 46 After multiplying it with the first equation of Eq. (2.135), we get: λα (σi )e λα̇ (σi ) ( λi )α ( e λi )α̇ = ∏ j6=i (σi − σj ) i = 1, 2 . . . n , (2.147) which is nothing but the four dimensional scattering equation [32]. The relation with the canonical form (2.138) is derived in Appendix C. However, Eq. (2.138) is more general than this equation since Eq. (2.138) is valid in arbitrary dimensions. CHAPTER 3 CACHAZO-HE-YUAN FORMALISM Starting from 2013, Cachazo, He, and Yuan published a series of papers [33-37], proposing a unified recipe for calculating n-point tree level massless field theory amplitude in arbitrary spacetime dimensions:1 Z dz1 . . . dzn Vol [SL(2, C)] " n ∏ 0 δ ( f a ) In # . (3.1) a =1 It is now called the CHY formalism. The meaning of each building block in this equation is listed below: Symbol Meaning Note In the CHY integrand specifies the theory see Section 3.3 fa the scattering equation see Eq. (3.5) ∏ 0δ ( fa) the permutation invariant delta function see Eq. (3.20) dz1 . . . dzn Vol [SL(2, C)] integration measure see Eq. (3.24) n a =1 They will be discussed in detail later in this chapter. Currently, the CHY formalism exists for the following theories: Yang-Mills, pure gravity, Einstein-Yang-Mills, Yang-Mills-scalar, φ3 , φ4 , Dirac-Born-Infeld, nonlinear sigma model, and special Galileon [37], F3 and R3 theories [64]. All of them are formulated in a way that is valid in arbitrary spacetime dimensions. The scattering equations f a = 0 depend only on the external kinematic data, whose solutions provide the support for all physical amplitudes. Then the integrand In specifies the dynamics of different theories. Moreover, the requirement of world sheet SL(2, C) invariance of the form: 1 The integration should be understood as a contour integral in the n-punctured moduli space of the Riemann sphere. 48 n (dz1 ∧ dz2 ∧ . . . ∧ dzn ) ∏ 0 δ ( f a ) In (3.2) a =1 significantly constrains the possible forms of the integrand. In this sense, building amplitudes is like the game of mix-and-match guided by a symmetry principle. Many unexpected elegant structures have been discovered in this way in a large class of theories. The CHY formalism is thus a firm step towards conquering the territory beyond four dimensional Yang-Mills and gravity, and their supersymmetrization. 3.1 The scattering equations Long before the scattering equation was rediscovered and endowed new physical importance by Cachazo, He, and Yuan, it has already made the appearance in the early days of string theory [65] (for a review, see [66]). The original motivation is to find a null map Pµ (z) from the world sheet to the lightcone, such that the momenta of physical massless particles are associated to the punctures on the world sheet. For n scattering particles with momentum conservation k2a = 0 . k1 + k2 + . . . + k n = 0 , µ we require that each k a is the residue of Pµ (z) at a puncture σa :2 µ ka = I |z−σa |=e dz µ P (z) . 2πi The map Pµ (z) can thus be written as: n µ ka P (z) = ∑ . z − σa a =1 µ (3.3) Furthermore, the null condition we have imposed on Pµ (z) requires that: P2 ( z ) = n n ka · k ∑ ∑ (z − σa )(zb− σb ) = 0 . (3.4) a =1 b =1 As a holomorphic function in z, it is equivalent to imposing the zero residue at each puncture σa , which leads to the scattering equation (SE): n fa ≡ s ab = 0, σ − σb b =1 a ∑ a ∈ {1, 2, . . . , n} . (3.5) b6= a 2 For tree level scattering, the world sheet is a Riemann sphere, or CP1 . At loop level, the world sheet would be some higher genus Riemann surface, for example, torus for one loop. Consequently, the scattering equation will take another form, which will not be discussed in this dissertation. 49 In 1987, Gross and Mende [67] found that high energy string amplitudes are dominated by the saddle point of the Koba-Nielsen exponent, determined just by the SE (3.5). The high-energy limit can be effectively taken by setting α0 → ∞, while the usual field theory emerges at low energy with α0 → 0. To answer the question why the SE, as an object derived in the opposite limit, should have anything to do with the field theory amplitudes, people have developed new types of string theories that possibly have the CHY formula as the field theory limit, like the ambitwistor string [68-70] and the chiral string [71, 72]. 3.1.1 Properties of the scattering equation In this subsection, we investigate some crucial properties of the solutions to the SE. First, the SE is invariant under the world sheet SL(2, C) transformation in the sense that given a solution {σa }, the following set {ζ a } aσa + b a, b, c, d ∈ C, ad − bc = 1 ζa = cσa + d is also a solution. Indeed, it is easy to verify that: (cσb + d)s ab s ab s ab = (cσa + d) ∑ = (cσa + d)2 ∑ = 0. ζ − ζ σ − σ σ − σb a b b b6= a a b6= a b6= a a ∑ To obtain the second equality, we have used the fact that: cσb + d = c(σb − σa ) + (cσa + d) , ∑ sab = −k2a = 0 . b6= a The consequence of this SL(2, C) invariance is that we can "fix the gauge" by specifying the positions of any three punctures. For example, a very convenient choice is σn−2 = 0 σn−1 = 1 σn = ∞ . Now that the SE (3.5) consists of n polynomial equations, it seems that the SE were overdetermined and should not have any solution. Actually, there are three linear relations between the set { f a }: n P( f ) ≡ ∑ a =1 n fa = 0 Q( f ) ≡ ∑ σa f a = 0 a =1 n R( f ) ≡ ∑ σa2 f a = 0 , (3.6) a =1 such that there are only n − 3 independent equations in Eq. (3.5), agreeing with the number of variables. 50 proof of Eq. (3.6). The first relation in Eq. (3.6): n ∑ n fa = a =1 s ab =0 σ − σb a =1 b 6 = a a ∑∑ comes from the fact that s ab /(σa − σb ) is antisymmetric while the summation ∑na=1 ∑b6=a is symmetric. The second relation Q( f ) = 0 can be proved as: n ∑ n σa f a = a =1 σa s ∑ ∑ σa −abσb = a =1 b 6 = a n n σb s ab = − ∑ ∑ σ − σb ∑ ∑ σa f a = 0 . a =1 b 6 = a a a =1 b 6 = a To obtain the second equality, we need to use the identity σa = (σa − σb ) + σb and the momentum conservation. Then we exchange the dummy index a and b to reach the third equality. For the third relation R( f ) = 0, we have n n σa2 s ab = σ − σb a =1 b 6 = a a ∑ σa2 f a = ∑ ∑ a =1 n n −σb2 s ab (σ2 − σb2 )s ab (σa + σb )s ab = ∑ ∑ a = ∑ ∑ . σ − σ 2 ( σ − σ ) 2 a b b a =1 b 6 = a a a =1 a 6 = b a =1 b 6 = a n ∑∑ Again we have used the renaming trick to derive the second equality. This result is identically zero since 1 n ∑ (σa + σb )sab = 2 a∑ =1 b 6 = a n ∑ σa ∑ sab = 0 , a =1 b6= a where we have renamed b to a in the second term. The SE has a very simple and compact form, as one can tell from Eq. (3.5), but it is very difficult to solve. Actually, there are in all (n − 3)! solutions. The first derivation is given in [33], using a induction at the soft limit. To start the induction, we first solve the SE at n = 4. It is easy to show that all the four equations gives the same solution: σ1 = − s12 s23 if we fix the gauge as {σ2 , σ3 , σ4 } = {0, 1, ∞}. Thus at n = 4 there is only one solution. Now suppose we have n − 1 particles satisfying the momentum conservation k1 + k2 + . . . + k n−1 = 0, we can write down and solve the SE with n − 1 particles: n −1 fa = s ab = 0, σ − σb b =1b 6 = a a ∑ a ∈ {1, 2, . . . , n − 1} . (3.7) We assume that this set of equations has (n − 4)! solutions, as the inductive assumption. Now the punchline is that we only add the particle n softly as: snb = eŝnb e → 0, 51 such that the momentum conservation of the previous n − 1 particles is preserved at the zeroth order of e. Consequently, the previous (n − 4)! solutions of {σ1 , . . . , σn−1 } are also unchanged. With this new particle, we get one more equation on σn to solve: n −1 fn = e ∑ b =1 n −1 −1 e ∑nb= ŝnb 1 ŝnb ∏k 6=b ( σn − σk ) = = 0. σn − σb ∏nj=−11 (σn − σj ) (3.8) −1 Due to the momentum conservation ∑nb= 1 ŝnb = 0, the numerator is a degree n − 3 polyno- mial in σn : n −1 n −1 b =1 k6=b ∑ ŝnb ∏ (σn − σk ) = σnn−2 n −1 ∑ ŝnb ! + σnn−3 n −1 ∑ ∑ ŝnb σk ! + . . . = σnn−3 (· · · ) + . . . . b =1 k 6 = b b =1 Therefore, for each solution {σ1 , . . . , σn−1 }, there are n − 3 solutions of σn from Eq. (3.8), such that the total number of solutions to the n-particle SE is: (n − 3)(n − 4)! = (n − 3)! . (3.9) We note that this result derived at the soft limit is generally true since the number of solutions to a system of polynomial equations must be an integer, such that it does not change when we continuous move the kinematics away from the soft limit. Of course, when we hit the multilinear boundaries of the kinematics space, the solutions will become degenerate and factorize into two sectors. Finally, we note that in four dimensions, the (n − 3)! solutions fall into (n − 3) sectors labeled by d = 1, 2 . . . n − 3. The number of solutions in sector d is the Eulerian number A(n − 3, d − 1). The derivation will be given in Appendix C. More interestingly, only those solutions in the sector d = k + 1 give nonzero contribution to the Nk MHV Yang-Mills and gravity amplitudes. This feature will be further studied in Chapter 6. 3.1.2 Polynomial form The SE given in the form of Eq. (3.5) has all the variables in the denominator, which is not convenient for further mathematical manipulation. Following Dolan and Goddard [73], we can write the SE into a polynomial form: h̃m ≡ ∑ k2S σS = 0 2 6 m 6 n −2, |S|=m, S⊂N where the summation is over order m subsets of N = {1, 2, . . . , n}, and kS ≡ ∑ ki i ∈S σS ≡ ∏ σi . i ∈S (3.10) 52 Each h̃m is a degree m polynomial in σ's. More interestingly, each monomial of h̃m has degree m and it is multilinear in σ's. To derive Eq. (3.10), we first remind that the SE is equivalent to P2 (z) = 0, such that it is also equivalent to the vanishing of the following polynomial: n F (z) ≡ P2 (z) ∏(z − σi ) = i =1 n n n ∑ ∑ (k a · kb ) ∏ (z − σi ) = 0 . a =1 b =1 (3.11) i 6= a,b From a simple power counting, we find that F (z) is at most degree n − 2 in z. However, we can show that the coefficients of zn−2 and zn−3 vanish identically due to the momentum conservation, and F (z) can be transformed into the form: n −2 F (z) = ∑ (−1)m zn−m−2 h̃m . (3.12) m =2 Thus F (z) = 0 imposes n − 3 equations in Eq. (3.10), which should be equivalent to the SE. Now we can partially fixed the gauge by choosing z1 = ∞ and zn = 0 such that Eq. (3.10) becomes: hm ≡ h̃m+1 = ∑ 0 (k1 + kS )2 σS = 0 z1 →∞, zn →0 z1 |S|=m, S⊂N lim 1 6 m 6 n −3, (3.13) where N0 = {2, 3, . . . , n − 1}. Then Eq. (3.13) is a set of polynomial equations defined on the projective space CPn−3 with homogeneous coordinates (σ2 , σ3 , . . . , σn−1 ). Each hm is a special degree m homogeneous polynomial in that each monomial of hm is degree m and multilinear in the σa 's. Following the Bézout theorem, the number of solutions is n −3 ∏ deg(hm ) = (n − 3)! , (3.14) m =1 which agrees with the result (3.9) derived from soft limit recursion. The polynomial form (3.10) is very useful to numerically evaluate the solutions to SE. Besides, it makes manifest several interesting algebraic properties of the SE, for example, {h̃m } being a representation of the SL(2, C) algebra [73]. More details in direction can be found in [74]. Moreover, Eq. (3.10) forms a basis that possibly can be used to reduce the CHY integrand. Some recent development can be found in [75, 76]. 3.2 The integrated CHY formula Having studied some general properties of the SE, our next job is to localize the integrand In on to the solutions. This can be formally done by a series of delta functions 53 ∏ δ( f a ). However, since three f a 's are redundant according to Section 3.1.1, we need to delete them from the chain. This has to be done in an elegant way: we need to define a permutation invariant delta function: n ∏ 0 δ( f a ) (3.15) a =1 such that this quantity is independent of which three a's we choose to delete. The reason is that the solutions to the SE certainly do not depend on such a choice, so that we do not want anything in our formalism to depend on it. To derive the form of Eq. (3.15), we start from inspection the two chain of delta functions: ∏ δ( f a ) a6=i,j,k ∏ δ( f a ) (i < j < k, α < β < γ) , a6=α,β,γ and {i, j, k } 6= {α, β, γ} as sets. They actually only differ by a change of variables: ∂( f , f , . . . , fˆ , fˆ , fˆ , . . . , f ) α γ n 2 1 β ∏ δ( f a ) = ∂( f , f , . . . , fˆ , fˆ , fˆ , . . . , f ) ∏ δ( f a ) . n 1 2 i j k a6=α,β,γ a6=i,j,k (3.16) In the Jacobian, we permute { f i , f j , f k } in the numerator and { f α , f β , f γ } in the denominator to the front, which leads to: ∂( f , f , . . . , fˆ , fˆ , fˆ , . . . , f ) α β γ n 1 2 i + j+k+α+ β+γ ∂ ( f i , f j , f k ) = −(− 1 ) . ∂( f 1 , f 2 , . . . , fˆi , fˆj , fˆk , . . . , f n ) ∂( f α , f β , f γ ) (3.17) There is a sign dependence since each permutation of two f 's leads to a minus sign. This sign dependence holds if the orders of {i, j, k } and {α, β, γ} are preserved. Now using the linear relations (3.6) and the implicit function theorem, we get: −1 ∂( f i , f j , f k ) = − ∂( P, Q, R) ∂( P, Q, R) ∂( f α , f , f γ ) ∂( f , f , f ) ∂( f α , f , f γ ) β β i j k 1 1 1 1 1 1 −1 zαβ z βγ zγα = − zα z β zγ zi z j zk = − , zij z jk zki z2α z2 z2γ z2 z2 z2 i j β k (3.18) such that Eq. (3.16) reaches the form (−1)i+ j+k zij z jk zki ∏ δ( f a ) = (−1)α+ β+γ zαβ z βγ zγα ∏ a6=α,βγ a6=i,j,k Here we have used the abbreviation: zij ≡ zi − z j , δ( f a ) . (3.19) 54 which will be used throughout the following chapters. This result legitimates our definition for the permutation invariant delta function: n ∏ 0 n δ ( f a ) ≡ (−1)i+ j+k zij z jk zki ∏ δ ( fa) , (3.20) a =1 a6=i,j,k a =1 which must be independent of the choice of {i, j, k }. Before we can explicitly perform the integral, we have to divide out the volume of the world sheet gauge group SL(2, C). This is taken care by a standard Fadeev-Popov trick. We first pick out three arbitrary indices { p, q, r } with p < q < r, and rewrite Eq. (3.2) as: # ! " n ^ dz p ∧ dzq ∧ dzr (3.21) dzc (−1) p+q+r z pq zqr zrp ∏ 0 δ ( f a ) In . z pq zqr zrp a =1 c6= p,q,r We can easily verify that the measure (dz p dzq dzr )/(z pq zqr zrp ) is SL(2, C) invariant, such that the combination in the bracket should also be invariant under the SL(2, C) transformations on z p , zq and zr . Therefore, we can use this degree of freedom to fix them at three arbitrary points: {z p , zq , zr } = {σp , σq , σr }, namely: " # " n n a =1 a =1 z pq zqr zrp ∏ 0 δ ( f a ) In = σpq σqr σrp ∏ 0 δ ( f a ) In # . (3.22) {z p ,zq ,zr }={σp ,σq ,σr } Here {σp , σq , σr } can be chosen as the three gauge fixed punctures in the solutions of the SE. Subsequently, the integrand is independent of z p , zq , and zr , so that they can be trivially integrated and cancel the volume of SL(2, C): Z dz p dzq dzr = Vol [SL(2, C)] . z pq zqr zrp (3.23) In this sense, we can effectively define the measure as: dz1 . . . dzn = (−1) p+q+r σpq σqr σrp Vol [SL(2, C)] n ∏ dzc . (3.24) c =1 c6= p,q,r Now collecting Eq. (3.20), Eq. (3.22), and Eq. (3.24), we can perform the integration in Eq. (3.1) to explicitly localize the amplitude on the solutions of the SE: ! Z n dz1 . . . dzn 0 An = δ ( f a ) In Vol [SL(2, C)] a∏ =1 = (−1)i+ j+k+ p+q+r (σij σjk σki )(σpq σqr σrp ) In (σ ) , ijk det Φ pqr {σ}∈sol. ∑ (3.25) 55 where the n × n symmetric matrix Φ is defined as:  s ab    σ2 ab = s ac   −∑ 2  σ c6= a ac Φ ab a 6= b a=b . (3.26) ijk By Φ pqr , we mean the (n − 3) × (n − 3) submatrix of Φ with row {i, j, k } and column { p, q, r } deleted. It is useful to define the reduced determinant of Φ as: ijk (−1)i+ j+k+ p+q+r det Φ pqr det (Φ) ≡ , (σij σjk σki )(σpq σqr σrp ) 0 (3.27) since this quantity is independent of which three rows and columns are deleted. This property is expected from how we perform the world sheet integration to obtain Eq. (3.25). Nevertheless, we can directly prove it using the SE. This is done in Appendix A, together with proving det0 (Φ) being permutation invariant. Finally, the integrated form of the CHY formula can be simply written as: In (σ ) , 0 {σ}∈sol. det ( Φ ) ∑ An = (3.28) which is the main result of this subsection. 3.3 The CHY integrands In the long list of theories that have a CHY formalism, we are goint to study in detail the amplitudes of Yang-Mills, pure gravity and EYM in this work. Now we present the CHY integrands for these theories. 3.3.1 Yang-Mills According to [34, 35], the integrand for Yang-Mills is: InYM (k, e, σ) Tr( T a1 T a2 . . . T an ) = + ( noncyclic perm. ) Pf 0 (Ψ) , σ12 σ23 . . . σn1 (3.29) where Ψ is a 2n × 2n matrix composed of four n × n matrices: Ψ= A −C T C B . (3.30) 56 These matrices have the following forms: A ab ( s ab σab = 0 a 6= b a=b Bab ( e a · eb σab = 0 a 6= b a=b Cab  √  2 ea · k b    − √σab = 2 ea · k c     ∑ σac c6= a a 6= b . a=b (3.31) Finally, the reduced Pfaffian of Ψ is defined as: Pf 0 (Ψ) ≡ (−1)i+ j ij Pf Ψij σij 1 6 i < j 6 n, (3.32) ij where Ψij is the submatrix of Ψ with the row {i, j} and column {i, j} deleted. The reduced Pfaffian is permutation invariant and independent of the choice of {i, j}, and this is the reason why it stays outside the sum over permutations in Eq. (3.29). The proof is given in Appendix A. However, if we do not delete any rows and columns, we have Pf (Ψ) = 0 since the upper half ( A, −C T ) has two null vectors (1, . . . , 1) and (σ1 , . . . , σn ). In particular, the color ordered Yang-Mills amplitude in arbitrary dimensions has the CHY representation: 1 Pf 0 (Ψ) An (12 . . . n) = ∑ . 0 σ12 σ23 . . . σn1 {σ}∈sol. det ( Φ ) (3.33) The gauge invariance of this formula is very easy to verify: if we replace one ea by k a in ij Pf 0 (Ψ), then there must be two rows and columns identical in Ψij such that the reduced Pfaffian must vanish. The correctness of this formula has first been proved in [77] using the BCFW recursive relation [44]. 3.3.2 Pure gravity With the color ordered Yang-Mills amplitudes in hand, it is straightforward to write down the CHY formula for the pure gravity: just perform the KLT product. Moreover, since only the Parke-Taylor factors depend on permutations, the summation over permutations in the KLT product only acts them: Mn = Pf 0 [Ψ(σ(i) )]Pf 0 [Ψ(σ( j) )] det0 [Φ(σ(i) )] det0 [Φ(σ( j) )] { σ (i ) } { σ ( j ) } " ∑ ∑ × − ∑ β ∈ S n −3 α ,β (i ) # ( j) PT (1, α , n, n − 1)S[α |β ] PT (1, β , n − 1, n) , (3.34) 57 where the two Parke-Taylor factors are: PT (1, α , n, n − 1) = σ1α2 σα2 α3 . . . σαn−2 n σn,n−1 σn−1,1 σ=σ(i) 1 ( j) β PT (1, , n − 1, n) = . σ σ ...σ σ σ 1 (i ) 1β 2 β 2 β 3 β n−2 ,n−1 n−1,n n1 σ=σ( j) However, Eq. (3.34) does not seems to be compatible with our scheme (3.28) since it involves a double summation over the solutions. In a seminal paper that established the beauty of this formalism [33], CHY proved the KLT orthogonality of the solutions: − ∑ PT (i) (1, α , n, n − 1)S[α |β ] PT ( j) (1, β , n − 1, n) = det0 [Φ(σ(i) )]δij , (3.35) α ,β β ∈ S n −3 namely, if we view each Parke-Taylor factor as a vector in the (n − 3)! dimensional solution space, they are orthogonal with respect to the KLT kernel. As a result, the CHY formula for pure gravity significantly simplifies to: Mn = Pf 0 (Ψ) × Pf 0 (Ψ) . det0 (Φ) {σ}∈sol. ∑ (3.36) This resembles the double copy relation (2.105): we just need to trade one "color factor" Tr( T a1 T a2 . . . T an ) σ12 σ23 . . . σn1 by another "kinematic numerator" Pf 0 (Ψ) to obtain a gravity amplitude. Recent studies on the C-K duality in the context of CHY can be found in [35, 78, 79]. 3.3.3 Einstein-Yang-Mills The CHY formalism for general multitrace EYM amplitudes can be found in [37]. In this work, we only focus on the n-point single trace EYM amplitudes with r gluons and s = n − r gravitons. This prescription is first given in [36]. In our notation, we use H and G to denote the set of gravitons and gluons: G ≡ {1, 2, . . . , r } H ≡ {r + 1, r + 2, . . . , r + s} . (3.37) The color ordered single trace EYM amplitude can then be written as: Mn,r (12 . . . r |H) = 1 Pf (ΨH ) Pf 0 (Ψ) 0 σ12 σ23 . . . σr1 {σ}∈sol. det ( Φ ) ∑ (2 6 r 6 n ) . (3.38) 58 µ Each gluon carries a polarization vector ei , while each graviton carries a polarization µ tensor ei e eiν . Thus we have two sets of polarizations involved in EYM amplitudes: { e } ≡ { e1 , . . . , e n } {e e} ≡ {e er + 1 . . . , e er + s } . According to Eq. (3.38), the n polarizations in the set {e} are encoded in the matrix Ψ, defined as in Eq. (3.30), while the other s polarizations in {e e} are contained in the 2s × 2s matrix ΨH , defined as: ΨH = AH −CHT CH BH , (3.39) where each component is just the submatrix of the corresponding one in Ψ whose indices are in the graviton set H: AH ≡ A12...r 12...r 12...r BH ≡ B12...r 12...r CH ≡ C12...r . (3.40) Gluon polarizations do not appear in ΨH , while gluon momenta appear only in the diagonal elements of CH : n (CH ) aa = ∑ b =1 b6= a √ 2e ea · k b . σab The Bose symmetry in the gravitons is realized trivially in Eq. (3.38). Exchanging two gravitons only leads to a switch of two pairs of rows and columns in Ψ and ΨH , which apparently leaves the amplitude invariant. If we have n − 1 gravitons, Eq. (3.38) is not well defined because no proper ParkeTaylor factor exists for a single gluon. This amplitude should vanish since emitting a single gluon is forbidden by the color conservation. Thus, we can define that the CHY integrand vanishes for n − 1 gravitons. Finally, if all the n particles are gravitons, we just return to Eq. (3.36). There is no consistent way to reduce the number of gluons to zero in EYM. This can be understood from the fact that heterotic and closed string have different world sheet topology. CHAPTER 4 SPECIAL RATIONAL SOLUTION AND MHV AMPLITUDES In principle, the solutions to the SE are very complicated algebraic functions of the Mandelstam variables. However, in four dimensions, two special solutions exist that are rational in spinor variables, and related through a complex conjugation: (1) σa (2) σa h aη ihθξ i , h aξ ihθη i [ aη ][θξ ] = , [ aξ ][θη ] (1) h abihθξ ihηξ i , h aξ ihbξ ihθη i [ ab][θξ ][ηξ ] = , [ aξ ][bξ ][θη ] σab = = (2) σab (4.1) (4.2) where the projective spinor η, θ and ξ encode the SL(2, C) redundancy in this system, as discussed in Section 3.1.1. For example, if we choose η = n − 2, θ = n − 1 and ξ = n, we can put: (1) (2) σn−2 = σn−2 = 0 (1) (2) σn−1 = σn−1 = 1 (2) (2) σn = σn = ∞ . Written in terms of momentum components, this solution was already known in the first appearance of the SE [65, 66]. However, this spinor form was first given in [80]. Since Eq. (4.1) and Eq. (4.2) depend only on angular and square brackets, respectively, people immediately conjecture that they should correspond to the MHV and anti-MHV amplitudes, for example, see [63, 81]. In our work [39], we proved this connection analytically: Eq. (4.1) supports the MHV amplitudes while Eq. (4.2) supports the anti-MHV amplitudes. We will see in this chapter that once plugged in by the solution (4.1), both Pf 0 (Ψ) and det0 (Φ) will take a very nice form: the world sheet SL(2, C) dependency factorizes out of the gauge invariant building blocks of physical amplitudes completely. Then using the prescription Eq. (3.29) and Eq. (3.36), we can explicitly derive the Parke-Taylor formula for Yang-Mills MHV and Hodges formula [50] for gravity MHV amplitudes. This chapter presents the derivation, which gives essential experience and insight to work with other solutions and amplitudes. In this hindsight, we call Eq. (4.1) the MHV solution and Eq. (4.2) 60 the anti-MHV solution. In Section 4.1 and Section 4.2, we derive the expressions for det0 (Φ) and Pf 0 (Ψ) evaluated on the MHV solution. Then in Section 4.3, we present the resultant MHV amplitudes. Finally, we discuss and remark on the contribution of other solutions in Section 4.4. 4.1 Calculation of the reduced determinant If we plug in the MHV solution to matrix Φ given in Eq. (3.26), we have: [ ab]h aξ i2 hbξ i2 hθη i2 h abihηξ i2 hθξ i2 h aξ i2 hθη i2 [ al ]hlξ i2 = hηξ i2 hθξ i2 l∑ h al i 6= a Φ ab = − ( a 6= b) Φ aa (diagonal elements) . (4.3) In the off-diagonal elements, there are a common factor h aξ i2 and hbξ i2 to each row and column, such that it is tempting to pull them out of the determinant. To do this, we need a factor h aξ i4 in the diagonal elements, so that we need to rewrite the summation into a better form to make the factor h aξ i4 manifest: [ al ]hlξ i2 [ al ]hlξ i2 h aη i [ al ]hlξ ihηξ i [ al ]hlξ ihlη i =∑ =∑ + h aξ i2 ∑ h al i h al ih aη i h aη i h al ih aξ ih aη i l 6= a l 6= a l 6= a l 6= a ∑ [ al ]hlξ ihlη i , h al ih aξ ih aη i l 6= a = h aξ i2 ∑ where the first summation over l vanishes due to the momentum conservation. Therefore, the matrix elements of Φ can be related to those of the Hodges matrix φ defined in Eq. (2.94) as: Φ ab = − h aξ i2 hbξ i2 hθη i2 φab hηξ i2 hθξ i2 ( a 6= b) , Φ ab = h aξ i4 hθη i2 φaa , hηξ i2 hθξ i2 (4.4) such that all the prefactors can be pulled out when calculating the determinant. Consequently, we have: ijk ijk det(Φ pqr ) = (−1)n−3 ( Fηθξ )2n−6 ( Pξ )4 (dijk d pqr )2 det(φ pqr ) , (4.5) where the F, P and d symbols are defined as Fηθξ = hθη i hηξ ihθξ i n Pξ = ∏ haξ i d abc = d abc = a =1 1 . h aξ ihbξ ihcξ i (4.6) As another ingredient of the reduced determinant, we have: σij σjk σki σpq σqr σrp = ( Fηθξ )−6 (cijk c pqr )−1 (dijk d pqr )2 , (4.7) 61 where c symbols have been defined in Eq. (2.93). Combining Eq. (4.5) and Eq. (4.7), we get the final result of the reduced determinant det0 (Φ) evaluated on the MHV solution (4.1): ijk (−1)i+ j+k+ p+q+r det Φ pqr = ( Fηθξ )2n ( Pξ )4 M̄(12 . . . n) , det (Φ) = (σij σjk σki )(σpq σqr σrp ) 0 (4.8) where the reduced gravity amplitude M̄ has been defined in Eq. (2.92). In this equation, we can clearly see that the world sheet SL(2, C) dependent parts factorize out of the physical gauge invariant quantity M̄. Since we have proved in Appendix A that det0 (Φ) is independent of the deleted rows and columns, so does the reduced gravity amplitude M̄. For other solutions, det0 (Φ) can be viewed as the generalization of M̄ to other helicity configurations. 4.2 Calculation of the reduced Pfaffian Before we start to calculate Pf 0 (Ψ), we need to fix the gauge in the polarization vectors such that the structure of matrix Ψ is made as simple as possible. For the MHV configuration (1− 2− 3+ . . . n+ ), the best gauge choice is: h a|σµ |n] (e− a )µ = √ 2 [na] [ a|σµ |1i (e+ a )µ = √ 2 h1ai ( a = 1, 2) , ( a = 3, 4 . . . n) , (4.9) such that the only nonzero inner products between polarization vectors are e2 · e+ a (except for a = n). In other words, all nonzero elements in the matrix B are in the second row and − column. Moreover, this gauge choice also leads to k1 · e+ a = 0 and k n · e a = 0, which leads to additional zero elements in the matrix C. Therefore, before any manipulation, the shape of matrix Ψ is shown in Figure 4.1. For convenience, we deleted the (n − 1)-th and n-th row and column of Ψ when calculating the reduced Pfaffian: Pf 0 (Ψ) = −1 σn−1,n 1,n Pf Ψnn− −1,n ≡ −1 σn−1,n e) . Pf (Ψ (4.10) As proved in Appendix A, the value of Pf 0 (Ψ) does not depend on such a choice. e ), we can expand it along one row using the formula: To evaluate Pf (Ψ 2N Pf ( X ) = ∑ (−1) j =1 j 6 =i i + j +1+ θ ( i − j ) xij Pf ij Xij , θ (i − j ) = 1 i>j , 0 i<j (4.11) for a 2N × 2N antisymmetric matrix X = ( xij ). After this expansion, we would like the remaining submatrix to contain as many zeros as possible, so that further simplification 62 0 Ψ= A −C T C B 0 0 0 0 0 0 0 0 0 0 = B2b Ba2 0 0 0 0 0 0 0 0 0 Figure 4.1. The shape of matrix Ψ. Here we show the structure of Ψ after we fixed the gauge (4.9). Only the shaded regions are nonzero. e along its n-th row, which can be straightforward. Following this guideline, we expand Ψ is the 2nd row of the matrix B and C: e) = Pf (Ψ 2n−2 ∑ (−1)n+b+1+θ(n−b) Ψe nb Pf e nb . Ψ nb (4.12) b =1 b6=n Depending on the value of b, we have: C2b 1 6 b 6 n−2 e nb = Ψ . B2,b−n+2 n − 1 6 b 6 2n − 2 (4.13) Clearly, when 1 6 b 6 n − 2, the b-th row and column deleted are in the C part, such e nb has the shape displayed in Figure 4.2a. In this case, the lower left that the submatrix Ψ C part has dimension (n − 1) × (n − 3). As a result, we can always find an elementary transformation that makes at least two rows in this region zero, such that we have at least e nb . Therefore, we have Pf (Ψ e nb ) = 0 in this region, and we can rewrite two rows of zero in Ψ nb the expansion (4.12) as: e) = Pf (Ψ n n,m+n−2 ∑ (−1)m+1 B2m Pf Ψe n,m + n −2 ≡ m =3 n ∑ (−1)m+1 B2m Pf (ψm ) , (4.14) m =3 where the shape of ψm is shown in Figure 4.2b. Next, we apply the expansion (4.11) to each ψm along the (n − 1)-th row, which is the 1-st row of the matrix C: n −2 Pf (ψm ) = ∑ (−1)n+s+1 C1s Pf s =1 1,s [ψm ]nn− −1,s . (4.15) 0 0 0 0 n−2 n−3 63 0 n−1 0 n−2 0 0 0 0 n−3 n−2 n−1 (a) 1 6 b 6 n − 2 n−2 (b) n − 1 6 b 6 2n − 2 e nb . In this figure, we display the structure of Ψ e nb : (a) The Figure 4.2. The shape of matrix Ψ nb nb e when the deleted column is in the C part during the expansion of Pf (Ψ e ). submatrix of Ψ e obtained when the deleted The Pfaffian of this submatrix is zero. (b) The submatrix of Ψ e column is in the B part during the expansion of Pf (Ψ). 1,s In this expansion, we can similarly show that all the submatrix [ψm ]nn− −1,s with s > 2 must have zero Pfaffian. We can perform the elementary transformation shown in Figure 4.3 for those submatrices, which only changes their Pfaffians by a sign. After that, we see an (n − 2) × (n − 2) block of zeros at the right bottom corner, while the off-diagonal block at the left bottom is (n − 2) × (n − 4) dimensional. Therefore, we can always find another elementary transformation in the lower part of the matrix that makes at least two entire rows zero, which consequently leads to a zero Pfaffian. Then the summation in Eq. (4.15) only contains one term: 1,1 n 0 Pf (ψm ) = (−1)n C11 Pf [ψm ]nn− −1,1 ≡ (−1) C11 Pf ( ψm ) , (4.16) 0 is an (2n − 6) × (2n − 6) matrix. Now it is a good time to see how ψ0 is made where ψm m from the original A, B and C matrix:  0 ψm  A1,n−1,n  1,n−1,n =   1,2,m C1,n−1,n  1,2,m − C1,n −1,n 0 T   .   (4.17) 1,2,m If the lower left block C1,n −1,n does not have full rank, we can find an elementary transfor- 0 ) = 0. On the other hand, mation to make one entire row zero in this part such that Pf (ψm 64 switch 0 n−4 0 switch n−3 0 0 0 0 0 0 0 0 0 0 n−2 n−3 0 0 0 0 0 0 n−3 n−3 n−4 n−2 1,s Figure 4.3. The shape of [ψm ]nn− −1,s with s > 2. The (2n − 6) × (2n − 6) submatrix of ψm when the deleted column is not the first one during the expansion. To tell the Pfaffian of this submatrix is zero, one can switch the two rows and columns as indicated. 1,2,m if C1,n −1,n has full rank, we can find another elementary transformation to make the entire −1,n A1,n 1,n−1,n matrix zero. In both cases, we can write: 0 Pf (ψm ) = (−1) (n−2)(n−3) 2 12m det C1,n −1,n . (4.18) Now collecting Eq. (4.10), Eq. (4.14), Eq. (4.16), and Eq. (4.18), we can express the expansion of Pf 0 (Ψ) as: 0 Pf (Ψ) = (−1) (n−2)(n−3) 2 n ∑ (−1) m + n +1 −1 B2m C11 m =3 σn−1,n det 12m C1,n −1,n . (4.19) 1,2,m If we divide the matrix C into two parts C± according to the helicities, we find that C1,n −1,n is actually a submatrix of C+ . Then Eq. (4.19) implies that to make Pf 0 (Ψ) 6= 0 at MHV, we must have rank(C+ ) = n − 3. This is the first hint that in four dimensions, the solutions to SE and the rank of the matrix C may have an interesting relation. We will discuss this point in detail in Chapter 6. The next task is thus to plug in the MHV solution (4.1) into Eq. (4.19). First, the relevant C matrix elements are: [ ab]h aξ ihbξ ihb1ihθη i h abih a1ihθξ ihηξ i h aξ i2 hθη i n [ al ]hlξ ihl1i Caa = − hθξ ihηξ i l∑ h al ih aξ ih a1i =1 (3 6 a 6 n and b 6= a) , Cab = (3 6 a 6 n ) . l 6= a 1,2,m Plugging them into the determinant of C1,n −1,n , we get: 12m n −3 det C1,n ( Pξ )2 d12m d1,n−1,n −1,n = ( Fηθξ ) h12ih1mi 12m det φ1,n −1,n . h1, n − 1ih1ni (4.20) 65 1,2,m The origin of those coefficients in front of the Hodges minor det(φ1,n −1,n ) is not difficult to identify: first, Fηθξ is a factor common to all the n − 3 rows, so that it is raised to this power; ( Pξ )2 d12m d1,n−1,n comes from pulling out h aξ i and hbξ i that are common to each row and column respectively, where the d's are to compensate the deleted rows and columns; finally, when pulling out 1/h a1i and hb1i from each row and column, we need the factor h12ih1mi/(h1, n − 1ih1ni) to compensate the deleted rows and columns. Now grouping the above result with −1/σn−1,n , we can trade one d factor into one of the c's that is required for a reduced gravity amplitude M̄: 12m − det(C1,n h12ih1mi −1,n ) 12m = ( Fηθξ )n−2 ( Pξ )2 d12m c1,n−1,n det φ1,n . −1,n σn−1,n h1ξ i The other d is actually transformed into c with the help of B2m : (−1)m+n+1 B2m = (−1)m+n+1 e2 · e m c [mn] = Fθηξ h12i2 (−1)m+n+1 12m . σ2m d12m h1ξ i[n2] (4.21) Interestingly, the alternating sign (−1)m gets grouped into the m-independent M̄ such that the summation over m resutls in: n ∑ (−1) m + n +1 B2m m =3 −1 σn−1,n det 12m C1,n −1,n = −( Fθηξ )n−1 ( Pξ )2 n h1mi[mn] h12i3 M̄ ( 12 . . . n ) ∑ h1ξ i2 [n2] m =3 = −( Fθηξ )n−1 ( Pξ )2 h12i4 M̄(12 . . . n) . h1ξ i2 (4.22) Finally, combining with C11 : n C11 = ∑ b =2 √ 2 e1 · k b = ( Fθηξ )h1ξ i2 , σ1b we arrive at the final result of Pf 0 (Ψ) evaluated on the MHV solution (4.1): Pf 0 (Ψ) = −(−1) (n−2)(n−3) 2 ( Fθηξ )n ( Pξ )2 h12i4 M̄(12 . . . n) . (4.23) Similar to Eq. (4.8), The final result features a complete separation of the world sheet SL(2, C) dependent factor and the physical gauge invariant component. In particular, Pf 0 (Ψ) reproduces the correct Parke-Taylor numerator h12i4 . If we had started with a less convenient gauge, we would reach the same result. The gauge freedom is encoded in φaa contained in M̄, and we have already proved it in Eq. (2.95). 66 Finally, we note that if the two negative helicity gluons are at the position i and j, Eq. (4.19) has a generalized expression [39]: 0 Pf (Ψ) = −(−1) (n−2)(n−3) 2 n ∑ perm(ijm)perm(ipq) Bjm Cii m =1 m6=i,j 1 ijm det Cipq , σpq (4.24) where perm(ijm) is the permutation signature of {i, j, m, 1, 2 . . .} with respect to the identity {1, 2 . . . n}. After plugging in the MHV solution (4.1), the result is: Pf 0 (Ψ) = (−1) (n−1)(n−4) 2 ( Fθηξ )n ( Pξ )2 hiji4 M̄(12 . . . n) . (4.25) Comparing with Eq. (4.23), we only need to replace the Parke-Taylor numerator h12i4 by the general one hiji4 . We have also absorbed a minus sign into the power of (−1). 4.3 Yang-Mills and gravity MHV from CHY The final piece in the Yang-Mills integrand is the Parke-Taylor factor. When evaluated on the MHV solution (4.1), it simply equals: PT (12 . . . n) = ( Fθηξ )n ( Pξ )2 1 = . σ12 σ23 . . . σn1 h12ih23i . . . hn1i (4.26) Noticeably, the world sheet SL(2, C) dependence of PT is exactly the same as that of Pf 0 (Ψ), see Eq. (4.23). Then using the CHY formula for color ordered Yang-Mills amplitude (3.33) with Eq. (4.8) Eq. (4.23), and Eq. (4.26), we can reproduce the correct Parke-Taylor formula for MHV, up to an inconsequential overall factor: A n (1− 2− 3+ . . . n + ) = (n−1)(n−4) PT (12 . . . n)Pf 0 (Ψ) h12i4 2 . = (− 1 ) h12ih23i . . . hn1i det0 (Φ) (4.27) In this equation, the world sheet SL(2, C) dependent factors Fθηξ and Pξ in the three building blocks nicely cancel each other. Moreover, the reduced gravity amplitude M̄, contained in both Pf 0 (Ψ) and det0 (Φ), also gets canceled. Since PT and Pf 0 (Ψ) have the same world sheet SL(2, C) dependence, replacing the former by the latter does not spoil the cancellation of such a dependence. According to the CHY formulism, this operation results in the gravity MHV amplitude: Mn ( 1 − 2 − 3 + . . . n + ) = Pf 0 (Ψ) × Pf 0 (Ψ) = h12i8 M̄(12 . . . n) . det0 (Φ) Indeed, we obtain correctly the Hodges formula. (4.28) 67 We note that by exchanging all the angular and square brackets, we can obtain the anti-MHV Yang-Mills and gravity amplitudes along the same line of derivation. In other words, the anti-MHV solution (4.2) can reproduce the correct anti-MHV Yang-Mills and gravity amplitudes in the same way. 4.4 Remarks and summary As we have argued in Section 4.2, we have Pf 0 (Ψ) 6= 0 at MHV configurations if and only if rank(C+ ) = n − 3. The MHV solution explicitly does so by leading to the correct Parke-Taylor formula and Hodges formula for Yang-Mills and gravity MHV amplitudes. It is thus tempting to check what happens if we use the anti-MHV solution (4.2) into Eq. (4.19). In this case, the C+ part elements are: [ aξ ][bξ ]hb1i[θη ] h a1i[θξ ][ηξ ] [ aξ ]2 [θη ] Caa = [θξ ][ηξ ] Cab = (3 6 a 6 n and b 6= a) (3 6 a 6 n ) . (4.29) It is now easy to verify that all the rows are proportional, since h a1i × Cab is independent of a. [ aξ ] Therefore, the anti-MHV solution leads to rank(C+ ) = 1, and it does not contribute to MHV amplitudes. Similarly, all the other solutions make rank(C+ ) < n − 3 such that only the MHV solution (4.1) contributes to MHV amplitudes, as the name indicates. To summarize, in this chapter, we have proved analytically that the MHV solution (4.1) does lead to the correct Parke-Taylor formula for Yang-Mills and Hodges for gravity at MHV, as in Eq. (4.27) and Eq. (4.28). We claim that all the other solutions do not contribute to MHV amplitudes since they make rank(C+ ) < n − 3. This point will be further clarified later in Chapter 6. CHAPTER 5 SINGLE TRACE EINSTEIN-YANG-MILLS MHV AMPLITUDES In the previous chapter, we derived the Parke-Taylor and Hodges formula for YangMills and gravity MHV amplitudes. We expect that such techniques may also lead to simple and compact expression for MHV amplitudes of other theories. Einstein-YangMills (EYM) is such an immediate generalization. Indeed, we show in [40] that the CHY formalism can give a much simpler expression for this amplitude, compared to what exists in the literature. To be specific, we are looking at the single trace (gluon) color ordered amplitude Mn,r (12 . . . r |H), in which gluon and graviton set are: G = {1, 2 . . . r } , H = {r + 1, r + 2 . . . r + s} , (n = r + s) . (5.1) As before, we use N = {1, 2 . . . n} = G ∪ H to denote the set of all external particles. In addition, it is convenient to define the sets of positive and negative helicity gravitons H± and similarly gluons G± , whose orders are |H± | = s± and |G± | = r ± . The CHY integrand in this case is given by Eq. (3.38), repeated here as: Mn,r (12 . . . r |H) = Pf (ΨH ) Pf 0 (Ψ) 1 ∑ 0 σ12 σ23 . . . σr1 {σ}∈sol. det ( Φ ) (2 6 r 6 n ) , (5.2) where Pf (ΨH ) is defined in Eq. (3.39). At MHV, since the integrand contains Pf 0 (Ψ), the support can only be the MHV solution (4.1). As calculated in the previous chapter, we already have: (n−1)(n−4) Pf 0 (Ψ) (−1) 2 hiji4 = ( Fθηξ )n ( Pξ )2 det0 (Φ) ( Fθηξ )r 1 = σ12 . . . σr1 h12i . . . hr1i r ∏ haξ i2 , (5.3) a =1 if the two negative helicity particles are i and j, be it gluons or gravitons.1 Thus, the main subject of this chapter is to calculate Pf (ΨH ). The outcome will be different depending on the nature of negative helicity particles, and we have the following three cases: 1 Namely, we have 1 6 i < j 6 n. 69 1. Two negative helicity gluons, hereafter, ( g− g− ) amplitude. 2. One negative helicity gluon and the other graviton, hereafter, ( g− h− ) amplitude. 3. Two negative helicity gravitons, hereafter, (h− h− ) amplitude. We first derive the expressions for ( g− g− ) and ( g− h− ) MHV amplitudes in Section 5.1. Then in Section 5.2, we prove that the (h− h− ) amplitude must identically vanish. Finally, we prove that our new results, which are much simpler, agree with the existing ones in the literature, using a graph theoretical approach discussed in Section 5.3. 5.1 ( g− g− ) and ( g− h− ) MHV amplitudes According to our convention i < j, we must have i ∈ G, and j ∈ G for ( g− g− ) amplitude but j ∈ H for ( g− h− ) amplitude. To make the matrix BH as simple as possible, we choose the gauge in {e e} as: ( g− g− ) : ( g− h− ) : h j|σµ | a] (e e+ a )µ = √ 2h jai h j|σµ | a] (e e+ a )µ = √ 2h jai ( a ∈ H) h j | σ µ |1] (e e− j )µ = √ 2[1j] ( a ∈ H , j ∈ H) . (5.4) This choice leads to BH = 0 for both ( g− g− ) and ( g− h− ) amplitude. Therefore, independent of the solutions, we have: Pf (ΨH ) = Pf AH −CHT CH 0 = (−1)s(s+1)/2 det(CH ) . (5.5) The evaluation of det(CH ) is slightly different for the two cases: ( g− g− ) amplitude After we plug in the MHV solution (4.1), the matrix elements of CH have the form: [ ab]hbjih aξ ihbξ i h abih aji n [ ab]hbjihbξ i = −( Fθηξ )h aξ i2 ∑ h abih ajih aξ i b =1 (CH ) ab = ( Fθηξ ) ( a, b ∈ H , a 6= b) (CH ) aa ( a ∈ H , diagonal) . (5.6) b6= a We note that in the diagonal elements (CH ) aa , the summation is from 1 to n, not just within the graviton set H. After we take the determinant, the common factors can be pulled out such that: det(CH ) = ( Fθηξ )s n ∏ a =r +1 ! h aξ i2 det(φH ) , (5.7) 70 where φH is an s × s diagonal submatrix of the Hodges matrix φ: 12...r φH ≡ φ12...r {1, 2 . . . r } = H (complement of H in N) . (5.8) Namely, φH is obtained from φ by deleting all gluon rows and columns. In this case, H = H+ since all gravitons have positive helicities. Combining it with Eq. (5.3), we get the following expression for the ( g− g− ) amplitude: Mn,r 12 . . . r |H; i− j− ∝ hiji4 det(φH ) h12ih23i . . . hr1i Here we have neglected an overall coefficient (−1) (n−1)(n−4)+s(s+1) 2 (i, j ∈ G) . (5.9) . Finally, we note that for H = ∅, we just define det(φ∅ ) = 1 such that Eq. (5.9) returns to the Parke-Taylor formula. ( g− h− ) amplitude In this case, the matrix CH is identical to that of Eq. (5.6), except for the ( j − r )-th row, which is the row corresponding to the negative helicity graviton j. Moreover, since e e+ a · k j = 0 according to our gauge choice (5.4), the only nonzero element in the column j is (CH ) jj , which equals: n (CH ) jj = √ ∑ b =1 b6= j 2e e− j · kb σjb = ( Fθηξ )h jξ i2 . We can then expand det(CH ) along the column j, which leads to: ! n j s 2 det(CH ) = (CH ) jj det (CH ) j = ( Fθηξ ) ∏ haξ i det(φH+ ) , (5.10) (5.11) a =r +1 where φH+ is the (s − 1) × (s − 1) submatrix of the Hodges matrix φ that corresponds to positive helicity gravitons: 12...rj φH+ ≡ φ12...rj {1, 2 . . . r, j} = H+ . (5.12) After we combine it with Eq. (5.3), the final result for the ( g− h− ) amplitude is: Mn,r 12 . . . r |H; i− j− ∝ hiji4 det(φH+ ) h12ih23i . . . hr1i ( i ∈ G , j ∈ H) . (5.13) According to Eq. (5.9) and Eq. (5.13), We can actually write a unified expression for both the ( g− g− ) and ( g− h− ) amplitude: Mn,r 12 . . . r |H; i− j− ∝ hiji4 det(φH+ ) h12ih23i . . . hr1i where for H+ = ∅, we define det(φH+ ) = 1. (i ∈ G , j ∈ G or H) , (5.14) 71 It is interesting to note that according to Eq. (5.14), the ( g− g− ) amplitude with only two gluons must vanish identically: Mn,2 (1− 2− |H+ ) = 0, since the Hodges matrix only has rank n − 3 while in this case φH+ is always n − 2 dimensional. On the other hand, the ( g− h− ) amplitude with only two gluons do not vanish, since in this case φH+ is always n − 3 dimensional. This pattern is correct even at 3-point: h12i4 det(φ{3} ) = 0 h12ih21i h13i4 h13i4 M3,2 (1− 2+ |3− ) ∝ det(φ∅ ) ∝ . h12ih21i h12i2 M3,2 (1− 2− |3+ ) ∝ (5.15) For 3-point MHV, the Hodges matrix has rank zero since all the square brackets vanish, such that we have det(φ{3} ) = φ33 = 0. This result agrees with the analysis in [82, 83]. Before moving on, we note that Selivanov [84, 85], Bern, De Freitas, and Wong [49] have given a generating function for the ( g− g− ) and ( g− h− ) amplitude (hereafter SBDW formula). We are going to prove in Section 5.3 that their results exactly agrees with Eq. (5.14), while our new expression is much simpler. 5.2 The vanishing of (h− h− ) MHV amplitudes In this section, we are going to show that Pf (ΨH ) = 0 when evaluated on the MHV solution (4.1) for the helicity configurations with s− > 2. The (h− h− ) MHV amplitude corresponds to s− = 2, such that it vanishes identically. We start with the following gauge choice: h a|σµ |q] (e e− a )µ = √ 2[qa] ( a ∈ H− ) , h p|σµ | a] (e e+ a )µ = √ 2h pai ( a ∈ H+ ) , (5.16) where p and q are two arbitrary reference spinors, which in general are not any of the graviton momenta. After we plug in the MHV solution (4.1) into Pf (ΨH ), we can pull out as many common factors to rows and columns as possible, and write: ! ! h api s 2 Pf (ΨH ) = ( Fθηξ ) ∏ h aξ i ∏ [aq] Pf (Ψe H ) , a ∈H a ∈H− e H is another 2s × 2s matrix with the form: where Ψ e −C eT A e ΨH = . e e C B e B e have the following forms:2 e and C The s × s matrices A, 2 In ea−r,b−r ≡ A eab , for example. the following, we define A (5.17) (5.18) 72 • A-part: eab = A [ ab] h apihbpi ( a, b ∈ H) . (5.19) • B-part: For a ∈ H− : eab B For a ∈ H+ : eab B   0 h api[bq] =  h abi   [ aq]hbpi = h abi  0 b ∈ H− b ∈ H+ b ∈ H− (5.20) b ∈ H+ • C-part: For a ∈ H− : eab = [bq] , C hbpi For a ∈ H+ : eab = φab . C (5.21) In particular, those rows in H− are identical. Now we choose an arbitrary particle i ∈ H− and then perform the following elementary e H: transformations on to Ψ 1. For all j ∈ H− and j 6= i, we subtract the (s + j)-th row and column by the (s + i )e all the rows corresponding to H− are th row and column. Then in the matrix C, identically zero except for the i-th. 2. We substract the first s rows and columns by a multiple of the i-th: hipi[bq] [iq]hbpi e H ) a× − ( Ψ e H )i× hipi[ aq] → (Ψ [iq]h api e H )×b → ( Ψ e H )×b − ( Ψ e H ) ×i (Ψ ( b ∈ H) e H ) a× (Ψ ( a ∈ H) . e zero, except for the element C eii . This operation further makes the i-th row of C e zero, except for the i-th row and column. Moreover, it also makes the entire matrix A This can be shown by using the Schouten identity: eab = A [ ab] [ ab] [ ai ][bq] [ aq][ib] → − − = 0. h apihbpi h apihbpi [iq]h apihbpi [iq]h apihbpi 3. Finally, we subtract the first s row and column by a multiple of the (s + i )-th: [bi ] [iq]hbpi e H ) a× − ( Ψ e H )s+i ,× [ ai ] → (Ψ [iq]h api e H )×b → ( Ψ e H )×b − ( Ψ e H )×,s+i (Ψ ( b ∈ H) e H ) a× (Ψ ( a ∈ H) , 73 e is made zero. Then for those rows and columns corresuch that the entire matrix A sponding to H+ , we perform: hipi[bi ] [iq]hbi i e H ) s + a ,× − ( Ψ e H )s+i ,× hipi[ ai ] → (Ψ [iq]h ai i e H )×,s+b → (Ψ e H )×,s+b − (Ψ e H )×,s+i (Ψ b ∈ H+ e H ) s + a ,× (Ψ a ∈ H+ , eii = [iq]/hipi. such that the only nonzero element in the i-th row and column is C eii and write: Now we can just pull out C Pf (ΨH ) = (−1)s [iq] 0 eH Pf (Ψ ), hipi (5.22) e 0 has the shape: where the (2s − 2) × (2s − 2) matrix Ψ H s + s− − 2 e0 = Ψ H 0 s+ s + s− − 2 e+ part C T −Ce+ part . (5.23) e part B e+ block has The exact form of each matrix element here is not important. The lower left C dimension s+ × (s + s− − 2), such that its columns are more than the rows if s− > 2. For these cases, we can always find an elemetary transformation to make at least one entire row and column zero. As a result, we have proved that Pf (ΨH ) = 0 for s− > 2 when evaluated on the MHV solution (4.1). In particular, the (h− h− ) MHV amplitude is identically zero. Moreover, we will show in Chapter 6 that using the similar technique, we can prove that if gluons have the same helicity, all the single trace tree level EYM amplitudes must vanish, independent of the graviton helicity configurations. This statement is first conjectured in [49]. 5.3 Spanning forests and MHV amplitudes Using the CHY formalism for EYM, we have derived in the previous section a set of very simple and compact expressions for single trace tree level MHV amplitudes:  hiji4    det(φH+ ) (i ∈ G , j ∈ G or H) h12ih23i . . . hr1i Mn,r (12 . . . r |H; i− j− ) = .    0 (i, j ∈ H) (5.24) 74 On the other hand, the SBDW formula [49, 84, 85] states that the ( g− g− ) and ( g− h− ) MHV amplitude have the following form: Mn,r (12 . . . r |H; i− j− ) ∝ (−1)s + hiji4 S(H+ ; N) h12ih23i . . . hr1i (i ∈ G , j ∈ G or H) , (5.25) where the graviton factor S(H+ ) can be obtained from a generating function:3 G ( a µ , H+ ; N ) (µ ∈ H+ and H + = the set complement of H+ in N )   !# "    a ψ exp ( . . . ) (5.26) exp a ψ ψ exp = exp a ∑ n3 n3 n2 ∑ n2 n2 n1 ∑ n1 ∑ n1 l     n 1 ∈H+ n ∈ H n ∈ H + + 3 2 l ∈H+ n3 6=n1 ,n2 n2 6 = n1 in which the matrix element ψab is defined as: ψab = φab hbξ ihbη i . h aξ ih aη i (5.27) Here ξ and η are the two reference spinors used in the diagonal elements φaa : n [ al ]hlξ ihlη i . h al ih aξ ih aη i l =1 φaa = − ∑ l 6= a Then S(H+ ; N) is just a certain Taylor expansion coefficient of G ( aµ , H+ ; N): S(H+ ; N) = ∏ m ∈H+ ∂ ∂am ! G ( a µ , H+ ; N ) . (5.28) a m =0 The SBDW formula looks very different from what we have obtained from CHY at the first glance. In this section, we are going to prove that they are exactly equivalent, namely, + S(H+ ; N) = (−1)s det(φH+ ) , (5.29) using a graph theory technique. 3 In [49], the second summation is over l ∈ G instead of l ∈ H . The reason is that [49] has used a special + + gauge choice ξ = i and η = j as in Eq. (5.27). The final amplitude is of course gauge invariant so that our definition is equivalent to that of [49]. 75 5.3.1 Generating the spanning forests We can construct a weighted complete graph Kn with the vertex set Vn = {v1 , v2 . . . vn }, and ψab as the weight on the edge v a vb .4 The information of such a graph Kn is encoded in the n × n Laplace matrix Wn , defined as: (Wn ) ab =     −ψab a 6= b n    ∑ ψab diagonal . (5.30) b=1 ,b6= a The matrix Wn is useful since all its diagonal minors must equal that of φ. In particular, we have: ∏b∈H+ hbξ ihbη i + + (−1)s det(φH+ ) = (−1)s det(φH+ ) . det (Wn )H+ = ∏ a∈H+ h aξ ih aη i (5.31) Therefore, to prove Eq. (5.29), we can equivalently show that: S(H+ ; N) = det (Wn )H+ . (5.32) The benefit of studying Wn is that now we can use the graph theory to our favor. First, we have a graph combinatorical interpretation for S , according to the following theorem: Theorem 5.1: Suppose Ir = {i1 , i2 . . . ir } is an r-element subset of the vertex set Vn , then the function G ( aµ , I r ; N) is the generating function of all the weighted spanning forests of Kn rooted in Ir in the following sense: S( I r ; N) = ∏ m∈ I r ∂ ∂am ! G ( a µ , I r ; N) = a m =0 ∑ F ∈F Ir (Kn ) ∏ ! ψab , (5.33) v a vb ∈ E( F ) where F Ir (Kn ) denotes the set of the spanning forests of Kn rooted in Ir . The edge set E( F ) of a forest F consists of all edges v a vb that are directed into the roots. Several spanning forest examples are given in Figure 5.1. We will give another 7-point example in Section 5.3.2 to demonstrate the relation (5.33). Then we will give the general proof in Section 5.3.3. 4 If ψab is symmetric, the graph is undirected. Otherwise, the graph is directed (our case). 76 3 4 1 5 3 2 1 4 5 3 2 1 4 5 3 2 1 4 5 2 Figure 5.1. Some spanning forest examples. Here we show several spanning forests of K5 rooted in I2 = {1, 2} (the vertices enclosed by the dashed boxes). On the other hand, the evaluation of det (Wn ) I r has the same graph combinatorical interpretation [86]:5 det (Wn ) I r = ∑ F ∈F Ir (Kn ) ! ∏ ψab . (5.34) v a vb ∈ E( F ) Therefore, according to Theroem 5.1, we have established the identity: S( I r ; N) = det (Wn ) I r . (5.35) Then after we choose I r = H+ , the desired relation (5.32) follows immediately. In this way, we have proved that our new expression Eq. (5.14) is equivalent to the SBDW formula. However, our expression is much simpler and easier to evaluate, as the example in Section 5.3.2 will demonstrate. 5.3.2 A seven-point example Now we demonstrate that the graviton factor of the 7-point amplitude: − + + + + + M7,4 1− g 2 g 3 g 4 g |5h 6h 7h indeed satisfies the relation (5.33). The root set is H+ = {1, 2, 3, 4}. However, we can choose the reference spinors in Eq. (5.27) as ξ = 1 and η = 2, such that: ψa1 = ψa2 = 0 for all a ∈ {3, 4, 5, 6, 7} . In other words, the root 1 and 2 do not connect to any other vertices. The problem is thus reduced to summing over all the spanning forests with root set {3, 4} of the graph K5 whose vertex set is V5 = {3, 4, 5, 6, 7}. The SBDW generating function is: 5 This relation is called matrix-tree theorem II in [86]. 77 n G ( a5 , a6 , a7 ; V5 ) = exp a5 (ψ53 + ψ54 ) exp [ a6 ψ65 exp( a7 ψ76 ) + a7 ψ75 exp( a6 ψ67 )] + a6 (ψ63 + ψ64 ) exp [ a7 ψ76 exp( a5 ψ57 ) + a5 ψ56 exp( a7 ψ75 )] o + a7 (ψ73 + ψ74 ) exp [ a5 ψ57 exp( a6 ψ65 ) + a6 ψ67 exp( a5 ψ56 )] . Our main task here is to verify that the graviton factor ∂ ∂ ∂ S({5, 6, 7}; V5 ) = G ( a5 , a6 , a7 ; V5 ) ∂a5 ∂a6 ∂a7 a5 = a6 = a7 =0 (5.36) indeed gives the graph combinatorical result ∑ F ∈F{3,4} (K5 ) ∏ ! ψab . (5.37) v a vb ∈ E( F ) Our strategy is to expand both Eq. (5.36) and Eq. (5.37) with respect to ψl3 and then compare them order by order. First, Eq. (5.37) can be expanded as: ∑ F ∈F{3,4} (K5 ) ∏ v a vb ∈ E( F ) ! ψab = A+ ∑ l ∈{5,6,7} ψl3 Bl + ∑ ψl3 ψk3 Clk + ψ53 ψ63 ψ73 D . (5.38) l,k ∈{5,6,7} Effectively, this expansion puts all the spanning forests into four categories A , B , C , and D . In category A , which contributes only to A, ψl3 does not appear, which means that all the leaves are grown from the root 4. Therefore, A is contributed by the spanning trees of V4 = {4, 5, 6, 7} with root 4. There are in all 16 of them, as shown in Figure 5.2. Similarly in category B , the coefficient B5 is contributed by those forests in which if the leaves {6, 7} belong to the root 3, they must first converge to 5. In other words, B5 is contributed by all the spanning forests of V4 = {4, 5, 6, 7} with root set {4, 5}. In general, each category corresponds to a set of spanning forests of V4 = {4, 5, 6, 7} with different root sets, which is collected in Table 5.1. Therefore, each of the coefficients corresponds to one of the sub-problems with one less vertex. This feature strongly suggests an inductive proof for the general relation Eq. (5.33). Next, we need to verify that the Taylor expansion of S({5, 6, 7}; V5 ) with respect to ψl3 exactly reproduces the same terms as in Eq. (5.38). To start with, the zeroth order can be simply obtained by setting all ψl3 = 0 in the generating function: 78 Figure 5.2. The 16 spanning trees of K4 . The direction on each edge is understood as pointing towards the root, which is the bottom vertex. Table 5.1. The corresponding spanning forests of V4 = {4, 5, 6, 7}. They are separated into category A , B , C and D according to Eq. (5.38). category root set # of diagrams A Bl Clk D {4} {4, l } {4, l, k} {4, 5, 6, 7} 16 8 × 3 = 24 3×3 = 9 1 G ( a5 , a6 , a7 ; V4 ) = G ( a5 , a6 , a7 ; V5 )|ψ53 =ψ63 =ψ73 =0 n = exp a5 ψ54 exp [ a6 ψ65 exp( a7 ψ76 ) + a7 ψ75 exp( a6 ψ67 )] + a6 ψ64 exp [ a7 ψ76 exp( a5 ψ57 ) + a5 ψ56 exp( a7 ψ75 )] o + a7 ψ74 exp [ a5 ψ57 exp( a6 ψ65 ) + a6 ψ67 exp( a5 ψ56 )] . By a straightforward but tedious calculation, we obtain: ∂ ∂ ∂ A = S({5, 6, 7}; V4 ) = G ( a5 , a6 , a7 }; V4 ) , ∂a5 ∂a6 ∂a7 a5 = a6 = a7 =0 (5.39) namely, S({5, 6, 7}; V4 ) contains 16 terms that exactly correspond to the 16 spanning trees as shown in Figure 5.2. For the coefficients in category B , it is sufficient to calculate just B5 as an example. On the graph theory side, we have: 79 7 6 5 B5 = 7 6 5 + 4 3 7 6 5 4 3 6 5 + 4 3 7 6 5 + + 7 3 6 4 3 7 6 4 3 5 + 4 3 7 6 5 + 4 7 5 + 4 3 = ψ74 ψ64 + ψ67 ψ74 + ψ76 ψ64 + ψ74 ψ65 + ψ76 ψ65 + ψ75 ψ65 + ψ67 ψ75 + ψ75 ψ64 . (5.40) According to Table 5.1, the solid edges just consist of the spanning forests of V4 with the root set {4, 5}. In G ( a5 , a6 , a7 ; V5 ), we find that ψ53 only appears in the outmost level of the exponents. Then the Taylor expansion coefficient of ψ53 can be extracted by acting ∂/∂a5 only on a5 ψ53 , and then set ψ63 = ψ73 = 0.6 The result is: G ( a6 , a7 ; V4 ) = exp [ a6 (ψ64 + ψ65 ) exp( a7 ψ76 ) + a7 (ψ74 + ψ75 ) exp( a6 ψ67 )] . (5.41) Now we can explicitly verify that: ∂ ∂ B5 = S({6, 7}; V4 ) = G ( a6 , a7 ; V4 ) ∂a6 ∂a7 a6 = a7 =0 a (ψ +ψ ) ∂ a6 ψ67 (ψ74 + ψ75 )e + a6 (ψ64 + ψ65 )ψ76 e 6 64 65 = ∂a6 a6 =0 = (ψ74 + ψ75 )ψ67 + (ψ64 + ψ65 )ψ76 + (ψ64 + ψ65 )(ψ74 + ψ75 ) . (5.42) The coefficient B6 and B7 can also be calculated in this way, and they do agree with a graph summation similar to Eq. (5.40). Then for the coefficients in category C , again we only calculate C56 as an example. On the graph theory side, we have the spanning forests of V4 with the root set {4, 5, 6}: 7 6 C56 = 5 7 6 + 4 3 5 7 6 + 4 3 5 = ψ74 + ψ75 + ψ76 . 4 (5.43) 3 6 The other ways of acting the derivative give only lower order terms in ψ . We set ψ 53 63 = ψ73 = 0 since they only contribute to higher order terms in the ψl3 expansion. 80 In the graviton factor S({5, 6, 7}; V5 ), the coefficient of ψ53 ψ63 is generated by G ( a7 ; V4 ), which is obtained from G ( a5 , a6 , a7 ; V5 ) by acting the derivative (∂/∂a5 )(∂/∂a6 ) on to a5 ψ53 and a6 ψ63 only, and then setting a5 = a6 = ψ73 = 0: G ( a7 ; V4 ) = exp [ a7 (ψ74 + ψ75 + ψ76 )] . (5.44) Indeed, the two coefficients match: C56 ∂ = S({7}; V4 ) = G ( a7 ; V4 ) = ψ74 + ψ75 + ψ76 . ∂a7 a7 =0 (5.45) Finally, the category D only contains one trivial graph: 6 7 5 = 1. D= (5.46) 3 4 On the other hand, by acting all the three derivatives onto a5 ψ53 , a6 ψ63 , and a7 ψ73 , we get the same result: D = S(∅; V4 ) = G (∅; V4 ) = 1 . (5.47) Now we have checked that each term in the Taylor expansion of S({5, 6, 7}; V5 ): S({5, 6, 7}; V5 ) = S({5, 6, 7}; V4 ) + ∑ ψl3 S({5, 6, 7}\{l }; V4 ) l ∈{5,6,7} + ∑ ψl3 ψk3 S({5, 6, 7}\{l, k }; V4 ) + ψ53 ψ63 ψ73 S(∅; V4 ) (5.48) l,k ∈{5,6,7} exactly matches that of the graph expansion Eq. (5.38), such that the desired equality holds: ! S({5, 6, 7}; V5 ) = ∑ F ∈F{3,4} (K5 ) ∏ ψab . (5.49) v a vb ∈ E( F ) However, when performing this calculation, we are encountered by the summation over: | A | + |B | + |C | + |D | = 16 + 8 × 3 + 3 × 3 + 1 = 50 terms generated by different ways of acting derivatives. Therefore, the SBDW style evaluation is very calculationally heavy, which will quickly grow out of control for more points. In fact, those 50 terms can be nicely grouped into 3 × 3 determinant, as our new formula Eq. (5.14) makes manifest. This example also demonstrates that the CHY formalism is superior than the SBDW generating function in evaluating the MHV amplitudes for EYM. 81 5.3.3 General relation Gaining enough experience from the explicit 7-point example, we are now ready to present the general proof of Theorem 5.1. First, at n = 2, we have only two vertices. We can just choose the root to be I = {1} such that I = {2}. Then we can show that v2 ψ21 = ψ21 v1 d . = S({2}; {1, 2}) = exp ( a2 ψ21 ) da2 a2 =0 (5.50) such that Eq. (5.33) holds at n = 2. Next, we assume that Eq. (5.33) holds at (n − 1)-point and check whether it also holds at n-point. For a forest F with roots Ir = {i1 , i2 . . . ir }, we can classify it by the set of vertices that are immediately connected to the root ir . Suppose the vertex set Pt = { p1 , p2 . . . pt } ⊂ I r is directly connected to ir , then we have: ! t ∏ ψab = ∏ ψp i ∏ ψab Fe ∈ F I (Kn−1 ) , (5.51) r −1+ t k r v a vb ∈ E( F ) k =1 v a vb ∈ E( Fe) where Fe is an (n − 1)-point forest with roots Ir−1+t = {i1 , i2 . . . ir−1 , p1 , p2 . . . pt }. This expansion is depicted in Figure 5.3. Therefore, the right hand side of Eq. (5.33) can be expanded as: ∑ F ∈F Ir (Kn ) ∏ ! ψab n −r = ∑ ∑ t ∑ k r t=0 Pt ⊂ I r v a vb ∈ E( F ) ∏ ψp i  ! k =1 = S( I r0−1 ; N0 ) + ∑ ∑ t=1 Pt ⊂ I r ∏  t ∏ ψp i k r ψab  v a vb ∈ E( Fe) Fe∈F Ir−1+t (Kn−1 ) n −r  ! S( I r0−1+t ; N0 ) , (5.52) k =1 where N0 = N\{ir } and I 0 is defined as the complement of I in the set N0 . Consequently, we must have I r = I r0−1 . We note that the first line is nothing but the generalization of Eq. (5.38) in our example, and the second line is obtained from the induction assumption. Our next job is to verify that Eq. (5.52) agrees with the Taylor expansion of S( I r ; N) with respect to ψ pk ir . If this is the case, then we just prove that Eq. (5.33) holds at n-point, which completes the inductive proof of Eq. (5.33) and thus Theorem 5.1. To start with, the zeroth order of S( I r ; N) can be extracted by setting all ψ pk ir = 0 in the generating function G ( aµ , I r ; N). According to Eq. (5.26), all ψ pk ir appear only in the outmost level of exponent in G ( aµ , I r ; N) in the form: ∑ n1 ∈ I r a n1 ∑ ψn l = ∑ 1 l ∈ Ir n1 ∈ I r0−1 a n1 ψn1 ir + ∑ l ∈ Ir−1 ! ψn1 l . (5.53) 82 F: solid and dashed edges e solid edges only F: ... p1 pt ... i1 i2 i3 ir Figure 5.3. A classification of spanning forests. Here, we show that an n-point forest rooted in the set {i1 , i2 . . . ir } can be constructed from an (n − 1)-point forest rooted in {i1 , i2 . . . ir−1 , p1 . . . pt }, with { p1 . . . pt } connected to ir . Then by setting all ψn1 ir to zero, the exponent becomes:    ∑ a n1 ∑ ψn l exp(. . .) ∑ = 1 l ∈ Ir n1 ∈ I r a n1 ψn1 ir =0 ∑ ψn1 l exp(. . .) , (5.54) l ∈ Ir−1 n1 ∈ I r0−1 such that G ( aµ , I r ; N) reduces to:  G ( a µ , I r ; N) ψ n1 ir =0  = exp    ∑ ∑ a n1 ∑ ψn1 l exp  an2 ψn2 n1 exp(. . .)   n2 ∈ I r0−1 n2 6 = n1 l ∈ Ir−1 n1 ∈ I r0−1 ! = G ( aµ , I r0−1 ; N0 ) . (5.55) Therefore, the zeroth order in S( I r ; N) agrees with the first term of Eq. (5.52):   ∂ 0 0  ∏  G ( a µ , I r −1 ; N ) O(1) : = S( I r0−1 ; N0 ) . ∂a m m∈ I r0−1 (5.56) a m =0 Next, we go to a generic order t with fixed Pt = { p1 , p2 . . . pt }, and study the Taylor expansion at the order O(ψ p1 ir . . . ψ pt ir ). In this case, the derivative part can be separated into two groups: ∏ m∈ I r   ∂ = ∂am ∂  ∂am ∏ m∈ I r0−1+t t ∂ ∏ ∂a p k k =1 ! . (5.57) Now we can put the outmost exponent into four groups: ∑ n1 ∈ I r a n1 ∑ ψn l = ∑ a n1 1 l ∈ Ir n1 ∈ I r0−1+t t + ∑ k =1 a pk ∑ ψn1 l + l ∈ Ir−1 ∑ l ∈ Ir−1 ∑ an1 ψn1 ir n1 ∈ I r0−1+t t ψ pk l + ∑ a p ψp i k k =1 k r . (5.58) 83 We can first set all ψn1 ir = 0 since they do not contribute to the order O(ψ p1 ir . . . ψ pt ir ) currently under consideration. Then when acting the underlined derivatives in Eq. (5.57) onto G ( aµ , I r ; N), we will reproduce the desired O(ψ p1 ir . . . ψ pt ir ) term only if all of them are acted on the underlined term in Eq. (5.58). All the other ways of distributing the derivatives will result in less ψ pk ir than required, so that they give lower order terms in the expansion. In this sense, we can then define an "effective" generating function by setting all the other a pk = 0 except for the underlined ones in Eq. (5.58): " eff G ( aµ , I r ; N) = exp ∑ ∑ a n1 ∑ a p ψp i k r k an2 ψn2 n1 exp(. . .) n2 ∈ I r0−1+t n2 6 = n1 t + ∑ ψn1 l exp l ∈ Ir−1 n1 ∈ I r0−1+t ! !# ∑ exp an2 ψn2 pk exp(. . .) , (5.59) n2 ∈ I r0−1+t k =1 where in the higher level exponents, we have used, for example: ∑ n2 ∈ I r n2 6 = n1 ∑ n2 ∈ I r n2 6 = n1 an2 ψn2 n1 exp(. . .) an2 ψn2 pk exp(. . .) = a p k =0 = a p k =0 ∑ an2 ψn2 n1 exp(. . .) ∑ an2 ψn2 pk exp(. . .) . n2 ∈ I r0−1+t n2 6 = n1 n2 ∈ I r0−1+t After acting the underlined derivatives in Eq. (5.57) onto this Geff , we get nothing but the generating function G ( aµ , I r0−1+t ; N0 ): t ∂ ∏ ∂a p k k =1 ! t = ∏ ψp i Geff ( aµ , I r ; N) ! " exp k r = ∏ ψp i ! exp k r = ∏ ψp i k r ∑ n2 ∈ I r0−1+t k =1 t " ∑ ψn p a n2 n2 ∈ I r0−1+t k =1 t ∑ t 2 k # " exp(. . .) exp a n1 ∑ ∑ n1 ∈ I r0−1+t k =1 a n1 ∑ # ψn1 l exp(. . .) l ∈ Ir−1 # ψn2 l exp(. . .) l ∈ Ir−1+t ! G ( aµ , I r0−1+t ; N0 ) . (5.60) k =1 To obtain the third line, we have renamed the index n2 to n1 , n3 to n2 , etc, in the first exponential factor. Therefore, at the order O(ψ p1 ir . . . ψ pt ir ), the Taylor expansion of S( I r ; N) agrees with that of Eq. (5.52): 84  O(ψ p1 ir . . . ψ pt ir ) :  ∏ m∈ I r0−1+t  ∂  G ( aµ , I r0−1+t ; N0 ) = S( I r0−1+t ; N0 ) . ∂am (5.61) Since our choice of Pt = { p1 , p2 . . . pt } is completely general, we have just proved that S( I r ; N) has the Taylor expansion Eq. (5.52): S( I r ; N) = S( I r0−1 ; N0 ) + n −r ∑ ∑ t=1 Pt ⊂ I r t ∏ ψp i k r ! S( I r0−1+t ; N0 ) . k =1 This completes our inductive proof of Eq. (5.33) and thus Theorem 5.1. (5.62) CHAPTER 6 SOLUTIONS AND HELICITY CONFIGURATIONS We have already seen in Chapter 4 that the MHV solution Eq. (4.1) alone can reproduce the correct Yang-Mills and gravity MHV amplitudes in four dimensions. We have also suggested in Section 4.4 that all the other solutions do not contribute to MHV amplitudes since they all make rank(C+ ) < n − 3 at MHV. In this chapter, we are going to address a more general problem: how to characterize those solutions that contribute to the Nk MHV amplitudes. Actually, the study in Chapter 4 has already given us a hint: we can use the rank of some matrix. Moreover, this matrix must be closely related to the C matrix defined in Eq. (3.31). The outline of this chapter is as follows. In Section 6.1, we define two discriminant matrices C± and give a general argument that their ranks can link the solutions to SE to helicity configurations. The details on how to characterize the solutions are then presented in Section 6.2. Next, we prove that those solutions that belong to the sector k in this characterization can only support the Nk MHV Yang-Mills and gravity amplitudes, not any other k0 6= k, in Section 6.4. Finally, we give some applications of this technique to single trace EYM amplitudes in Section 6.5. In particular, we can prove from the CHY's perspective that if gluons have the same helicity, then the single trace EYM amplitudes must vanish, independent of the graviton helicities. Many of these results can be found in our work [41]. 6.1 Discriminant matrices We find that the ranks of the following two n × n matrices can be used to characterize the solutions to SE: 86 (C− ) ab  h abi    σ ab = h abi[bq]    − ∑ σ [ aq] ab b6= a a 6= b (C+ ) ab a=b  [ ab]    σ ab = [ ab]hbpi    − ∑ σ h api b6= a ab a 6= b , a=b (6.1) where p and q are two arbitrary reference spinors. Like the Hodges matrix (2.94), the choice of these reference spinors does not affect the diagonal elements as long as {σ} is a solution to SE. For example, with another spinor q̃, we have: h abi[bq̃] h abi[bq̃][ aq] ∑ σab [aq̃] = ∑ σab [aq̃][aq] = ∑ b6= a b6= a b6= a h abi[ ab][qq̃] h abi[bq] + σab [ aq̃][ aq] σab [ aq] = h abi[bq] . σab [ aq] b6= a ∑ (6.2) These two matrices can be understood as the generalization of the Hodges matrix. For a submatrix of C− with rows Ir = {i1 . . . ir } and columns Jr = { j1 . . . jr } deleted, its determinant scales under | ai → t| ai as:   1 a ∈ Ir and a ∈ Jr i1 ...ir r t a ∈ Ir or a ∈ Jr (not both) det[(C− )ij11 ...i ] → det [( C ) ] × − j1 ...jr ...jr  2 t a∈ / Ir and a ∈ / Jr . r If det[(C− )ij11 ...i ...jr ] 6 = 0, the rescaling will not make it vanish. On the other hand, if we have r det[(C− )ij11 ...i ...jr ] = 0 instead, it remains so after the rescaling. Therefore, the rank of C± is invariant under the little group rescaling and thus depends only on the kinematics. According to Eq. (3.31), the C matrix in the CHY integrand Ψ splits into positive and negative helicity parts as: C= C− C+ , (6.3) at general Nk MHV configurations.1 If we choose the polarization vectors as: h a|σµ |q] (e− a )µ = √ 2[qa] h p | σµ | a] (e+ a )µ = √ 2h pai a ∈ N− = {1, 2 . . . k + 2} a ∈ N+ = {k + 3, k + 4 . . . n} , (6.4) the (k + 2) × n matrix C− and (n − k − 2) × n matrix C+ can be related to C± as: (C− ) ab = (C− ) ab [bq] [ aq] a ∈ N− , (C+ ) ab = (C+ ) ab hbpi h api a ∈ N+ . (6.5) 1 Since Pf 0 ( Ψ ) is invariant under permutation, we can always rearrange and relabel the particles into this configuration. 87 In the following discussion, we will use N± to denote the set of positive (negative) helicity particles. The rank of C± interplays with both the rank of C± and the helicity configurations. On one hand, the helicity configuration gives the upper bounds for rank(C± ): rank(C− ) 6 k + 2 rank(C+ ) 6 n − k − 2 . On the other hand, if C± is not of full rank, we must have: rank(C− ) = rank(C− ) if rank(C− ) < k + 2 rank(C+ ) = rank(C+ ) if rank(C+ ) < n − k − 2 . (6.6) The reason is that when calculating the minors of C± , the gauge dependent part can always be pulled out of the determinant. Schematically, we have: ! ! 1 minor (C− ) = ∏ ∏ [bq] minor (C− ) , row [ aq ] column ! ! 1 minor (C+ ) = ∏ ∏ hbpi minor (C+ ) . row h ap i column As a result, if the minor of C± vanishes, the corresponding minor of C± will also vanish, which leads to Eq. (6.6). Altogether, we have: rank(C− ) = min{k + 1, rank(C− )} rank(C+ ) = min{n − k − 2, rank(C+ )} . (6.7) In general, rank(C ) can be smaller that the sum of rank(C± ) if there exists a linear relation between rows in C± . However, we can break such relations by choosing a different gauge, such that we can always make rank(C ) = rank(C− ) + rank(C+ ) 6 n − 2 , (6.8) since C always has two null vectors, as discussed in Section 3.3. By simple observation, we also find that rank(C± ) 6 n − 2 since they respectively have two null vectors independent of the solutions:  null vectors of C− :     [1q] [2q] .. .           [nq]  null vectors of C+ :     h1pi h2pi .. . hnpi [1q]σ1 [2q]σ2 .. .      [nq]σn           h1piσ1 h2piσ2 .. . hnpiσn    .  (6.9) 88 Actually, using Eq. (6.8), we can further prove as follows that rank(C± ) 6 n − 3 for any solution: we choose n − 1 negative helicity particles and only one positive helicity particle, such that in C the C− part has n − 1 rows while the C+ part has only one row. In this case, we must have rank(C+ ) = 1 and consequently rank(C− ) 6 n − 3. Since C− is not of full rank, we must have rank(C− ) = rank(C− ) 6 n − 3. The rank(C+ ) 6 n − 3 can be derived in exactly the same way. We can explicitly check that the two special solutions (4.1) and (4.2) lead to: rank[C− (σ(1) )] = 1 rank[C− (σ(2) )] = n − 3 , rank[C+ (σ(1) )] = n − 3 rank[C+ (σ(2) )] = 1 . (6.10) Thus, the lower bound 1 and the upper bound n − 3 of rank(C± ) can indeed be reached by some solutions. 6.2 Rank characterization The discussion in the previous section enables us to define a partition of the solution set using rank(C− ): solution set = n[ −4 P − (n − 3, m) P − (n − 3, i ) ∩ P − (n − 3, j) = ∅ if i 6= j , (6.11) m =0 such that for each {σ } ∈ P − (n − 3, m), we have rank[C− (σ)] = m + 1 . Similarly, rank(C+ ) gives another partition: solution set = n[ −4 P + (n − 3, m) P + (n − 3, i ) ∩ P + (n − 3, j) = ∅ if i 6= j , (6.12) m =0 such that for each {ω } ∈ P + (n − 3, m), we have rank[C+ (ω )] = m + 1 . Intuitively, these two partitions should coincide with each other in the sense that P + should just be a relabeling of the sets in P − . Our next task is thus to derive this identification. Given a solution {σ} ∈ P − (n − 3, m), we can extract m + 1 linearly independent rows of C− and make them into an (m + 1) × n matrix C− with rank(C− ) = m + 1. Then the C+ 89 part must have co-rank more then two: rank(C+ ) = rank(C+ ) 6 n − m − 3. Namely, the partition P − has the following property: for each solution {σ } ∈ P − (n − 3, m), we must have: rank[C− (σ)] = m + 1 rank[C+ (σ)] 6 n − m − 3 . (6.13) Similarly, for each {ω } ∈ P + (n − 3, m), we can derive: rank[C+ (ω )] = m + 1 rank[C− (ω )] 6 n − m − 3 . (6.14) Using this piece of information, we can prove that Nk MHV Yang-Mills and gravity amplitudes are locked with only one subset of solutions: Theorem 6.1: Only those solutions in the subset P − (n − 3, k ) [or P + (n − 3, k)] can support Pf 0 (Ψ) at Nk MHV (or Nn−k−4 MHV) configurations. This theorem will be proved in the next section. Under the 3 + 1 signature, we have [C− (σ)]∗ = C+ (σ∗ ) since (λi )α and (e λ)α̇ are complex conjugate to each other. Therefore, for each {σ} ∈ P − (n − 3, m), we must have: rank[C+ (σ∗ )] = m + 1 . (6.15) It means that P ∗− (n − 3, m), the complex conjugate of P − (n − 3, m), must be a subset of P + (n − 3, m). We can similarly derive that P ∗+ (n − 3, m) must be a subset of P − (n − 3, m). Therefore, we must have the relation: P ∗− (n − 3, m) = P + (n − 3, m) . (6.16) Then, according to Theorem 6.1, both P − (n − 3, k ) and P + (n − 3, n − k − 4) support the Nk MHV configuration, such that they must equal. Then using Eq. (6.16), we get: P + (n − 3, n − k − 4) = P ∗− (n − 3, n − k − 4) = P − (n − 3, k) . (6.17) This leads to the identification of the two partitions: for all {σ} ∈ P − (n − 3, m) : rank[C− (σ)] = m + 1 rank[C+ (σ)] = n − m − 3 . (6.18) We note that this result can be viewed as a corollary of Theorem 6.1, and it is, of course, not used in the proof of Theorem 6.1. 90 6.3 Rank and Eulerian sectors Now we need to address the last question: how many solutions does each subset P − (n − 3, k ) contain? In this section, we provide a derivation2 showing that this rank characterization can be related to the degree characterization discussed in Appendix C through deg[λα (z)] = d deg[e λα̇ (z)] = n − d − 2 ⇐⇒ rank(C− ) = n − d − 2 , rank(C+ ) = d (6.19) with d = n − k − 3. The number of solutions in each partition is thus |P ± (n − 3, k)| = A(n − 3, d − 1) = A(n − 3, k) . (6.20) We start with a generic solution {σ} that makes deg[λα (z)] = d deg[e λα̇ (z)] = de = n − d − 2 . More specifically, we have d (λi )α = ti λα (σi ) λα (σi ) = ∑ (ρl )α σil , l =0 (e λi )α̇ = de e λα̇ (σi ) ti ∏ j6=i (σi − σj ) e λα̇ (σi ) = ∑ (ρel )α̇ σil , (6.21) l =0 according to Section 2.5 and Appendix C. In the diagonal elements of C− , we have d 1 [ jq] = ∑ [ρl q]σjl = [ρd q](σj − σp1 ) . . . (σj − σpd ) tj l =0 d 1 [iq] = ∑ [ρl q]σil = [ρd q](σi − σp1 ) . . . (σi − σpd ) , ti l =0 (6.22) where {σp1 . . . σpd } are nothing but the d zeros of the degree d polynomial ∑dl=0 [ρl q]zl . As a result, (C− )ii becomes: hijit j σ t j6=i ij i (C− )ii = − ∑ 2 This d (σj − σpa ) ∏ (σi − σp ) . a =1 a derivation is partly inspired by private communication with Freddy Cachazo. (6.23) 91 Now the gauge freedom in (C− )ii can be rephrased as the following: the value of (C− )ii is independent of the choice of the reference points σpa . To prove this, we change σp1 to e σp1 , and then we have: σp1 hijit j σj − e σ t σ −e σp1 j6=i ij i i −∑ d (σj − σpa ) ∏ (σi − σp ) a =2 a σp1 ) d (σj − σpa ) hijit j (σi − σp1 )(σj − e . σ t (σi − σp1 )(σi − e σp1 ) a∏ =2 ( σi − σp a ) j6=i ij i = −∑ (6.24) Using the Schouten identity (σi − σp1 )(σj − e σp1 ) = (σi − σj )(σp1 − e σp1 ) + (σi − e σp1 )(σj − σp1 ) , we can write the second line of the above equation as (C− )ii − σp1 − e σp1 ti (σi − e σp1 ) ∏da=1 (σi − σpa n d j =1 a =2 ∑ hijit j ∏ (σj − σp ) . ) a (6.25) In the summation over j, we replace (e λ j )α̇ t j according to Eq. (6.21), such that we have: hi e λ(σj )i ∏da=2 (σj − σpa ) . ∑ ∏k6= j (σj − σk ) j =1 n (6.26) e while ∏d (σj − σp ) is of degree d − 1 in σj , such In this numerator, hi e λ(σj )i is of degree d, a =2 a that the numerator is a degree de + d − 1 = n − 3 polynomial of σj . Then, according to Eq. (2.139), this summation gives zero. Moreover, this calculation can still go through if we have one more factor σj −σp0 σi −σp0 in (C− )ii . Therefore, our conclusion is that hijit j σ t j6=i ij i (C− )ii = − ∑ d (σj − σpa ) ∏ (σi − σp ) a =0 (6.27) a is independent of the choice of the d + 1 reference points {σp0 . . . σpd }. Eq. (6.23) can then be viewed as a special case of Eq. (6.27) with σp0 → ∞. Now we define another matrix: e ij = (C− )ij ti t j , Φ (6.28) such that e ij = hiji (ti t j ) Φ σij (i 6 = j ) d (σ − σ ) pa e ii = − ∑ hiji (ti t j ) ∏ j Φ . σ a=0 ( σi − σp a ) j6=i ij e ij defined in [27, 28], and it has the same rank as C− . This matrix agrees with the Φ (6.29) 92 Next, we come to prove our main result of this section, Eq. (6.19). First, to prove that e) = n − d − 2, rank(C− ) = rank(Φ e In we only need to find the d + 2 linearly independent vectors that are annihilated by Φ. hindsight, we claim that these vectors are: v j (m) = σjm (0 6 m 6 d + 1) , (6.30) and show that indeed n ∑ Φe ij v j (m) = 0 . (6.31) j =1 First, we have n ∑ Φe ij v j = ∑ Φe ij v j + Φe ii vi j 6 =i j =1 =∑ hijiti t j σjm σij j 6 =i hijiti t j σim −∑ σij j 6 =i d (σj − σpa ) ∏ (σi − σp ) . a =0 (6.32) a In the first term of Eq. (6.32), we substract and add σim to the numerator, such that ∑ hijiti t j σjm σij j 6 =i =∑ hijiti t j (σjm − σim ) σij j 6 =i = ti ∑ j 6 =i hijiti t j σim σij j 6 =i +∑ hi e λ(σj )i(σjm − σim ) σij ∏k6= j (σj − σk ) hijiti t j σim . σij j 6 =i +∑ (6.33) Now in the first term of Eq. (6.33), since lim hi e λ(σj )i = hi e λ(σi )i = hii iti ∏ σij = 0 , σj →σi j 6 =i we can factorize out a factor of (σi − σj ) from it, namely hi e λ(σj )i = (σi − σj )hi e λ? (σj )i det[e λ? (σj )] = n − d − 3 . (6.34) Therefore, the summation in gives ∑ j 6 =i hi e λ(σj )i(σjm − σim ) σij ∏k6= j (σj − σk ) n = ∑ j =1 hi e λ? (σj )i(σjm − σim ) ∏k6= j (σj − σk ) , (6.35) 93 which is zero according to Eq. (2.139) since the numerator always has degree no more than n − 2 in σj . In addition, we can restore j = i in the summation because this numerator vanishes at j = i. Therefore, Eq. (6.33) leads to ∑ hijiti t j σjm σij j 6 =i hijiti t j σim hi e λ(σj )i = ti σim ∑ σij σ (σ − σk ) j 6 =i j6=i ij ∏k 6= j j =∑ hi e λ? (σj )i (σ − σk ) j 6 =i ∏ k 6 = j j = ti σim ∑ = −ti σim hi e λ? (σi )i , ∏k6=i (σi − σk ) (6.36) where we have used again Eq. (2.139) in the last line. This gives the final form of the first term in Eq. (6.32). The second term in Eq. (6.32) can be transformed as hijiti t j σim ∑ σij j 6 =i (σj − σpa ) hi e λ(σj )i ∏ a (σj − σpa ) ti σim = ∏ (σi − σp ) ∏ (σi − σp ) ∑ σij ∏ (σj − σk ) a k6= j a a a =0 j 6 =i d = hi e λ? (σj )i ∏ a (σj − σpa ) ti σim ∑ ∏ a (σi − σpa ) j6=i ∏k6= j (σj − σk ) = −ti σim hi e λ? (σi )i . ∏k6=i (σi − σk ) (6.37) Therefore, the first and second term in (6.32) equal to each other, such that we have proved that n ∑ Φe ij v j (m) = 0 (0 6 m 6 d + 1) . (6.38) j =1 e has d + 2 null vectors v j (m), which leads to Now we have shown that Φ deg[λα (z)] = d deg[e λα̇ (z)] = n − d − 2 e ) = rank(C− ) = n − d − 2 . =⇒ rank(Φ (6.39) The opposite direction also holds since every step can be reversed. For C+ , we can similarly define the matrix Φij = (C+ )ij (ti t j )−1 (6.40) such that Φij = [ij] σij ti t j (i 6 = j ) e [ij] ∏k6=i (σi − σk ) d (σj − σpa ) . σ tt (σ − σk ) a∏ =0 ( σi − σp a ) j6=i ij i j ∏k 6= j j Φii = − ∑ (6.41) 94 This matrix agrees with the Φ in [27, 28]. It has the following independent n − d null vectors: v j (m) = σjm (0 6 m 6 n − d − 1) , ∏k6= j (σj − σk ) (6.42) such that rank(Φ) = rank(C+ ) = d . (6.43) This completes our proof of Eq. (6.19). 6.4 Relating solution sectors to helicity sectors In this section, we give a proof to Theorem 6.1. In the first part of the proof, we consider a solution {σ} ∈ P − (n − 3, m) with m 6 k, which gives C− nonzero co-rank. We will show that Pf 0 (Ψ) 6= 0 only when m = k. For convenience, we use r ≡ m + 1 to denote the rank. Under the gauge choice Eq. (6.4), the matrix Ψ has the following form at Nk MHV:   T −C T A −C− + 0 B . Ψ =  C− (6.44) T C+ −B 0 While C± has already been given in Eq. (6.5), the n × n matrix A is given by: A ab = − h abi[ ab] σab a, b ∈ N , (6.45) and the (k + 2) × (n − k − 2) matrix B is given by: B ab = − h api[bq] [ aq]hbpiσab a ∈ N− b ∈ N+ . Since rank(C− ) = r, we can choose in the C− part an r × r reference matrix:   (C− )i1 j1 (C− )i1 j2 · · · (C− )i1 jr   .. .. R =  ... , . . (C− )ir j1 (6.46) (6.47) (C− )ir j2 · · · (C− )ir jr where the row and column indices are in the set: Ir = { i 1 , i 2 . . . i r } Jr = { j1 , j2 . . . jr } . The determinant of R can be related to the corresponding minor of C− through:   ! (C− )i1 j1 (C− )i1 j2 · · · (C− )i1 jr r [ j q]  .. .. ..  . det[(C− )Ir Jr ] , (C− )Ir Jr ≡  det(R) = ∏ k .  . . [ i q ] k =1 k (C− )ir j1 (C− )ir j2 · · · (C− )ir jr (6.48) 95 Next, we delete the (n − 1)-th and n-th row and column in Ψ. Before any further manipulation, this matrix looks like: n n−2 1,n ψ ≡ Ψnn− −1,n = k+2 n−2 −R T A T −C+ , R 0 (6.49) B −B T 0 C+ where we have already moved R to the upper left corner of C− through permutations. Since R has full rank, we can use it to eliminate all the elements in the gray shaded region in Eq. (6.49) by the following two elementary transformations: ψ → (P 2T P 1 )ψ (P 1T P 2 ) . The (2n − 2) × (2n − 2) matrix P 1 and P 2 have the following form:     1 ( n −2) 1r −x   1r 0  , P1 =  P 2 =  0 1 n −2−r   −y 1 (k+2−r) 1n 1 ( n − k −2) (6.50) (6.51) where 1 (n−2) is the (n − 2) × (n − 2) unit matrix, for example. The matrix x and y are solved from the following two linear equations:   (C− )i1 ,r+1 · · · (C− )i1 ,n−2   .. .. Rx =  , . . (C− )ir ,r+1 · · · (C− )ir ,n−2   (C− )r+1,j1 · · · (C− )r+1,jr   .. .. yR =  , . . (C− )k+2,j1 · · · (C− )k+2,jr (6.52) (6.53) where the underlined and overlined indices are taken from the range: {r + 1, . . . , n − 2} = {1, 2 . . . n − 2}\ Jr , {r + 1, . . . , k + 2} = {1, 2 . . . k + 2}\ Ir . The expression for y = (yij ) will not be used in the following, while the expression for x can be derived from the Cramer's rule: 1 [ aq] jk xik a = det , ca det(Cr− ) [ jk q] i k ∈ Ir , a ∈ {1, 2, . . . , n − 2}\ Jr . (6.54) 96 Here and later in this chapter, we use the notation: j1 j2 · · · c a1 c a2 · · · (6.55) to stand for the matrix obtained from (C− )Ir Jr with the columns     (C− )i1 j1 (C− )i1 j2  (C− ) i j   (C− ) i j  2 1  2 2    c j1 ≡  c ≡    .. .. j2     . . (C− )ir j1 ··· (C− )ir j2 by the columns {c a1 , c a2 . . .}. After the transformation (6.50), the gray shaded region becomes zero as designed. However, the lower right corner of A part also becomes zero, due to the identity: r A ab − ∑ r A ajk xik b + A jk b xik a + k =1 ∑ a, b ∈ {1, 2 . . . n − 2}\Jr . (6.56) xik a A jk js xis b = 0 k,s=1 The detailed proof of this identity can be found in our work [41]. Therefore, after the elementary transformation (6.50), the shape of ψ becomes: n n−2 ψ= k+2 n−2 −R T 0 T −C+ R 0 C+ 0 , (6.57) B −B T 0 where only the shaded regions are in general nonzero. Finally, after using R to make the first r rows and columns entirely zero except for R and −R T themselves, we can factorize the Pfaffian into two parts: −R T R 0 = det(R) Pf 2n − 2 − 2r Pf (ψ ) = Pf n + k − 2r 0 , (6.58) (up to a minus sign) where the dashed lines separate the original B and C parts. However, the elements in these regions in general have very complicated expressions. In this result, the nonzero lower left 97 block has the dimension (n − k − 2) × (n + k − 2r ), which has more columns than rows when r 6 k. Therefore, we must have Pf 0 (Ψ) ∝ Pf (ψ ) = 0 for r 6 k. Only when r = k + 1, we can possibly have a nonzero result: n−k−2 Pf (ψ ) = det(R) det (up to a minus sign). (6.59) Up to now, we have proved that if m 6 k, we have Pf 0 (Ψ) 6= 0 only with m = k. As the second part of the proof, we study the case with m > k. Now the C− part always has the full rank, while the C+ part at least has co-rank 2: rank(C+ ) = r 0 6 n − k − 4 . After permuting the C+ with C− part, we can perform exactly the same operation as the first part on C− , with angular and square brackets exchange. The final result still has the shape of Eq. (6.58), but with different dimensions: upper left zero block : (2n − k − 2r 0 − 4) × (2n − k − 2r 0 − 4) lower left block : (k + 2) × (2n − k − 2r 0 − 4) . With r 0 6 n − k − 4, the lower left block always have more columns than rows such that Pf 0 (Ψ) ∝ Pf (ψ ) = 0 for all m > k. Therefore, we have completed the proof that Pf 0 (Ψ) 6= 0 only with m = k at Nk MHV. The relation between P + and Nn−k−4 MHV can also be prove identically, which completes the proof for Theorem 6.1. 6.5 General single trace Einstein-Yang-Mills amplitudes In the single trance EYM integrand, Pf (ΨH ) shares a very similar structure to Pf 0 (Ψ). Using the technique we have just described, we can also study the support of Pf (ΨH ) on the solution set at different graviton helicity configurations. According to the graviton helicities, we can divide the matrix CH in ΨH into two parts: CH = (CH )− (CH )+ (CH )− : s− × s matrix , (CH )+ : s+ × s matrix (6.60) 98 where s± is the number of positive (negative) helicity gravitons. For a solution {σ} in the subset P − (n − 3, m), the ranks of (CH )± must satisfy: r = rank[(CH )− ] = min{m + 1, s− } r 0 = rank[(CH )+ ] = min{n − m − 3, s+ } . (6.61) Note that we have used Eq. (6.18), the corollary of Theorem 6.1. Then we choose an r × r reference matrix RH in the (CH )− part, and perform the same transformation as in Section 6.4. The result is: Pf (Ψh ) = det(Rh ) Pf 2s − 2r s + s− − 2r 0 (up to a minus sign) . (6.62) The lower left block has the dimension s+ × s + s− − 2r such that there are more columns than rows if r 6 s− − 1. It means that Pf (ΨH ) = 0 if the (CH )− part has a nonzero co-rank. We can also perform the same transformation onto the (CH )+ part, and the result is that Pf (ΨH ) = 0 also when the (CH )+ part has a nonzero co-rank: r 0 6 s+ − 1. Therefore, we have Pf (ΨH ) = 0 if m 6 s− − 2 or m > n − s+ − 2. The support of Pf (ΨH ) is thus: Pf (ΨH ) 6= 0 on the solution set n−[ 3− s + P − (n − 3, m) . (6.63) m = s − −1 In Table 6.1, we show this pattern for n = 7 and n = 8 with all possible graviton helicity configurations. In this table, bold face numbers denote the order of the subsets. The barred subsets are the complex conjugate of the unbarred ones. Knowing the support of Pf (ΨH ), we can very easily prove that if gluons have the same helicity, the single trace EYM amplitudes must vanish. This is actually true for all Nk MHV configurations. Suppose all gluons have positive helicity at Nk MHV, then we must have s− = k + 2 such that the support of Pf (ΨH ) becomes: Pf (ΨH ) 6= 0 on the solution set n−[ 3− s + P − (n − 3, m) if gluons are all-plus. (6.64) m = k +1 Since the set P − (n − 3, k ), the support of Pf 0 (Ψ) at Nk MHV, is not in the support of Pf (ΨH ), we must have Pf (ΨH ) Pf 0 (Ψ) = 0 such that the amplitude must vanish. Similarly, if gluons 99 Table 6.1. The solution sets that support Pf (Ψh ) at different graviton helicity configurations at n = 7 and n = 8. The s+ and s− are the numbers of positive and negative helicity gravitons. The amplitude vanishes if the graviton number is more than n − 2. n = 7, Pf (Ψh ) 6= 0 s− =0 s− =1 s− = 2 s− = 3 s− = 4 s− = 5 ∅ s+ = 0 24 24 11 + 11 + 1 11 + 1 1 s+ 24 24 11 + 11 + 1 11 + 1 1 1 + 11 + 11 1 + 11 + 11 11 + 11 11 11 =1 s+ = 2 s+ =3 1 + 11 1 + 11 s+ =4 1 1 s+ = 5 ∅ n = 8, Pf (Ψh ) 6= 0 s− = 0 s− = 1 s+ = 0 120 120 s+ = 1 120 120 1 + 26 1 + 26 26 + 66 + 26 + 66 + 26 + 66 + 26 1 + 26 + 66 1 + 26 + 66 26 + 66 1 + 26 1 + 26 26 s+ = 5 1 1 s+ ∅ s+ = 2 s+ = 3 s+ =4 =6 s− = 2 s− = 3 26 + 66 66 + 26 + 1 + 26 + 1 26 + 66 66 + 26 + 1 + 26 + 1 66 + 26 s− = 4 s− = 5 s− = 6 26 + 1 1 ∅ 26 + 1 1 26 66 are all-minus, then the number of positive helicity gravitons must be s+ = n − k − 2 such that the support of Pf (ΨH ) becomes: Pf (ΨH ) 6= 0 on the solution set k[ −1 P − (n − 3, m) if gluons are all-minus. (6.65) m = s − −1 Again, the set P − (n − 3, k ) is not in the support of Pf (ΨH ) and this amplitude must also vanish. Therefore, we have completed the proof that the single trace tree amplitudes of EYM with gluons have the same helicity must vanish. This result is first argued in [49] using the factorization properties at soft and collinear limit. Using the BCFW recursive relation [44], we should also be able to give an inductive proof. CHAPTER 7 DISCUSSION AND CONCLUSION We first mention some possible future directions in the CHY formalism. For string theorists, a question they may ask is what the UV completion of CHY is. The textbook bosonic and fermionic strings do not reproduce the CHY formula for amplitudes. In four dimensions, the UV completion may be the topological string as proposed in Witten-RSV. However, CHY is written in a form that is valid in any dimensions, while Witten-RSV, using explicitly the spinor helicity variables, only works in four dimensions.1 The ambitwistor string theory [68, 70], which can be defined in any dimension, seems to be a promising candidate. The field theory limit in the ambitwistor string comes at α0 → ∞, opposite to the usual α0 → 0 limit. One may ask whether there exists any kind of duality between these string theories. Moreover, at strong gauge coupling, we have the famous gauge/gravity, or AdS/CFT duality [88]. At weak gauge coupling, we have instead double copy relations, as we have intensively explored in the thesis. It is then interesting to ask how these relations interplay, or even get integrated into one unified treatment. Another interesting direction is to study the properties of the solutions to SE. Although the SE has a very compact form, it is actually very hard to solve. Except for the two rational solutions (4.1) and (4.2) in four dimensions, the analytical expressions for the other solutions are not available. On the other hand, solving the SE is equivalent to finding the equilibrium positions of a system of two dimensional hard core gas. At this preliminary stage, one can ask whether we can make all the punctures {σ} real in one solution. Along this line of thought, Cachazo's group proposed that the positive kinematics can make all the (n − 3)! solutions real [89]. However, the positive kinematics cannot be realized in the Minkowskian signature, and thus it is not physical. We find that some of the solutions can still be real even inside the physical kinematic space, and it seems that the number of 1 The attempt to prove the equivalence in four dimensions can be found in [64, 87]. 101 real solutions can be used to characterize certain region of the kinematic space bounded by multilinear configurations. This investigation is still ongoing. As to the big picture, we hope the study of scattering amplitude can provide us some hints on the nature of quantum field theory. The best hope is to finally reformulate the field theory in a form that respect the full mathematical structure and symmetry of scattering amplitude. There are some success in our spherical cow: N = 4 super Yang-Mills, while a lot of work still need to be done in more general and less symmetric field theories. The CHY formalism will be very useful since it can cope with a large class of theories beyond the gauge and gravity. To conclude, we have shown that the CHY style direct evaluation can lead to the correct MHV gauge and gravity amplitudes. Each component involved in the CHY integrand can factorize into a gauge independent physical part and an SL(2, C) world sheet gauge dependent part. The SL(2, C) dependence cancels explicitly. Using the same technique, we derived a new formula for the single trace MHV amplitudes for EYM, which is equivalent to, but much simpler than, the SBDW prescription in the literature. Finally, we studied why in four dimensions, solutions can be categorized by the helicity configuration. The link is made explicit by the rank of our discriminant matrices. APPENDIX A PERMUTATIONS IN CHY This appendix is devoted to verify that the reduced determinant det0 (Φ) defined in Eq. (3.27) and the reduced Pfaffian Pf 0 (Ψ) defined in Eq. (3.32) are both invariant under the choice of which rows and column being deleted. As a result, they are both invariant under permutations. We start with the det0 (Φ): Proposition A.1: The reduced determinant det0 (Φ) is independent of the choice on which three rows and columns being deleted. According to the way det0 (Φ) being derived in Section 3.2, it should not depend on such a choice. However, it is instructive to check it afterwards. proof of Proposition A.1. Since Φ is symmetric, it is sufficient to fix the deleted columns {i, j, k} and show that ij,k +1 ijk det Φ pqr det Φ pqr =− σij σjk σki σij σj,k+1 σk+1,i (i < j < k ) , (A.1) since the generic case is generated by succeesive neighboring permutations. Next, we ijk multiply the (k − 2)-th row of Φ pqr by σj,k+1 σk+1,i and define it as Φ0 . Similarly, we multiply ij,k+1 the (k − 2)-th row of Φ pqr by σjk σki and define it as Φ00 . The determinant of these two new matrices satisfy: ijk det(Φ0 ) = σj,k+1 σk+1,i det Φ pqr ij,k +1 det(Φ00 ) = σjk σki det Φ pqr The matrix Φ0 and Φ00 differ only by their (k − 2)-th row: s sk+1,2 sk+1,3 k +1,1 0 ··· [ Φ ] k −2 : σj,k+1 σk+1,i × 2 σk+1,1 σk2+1,2 σk2+1,3 skn sk1 sk2 sk3 00 ··· [ Φ ] k −2 : σjk σki × . 2 2 2 2 σk1 σk2 σk3 σkn . sk+1,n σk2+1,n 2 by σ σ and add it up to the ( k − 2)-th In Φ0 , we multiply the row starting with sm1 /σm1 jm mi row. This operation does not change the determinant while the (k − 2)-th row becomes: 103 2 and s /σ2 do not contain the original diagonal elements, • The columns of smi /σmi mj mj such that the result is: ∑ m6=i,j,k ∑ m6=i,j,k smi s smi s − smi = σij ki + ski = −σjk σki ki2 σjm σmi = ∑ σji 2 σmi σki σmi σki m6=i,j,k skj skj smj smj σjm σmi = ∑ − smj = σji + skj = −σjk σki 2 . σij 2 σmj σkj σmj σkj m6=i,j,k 2 does not contain the origianl diagonal elements either, but it • The column of smk /σmk turns out to be: ∑ σjm σmi m6=i,j,k smk smk smk smk = σ σ + + σ − s σ jk ki ∑ ik mk jk ∑ 2 2 σmk σmk σmk m6=i,j,k σmk m6=i,j,k s = σjk σki ∑ km . 2 m6=k σkm • For the other columns with label l, there must be one original diagonal element Φll in the summation, such that the result is: −σjl σli ∑ m6=l s s s slm + ∑ σjm σmi ml = −σjk σki kl2 + ∑ σjm σmi − σjl σli ml 2 2 2 σlm m6=k,l σml σkl m6=l σml The second term yiels zero since σjm σmi = (σjl − σml )(σml + σli ) and ∑ m6=l sml sml σjm σmi − σjl σli 2 = ∑ (σjl − σli ) − sml = 0 σml σml m6=l After this operation, we find that the (k − 2)-th row of Φ0 becomes the negative of that of Φ00 , while all the other elements are identical. Since this elementary transformation does not change the determinant, we have just proved that: det(Φ0 ) = − det(Φ00 ) . Eq. (A.1) thus follows immediately such that we have proved that det0 (Φ) is independent of the choice of deleted rows and columns. Similar property holds for Pf 0 (Ψ): Proposition A.2: The reduced Pfaffian Pf 0 (Ψ) is independent of the choice on which two rows and columns being deleted, as long as they are in the range 1 to n. 104 Proof of Proposition A.2. The proof is carried out in the same manner as the previous one. It is sufficient to just prove that 1 1 ij i,j+1 Pf Ψij = − Pf Ψi,j+1 σij σi,j+1 (1 < i < j 6 n − 1) . (A.2) ij Then we multiply σi,j+1 to the ( j − 1)-th row and column of Ψij , and similarly multiply σij i,j+1 to the ( j − 1)-th row and column of Ψi,j+1 . The new matrices obtained this way are called Ψ0 and Ψ00 , and their Pfaffians equal to each other: i,j+1 Pf (Ψ00 ) = σij Pf Ψi,j+1 . ij Pf (Ψ0 ) = σi,j+1 Pf Ψij Then Ψ0 and Ψ00 only differ by the ( j − 1)-th row and column. Since both of them are antisymmetric, we only display the rows explicitly: [ Ψ 0 ] j −1 : σi,j+1 × [Ψ00 ] j−1 : σij × a j +1 aj c j +1 cj , where a j and a j+1 are (n − 2)-component row vectors:  aj a j +1 s j1 σj1  =  s j+1,1 σj+1,1 s j2 σj2 ··· s j+1,2 σj+1,2 ··· c s ji σji sd j+1,i σj+1,i s j,j−1 σj,j−1 ··· s j+1,j−1 σj+1,j−1 ··· s j,j+1 σj,j+1 s j+1,j σj+1,j s j,j+2 σj,j+2 ··· s j+1,j+2 σj+1,j+2 ··· s jn σjn s j+1,n σj+1,n   .  Meanwhile, c j and c j+1 are n-component row vectors: (ek)1j  σ1j cj =  (ek )1,j+1 c j +1 σ1,j+1  ··· ··· (ek) j−1,j σj−1,j (ek) j−1,j+1 σj−1,j+1 Σj (ek) j+1,j σj+1,j (ek) j,j+1 σj,j+1 Σ j +1 (ek) j+2,j σj+2,j (ek) j+2,j+1 σj+2,j+1  ···  ,  ··· where we have used the shorthand notation: (ek) ab = −2ea · k b Σa = Now in Ψ0 , we multiply the row started with 2ea · k b (ek) ab =−∑ . σab σab b6= a b6= a ∑ sk1 σk1 by σik , adding it to the ( j − 1)-th row, and then do the same to the columns. The result is: • The a part becomes: σi,j+1 s j+1,l s jl s jl s s + ∑ σik kl = σil ∑ kl − ∑ skl = −σil + s jl = −σij . σj+1,l k6=i,j,j+1,l σkl σ σjl σjl k 6= j,i,l kl k 6= j,i,l Namely, we have σi,j+1 × a j+1 → −σij × a j . 105 • Now we study the c part. For the column containing (ek) j,j+1 /σj,j+1 , there is no diagonal elements in the summation, such that we have: −σi,j+1 2e j · k p 2e j · k p e j · k j +1 − ∑ σip = −σij ∑ + 2e j · k i = −σij Σ j . σj,j+1 σj,p σjp p6=i,j,j+1 p6=i,j For the other columns with label l, there must be one diagonal element Σl involved in the summations, such that the result turns out to be: σil Σl − ∑ p6=i,j,l σip 2el · k j 2el · k j 2el · k j 2el · k p = σil − ∑ 2el · k p = σil + 2el · k j = σij . σl p σlj σlj σlj p6= j,l Therefore, we have σi,j+1 × c j+1 → −σij × c j in the c part. • The ( j − 1)-th column transforms identically as the row. After this operation, we find that Ψ0 and Ψ00 are identical except for a minus sign in the ( j − 2)-th row and column. Since the elementary transformation does not change the Pfaffian, we must have: Pf (Ψ0 ) = −Pf (Ψ00 ) , such that Eq. (A.2) follows immediately. In this way, we have proved that Pf 0 (Ψ) does not depend on the choice of the deleted rows and the corresponding columns. With these two propositions being proved, it is very straightforward to see that both det0 (Φ) and Pf 0 (Ψ) are invariant under permutations. Suppose we exchange two particles a and b, if neither of them belongs to those deleted rows and columns, the permutation amounts to exchange a pair of rows and columns in Φ, which leaves the determinant invariant; if any of a and b coincides with the deleted rows and columns, the permutation amounts to choose another set of deleted rows or columns, which again leaves det0 (Φ) invariant due to Proposition A.1. Exactly the same argument holds to show that Pf 0 (Ψ) is invariant. The only difference is that we now need to exchange two pairs of rows and columns, since each particle index appears twice in Ψ. APPENDIX B THE PARKE-TAYLOR FACTOR AND AMPLITUDE RELATIONS As demonstrated in Appendix A, both det0 (Φ) and Pf 0 (Ψ) in the Yang-Mills integrand are invariant under permutatioins, such that the only ingredient that can encode the amplitude relations is the Parke-Taylor factor (2.137), repeated here as: PT (I ) = 1 1 ≡ , σ12 σ23 . . . σn1 (12)(23) . . . (n1) (B.1) where we have further simplified the notation by using ( ab) ≡ σab . The main purpose of this appendix is to verify that PT indeed manifestly satisfies all the amplitude relations listed in Section 2.2.3. First, the cyclic symmetry and reflection are trivially satisfied as complex number identities: PT (12 . . . n) = PT (2 . . . n1) PT (12 . . . n) = (−1)n PT (n . . . 21) . (B.2) Namely, we do not need to require {σ} be a solution to the SE (3.5). Less trivially, the KK relation is also just a complex number identity of PT, with no requirement on {σ}: Theorem B.1: PT satisfies the KK relation: (−1)|β | PT (1, α , n, β ) = ∑ PT (1, σ , n) (B.3) βT ) σ ∈OP(α ,β for any set of complex numbers {σ} Next, we demonstrate this point by two explicit 6-point calculations. The general n-point proof will be given later. Example 1: We first choose α = {2, 3, 4} and β = {5}. When |β | = 1, the KK relation reduces to the U (1) decoupling identity. The left hand side of Eq. (B.2) becomes: − PT (123465) = − (61) 1 = − PT (12346) × . (12)(23)(34)(46)(65)(51) (65)(51) 107 Next, we rewrite the last factor as:1 ( ab) = ( ac)(cb) Z σb σa dw ≡ (w − σc )2 Z (b) ( a) dw , (wc)2 (B.4) such that we have: − PT (123465) = − PT (12346) Z (1) (6) dw = PT (12346) (w5)2 Z (6) (1) dw . (w5)2 (B.5) For this case, the right hand side of Eq. (B.2) reads (we underline the numbers in β ): PT (123456) + PT (123546) + PT (125346) + PT (152346) (46) (34) (23) (12) = PT (12346) + + + (45)(56) (35)(54) (25)(53) (15)(52) Z (6) Z (4) Z (3) Z (2) dw + + + = PT (12346) (w5)2 (4) (3) (2) (1) = PT (12346) Z (6) (1) dw , (w5)2 (B.6) which indeed equals Eq. (B.5). We do not need to specify the integration path since the integrand has no simple pole. Example 2: Next, we consider α = {2, 3} and β = {5, 4}. This is a less trivial but more illustrative example. The left hand side of Eq. (B.2) becomes: PT (123654) = PT (1236) (61) (51) = PT (1236) (65)(51) (54)(41) Z (6,5) (1,1) Ω(54) , (B.7) where the 2-form Ω(54) is: Ω(54) = dw1 dw2 . ( w1 5 ) 2 ( w2 4 ) 2 (B.8) In particular, the integration area is the cube with principal diagonal (1, 1) ≡ (σ1 , σ1 ) to (6, 5) ≡ (σ6 , σ5 ) in the (w1 , w2 ) space: (1, 5) Z (6,5) (6, 5) = (B.9) (1,1) (1, 1) 1 This (6, 1) identity is first used by Hodges to prove the cancellation of spurious poles in the BCFW results [24]. 108 Now the right hand side of Eq. (B.2) becomes: PT (123456) + PT (124356) + PT (142356) + PT (124536) + PT (142536) + PT (145236) . (B.10) The reason for such an arrangement will be clear soon. Each of these six terms corresponds to an integration of Ω(54): ( A) PT (123456) = PT (1236) ( B) PT (124356) = PT (1236) (C ) PT (142356) = PT (1236) (D) PT (124536) = PT (1236) ( E) PT (142536) = PT (1236) ( F) PT (145236) = PT (1236) Z (6,5) (3,3) Z (6,3) (3,2) Z (6,2) (3,1) Z (3,5) (2,2) Z (3,2) (2,1) Z (2,5) (1,1) Ω(54) Ω(54) Ω(54) Ω(54) Ω(54) Ω(54) . The total integration region is exactly (B.9): (2, 5) (3, 5) (6, 5) ( A) (6, 3) (D) Z ( B) ( F) = = (3, 3) = (2, 2) A+ B+C + D + E+ F (B.11) (6, 2) ( E) (1, 1) (2, 1) = (3, 2) (C ) (3, 1) (6, 1) which establishes the equality between Eq. (B.7) and Eq. (B.10). We note that each line in Eq. (B.10) gives a vertical stripe in Eq. (B.11). Now we are ready for generic cases. Suppose we have α = {α1 , α2 , . . . , αs } and β = { β 1 , β 2 , . . . , β r } with r + s = n − 2, the two examples above inspire us to define the following r-form: Ω( β1 β2 . . . β r ) ≡ r dwi ∏ ( ωi β i )2 . i =1 (B.12) 109 Then the left hand side of Eq. (B.2) just corresponds to the integration of Ω over a cube with principal diagonal (1, 1 . . . 1) to (n, β 1 , β 2 . . . β r−1 ) in the space of (ω1 , ω2 . . . ωr ): (−1)|β | PT (1, α , n, β ) = PT (1, α , n) = PT (1, α , n) ( β 1 1) ( β r −1 1 ) (n1) ··· (nβ 1 )( β 1 1) ( β 1 β 2 )( β 2 1) ( β r−1 β r )( β r 1) Z (n, β1 ...βr−1 ) (1, 1... 1) Ω( β 1 β2 . . . β r ) , (B.13) where the (−1)|β | factor is absorbed by a change of the integral orientation. Again there is no need to specify the integration path since Ω has no simple pole. Therefore, to prove Theorem B.1, we only need to show that the right hand side of Eq. (B.2) gives the same integration. This can be done with the help of the following lemma, to be proved later: Lemma B.1: If we have a string { β r , β r−1 . . . β p } strictly before αi ,2 then ∑ PT (1, σ , αi . . .) = PT (1, α i . . .) σ Z (αi , β p ... βr−1 ) (αi−1 , 1... 1) Ω( β p . . . β r ) , (B.14) where α i = {α1 . . . αi−1 , αi }. The summation of σ is over the order preserved permutations: σ ∈ OP (α i−1 , βe p+1 ) ∪ { β p } , α i −1 = { α 1 . . . α i −1 } βe p+1 = { β r . . . β p+1 } , namely, σ must have β p come last, such that β p must be right before αi . If i = 1, we just define α0 ≡ 1. Finally, if p = r, then the summation reduces to only one term σ = {α1 . . . αi−1 , β p } ( σ = { β p } if i = 1). We now present the proof of Theorem B.1 with the help of Lemma B.1: Proof of Theorem B.1. In Eq. (B.14), we choose p = 1: ∑ PT (1, eσ , αi . . . αs , n) = PT (1, α , n) σ e Z (αi , β1 ...βr−1 ) (αi−1 , 1 ... 1) Ω( β 1 . . . β r ) , (B.15) where e σ ∈ OP(α i−1 , βe2 ) ∪ { β 1 }. It is easy to realize that Eq. (B.15) gives the integration over a stripe with principal diagonal (αi−1 , 1 . . . 1) to (αi , β 1 . . . β r−1 ). In Eq. (B.11), the three vertical stripes ( A + B + C ), ( D + E) and ( F ) are just special examples of our general 2 The "strictly before" means that there must be elements in { β . . . β } between α and α p r i i −1 . It is equivalent to putting β p right before αi . 110 expresseion (B.15). Finally, to sum over σ ∈ OP(α , β T ), we just need to sum over all possible positions that β T can be strictly before: s +1 ∑ PT (1, σ , n) = βT ) σ ∈OP(α ,β ∑ ∑ PT (1, eσ , αi . . . αs , n) i =1 e σ s+1 Z (αi , β 1 ...β r−1 ) = PT (1, α , n) ∑ = PT (1, α , n) i =1 (αi−1 , 1 ... 1) Z (n, β1 ...βr−1 ) Ω( β 1 . . . β r ) Ω( β1 . . . β r ) , (1 ... 1) (B.16) where we have identified that α0 ≡ 1 and αs+1 ≡ n. Again Eq. (B.11) is a good way to visualize how these stripes add up to the cubic region we want. Since Eq. (B.16) exactly equals Eq. (B.13), Theorem B.1 is thus proved. Finally, we come to prove Lemma B.1. The following result will be used in the proof: if a consecutive string of β's is sandwiched between two adjacent α's, we have: PT (. . . αi−1 , β q . . . β p , αi . . .) ( α i −1 β q −1 ) ( α i −1 β q −2 ) ( α i −1 α i ) = PT (. . . αi−1 , αi . . .) ··· (αi−1 β q )( β q β q−1 ) (αi−1 β q−1 )( β q−1 β q−2 ) (αi−1 β p )( β p αi ) = PT (. . . αi−1 , αi . . .) Z (αi , β p ··· β q−1 ) (αi−1 , αi−1 ··· αi−1 ) Ω ( β p β p +1 . . . β q ) . (B.17) Then Lemma B.1 can be proved using induction: Proof of Lemma B.1. We first look at the starting point of the induction. For r − p = 0, Eq. (B.14) reads: PT (1, α i−1 , β r , αi . . .) = PT (1, α i . . .) Z ( αi ) ( α i −1 ) Ω( β r ) . For r − p = 1, Eq. (B.14) reads: i −1 ∑ PT (1, α j−1 , βr , α j . . . βr−1 , αi . . .) + PT (1, α i−1 , βr , βr−1 , αi . . .) j =1 ( α i −1 α i ) = PT (1, α i . . .) (αi−1 β r−1 )( β r−1 αi ) = PT (1, α i . . .) Z ( α i , β r −1 ) ( α i −1 , 1 ) " i −1 ( α j −1 α j ) ( α i −1 β r −1 ) ∑ (α j−1 βr )( βr α j ) + (αi−1 βr )( βr βr−1 ) j =1 Ω ( β r −1 β r ) . Therefore, Eq. (B.14) has been proved for the first two cases. # 111 Now we assume that Eq. (B.14) holds up to an arbitrary r − p. In other words, Eq. (B.14) holds for any βe shorter or equal to βe p+1 . Then we examine the case of r − p + 1. The problem is thus to sum over all order preserved permutations with the string { β r , . . . , β p−1 } strictly before αi , namely, β p−1 is right before αi : ∑ PT (1, σ , αi . . .) σ ∈ OP (α i−1 , βe p ) ∪ { β p−1 } . σ The permutation {1, σ , αi } can be further divided according to: 1. How many β's are between αi and αi−1 ; 2. The αk that the rest β's are strictly before. We can thus rewrite the summation over σ as: ∑ PT (1, σ , αi . . .) = σ r i −1 ∑ ∑ ∑ PT (1, eσ , αk . . . αi−1 , β j . . . β p , β p−1 , αi . . .) , (B.18) j = p −1 k =1 e σ where e σ ∈ OP (α k−1 , βej+2 ) ∪ { β j+1 }. By construction, there is no β between αk and αi−1 , and there is no α between β j and β p−1 . For j = r − 1, the summation over e σ contains only σ = { α 1 . . . α k −1 , β r } ( e σ = { β r } if k = 1). Similarly, for k = 1, the summation over one term e e σ = { β r . . . β j+1 }. Finally, for j = r, the double summation over k σ also reduces to only e and e σ contains only the single term: PT (1, α i−1 , β r . . . β p , β p−1 , αi . . .) . In any situation, the length of βej+2 is at most that of βe p+1 , such that we can use the induction assumption, together with Eq. (B.17), to write: i −1 ∑ ∑ PT (1, eσ , αk . . . αi−1 , β j . . . β p , β p−1 , αi . . .) k =1 e σ = PT (1, α i . . .) = PT (1, α i . . .) Z (αi , β p−1 ... β j−1 ) (αi−1 ... αi−1 ) Ω ( β p −1 . . . β j ) Z (αi , β p−1 ... β j−1 |αi−1 , β j+1 ... βr−1 ) (αi−1 ... αi−1 |1 ... 1) i −1 ∑ Z (αk , β j+1 ... βr−1 ) k=1 (αk−1 , 1 ... 1) Ω ( β j +1 . . . β r ) Ω ( β p −1 . . . β r ) ≡ PT (1, α i . . .)F ( j) . (B.19) The integration is taken in an r − p + 2 dimensional cube, and the vertical line separates the first j − p + 2 and the last r − j coordinates. The last step is to take the j-sum over the F ( j). This can be done using another induction. First, we have: 112 F (r ) + F (r − 1) = = (αi , β p−1 ... β r−2 , β r−1 ) Z (αi−1 ... αi−1 , αi−1 ) Z (αi , β p−1 ... βr−2 , βr−1 ) (αi−1 ... αi−1 , 1) + Z (αi , β p−1 ... βr−2 , αi−1 ) (αi−1 ... αi−1 , 1) Ω ( β p −1 . . . β r ) Ω ( β p −1 . . . β r ) . Now if the sum is taken from j = m, we assume that: r ∑ F ( j) = Z (αi , β p−1 ... β m−1 | β m ... βr−1 ) j=m (αi−1 ... αi−1 |1 ... 1) Ω ( β p −1 . . . β r ) , where the vertical line separates the first m − p + 2 coordinates and the last r − m coordinates. Then adding F (m − 1) gives: r ∑ F ( j) = j = m −1 = Z (αi , β p−1 ... β m−1 | β m ... β r−1 ) + Z (αi , β p−1 ... β m−2 , αi−1 | β m ... βr−1 ) (αi−1 ... αi−1 |1 ... 1) (αi−1 ... αi−1 , 1|1 ... 1) Z (αi , β p−1 ... β m−2 , β m−1 | β m ... βr−1 ) (αi−1 ... αi−1 , 1|1... 1) Ω ( β p −1 . . . β r ) Ω ( β p −1 . . . β r ) , namely, in the lower limit the length of 1's grows by one to the left. Therefore, by induction, we can write down the result: r ∑ F ( j) = Z (αi , β p−1 ... βr−1 ) j = p −1 (αi−1 , 1 ... 1) Ω ( β p −1 . . . β r ) . (B.20) Plugging it into Eq. (B.19) and then Eq. (B.18), we get the final result: ∑ PT (1, σ , αi . . .) = PT (1, α i . . .) σ Z (αi , β p−1 ... βr−1 ) (αi , 1 ... 1) Ω ( β p −1 . . . β r ) , (B.21) where σ = OP (α i−1 , βe p ) ∪ { β p−1 }. Eq. (B.14) and the Lemma B.1 is thus proved by the principle of induction. At last, we show that PT also satisfies the BCJ relation: n i i =3 j =3 ∑ ∑ s j2 ! PT (1 . . . i, 2, i + 1 . . . n) = 0 . (B.22) Unlike the previous cases, the BCJ relation does not hold for arbitrary σ's as a complex number identity. It is true if and only if {σ} is a solution to the SE. Namely, the n-particle kinematics is involved. We present the proof as the following: 113 Proof of the BCJ relation: We can prove the relation by directly rewrite the left hand side of Eq. (B.22) as: n i i =3 j =3 ∑ ∑ s j2 n = ! PT (1 . . . i, 2, i + 1 . . . n) i s j2 ∑ ∑ (13)(34) . . . (i2)(2, i + 1) . . . (n1) i =3 j =3 n i n i s j2 (i, i + 1) (i2)(2, i + 1) i =3 j =3 = PT (134 . . . n) ∑ ∑ = PT (134 . . . n) ∑ ∑ s j2 i =3 j =3 Z ( i +1) (i ) dw . (w2)2 (B.23) We then can change the order of summation: ∑in=3 ∑ij=3 = ∑nj=3 ∑in= j , such that: n i ∑ ∑ s j2 Z ( i +1) (i ) i =3 j =3 dw = (w2)2 n n ∑ ∑ s j2 Z ( i +1) (i ) j =3 i = j n Z (1) j =3 ( j) ∑ s j2 = n = dw (w2)2 dw (w2)2 s j2 σj1 ∑ σj2 σ21 . (B.24) j =3 Using σj1 = σj2 + σ21 , we can easily verify that: n s j2 σj1 ∑ σj2 σ21 = j =3 n n s s j2 j2 + ∑ σ21 ∑ σj2 = j =3 j =3 n s j2 = 0. σ j=1, j6=2 j2 ∑ (B.25) Therefore, we have proved that the BCJ relation: ! n i i =3 j =3 ∑ ∑ s j2 PT (1 . . . i, 2, i + 1 . . . n) = 0 (B.26) holds if and only if the {σ} satisfies the SE. Actually, this relation, first noticed by Cachazo in [31], motivated the establishment of the CHY formalism. APPENDIX C EULERIAN SECTORS In this appendix, we are going to show that in four dimensions, the solutions to SE fall into sectors. The number of solutions in each sector displays an Eulerian number pattern. The 4d spacetime is critical here since we need the spinor helicity formalism to derive this property. The proof is carried out using induction at the soft limit, like the one used to prove that there are in all (n − 3)! solutions to SE shown in Section 3.1.1. We start with the map Pµ (z) defined in Eq. (3.3): µ ∑na=1 k a ∏b6=a (z − σb ) ka = . P (z) = ∑ z − σa ∏na=1 (z − σa ) a =1 µ n µ (C.1) Due to the momentum conservation, the numerator of Pµ (z) is a degree n − 2 polynomial in z. As discussed in Section 3.1, the SE (3.5) is equivalent to imposing the null condition P2 (z) = 0 on the entire Riemann sphere. In four dimensions, we can equivalently impose this null condition by requiring that Pαα̇ (z) = λα (z)e λα̇ (z) , n ∏ a=1 (z − σa ) (C.2) where the degree of λ(z) and e λ(z) should add to n − 2: deg[λα (z)] = d deg[e λα̇ (z)] = n − 2 − d d ∈ {1, 2 . . . n − 3} . (C.3) These two polynomials, together with the puncture positions {σ}, can be determined by µ requiring that the residue of Pµ (z) at σa equals k a : (k a )αα̇ = λα (σa )e λα̇ (σa ) . ∏b6=a (σa − σb ) (C.4) The unknowns of Eq. (C.4) are the σ's, and the coefficients of λ(z) and e λ(z):1 d λα (z) = ∑ m =0 ραm zm n −2− d e λα̇ (z) = ∑ m =0 ρeα̇m zm . (C.5) 1 For generic external momentum data, d = 1 or d = n − 2 are NOT allowed. This would only lead to k a · k b = 0 for any pairs, according to Eq. (C.4). 115 In all, there are 2(d + 1) + 2(n − 1 − d) − 1 = 2n − 1 independent ρ and ρe, taking into account the rescaling freedom in λ and e λ. Together with n − 3 independent σ's after the SL(2, C) gauge fixing, the total number of independent unknowns is 3n − 4. On the other hand, there are in all 4n equations in Eq. (C.4), in which only 3n − 4 are independent.2 Since the number of unknowns exactly matches the number of equations, the unknowns {σ, ρ, ρe} always have solutions. In addition, the set {σ} solved from Eq. (C.4) must agree with that solved from the original SE (3.5). In Section 3.1.1, we proved that the SE has (n − 3)! solutions in {σ}. Since we expect that Eq. (C.4) is equivalent to the original form (3.5), we need to count that Eq. (C.4) also has (n − 3)! solutions. As we are going to show below, this counting naturally put the solutioins into sectors labelled by the degree of the polynomial λ(z), or equivalently, the degree of e λ ( z ). We first look at those solutions that make deg[λ(z)] = d. We assume that the number µ of such solutions is Nn, d . In the soft limit k n → 0, we have: (k n )αα̇ = λα (σn )e λα̇ (σn ) → 0. −1 ∏nb= 1 ( σn − σb ) (C.6) This means we either have λα (σn ) = 0 or e λα̇ (σn ) = 0. For the first case, we can parameterize λα (z) at the soft limit as: λα (z) → (z − σn )λα? (z) det[λα? (z)] = d − 1 . (C.7) If we plug this into the other equations, the outstanding factor (z − σn ) will get cancelled by the same factor in the denominator, such that we get: (k a )αα̇ = λα? (σa )e λα̇ (σa ) λα (σa )e λα̇ (σa ) → −1 ∏nb=1 ,b6=a (σa − σb ) ∏nb= 1 ,b6= a ( σa − σb ) a ∈ {1, 2 . . . n − 1} , (C.8) which is the same set of equations as Eq. (C.4) with n − 1 particles and det[λα? (z)] = d − 1. The number of solutions for Eq. (C.8) is Nn−1, d−1 according to our notation. Next, we come study that for each solution of Eq. (C.8), how many solutions are there for σn . In the soft limit, the spinor helicity form of k n is: (k n )αα̇ = e(λn )α (e λn )α̇ . 2 This is the number of the degrees of freedom in the external data {k a }. 116 Since we have assumed in Eq. (C.6) that (k n )αα̇ → 0 is driven by λα (σn ) → 0, we must have λα (σn ) ∝ e(λn )α and (e λn )α̇ ∝ e λα̇ (σn ) ⇐⇒ λ(σn )i = 0 . he λn e (C.9) Since det[e λ(z)] = n − d − 2, Eq. (C.9) will give n − d − 2 solutions for σn . Therefore, the number of solutions that lead to λα (σn ) → 0 at the soft limit is (n − d − 2)Nn−1, d−1 . Similarly analysis applies to the case e λα̇ (σn ) → 0. We can write e λα̇ (z) → (z − σn )e λα̇? (z) at the soft limit, which leads to: (k a )αα̇ → λα (σa )e λα̇? (σa ) −1 ∏nb= 1 ,b6= a ( σa − σb ) a ∈ {1, 2 . . . n} . (C.10) Since now deg[λα (z)] = d, the solutions to this equation are in the sector Nn−1, d . Then for each solution of Eq. (C.10), the equation for σn is [λn λ(σn )] = 0, which gives d solutions since λ(z) has degree d. Collecting both contributions, we have the following recursive relation for Nn, d : Nn, d = (n − 2 − d)Nn−1, d−1 + dNn−1, d . (C.11) Now Nn, d can be completely determined if we can fix the initial condition at n = 4. First, since we do not allow deg[λ(z)] = 0 or deg[e λ(z)] = 0, which makes all Mandelstam variables vanish, we have N4, 0 = N4, 2 = 0. For d = 1, we can write down the following equations: (ρα0 + ρα1 σ1 )(ρeα̇0 + ρeα̇1 σ1 ) ( λ1 ) α ( e λ1 )α̇ = σ12 σ13 σ14 (ρα0 + ρα1 σ2 )(ρeα̇0 + ρeα̇1 σ2 ) ( λ2 ) α ( e λ2 )α̇ = σ21 σ23 σ24 eα̇0 + ρeα̇1 σ3 ) ( ρ + ρ α1 σ3 )( ρ e 3 )α̇ = α0 ( λ3 ) α ( λ . σ31 σ32 σ34 (C.12) Then we take the spinor inner products and get: s12 = hρe0 ρe1 i[ρ0 ρ1 ] σ13 σ14 σ23 σ24 s23 = hρe0 ρe1 i[ρ0 ρ1 ] , σ21 σ24 σ31 σ34 (C.13) under the SL(2, C) gauge σ2 = 0, σ3 = 1 and σ4 = ∞, we have: − s12 σ σ σ = 24 34 12 → σ1 . s23 σ14 σ24 σ32 (C.14) 117 Namely, we only have one solution σ1 = −s12 /s23 , such that N4, 1 = 1. With this initial condition, the recursive relation generates an Eulerian number pattern: Nn, d = A(n − 3, d − 1) . (C.15) If we sum over the d sectors, we recover the correct counting for the total number of solutions to SE, because of the identity: n −3 ∑ A(n − 3, d − 1) = (n − 3)! . (C.16) d =1 The first few Eulerian numbers series are shown in Table C.1 on the current page. 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