The measured partial widths of the semileptonic decays D to K l nu and D to
pi l nu can be combined with the form factors calculated on the lattice to
extract the CKM matrix elements |V_cs| and |V_cd|. The lattice calculations can
be checked by comparing the form factor shapes from the lattice and experiment.
We have generated a sizable data set by using heavy clover quarks with the
Fermilab interpretation for charm and asqtad staggered light quarks on 2+1
flavor MILC ensembles with lattice spacings of approximately 0.12, 0.09, 0.06,
and 0.045 fm. Preliminary fits to staggered chiral perturbation theory suggest
that we can reduce the uncertainties in the form factors at zero invariant mass
to below 5%.Comment: 7 pages, 4 figures, proceedings of the XXIX International Symposium
  on Lattice Field Theory - Lattice 2011, July 10-16, 2011, Squaw Valley, Lake
  Tahoe, Californi

Bailey, Jon A.

Bazavov, A.

El-Khadra, A. X.

Gottlieb, Steven

Jain, R. D.

Kronfeld, A. S.

Van de Water, R. S.

Zhou, R.

English

arXiv

The measured partial widths of the semileptonic decays D {yields} K{ell}{nu} and D {yields} {pi}{ell}{nu} can be combined with the form factors calculated on the lattice to extract the CKM matrix elements |V{sub cs}| and |V{sub cd}|. The lattice calculations can be checked by comparing the form factor shapes from the lattice and experiment. We have generated a sizable data set by using heavy clover quarks with the Fermilab interpretation for charm and asqtad staggered light quarks on 2+1 flavor MILC ensembles with lattice spacings of approximately 0.12, 0.09, 0.06, and 0.045 fm. Preliminary fits to staggered chiral perturbation theory suggest that we can reduce the uncertainties in the form factors at q{sup 2} = 0 to below 5%

El-Khadra, A.X.

Jain, R.D.

Kronfeld, A.S.

Van de Water, R.S.

UNT Digital Library

D semileptonic form factors and |Vcs(d)| from 2+1flavor lattice QCDJon A. Bailey∗a,b, A. Bazavovc, A.X. El-Khadrad , Steven Gottliebe, R.D. Jaind ,A.S. Kronfelda, R.S. Van de Waterc, and R. ZhoueEmail: jabailey@fnal.govaTheoretical Physics Department, Fermilab, Batavia, IL 60510, USAbDepartment of Physics and Astronomy, Seoul National University, Seoul, 151-747, ROKcDepartment of Physics, Brookhaven National Laboratory, Upton, NY 11973, USAdPhysics Department, University of Illinois, Urbana, IL 61801, USAeDepartment of Physics, Indiana University, Bloomington, IN 47405, USAFermilab Lattice and MILC CollaborationsThe measured partial widths of the semileptonic decays D→K`ν and D→ π`ν can be combinedwith the form factors calculated on the lattice to extract the CKM matrix elements |Vcs| and |Vcd |.The lattice calculations can be checked by comparing the form factor shapes from the lattice andexperiment. We have generated a sizable data set by using heavy clover quarks with the Fermilabinterpretation for charm and asqtad staggered light quarks on 2+1 flavor MILC ensembles withlattice spacings of approximately 0.12, 0.09, 0.06, and 0.045 fm. Preliminary fits to staggeredchiral perturbation theory suggest that we can reduce the uncertainties in the form factors atq2 = 0 to below 5%.XXIX International Symposium on Lattice Field TheoryJuly 10-16 2011Squaw Valley, Lake Tahoe, California∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/FERMILAB-CONF-11-623-TD→ K(π)`ν form factors and |Vcs(d)| from lattice QCD Jon A. Bailey1. IntroductionPerforming a global fit under the assumption of a unitary CKM matrix [1] yields precise valuesfor |Vcs| and |Vcd | [2]. If new physics in flavor introduces deviations from unitarity, however, valuesof the CKM matrix elements from direct determinations will in general differ from those predictedby the global fit. Moreover, improving tests of second row and column unitarity requires improveddirect determinations of |Vcs| [2].In the limit of massless leptons, the rates for the semileptonic decays D→ K(π)`ν become1dΓ(D→ K(π)lν)dq2=G2F24π3|pK(π)|3 |Vcs(d)|2| fD→K(π)+ (q2)|2, (1.1)where q2 = (pD− pK(π))2, pK(π) is the momentum of the K(π) in the rest frame of the D, andf D→K(π)+ (q2) is defined in terms of the hadronic matrix element of the current Vµ = is̄γµc (id̄γµc):〈K(π)|Vµ |D〉= f D→K(π)+ (q2)(pD + pK(π)−m2D−m2K(π)q2q)µ+ f D→K(π)0 (q2)m2D−m2K(π)q2qµ .(1.2)Given the normalization of the form factors f D→K(π)+ (q2) from lattice QCD, the CKM matrix ele-ments |Vcs(d)| can be extracted from experimental measurements of the branching fractions.Agreement with the Standard Model values provides important validation of our methods,which we also use to calculate the form factors for B→ π`ν [4] and B→ K`+`− [5]. Both decaysare central in searches for new physics; the former allows direct extraction of |Vub|, while thelatter is loop-suppressed in the Standard Model. New physics seems unlikely to enter the tree-leveldecays D→ K(π)`ν before the B decays. The Fermilab method applies to charm and bottom, soconsistency between our D form factors and the normalizations implied by the global fit is directevidence of our ability to precisely extract the B form factors.2. MethodFor the up, down, and strange quarks we use the asqtad-improved staggered action [6], for thecharm quark we use the clover action with the Fermilab interpretation [7], and for the gluons weuse a one-loop Symanzik improved gauge action [8]. We set the scale with fπ , tune the masses ofthe light quarks using the experimental values of mπ and mK , and tune the hopping parameter ofthe charm quark with the spin-averaged kinetic mass of the Ds.Table 1 summarizes our data set. We vary the valence light-quark masses on each ensemblefrom near the tuned strange mass ms down to ∼ 0.1ms, and the lattice spacings from ≈ 0.12 fm to≈ 0.045 fm. To increase statistics and reduce autocorrelations, we average over four source timesand randomize the source spatial locations.For calculations in the rest frame of the D and in heavy-meson chiral perturbation theory, thehadronic matrix elements are conveniently parametrized by form factors f⊥ and f‖:〈K(π)|Vµ |D〉=√2mD[vµ fD→K(π)‖ (q2)+ p⊥µ fD→K(π)⊥ (q2)], (2.1)1For decays to the π0, there is an additional isospin factor of 1/2 on the right-hand side of Eq. (1.1).2D→ K(π)`ν form factors and |Vcs(d)| from lattice QCD Jon A. Bailey≈ a (fm) L3×nt Nconf nsrc nsnk aml/ams amval κc0.12 203×64 2052 4 4 0.02/0.05 {0.005, 0.007, 0.1259203×64 2259 4 4 0.01/0.05 0.01, 0.02, 0.1254203×64 2110 4 4 0.007/0.05 0.03, 0.0415, 0.1254243×64 2099 4 4 0.005/0.05 0.05, 0.0349} 0.12540.09 283×96 1996 4 4 0.0124/0.031 {0.0031, 0.0047, 0.1277283×96 1931 4 4 0.0062/0.031 0.0062, 0.1276323×96 984 4 4 0.00465/0.031 0.0093, 0.0124, 0.1275403×96 1015 4 4 0.0031/0.031 0.031, 0.0261} 0.1275643×96 791 4 4 0.00155/0.031 0.12750.06 483×144 593 4 4 0.0072/0.018 {0.0018, 0.0025, 0.1295483×144 673 8 4 0.0036/0.018 0.0036, 0.1296563×144 801 4 4 0.0025/0.018 0.0054, 0.0072, 0.1296643×144 827 4 4 0.0018/0.018 0.0160, 0.0188} 0.12960.045 643×192 801 4 4 0.0028/0.014 {0.0018, 0.0028, 0.13100.0040, 0.0056, 0.0084, 0.0160, 0.0130}Table 1: Data on the 2+1 flavor asqtad staggered MILC ensembles for various valence masses and sourcetimes. The columns are, respectively, the lattice spacing, lattice dimensions, number of configurations,number of source times, number of 3-point sink times, light/strange sea-quark masses, valence light-quarkmasses, and charm hopping parameter. The analysis to date includes the full QCD points; we are consideringgenerating data at additional source times.where v = pD/mD, and p⊥ = pK(π)−(pK(π) ·v)v. f⊥ and f‖ can be extracted from correlator ratios.We considerRD→K(π)3,µ (t,T ;q2)≡ 1φK(π)µCD→K(π)3,µ (t,T ;pK(π))√CK(π)2 (t;pK(π))CD2 (T − t)√2EK(π)e−EK(π)te−mD(T−t), (2.2)where φK(π)µ ≡ (1, pK(π)), EK(π) = (m2D +m2K(π)−q2)/(2mD), and C3, C2 are averages of corre-lators constructed to eliminate oscillations from opposite-parity states [4]. T and t are respectivelythe source-sink separation and current insertion time in the vector-current 3-points.We use local operators for the K(π) 2-points, smear the D interpolators with a charmoniumwavefunction, and construct the currents out of light staggered and heavy clover fields [9]. Forinsertion times t far from 3-point source and sink, the ratios plateau to the form factors:RD→K(π)3,0 ∼ fD→K(π)‖ and RD→K(π)3,i ∼ fD→K(π)⊥ for 1 t T . (2.3)The averages C3 require raw 3-points at successive source-sink separations T and T + 1; to min-imize our errors as a function of momentum pK(π), we generate the 3-point correlators at twophysical separations on each ensemble and for each set of valence quark masses [10].We inject the 3-points and K(π) 2-points with momenta apK(π)/(2π/L) = (0,0,0), (1,0,0),(1,1,0), (1,1,1), and (2,0,0) (and permutations and negatives of these components). We aver-age the correlators over equivalent momenta (up to axis interchange) and have checked that the3D→ K(π)`ν form factors and |Vcs(d)| from lattice QCD Jon A. Baileywavefunction overlap factors of the K(π) 2-points are independent of momentum. We then replaceCK(π)2 (pK(π)) with the less noisy CK(π)2 (0) in the ratios of Eq. (2.2) and useR′D→K(π)3,µ (t,T ;q2)≡ 1φK(π)µCD→K(π)3,µ (t,T ;pK(π))√CK(π)2 (t;0)CD2 (T − t)EK(π)e−EK(π)t√2e−mK(π)te−mD(T−t). (2.4)We have checked that our data obeys the continuum dispersion relation, and we substitute thisrelation for EK(π) in the above ratio.To extract the masses of the K(π) and D from the 2-point correlators, we fit to sums of ex-ponentials with oscillating terms to account for contributions from opposite-parity states. We usethe masses to construct the ratios, and we propagate the errors via 500 bootstraps. To extract theplateaus we fit the ratios; varying the fit function and time intervals does not significantly changethe results.The vector currents undergo renormalization. We match to the continuum by writing the renor-malization factors as products of the degenerate-mass vector-current normalization factors, whichwe compute nonperturbatively, and correction factors whose deviations from one are perturbativelycalculable [11]:〈K(π)|Vµ |D〉= Zcs(d)Vµ 〈K(π)|Vlatµ |D〉, Zcs(d)Vµ = ρcs(d)Vµ√ZccV4 Zss(dd)V4 . (2.5)We blind the analysis by introducing an offset in ρ .After operator renormalization, the results for the form factors on all ensembles and for allcombinations of valence masses and momenta are simultaneously fit to staggered chiral perturba-tion theory (SχPT) [12]. Fits to SU(3) SχPT are shown in Fig. 1. The data are for the D→ π decayon a subset of the ensembles of Table 1: the four coarse (a≈ 0.12 fm) ensembles, four of the fine(a ≈ 0.09 fm) ensembles (excluding the 0.05ms ensemble), and three of the superfine (a ≈ 0.06fm) ensembles (excluding the 0.14ms ensemble). Data with momenta up through pπ = (1,1,0) areincluded in the fits.2 The fit function includes leading chiral logarithms (NLO loops) and analyticterms through NNLO.The SU(3) SχPT fits for f D→π⊥ and fD→K⊥ are good, much better than SU(3) SχPT fits forf D→π‖ and fD→K‖ . As exemplified in Fig. 1, fits of fD→π‖ data to SU(3) SχPT are marginal inquality. We understand this behavior in terms of the features of the χPT description of the energydependence of the form factors.The pole from the resonance dominates the energy dependence of f⊥, but the χPT expressionsfor f‖ contain no pole, and the fits suffer. The fit quality for f‖ may also reflect the limitationsof χPT. For energies comparable to the chiral symmetry breaking scale, EK(π) ∼ ΛχSB, we expectthe χPT description of the energy dependence to break down. The largest-momentum data infits to date have pK(π) = (1,1,0), which corresponds to Eπ ∼ ΛχSB; on the coarse ensembles,pπ = (1,1,1) corresponds to energies Eπ & ΛχSB.SU(3) SχPT fits for f D→K‖ support the idea that the χPT description of the energy dependenceof f‖ is breaking down. For the same momenta, the energies EK are larger than the energies Eπ , the2Hereafter all momenta are given in units of 2π/(aL) unless otherwise specified.4D→ K(π)`ν form factors and |Vcs(d)| from lattice QCD Jon A. Bailey0 0.5 1 1.5 2(r1E)20.511.52r 1-1/2f ⊥0.4ms coarse0.2ms coarse0.14ms coarse0.1ms coarse0.4ms fine0.2ms fine0.15ms fine0.1ms fine0.4ms superfine0.2ms superfine0.1ms superfinep = 0.29, χ2/22 = 1.10 0.5 1 1.5 2(r1E)20.511.52r 11/2f ||p = 0.027, χ2/33 = 1.5Figure 1: Fits of D→ π data to SU(3) SχPT; the legend applies to both plots. Included are full QCD datafrom 11 ensembles at three lattice spacings and four light-quark masses. The fit quality for f D→π⊥ (left) isgood, but the fit quality for f D→π‖ is marginal (right).χPT expansion breaks down for smaller momenta, and fits for f D→K‖ should be worse than fits forf D→π‖ . In fact, SU(3) SχPT fits for fD→K‖ have unacceptably small p-values (p < 10−3).To better parametrize the f‖ data, we are investigating alternative fit functions, including SU(2)SχPT. An improved treatment of the energy dependence of f‖ would be especially desirable foradding data at higher momenta. (Cf. Sec. 4.) At the same time, f⊥ dominates the desired formfactor f+, and SU(3) SχPT fits for f D→π⊥ and fD→K⊥ are good. Any model dependence in fD→π+introduced by the χPT description of the energy dependence of f D→π‖ is probably small. Wetherefore proceed to compare the shape of the form factor f D→π+ obtained from the fits shown inFig. 1 to the shape measured by CLEO.3. Comparison of lattice QCD and experimentIf lattice calculations are to be combined with experimental results, the lattice and experimen-tal results must be consistent. Experiments measure shapes of form factors but cannot fix theirnormalizations without the CKM matrix elements (Eq. (1.1)). For D→ K(π)`ν , experiments andlattice calculations access the same q2 region, and comparisons of the lattice and experimental formfactor shapes provide stringent tests of lattice QCD.In Fig. 2 we overlay our calculated f D→π+ , normalized to the point q̃2 = 0.15 GeV2, with thesame ratio from CLEO [3]. The orange (dark grey) error band is the statistical error obtained byincluding data from two coarse and three fine ensembles [13]. The yellow (light gray) error bandis the statistical error obtained by including data from the 11 ensembles of Fig. 1. The curves arefrom SU(3) SχPT, and the errors are from 500 bootstraps. The errors scale as naively expected.The form factor shapes from CLEO and the lattice calculation agree well.4. Projected errorsTo conservatively estimate our errors, we begin with the error budget of our B→ π`ν calcula-tion [4]. At q2 = 0 naive scaling to the full QCD data set of Table 1 gives a statistical error of 4.2%5D→ K(π)`ν form factors and |Vcs(d)| from lattice QCD Jon A. Bailey0 0.5 1 1.5 2 2.5 3q2 (GeV2)123456f +(q2) / f +(0.15 GeV2)5 ensembles of full QCD lattice data (stat. errors only)11 ensembles of full QCD lattice data (stat. errors only)CLEO-c, PRD (2009), arXiv:0906.2983Figure 2: Overlay of the ratiof D→π+ (q2)/ f D→π+ (q̃2) from the lattice (curves anderror bands) and CLEO (blue points) [3]. Thelarger lattice errors are from 5 ensembles, andthe smaller are from 11 ensembles. The formfactor shapes from CLEO and the lattice agree;the errors scale as expected.0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3q2 (GeV2)00.511.522.533.5f (q2)f+f00123456% errorf+ errorf0 errorFigure 3: The form factors f D→π+ (q2) and f D→π0 (q2)and their statistical errors. The curves are from SχPT,and the errors are from the Hessian. The percent er-rors are plotted against the right-hand axis. The largererrors at small q2 probably reflect excluded data atcorresponding momenta on the finer, more chiral en-sembles.and an error from the degenerate-mass vector-current normalization factors of 0.6%. Updating theheavy-quark and ρ-factor power counting estimates to account for the ultrafine data gives errors of2.5% and 0.5%, respectively. For the remaining errors we adopt our previous estimates [10]. Thisleads to a total error of 6.1% for f D→K(π)+ (0). However, this estimate may be overly conservative.In Fig. 3 we plot the form factors (with the ρ-factors set to one) f D→π+ (q2) and f D→π0 (q2) and theirstatistical errors as functions of q2. The data are from the 11 ensembles of Fig. 1.The errors at q2max are much smaller for f0 than for f+ because f+ ( f0) is dominated by f⊥ ( f‖),and we have data for f⊥ ( f‖) for |pπ | ≥ 0.33 GeV↔ q2 ≤ 2.0 GeV2 (|pπ | ≥ 0↔ q2 ≤ q2max). As q2decreases, the errors reflect the addition of data and the hyperbolic behavior of the form factors; theerrors in f+ ( f0) grow approximately linearly in the region 1.6 GeV2 & q2 & 0.8 GeV2 (2.0 GeV2 &q2 & 0.6 GeV2). The largest momentum of data in the fit is pπ = (1,1,0), which correspondsto q2 ∈ [0.43,1.58] GeV2 on the ensembles with ml ≤ 0.2ms, and to q2 ∈ [0.69,1.58] GeV2 onthe superfine ensembles. Without data points below q2 = 0.43 GeV2 on the finer, more chiralensembles, the errors increase rapidly as q2 decreases in the region 0.3 GeV2 > q2 > 0.Extrapolating the curve for the statistical error in f+ to q2 = 0, the error grows to about 3.6%,somewhat smaller than that expected from naive scaling to the entire full QCD data set. Includingdata at smaller q2 would allow interpolation to q2 = 0 and might improve the errors significantly.We can appreciate the potential of the additional data by linearly extrapolating the curve for theerror in f+ for q2 ∈ [0.8,1.6] GeV2 to q2 = 0. The resulting expected error is about 2.0%. Addinga statistical error of 3.6% (2.0%) to our systematics yields a total error of 5.7% (4.8%). Corre-sponding error budgets are in Table 2. We conclude that including data at momenta greater thanpK(π) = (1,1,0) may improve the error in f+(0) to better than 5%. This prospect further motivatesus to consider alternatives to SU(3) SχPT for describing the energy dependence of f‖ at small q2.(Cf. the last paragraph of Sec. 2.)6D→ K(π)`ν form factors and |Vcs(d)| from lattice QCD Jon A. BaileyStat. gπ r1 m̂ ms κc pπ HQ ZV ρ FV Sys. Tot.(a) 3.6 2.9 1.4 0.3 1.3 0.2 0.1 2.5 0.6 0.5 0.5 4.4 5.7(b) 2.0 2.9 1.4 0.3 1.3 0.2 0.1 2.5 0.6 0.5 0.5 4.4 4.8Table 2: Projected error budgets for the form factors at q2 = 0, assuming we (a) exclude data at momentagreater than pK(π) = (1,1,0) and (b) include data at momenta greater than pK(π) = (1,1,0). Errors are dueto limited statistics and the truncation of SχPT; uncertainties in the D∗Dπ coupling, scale, average up-downquark mass, strange quark mass, and charm hopping parameter; momentum-dependent discretization effectsof the light quarks and gluons; heavy-quark discretization effects; uncertainties in the matching factors ZVand ρ; and finite volume effects. The last two entries are the total systematics and total error.Computations were carried out with resources provided by the USQCD Collaboration, Ar-gonne Leadership Computing Facility, and National Energy Research Scientific Computing Center,which are funded by the Office of Science of the U.S. DOE; and with resources provided by theNational Institute for Computational Science, Pittsburgh Supercomputer Center, San Diego Super-computer Center, and Texas Advanced Computing Center, which are funded through the NationalScience Foundation’s Teragrid/XSEDE Program. This work was supported in part by the U.S. DOEunder grant Nos. DE-FG02-91ER40661 (S.G.), No. DE-FG02-91ER40677 (R.D.J., A.X.K.). Thismanuscript was co-authored by employees of Brookhaven Science Associates, LLC, under Con-tract No. DE-AC02-98CH10886 with the U.S. DOE. R.S.V. acknowledges support from BNL viathe Goldhaber Distinguished Fellowship. J.A.B. is supported by the Creative Research Initiativesprogram (3348-20090015) of the NRF grant funded by the Korean government (MEST). Fermilabis operated by Fermi Research Alliance, LLC, under Contract No. DE-AC02-07CH11359 with theU.S. DOE.References[1] J. Charles et al. [CKMfitter], Phys. Rev. D 84, 033005 (2011)[2] K. Nakamura et al. [PDG], J. Phys. G 37, 075021 (2010)[3] D. Besson et al. [CLEO], Phys. Rev. D 80, 032005 (2009)[4] J. Bailey et al. [Fermilab Lattice and MILC], Phys. Rev. D 79, 054507 (2009)[5] R. Zhou et al. [Fermilab Lattice and MILC], arXiv:1111.0981 [hep-lat][6] K. Orginos, D. Toussaint and R. L. Sugar [MILC], Phys. Rev. D 60, 054503 (1999), and referencestherein.[7] A. X. El-Khadra, A. S. Kronfeld and P. B. Mackenzie, Phys. Rev. D 55, 3933 (1997), A. S. Kronfeld,ibid. 62, 014505 (2000)[8] C. W. Bernard et al. [MILC], Phys. Rev. D 58, 014503 (1998), and references therein.[9] M. Wingate et al. [HPQCD], Phys. Rev. D 67, 054505 (2003)[10] J. A. Bailey et al. [Fermilab Lattice and MILC], PoS LAT2009, 250 (2009)[11] A. X. El-Khadra et al., Phys. Rev. D 64, 014502 (2001)[12] C. Aubin and C. Bernard, Phys. Rev. D 76, 014002 (2007)[13] J. A. Bailey et al. [Fermilab Lattice and MILC], PoS LATTICE2010, 306 (2010)7

$D$ semileptonic form factors and $|V_{cs(d)}|$ from 2+1 flavor lattice QCD

D semileptonic form factors and |V_cs(d)| from 2+1 flavor lattice QCD

D semileptonic form factors and |V_cs(d)| from 2+1 flavor lattice QCD

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