The B → D∗ℓν form factor at zero recoil from three-flavor lattice QCD: A model independent determination of |V | cb C. Bernard,1 C. DeTar,2 M. Di Pierro,3 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. Hetrick,8 A. S. Kronfeld,9 J. Laiho,1,9 L. Levkova,2 P. B. Mackenzie,9 M. Okamoto,9 9 J. Simone,9 R. Sugar,10 D. Toussaint,11 and R. S. Van de Water9 0 0 2 (Fermilab Lattice and MILC Collaborations) n a 1Department of Physics, Washington University, St. Louis, Missouri, USA J 6 2Physics Department, University of Utah, Salt Lake City, Utah, USA 2 3School of Computer Science, Telecommunications and Information Systems, ] t a DePaul University, Chicago, Illinois, USA l - p 4Physics Department, University of Illinois, Urbana, Illinois, USA e h 5Liberal Arts Department, The School of the Art Institute of Chicago, Chicago, Illinois, USA [ 6Department of Physics, Indiana University, Bloomington, Indiana, USA 2 v 9 7American Physical Society, Ridge, New York, USA 1 5 8Physics Department, University of the Pacific, Stockton, California, USA 2 . 9Fermi National Accelerator Laboratory, Batavia, Illinois, USA 8 0 10Department of Physics, University of California, Santa Barbara, California, USA 8 0 11Department of Physics, University of Arizona, Tucson, Arizona, USA : v i X (Dated: January 26, 2009) r a 1 Abstract We present the first lattice QCD calculation of the form factor for B D∗ℓν 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, B→D∗(1) = 0.921(13)(20), where the first error is statistical and the second is the F sum of all systematic errors in quadrature. Applying a 0.7% electromagnetic correction and taking the latest PDG average for B→D∗(1)Vcb leads to Vcb =(38.7 0.9exp 1.0theo) 10−3. F | | | | ± ± × PACS numbers: 12.38.Gc, 13.25.Hw, 12.15.Hh 2 I. INTRODUCTION TheCabibbo-Kobayashi-Maskawa matrixelement V playsanimportantroleinthestudy cb of flavor physics [1]. Since V is one of the fundamental parameters of the Standard cb | | Model, its value must be known precisely in order to search 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 ǫ depends K sensitively on V (as the fourth power), and the present errors on this quantity contribute cb | | errors to the theoretical prediction of ǫ that are around the same size as the errors due to K B , the kaon bag parameter, which has been the focus of much recent work [2, 3, 4, 5]. It is K possible to obtain V from both inclusive and exclusive semileptonic B decays, and both cb | | determinations are limited by theoretical uncertainties. The inclusive method [6, 7, 8, 9, 10] makes use of the heavy-quark expansion and perturbation theory. The method also requires non-perturbative input from experiment, which is obtained from the measured moments of the inclusive form factor B X ℓν as a function of the minimum electron momentum. The c ℓ → dominant uncertainties in this method are the truncation of the heavy quark expansion and perturbation theory [11, 12]. In order to be competitive with the inclusive determination of V and thus serve as a cross-check, the exclusive method requires a reduction in the cb | | uncertainty of the B D∗ semileptonic form factor B→D∗, which has been calculated → F previously using lattice QCD in the quenched approximation [13]. Given the phenomenological importance of V , we have revisited the calculation of cb | | B→D∗ at zero recoil using the 2+1 flavor MILC ensembles with improved light staggered F quarks [14, 15]. The systematic error due to quenching is thus eliminated. The systematic error associated with the chiral extrapolation to physical light quark masses is also reduced significantly. Since staggered quarks are computationally less expensive than many other formulations, we are able to simulate at quite small quark masses; our lightest corresponds toapionmassofroughly240MeV.Giventhepreviousexperience oftheMILCCollaboration with chiral fits to light meson masses and decay constants [16], we are in a regime where we expect rooted staggered chiral perturbation theory (rSχPT) [17, 18, 19, 20, 21] to apply. We therefore use the rSχPT result for the B D∗ form factor [22] to perform the chiral → extrapolationandtoremovediscretizationeffectsparticulartostaggeredquarks. Inaddition, 3 weintroduceaset ofratiosthatallowsustodisentanglelight-andheavy-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. [13]. This new method requires many fewer three-point correlation functions, and has allowed for a savings of roughly a factor of ten in computing resources, while at the same time simplifying the analysis. The differential rate for the semileptonic decay B D∗ℓν is ℓ → dΓ G2 dw = 4πF3m3D∗(mB −mD∗)2√w2 −1 G(w)|Vcb|2|FB→D∗(w)|2, (1) where w = v′ v is the velocity transfer from the initial state to the final state, and · (w) B→D∗(w) 2 contains a combination of four form factors that must be calculated non- G |F | perturbatively. At zero recoil (1) = 1, and B→D∗(1) reduces to a single form factor, G F h (1). Given h (1), the measured decay rate determines V . A1 A1 | cb| The quantity h is a form factor of the axial vector current, A1 hD∗(v,ǫ′)|Aµ|B(v)i = i√2mB2mD∗ ǫ′µhA1(1), (2) where µ is the continuum axial-vector current and ǫ′ is the polarization vector of the D∗. A Heavy-quark symmetry plays a useful role in constraining h (1), leading to the heavy-quark A1 expansion [23, 24] ℓ 2ℓ ℓ V A P h (1) = η 1 + , (3) A1 A − (2m )2 2m 2m − (2m )2 (cid:20) c c b b (cid:21) up to order 1/m2, and where η is a factor that matches heavy-quark effective theory Q A (HQET) to QCD [25, 26]. The ℓ’s are long distance matrix elements of the HQET. Heavy- quark symmetry forbids terms of order 1/m at zero recoil [27], and various methods have Q been used to compute the size of the 1/m2 coefficients, including quenched lattice QCD [13]. Q The earlier work by Hashimoto et al. [13] used three double ratios in order to obtain separately each of the three 1/m2 coefficients in Eq. (3). These three double ratios also Q determine three out of the four coefficients appearing at 1/m3 in the heavy-quark expansion. Q It was shown in Ref. [28] that, for the Fermilab method matched to tree level in α and to s next-to-leading order in HQET, the leading discretization errors for the double ratios for this quantity are of order α (Λ/2m )2f (am ) and (Λ/2m )3f (am ), where Λ is a QCD scale s Q B Q Q i Q 4 stemming from the light degrees of freedom, such as that appearing in the HQET expansion for the heavy-light meson mass, m = m +Λ+ . The functions f (am ) are coefficients M Q i Q ··· depending on am and α , but not on Λ. When am 1, the f (am ) are of order one; Q s Q i Q ∼ when am 1, they go like a power of am , such that the continuum limit is obtained. Q Q ≪ The powers of 2 are combinatoric factors. As discussed in Ref. [13], all uncertainties in the double ratios used in that work R scale as 1 rather than as . Statistical errors in the numerator and denominator are R− R 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 one which can be computed reliably in perturbation theory. Finally, the quenching error, relevant to Ref. [13] but not to the present unquenched calculation, scales as 1 rather than as . R− R This scaling of the error occurs because the double ratios constructed in Ref. [13] become the identity in the limit of equal bottom and charm quark masses. In the calculation reported here, the form factor h (1) is computed more directly using A1 only one double ratio, D∗ cγ γ b B B bγ γ c D∗ = h | j 5 | ih | j 5 | i = h (1) 2, (4) RA1 D∗ cγ c D∗ B bγ b B | A1 | 4 4 h | | ih | | i 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 [α (Λ/2m )2 and (Λ/2m )3], but which again vanish in the s Q Q continuum limit. The errors in the ratio introduced in Eq. (4) do not scale rigorously as 1 because is not one in the limit of equal bottom and charm quark masses. RA1 − RA1 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 1, including the quenching error, is no longer crucial. The R − 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/mn. In Q 1 Note that the notation RA1 stands for a different double ratio in Ref. [13]. 5 short, for an unquenched QCD calculation, the method using Eq. (4) gives a smaller total error than the method used in Ref. [13] for a fixed amount of computer time . The currents of lattice gauge theory must be matched to the normalization of the contin- uum to obtain . The matching factors mostly cancel in the double ratio [29, 30], leaving RA1 h (1) = = ρ R , where R is the lattice double ratio and ρ, the ratio of matching A1 RA1 A1 A1 factors, ispvery closepto 1. (For the remainder of this paper we shall use the convention that a script letter corresponds to a continuum quantity, while a non-script letter corresponds to a lattice quantity.) This ρ 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 ρ factors to implement a blind analysis. Two of us involved in the perturbative calculation applied a common multiplicative offset to the ρ factors needed to obtain h (1) at different lattice A1 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 ofthefourth-rootprocedurefor eliminating the unwanted four-folddegen- eracy of staggered quarks. At non-zero lattice spacing, this procedure has small violations of unitarity [31, 32, 33, 34, 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 localtheory of QCD as a 0 [37, 38]. When → coupled with other analytical and numerical evidence (see Refs. [39, 40, 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 ρ 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 Section VIII. 6 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(a2) Symanzik improved gauge ≈ action and 2+1 flavors of “AsqTad” improved staggered sea quarks [42, 43, 44, 45, 46, 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 (m = m ), val sea 6 and our light quark masses range between m /10 and m /2. Table II shows the valence s s masses computed on each ensemble. In this work we follow the notation [16] where m is s the physical strange quark mass, m is the average u-d quark mass, and m′, m′ indicate s the nominal values used in simulations. In practice, the MILC ensembles choose m′ within s b b 10–30% of m and a range of m′ to enable a chiral extrapolation. s The heavy quarks are computed using the Sheikholeslami-Wohlert (SW) “clover” action b [48] with the Fermilab interpretation via HQET [49]. The SW action includes a dimension- five interaction with a coupling c that has been adjusted to the value u−3 suggested by SW 0 tadpole-improved, tree-level perturbation theory [50]. The value of u is calculated either 0 from the plaquette (a 0.15 fm and a 0.09 fm), or from the Landau link (a 0.12 fm). ≈ ≈ ≈ The adjustment of c is needed to normalize the heavy quark’s chromomagnetic moment SW correctly [49]. The tadpole-improved bare quark mass for SW quarks is given by 1 1 1 am = , (5) 0 u 2κ − 2κ 0 (cid:18) crit(cid:19) where tuning the parameter κ to the critical quark hopping parameter κ would lead to crit a massless pion. The spin averaged B and D kinetic masses are computed on a subset s s of the ensembles in order to tune the bare κ values for bottom and charm (and hence the corresponding bare quark masses) to their physical values. These tuned values were then used in the B D∗ℓν form-factor production run. → The relative lattice scale is determined by calculating r /a on each ensemble, where r is 1 1 related to the force between static quarks by r2F(r ) = 1.0 [51, 52]. To avoid introducing 1 1 implicit dependence on m′, m′ via r (m′,m′,g2) (where, as above, primes denote simulation s 1 s masses), we interpolate in m′ and extrapolate in m′ to obtain r (m,m ,g2)/a at the physical s 1 s b b masses. We then convert from lattice units to r units with r (m,m ,g2)/a. Below we shall 1 1 s b b 7 b TABLEI:Parametersofthesimulations. Thecolumnsfromlefttorightaretheapproximatelattice spacing in fm, the sea quark masses am′/am′, the linear spatial dimension of the lattice ensemble s in fm, the dimensionless factor m L (m corresponds to the taste-pseudoscalar pion composed of π π b 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 c used for both bottom SW and charm quarks. a(fm) am′/am′ L(fm) m L 10/g2 Volume # Configs κ κ c s π b c SW 0.15 0.0194/0.0484 2.4 5.5 6.586 163 48 628 0.076 0.122 1.5673 b × 0.15 0.0097/0.0484 2.4 3.9 6.572 163 48 628 0.076 0.122 1.5673 × 0.12 0.02/0.05 2.4 6.2 6.79 203 64 460 0.086 0.122 1.72 × 0.12 0.01/0.05 2.4 4.5 6.76 203 64 592 0.086 0.122 1.72 × 0.12 0.007/0.05 2.4 3.8 6.76 203 64 836 0.086 0.122 1.72 × 0.12 0.005/0.05 2.9 3.8 6.76 243 64 528 0.086 0.122 1.72 × 0.09 0.0124/0.031 2.4 5.8 7.11 283 96 516 0.0923 0.127 1.476 × 0.09 0.0062/0.031 2.4 4.1 7.09 283 96 556 0.0923 0.127 1.476 × 0.09 0.0031/0.031 3.4 4.2 7.08 403 96 504 0.0923 0.127 1.476 × call this procedure the mass-independent determination of r . 1 In order to fix the absolute lattice scale, one must compute a physical quantity that can be compared directly to experiment; we use the Υ 2S–1S splitting [53] and the most recent MILC determination of f [54]. The difference between these determinations results in a π systematic error that turns out to be much smaller than our other systematics. When the Υ scale determination is combined with the continuum extrapolated r value at physical 1 quark masses, a value rphys = 0.318(7) fm [55] is obtained. The f determination is rphys = 1 π 1 0.3108(15)(+26)fm[54]. Givenrphys,itisthenstraightforwardtoconvert quantitiesmeasured −79 1 in r units to physical units. 1 The dependence on the lattice spacing a is mild in this analysis. Since a only enters the calculation through the adjustment of the heavy and light quark masses, the dependence of 8 TABLE II: Valence masses used in the simulations. The columns from left to right are the approx- imate lattice spacing in fm, the sea quark masses am′/am′ identifying the gauge ensemble, and the s valence masses computed on that ensemble. b a(fm) am′/am′ am s x 0.15 0.0194/0.0484 0.0194 ≈ b 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 ≈ h (1) on a is small. Staggered chiral perturbation theory indicates that the a dependence A1 coming from staggered quark discretization effects is small [22], and this is consistent with the simulation data. In this work, we construct lattice currents as in Ref. [49], Jµhh′ = ZVh4hZVh4′h′ΨhΓµΨh′, (6) q where Γ is either the vector (iγµ) or axial-vector (iγµγ ) current. The rotated field Ψ is µ 5 h defined by Ψ = (1+ad γ D )ψ , (7) h 1 lat h · where ψ is the (heavy) lattice quark field in the SW action. D is the symmetric, nearest- h lat neighbor, covariant difference operator; the tree-level improvement coefficient is 1 1 1 d = . (8) 1 u 2+m a − 2(1+m a) 0 (cid:18) 0 0 (cid:19) In Eq. (6) we choose to normalize the current by the factors of Zhh (h = c,b) since even V4 for massive quarks they are easy to compute non-perturbatively. The continuum current is 9 related to the lattice current by hh′ = ρ Jhh′ (9) Jµ JΓ µ up to discretization effects, where ZbcZcb ρ2 = JΓ JΓ, (10) JΓ ZccZbb V4 V4 and the matching factors Zhh′’s are defined in Ref. [30]. Note that the factor ZbbZcc JΓ V4 V4 multiplying the lattice current in Eq. (6) cancels in the double ratio by design, leavqing only the ρ factor, which is close to one and can be computed reliably using perturbation theory. The perturbative calculation of ρ is described in more detail in Section IV. JΓ Interpolating operators are constructed from four-component heavy quarks and staggered quarks as follows. Let D∗(x) = χ(x)Ω†(x)iγjψc(x), (11) O j † (x) = ψ (x)γ Ω(x)χ(x), (12) OB b 5 where χ is the one-component field in the staggered-quark action, and Ω(x) = γx1/aγx2/aγx3/aγx4/a. (13) 1 2 3 4 The left (right) index of Ω† (Ω) can be left as a free taste index [41] or χ can be promoted to a four-component naive-quark field to contract all indices [56]. The resulting correlation functions are the same if the initial and final taste indices are set equal and then summed. The same kinds of operators have been used in previous calculations [57, 58, 59]. Lattice matrix elements are obtained from three-point correlation functions. The three- point correlation functions needed for the B D∗ transition at zero-recoil are → CB→D∗(ti,ts,tf) = h0|OD∗(x,tf)Ψcγjγ5Ψb(y,ts)OB† (0,ti)|0i, (14) x,y X CB→B(t ,t ,t ) = 0 (x,t )Ψ γ Ψ (y,t ) † (0,t ) 0 , (15) i s f h |OB f b 4 b s OB i | i x,y X CD∗→D∗(ti,ts,tf) = h0|OD∗(x,tf)Ψcγ4Ψc(y,ts)OD† ∗(0,ti)|0i. (16) x,y X 10