B→πlν semileptonic form factor from three-flavor lattice QCD: a model-independent determination of [Vub]

We calculate the form factor f+(q2) for B-meson semileptonic decay in unquenched lattice QCD with 2 + 1 flavors of light sea quarks. We use Asqtad-improved staggered light quarks and a Fermilab bottom quark on gauge configurations generated by the MILC Collaboration. We simulate with several light-quark masses and at two lattice spacings, and extrapolate to the physical quark mass and continuum limit using heavy-light meson staggered chiral perturbation theory. We then fit the lattice result for f+(q2) simultaneously with that measured by the BABAR experiment using a parameterization of the form-factor shape in q2, which relies only on analyticity and unitarity in order to determine the Cabibbo-Kobayashi-Maskawa matrix element [Vub]. This approach reduces the total uncertainty in [Vub] by combining the lattice and experimental information in an optimal, model-independent manner.We find a value of [Vub] X 103 = 3:38 + 0:36.

PHYSICAL REVIEW D 79, 054507 (2009) B -* ttI v semileptonic form factor from three-flavor lattice QCD: A model-independent determination of |FUbl Jon A. Bailey,1 C. Bernard,2 C. DeTar,J M. Di Pierro,4 A. X. El-Khadra,5 R. T. Evans,5 E. D. Freeland,6 E. Gamiz,5 Steven Gottlieb,7 U. M. Heller,8 J. E. Hetrick,9 A. S. Kronfeld,1 J. Laiho,2 L. Levkova,J P. B. Mackenzie,1 M. Okamoto,1 J. N. Simone,1 R. Sugar,10 D. Toussaint,11 and R.S. Van de Water1'* (Fermilab Lattice and MILC Collaborations) 1 Fenni National Accelerator Laboratory, Batavia, Illinois, USA 1 Department of Phvsics, Washington University, St. Louis, Missouri, USA 3Physics Department, University of Utah, Salt Lake City, Utah, USA 4School of Computer Sci., Telecom, and Info. Systems, DePaul University, Chicago, Illinois, USA 5Physics Department, University of Illinois, Urbana, Illinois, USA 6Liberal Arts Department, The School of the Art Institute of Chicago, Chicago, Illinois, USA 7Department of Physics, Indiana University, Bloomington, Indiana, USA 8American Physical Society, One Research Road, Ridge, New York, USA 9Physics Department, University of the Pacific, Stockton, California, USA 10Department of Physics, University of California, Santa Barbara, California, USA 11 Department of Phvsics, Universitv of Arizona, Tucson, Arizona, USA (Received 20_ December 2008; published 30 March 2009) We calculate the form factor f+(q2) for B-meson semileptonic decay in unquenched lattice QCD with 2 + 1 flavors of light sea quarks. We use Asqtad-improved staggered light quarks and a Fermilab bottom quark on gauge configurations generated by the MILC Collaboration. We simulate with several light- quark masses and at two lattice spacings, and extrapolate to the physical quark mass and continuum limit using heavy-light meson staggered chiral perturbation theory. We then fit the lattice result for f+(q2) simultaneously with that measured by the BABAR experiment using a parameterization of the form-factor shape in q2, which relies only on analyticity and unitarity in order to determine the Cabibbo-Kobayashi- Maskawa matrix element |Vub|. This approach reduces the total uncertainty in |Vub| by combining the lattice and experimental information in an optimal, model-independent manner. We find a value of | Vub | X 103 = 3.38 ± 0.36. DOI: 10.1103/PhysRevD.79.054507 PACS numbers: 12.38.Gc, 12.15.Hh, 13.20.He I. INTRODUCTION The semileptonic decay B -> rriv is a sensitive probe of the heavy-to-light-quark-flavor-changing b -> u transition. When combined with an experimental measurement of the differential decay rate, a precise QCD determination of the B -> Trtv form-factor allows a clean determination of the Cabibbo-Kobayashi-Maskawa (CKM) matrix element |Vub| with all sources of systematic uncertainty under control. In the standard model, the differential decay rate for this process is dns-vt?) G!fivj!r, ,, , , ,,,,, -i?- - i9SS5[(m«+ m' -rY ~ x l/+(r)l2. (i) where q = pB - is the momentum transferred from the B meson to the outgoing leptons. The form factor f+(q2) parameterizes the hadronic contribution to the weak decay, * [e-mail] and must be calculated nonperturbatively from first prin­ciples using lattice QCD. A precise knowledge of CKM matrix elements such as |Vub| is important not only because they are fundamental parameters of the standard model, but because inconsis­tencies between independent determinations of the CKM matrix elements and CP-violating phase would provide evidence for new physics. Although the standard model has been amazingly successful in describing the outcome of most particle physics experiments to date, it cannot account for gravity, dark matter, and dark energy, or the large matter-antimatter asymmetry of the Universe. Thus, we know that it is incomplete, and expect new physics to affect the quark-flavor sector to some degree, although we do not know a priori what experimental and theoretical precision will be needed to observe it. The determination of |Vub| from B -> tt€v semileptonic decay relies upon the assumption that, because the leading standard model decay amplitude is mediated by tree-level W-boson exchange, it will not be significantly affected by new physics at the current level of achievable precision. Recently, however, hints of new physics have appeared in 1550-7998/ 2009 /79(5 )/054507(27) 054507-1 © 2009 The American Physical SocietyJON A. BAILEY et al. PHYSICAL REVIEW D 79, 054507 (2009) various regions of the heavy-quark-flavor sector such as CP asymmetries in Z? -*> Ktt TIL constraints on sin(2/3) from AF = 2 neutral meson mixing and 1-loop penguin- induced decays [2], and the phase of the Bs-mixing ampli­tude [3-5]. The unexpected inconsistency most relevant to our new lattice QCD calculation of the B ->«■ tt(v form factor and |Vub| is the current "/0j puzzle" [6]. The HPQCD Collaboration's lattice QCD calculation of the Ds-meson leptonic decay constant fn [7] disagrees with the latest results from the Belle, BABAR, and CLEO ex­periments [8-121 at the 3-a level, although HPQCD's determinations of the masses mD+ and mn^ and the decay constants /„., /*-, and fD+ all agree quite well with experimental measurements [13,141. Furthermore, because the significance of the discrepancy is dominated by the statistical experimental uncertainties, it cannot easily be explained by an underestimate of the theoretical uncertain­ties. Additional lattice QCD calculations ofare needed to either confirm or reduce the inconsistency. If the dis­agreement holds up, however, it is evidence for a large new physics contribution to a tree-level standard model process at the few percent level. Therefore, although B ->«■ tt(v semileptonic decay provides a theoretically clean method for determining | Vub | within the framework of the standard model, we should keep in mind that new physics could appear in b -> u transitions. Understanding and controlling all sources of systematic uncertainty in lattice QCD calculations of hadronic weak matrix elements is essential in order to allow accurate determinations of standard model parameters and reliable searches for new physics. The hadronic amplitudes for B -> rrfv, in particular, can be calculated accurately using current lattice QCD methods because the decay process is "gold plated," i.e., there is only a single stable hadron in both the initial and final states. Lattice calculations with staggered quarks allow for realistic QCD simulations with dynamical quarks as light as mj 10, multiple lattice spac- ings, large physical volumes, and high statistics. The re­sulting simulations of many light-light and heavy-light meson quantities with dynamical staggered quarks are in excellent numerical agreement with experimental results [151. This includes both postdictions, such as the pion decay constant [161, and predictions, as in the case of the Bc meson mass [171. Such successes show that the system­atic uncertainties in these lattice QCD calculations are under control, and give confidence that additional calcu­lations using the same methods are reliable. The publicly available MILC gauge configurations with three flavors of improved staggered quarks [181 that have enabled these precise lattice calculations make use of the "fourth-root" procedure for removing the undesired four­fold degeneracy of staggered lattice fermions. Although this procedure has not been rigorously proven correct, Shamir uses plausible assumptions to argue that the con­tinuum limit of the rooted theory is in the same universality class as QCD [19,201. The rooting procedure leads to violations of unitarity that vanish in the continuum limit; both theoretical arguments [21,221 and numerical simula­tions [23-251, however, show that the unitarity-violating lattice artifacts in the pseudo-Goldstone boson sector can be described and hence removed using rooted staggered chiral perturbation theory (rS^PT), the low-energy effec­tive description of the rooted staggered lattice theory [26­281. Given the wealth of numerical and analytical evidence supporting the validity of the rooting procedure, most of which is reviewed in Refs. [29-311, we work under the plausible assumption that the continuum limit of the rooted staggered theory is QCD. We note, however, that it is important to have cross-checks of lattice calculations of phenomenologically important quantities using a variety of fermion formulations, since they all have different sources of systematic uncertainty. Both existing unquenched lattice calculations of the B -> tt(v form factor use the MILC configurations. When combined with the Heavy Flavor Averaging Group's latest determination of the experimental decay rate from ICHEP 2008 [321, they yield the following values for |Vub|: I Vub I X 103 = 3.40 ± 0.201^9 HPQCD [33]. (2) I Vub I X 103 = 3.62 ± 0.22:;];® Fermilab-MILC [34]. (3) where the errors are experimental and theoretical, respec­tively. Both analyses primarily rely upon data generated at a "coarse" lattice spacing of a ~ 0.12 fm, and use a smaller amount of "fine" data at a ~ 0.09 fm to check the estimate of discretization errors. Neither is therefore able to extrapolate the B -> tt(.v form factor to the con­tinuum (a -> 0). The most significant difference in the two calculations is their use of different lattice formulations for the bottom quarks. The HPQCD Collaboration [33] uses nonrelativistic heavy quarks [34], whereas we use relativ- istic clover quarks with the Fermilab interpretation [35] via heavy-quark effective theory (HQET) [36-38]. Both meth­ods work quite well for heavy bottom quarks. The Fermilab treatment, however, has the advantage that it can also be applied to charm quarks; we can therefore use the same method for other semileptonic form factors such as D ->«■ ttCv, D - Ktv, and B-> D*tv [39,40]. The two un­quenched lattice calculations of the B -> tt(.v form factor, which have largely independent sources of systematic uncertainty, nevertheless lead to consistent values of | VubI with similar total errors of ~15%. In this paper we present a new model-independent un­quenched lattice QCD calculation of the B -> tt(.v semi­leptonic form factor and |Vub|. Our work builds upon the previous Fermilab-MILC calculation and improves upon it in several ways. We now include data on both the coarse and fine MILC lattices, and can therefore take the a-* 0 054507-2 B - ttCv SEMILEPTONIC FORM FACTOR FROM PHYSICAL REVIEW D 79, 054507 (2009) limit of our data, which is generated at nonzero lattice spacing. We also have additional statistics on a subset of the coarse ensembles. The most important improvements, however, are in the analysis procedures. We have removed all model-dependent assumptions about the shape in q2 of the form factor from the current analysis. Our result is therefore theoretically cleaner and more reliable than those of previous lattice QCD calcula­tions. The first refinement over previous unquenched lat­tice B-* ttC.v form-factor calculations is in the treatment of the chiral and continuum extrapolations. We simulta­neously extrapolate to physical quark masses and zero lattice spacing and interpolate in the momentum transfer q2 by performing a single fit to our entire data set (all values of m,r a, and q2) guided by functional forms derived in heavy-light meson staggered chiral perturbation theory (HMS^PT) [41], We thereby extract the physical form factor j\{q2) in a controlled manner without introducing a particular ansatz for the form factor's q2 dependence. The second refinement over previous unquenched B-> lattice form-factor calculations is in the combination of the lattice form-factor result and experimental data for the decay rate to determine the CKM matrix element | \/ub|. We fit our lattice numerical Monte Carlo data and the 12-bin BABAR experimental data [42] together to the model- independent expansion" of the form factor given in Ref. [43], in which the form factor is described by a power series in a small quantity ; with the sum of the squares of the series coefficients bounded by unitarity constraints. We leave the relative normalization factor | V7ub | as a free parameter to be determined by the fit, thereby extracting | \/ub | in an optimal, model-independent way. Others have also fit lattice and experimental results together using different, equally valid, parameterizations [44,45]. This work, however, is the first to use the full correlation ma­trices, derived directly from the data, for both the lattice calculation and experimental measurement. This paper is organized as follows: In Sec. II, we de­scribe the details of our numerical simulations. We discuss the gluon, light-quark, and heavy-quark lattice actions, and present the parameters used, such as the quark masses and lattice spacings. We then define the matrix elements needed to calculate the semileptonic form factors and discuss the method for matching the lattice heavy-light current to the continuum. Next, we describe our analysis for determining the form factors in Sec. ITT. This is a three- step procedure. We first fit pion and B-meson 2-point correlation functions to extract the meson masses. We then fit the B -► tt 3-point function, using the masses and amplitudes from the 2-point fits as input, to extract the lattice form factors at each value of the light-quark mass and lattice spacing. Finally, we extrapolate the results at unphysical quark masses and nonzero lattice spacing to the physical light-quark masses and zero lattice spacing using HMS^PT. In Sec. IV, we estimate the contributions of the various systematic uncertainties to the form factors, discussing each item in the error budget separately. We then present the final result for j\{q2) with a detailed breakdown of the error by source in each q2 bin. We combine our result for the form factor with experimental data from the BABAR Collaboration to determine the CKM matrix element | V7ub | in Sec. V. We also define the model- independent description of the form-factor shape that we use in the fit and discuss alternative parameterizations of the form factor. Filially, in Sec. VI, we compare our results with those of previous unquenched lattice calculations. We also compare our determination of | Vub | with inclusive determinations and to the preferred values from the global CKM unitarity triangle analysis. We conclude by discus­sing the prospects for improvements in our calculation and its impact oil searches for new physics in the quark-flavor sector. II. LATTICE CALCULATION In this section we describe the details of our numerical lattice simulations. We first present the actions and pa­rameters used for the light (up, down, strange) and heavy (bottom) quarks in Sec. II A. We then define the procedure for constructing lattice correlation functions with both staggered light and Wilson heavy quarks in Sec. TIB. Finally, in Sec. IIC, we show how to match the lattice heavy-light vector currents to the continuum with a mostly nonperturbative method, so that lattice perturbation theory is only needed to estimate a small correction. A. Actions and parameters We use the ensembles of lattice gauge fields generated by the MILC Collaboration and described in Ref. [18] at two lattice spacings, a ~ 0.12 and 0.09 fm, in our numeri­cal lattice simulations of the B -► tt(.v semileptonic form factor. The ensembles include the effects of three dynami­cal staggered quarks-two degenerate light quarks with masses ranging from mj8 - mj2 and one heavier quark tuned to within 10-30% of the physical strange quark mass. This allows us to perform a controlled extrapolation to both the continuum and the physical average u-d quark mass. The physical lattice volumes are all sufficiently large (m^L s 4) to ensure that effects due to the finite spatial extent remain small. For each independent ensemble we compute the light valence quark in the 2-point and 3-point correlation func­tions only at the same mass, mh as the light quark in the sea sector. Thus all of our simulations are at the "full QCD" point. Note, however, that we still have many correlated data points on each ensemble because of the multiple pion energies. Table I shows the combinations of lattice spac- ings, lattice volumes, and quark masses used in our calculation. For bottom quarks in 2-point and 3-point correlation functions we use the Sheikholeslami-Wohlert (SW) "clo­054507- 3 JON A. BAILEY et al. PHYSICAL REVIEW D 79, 054507 (2009) TABLE I. Lattice simulation parameters. The columns from left to right are the approximate lattice spacing in fm, the bare light- quark masses a/n(/ams, the linear spatial dimension of the lattice in fm, the dimensionless factor m^L (corresponding to the taste- pseudoscalar pion composed of light sea quarks), the dimensions of the lattice in lattice units, the number of configurations used for this analysis, the clover term csw and bare k value used to generate the bottom quark, and the improvement coefficient used to rotate the bottom quark field in the b -» u vector current. a (fm) ami/ams L (fm) m^L Volume # Configs. £'sw «b d\ 0.09 0.0062/0.031 2.4 4.1 283 X 96 557 1.476 0.0923 0.09474 0.09 0.0124/0.031 2.4 5.8 283 X 96 518 1.476 0.0923 0.09469 0.12 0.005/0.05 2.9 3.8 243 X 64 529 1.72 0.086 0.09 372 0.12 0.007/0.05 2.4 3.8 203 X 64 836 1.72 0.086 0.09 372 0.12 0.01/0.05 2.4 4.5 203 X 64 592 1.72 0.086 0.09 384 0.12 0.02/0.05 2.4 6.2 203 X 64 460 1.72 0.086 0.09 368 ver" action [46] with the Fermilab interpretation via HQET [35,36], which is well suited for heavy quarks, even when amg ^ 1. Because the spin-flavor symmetry of heavy-quark systems is respected by the lattice regula­tor, the expansion in \/mg of the heavy-quark lattice action has the same form as the 1 /mg expansion of the heavy-quark part of the QCD action. Discretization effects in the lattice heavy-quark action are therefore parameter­ized order by order in the heavy-quark expansion by devi­ations of effective operator coefficients from their values in continuum QCD. Thus, in principle, the lattice heavy- quark action can be improved to arbitrarily high orders in HQET by tuning a sufficiently large number of parameters in the lattice action. In practice, we tune the hopping parameter k, and the clover coefficient csw, of the SW action, to remove discretization effects through next-to- leading order (NLO), 0{\/mg), in the heavy-quark expansion. The SW action includes a dimension-five interaction with a coupling csw that must be adjusted to normalize the heavy quark's chromomagnetic moment correctly [35]. In our calculation we set the value of csw = «q3, as suggested by tadpole-improved, tree-level perturbation theory [47]. We determine the value of u0 either from the plaquette {a ~ 0.09 fm) or from the Landau link (a ~ 0.12 fm). The tadpole-improved bare quark mass for SW quarks is given by 1 (1 M amQ=-[--------1, (4) «0 \2k 2/ccrit/ such that tuning the parameter k to the critical quark hopping parameter /ccrjt leads to a massless pion. Before generating the correlation functions needed for the B -> 7t(.v form factor, we compute the spin-averaged Bs kinetic mass on a subset of the available ensembles in order to tune the bare k value for bottom (and hence the corresponding bare quark mass) to its physical value [35]. We then use the tuned value of Kb for the B -> ir(v form-factor production runs. Table I shows the values of the clover coefficient and tuned Kb used in our calculation. In order to take advantage of the improved action in the calculation of the B -> ir(v form factor, we must also improve the flavor-changing vector current to the same order in the heavy-quark expansion. We remove errors of 0(1 /mg) in the vector current by rotating the heavy-quark field used in the matrix element calculation as 4>b = (1 + ad^y- D]at)if/b, (5) where Dhl is the symmetric, nearest-neighbor, covariant difference operator. We set d\ to its tadpole-improved tree- level value of [35] «0 \2 + m0a 2(1 + m0a)/' The values of the rotation parameter used in our calculation are given in Table I. In order to convert dimensionful quantities determined in our lattice simulations into physical units, we need to know the value of the lattice spacing, a, which we find by computing a physical quantity that can be compared di­rectly with experiment. We first determine the relative lattice scale by calculating the ratio r^/a on each en­semble, where is related to the force between static quarks, = 1.0 [48]. These r^/a estimates are then smoothed by fitting to a smooth function of the gauge coupling and quark masses. This scale-setting method has the advantage that the ratio r^/a can be determined pre­cisely from a fit to the static quark potential [49,50]. We convert all of our data from lattice spacing units into units before performing any chiral fits in order to account for slight differences in the value of the lattice spacing between ensembles. In this work we use the value of rphys _ 0.3108(15)(i"g) obtained by combining a recent lattice determination of rhf^ [51] with the PDG value of fw = 130.7 ± 0.1 ± 0.36 McV [52] to convert lattice re­sults from f] units to physical units. B. Heavy-light meson correlation functions The B -> it I v semileptonic form factors parameterize the hadronic matrix element of the b -> u quark flavor- 054507-4 B - itCp SEMILEPTONIC FORM FACTOR FROM ... changing vector current 'V** = iuy^b: {tt\Y»\B) = f+(q2)(p% + p* ~ mi m% , t. m% + Mg2) - mt -gM- (7) where q2 is the momentum transferred to the outgoing lepton pair. For calculations oil the lattice and in HQET, it is more convenient to write the matrix element as [53] (7r\y^\B) = y/2m^[v>* + P±.f±(E„)]. (8) where v>* = pB/mB is the four-velocity of the B meson, P± = P% - (Pn ' v)v>i is the component of the pion mo­mentum orthogonal to v. and Ew = pw ■ v = (m2B + - q2)/(2mB) is the energy of the pion in the B-meson rest frame (pB = 0). In this frame the form factors /||(EJ and f±(E„) are directly proportional to the hadronic matrix elements /|| (EJ w\y°\B) •J2 mB <My‘\B) i Pv (9) (10) We therefore first calculate the hadronic matrix elements in Eqs. (9) and (10) in the rest frame of the B meson to obtain f\\{E„) and f±(E1T). and then extract the standard form factors fo(q2) and f+(q2) using the following relations: Mg2) f+(g2) •J2 mB mi mz ■[(mB - EJMEJ + (El - mDf^Ej], (11) ■J2 mB [/l,(Ej + (mB ~ Ej/±(Ej]. (12) These relations automatically satisfy the kinematic con­straint /+(0) = /o(0). The 2-point and 3-point correlation functions needed to extract the lattice matrix element for B -> ir( v decay are CTO: pj = 0)0l(t. -?)). (13) X Cf(r) = X<^(O.6)0l(r.x)). (14) X T■ P-rr) = °. 0)VM (t. y)Ot(T..?)). x,y (15) where 0B and Ow are interpolating operators for the B meson and pion, respectively, and VM is the heavy-light vector current on the lattice. In practice, to construct the heavy-light bilinears we must combine a staggered light quark, which is a 1- componeiit spinor, with a 4-component Wilson-type bot­tom quark; we do so using the method established by Wingate et al. in Ref. [54]. For the B meson we use a mixed-action interpolating operator: ®b. h(-0 = ij'cl(x)yiptopz.(x)x(x)' (16) where a, /3 are spin indices and il(x) = To'Ti' Y^yV- The fields ;// and x are the 4-component clover quark field and 1-component staggered field, respectively. Based on the transformation properties of ObB(x) under shifts by one lattice spacing, E plays the role of a (fermionic) taste index [31,54]. Once ObB(x) is summed over 24 hypercubes in the correlation functions that we compute, E also takes on the role of a taste degree of freedom, in the sense of Refs. [55,56]. Because the heavy-quark field ^a(x) is slowly varying over a hypercube, it does not affect the construction of Refs. [55,56]. For the pion we use the local pseudoscalar interpolating operator PHYSICAL REVIEW D 79, 054507 (2009) Ow(x) = s(x)x(x)x(x). (17) where e(.r) = (-1 )( V|+v2+v>+v+)_ \ye take the vector current to be V^(x) = Va(x)yZpilf)B(x)x(x). (18) where 'P is the rotated heavy-quark field given in Eq. (5). When forming Cf (?) and C?^(t, T; p„), we sum over the taste index. This yields the same correlation functions, with respect to taste, as in Ref. [54]. Our principal differ­ence with Ref. [54] is to use 4-component heavy quarks instead of 2-component lioiirelativistic (NRQCD) quarks, and to derive the correlators in the staggered formalism, without the introduction of naive fermions. We work in the rest frame of the B meson, so only the pious carry momentum. We compute both the 2-point function CJ(t: pw) and the 3-point function T; p„) at discrete values of the momenta p„ = 2-77(0, 0. 0)/L. 277(1.0. 0)/L.277(1. 1. 0)/L. 2ir(1. 1. 1 )/L, and 2-77(2, 0, 0)/L allowed by the finite spatial lattice vol­ume. We use only data through momentum pw = 2-77(1, 1, 1 )/L. however, because the statistical errors in the correlators increase significantly with momentum. We use a local source for the pious throughout the calculation, while we smear the B-meson wave function in both the 2-point function C?(?) and the 3-point function Cf^(t, T: pj: 0B.=(?. x) = ^S(y)ij/a(t. x + y)yipilpB({- *)■ y (19) where S(y) is the spatial smearing function. This reduces contamination from heavier excited states and allows a 054507-5 JON A. BAILEY et al. PHYSICAL REVIEW D 79, 054507 (2009) better determination of the desired ground-state amplitude. In our study of choices for how to smear the B meson, we found that a wall source, S{y) = 1, worked extremely well for suppressing excited-state contamination, but at the cost of large statistical errors in the 2-point and 3-point corre­lation functions. In contrast, use of a IS wave function S{y) = cxp( -/i|v|), optimized to have good overlap with the charmonium ground state led to smaller statistical errors at the cost of undesirably large excited-state contri­butions to the 3-point function that would make it difficult to extract the ground-state amplitude. In order to achieve a balance between small statistical errors and minimal excited-state contamination, we tune the coefficient of the exponential in the IS wavefunction to the smallest value (i.e., the widest smearing) for which the 5-meson 2-point effective mass is still well behaved; we find a value of cifji = 0.20 for the coarse ensembles. We note that our determination of the optimal 5-meson smearing function is consistent with the theoretical expectation that the 5-meson wave function should be wider than the charmo­nium wave function. For the calculation of the 3-point function, we fix the location of the pion source at f, = 0 and the location of the 5-meson sink at tf = T, and vary the position of the operator over all times t in between. If the source-sink separation is too small, then the entire time range 0 < t < T is contaminated by excited states, but if the source-sink separation is too large, then the correlation function be­comes extremely noisy. In practice, we set the sink time to T = 16 on the coarse lattices; we have checked, how­ever, that our results using this choice are consistent with those determined from using T = 12 and T = 20. On the fine lattices we scale the source-sink separation by the approximate ratio of the lattice spacings anne/«coarse aild use T = 24. In order to minimize the statistical errors given the available number of configurations in each ensemble, we compute the necessary 2-point and 3-point correlations not only with a source time of tj = 0, but also with source times of f, = rt,/4, nj2, and 3n,/4 (n, is the temporal extent of the lattice) and the sink time T shifted accord­ingly. We then average the results from the four source times; this effectively increases our statistics by a factor of 4. C. Heavy-light current renormalization In order to recover the desired continuum matrix ele­ment, the lattice amplitude must be multiplied by the appropriate renormalization factor Z1^: <tt|%|B) = Z« X<7t|^|B), (20) where VM and are the lattice and continuum b -► u vector currents, respectively. This removes the dominant discretization errors from the lattice current operator. In terms of the form factors, Eq. (20) can be rewritten as /||=41x/if' (21) f±=ZhvltXf'*, (22) where explicit expressions relating /j"l and /j" to correla­tion functions are given in Eqs. (40) and (41). In this work, we calculate Zy ( via the mostly nonper­turbative method used in the earlier quenched Fermilab calculation [531. We first rewrite Zy as V [X A\, = p'D^Z^Z". (23) The flavor-conserving renormalization factors Zyb and Zy account for most of the value of Zy [37]. They can be determined from standard heavy-light meson charge nor­malization conditions Z% X (D\Vilo\D) = 1, (24) Zbvb X {B\Vbb-°\B) = 1, (25) where the light-light and heavy-heavy lattice vector cur­rents are given by = A'tU)^U)|a.7a/3ilW/3='A'U). (26) V^.m{x) = %a(x)y^hp(x), (27) respectively. In order to reduce the statistical errors in Z^, we compute the lattice matrix element (D\VII-°\D) using a clover charm quark as the spectator in the 3-point correla­tion function. We eliminate contamination from staggered oscillating states in the determination of Zyb by using a clover strange quark for the spectator in the 3-point corre­lation function (B\Vbb0\B). Once Z$ and Z!yb have been determined nonperturbatively, the remaining correction factor in Eq. (23), py , is expected to be close to unity because most of the radiative corrections, including con­tributions from tadpole graphs, cancel in the ratio [37]. We therefore estimate py from 1-loop lattice perturbation theory [47]. The matching factor pty has been calculated by a subset of the present authors, and a separate publication describ­ing the details is in preparation [57]. The corrections to py ^ can be expressed as a perturbative series expansion in powers of the strong coupling Pv/t = 1 + 4Trav{q*)py^] + O(ap). (28) where av(cf") is the renormalized coupling constant in the V-scheme and is determined from the static quark potential with the same procedure as is used in Ref. [58]. The scale £/*, which should be the size of a typical gluon loop momentum, is computed via an extension of the methods outlined by Brodsky, Lepage, and Mackenzie [47,59] and Hornbostel, Lepage, and Morningstar [60]. The value of <f 054507-6 B - 7 7 (/-' SEMILEPTONIC FORM FACTOR FROM PHYSICAL REVIEW D 79, 054507 (2009) ranges from 2.0-4.5 GeV for the parameters used in our simulations. The 1-loop coefficient and higher mo­ments are calculated using automated perturbation theory and numerical integration as described in Refs. [61,621. We find that the perturbative corrections to matrix elements of the temporal vector current, V0, are less than a percent, while the corrections to matrix elements of the spatial vector current Vt are 3-4%. III. ANALYSIS In this section, we describe the three-step analysis pro­cedure used to calculate the B -> tt(v semileptonic form factor j\{q2). In the first subsection. Sec. Ill A, we de­scribe how we fit the pion and B-meson 2-point correlation functions in order to determine the pion energies and B-meson mass. We use both of these quantities in the later determination of the lattice form factors fwiE^) and Next, in Sec. IIIB, we construct a useful ratio of the 3-point correlation function (7t|V|B) to the 2-point functions. We then fit this ratio to a simple plateau ansatz to extract the desired form factors. Finally, in Sec. Ill C, we extrapolate the form factors calculated at unphysically heavy-quark masses and finite lattice spacing to the physi­cal light-quark masses and zero lattice spacing using NLO HMS^PT expressions extended with next-to-next-to- leading order (NNLO) analytic terms. (We perform a si­multaneous extrapolation in mq and a and interpolation in Ew.) We then take the appropriate linear combination of fwiEjr) and f±{Ew) to determine the desired form factor, /V(i/2), with statistical errors. A. Two-point correlator fits The pion and B-meson 2-point correlators obey the following functional forms: C^Prr) = £(-in<0| (29) m 2En p-4m)* cfw = Xi-irKGieyB*'"')!2--. po) m 2 tllg In the above expressions, terms with odd m contain the prefactor (-1)'. This leads to visible oscillations in time in the meson propagators; such behavior arises with stag­gered quarks because the parity operator is a composition of spatial inversion and translation through one timeslice [63,641. The contributions of the opposite-parity oscillat­ing states are found to be significant throughout the entire time range and must therefore be included in fits to extract the desired ground-state energy. Because the statistical errors in the pion energies and B-meson mass contribute very little to the total statistical error in the B -► rrl v form factor, we use a simple proce­dure to fit the 2-point functions. Although this does not optimize the determinations of Ew and mB, it is sufficient for the purpose of this analysis. We first select a fit range, Anin-Anax" which allows a good correlated, unconstrained fit including only contributions from the ground state and its opposite-parity partner. We then reduce /mjn by one time­slice and redo the fit. If the correlated confidence level is too low ( :£ 10%), we increase the number of states and try the fit again with the same time range. Otherwise, if the fit is good, we reduce /mjn by one more timeslice and repeat the fit. We repeat this procedure until we can no longer get a good fit without using a large number (greater than 4) of states. We note that, by including only as many states as the data can determine, we minimize the possibility of spuri­ous solutions in which the fitter exchanges the ground state with one of the same-parity excited states. We have, how­ever, checked that this method yields the same results a = 0.12 fm; ami/ams = 0.02/0.05; imax = 20 a = 0.12 fm; ami/ams = 0.02/0.05; imax = 15 0.3025 --------1--------------1--------------1--------------1--------------1--------------1--------------r-i 1.945 r 0.302 0.3015 0.301 0.3005 1.935 1.93 1.925 1.91 ■'min mm FIG. 1 (color online). Pion mass (left plot) and B-meson mass (right plot) versus minimum timcslicc in 2-point correlator fit. The red (filled) data points show the fit ranges selected for use in the B -» tt(v form-factor analysis. 054507-7 JON A. BAILEY et al. a - 0.12 fm; ami/ams - 0.02/0.05 PHYSICAL REVIEW D 79, 054507 (2009) a - 0.12 fm; ami/am3 - 0.02/0.05 o.i O -0.05 -0.1 OO () o o oo C) 10 20 30 t FIG. 2 (color online). Pion correlator fit corresponding to the red data point in the left-hand graph of Fig. 1. The left plot shows the fit (red line) to the zero-momentum pion propagator on a log scale, while the right plot shows the deviation of the fit from the data point for each timeslice. On both plots the dashed vertical line indicates rmin. Single-elimination jackknife statistical errors arc shown. within statistical errors as a constrained fit that includes up to three or four pairs of states. Figure 1 shows examples of both m„ vs /mjn (left plot) and mB vs /mjn (right plot) on the amfam, = 0.02/0.05 coarse ensemble, which has the largest light-quark mass of the coarse ensembles. The masses are stable as /mjn is reduced, and the statistical errors in m,s become smaller as additional timeslices are added to the fit. The statistical errors are determined by performing a separate fit to 500 bootstrap ensembles; each fit uses the full single­elimination jackknife correlation matrix, which is remade before every fit. The size of the statistical errors does not change when the number of bootstrap ensembles is in­creased by factors of two or four. We select the time range to use in the B -> u-C v analysis based on several criteria: a good correlated confidence level, relatively symmetric upper and lower bootstrap error bars, no 5-cr or greater outliers in the bootstrap distribution, and no sign of excited-state contamination. The red (filled) data points in Fig. 1 mark the chosen fit ranges for the ensemble in the example plots. Figures 2 and 3 show the corresponding pion and fi-meson correlator fits, respectively, which go through the data points (shown with jackknife errors) quite well. The gauge configurations have been recorded every six trajectories, and the remaining autocorrelations between consecutive configurations cannot be neglected. We ad­dress this by averaging a block of successive configurations , bO o a = 0.12 fm; ami/ams = 0.02/0.05 a - 0.12 fm; ami/ams - 0.02/0.05 ii- o.i O -o.i -0.2 ~\-r j___L FIG. 3 (color online), fl-meson correlator fit corresponding to the red data point in the right-hand graph of Fig. 1. The left plot shows the fit (red line) to the fl-meson propagator on a log scale, while the right plot shows the deviation of the fit from the data point for each timeslice. On both plots the dashed vertical line indicates rmin. Single-elimination jackknife statistical errors arc shown. 054507-8 B - itCp SEMILEPTONIC FORM FACTOR FROM ... a - 0.12 fm; ami/ams - 0.02/0.05 block size FIG. 4 (color online). Singlc-climination jackknifc error ver­sus block size in the zcro-momcntum pion propagator at t = 6. The statistical errors in the errors arc calculated with an addi­tional singlc-climination jackknifc loop. The red line is an average of the errors for block sizes 5-8 and is only to make it easier to sec that the statistical error plateaus after a block size of 5; it is not used in the form-factor analysis. together before calculating the correlation matrix and per­forming the fit. Wc determine the optimal block size by increasing the number of configurations in a block until the singlc-climination jackknifc statistical error in the correla­tor data remains constant within errors. This is shown for a representative timcslicc of the pion propagator on a coarsc ensemble in Fig. 4. Wc find that it is ncccssary to use a block size of 5 on the coarsc ensembles and 8 on the fine ensembles, and wc use these values for the rest of the form- factor analysis. Wc note that the size of the statistical errors that arises from blocking by 5 on the coarsc ensemble is consistent with that estimated based on a calculation of the integrated autocorrelation time. The pion energy Ethat is cxtractcd from fitting the 2- point function, Cf(?; p^), should satisfy the dispersion relation E% = [/3^[2 + ml in the continuum limit due to the restoration of rotational symmetry. Similarly, the pion amplitude = |(0|Off|7r)| should be independent of as a -► 0. As shown in Fig. 5, our results arc consistent with these continuum relations within statistical errors.1 Wc therefore rcplacc the pion energy Ew by Vi PJ1 + ml when calculating the latticc form factors f\\{E^) and /j_ (E^S) in order to rcducc the total statistical uncertainty. The pion amplitude drops out of the form-factor calcula­tion, however, bccausc wc take suitable ratios of 3-point to 2-point correlators. B. Three-point correlator fits The B -► tt 3-point correlator obeys the following func­tional form: C^U, T) = Xi-1)'"'!-1 m,n (31) where PHYSICAL REVIEW D 79, 054507 (2009) a = 0.12 fm; ami/ams = 0.02/0.05 a = 0.12 fm; ami/ams = 0.02/0.05 IP* TO *31 1.15 1.05 1 " - as{a\plt\)'1 1 1 1 1 c 1 1 . 1 IpttI FIG. 5 (color online). Comparison of pion energy Ew (left plot) and amplitude (right plot) with the prediction of the continuum dispersion relation. Wc also show a powcr-counting estimate for the size of momentum-dependent discretization errors, which arc of 0(as(a\pn\)2), as dashed black lines. 'As this analysis was being completed wc generated data with 4 times the statistics on the amj/ams, = 0.02/0.05 coarsc ensemble. In order to make the comparison to the continuum expectation dearer, wc use the higher statistics data in Fig. 5. 054507-9 JON A. BAILEY et al. <0 e> k,ra)>, , , , ,(B(n)0*0) . 'Wm)\Vu\BM)------!-(32) 2E!:n) M 2 m{"} A mn = M 2Eim) Writing out the first four terms of C^w(t, T) makes the behavior of the 3-point correlator as a function of both t and T more transparent: Cf^(r, T) = A™e-rf*'e-mBiT-<) + (_ iyT-t)A0\ e-m"\T-t) _)_ lyAWg-rfi'tg-m'giT-t) + (- {)T A1^ e^r}-< e^m{BiT-,] + ... (33) As in the case of the pion and B-meson propagators, the leading contributions from the opposite-parity excited states (the A1® and A®1 terms) change sign when t-> t + 1; these produce visible oscillations in the correlation function along the time direction. The subleading contri­bution from the opposite-parity excited states (the A" term), however, only changes sign when the source-sink separation is varied, e.g., T -> T + 1; this contribution is not as clearly visible in the data as those that oscillate with the time slice t. The lattice form factors are related to the ground-state amplitude of the 3-point function C^w(t, T) as follows: rV&t - J II aoo/2E£V^\ (34) ft = Aoo 1 \ Z r-7. ft } P Jl (35) where, as before, Z„ = |{O|0j77)| and ZB = |{0|(9B|fi)|. The pion and B-meson energies and amplitudes are known from the 2-point fits described in the previous subsection. Thus, the goal is to determine the 3-point amplitude A°° for li along both the spatial and temporal directions. In principle, the easiest way to determine the coefficient Af is to divide the 3-point function C^w(t, T) by the appropriate 2-point functions and fit to a constant (plateau) ansatz in a region of time slices 0 t T that are sufficiently far from both the pion and B-meson sources, such that excited-state contamination can be neglected. In practice, however, oscillating excited-state contributions are significant throughout the interval between the pion and B-meson, so our raw correlator data cannot be fit to such a simple function. Therefore, we construct an average correlator in which the oscillations are reduced before performing any fits. This method for determining the form factors requires knowledge of Ew and ma\ we use the values determined in the 2-point fits described in the previous subsection and propagate the bootstrap uncertain­ties in order to properly account for correlations. The final ratio of correlators used to determine A®° entails several pieces. To begin consider the carefully constructed average of the value of the B-meson propaga­tor at time slice t with that at t + 1: PHYSICAL REVIEW D 79, 054507 (2009) Cf (t) - Cf{t) = Cf(r) Cf (r + 1)' 72 r 2m (0)' 2m (i)' X (I - {--y.... (36) where Amn = m^ - m1®. By removing the leading ex­ponential behavior from the correlator before taking the average we suppress the leading oscillating contribution by a factor of the mass-splitting Ams/2 while leaving the desired ground-state amplitude unaffected. Note also that, while this procedure affects the size of the excited- state amplitudes, it does not alter the functional form of the correlator, nor does it alter the energies in the exponentials. Therefore, the average in Eq. (36) is equivalent to using a smeared source that has a smaller coupling to the opposite- parity excited states. This averaging procedure can be iterated in order to make the oscillating terms arbitrarily small. Empirically, we find that two iterations are sufficient for all of our numerical data: Cf(t) - Cf(r) 2Cf(r+ 1) Cf (t + 2) zi mf(t~ 1) •a (0) 2m f + O(Aml). 2m m' mf(t~ 2) ,^_A m) (37) At our various light-quark masses and lattice spacings the mass splittings lie in the range 0.1 S AmB ■& 0.3 in lattice units; thus, use of the iterated average in Eq. (37) reduces the leading oscillating state amplitude by a factor of -50^100 such that it can be safely neglected. In the case of the -> 77 3-point correlation function, we wish to reduce both the oscillating contributions and the less visible lionoscillating contributions that arise from the cross term between the lowest-lying pion and B-meson opposite-parity states. If these contributions are reduced sufficiently, we can safely neglect all of them when ex­tracting the ground-state amplitude A°°. We therefore con­struct a slightly more sophisticated average, which combines the correlator both at consecutive time slices (t and t + 1) and at consecutive source-sink separations (T and T + 1): 054507-10 B - itCp SEMILEPTONIC FORM FACTOR FROM ... Cf-Ht, T) cf-nt, t +1) e-Efu)e-mf(T+l-t) PHYSICAL REVIEW D 79, 054507 (2009) 2 C f ^ ( r + 1, T) 2Cf^(r + 1, T + 1) „-Ef(t+n„-mf(T-t) CMr + 2, T) CMr + 2, T + 1) j __________________________ _|_____3>bL Afe-E{n>e-<iT-,) + ^-l)TAUe-ei'te-«,''\T- „<0!. -Ef(1 + 2) p-m{'hT-1-2) f7 Oh e + 0{AE%. AE„AmB, Am2). (38) } This average reduces the unwanted parity states' contami­nation significantly. It eliminates both the leading 0(1) and subleading O(AE^) contributions to the oscillating <410 term, the two lowest-order 0(1, AmB) contributions to the oscillating /\01 term, and the 0(1, AET) contributions to the nonoscillating ,411 term. The size of the remaining ,411 term is a factor of ~7-20 times smaller than in the unsmeared 3-point correlator. We can now safely ignore contamination from opposite- parity states and determine the lattice form factors in a simple manner. We construct the following ratio of the smeared correlators: T) rz------:-----------1/ ~Fmt YCf(r)Cf(T - t) Ve ' e The lattice form factors are then (39) iat ff (40) (41) We fit /j|at and /'f as defined in Eqs. (40) and (41) to a plateau in the region 0 « t « T, where ordinary excited- state contributions can be neglected. Figure 6 shows the determinations of/["' (left plot) and /'f (right plot) for all of the momenta that we use in the chiral extrapolation on the coarse ensemble with am[/ams = 0.02/0.05. In prac­tice, we fit a range of four time slices, choosing the interval that results in the best correlated confidence level. We have cross-checked the determination of the form factors via Eqs. (40) and (41) against determinations of the form factor that explicitly include excited-state dependence in the fit ansatz and find that the results agree within errors. Our preferred method, however, yields the smaller statis­tical uncertainty in the form factors. C. Continuum and chiral extrapolation The quark masses in our numerical lattice simulations are heavier than the physical up and down quark masses. The effects of nonzero lattice spacings in Asqtad simula­tions are also too large to be neglected. In order to account for these facts, we calculate the desired hadronic matrix elements for multiple values of the light-quark masses and lattice spacing, and then extrapolate to the physical quark masses and continuum using functional forms from heavy- 1.5 To 'OS a - 0.12 fm; ami/ams - 0.02/0.05 sc m - fin = ^(0,0,0); C.L. = 0.25 pOT = x( 1,0,0); C.L. = 0.29 - fin = ^(1,1,0); C.L. = 0.29 © m 15 a - 0.12 fm; ami/ams - 0.02/0.05 FIG. 6 (color online). Determination of the form factors /y (left plot) and f± (right plot) from plateau fits to the ratios defined in Eqs. (40) and (41). The statistical errors on the data points are from a single-elimination jackknife. The statistical errors in the plateau determination are from separate fits of 500 bootstrap ensembles. 054507-11 JON A. BAILEY et al. PHYSICAL REVIEW D 79, 054507 (2009) light meson staggered chiral perturbation theory (HMS^PT) [41]. The HMS^PT expressions are derived using the symmetries of the staggered lattice theory, and therefore contain the correct dependence of the form fac­tors on the quark mass and lattice spacing. In the case of the B -> tt('v form factors, the HMS^PT expressions are also functions of the pion energy (recall that we work in the frame where the R meson is at rest). HMS^PT is a systematic expansion in inverse powers of the heavy-quark mass. In the chiral and soft pion limits (mt -> 0 and Ew -> 0), the leading-order continuum HM^PT expressions for f)\ and f\ take the following simple forms: <h fir (42) (43) where (f>B = fB^m^, f b is the B-meson decay constant, and is the pion decay constant. The coefficient gB-Bv parameterizes the size of the B*-B-tt coupling. In the static heavy-quark limit, heavy-quark spin symmetry does not distinguish between the pseudoscalar B meson and the vector B* meson, which implies that the decay constant 0b* = 0b and the mass difference = mB< - mB -> 0. Inclusion of the parameter AJj, however, ensures the proper location of the pole at in the physical form factor f\(q2). At the next order in the heavy-quark expansion, 0(1 /mh) corrections split the degeneracy between the B­and /T-meson masses and decay constants. Furthermore, in the chiral and soft pion limits, all \/mh corrections can be absorbed into the values of the parameters 4>B, </>B*, 8b° Biti and AJj [65]; thus, f)\ and f\ retain the functional forms in Eqs. (42) and (43) even at NLO in HM^PT. At lowest order in S^PT, discretization effects split the degeneracies among the 16 tastes of pseudo-Goldstone mesons m‘v= = ii(mx + my) + cr A=, (44) where .r and v indicate the quark flavors, /x is a continuum low-energy constant, and A= is the mass splitting of a meson with taste E. An exact £7(1 )A symmetry protects the taste-pseudoscalar meson from receiving a mass shift to all orders in S^PT, implying that AP = 0. In addition, at 0(a2), a residual 5(9(4) taste symmetry preserves the degeneracies among mesons that are in the same irreduc­ible representation: P, V, A,T,1 \26]. Numerically, the size of the taste splittings turn out to be comparable to those of the pion masses for the a = 0.09 fm and a = 0.12 fm Asqtad staggered lattices used in this work [16]. We extrapolate our numerical form-factor data using HMS^PT expressions derived to zeroth order in \/mh. The fit functions therefore depend upon the three remain­ing expansion parameters: mh a, and Ew. The HMS^PT expressions for the form factors to 0(nj/, a2, E2.) are given explicitly in Eqs. (65)-(67) of Ref. [41]. Schematically, they read J0) II f\\(mh Ew, a) = --[1 + logs + c^m, + + ms) J 17 -<2>/ + + cfEl + cJ^Vr], (4) 2 , (?) 21 (45) c<°> f±(mh a) = -±- J 7T + • X logs + cf/U E^ + At X [c^'nj/ + + ms) + c'f E%. + (46) where "logs" indicate nonanalytic functions of the pseudo-Goldstone meson masses, e.g., m|ln(m|/A|). The continuum low-energy constant gB'btt enters these expressions in the coefficients of the chiral logarithms, which are completely fixed at this order. We use the phenomenological value of gB^Bv = 0.51 [43] for the cen­tral value and vary gB*B7r by a reasonable amount (see Sec. IV B) to estimate its contribution to the systematic uncertainty. Because the size of the mass splitting AJj is poorly determined from the lattice data and is consistent with the physical value within statistical errors, we fix AJj to the PDG value, 45.78 MeV [13], in our fits. The chiral logarithms also depend upon six extra constants that pa­rameterize discretization effects due to the light staggered quarks: the four taste splittings a2Ay, a2Aa, a2Ay, a2A, and the two flavor-neutral "hairpin" coefficients a28[, and a2 8'a [27]. These parameters can be determined separately from fits to light pseudoscalar meson masses and decay constants; we therefore hold them fixed to the values determined in Ref. [66], while performing the continuum-chiral extrapolation. The variation of these pa­rameters within their statistical errors results in a negligible change to the extrapolated form factors. The five terms analytic in mh a2, and Ew absorb the dependence upon the scale in the chiral logarithms, Aa„ such that the form factor is scale independent. We leave the tree-level coefficients c™ and the NLO analytic term coefficients as free parameters to be determined via the fit to the lattice form-factor data. In practice, we omit the analytic term proportional to (2mt + ms) from our fits because the strange sea-quark mass is tuned to approximately the same value on each of our ensembles and we have simu­lated only full QCD points. This term is therefore largely indistinguishable from the analytic term proportional to ni[. We have checked that omission of the sea-quark mass 054507-12 B - i t C p SEMILEPTONIC FORM FACTOR FROM ... PHYSICAL REVIEW D 79, 054507 (2009) analytic term has a negligible impact on the form factors in the chiral and continuum limits. In both earlier unquenched analyses of the B -> tt(v semileptonic form factor [33,671, the chiral extrapolation is performed as a two-step procedure: first interpolate the lattice data to fiducial values of Ew and then extrapolate the results to the physical quark masses and continuum independently at each value of Ew. The function used for the interpolation (which is different in the two analyses) introduces a systematic uncertainty that is difficult to esti­mate, In both cases, the chiral-continuum extrapolation makes use of the correct functional forms derived in HMS^ PT, but, by extrapolating the results for each value of Ew separately, the constraint that the low-energy con­stants of the chiral effective Lagrangian are independent of the pion energy is lost. This omission of valuable informa­tion about the form-factor shape introduces a further error that is unnecessary. The new analysis presented here in­stead employs a simultaneous fit using HMS^PT to our entire data set (all values of mh a, and Ew) to extrapolate to physical quark masses and the continuum and interpolate in the pion energy [68]. This improved method eliminates the systematic uncertainties introduced in the two-step iiiterpolate-theii-extrapolate procedure, and exploits the available information in an optimal way. We perform our combined chiral and continuum ex­trapolation using the method of constrained curve fitting [69]. Although we know that lattice data generated with sufficiently small quark masses and fine lattice spacings, and, in the case of the B -*«■ rrl v form factor, sufficiently low pion energies, must be described by lattice ^PT, we do not know precisely the range of validity of the effective theory. Furthermore, the order in ^ PT to which we must work and the allowed parameter values depend upon both the quantity of interest and the size of the statistical errors. We therefore need a fitting procedure that both incorpo­rates our general theoretical understanding of the suitable chiral effective theory and accounts for our limited knowl­edge of the values of the low-energy constants and sizes of the higher-order terms. Constrained curve fitting provides just such a method. Next-to-leading order ^PT breaks down for pion ener­gies around and above the kaon mass. Less than half of our numerical form-factor data, however, is below this cutoff. Therefore, although we do not expect NLO HMS^PT to describe our data through momentum p = 277(1, 1,0)/L, we cannot remove those points without losing the majority of our data. Nor can we abandon the NLO HMS^PT expressions for j)\ and /j_, Eqs. (45) and (46), which are the only effective field theory guides that we have for extrapolating the numerical lattice form-factor data to the continuum and physical quark masses. We therefore per­form the continuum-chiral extrapolation using the full NLO HMS^'PT expressions for /|| and j\, including the 1-loop chiral logarithms, plus additional NNLO analytic terms to allow a good fit to the data through p = 277(1, 1, 0)/L. The NNLO terms smoothly interpolate be­tween the region in which ^ PT is valid and the region in which the pion energies are too large and the higher-order chiral logarithms in Ew can be approximated as polynomials. We express the analytic terms in the formulae for /n and /±- Eqs. (45) and (46), as products of dimensionless ex­pansion parameters Xn Xir Xe, 2 fim 1 a2 A %TT2fi \f2Ew ■0.05-0.19, 0.03-0.09, • 0.22-0.78, (47) (48) (49) where A is the average staggered taste splitting and we show the range of values for each of these parameters corresponding to our numerical lattice data. (Note that we omit the p = 277(1, 1, 1)/L data points from our chiral fits because these would lead to xe. ~ 1 •) Because each of the above expressions is normalized by the chiral scale A^- 477/^, the undetermined coefficients should be of 0(1) in these units. We therefore constrain the values of the low-energy constants cj|°j_-c{|3j_ in our fits with Gaussian priors of width 2 centered about 0. The statistical errors in the numerical lattice data come from the 3-point fits described in the previous subsection. In order to account for the correlations among the various pion energies 011 the same sea-quark ensemble in the chiral-continuum extrapolation, we preserve the bootstrap distributions. We perform a separate correlated fit to each of the 500 bootstrap ensembles in which we remake the full bootstrap covariance matrix for each fit. We average the 68% upper and lower bounds on the form-factor distribu­tions to determine the statistical and systematic errors in /]| and that are plotted in Fig. 7 and presented in Table II below. Because we do not know a priori how many terms are necessary to describe the available lattice data, we begin with strictly NLO fits using the formulae for j)\ and j\ in Eqs. (45) and (46). We fit the lattice data for j)\ and j\ separately even though the ratio of leading-order coeffi­cients c^'/cy is predicted to be equal gB'Bw t0 NLO in ^PT; this is because the value of gB*B?r is known to only -50% from phenomenology. We obtain a good fit of the lattice data to the NLO expression without the inclusion of higher-order NNLO terms. This is probably because the shape of/i is dominated by the 1 /(E„ + A*) behavior and therefore largely insensitive to the other terms. We cannot, however, obtain a good fit of /|| to the strictly NLO 054507-13 JON A. BAILEY et a l 3 - 1 - X2/d.o.f. = 1.1 n--1--1--1--1--1--1--1--1--1-- ami/am3 = 0.0062/0.031 fine ami/am8 = 0.0124/0.031 fine ami/am8 = 0.005/0.05 coarse ami/ama = 0.007/0.05 coarse ami/ama = 0.01/0.05 coarse ami/ams = 0.02/0.05 coarse - continuum QCD (ri E,r X2/d.o.f. = 0.32 1--1--1--1--1--1--'--'--1--I" ami/ams = 0.0062/0.031 fine ami/ams = 0.0124/0.031 fine ami/ama = 0.005/0.05 coarse ami/ams = 0.007/0.05 coarse ami/ama = 0.01/0.05 coarse amilama = 0.02/0.05 coarse continuum QCD (r^)2 factors stabilize and the statistical errors in the form factors reach a maximum. This indicates that any further terms are of sufficiently high order, that they do not affect the fit, and can safely be neglected. We find that this occurs once the extrapolation formulae for/n and/i contain all eight sea- quark mass-independent NNLO analytic terms. The inclu­sion of NNNLO analytic terms does not further increase the size of the error bars. Figure 7 shows the preferred constrained fits of f)\ (upper plot) and f± (lower plot) versus E2W, where both the x and v axes are in r, units.2 Each fit is to the NLO HMS^PT expression, Eqs. (45) and (46), plus all sea-quark mass-independent NNLO analytic terms. The square sym­bols indicate fine lattice data, while the circles denote coarse data. The six colored curves show the fit result projected onto the masses and lattice spacings of the six sea-quark ensembles; the red line should go through the red circles, and so forth. The thick black curve shows the form factor in the continuum at physical quark masses with symmetrized bootstrap statistical errors. We use functions and constraints based oil HMS^PT to perform the chiral-continuum extrapolation because we know that HMS^PT is the correct low-energy effective description of the lattice theory. Nevertheless, we must compare various properties with theoretical expectations in order to check for overall consistency. Ail essential first test is that we can successfully fit the data with good confidence levels and obtain low-energy coefficients that are of the predicted size. We can also verify the conver­gence of the series expansion by calculating the ratios of the higher-order contributions to the leading-order form- factor contributions PHYSICAL REVIEW D 79, 054507 (2009) FIG. 7 (color online). Chiral-continuum extrapolation of /y (upper) and (lower) using constrained NLO HMS^PT plus all NNLO analytic terms with gB*BlT = 0.51 and = 0.311 fm. The square symbols indicate a ~ 0.09 fm latticc data points, while the circular symbols indicate a ~ 0.12 fm coarsc data points. The black curvc is the chiral-continuum extrapolated form-factor symmetrized bootstrap statistical errors only. expression, and must add higher-order terms in order to obtain a successful fit. Specifically, NNLO analytic terms proportional to m,Ew and E\ are both necessary to achieve a confidence level better than 10%. Although we could, at this point, choose to truncate the HMS^PT extrapolation formulae to include only those terms necessary for a good confidence level, we instead include "extra" NNLO analytic terms to both the f\\ and /_l fits, constraining the values of their coefficients with Gaussian priors of 0 ± 2. The introduction of more free parameters increases the statistical errors in the extrapo­lated values of the form factors; these larger errors reflect the uncertainty in the size of the newly included higher- order contributions. We continue to add higher-order ana­lytic terms until the central values of the extrapolated form /] ■NLO /1.0 J\\ 47%, =500 MeV /] ■NNLO fLO J\\ 3%, F.„=500 MeV jNLO sLO J1 /•NNLO J± j.'LO =500 MeV 48%, (50) =500 MeV 4%, (51) where we choose a nominal value of Ew = 500 MeV for illustration because it is on the high end of the expected range of validity of ,yPT. Finally, because the leading-order coefficient cj|0> is expected to be equal to 4>b = iu HM^PT, we can compare its value with that of <f>B deter­mined from our preliminary decay constant analysis. ‘As a cross-chcck of the constrained fits, wc also perform unconstrained fits of /y and f± with only the minimal number of analytic terms needed for a good fit. The results arc consistent, but the unconstrained fit results have smaller statistical errors bccausc they includc 6-8 fewer fit parameters. 054507-14 B - 7tCv SEMILEPTONIC FORM FACTOR FROM PHYSICAL REVIEW D 79, 054507 (2009) TABLE II. Statistical and systematic error contributions to the B -> ttCv form factor. Each source of uncertainty is discussed in Sec. IV. For each of the 12 q2 bins, the error is shown as a percentage of the total form factor. /+. (q2). which is given in the second row from the top. Because the bootstrap errors in the form factor are asymmetric, the errors shown are the average of the upper and lower bootstrap errors. In order to facilitate the use of our result, we also present the normalized statistical and systematic bootstrap correlation matrices in Table IV and the total bootstrap covariance matrix in Table V. q2 (GeV2) 26.5 25.7 25.0 24.3 23.5 22.8 22.1 21.3 20.6 19.8 19.1 18.4 9.04 6.32 4.75 3.75 3.06 2.56 2.19 1.91 1.69 1.51 1.37 1.27 statistics + /yPT(%) 24.4 18.5 13.5 9.6 7.1 6.3 6.5 6.9 7.2 7.5 8.2 9.8 g,rBv uncertainty 1.1 0.3 0.8 1.8 2.4 2.8 2.9 2.8 2.6 2.5 2.6 2.9 r i 0.4 0.7 0.9 1.1 1.2 1.3 1.4 1.4 1.5 1.5 1.4 1.4 m 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.3 ms 0.6 0.6 0.6 0.7 0.7 0.8 0.8 0.9 1.0 1.1 1.2 1.3 mb 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 Heavy-quark discretization 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.4 3.4 Nonperturbative Zv 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 Perturbative p 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 «o 2.9 2.1 1.2 0.5 0.3 0.8 1.1 1.3 1.4 1.3 1.3 1.3 Finite volume 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 Total systematics (%) 5.9 5.4 5.3 5.4 5.7 5.9 6.0 6.1 6.0 6.0 6.0 6.2 Although the B-meson decay constant calculation uses the same staggered gauge configurations, it employs different heavy-light meson 2-point correlation functions with the axial current, a different HMS^PT fit function, and differ­ent perturbative renormalization factors, and is therefore largely independent of the B -► rrl v semileptonic form- factor calculation. For the preferred f\\ fit shown in Fig. 7, we find cj|°> =0.81 ± 0.07, where the errors are statistical only. This is quite close to our current preliminary result, r'\24>n = 0-92 ± 0.03 (statistical error only) [70,711, es­pecially considering that the HMS^PT extrapolation for­mula for /ii neglects some of the 0(1 /mh) contributions. An interesting use of our numerical B -► rrl v form- factor data is to determine the approximate value of the B*-B-77 coupling, girnw from lattice QCD. For the pre­ferred fits shown in Fig. 7, we find that the ratio of leading- order coefficients is c!0) gB'Brr ~ -jfcj = 0.22 ± 0.07, (52) cll and is independent of the choice for g/j*/jrr in the chiral logarithms within statistical errors. This determination omits the 0(1 /mh) corrections to the chiral logarithms in the HMS^PT extrapolation formulae for /|| and f±, Eqs. (45) and (46), and neglects the difference between 4>n and we do not attempt to estimate the systematic uncertainty introduced by these or other effects. The value is lower than the determination of Stewart, gn-nrr = 0.51, which comes from a combined analysis of several experi­mental quantities, including the D*-meson decay width, through 0(1 /mc) in HM^PT [43]. It is consistent, how­ever, with the range of values determined by the HPQCD Collaboration, who allowed g/rBw t0 be a free parameter in their chiral-continuum extrapolation and found 0 < gB'Bw ~ 0-45 [33]. IV. ESTIMATION OF SYSTEMATIC ERRORS In this section, we discuss all of the sources of system­atic uncertainty in our calculation of the B -> rr(v form factor f^(q2). We present each error in a separate subsec­tion for clarity. The value of the form-factor f^(q2), along with the total error budget, is given in Table II. A. Chiral-continuum extrapolation fit ansatz We use the method of constrained curve fitting to esti­mate the effect of neglected higher-order terms in the HMSa'PT chiral-continuum extrapolation formulae. Our fit procedure is described in detail in Sec. Ill C. Therefore, the errors in and f± extrapolated to physical quark masses and zero lattice spacing shown in Fig. 7 reflect both the statistical errors in the Monte Carlo data and the systematic errors due to our limited knowledge of the higher-order terms, which we specified with priors. We do not need to include a separate systematic uncertainty due to the choice of fit function, as would be the case had we used an unconstrained fit with fewer terms. B. gn^nr, uncertainty We fix the size of the coupling to g^Bw = 0-51 in the coefficients of the chiral logarithms, while extrap­olating to the physical light-quark masses and continuum. Our choice is based upon the following considerations: Because the coupling g/j*/jrr is expected to be approxi­mately equal to g/rnw due to heavy-quark symmetry, we use the phenomenological value of the D*-D-tt coupling determined by Stewart in Ref. [43], which comes from a 054507-15 JON A. BAILEY et al. PHYSICAL REVIEW D 79, 054507 (2009) combined analysis of several experimental quantities that includes the D*-meson decay width [72]. This value is presented without errors, and is an update of Stewart's earlier analysis in Ref. [731, which incorporates additional experimental results. His earlier calculation finds a signifi­cantly lower value of go'-mr = 0.27ioo?-o(p, where the first errors are experimental and the second errors come from an estimate of the sizes of the 1-loop counterterms [731. A more recent phenomenological determination of the D -D-tt coupling by Kamenik and Fajfer, which also includes up-to-date experimental data, improves upon the analysis method of Stewart by including contributions from both positive and negative parity heavy mesons in the loops [741. They find an even higher value of girnw = 0.661qq|, where the uncertainty only reflects the error due to counterterms. We therefore conclude that, although recent experimental measurements of the D* width may constrain the coupling [721 at tree-level, the size of gB'B„ is not well determined in the literature. In order to determine the error in the form factor from the uncertainty gB>B„ we vary the parameter over a gen­erous range. The smallest value of gB'Bjr that we have seen in the literature is g,yDw = 0.27 [731. The largest is Sd'Dtt = 0.67, which comes from a quenched lattice cal­culation [751. (There has not yet been an unquenched "2 + 1" flavor determination of gB'Bjr.) We therefore vary gB'Bjr over the entire range from 0.27-0.67 and take the largest difference from the preferred determination of f~{q2) using gB'Bjr = 0.51 as the systematic error due to the uncertainty in the B*-B-rr coupling. The lattice data is largely insensitive to the value of gB' B„ in the coefficient of the chiral logarithms; all values of the parameter yield similar fit confidence levels. The resulting systematic un­certainty in f~{q2) is less than 3% for all q2 bins despite varying gB'Bjr by almost 50%. C. Scale (r|) uncertainty We use the MILC Collaboration's determination of the scale from their calculation of/„., rj = 0.311 fm, to con­vert between lattice and physical units [511. The parameter rj enters the form-factor calculation in a number of places: we use the PDG values of fw and in the chiral- continuum extrapolation formulae [131, we set m„ to the PDG value in the resulting fit functions to determine the form factors at the physical point, and we convert /y and /_l to physical units via rx before combining them to extract f+(q2). An alternative determination using the HPQCD Collaboration's lattice data for the Y 25-1S [761 splitting yields a result that is ~2% larger, rx = 0.317 fm. We therefore repeat the chiral-continuum extrapolation of /ll and f± using this higher value of rj, combine them into the dimensionless form factor f~(q2) using this higher value of rj, and take the difference from the preferred form-factor result as the systematic error due to uncertainty in the overall lattice scale. The difference ranges from 1- 1.5% for most q2 values. This is consistent with our naive expectation that a ~2% difference in rx will result in a ~1% difference in f~{q2) because /y has dimensions of GeV1/<2 and f± has dimensions of GeV-1/2. D. Light-quark mass (in, ins) determinations We obtain the form factors /y and f± in continuum QCD by setting the lattice spacing to zero and the light-quark masses to their physical values in the HMS^PT expres­sions, once the coefficients have been determined from fits to the numerical lattice data. We use the most recent calculations of the bare quark masses by the MILC Collaboration from fits to light pseudoscalar meson masses r,m X 103 = 3.78(16), (53) rxms X 103 = 102(4), (54) where m is the average of the up and down quark masses and the quoted errors include both statistics and system- atics [511. We vary the bare light-quark mass, rxm. within its stated uncertainty and take the maximal difference from the preferred form-factor result to be the systematic error; we find that the error is 0.3% or less for all values of q2. We perform the same procedure for the bare strange quark mass, and find that the resulting error ranges from ~0.5-1.5% over the various q2 bins. E. Bottom quark mass (mb) determination The value of the form factor f~{q2) depends upon the Z?-quark mass, which we fix to its physical value through­out the calculation. Specifically, we first determine the value of the hopping parameter, k, in the SW action for which the lattice kinetic mass agrees with the experimen­tally measured -meson mass. We then use this tuned xb when calculating all of the 2- and 3-point heavy-light correlators needed for the B -> rrf v form factor. With our current tuning procedure we are able to determine Kh to ~6% accuracy. This uncertainty in Kb is conservative; it is primarily due to poor statistics, and will decrease con­siderably after the analysis of the larger data set that is currently being generated. The uncertainty in xb produces an uncertainty in the form factor. We estimate this by calculating the form factor ,/V at two additional values of k (one above and one below the tuned value) on the amt/ams = 0.02/0.05 coarse en­semble. This is sufficient because the heavy-quark mass- dependence of the form factor is largely independent of the sea-quark masses and lattice spacing. We find the largest dependence upon Kb at momentum p = 277(1, 1,0)/L, shown in Fig. 8, for which a 6% uncertainty in xh produces a 1.2% uncertainty in the form factor. We therefore take 1.2% to be the systematic error in f~{q2) due to uncer­tainty in the determination of the /;-quark mass. 054507-16 B - t tC v SEMILEPTONIC FORM FACTOR FROM ... a = 0.12 fm; ami/ams = 0.02/0.05; p= ^(1,1,0) K FIG. 8 (color online). Normalized form factor at momen­tum p = 1,0)/L as a function of k on the am,/ams = 0.02/0.05 coarse ensemble. The central data point corresponds to the tuned Kb. and the thick red line shows a linear fit to the three data points. The two dashed vertical lines indicate the upper and lower bounds on Kb. F. Gluon and light-quark discretization errors Wc estimate the size of discretization errors in the form factor f~{q2) with power counting. Wc choose conserva­tive values for the parameters that enter the estimates: A = 700 MeV and av(q") = 1/3, which is a typical value on the fine latticc spacing [57,621. Wc calculate the B -► tt(.v scmilcptonic form factor using a onc-loop Symanzik-improvcd gauge action for the gluons [77-801 and the Asqtad-improvcd staggered action for the light up, down, and strange quarks [81,821. Bccausc both the gluon and light-quark actions arc 0(a2) improved, the leading discretization cffccts arc of 0(as(aA)2). Wc parameterize these errors in the fit to numerical latticc form-factor data by including analytic terms proportional to a2 in the HMS^PT extrapolation formulae for /|| and f±. Bccausc wc only have data at two latticc spacings, however, wc do not includc a separate term proportional to as(aA)2 to account for the fact that as differs by a few percent between the latticc spacings. Wc then remove the majority of light-quark and gluon discre­tization cffccts from the final result by taking a -► 0. Similarly, wc identify and remove higher-order discretiza­tion cffccts in the chiral-continuum extrapolation through the NNLO analytic terms in the fit functions. The remain­ing gluon and light-quark discretization errors arc negligible. The calculation of the B -► tt(.v form factor requires 2- point and 3-point functions with nonzero momenta, which introduces momentum-dependent discretization errors. The leading /j-dcpcndcnt discretization error is of 0{as{ap)2). Wc parameterize these errors, up to variations in as at the two latticc spacings, with the two NNLO analytic terms proportional to a2E% and a2m/ in the cx- trapolation formulae for /|| and f± and remove them by taking the continuum limit of the resulting fit functions. This also largely removes errors of 0(a2(ap)2). The re­maining momentum-dependent discretization cffccts arc of 0{as{ap'f ). On the 283 fine lattices, as{ap'f = 0.003 for our highcst-momcntum data points with ap = 277(1, 1, 0)/28. Therefore, the uncertainty in f~(q2) due to momentum-dependent discretization cffccts is negli­gible compared with our other systematic errors. G. Heavy-quark discretization errors Wc use HQET as a theory of cutoff cffccts to estimate the size of discretization errors due to use of the Fermilab action for the heavy bottom quark. Bccausc both the latticc and continuum theories can be described by HQET, heavy- quark discretization cffccts can be classified as a short- distancc mismatch of highcr-dimcnsion operators [36-381. Each contribution to the error is given by [831 error, = |[Cyx(mQ, m0a) - Cfnx(mQ)]{0,)\, (55) where 0, is an effective operator, and Cf'imQ, m0a) and ) arc corresponding short-distancc coefficients when HQET is used to describe latticc gauge theory or continuum QCD, respectively. The coefficient mismatch can be written as C)M(mQ, m0a) ~ Cfmi(mQ) = (56) and the relative error in our matrix elements can be esti­mated by setting (O,) ~ Aq™® 1 4. Then cach contribution to the error is error, = /j(m0a)(aAQCD)d,me'"4, (57) recovering the counting in powers of a familiar from Symanzik, while maintaining the full m0a dcpcndcncc. The functions /, can be dcduccd from Refs. [35,841 and arc compiled in Appendix A. Adding all contributions of 0{asa) and 0(a2) from the action and the current, wc obtain a relative error of 2.84% (4.16%) for f\\ and 3.40% (4.98%) for /_l on the fine (coarse) lattices. Wc therefore take 3.4% to be the error in f~{q2) due to heavy-quark discretization cffccts. H. Heavy-light current renormalization Wc determine the majority of the heavy-light current renormalization nonpcrturbativcly. The dcpcndcncc of Z\P on the sca-quark masses and on the mass of the light spectator quark in the 3-point correlator arc both negli­gible; the statistical error in Zyb is -1%. The dcpcndcncc of Zy on the sca-quark masses is also negligible, and the statistical error in Zy is -1%. Wc therefore includc V(l%2 + 1%2) « 1.4% as the systematic uncertainty in PHYSICAL REVIEW D 79, 054507 (2009) 054507-17 the form-factor f+(q2) due to the uncertainty in the non­perturbative renormalization factors Zyh and Zy for all values of q2. We determine the remaining renormalization of the heavy-light current using lattice perturbation theory. The 1 -loop correction to is ~3% on the fine ensembles and ~4% on the coarse ensembles. Because we calculate p\l v p, to 0(as), the leading corrections are of 0(a2). We might therefore expect the 2-loop corrections to pf/ to be a factor of as smaller, or ~1%. In order to be conservative, how­ever, we take the entire size of the 1 -loop correction on the fine lattices, or 3%, as the systematic uncertainty in f+(q2) due to higher-order perturbative contributions for all q2 bins. JON A. BAILEY et al. I. Tadpole parameter (h0) tuning In order to improve the convergence of lattice perturba­tion theory, we use tadpole-improved actions for the glu­ons, light quarks, and heavy quarks [47]. We take «0 from the average plaquette for the gluon and sea-quark action [18]. On the fine lattice, we make the same choice for the valence quarks. For historical reasons, however, we use u0 determined from the average link in Landau gauge for the valence quarks on the coarse ensembles. The difference between u0 from the two methods is 3-4% on the coarse ensembles. We must, therefore, estimate the error in the form factor due to this poor choice of tuning. The tadpole-improvement factor enters the calculation of f+(q2) in several ways. The factor of u0 that enters the normalization of the heavy Wilson and light staggered quark fields cancels exactly between the (-n-IV^lB) lattice matrix element and the nonperturbative renormalization factor The most significant effect of the mixed Uq values is in the chiral-continuum extrapolation of /y and f i. The different choices for valence and sea-quark actions imply that the coarse lattice data is partially quenched. We study this effect by performing the chiral extrapolation in two ways: one assuming that both valence and sea quarks have the mass of the sea quark and the other assuming that both have the mass of the valence quark. This leads to a 3% error in the highest q2 bin, and a ~ 1-1.5% error in the bins that affect the determination of |Vub|. Most of the other effects of changing «0 in the lattice action and current can be absorbed into our estimate of the uncertainty from higher-order perturbative corrections to p!{L to discretiza­tion errors, and to the normalization of the Naik term. All but the last are already budgeted in Table II. The Naik term in the Asqtad action ensures that the leading discretization errors in the pion dispersion relation are 0(asa2p2). We therefore estimate the error in /+ (q2) due to different Naik terms in the valence and sea sectors to be equal to the largest value of asa2p2 on the coarse lattice times the ratio of the Landau link over plaquette u0 cubed, or ~0.2%. We add the flat error from the Naik term to the bin-by- bin error due to the light-quark mass used in the chiral extrapolation in quadrature to obtain the total uncertainty. Although our estimate is of necessity rather rough, we find that the errors due to «0 tuning are much smaller than the dominant errors in f+(q2). Our error estimate is therefore adequate for the determinations of the B-* ttC.v form factor and |Vub| presented in this work. J. Finite-volume effects We estimate the uncertainty in the form factor f+(q2) due to finite-volume effects using 1-loop finite volume HMS^PT. The finite-volume corrections to the HMS^PT expressions for/y and f± are given in Ref. [41 ] in terms of integrals calculated in Ref. [85]. It is therefore straightfor­ward to find the relevant corrections for our simulation parameters. We find that the 1 -loop finite-volume correc­tions are well below a percent for all of our lattice data points. Because finite-volume errors increase as the light- quark mass decreases, they are largest on the ami/am5 = 0.007/0.05 coarse ensemble. The biggest correction is to fi at p = 277(1. 1. 0)/L, and is 0.5%. We therefore take this to be the uncertainty in f+(q2) due to finite-volume errors for all q2 bins. V. MODEL-INDEPENDENT DETERMINATION OF |Fub| It is well established that analyticity, crossing symmetry, and unitarity largely constrain the possible shapes of semi­leptonic form factors [86-89]. In this section we apply constraints based on these general properties to our lattice result for the form factor f+(q2) and thereby extract a model-independent value for the CKM matrix element \VJ. Until now the standard procedure used to extract |Vub| from B -► tt(.v semileptonic decays has been to integrate the form factor \f+{q2)\2 over a region of q2, and then combine the result with the experimentally measured de­cay rate in this region: PHYSICAL REVIEW D 79, 054507 (2009) /• J q: 1927r'mB j(]- - 4m2Bmif/2\f+(q2)\2. dq~[(m~B + m^. - q~) 2\2 (58) The integration, however, necessitates a continuous pa­rameterization of the form factor over the full range from C,in t0 ‘/max- In our earlier, preliminary unquenched analysis, we determine f+(q2) by fitting the lattice data points to the Becirevic-Kaidalov (BK) parameterization [90], f+(q2) /+<0) (1 - q2){\ - acj2) ' (59) 054507-18 B - itCp SEMILEPTONIC FORM FACTOR FROM ... PHYSICAL REVIEW D 79, 054507 (2009) fair) /+(0) (1 -r/PY (60) where q2 = q2/m2R„. The BK ansatz contains three free parameters and incorporates many of the known properties of the form factor such as the kinematic constraint at q2 = 0, heavy-quark scaling, and the location of the B* pole. The HPQCD Collaboration instead uses the four-parameter Ball-Zwicky (BZ) parameterization [91], which is the same as the BK function in Eq. (59) plus an additional pole to capture the effects of multiparticle states. In both cases, however, the choice of fit function introduces a systematic uncertainty that is difficult to quantify. It is likely the BK and BZ parameterizations can be safely used to interpolate between data points, whether they be at high q1 from lattice QCD or at low q1 from experiment. It is less clear, however, how well these ansa- tze can be trusted to extrapolate the form-factor shape beyond the reach of the data points. Furthermore, compari­sons of lattice and experimental determinations of BK or BZ fit parameters are not necessarily meaningful. For example, if the slope parameters a from experiment and lattice QCD were found to be inconsistent, we would not know whether theory and experiment disagree, or whether the parameterization is simply inadequate. A parameteri­zation that circumvents this issue is therefore desirable. In this work we pursue an analysis based on the model- independent z parameterization, which is pedagogically reviewed in Ref. [87]. A. Analyticity, unitarity, and heavy-quark constraints on heavy-light form factors All form factors are analytic functions of q1 except at physical poles and threshold branch points. In the case of the B -* ttIv form factors, f(q2) is analytic below the Bit production region except at the location of the B* pole. The fact that analytic functions can always be expressed as convergent power series allows the form factors to be written in a particularly useful manner. Consider mapping the variable q2 onto a new variable, s, in the following way: z(q2. t0) Vi - q2/t+ - Vi - tQ!u Vi - <rA+ + V1 " h/u (61) where t+ = [mB + m^)2, r_ = [mB - m„Y- and t0 is a free parameter. Although this mapping appears compli­cated, it actually has a simple interpretation in terms of q2: this transformation maps q2 > t+ (the production re­gion) onto |z| = 1 and maps q2 < t+ (which includes the semileptonic region) onto real z €E [- 1.1]. In terms of s, the form factors have a simple form: fiq2) 1 P(q-)(l>(q-, T0) X ak(to)z(q2. t0)k. (62) where the Blaschke factor P(q2) is a function that contains subthreshold poles and the outer function ^{q2, t0) is an arbitrary analytic function (outside the cut from t+ < q2 < oo) whose choice only affects the particular values of the series coefficients ak. For the case of the B -> tt('v form factor f+(q2), the Blaschke factor P+(q2) = z(q2,mjr) accounts for the B* pole. In this work we use the same outer function as in Ref. [43]: <t> + (q2. t0) \ 3 967TXj°‘ x -Y/2(Jt+ - q2 + Jn)-5 (t+ ~ q2) (n -r0y/4' (63) where Xj0) is a numerical factor that can be calculated via the operator product expansion [88,92]. This choice of (f} + {q2, t0), when combined with unitarity and crossing- symmetry, leads to a particularly simple constraint on the series coefficients in Eq. (62). Although the t dependence of Eq. (63) appears complicated, it is designed so that the sum over the squares of the series coefficients is t inde­pendent: : _L = A' (64) 2iti J z k=0 where the value of the constant A depends upon the choice of Xj0) in Eq. (63). Because the decay process B -> ir(v is related to the scattering process Cv-* Btt by crossing symmetry, the sum of the series coefficients is bounded by unitarity, i.e., the fact that the production rate of Bit states is less than or equal to the production of all final states that couple to the -* u vector current. In particular, if one chooses the numerical factor Xj0) to be equal to the appropriate integral of the inclusive rate (v-> the sum of the coefficients is bounded by unity: 1. (65) k=0 k=0 where this constraint holds for any value of N and the symbol indicates higher-order corrections to xT in as a°d the operator product expansion. Such higher-order corrections turn out to be negligible for the B -> tt(, v form factor because the bound in Eq. (65) is far from saturated, i.e., the sizes of the coefficients turn out to be much less than one. Becher and Hill [93] have pointed out that this is due to the fact that the fc-quark mass is so large. In the heavy-quark limit, the leading contribu­tions to the integral in Eq. (64) are of 0(A3/m|), where A is a typical hadronic scale. Assuming that the ratio A /mb ~ 0.1, the heavy-quark bound on the ak's is approxi- 054507-19 JON A. BAILEY et al. PHYSICAL REVIEW D 79, 054507 (2009) mately 30 times more constraining than the bound from unitarity alone: 0.001. (66) We point out that the authors of Ref. [45] have recently proposed a slightly different parameterization of the B -> tt(- i' form factor with a simpler choice of outer function, 0 = 1: i 1 - q2/m2. X bk(to)z(g2. t0)k. (67) B* k=0 This choice enforces the correct scaling behavior, t\{q2) ~ 1 /q2 as q2 -> oo. It leads, however, to a more complicated constraint on the series coefficients: I Bjkbjbk S 1. j.k=o (68) where the elements of the symmetric matrix Bjk are cal­culable functions of t0. Because B -> tt(i> semileptonic decay is far from q2 -> oo, and because the unitarity bound is so far from being saturated, the choice of outer function should make a negligible impact on the resulting determi­nation of |Vub|. We therefore use the more standard outer function given in Eq. (63) because the constraint in Eq. (65) is independent of the number of terms in the power series, and is therefore simpler to implement. The free parameter t0 can be chosen to make the maxi­mum value of U| as small as possible in the semileptonic region; we choose t0 = 0.65/_ as in Ref. [43]. For B -> rrlv semileptonic decays this maps the physical region onto 0 < t < t- 0.34 < z < 0.22. (69) The bound on the coefficients in the z expansion combined with the small numerical values of |z| in the physical region ensures that one needs only the first few terms in the £ expansion to accurately describe the form-factor shape. Moreover, as the precision of both the lattice cal­culations and experimental measurements improve, one may easily include higher-order terms as needed. B. Determination of | Vub | using z parameterization In 2007 the BABAR Collaboration published a measure­ment of the shape of the B -> ttCv semileptonic form factor with results for 12 separate q2 bins between t/2jn ~ 1 GeV2 and t/2 ax ~ 24 GeV2 [42], This suggests that lat­tice QCD calculations are now needed primarily to provide a precise form-factor normalization at one value of q2 in order to determine I Vub|. The minimal error in | Vub| can, of course, still be attained by using all of the available infor­mation on the form-factor shape and normalization, pro­vided that one analyzes the data in a model-independent way. Because as many terms can be added as are needed to describe the B -* tt(i> form factor to the desired accuracy, use of the convergent series expansion allows for a sys­tematically improvable determination of I Vub|. We fit our lattice numerical Monte Carlo data and the 12-bin BABAR experimental data together to the z expansion, leaving the relative normalization factor | Vub| as a free parameter to be determined by the fit. In this way we determine | Vub| in an optimal, model-independent way. We first fit the lattice numerical Monte Carlo data and the 12-bin BABAR experimental data separately to the z expansion in order to check for consistency. We use Gaussian priors with central value 0 and width 1 on each coefficient in the z expansion to impose the unitarity con­straint. Although this manner of constraining the coeffi­cients is less stringent than the strict bound given in Eq. (65), the choice does not matter because the unitarity bound is far from saturated and the individual coefficients all turn out to be much less than 1. We obtain identical fit results even when the coefficients are completely unconstrained. The left-hand plot in Fig. 9 shows the BABAR measure­ment of the B -> ttCv semileptonic form factor, f\(q2) [42], The right-hand plot shows the same data multiplied by the functions P^(q2) and cf>^(q2, t0) and plotted versus the variable s. After remapping from q2 to z there is almost no curvature in the experimental data. This indicates that most of the curvature in the data is due to well-understood QCD effects that are parameterized by the functions P^(q2) and 4>^{q2,t0). Consequently, the experimental data is well described by a normalization (a0) and slope (ai/ao). as shown in Fig. 9. The slope of the BABAR experimental B -> rrl v form-factor data is - = -1.60 ± 0.26. «o (70) If one includes a curvature term in the z fit, the coefficient a2 is poorly determined, but is found to be negative at -1.5£7". The value of a\ is consistent with the result of the linear fit. Figure 10 shows the lattice determination of the B -* 77tv semileptonic form factor, vs q2 (left plot) and the remapped form factor, P^cf>^f\ vs z (right plot). As is the case for the experimental data, the shape of the lattice form factor is less striking after taking out the B* pole and other known QCD effects. When the lattice calculation of the form factor is fit to the z parameterization, however, it determines both a slope and a curvature. One cannot, in fact, successfully fit the lattice data without including a curvature term. The slope and curvature of the lattice determination of the B -> tt( v form factor are aj_ «o = -1.75 ± 0.91. (71) 054507-20 B - ttCv SEMILEPTONIC FORM FACTOR FROM .. 12-bin BABAR data PHYSICAL REVIEW D 79, 054507 (2009) 12-bin BABAR data g2(GeV2) z FIG. 9 (color online). Experimental data for the B -» ttIv form factor times the CKM element | Vub| from the BABAR Collaboration [42], The left plot shows | V^l X f+ versus q2, while the right plot shows | V^l X f+ multiplied by the functions P+<p+ and plotted against the new variable z- Both the 2-parameter fit (dashed blue line) and 3-parameter fit (solid red curve) have good ^'2/d.o.f.'s. -5.22 ± 1.39. (72) The above uncertainties are the standard errors computed from the inverse of the parameter Hessian matrix that result from a fit using the full covariance matrix determined from the bootstrap distributions of chiral-continuum extrapo­lated values of /|| and/j_, including systematics. Because the shapes of the lattice calculation and experi­mental measurement of the form factor are consistent, we now proceed to fit them simultaneously to the z expansion and determine |Vub|. The numerical lattice and measured experimental data are independent, so we construct a block-diagonal covariance matrix, where one block is the total lattice error matrix and the other is the total experi­mental error matrix. The combined fit function includes the series coefficients (ak's) plus an additional parameter for the relative normalization between the lattice and experi­mental results (|Vub|). In order to account for the system­atic uncertainty in |Vub| due to poorly constrained higher- order terms in z, we continue to add terms in the series until the error in | Vub| reaches a maximum. This occurs once we include the term proportional to z3. The resulting combined Fermilab-MILC lattice data Fermilab-MILC lattice data g2(GeV2) 2 FIG. 10 (color online). Lattice calculation of the B -» tt!v form factor. The left plot shows /+ vs q2, while the right plot shows P+<f>+f+ vs z. The inner error bars indicate the statistical error, while the outer error bars indicate the sum of the statistical and systematic added in quadrature. A 3-parameter z fit is needed to describe the lattice data with a good ^2/d.o.f. 054507-21 JON A. BAILEY et al. X2/d.o.f. = 0.59 1 1 i 1 1 1 1 i 1 1 1 1 i 1 1 1 1 i 1 1 1 1 i 1 1 1 1 - simultaneous 4-parameter z-fit 11- O Fermilab-MILC lattice data * BABAR data rescaled by |V,(,| from z-fit O o Ctll 0.02 - o.oi -0.3 -0.2 -0.1 0.1 0.2 PHYSICAL REVIEW D 79, 054507 (2009) %2/d.o.f. = 0.59 g2(GeV2) FIG. 11 (color online). Model-independent determination of ll^J from a simultaneous fit of latticc and experimental B-* tt-Cp semileptonic form-factor data to the c paramctcrizaton. The left plot shows P+4>+f+ vs ", while the right plot shows /+ vs q2. Inclusion of terms in the powcr-scrics through -3 yields the maximum uncertainty in | V^l; the corresponding 4-parameter - fit is given by the red curve in both plots. The circles denote the Fermilab-MILC latticc data, while the stars indicate the 12-bin BABAR experimental data, rescaled by the value of ll^l determined in the simultaneous --fit. z fit is shown in Fig. 11, and the corresponding fit parame­ters arc | Vub| X 103 = 3.38 ± 0.36, (73) a0 = 0.0218 ± 0.0021, (74) a, = ^0.0301 ± 0.0063, (75) a2 = -0.059 ± 0.032, (76) a3 = 0.079 ± 0.068. (77) The values of the cocfficicnts arc all much smaller than 1, as cxpcctcd from heavy-quark powcr-counting. The sum of the squares of the cocfficicnts is X al = 0.011 ± 0.012, and is consistent with the prediction of Bcchcr and Hill within uncertainties in the series cocfficicnts and in the choicc of the hadronic scalc in Eq. (66) [93]. By combining all of the available numerical latticc Monte Carlo data and 12-bin BABAR experimental data for the B -> Trtv form factor in a simultaneous fit wc arc able to determine |Vub| to ~1 \% accuracy. This error is independent (within :S 0.5%) of the choicc of the parame­ter t0 used in the changc of variables from q2 to z(q2, t0) and in the outer function 0 + (f/2, /0). In order to demon­strate the advantage of the combined fit method, wc com­pare the error in |Vubl given in Eq. (73) with that obtained from separate z fits of the latticc and experimental data. A z fit to the 12-bin BABAR experimental data alone deter­mines the normalization ae0xp to ~8%, while a z fit to our numerical latticc data determines ajf to ~14%. Thus, separate fits lead to a determination of |Vubl = aSQV/a^x with an approximately 16% total uncertainty.3 The com­bined fit yields a significantly smaller error and is thus preferred. When the numerical latticc data and experimental data arc fit simultaneously, utilizing all of the available data points is of secondary importance for reducing the total uncertainty in |Vubl- For example, wc can evaluate the importance of the low q2 experimental points to the ex­traction of | VUbI by removing them from the combined z fit. Including only the three experimental data points with q2 > 18 GeV2, wc find a consistent value of |Vub| with only a ~1% larger uncertainty. Similarly, wc can evaluate the importance of having many latticc data points, rather than only a single point, by using only the most prccisc latticc point with a total error of ~9%. This allows the form-factor shape to be completely determined by the 3Bccausc the values of the cocfficicnts of the power scries in c depend upon the choicc of the parameter f0 in Eqs. (61)-(63), wc could, in principle, choose a different value of tf) in order to minimize the error in cither OyXp or For example, use of f0 = 22.8 GeV2 reduces the uncertainty in the latticc normalization because the error in the latticc form factor is smallest at this q2 valuc. Use of f0 = 22.8 GeV2 greatly increases the uncertainty in the experimental normalization, however, because the experi­mental data is poorly determined at large values of q2. Ultimately, this choicc of f0 leads to an even worse determination of | than from our standard choicc of tf) = 0.65/_. Although wc did not attempt to determine the value of f0 that minimizes the total error in | V^l, the errors resulting from separate fits were greater than that obtained with the simultaneous fit for all values of f0 that wc tried. 054507-22 B - t tC v SEMILEPTONIC FORM FACTOR FROM PHYSICAL REVIEW D 79, 054507 (2009) experimental data. We find a consistent value of | V7ub | but with an even larger error of -13%. We therefore conclude that combining all of the numerical lattice data with all of the experimentally measured BABAR data minimizes the total uncertainty in IV^I. Because the small error in our final determination of | V7ub | is primarily due to the power of the combined e-fit method, one could easily use the pro­cedure outlined in this section to improve the exclusive determination of |\/ub| from existing lattice QCD calcula­tions of the B -> irtv form factor such as that by the HPQCD Collaboration [33]. VI. RESULTS AND CONCLUSIONS Combining our latest unquenched lattice calculation of the B -> 7t€v form factor with the 12-bin BABAR experi­mental data, we find the following model-independent value for I^J4: |Vub| X 103 = 3.38 ± 0.36. (78) The total error is - 11%, and it is nontrivial to separate the error precisely into contributions from statistical, system­atic, and experimental uncertainty because of the com­bined s-fit procedure used. If we assume, however, that the error in |\/ub| is dominated by the most precisely determined lattice point (which is not quite true, as shown in the previous section), we can estimate that the contri­butions are roughly equally divided as -6% lattice statis­tical, -6% lattice systematic, and -6% experimental. Our result is consistent with, although slightly lower than, our earlier, preliminary determination of | \/ub |. The reduction in central value is primarily due to a change in the lattice determination of the form factor, not the proce­dure used to determine | V7ub |. Because our new analysis uses a second lattice spacing, we are able to take the continuum limit of the form factor. We find that the con­tinuum extrapolation increases the overall normalization of f j. (i/2), and hence decreases the value of | V7ub |. Our errors are smaller than those of pre vious exclusive determinations primarily because we have reduced the size of the discre­tization errors, which are significantly smaller than in the previous Fermilab-MILC calculation ( ~ 7% -> 3%) be­cause of the additional finer lattice spacing. Our new result is -1-2a lower than most inclusive determinations of | \/ub|, which typically range from 4.0 - 4.5 X 10-3 [321. Much of the variation among the inclu­sive values is due to the choice of input parameters-in 4At three conferences during Summer 2008 wc presented a version of this model-independent analysis with a numerical value for |VUbI that is l-cr lower than that given here in Eq. (78). Wc have sincc improved several aspects of the latticc calculation, most notably reducing the statistical errors that enter the chiral and continuum extrapolations of/j_ and/n and, hcncc, /+. Equation (78) is our final result for |l/ub| based on the latticc data from the ensembles in Table I and the methodology of Secs. Ill and V. particular that of the &-quark mass [941. The recent deter­mination of mh by Kiihn, Steinhauser, and Sturm using experimental data for the cross section for -> had­rons in the bottom threshold region yields the value of mb to percent-level accuracy [951, and is consistent with the PDG average [131. Neubert has shown, however, that an updated extraction of mb from fits to B -> XtZv moments using only the theoretically cleanest channels (excluding b -> Xsy) results in a larger fr-quark mass and hence smaller inclusive value of | V7ub |, thereby reducing the ten­sion between inclusive and exclusive determinations [961. Our result is consistent with the currently preferred values for |\/ub| determined by the global CKM unitarity triangle analyses of the CKMfitter Collaboration, |\/ub| X 103 = 3.44lg;22 [971, and UTFit Collaboration, jl/J X 103 = 3.48 ± 0.16, [981. Further reduction in the errors is therefore essential for a more stringent test of the CKM framework and a more sensitive probe of physics beyond the standard model. The dominant uncertainty in our lattice calculation of the B -> 77-tV form factor comes from the statistical errors in the 2-point and 3-point correlations. This error can be reduced in a straightforward manner with use of an im­proved source for the pion and/or additional gauge con­figurations. The statistical errors in the nonperturbative renormalization factors Zy? and can be brought to below a percent in the same way. The chiral-continuum extrapolation error, which is inextricably linked to the statistical errors in the correlation functions, can also be improved by simulating at more light-quark masses and an additional finer lattice spacing of a - 0.06 fm. Presumably a better constrained chiral and continuum extrapolation will reduce the size of other ^2-dependent errors such as those from ginsjr, ru and the light-quark masses by some unknown amount as well. Use of a finer lattice with a - 0.06 fm will further decrease the momentum-dependent and heavy-quark discretization errors, which we now esti­mate with power counting. The extraction of | V7ub | can also be improved by including more experimental measure­ments of the B -*«■ 7T-C' v branching fraction. This, however, will require understanding the correlations among the vari­ous systematic uncertainties. Given these refinements of the current calculation, an even more precise, model- independent value of | V7Llb | can be obtained in the near future. ACKNOWLEDGMENTS We thank R. Hill and E. Lunghi for helpful discussions, and T. Becher for valuable comments on the manuscript. Computations for this work were carried out in part on facilities of the USQCD Collaboration, which are funded by the Office of Science of the U.S. Department of Energy, and on facilities of the NSF Teragrid under Allocation No. TG-MCA93S002. This work was supported in part by the United States Department of Energy under Grant 054507-23 Nos. DE-FC02-06ER41446 (C.D., L.L.), DE-FG02- 91ER40661 (S.G.), DE-FG02-91ER40677 (A.X.K., R.T.E., E.G.), DE-FG02-91ER40628 (C.B., J.L.), DE- FG02-04ER41298 (D.T.) and by the National Science Foundation under Grant Nos. PHY-0555243, PHY- 0757333, PHY-0703296 (C.D., L.L.), PHY-0555235 (J.L.), PHY-0456556 (R.S.). R.T.E. and E.G. were sup­ported in part by URA. Fermilab is operated by Fermi Research Alliance, LLC, under Contract No. DE-AC02- 07CH11359 with the United States Department of Energy. APPENDIX A: ESTIMATE OF HEAVY-QUARK DISCRETIZATION ERRORS In this appendix we collect the short-distance functions /, used to estimate the heavy-quark discretization effects. For more background, see Refs. [35-38,83,84]. 1. 0{a2) errors We start with these because explicit expressions for the functions fj(m0a) are available. a. 0(a2) errors from the iMgrangian There are two bilinears, hD -Eh and hiX • [/3 X E]h, and many four-quark operators. At tree level the coeffi­cients of all four-quark operators vanish and the coeffi­cients of the two bilinears are the same. The mismatch function is given by JON A. BAILEY et al. fE(m0a) = 1 1 Striper 2(2 m2af (Al) Using explicit expressions for l/m2 [35] and 1 fm2E [84], one finds fE(m0a) _ 11" c/j(l + m0a) 1 2Lm0a(2 + m0d)(l + m0d) 4(1 + mod) (A2) We use c/7 = 1 in our numerical simulations. b. 0(a2) errors from the current There are three terms with nonzero coefficients, qTD2h, qYiX • Bh, and qYa • Eh. which can be deduced from Eq. (A17) of Ref. [35]. Their coefficients can be read off from Eqs. (A19) [35]. When cB = rs the first two share the same coefficient: fx(m0a) = PHYSICAL REVIEW D 79, 054507 (2009) l <T^i(i + m oa ) 1 8m\ct m0a(2 + m0a) 2(2m2a)2' 1 + 1 (2 + mQa)(l + mod) 2(1 + mod) 1 1 4(1 + m0d)2 (2 + m0d)2 11" 1 ( m0a 2 L 2( 1 + mod) \2(2 + mod)(l + mod) r (A3) where the last term on the second line comes from using the tree-level d\. For the third operator, qYa ■ Eh. fY(m0a) = £(1 ~ cg)(l + mpa)' m0a(2 + m0a) 2 + 4 m0a + (m0a)2 4(1 + m0a)2( 2 + m0a)2' (A4) where the last line reflects the choices made for cE and dx. 2. 0{asa) errors Because we improve both the action and current, the mismatch functions fj(m0a) start at order as. and we do not have explicit expressions for them. (The calculation of these functions would be needed to match at the one-loop level.) So we shall take unimproved tree-level coefficients as a guide to the combinatoric factors and consider asymp­totic behavior in the limits m0a -► 0, oo. a. 0(asd) errors from the IMgrangian There are two bilinears, the kinetic energy hD2h and the chromomagnetic moment hiX • Bh. There is no mismatch in the coefficient of the kinetic energy, by assumption, since we identify the kinetic mass with the heavy-quark mass. This tuning is imperfect, but the associated error is budgeted in Sec. IVE. At the tree level the chromomagnetic mismatch is /^OJ(m0a) = CB 1 2(1 + m0a) (A5) This has the right asymptotic behavior in both limits, so our ansatz for the one-loop mismatch function is simply A°W> = 2(1 + m0a)' (A6) and errorB is this function multiplied by aA. 054507-24 B -> Trtv SEMILEPTONIC FORM FACTOR FROM PHYSICAL REVIEW D 79, 054507 (2009) TABLE III. Relative error from mismatches in the heavy- quark Lagrangian and current for the bottom quark with A = 700 MeV. To obtain the totals given in the text, E and X are counted twice, and 3 is counted twice for /y and 4 times for f±. Entries are in percent. a (fm) av{q*) m0a B 3 E X Y 0.09 0.33 2.018 1.76 1.32 0.28 0.80 0.24 0.12 0.41 2.617 2.48 1.94 0.39 1.26 0.33 b. 0{asa) errors from the current There is only one correction at tree level, but more generally there are two for the temporal current and four for the spatial current. (See Eqs. (2.27)-(2.32) of Ref. [37].) The tree-level mismatch function ends up being the same as dx: ffl(m0a) m0a 2(2 + njo«)(l + niQ a) (A7) It is, however, an accident that it vanishes as m0a Therefore, we instead take 0. Mm0a) av (A8) 2(2 + niQ(i)' which has the right asymptotic behavior. 3. Numerical estimates The relative errors due to mismatches in the heavy-quark Lagrangian and current on the MILC coarse and fine ensembles are tabulated in Table III. At the fine lattice spacing we take the typical av{q*) to be 1/3, and we use one-loop running to obtain av{q*) at the coarse lattice spacing. The contribution errorK from the a • E error in the current is so small both because cE = 1 in our simulation and because d\ is small. Adding the individual errors given in Table III in quadrature, and taking into account multiple contributions of the same size, we find the total error to be 2.84% (4.16%) for /„ and 3.40% (4.98%) for f\ on the fine (coarse) lattices. APPENDIX B: STATISTICAL AND SYSTEMATIC ERROR MATRICES In this appendix we present the normalized statistical and systematic bootstrap correlation matrices for the TABLE IV. Normalized statistical (upper) and systematic (lower) bootstrap correlation matrices for the B -> vtv form factor, f+(q2). These should be combined with the values of f+(q2) presented in Table II to reconstruct the full correlation matrices. q2 (GeV2) 26.5 25.7 25.0 24.3 23.5 22.8 22.1 21.3 20.6 19.8 19.1 18.4 26.5 1.00 0.99 0.97 0.88 0.66 0.29 -0.01 -0.16 -0.19 -0.14 -0.04 0.05 25.7 0.99 1.00 0.99 0.92 0.73 0.38 0.07 -0.09 -0.13 -0.09 0.00 0.08 25.0 0.97 0.99 1.00 0.97 0.82 0.51 0.21 0.04 -0.02 0.01 0.07 0.13 24.3 0.88 0.92 0.97 1.00 0.93 0.69 0.42 0.25 0.18 0.17 0.20 0.21 23.5 0.66 0.73 0.82 0.93 1.00 0.91 0.72 0.56 0.48 0.43 0.39 0.32 22.8 0.29 0.38 0.51 0.69 0.91 1.00 0.94 0.85 0.77 0.69 0.58 0.42 22.1 -0.01 0.07 0.21 0.42 0.72 0.94 1.00 0.98 0.92 0.84 0.69 0.48 21.3 -0.16 -0.09 0.04 0.25 0.56 0.85 0.98 1.00 0.98 0.92 0.78 0.55 20.6 -0.19 -0.13 -0.02 0.18 0.48 0.77 0.92 0.98 1.00 0.97 0.86 0.66 19.8 -0.14 -0.09 0.01 0.17 0.43 0.69 0.84 0.92 0.97 1.00 0.95 0.80 19.1 -0.04 0.00 0.07 0.20 0.39 0.58 0.69 0.78 0.86 0.95 1.00 0.94 18.4 0.05 0.08 0.13 0.21 0.32 0.42 0.48 0.55 0.66 0.80 0.94 1.00 q2 (GeV2) 26.5 25.7 25.0 24.3 23.5 22.8 22.1 21.3 20.6 19.8 19.1 18.4 26.5 1.00 0.98 0.95 0.89 0.85 0.88 0.9 0.91 0.91 0.92 0.91 0.90 25.7 0.98 1.00 0.98 0.92 0.87 0.87 0.88 0.89 0.90 0.91 0.9 0.88 25.0 0.95 0.98 1.00 0.97 0.94 0.94 0.94 0.94 0.95 0.95 0.95 0.94 24.3 0.89 0.92 0.97 1.00 0.99 0.99 0.98 0.98 0.98 0.98 0.98 0.97 23.5 0.85 0.87 0.94 0.99 1.00 0.99 0.99 0.98 0.98 0.98 0.98 0.98 22.8 0.88 0.87 0.94 0.99 0.99 1.00 1.0 1.00 0.99 0.99 0.99 0.99 22.1 0.90 0.88 0.94 0.98 0.99 1.00 1.00 1.00 1.00 1.00 1.00 1.00 21.3 0.91 0.89 0.94 0.98 0.98 1.00 1.00 1.00 1.00 1.00 1.00 1.00 20.6 0.91 0.90 0.95 0.98 0.98 0.99 1.00 1.00 1.00 1.00 1.00 1.00 19.8 0.92 0.91 0.95 0.98 0.98 0.99 1.00 1.00 1.00 1.00 1.00 1.00 19.1 0.91 0.9 0.95 0.98 0.98 0.99 1.00 1.00 1.00 1.00 1.00 1.00 18.4 0.90 0.88 0.94 0.97 0.98 0.99 1.00 1.00 1.00 1.00 1.00 1.00 054507-25 JON A. BAILEY et al. PHYSICAL REVIEW D 79, 054507 (2009) TABLE V. Total bootstrap covariance matrix for the B -> irtv form factor, /+ (q2), derived by adding the statistical and systematic errors in quadrature. The elements of the matrix are given by Mit = X af where «-/ ..[,,?) is the total uncertainty in /+ (q2) in the /' th q2 bin. q2 (GeV2) 26.5 25.7 25.0 24.3 23.5 22.8 22.1 21.3 20.6 19.8 19.1 18.4 26.5 5.13 2.74 1.49 0.79 0.39 0.18 0.06 0.01 -0.0 0.01 0.03 0.05 25.7 2.74 1.48 0.82 0.45 0.24 0.12 0.05 0.02 0.01 0.02 0.03 0.04 25.0 1.49 0.82 0.47 0.27 0.15 0.09 0.05 0.03 0.02 0.02 0.02 0.03 24.3 0.79 0.45 0.27 0.17 0.11 0.07 0.05 0.03 0.03 0.02 0.02 0.02 23.5 0.39 0.24 0.15 0.11 0.08 0.06 0.04 0.04 0.03 0.03 0.02 0.02 22.8 0.18 0.12 0.09 0.07 0.06 0.05 0.04 0.04 0.03 0.03 0.02 0.02 22.1 0.06 0.05 0.05 0.05 0.04 0.04 0.04 0.03 0.03 0.03 0.02 0.02 21.3 0.01 0.02 0.03 0.03 0.04 0.04 0.03 0.03 0.03 0.02 0.02 0.02 20.6 0.00 0.01 0.02 0.03 0.03 0.03 0.03 0.03 0.03 0.02 0.02 0.02 19.8 0.01 0.02 0.02 0.02 0.03 0.03 0.03 0.02 0.02 0.02 0.02 0.02 19.1 0.03 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 18.4 0.05 0.04 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 B -> 7r-CV form factor, /+(<y2), which were used in our facilitate the use of our result, we also show the resulting model-independent determination of I Vub|. 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