B̄→D*ℓν̄ form factor at zero recoil from three-flavor lattice QCD: a model independent determination of [Vcb]

We present the first lattice QCD calculation of the form factor for B→ D* lv with three flavors of sea quarks. We use an improved staggered action for the light valence and sea quarks (the MILC configurations), and the Fermilab action for the heavy quarks. The form factor is computed at zero recoil using a new double ratio method that yields the form factor more directly than the previous Fermilab method. Other improvements over the previous calculation include the use of much lighter light-quark masses, and the use of lattice (staggered) chiral perturbation theory in order to control the light-quark discretization errors and chiral extrapolation. We obtain for the form factor, FB→D*(1) = 0:921(13)(20), where the first error is statistical and the second is the sum of all systematic errors in quadrature. Applying a 0.7% electromagnetic correction and taking the latest PDG average for FB→D*(1)[Vcb] leads to [Vcb] = (38:7+ 0:9exp + 10theo) X 10-3.

B -* D*iv form factor at zero recoil from three-flavor lattice QCD: A model independent determination of | Vcb | C. Bernard,1 C. DcTar,2 M. Di Picrro,J A. X. El-Khadra,4 R.T. Evans,4 E. D. Freeland,5 E. Gamiz,4 Steven Gottlieb,6 U.M. Heller,7 J.E. Hctrick,8 A. S. Kronfcld,9 J. Laiho,1'9 L. Lcvkova,2 P.B. Mackenzie,9 M. Okamoto,9 J. Simone,9 R. Sugar,10 D. Toussaint,11 and R.S. Van dc Water9 (Fermilab Latticc and MILC Collaborations) 1 Department of Physics, Washington University, St. lx>uis, Missouri, USA 2Pliysics Department, University of Utah, Salt Ixike City, Utah, USA 3School of Computer Science, Telecommunications and Infonnation Systems, DePaul University, Chicago, Illinois, USA 4Physics Department, University of Illinois, Urbana, Illinois, USA 5Liberal Arts Department, The School of the Art Institute of Chicago, Chicago, Illinois, USA 6Department of Physics, Indiana University, Bloomington, Indiana, USA 1 American Physical Society, Ridge, New York, USA 8Physics Department, University of the Pacific, Stockton, California, USA 9Fermi National Accelerator Laboratoty, Batavia, Illinois, USA 10Department of Physics, University of California, Santa Barbara, California, USA 11 Department of Physics, University of Arizona, Tucson, Arizona, USA (Received 28 August 2008; published 22 January 2009) We present the first lattice QCD calculation of the form factor for B -» D 'i'P with three flavors of sea quarks. We use an improved staggered action for the light valence and sea quarks (the MILC configu­rations), and the Fermilab action for the heavy quarks. The form factor is computed at zero recoil using a new double ratio method that yields the form factor more directly than the previous Fermilab method. Other improvements over the previous calculation include the use of much lighter light-quark masses, and the use of lattice (staggered) chiral perturbation theory in order to control the light-quark discretization errors and chiral extrapolation. We obtain for the form factor, T= 0.921(I3)(20), where the first error is statistical and the second is the sum of all systematic errors in quadrature. Applying a 0.7% electromagnetic correction and taking the latest PDG average for jF«_/j‘(l)|VL.h\ leads to \Vch\ = (38.7 ± 0.9exp ± 1.0theo) X 10"3. DOI: 10.1103/PhysRevD.79.014506 PACS numbers: 12.38.Gc, 12.15.Hh, 13.25.Hw PHYSICAL REVIEW D 79, 014506 (2009) I. INTRODUCTION The Cabibbo-Kobayashi-Maskawa matrix element Vcb plays an important role in the study of flavor physics [1]. Sincc IV^I is one of the fundamental parameters of the standard model, its value must be known prcciscly in order to scarch for new physics by looking for inconsistencies between standard model predictions and experimental measurements. For example, the standard model contribu­tion to the kaon mixing parameter eK depends sensitively on | Vch | (as the fourth power), and the present errors on this quantity contribute errors to the theoretical prediction of eK that arc around the same size as the errors due to BK, the kaon bag parameter, which has been the focus of much rcccnt work [2-5]. It is possible to obtain IV^I from both inclusive and cxclusivc scmilcptonic B dccays, and both determinations arc limited by theoretical uncertainties. The inclusive method [6-10] makes use of the heavy-quark expansion and perturbation theory. The method also re­quires nonperturbative input from experiment, which is obtained from the measured moments of the inclusive form factor B->X(.-CP( as a function of the minimum clcctron momentum. The dominant uncertainties in this method arc the truncation of the heavy-quark expansion and perturbation theory [11,12]. In order to be competitive with the inclusive determination of and thus serve as a cross-chcck, the cxclusivc method requires a reduction in the uncertainty of the B -> D* scmilcptonic form factor ^ B->D* > which has been calculated previously using latticc QCD in the qucnchcd approximation [13]. Given the phcnomcnological importance of |V(.^|, wc have revisited the calculation of J7B_D* at zero recoil using the 2+1 flavor MILC ensembles with improved light staggered quarks [14,15]. The systematic error due to qucnching is thus eliminated. The systematic error associ­ated with the chiral extrapolation to physical light-quark masses is also rcduccd significantly. Sincc staggered quarks arc computationally less expensive than many other formulations, wc arc able to simulate at quite small quark masses; our lightest corresponds to a pion mass of roughly 240 McV. Given the previous cxpcricncc of the MILC Collaboration with chiral fits to light meson masses and 1550-7998/ 2009/79(1 )/014506(20) 014506-1 © 2009 The American Physical SocietyC. BERNARD et al. PHYSICAL REVIEW D 79, 014506 (2009) decay constants [161, we are in a regime where we expect rooted staggered chiral perturbation theory (rS^PT) [17­211 to apply. We therefore use the rS^PT result for the B -> D* form factor [221 to perform the chiral extrapolation and to remove discretization effects particular to staggered quarks. In addition, we introduce a set of ratios that allows us to disentangle light- and heavy-quark discretization effects, and we suggest a strategy for future improvement. Finally, we extract the B -> D* form factor using a differ­ent method from that originally proposed in Ref. [131. This new method requires many fewer three-point correlation functions, and has allowed for a savings of roughly a factor of 10 in computing resources, while at the same time simplifying the analysis. The differential rate for the semileptonic decay B -> D*(,v( is dr g2 - = - Air' ° dw ]r{mB - mD.)2V>v2 - 1 x g(W)\vchmB^(W)\2. (1) where = v' ■ v is the velocity transfer from the initial state to the final state, and (j(m')|J7b^0*(m,)|2 contains a combination of four form factors that must be calculated nonperturbatively. At zero recoil (j(l) = 1, and .yB^D*(l) reduces to a single form factor hAi( 1). Given /i4|(1), the measured decay rate determines |V7f/,|. The quantity hAl is a form factor of the axial-vector current (D*(v, eOlJ^ |B(u)) = isjlmB2m[yeltlhA] (1). (2) where is the continuum axial-vector current and e is the polarization vector of the D*. Heavy-quark symmetry plays a useful role in constraining /^(l), leading to the heavy-quark expansion [23,241 Va 1 - 2€4 (2 mcY 2mc2mh (2 mbY (3) up to order 1/m2,, and where r]A is a factor that matches heavy-quark effective theory (HQET) to QCD [25,261. The •Ts are long-distance matrix elements of the HQET. Heavy- quark symmetry forbids terms of order I /m ,j at zero recoil [271, and various methods have been used to compute the size of the 1 /ntg coefficients, including quenched lattice QCD [131. The earlier work by Hashimoto et al. [131 used three double ratios in order to obtain separately each of the three 1 /m2Q coefficients inEq. (3). These three double ratios also determine three out of the four coefficients appearing at 1 /nig in the heavy-quark expansion. It was shown in Ref. [281 that, for the Fermilab method matched to tree level in as and to next-to-leading order in HQET, the leading discretization errors for the double ratios for this quantity are of order as(A/2mg)2f%(amg) and (A/2mg'yfjiamg). where A is a QCD scale stemming from the light degrees of freedom, such as that appearing in the HQET expansion for the heavy-light meson mass mM = nig + A + • • •. The functions fj(amg) are coeffi­cients depending on anig and as. but not on A. When aniQ ~ 1, the fj(amg) are of order one; when anig 1, they go like a power of anig, such that the continuum limit is obtained. The powers of 2 are combinatoric factors. As discussed in Ref. [131, all uncertainties in the double ratios 31 used in that work scale as 31 - 1 rather than as %. Statistical errors in the numerator and denominator are highly correlated and largely cancel in these double ratios. Also, most of the normalization uncertainty in the lattice currents cancels, leaving a normalization factor close to 1, which can be computed reliably in perturbation theory. Finally, the quenching error, relevant to Ref. [131 but not to the present unquenched calculation, scales as 31 - 1 rather than as 31. This scaling of the error occurs because the double ratios constructed in Ref. [131 become the identity in the limit of equal bottom and charm quark masses. In the calculation reported here, the form factor hAi (1) is computed more directly using only one double ratio T? - (D*\c7j75b\B)(B\byjy5c\D*) _ |; 2 '4| {D*|cy4c|D*){B|Z?y4Z?|B) ' which is exact to all orders in the heavy-quark expansion in the continuum.1 The lattice approximation to this ratio still has discretization errors that are suppressed by inverse powers of heavy-quark masses \as(A/2mg)2 and {A/2nig)3}. but which again vanish in the continuum limit. The errors in the ratio introduced in Eq. (4) do not scale rigorously as 31j4| - 1 because 3l4| is not one in the limit of equal bottom and charm quark masses. Nevertheless, this double ratio still retains the desirable features of the previous double ratios, i.e., large statistical error cancella­tions and the cancellation of most of the lattice current renormalization. Because the quenching error has been eliminated, the rigorous scaling of all the errors as 31 - 1, including the quenching error, is no longer crucial. The more direct method introduced here has the significant advantage that extracting coefficients from fits to HQET expressions as a function of heavy-quark masses is not necessary, and no error is introduced from truncating the heavy-quark expansion to a fixed order in 1 /nig. In short, for an unquenched QCD calculation, the method using Eq. (4) gives a smaller total error than the method used in Ref. [131 for a fixed amount of computer time. The currents of lattice gauge theory must be matched to the normalization of the continuum to obtain 31j4|. The matching factors mostly cancel in the double ratio [29,301, *Notc that the notation stands for a different double ratio in Ref. [13], 014506-2 B - D*iv FORM FACTOR AT PHYSICAL REVIEW D 79, 014506 (2009) leaving hA[ (1) = \ Ps/Ra, - where RAf is the lattice double ratio and p, the ratio of matching factors, is very close to 1. (For the remainder of this paper we shall use the convention that a script letter corresponds to a continuum quantity, while a nonscript letter corresponds to a lattice quantity.) This p factor has been calculated to one-loop order in perturbative QCD, and is found to contribute less than a 0.5% correction. We have exploited the p factors to implement a blind analysis. Two of us involved in the perturbative calculation applied a common multiplicative offset to the p factors needed to obtain /i^Q) at different lattice spacings. This offset was not disclosed to the rest of us until the procedure for determining the systematic error budget for the rest of the analysis had been finalized. The unquenched MILC configurations generated with 2 + 1 flavors of improved staggered fermions make use of the fourth-root procedure for eliminating the unwanted four-fold degeneracy of staggered quarks. At nonzero lat­tice spacing, this procedure has small violations of unitar­ity [31-35] and locality [36]. Nevertheless, a careful treatment of the continuum limit, in which all assumptions are made explicit, argues that lattice QCD with rooted staggered quarks reproduces the desired local theory of QCD as a -> 0 [37,38]. When coupled with other analyti­cal and numerical evidence (see Refs. [39-41] for reviews), this gives us confidence that the rooting procedure is in­deed correct in the continuum limit. The outline of the rest of this paper is as follows: Section II describes the details of the lattice simulation. Section III discusses the fits to the double ratios accounting for oscillating opposite-parity states. Section IV summa­rizes the lattice perturbation theory calculation of the p factor. Section V introduces the rooted staggered chiral perturbation theory formalism and expressions used in the chiral extrapolations. Section VI then discusses our treatment of the chiral extrapolation and introduces our approach for disentangling heavy- and light-quark discre­tization effects. Section VII provides a detailed discussion of our systematic errors, and we conclude in Sec. VIII. II. LATTICE CALCULATION The lattice calculation was done on the MILC ensembles at three lattice spacings with a ~ 0.15, 0.125, and 0.09 fm; these ensembles have an O(cr) Symanzik improved gauge action and 2+1 flavors of "AsqTad" improved staggered sea quarks [42-47]. The parameters for the MILC lattices used in this calculation are shown in Table I. We have several light masses at both full QCD and partially quenched points (mval # mse.,), and our light-quark masses range between mj 10 and mj2. Table II shows the valence masses computed on each ensemble. In this work we follow the notation [16], where ms is the physical strange quark mass, m is the average u-cl quark mass, and m', m's indicate the nominal values used in simulations. In prac­tice, the MILC ensembles choose m's within 10-30% of ms and a range of m' to enable a chiral extrapolation. The heavy quarks are computed using the Sheikholeslami-Wohlert (SW) "clover" action [48] with the Fermilab interpretation via HQET [49]. The SW action includes a dimension-five interaction with a coupling csw that has been adjusted to the value Mq3 suggested by tadpole-improved, tree-level perturbation theory [50]. The value of u0 is calculated either from the plaquette (a ~ 0.15 fm and a ~ 0.09 fm), or from the Landau link (a ~ 0.12 fm). The adjustment of csw is needed to normalize the heavy quark's chromomagnetic moment correctly [49]. The tadpole-improved bare quark mass for SW quarks is given by 1/1 1 \ am0 = - - - ------ . (5) u0 \2k 2/ccrity where tuning the parameter x to the critical quark hopping parameter Kcrjt would lead to a massless pion. The spin- averaged Bs and Ds kinetic masses are computed on a TABLE I. Parameters of the simulations. The columns from left to right are the approximate lattice spacing in fm, the sea quark masses am'/ani's, the linear spatial dimension of the lattice ensemble in fm, the dimensionless factor m^L (mCT corresponds to the taste-pseudoscalar pion composed of light sea quarks), the gauge coupling, the dimensions of the lattice in lattice units, the number of configurations used for this analysis, the bare hopping parameter used for the bottom quark, the bare hopping parameter used for the charm quark, and the clover term csw used for both bottom and charm quarks. a (fm) am'/am's L (fm) io/r Volume # Configs *h Kc £'sw 0.15 0.0194/0.0484 2.4 5.5 6.586 163 X 48 628 0.076 0.122 1.5673 0.15 0.0097/0.0484 2.4 3.9 6.572 163 X 48 628 0.076 0.122 1.5673 0.12 0.02/0.05 2.4 6.2 6.79 203 X 64 460 0.086 0.122 1.72 0.12 0.01/0.05 2.4 4.5 6.76 203 X 64 592 0.086 0.122 1.72 0.12 0.007/0.05 2.4 3.8 6.76 203 X 64 836 0.086 0.122 1.72 0.12 0.005/0.05 2.9 3.8 6.76 243 X 64 528 0.086 0.122 1.72 0.09 0.0124/0.031 2.4 5.8 7.11 283 X 96 516 0.0923 0.127 1.476 0.09 0.0062/0.031 2.4 4.1 7.09 283 X 96 556 0.0923 0.127 1.476 0.09 0.0031/0.031 3.4 4.2 7.08 403 X 96 504 0.0923 0.127 1.476 014506-3 TABLE II. Valence masses used in the simulations. The col­umns from left to right are the approximate lattice spacing in fm, the sea quark masses atrt'/am's identifying the gauge ensemble, and the valence masses computed on that ensemble. C. BERNARD et at. a (fm) am' / am's amx = 0.15 0.0194/0.0484 0.0194 = 0.15 0.0097/0.0484 0.0097, 0.0194 = 0.12 0.02/0.05 0.02 = 0.12 0.01/0.05 0.01, 0.02 = 0.12 0.007/0.05 0.007, 0.02 = 0.12 0.005/0.05 0.005, 0.02 = 0.09 0.0124/0.031 0.0124 = 0.09 0.0062/0.031 0.0062, 0.0124 = 0.09 0.0031/0.031 0.0031, 0.0124 subset of the ensembles in order to tunc the bare k values for bottom and charm (and hcncc the corresponding bare quark masses) to their physical values. These tuned values were then used in the B -> D*Cv form-factor production run. The relative latticc scale is determined by calculating r\/a on each ensemble, where r\ is related to the force between static quarks by rfF(rj) = 1.0 [51,52]. To avoid introducing implicit dependence on m', m's via r\(ni', m's, g2) (where, as above, primes denote simulation masses), wc interpolate in m's and extrapolate in in' to obtain rx{m,ms, g2)/a at the physical masses. Wc then convert from latticc units to rj units with r\(m,ms, g2)/a. Below wc shall call this procedure the mass-independent determination of rj. In order to fix the absolute latticc scale, one must com­pute a physical quantity that can be compared directly to experiment; wc use the Y 2S-1S splitting [53] and the most recent MILC determination of fw [54]. The difference between these determinations results in a systematic error that turns out to be much smaller than our other systcm- atics. When the Y scale determination is combined with the continuum extrapolated rj value at physical quark masses, a value rphys = 0.318(7) fm [55] is obtained. The f w determination is rphys = 0.3108( 15)(^||) fm [54]. Given rf1*' ', it is then straightforward to convert quantities measured in r\ units to physical units. The dependence on the latticc spacing a is mild in this analysis. Sincc a only enters the calculation through the adjustment of the heavy- and light-quark masses, the de­pendence of /iA, (1) on a>s small. Staggered chiral pertur­bation theory indicates that the a dependence coming from staggcrcd-quark discretization effects is small [22], and this is consistent with the simulation data. In this work, wc construct latticc currents as in Ref. [49], Jf = ^ZfZ^'^hY^h,, (6) Vh = (1 + adxy • Diat)ijth, (7) where tf/h is the (heavy) latticc quark field in the SW action. Diat is the symmetric, nearest-neighbor, covariant difference operator; the tree-level improvement cocfficicnt PHYSICAL REVIEW D 79, 014506 (2009) current. The rotated field is defined by cU = 1 1 1 uq\ 2 + m0a 2(1 + m0a) (8) In Eq. (6) wc choose to normalize the current by the factors of Zyh (h = c, b) sincc even for massive quarks they arc easy to compute nonpcrturbativcly. The continuum current is related to the latticc current by -j hh' _ „ ihh' J n ~ PjrJn up to discretization effects, where PI Zbc nrcb T /...J J J p J p yee nrbb y v4 (9) (10) and the matching factors Zj^'s arc defined in Ref. [30]. Note that the factor ^Z^Z^F multiplying the latticc current in Eq. (6) cancels in the double ratio by design, leaving only the p factor, which is close to 1 and can be computed reliably using perturbation theory. The perturbative calcu­lation of pJv is described in more detail in Sec. IV. Interpolating operators arc constructed from four- component heavy quarks and staggered quarks as follows: Let &l(x) = ifrb(x)y5{l(x)x(x), (11) (12) where x's the one-component field in the staggcrcd-quark action, and a(x) = y\s/"y\:/"y^/"yKi/" (13) The left (right) index of fit (fi) can be left as a free taste index [41] or x can t>c promoted to a four-component naivc-quark field to contract all indices [56]. The resulting correlation functions arc the same if the initial and final taste indices arc set equal and then summed. The same kinds of operators have been used in previous calculations [57-59], Latticc matrix elements arc obtained from three-point correlation functions. The three-point correlation functions needed for the B -> D* transition at zero-recoil arc CB^ir(,th ts, If) = £<O|0fl.(x, tf)-9cyjy5Vb(y, ts) x,y where f„ is either the vector (iy1*) or axial-vcctor [iy^y$) X 0\{0, /,)|0>, (14) 014506-4 CB^B(tj, Ts, tf) = X<O|0B(x, tf)^by4^b{y. Ts) vy x 0|(O, r,)|0), (15) Cn 'n (/,. ts, Tf) = X<O|0/r (x. //I'l', Ti'My. O vy x 0+,((U,)|O>. (16) In CB^/;) the polarization of the D* lies along spatial direction /'. If the source-sink separation is large enough, then we can arrange for both rs - tt and tf - ts to be large so that the lowest-lying state dominates. Then B - D *iv FORM FACTOR AT ... cB-n (t,. r, tf) = z;;~z 1/2 1/2r^75^ (17) where mB and m ,y are the masses of the B and D* mesons and ZH = |<O|0H|tf>|2. In practice, the meson source and sink are held at fixed tj = 0 and tf = T, while the operator time ts = t is varied over all times in between. Using the correlators defined in Eqs. c 14)-116) we form the double ratio CB^n (0. t. T)C,r^B(0. t. T) Cn -n (0. t. T)CB^B(0. t. T) (18) All convention-dependent normalization factors, including the factors of <JZH/2mH, cancel in the double ratio. In the window of time separations where the ground state domi­nates, a plateau should be visible, and the lattice ratio is simply related to the continuum ratio 'RAi by a renormal­ization factor Pa with pAi as in Eq. (10). The right-hand side of Eq. (17) is the first term in a series, with additional terms for each radial excitation, including opposite-parity states that arise with staggered quarks. Eliminating the opposite-parity states requires some care, and this is discussed in detail in the next section. In order to isolate the lowest-lying states we have chosen creation and annihilation operators, 0| and 0/y, which have a large overlap with the desired state. This was done by smearing the heavy quark and antiquark propagator sources with IS Coulomb-gauge wave functions. III. FITTING AND OPPOSITE-PARITY STATES Extracting correlation functions of operators with stag­gered quarks presents an extra complication because the contributions of opposite-parity states introduce oscilla­tions in time into the correlator fits [56]. Three-point functions obey the functional form PHYSICAL REVIEW D 79, 014506 (2009) Cx^ Y(0.t.T) = X A-0t=0 Xi-lY^Aae- 10. • (20) For odd k and (. the excited-state contributions change sign as the position of the operator varies by one time slice. Although they are exponentially suppressed, the parity partners of the heavy-light mesons are not that much heavier than the ground states in which we are interested, so the oscillations can be significant at the source-sink separations typical of our calculations. These separations cannot be too large because of the rapid decrease of the signal due to the presence of the heavy quark. Although one can fit a given three-point correlator to Eq. (20), in the calculation of hAi (1) we use double ratios in which numerator and denominator are so similar that most of the fitting systematics cancel, and it is convenient to preserve this simplifying feature. We do this by forming a suitable average over correlator ratios with different (even and odd) source-sink separations. It turns out that the amplitudes of the oscillating states in B -> D* correlation functions are much smaller than they are in many other heavy-light transitions [60,61], and that the oscillating states in B -> D* are barely visible at the present level of statistics. Even so, we introduce an average that reduces them still further, to the point where they are negligible. Although we shall take the average of the double ratio, let us first examine the average of an individual three-point function. Expanding Eq. (20) so that it includes the ground state and the first oscillating state, we have C^r(0. t, T) = + (- l)T~,A^¥ + (- 1 )r_Af pr e ~ ~ "'I-(r_ r) + ... = + cx^r(0. t, T) (21) where in the last line we have pulled out the ground state amplitude and exponential dependence. The function cA^r(0. t, T) is defined cx^Y(0. t, T) An] 01 Ax~*y ^00 Ax^r + -j£^7(-l)'erA"'*' ^o(T ax~*y + (22) ^0(T where AmXj = m'XY - mXj is the splitting between the lowest-lying desired-parity state and the lowest-lying wrong-parity state. Note that the first two terms produce oscillations as the position of the operator is varied over the 014506-5 time extent of the lattice. The third term, however, changes sign only when the total source-sink separation is varied. It is this term that our average is designed to suppress, since it will not be as clearly visible in the t dependence of the lattice data as those that oscillate in t. We define the average to be C. BERNARD et al. CA'^F(0. t. T) = ±CA^F(0. t. T) + ±C*^F(0. t. T + 1) + ±CA'^F(0. t + 1. T + 1). (23) Substituting the expression for CA_^} (0.1, T) from Eq. (21) into this definition gives CA^F(0. t. T) = AlfrY X [1 + cA'^F(0. t.T) + ...]. (24) where the function cx^} is cA^} (0. t. T) = ax-+y (-j) aX->Y v ' Am T- t g-Am y (T- t) X 1 1 , - + -(1 .2 4 -A my + A A'->}' 10 4X->}' Aoo (-l)V -A niyt - + -(1 .2 4 -A mx 4X->}' + _U___l)T -Amxl-Amr(T-l) aX^Yk v Am X - - - 2 4 -Am, (25) Note that Eq. (25) has the same exponential time depen­dence as Eq. (22), but with the size of the amplitudes reduced by the factors in square brackets. Thus, the aver­age is equivalent to a smearing that reduces the oscillating state amplitudes. It is possible to compute the Amx pre­cisely from fits to two-point correlators. We find values between about 0.2 and 0.4 in lattice units. Given these values, the first two factors in brackets reduce their respec­tive amplitudes by approximately a factor of 2, and the targeted, nonoscinating term is reduced by a factor of -6-10. Specializing to the B -> D* case, consider the double ratio RaMi.t) 4 B->[Y' aD*'-> ^00 ^00 aD'^D' a aqo aqo + cD'^B(0. t. T) - c - cB^B(0, t, T) + ...]. [1 + cB^D'(0.t.T) " ■" (0.1. T) (26) where we have again factored out the ground state contri­bution. Eq uation (26) follows from Eq. (18) treating the c s as small. Note that the c s are expected to be similar in numerator and denominator, and to the extent that they are the same they will cancel in this expression. Applying the average in Eq. (23) directly to the double ratio. PHYSICAL REVIEW D 79, 014506 (2009) FIG. 1 (color online). Double ratio RA on the am1 = 0.0124 fine (a = 0.09 fm) ensemble. The source was fixed to time slice 0. and the operator position was varied as a function of time. Two different sink points were used with even and odd time separa­tions between source and sink in order to study the effect of nonoscillating contributions from wrong parity states. R(0.1. T) = ifi(0.1. T) + ifi(0.1. T + 1) + ifi(0./ + 1 ,T+ 1). (27) we get * 4, (0. /. T) = ^:.[1 + cB-D (0. /. T) Aoo Aoo + c^r(0. t, T) - cD^D' (0. t, T) - cB^B'(0.t.T) + ...]. (28) where each of the oscillating state terms in the individual three-point functions is suppressed according to Eq. (25). Although A mB and A mD* can be obtained from fits to the two-point correlators, the oscillating state amplitudes FIG. 2 (color online). Averaged double ratio. RAr of Eq. (27) on the am' = 0.0124 fine (a = 0.09 fm) ensemble. The plateau fit is shown with 1 a error band. 014506-6 appearing in the three-point correlators must be determined directly from the three-point correlator data. Figure 1 shows the double ratio RA[ used to obtain hA[{ 1). The source is at time slice 0, the sink is at T, and the operator position is varied along t. Two different source-sink sepa­rations were generated that differed by a single time unit at the sink (T = 17, 18). The average of these two correlators was taken according to Eq. (27), and this average was fit (including the full covariance matrix) to a constant, as shown in Fig. 2. There is no detectable oscillation even before the average is taken, as can be seen in Fig. 1; according to Eq. (25) the oscillating contributions are reduced even further in the average so that their systematic errors can be safely neglected. B - D i v FORM FACTOR AT ... IV. PERTURBATION THEORY Lattice perturbation theory is needed in order to calcu­late the short-distance coefficient pA[ defined in Eq. (19). Although naive lattice perturbation theory ap­pears to converge slowly, the two main causes have been identified [50]: the bare gauge coupling is a poor expansion parameter, and coefficients are large when tadpole dia­grams occur. If a renormalized coupling is used as an expansion parameter, and one computes only those quan­tities for which the tadpole diagrams largely cancel, then lattice perturbation theory seems to converge as well as perturbation theory in continuum QCD. Only the vertex correction contributes to the p factor, as the wave-function renormalization (including all tadpoles) cancels by construction. Even the vertex correction par­tially cancels, and the one-loop coefficient is found to be small. The perturbative corrections to the p factor can be written as 7hh' „hh> Pjl jyhh yh'h' 1 + av{q'‘)4rrpj^ + .. (29) where Pj1'}^ is the coefficient of the one-loop correction, and the coupling av is the renormalized strong coupling constant in the V scheme [50,62], which is based on the static-quark potential. The coupling is determined follow­ing the procedure of Ref. [63]. The scale q* of the running coupling av{cf) should be chosen to be the typical mo­mentum of a gluon in the loop. A prescription for calculat­ing this scale was introduced by Brodsky, Lepage, and Mackenzie [50,62]. They defined cf by Ink/2) f d4qf{q)\a{q2) fd4qf{q) ' (30) where f{q) is the one-loop integrand and the numerator is the first log moment. This prescription was extended by Hornbostel, Lepage, and Morningstar (HLM) [64] to cases where the one-loop contribution is anomalously small leading to a break down of Eq. (30). The HLM prescription TABLE III. Computed values of pAl in the HLM prescription [64], The first three columns label each ensemble with the approximate lattice spacing in fm, the light sea quark mass am1, and the strange quark mass am's. The fourth column is t/i/i'u vt- where the error is calculated using the statistical error from VKGAS for the 0th, 1 st, and 2nd moments of the one-loop integrals. The fifth column is pi4| on that ensemble, and the errors are the statistical errors from the VKGAS evaluation, including the one-loop coefficients and i/HLV1. PHYSICAL REVIEW D 79, 014506 (2009) a (fm) am1 am's fl5HLM Pa, 0.15 0.0194 0.0484 2.03(10) 0.9966(2) 0.15 0.0097 0.0484 2.03(10) 0.9966(2) 0.12 0.02 0.05 1.96(10) 0.9964(2) 0.12 0.01 0.05 1.96(10) 0.9964(2) 0.12 0.007 0.05 1.96(10) 0.9964(2) 0.12 0.005 0.05 1.96(10) 0.9964(2) 0.09 0.0124 0.031 2.98(14) 1.00 298(9) 0.09 0.0062 0.031 2.98(14) 1.00 300(9) 0.09 0.0031 0.031 2.98(14) 1.00 301(9) for q* takes into account two-loop contributions to the gluon propagator via the inclusion of second log moments. Since we do encounter anomalously small one-loop cor­rections in pA[, the HLM prescription was used to deter­mine q*. Results for and pA[ needed for this calculation are given in Table III. The p factor varies somewhat as a function of lattice spacing, and is even slightly different from ensemble to ensemble at the same nominal lattice spacing, due to the slightly different /3 values used to generate the gauge fields. The calculation of pA[ is described in Refs. [65,66]. It uses automated perturbation theory techniques to generate the Feynman rules and VEGAS [67] for the numerical integration of the loop integrals. As a check, it was verified that this calculation reproduces known results for the heavy-heavy currents with the Wilson plaquette action [29] and for the V4 current in the massless limit with the Symanzik improved gauge action. As mentioned in the introduction, we have exploited the p factor to implement a blind analysis. Two of us applied a multiplicative offset close to 1 to the p factor, generated with a random key. The offset was not unlocked until the procedure for determining the systematic errors in the rest of the analysis had been finalized. V. STAGGERED CHIRAL PERTURBATION THEORY The simulation masses m'val and m'ea (for valence and sea) are all larger than the physical m. A controlled chiral extrapolation can be guided by an appropriate chiral effec­tive theory that includes the effect of staggered-quark discretization errors. Rooted staggered chiral perturbation theory (rS^PT), which has been formulated for heavy-light quantities in Ref. [68], is such a theory. In rS^PT, a replica j 014506-7 C. BERNARD et al. PHYSICAL REVIEW D 79, 014506 (2009) method is used to take into account the effect of rooting; this procedure has been justified in Refs. [33,69]. Because of taste-symmetry breaking, the staggered the­ory has 16 light pseudoscalar mesons instead of 1. The tree-level relation for the masses of light staggered mesons in the chiral theory is [17,18] way to estimate systematic errors. We do not include the NNLO logarithms because they are unknown and would require a two-loop calculation. The expression including analytic terms through NNLO is h NNLO ,41 (1)/Va = 1 + NLO + cxml + c2(2ml + ml) m Ho( mx + mx) + a2 A; (31) + c3fr. (33) where mx and my are staggered-quark masses, fi0 is the continuum low-energy constant, and a2As are the split­tings of the 16 pious of taste E. For staggered quarks there exists a residual SO(4) taste symmetry broken at (D(a2), such that there is some degeneracy among the 16 pions [17], and the taste index E runs over the multiplets P, A, J, V, I with degeneracies 1, 4, 6, 4, 1. The splitting a2AP vanishes because there is an exact (nonsinglet) lattice axial symmetry. Schematically, the next-to-leading-order (NLO) result for the relevant form factor is where the subscript P on the meson masses indicates the taste pseudoscalar mass. We use the notation from the rS^PT literature that mx= is a taste E meson made of two valence x quarks, mu= is a taste E meson made of two light sea quarks, and ms= is a taste E meson made of two strange sea quarks. By heavy-quark symmetry, the c, are suppressed by a factor of l/m2. Since the only free parameter through NLO is an overall constant, we include the NNLO analytic terms in the fit used for our central value. This leads to a larger statistical error and is more conservative. hf^W/VA = 1 + Xa(Ay) + X logs,_luup(A ). VI. TREATMENT OF CHIRAL EXTRAPOLATION 48tr2f- (32) where Xa(Ax) is a low-energy constant of the chiral effec­tive theory, and is therefore independent of light-quark mass and cancels the chiral scale dependence Av of the chiral logarithms. By heavy-quark symmetry, Xa(Ax) is proportional to l/m2 in the heavy-quark expansion. The term r]A is a factor that matches heavy-quark effective theory to QCD, and contains perturbative-QCD logarith­mic dependence on the heavy-quark masses; it is indepen­dent of light-quark mass. The term proportional to g2DDis shorthand for the one-loop staggered chiral logarithms, and is given in the appendix for ease of reference. The rooted staggered expression was derived in Ref. [22]. The one- loop staggered logarithms depend on both valence and sea quark masses, and include taste-breaking effects coming from the light-quark sector. This expression also contains explicit dependence on the lattice spacing a, and requires as inputs the parameters of the staggered chiral Lagrangian 8y, 8'A. in addition to the staggered taste splittings Ap.a.t.v.i n6]. These parameters can be obtained from chiral fits to the light pseudoscalar meson sector and are held fixed in the chiral extrapolation of hAl{ 1). The con­tinuum low-energy constant gDDappears, and below we take a generous range inspired by a combined fit to many different experimental inputs, including a leading-order analysis of the D* width. The D-D splitting A,f ' is well determined from experiment. The only other parameter that appears at NLO is the constant X^(A), and this is determined by our lattice data for hA) 1). Although the lattice data are well described by the NLO formula, it is useful to go beyond NLO and to include the next-to-next-to-leading-order (NNLO) analytic terms as a In this section, we discuss the approach we have devel­oped to disentangle the heavy- and light-quark discretiza­tion effects and to perform the chiral and continuum extrapolations. In the Fermilab method, heavy-quark dis­cretization errors can be estimated by comparing the heavy-quark expansions for lattice gauge theory and con­tinuum QCD [28-30,70]. The dependence on a is not simply a power series (unless ma 1), so power-counting estimates in HQET are used. On the other hand, some of the light-quark discretization effects are constrained by rS^PT. The heavy-quark errors are asymptotically con­strained by the Symanzik low-energy Lagrangian when mha 1 and by heavy-quark symmetry even when mha is close to 1. In the region in between, the errors smoothly interpolate the asymptotic behavior [49,70]. The errors in the SW action used for the heavy quarks decrease with lattice spacing as asa in the mha 1 region, as compared with the light-quark (improved staggered) discretization errors, which decrease much faster, as asa2. The first step of the method is to normalize the numeri­cal data for hA (1) to a fiducial point by forming the ratio 'R m1- m's■ a) hAl (mx, m', m's, a) hA (7?2^d. 7hud, i?4id. a)' (34) where /»l,d is a fiducial mass, mx is the light (spectator) valence quark, m' is the isospin averaged light sea quark on a particular ensemble, and m's is the strange sea quark on that ensemble. (Note that the factor of r]A in Eqs. (32) and (33) cancels in the ratio.) The principle advantage of this ratio is that heavy-quark discretization effects largely can­cel, since the heavy quarks are the same in numerator and denominator. This allows us to disentangle the heavy- quark discretization effects from those of the light-quark 014506-8 B - D*U> FORM FACTOR AT ... PHYSICAL REVIEW D 79, 014506 (2009) sector coming from staggered chiral logarithms, thus iso­lating the (taste-violating) discretization effects specific to the staggered light quarks. These light-quark discretization effects can appear in nonanalytic terms in rS^PT and are due to violations of taste symmetry. They can be removed to a given order in rS^ PT (we work to NLO) in fits to the numerical data at multiple lattice spacings using the ex­plicit rS^PT formula of Eq. (33), since this formula in­cludes the staggered lattice artifacts. The continuum limit of the ratio !Rfid can be obtained using our fitted values for parameters in rS^PT and taking a -> 0 in the rS^PT expression for 2lfid. We do not need a more explicit ansatz for the functional form of the heavy-quark discretization effects, since they largely cancel in the ratio. Normalizing the continuum extrapolated ratio !Rfid by hAi at the fiducial point on a very fine fiducial lattice where the heavy-quark discretization effects are small gives a value close to the physical continuum result, hAi [m, m, ms, 0) ~ hAi (mf, mfid, m[ld, afid) X m, ms, 0), (35) where the relation becomes exact as afld -> 0. Note that the requirement that the heavy-quark discretization effects must be small enforces the condition that the improved staggered light-quark discretization effects be even smaller (and likely negligible) because the staggered discretization effects decrease much faster with lattice spacing. The fiducial masses mjd, mfld, and m['d should be chosen large enough that it would be feasible to simulate this mass point on a very fine lattice (since the cost rises significantly as the mass of the light sea quarks is decreased), thus normalizing the lattice data to a point where the heavy-quark discreti­zation effects are small. The fiducial masses should not be chosen so large, however, that rS^PT would not be a reliable guide in performing the continuum and chiral extrapolation of 2lfid. This method can be considered the crudest form of step scaling, but it does illustrate that one does not need lattices, which are simultaneously fine enough for b quarks and large enough for light quarks in order to simulate, with high precision, quantities that in­volve both. In practice, we find m£d m fid m fid 0.4mv and ms are reasonable values for the fiducial masses. The fiducial lattice spacing should be chosen as fine as is practical; a succession of progressively finer fiducial latti­ces would be desirable for verifying that the a dependence is of the expected size. In this work we take our finest lattice (0.09 fm) as the fiducial lattice, but we apply Eq. (35) with the coarser lattices taken as fiducial lattices in order to estimate discretization errors. We note that the method presented above can be applied to all calculations involving the Fermilab treatment of heavy quarks and staggered light quarks, not only the B -> D* iv form factor hAr It may also be desirable to compute quantities at the fiducial point (or a succession of such points) using an even further improved action for the heavy quarks. Once the fiducial lattice spacing is of the order 0.03-0.01 fm, even the bottom quark may be treated as a "light" quark with the highly improved staggered action [71] or with chiral fermions, for which mass dependent discretization effects are small. Conserved currents could then be used for many simple heavy-light quantities, removing the need for a perturbative renormalization. For the chiral extrapolation of hAi we find it useful to form two additional ratios 31 sea(flY, m', a) = hA] (m *, m', m's, a) hAUnf,mM,mf\a)' (36) , v hA,{m„ m', m'K, a) n val(mr, m', ms, a) = ‘ 1 fid „ ~(37) n4| (m™, m, ms, a) whose product is clearly 2lfid, Eq. (34). !R.sea and !R.val separate the sea and valence quark mass dependence, which makes it easier to assess systematic errors. The values of hAl that enter Eqs. (36) and (37) are obtained from (38) PtJRa,' where RAl is the average of double ratios defined in Eq. (28). The ratios in Eqs. (36) and (37) are now qua­druple ratios, where the excited-state contamination is further suppressed over that of the double ratio. Performing the chiral extrapolation, taking the continuum limit of the two ratios, and multiplying them together we recover 2lfid(m, m, ms, 0) by construction. Thus, we can rewrite Eq. (35) as hftys * hA< (m" mfid, m?d, afid) X [K-sea(m> ms, 0) X !Rval(m, m, ms, 0)], (39) fid 0. where, again, the relation becomes exact as a To the extent that the extrapolation in sea quark masses is mild, the ratio !Rsea should be close to 1, since the valence light mass is the same in both numerator and denominator. !Rval contains less trivial chiral behavior. However, since the numerator and denominator are com­puted on the same ensemble (with different valence masses), they are correlated, and statistical errors tend to cancel in !Rval. The ratio !Rsea has small statistical errors because the valence mass /nfd in that ratio is relatively heavy. Of course, the heavy-quark discretization errors are significantly suppressed in both ratios, isolating the light- quark mass dependence and staggered discretization ef­fects. A direct chiral fit to the numerical data (not involving the ratios introduced here) would require a more explicit 014506-9 C. BERNARD et al. PHYSICAL REVIEW D 79, 014506 (2009) TABLE IV. Fiducial masses used at the three different lattice spacings. The first four columns are the approximate lattice spacing in fm, the fiducial valence quark mass, the fiducial light sea quark mass, and the fiducial strange quark mass. The fifth and sixth columns are the values of ^KA and h^, respectively, computed at that fiducial point. Lattice spacing (fm) affix'* amlld ami"1 3?* c. 0.15 0.0194 0.0194 0.0484 0.9211(73) 0.9180(73) 0.12 0.02 0.02 0.05 0.9112(73) 0.9079(73) 0.09 0.0124 0.0124 0.031 0.9210(85) 0.9237(85) ansatz for the treatment of the heavy-quark discretization effects than is needed in the ratio fits.2 Note that in the ratios the fiducial point need not be tuned to the same mass at every latticc spacing; differences can be accounted for in the fit itself. The fiducial points used at different latticc spacings arc m*"1 = »V'd = 0.4m' and m*"1 = m's. The ex­plicit values arc given in Table TV, along with the calcu­lated values of ^RA and at that fiducial point. The constant term X4(Aa,) in Eq. (32) cancels in the ratios 3lsea and 3lval, so the behavior of these ratios is completely predicted through NLO in the chiral expansion. Wc find good agreement between the predicted form and the numerical data. However, given that our fiducial spec­tator quark mass is rather large (around 0Ams), wc include the NNLO analytic terms in the ratio fits in order to estimate systematic errors associated with the chiral ex­pansion. There arc only two new continuum low-energy constants introduced at this higher order, and the ratios 3lsea and 3lval determine one each. There is also an ana­lytic term proportional to a2 appearing at this order, but it cancels in each of the 3lsea and 3lval ratios. In future calculations, it would be feasible to use a much finer latticc spacing for the fiducial point, thereby further reducing heavy-quark discretization errors. For now, how­ever, wc use »V'd, ml"1, 0.09 fm), with the fiducial masses in Table IV, in Eq. (39). As a way to estimate discretization errors wc use our results for hu^ at the two coarser latticc spacings in Eq. (39) also. At the latticc spacings used in this work the light-quark discretization effects may still be non-ncgligiblc compared with heavy-quark discretization effects. With rS^PT it is possible to remove from the discretization effects associated with staggered chiral logarithms, although purely analytic discretization errors remain. Removing this subset of staggered effects leads to a value for the 2A direct (correlated) chiral fit would still, however, reflect the correlations, which cause cancellations in the statistical errors in the ratios. FIG. 3 (color online). !Rva! on the am1 = 0.0062 fine en­semble. The valence mass in the numerator is the full QCD value of am[x = 0.0062 while the fiducial valence mass in the denominator is aml'd = 0.0124. The fit to a constant has a X2/A.o.i = 0.20. fiducial form factor, which wc call the "tastc-violations- out" value. Not removing them leads to the "tastc-viola- tions-in" value. The difference turns out to be negligible, less than 0.1% on our coarsest ensemble and less than 0.01% on the fine ensemble. Thus, the discretization effects in our latticc data coming from taste violations in the staggered chiral logarithms arc extremely small at the fiducial point mass, and wc neglect this difference in the analysis. Figure 3 shows the plateau fit to the ratio 3lval on the fine ensemble with (mh1, am's) = (0.0062, 0.031). The valence mass in the numerator is the full QCD value of am'x = 0.0062, while the fiducial valence mass in the denominator is amsf = 0.0124. Both numerator and denominator arc computed on the same ensemble, so they have the same sea quark masses, and correlated statistical errors largely can­cel in the ratio, as expected. Excited-state contamination is also reduced. Computed values for 3lsea on all of our ensembles arc given in Table V, and the computed values for !Rval arc given in Table VI. TABLE V. Computed values of !R„ca. The first three columns are the arguments of as defined in Eq. (36); they are the light sea quark mass m', the strange quark mass m's, and the approximate lattice spacing in fm. The fourth column is !R„ca. am1 ami a (fm) 0.0097 0.0484 0.15 1.009(12) 0.01 0.05 0.12 1.0070(98) 0.007 0.05 0.12 1.0027(91) 0.005 0.05 0.12 1.014(10) 0.0062 0.031 0.09 1.000(12) 0.0031 0.031 0.09 0.996(10) 014506-10 TABLE VI. Computed values of !Rva]. The first four columns are the arguments of 2lval as defined in Eq. (37); they are the light valence quark mass mx, the light sea quark mass m', the strange quark mass mj, and the approximate lattice spacing in fm. The fifth column is 3lva]. B - D i r FORM FACTOR AT ... am x am1 am's a (fm) Kval 0.0097 0.0097 0.0484 0.15 1.0056(65) 0.01 0.01 0.05 0.12 0.9994(41) 0.007 0.007 0.05 0.12 0.9900(57) 0.005 0.005 0.05 0.12 1.0081(90) 0.0062 0.0062 0.031 0.09 1.0005(50) 0.0031 0.0031 0.031 0.09 1.0043(62) VII. SYSTEMATIC ERRORS In the following subsections, we examine the uncertain­ties in our calculation due to fitting and excited states, the heavy-quark mass dependence, the chiral extrapolation of the light spectator quark mass, discretization errors, and perturbation theory. As mentioned in Sec. II, statistical uncertainties are computed with a single elimination jack- knife and the full covariance matrix. A. Fitting and excited states We have examined plateau fits to the time dependence of the double and quadruple ratios introduced in Secs I and V. The x2 in our fits is defined with the full covariance matrix. The fits to the ratios were done under a single elimination jackknife, after blocking the numerical data by 8 on the fine lattices and by 4 on the coarse and coarser lattices. The blocking procedure averages 4 (or 8) successive configu­rations before performing the single elimination jackknife. These values for the block size were chosen such that the statistical error on the double ratio fit did not increase when a larger block size was used. Statistical errors were deter­mined in fits that included the full correlation matrix, which was remade for each jackknife fit. The jackknife data sets on different ensembles were then combined into a larger block-diagonal jackknife data set in order to perform the chiral fits. In this way, the fully correlated statistical errors were propagated through to the final result. With our high statistics (several hundred lattice gauge field configurations for each ensemble), we are able to resolve the full covariance matrix well enough that we do not need to apply a singular value decomposition cut on the eigenvalues of the covariance matrix. The double ratio fit is needed to establish hAi{\) at the fiducial point (which was computed on the 0.0124/0.031 fine ensemble), while the quadruple ratios, 3lva) and !Rsea are computed on the other ensembles in order to perform the chiral extrapolation and to remove taste-breaking nonanalytic terms. We find that the fit to the double ratio at the fiducial point on the 0.0124/0.031 ensemble is well described by a constant over a range of seven time slices. The excited-state con­tamination in the quadruple ratios is even further sup­pressed, and we find that the correlated x2 values allow for a constant fit region of six to ten time slices, depending upon the lattice spacing. We take the good correlated ^2/d.o.f., ranging from 0.15 to 1.00, in our constant pla­teau fits as evidence that the excited-state contamination in these fits is negligible as compared with other errors. As an additional check of the jackknife fitting procedure, bootstrap fits were done to all of the double and quadruple ratios needed for this work. Close agreement was found for both central values and statistical errors. The statistical errors were typically the same size within 10%, and central values were well within 1 a. The jackknife procedure had slightly larger errors than that of the bootstrap. B. Heavy-quark mass dependence The value for hAi{\) depends on the heavy-quark masses, which are set by tuning the hopping parameters Kb and kc. The principal method starts by fitting the lattice pole energy to £(p) to the dispersion relation 2 3 £(p) = Mi + + bxp4 + ^2 X + '' '' ^0) j=i in order to obtain the kinetic mass M2 (as well as b{ and b2, which are unimportant here). In the Fermilab method [28,30,49], k is adjusted so that the kinetic mass agrees with experiment. Here we take the spin-average of kinetic masses of pseudoscalar and vector heavy-strange mesons and obtain our central values for Kb or kc, respectively, from the (spin-averaged) B{sl) and masses. Applying this procedure we find statistical and fitting errors of 5.6% for Kb and 1.2% for kc on the fine (a = 0.09 fm) ensem­bles. There is an additional error in k due to discretization effects. We determine this error by estimating the size of discretization effects for the Fermilab action (at a = 0.09 fm) as in Ref. [72]. This error is 1.3% for Kb and 0.3% for kc. Adding in quadrature the statistical and fitting error together with the discretization error leads to a total relative uncertainty of 5.7% for Kb and 1.2% for kc. This error budget is summarized in Table VII. Note that these errors are conservative and are likely to decrease substan­tially with more sophisticated fitting methods and the higher statistics data set currently being generated. We have computed ^,(1) at several different values of the bare charm and bottom quark masses, and these simu- TABLE VII. Errors in the Kh v parameters. The first column labels the heavy quark, the second gives the statistical and fitting error for the k parameter, the third gives the discretization error, and the fourth combines these in quadrature. PHYSICAL REVIEW D 79, 014506 (2009) K Statistics + fitting Discretization Total Kc 1.2% 0.3% 1.2% «h 5.6% 1.3% 5.7% 014506-11 C. BERNARD et al. 0.98 0.96 0.94 0.92 0.9 FIG. 4 (color online). hA ( 1) for different Kh values on the coarse m' = 0.02 ensemble (full QCD point). The points labeled Kb show how hA ( 1) depends on Kb when kc is fixed to its tuned value. For the points labeled kc the roles of Kb and kc arc reversed. lated points can be used to estimate the error in hA[ (1) from the above uncertainties in the tuning of the heavy-quark x values. Figure 4 illustrates the dependence of hA[{ 1) as a function of bottom and charm quark x values on one of the coarse (a = 0.12 fm) ensembles. The points labeled xb show ft4|(1) where we have fixed xc to the tuned charm value, but vary the bare xb along the x axis. The points labeled xc are similar, where the value of xb is fixed at its tuned value, and the bare xc is varied. The above uncer­tainties in the ks, combined with the variation of hA){ 1) with x, lead to a systematic error of 0.7% in hA) (1), labeled "kappa tuning" in Table X. C. Perturbation theory The perturbative calculation of pA[ is needed to match the heavy-quark lattice current, and the calculation has been carried out to one-loop order [0(«s)]. As discussed in Sec. IV, much of the renormalization cancels when forming the ratios of Z factors that define p [Eq. (29)], and the coefficients of the perturbation series are small, by construction. The one-loop correction is quite small, only 0.3-0.4% on the different lattice spacings. We take the entire one-loop correction of 0.3% on the fine lattices as an estimate of the error introduced by neglecting higher orders in the perturbative expansion. D. Chiral extrapolation We estimate our systematic error due to the chiral ex­trapolation by comparing fits with and without the addi­tional terms with coefficients c,- in Eq. (33), i.e. analytic terms of higher order than NLO in rS^PT, since the two- loop NNLO logarithms are unknown. We also compare with continuum ^PT, both NLO and (partial) NNLO. There are additional errors due to the uncertainties in the parameters that enter the NLO rS^PT formulas. By far the largest uncertainty of this kind is that due to the uncertainty in gD/r 7T. Finally, there is an error due to a mistiming of the parameter «0 on the coarse lattices. All of these errors are discussed below in more detail. In the discussion of chiral extrapolation errors, it is important to keep in mind that the chiral logarithms (either rS^PT or continuum) are tiny (~3X10-3) in the region where we have data. Nonanalytic behavior is important only near the physical pion mass where the ^PT should be a very good descrip­tion in the continuum. The main feature of the chiral extrapolation is a cusp that appears close to the physical pion mass (in the valence sector), due to the Dtt threshold and the fact that the D-D* splitting is very close to the physical pion mass. This cusp represents real physics, and must be included in any version of the chiral extrapolation used to estimate systematic errors. We extrapolate the light sea and light valence quark masses from the values used in the simulations, between ms/2 and mj 10, to the average physical light-quark mass, around mj21. We use staggered chiral perturbation theory and the prescription introduced in Sec. VI to remove the nonanalytic taste-breaking discretization effects coming from the staggered light-quark sector. Separate fits are performed for the two ratios introduced in Eqs. (36) and (37), 2lsea and !RV;I|. The chiral extrapolation is performed on these ratios, and the staggered discretization errors appearing in the NLO chiral logarithms are removed by taking a ->«■ 0 in the rS^PT expression. With the NNLO analytic terms given in Eq. (33) the chiral extrapolation formulas for the ratios are H va] = 1 + NLO]ugs + cxnrXt, (41) H sea = 1 + NLO]ugs + c2{2mlp + m2Sfi), (42) where NLOlugs is a schematic notation representing the chiral logarithms coming from numerator and denomina­tor. These terms are different for the two ratios, and can be obtained straightforwardly from the definitions of the ra­tios Eqs. (36) and (37), and the formula for the nonanalytic terms in Eq. (A1). The formula for 21 va] in the continuum is given explicitly in Eq. (A6), for the purposes of illustration. The NNLO term c3cr in Eq. (33) cancels in the ratios, and 2lsea and 2lval each determine one of the remaining two NNLO coefficients. Note that the factor of r/A in Eqs. (32) and (33) cancels in the chiral formulas for the two ratios. The only free parameters in our chiral fits are Cj and c2; the rest are determined from phenomenology or from rS^PT fits to the pseudoscalar sector. The ratios in Eqs. (36) and (37) are completely predicted through NLO in the continuum once fw, gmrw aild the D-D* splitting A(t 1 are taken from experiment. The con­stants fw and gmrw appear in an overall multiplicative PHYSICAL REVIEW D 79, 014506 (2009) 014506-12 B -> D i r FORM FACTOR AT ... X2/d.o.f. = 0.91. CL = 0.51 mx 2 (GeV2) FIG. 5 (color online). !Rval ratio versus valcncc pion mass squared on all ensembles for the three different latticc spacings. The curvc is the continuum prediction through NLO in contin­uum ^-PT for this quantity. (See appendix.) PHYSICAL REVIEW D 79, 014506 (2009) 1.04 o medium coarse (0.15 lm) ❖ coarse (0.12 fm) □ fine (0.09 fm) x extrapolated value 0.98 - 0.96 - ______i______I______i______I______i______I______ 0 0.1 0.2 0.3 mx 2 (GeV2) FIG. 6 (color online). !Rval ratio versus valcncc pion mass squared on all ensembles for the three different latticc spacings. The curvc is the fit with 1 sigma error band to the ratio for all three latticc spacings using rS^PT, extrapolated to the contin­uum by taking a -> 0 in the NLO staggered chiral logarithms. S~ * factor "Tf-, in front of the logarithmic term, as can be seen in Eq. (Al) and (A6). We take a fairly conservative range for the constant gnrfir determined from phenomenology, as discussed below, and the errors in this quantity are accounted for in our final error budget. In the mass region where we have data, the NLO continuum chiral logarithms contribute to /?A| (1) at the ~3 X 1CT3 level or less. Figure 5 illustrates this, where the NLO continuum ^ PT prediction Eq. (A6) is plotted over our data points for 2lva We find that the NLO continuum ^PT describes the data quite well, giving a ^2/d.o.f. = 0.91 and a corresponding CL = 0.51. This result is unchanged in the rS^PT fits; the effects of staggering are negligible in the region where we have data. We include the term proportional to cx in Eq. (41) in our fits used to obtain the central value for this quantity, as explained in Sec. V. (Since including a linear term proportional to r increases the statistical error in hA{, we take our central value and statistical error from this fit to be conservative.) This "partial NNLO" fit also has a good ^2/d.o.f. = 1.05, with a corresponding CL = 0.39. The constant linear term is small and consistent with zero [c‘i = -0.006(15)]. Figure 6 shows the fit to 3lvai versus m2x for all three lattice spacings using the rS^PT formula, Eq. (41). Although the data for 2lv.ai is consistent with a constant, the cusp appearing close to the physical pion mass is a prediction of NLO ^PT and has a physical origin, namely, the D-tt threshold, as we have remarked. Thus, any fits used to estimate systematic errors, even those that are somewhat ad hoc, such as those including higher-order polynomial terms, must include this cusp. Note that the cusp appears at the physical pion mass (in either SU(3) or SU(2) ^ PT). and is therefore in a region where ^PT is expected to be a reliable expansion. The cusp is a property of the function F(m, A{c)/m) given in Eq. (A2), and the position of the cusp as a function of m2Xji is determined by the D-D* splitting A(cl and the physical pion mass. We take these two quantities from experiment rather than from the lattice, since the experimental uncertainties are much smaller. We find that with or without the NNLO analytic terms, the ^PT (continuum or rooted staggered) describes the lattice data with ^2/d.o.f. close to 1 and correspondingly good confidence levels. We find a confidence level for the fit to !Rsea of 0.76 for the fit that includes NNLO analytic terms. The strictly NLO expression for the lattice ratio !Rsea has no free parameters, but it describes the data with a confidence level of 0.73. Similar fits to !Rva| are described above and yield reasonable confidence levels for both types of fits. Since the lattice data do not distinguish between these model fit functions, and the fit using only the NNLO analytic terms is not systematic in the chiral ex­pansion, we assign the difference between the two deter­minations, which is 0.9%, as the systematic error of leaving out higher-order terms when performing the chiral extrapo­lation. The final results for ^Rsea, ^.vai' and Hrid are given in Table VIII. The errors are statistical only; note that the strictly NLO values have no free parameters, and therefore no statistical errors. The final value of hAi still has statis­tical errors coming from the statistical errors in /?^d. The extrapolated results for 2lfid are consistent within the statistical errors of the NNLO fit. Again, we choose for our central value the result from the NNLO extrapolation with its larger errors to be conservative. The cyan (gray) band in Fig. 6 is the continuum extrapo­lation with a -> 0 in the rS^ PT formula. For this quantity. 014506-13 TABLE VIII. Continuum extrapolated values of R„ca, Rvai> Rnj, and hA[ (1) evaluated at the physical quark masses. The first column labels the quantity. The second is the computed value including NNLO analytic terms in the chiral fit. The third is the quantity evaluated in purely NLO ^PT, and has no free parame­ters (once gniy„, and Al<;) are taken from phenomenology) in the chiral fit. The final row shows which includes a statistical error coming from /?J4ld. The numbers are the same to the quoted precision using rS^PT or continuum ^PT. C. BERNARD et al. w/NNLO Strictly NLO 1.0059(90) 0.9983 2^val 0.9910(34) 0.9895 0.997(10) 0.9878 MU 0.921(13) 0.9124(84) the staggered lattice artifacts affecting the chiral loga­rithms in hA are negligible in the region where we have lattice data, which is due mainly to the small size of the chiral logarithms themselves. This is confirmed by the close agreement between the data points at each lattice spacing and the continuum curve. In fact, if we use con­tinuum A'PT to perform the chiral extrapolation, the result is unchanged. The primary difference between the rS^ PT expression and the continuum ^-PT expression is the re­duction of the cusp near the physical pion mass in rS^PT, though our lattice data are not near enough to the physical pion mass to demonstrate this effect. Figure 7 shows the fit to extrapolated to the con­tinuum and to the physical strange sea quark mass. Note that this ratio does not produce a cancellation of correla­tions between numerator and denominator and so has larger statistical errors than 2lvaj. Here again the discreti­zation effects due to staggered logarithms are negligibly small. Since the effects of including staggered discretiza­1.04 1.02 (3 0.98 0.96 0 0.1 0.2 0.3 mu2(GeV2) FIG. 7 (color online). R„ca ratio versus my for all ensembles and lattice spacings. The curve is the fit to all of the lattice data, extrapolated to the continuum. The curve is also extrapolated to the physical strange sea quark mass. \ 1 i o medium coarse (0.15 fm) ❖ coarse (0.12 fm) □ fine (0.09 fm) x extrapolated value tion effects in the chiral logarithms are negligible in the region where we have numerical data, and since the only nontrivial feature in the chiral extrapolation is the cusp near the physical pion mass, which we describe by con­tinuum ^-PT (our extrapolated curve has a -> 0 in the rS^'PT formula and thus reduces to the continuum form), we conclude that staggered taste-violating effects appear­ing in chiral logarithms are essentially removed in our ratio extrapolations. Figure 8 shows all of the full QCD points on the three lattice spacings. The curve is the quantity hv^(m') ~ h^(mxA, mUA, infJA, ciUA) x [K-sea(«V, ms, 0) X in', ms, 0)], (43) which again becomes an exact relation for the physical form factor when aUA -> 0. The curve is thus the product of the two continuum extrapolated ratio fits shown in Figs. 6 and 7, times the fiducial point, which we take to be amUA = 0.0124 at the fine lattice spacing (the solid square in Fig. 8). Because this is a full QCD curve, the valence mass mx equals the light sea mass m'. The other full QCD points are shown as open symbols in Fig. 8 for comparison, though the fits were performed on the ratios and normalized by the fiducial point at amUA = 0.0124. Note that the curve is already extrapolated in the strange sea quark mass, and so does not perfectly overlap with the amUA = 0.0124 point. As discussed above, when this quan- PHYSICAL REVIEW D 79, 014506 (2009) 0.98 0.96 _ 0.94 < ■" 0.92 0.9 0.88 m ' (GeV") 1C FIG. 8 (color online). The full QCD points versus mi on the three lattice spacings are shown in comparison to the continuum curve. The curve is the product of the two continuum extrapo­lated ratio fits shown in Figs. 6 and 7, times the fiducial point, which we have chosen to be the trt1 = 0.0124 fine lattice point (the filled square). The curve is already extrapolated to the physical strange sea quark mass, and so does not perfectly overlap with the lattice data point at the fiducial value. The cross is the extrapolated value, where the solid line is the statistical error, and the dashed line is the total systematic error added to the statistical error in quadrature. o medium coarse (0.15 fm) O coarse (0.12 fm) □ fine (0.09 fm) x extrapolated value 014506-14 tity is evaluated at m' = m it yields the value of hAl at physical quark masses. The cross is the extrapolated value, where the solid line is the statistical error, and the dashed line is the total systematic error added to the statistical error in quadrature. The low-energy constant gpp'w enters the chiral ex­trapolation formula and determines the size of the cusp near the physical pion mass. Our data do not constrain this constant, so we take a wide range for gp/y w that encom­passes the range of values coming from phenomenology and lattice calculations: fits to a wide range of experimental data prior to the measurement of the D* width by Stewart (gotr-tr = 0.27i(j;Q3 [731), an update of the Stewart analysis including the D* width (gpp'w = 0.51; no error quoted [741), quark models (gp/y w ~ 0.38 [751), quenched lattice QCD (gjyy^ = 0.67 ± O.OSlgg^ [761), two flavor lattice QCD in the static limit (gJJatic- = 0.516 ± 0.051 [771), and the measurement of the D" width (gpp'w = 0.59 ± 0.07 [781). There are as of yet no 2 + 1 flavor lattice calcula­tions of gp/y w. For this work we take gpp> w = 0.51 ± 0.2, leading to a parametric uncertainty of 0.9% in hA{{ 1) that is included as a systematic error. The additional low-energy constants that enter the chiral formulas are the tree-level continuum coefficients /jl0 and /, and the taste-violating parameters that vanish in the continuum. These are the taste splittings a2 A= with E = A\ 4, T, V, I, and the taste-violating hairpin-coefficients a2SA and a2S'v. We set / to the experimental value of the pion decay constant fn = 0.1307 GeV in the coefficient of the NLO logarithms. The pion masses used as inputs in the rS^'PT formulas are computed from the bare quark masses and converted into physical units using mly = (r1/rf-VS)2/titree(»lr + ™v). (44) where /xtK£ is obtained from fits to the light pseudoscalar mass squared to the tree-level form (in rj units), riAttree(mr + my)- This accounts for higher-order chiral corrections and is more accurate than using ^ obtained in the chiral limit, giving a better approximation to the pion mass squared at a given light-quark mass. Since the pa­rameters in our lattice simulations at different lattice spac­ings are expressed in rj units, we require the physical value of i~i to convert to physical units and take the physical pion mass and A(c) from experiment. Thus, the ~ 2.5% uncer­tainty in rjhys gives a parametric error in the chiral extrapo­lation. Because the chiral extrapolation is so mild, however, this error turns out to be negligible compared with other systematic errors. Since we are taking the pion mass from experiment there is a negligible error due to the light-quark mass uncertainty in the chiral extrapolation. The strange sea quark mass enters the chiral extrapolation formulas, but the dependence is weak, and the error in the bare strange quark mass leads to a negligible parametric error in hAl. The taste splittings As have been determined B - D *iv FORM FACTOR AT ... in Ref. [161, and their approximately 10% uncertainty also leads to a negligible error in /?A, (1)- The taste-violating hairpin coefficients have much larger fractional uncertain­ties, but these too lead to a negligible uncertainty in hAi (1). Even setting the rS^PT parameters to zero does not change our result for hA{ (1) significantly. As mentioned above, our result does not change if we use the continuum ^-PT formula in our chiral fits. In the calculation of the form factor, the tadpole- improved coefficient csw = 1/«q is obtained with «0 from the Landau link on the coarse lattices, but from the plaquette for «0 on the fine and coarser lattices. Though unintentional, there is nothing wrong with this, since it is not known a priori which provides the best estimate of the tadpole improvement factor. However, the «0 term for the spectator light (staggered) quark, which appears in the tadpole improvement of the Asqtad action, was taken from the Landau link on the coarse lattices, even though the sea quark sector used u0 from the plaquette. On the fine and coarser lattices, «0 was taken to be the same in the light valence and sea quark sectors. The estimates of u0 from plaquette versus Landau link differ only by 4% on the coarse lattices. Although the effect of this mistuning is expected to be small (correcting «0 would lead to a slightly different valence propagator and different tuned k values, thus leading to a small modification of the staggered chiral parameters in the valence sector for the coarse lattices used as inputs to the chiral fit), it is possible to study how much difference it makes using the hA{ lattice data. Including all three lattice spacings and using our preferred chiral fit, we find hAi (1) = 0.921 (13), where the error here is statistical only. If we neglect the coarse data points, we find /?A| (1) = 0.920(17), almost unchanged except for a somewhat larger statistical error. We can also examine the ratios !R.vai and K.sea- In our preferred fit to all the lattice data these are 0.9910(34) and 1.0059(90), respectively, where the errors are again only statistical. If we drop the coarse lattice data, these become 0.9960(56) and 0.999(13), respectively. Since the ratio 2lsea has very little valence quark mass dependence, we can combine 2lsea from the fit to all of the lattice data with !R.vai from the fit neglecting the coarse lattice data. This is useful, because 2lsea has the larger statistical error, so we would like to use the full lattice data set to determine this ratio, thus isolating the mistuning in the valence sector on the coarse lattices. When this is done we find that the central value of the final hAi( 1) is shifted upward by 0.4%, well within statistical errors and smaller than our other systematic errors. We assign a systematic error of 0.4% due to the u0 mistuning. E. Finite volume effects The finite volume corrections to the integrals which appear in heavy-light A'PT formulas, including those for B -> D* were given by Arndt and Lin [791. There are no PHYSICAL REVIEW D 79, 014506 (2009) 014506-15 new integrals appearing in the staggered case, and it is straightforward to use the results of Arndt and Lin in the rS^PT for hAi(\), as shown in Ref. [22]. We find that although the finite volume corrections in hAi(\) would be large near the cusp at the physical pion mass on the current MILC ensembles (ranging in size from 2.5-3.5 fm), for the less chiral data points at which we have actually simulated, the finite volume effects are negligible. For all data points in our simulations the finite volume corrections are less than 1 part in 104. We therefore assign no error due to finite volume effects. F. Discretization errors As shown in Refs. [28-30,49], the matching of lattice gauge theory to QCD is accomplished by normalizing the first few terms in the heavy-quark expansion. This is done by tuning the kinetic masses of the Ds and Bs mesons computed using the SW action (for the heavy quarks) to the experimental meson masses. Tree-level tadpole- improved perturbation theory is used to tune the coupling csw and the rotation coefficient d\ for the bottom and charm quarks. Once this matching is done, the discretiza­tion errors in hAi{\) are of order ots(h/2niQ)2 and (A/2mg)3 [28], where the powers of two are combinatoric factors. The leading matching uncertainty is of the order as(A/2mc)2. We estimate the size of this error setting cts = 0.3, A = 500 MeV, and mc = 1.2 GeV, which gives as(A/2mc)2 = 0.013. Since we have numerical data at three lattice spacings we are able to study how well the power-counting estimate accounts for observed discretization effects. Making use of Eq. (43), but varying the fiducial lattice spacing from our lightest to coarsest lattices, we are able to obtain hAi( 1) at physical quark masses, with discretization effects associ­ated with the staggered chiral logarithms removed in the ratios appearing in Eq. (43). The discretization effects that remain are: taste violations in h'A , taste violations at higher order than NLO in the ratios, the effect of the analytic term coming from light-quark discretization effects (propor­tional to asa2), and the heavy-quark discretization effects. The taste violations in h'A and the taste violations in the ratios appearing at higher order than NLO have been shown to be negligible. We now consider the remaining TABLE IX. hAi (1) at physical quark masses at different lattice spacings. where taste-violating effects have been removed, or shown to be negligible. Discretization effects due to analytic terms associated with the light-quark sector and heavy-quark discretization effects remain in the lattice data. C. BERNARD et al. a (fm) Mu 0.15 0.914(11) 0.12 0.907(14) 0.09 0.921(13) i 0.98 0.96 0.94 0.92 •c* 0.9 0.88 0.86 °'840 0.005 0.01 0.015 0.02 0.025 0.03 O a" FIG. 9. hA (1) at physical quark masses versus a2 (fm2). where taste-violating effects have been removed, or shown to be negligible. Discretization effects due to analytic terms associated with the light-quark sector and heavy-quark discretization effects remain in the lattice data. discretization errors coming from the light-quark analytic term and the heavy-quark discretization effects. Table IX presents the results for hAi(\) as obtained from Eq. (43), and Fig. 9 shows them plotted as a function of lattice spacing squared. Although the Fermilab action and cur­rents possess a smooth continuum limit, the MILC ensem­bles are not yet at small enough a to obtain simply 0(a) or 0(a2) behavior. The spread of the lattice data points gives some indication of the size of the remaining discretization effects, however, and we find that the fine (0.09 fm) lattice data point and the coarse (0.12 fm) lattice data point differ by 1.5%. This is similar to our power-counting estimate, and we assign the larger of the two, 1.5%, as the systematic error due to residual discretization effects. G. Summary Our final result, given the error budget in Table X, is hAx = 0.921 (13)(8)(8)(14)(6)(3)(4), (45) TABLE X. Final error budget for hA^l) where each error is discussed in the text. Systematic errors are added in quadrature and combined in quadrature with the statistical error to obtain the total error. PHYSICAL REVIEW D 79, 014506 (2009) Uncertainty Mu Statistics 1.4% Sod TT 0.9% NLO vs NNLO *PT fits 0.9% Discretization errors 1.5% Kappa tuning 0.7% Perturbation theory 0.3% u0 tuning 0.4% Total 2.6% 014506-16 B - D H v FORM FACTOR AT PHYSICAL REVIEW D 79, 014506 (2009) where the errors are statistical, parametric uncertainty in gDifw chiral extrapolation errors, discretization errors, parametric uncertainty in heavy-quark masses (kappa tun­ing), perturbative matching, and the u0 (mis)tuning on the coarse lattices. Adding all systematic errors in quadrature, we obtain /iAl(l) = 0.921(13)(20). (46) This final result differs slightly from that presented at Lattice 2007 [801, where a preliminary /iA| (1) = 0.924(12)(19) was quoted. There are three main changes in the analysis from the preliminary result: our earlier result used a value of as in the perturbative matching evaluated at the scale 2/a, while the present result uses the HLM [641 prescription to fix the scale. This causes a change of 0.1 %, well within the estimated systematic error due to the perturbative matching. In the previous result, the fine lattice data was blocked by 4 in the jackknife proce­dure; we now block by 8 to fully account for autocorrela­tion errors. This does not change the central value, but increases the statistical error slightly. Finally, we have chosen a value for gDD^ = 0.51 ± 0.2 instead of goo'-rr = 0.45 ± 0.15 to be more consistent with the range of values quoted in the literature. This causes a decrease in hAi (1) of 0.2%. VIII. CONCLUSIONS We have introduced a new method to calculate the zero- recoil form factor for the B -> D ;<> decay. We include 2 + 1 flavors of sea quarks in the generation of the gauge ensembles, so the calculation is completely unquenched. We have introduced a new double ratio, which gives the form factor directly, and leads to a large savings in the computational cost. The simulation is performed in a re­gime where we expect rooted staggered chiral perturbation theory to apply; we therefore use the rS^PT result for the B -> D* form factor [221 to perform the chiral extrapola­tion and to remove taste-breaking effects. To aid the chiral and continuum extrapolations, we introduced a set of ratios that has allowed us to largely disentangle light and heavy- quark discretization effects. Our new result JF(1) = hAi (1) = 0.921(13)(20) is consistent with the previous quenched result J(l) = 0.913lg;"| [131, but our errors are both smaller and under better theoretical control. This result allows us to extract |V(.6| from the experimental measurement of the B -> D*(.v form factor, which deter­mines jF(l)lK-fel- After applying a 0.7% electromagnetic correction to our value for JF(1) [811, and taking the most recent PDG average for I V^lJFCl) = (35.9 ± 0.8) X 10-3 [821, we find \Vcb\ = (38.7 ± 0.9exp ± 1.0theo) X 10-3. (47) This differs by about 2a from the inclusive determination IK J = (41.6"± 0.6) X 10^3 [821. Our new value super­sedes the previous Fermilab quenched number [131, as it should other quenched numbers such as that in Ref. [831.J Our largest error in JF(1) is the systematic error due to heavy-quark discretization effects, which we have esti­mated using HQET power counting and inspection of the numerical data at three lattice spacings. This error can be reduced by going to finer lattice spacings, or by using an improved Fermilab action [701. When using this improved action, it would be necessary to improve the currents to the same order. We have introduced a method for separating the heavy and light-quark discretization errors, where the physical hAi can be factorized into two factors hAd X !Rfid, such that the heavy-quark discretization errors are largely isolated in Combining our value of !R.fid = 0.997(10)(13) (where the first error is statistical, and the second is due to systematics that do not cancel in the ratio) with a determination of at finer lattice spacings and/or with an improved action would be a cost-effective way of reducing the heavy-quark discretization errors. The next largest error in our calculation of ^F(l) is statistical, and this error drives many of the systematic errors. This is mostly a matter of computing. It would also be desirable to perform the matching of the heavy-quark current to higher order in perturbation theory, or by using nonpertur­bative matching. With these improvements, it would be possible to bring the error in JF(1) to or below 1%, allow­ing a very precise determination of |Vf6| from exclusive semileptonic decays. ACKNOWLEDGMENTS We thank J. Bailey for a careful reading of 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 Contract No. TG-MCA93S002. This work was supported in part by the United States Department of Energy under Grant Nos. DE-FC02-06ER41446 (C. D., L.L.), DE-FG02-91ER40661 (S.G.), DE-FG02- 91ER40677 (A.X.K.), 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.), and PHY-0456556 (R.S.). R.T.E. and E.G. thank Fermilab and URA for their hospitality. Fermilab is operated by Fermi Research Alliance, LLC, under Contract No. DE-AC02-07CH11359 with the United States Department of Energy. 3Ref. [83] calculates the B-*D*€v form factor in the quenched approximation at zero and nonzero recoil momentum and uses a step-scaling method [84] to control the heavy-quark discretization errors. 014506-17 C. BERNARD et al. PHYSICAL REVIEW D 79, 014506 (2009) APPENDIX: CHIRAL PERTURBATION THEORY Eq. (34) of Ref. [22] gives the expression needed for hA (1) in partially quenched ^PT with degenerate up- and down- quark masses (the 2 + 1 case) in the rooted staggered theory: S=I, PAV. 4A, 67' 1 MMf>] J L Wm*>7 MM^} J + (V -A)}. (Al) where F{mj. zi) = - / ln^T + " 4;J + 277 " V'"/ " ^ + 2^lnt1 " " V'/ " " (7T^} (m t \ -f J + 0[(A(t»)3]. (A2) -£ with F(7?2j. ;;-) = F(7?2j. - Zj), and zj = Aic,/nij, where A(t l is the D-D* mass splitting. The residues R^n'k^ and D^ are defined in Refs. [18,19], and for completeness we quote them here: Rfl\{M}. {^}) = n_TDljf({M}. {^}) = - -A Rfk\{M}. {pi}). (A3) 11 (/;*r - mv "m7 i+i These residues are a function of two sets of masses, the numerator masses, {M} = {m^ m2.......m„} and the denominator masses, {/x} = {/x^ /x2.......In our 2 + 1 flavor case, we have {M*1} = {mv. mx}. {Mx]} = {mv, mv,. mx}. {fx} = {mv. ms). (A4) The masses mvr mVv, are given by [18] :Il + 2m^ tVi ^ ^ ‘'7v 2 ' 4 , mu, ^ms, , 1/ , , ,3 , \ , 1/ , , 3 , \ m;,. = -7T- -^±. = - I n*7/.. + rn. + -a S'v - Z 1. nr^ = -I + -a‘S'v + Z 1, a25'v. , , u9(crS'y)2 s¥ <"Uy> -{n% ~ n%} +----i6~ (A5) r? __ I/O 0 \‘J ** uy , ov z = v(™st. ~ mu>.r - mi,J The ratio R^° in the continuum through NLO in ^PT is RNU> = , 8bir 48^ LA. 3L ^ je- - I I D^liM^}:^})^.]}, (A6) j-u.il.s ~ A" jGiA/^'l where {M^.3,1} = {mv, mx>}. (A7) and where is a valence pion made of two quarks set to the fiducial valence quark mass, and the subscript x' refers to a valence quark at the fiducial mass. This ratio is one by construction when the valence quark mass equals the fiducial valence quark mass. 014506-18 [1] E. Barberio et al (Heavy Flavor Averaging Group (HFAG)), arXiv:0704.3575. [2] E. Gamiz et al. (HPQCD Collaboration), Phys. Rev. D 73, 114502 (2006). [3] T. Bae, J. Kim, and W. Lee, Proc. Sci., LAT2005 (2006) 335 [arXiv:hep-lat/0510008], [4] D. J. Antonio et al. (RBC Collaboration), Phys. Rev. Lett. 100, 032001 (2008). [5] C. Aubin, J. Laiho, and R. S. Van de Water, Proc. Sci., LAT2007 (2007) 375 [arXiv:0710.U21]. [6] J. Chay, H. Georgi, and B. Grinstein, Phys. Lett. B 247, 399 (1990). [7] I.I.Y. Bigi, N.G. Uraltsev, and A.I. Vainshtein, Phys. Lett. B 293, 430 (1992). [8] I.I.Y. Bigi, B. Blok, M.A. Shifman, N.G. Uraltsev, and A. I. Vainshtein, arXiv:hep-ph/9212227. [9] I.I.Y. Bigi, M.A. Shifman, N.G. Uraltsev, and A.I. Vainshtein, Phys. Rev. Lett. 71, 496 (1993). [10] I.I.Y. Bigi, M.A. Shifman, and N. Uraltsev, Annu. Rev. Nucl. Part. Sci. 47, 591 (1997). [11] O. Biichmuller and H. Flacher, Phys. Rev. D 73, 073008 (2006). [12] C.W. Bauer, Z. Ligeti, M. Luke, A. V. Manohar, and M. Trott, Phys. Rev. D 70, 094017 (2004). [13] S. Hashimoto, A. S. Kronfeld, P.B. Mackenzie, S.M. Ryan, and J.N. Simone, Phys. Rev. D 66, 014503 (2002). [14] C.W. Bernard et al., Phys. Rev. D 64, 054506 (2001). [15] C. Aubin et al, Phys. Rev. D 70, 094505 (2004). [16] C. Aubin et al. (MILC Collaboration), Phys. Rev. D 70, 114501 (2004). [17] W.-J. Lee and S.R Sharpe, Phys. Rev. D 60, 114503 (1999). [18] C. Aubin and C. Bernard, Phys. Rev. D 68,034014 (2003). [19] C. Aubin and C. Bernard, Phys. Rev. D 68,074011 (2003). [20] S.R. Sharpe and R.S. Van de Water, Phys. Rev. D 71, 114505 (2005). [21] C. Aubin and C. Bernard, Nucl. Phys. B, Proc. Suppl. 140, 491 (2005). [22] J. Laiho and R. S. Van de Water, Phys. Rev. D73, 054501 (2006). [23] A.F. Falk and M. Neubert, Phys. Rev. D 47, 2965 (1993). [24] T. Mannel, Phys. Rev. D 50, 428 (1994). [25] A. Czarnecki, Phys. Rev. Lett. 76, 4124 (1996). [26] A. Czarnecki and K. Melnikov, Nucl. Phys. B505, 65 (1997). [27] M.E. Luke, Phys. Lett. B 252, 447 (1990). [28] A. S. Kronfeld, Phys. Rev. D 62, 014505 (2000). [29] J. Harada et al., Phys. Rev. D 65, 094513 (2002). [30] J. Harada, S. Hashimoto, A. S. Kronfeld, and T. Onogi, Phys. Rev. D 65, 094514 (2002). [31] S. Prelovsek, Phys. Rev. D 73, 014506 (2006). [32] C.W. Bernard, C.E. DeTar, Z. Fu, and S. Prelovsek, Proc. Sci., LAT2006 (2006) 173. [33] C. Bernard, Phys. Rev. D 73, 114503 (2006). [34] C. Bernard, C.E. Detar, Z. Fu, and S. Prelovsek, Phys. Rev. D 76, 094504 (2007). [35] C. Aubin, J. Laiho, and R. S. Van de Water, Phys. Rev. D 77, 114501 (2008). [36] C. Bernard, M. Golterman, and Y. Shamir, Phys. Rev. D 73, 114511 (2006). [37] Y. Shamir, Phys. Rev. D 75, 054503 (2007). B - D*iv FORM FACTOR AT ... [38] Y. Shamir, Phys. Rev. D 71, 034509 (2005). [39] S. Diirr, Proc. Sci., LAT2005 (2005) 021 [arXiv:hep-lat/ 0509026], [40] S. R. Sharpe, Proc. Sci., LAT2006 (2006) 022 [arXiv:hep- lat/0610094], [41] A. S. Kronfeld, Proc. Sci., LAT2007 (2007) 016 [arXiv:0711.0699], [42] T. Blum et alPhys. Rev. D 55, RU33 (1997). [43] K. Orginos and D. Toussaint (MILC Collaboration), Phys. Rev. D 59, 014501 (1998). [44] J.F. Lagae and D. K. Sinclair, Phys. Rev. D 59, 014511 (1998)." [45] G.P. Lepage, Phys. Rev. D 59, 074502 (1999). [46] K. Orginos, D. Toussaint, and R.L. Sugar (MILC Collaboration), Phys. Rev. D 60, 054503 (1999). [47] C.W. Bernard et al (MILC Collaboration), Phys. Rev. D 61, 111502 (2000). [48] B. Sheikholeslami and R. Wohlert, Nucl. Phys. B259, 572 (1985). [49] A.X. El-Khadra, A.S. Kronfeld, and P.B. Mackenzie, Phys. Rev. D 55, 3933 (1997). [50] G. P. Lepage and P. B. Mackenzie, Phys. Rev. D 48, 2250 (1993). [51] R. Sommer, Nucl. Phys. B411, 839 (1994). [52] C.W. Bernard et al (MILC Collaboration), Phys. Rev. D 62, 034503 (2000). [53] A. Gray et al. (HPQCD Collaboration), Phys. Rev. Lett. 95, 212001 (2005). [54] C. Bernard (MILC Collaboration), Proc. Sci., LAT2007 (2007) 090 [arXiv:0710.1118]. [55] C. Bernard et al. (MILC Collaboration), Proc. Sci., LAT2005 (2006) 025 [arXiv:hep-lat/0509137], [56] M. Wingate, J. Shigemitsu, C. T. Davies, G. P. Lepage, and H. D. Trottier, Phys. Rev. D 67, 054505 (2003). [57] M. Okamoto et al., Nucl. Phys. B, Proc. Suppl. 140, 461 (2005). [58] C. Aubin et al (Fermilab Lattice), Phys. Rev. Lett. 94, 011601 (2005). [59] C. Aubin et al., Phys. Rev. Lett. 95, 122002 (2005). [60] E. Dalgic et al., Phys. Rev. D 73, 074502 (2006). [61] R. T. Evans, A. X. El-Khadra, and M. Di Pierro (Fermilab Lattice and MILC Collaborations), Proc. Sci., LAT2006 (2006) 081. [62] S.J. Brodsky, G.P. Lepage, and P.B. Mackenzie, Phys. Rev. D 28, 228 (1983). [63] Q. Mason et al (HPQCD Collaboration), Phys. Rev. Lett. 95, 052002 (2005). [64] K. Hornbostel, G.P. Lepage, and C. Morningstar, Phys. Rev. D 67, 034023 (2003). [65] A. X. El-Khadra, E. Gamiz, A. S. Kronfeld, and M. A. Nobes, Proc. Sci., LAT2007 (2007) 242 [arXiv: 0710.1437], [66] A. X. El-Khadra, E. Gamiz, A. S. Kronfeld, and M. A. Nobes (unpublished). [67] G.P. Lepage, J. Comput. Phys. 27, 192 (1978). [68] C. Aubin and C. Bernard, Phys. Rev. D 73, 014515 (2006). [69] C. Bernard, M. Golterman, and Y. Shamir, Phys. Rev. D 77, 074505 (2008). [70] M. B. Oktay and A. S. Kronfeld, Phys. Rev. D 78, 014504 (2008). PHYSICAL REVIEW D 79, 014506 (2009) 014506-19 [71] E. Follana et al. (HPQCD Collaboration), Phys. Rev. D 75, 054502 (2007). [72] A. S. Kronfeld, Nucl. Phys. B, Proc. Suppl. 53, 401 (1997). [73] I.W. Stewart, Nucl. Phys. B529, 62 (1998). [74] M.C. Arnesen, B. Grinstein, I.Z. Rothstein, and I.W. Stewart, Phys. Rev. Lett. 95, 071802 (2005). [75] R. Casalbuoni et al., Phys. Rep. 281, 145 (1997). [76] A. Abada et al., Nucl. Phys. B, Proc. Suppl. 119, 641 (2003). [77] H. Ohki, H. Matsufuru, and T. Onogi, Phys. Rev. D 77, 094509 (2008). C. BERNARD et al. [78] A. Anastassov et al. (CLEO Collaboration), Phys. Rev. D 65, 032003 (2002). [79] D. Arndt and C. J. D. Lin, Phys. Rev. D 70, 014503 (2004). [80] J. Laiho (Fermilab Lattice and MILC Collaborations), Proc. Sci., LAT2007 (2007) 358 [arXiv:0710.1111], [81] A. Sirlin, Nucl. Phys. B196, 83 (1982). [82] C. Amsler et al. (Particle Data Group), Phys. Lett. B 667,1 (2008). [83] G.M. de Divitiis, R. Petronzio, and N. Tantalo, Nucl. Phys. B807, 373 (2009). [84] M. Guagnelli, F. Palombi, R. Petronzio, and N. Tantalo, Phys. Lett. B 546, 237 (2002). PHYSICAL REVIEW D 79, 014506 (2009) 014506-20