We present a (2+1)-flavor lattice QCD calculation of the form factor ratio
between the semileptonic decays $\bar{B}^0_s \to D^+_sl^-\bar{\nu} $ and
$\bar{B}^0 \to D^+l^-\bar{\nu} $. This ratio is an important theoretical input
to the hadronic determination of the $B$ meson fragmentation fraction ratio
$f_s/f_d$ which enters in the measurement of $\mathrm{BR}(B^0_s\to
\mu^+\mu^-)$. Small lattice spacings and high statistics enable us to simulate
the decays with a dynamic final $D$ meson of small momentum and reliably
extract the hadronic matrix elements at nonzero recoil. We report our
preliminary result for the form factor ratio at the corresponding momentum
transfer of the two decays $f_0^{(s)}(M^2_\pi)/f_0^{(d)}(M^2_K)$.Comment: 7 pages, 6 figures. Talk presented at The XXIX International
  Symposium on Lattice Field Theory - Lattice 2011, July 10-16, 2011, Squaw
  Valley, Lake Tahoe, California; Minor errors corrected, references and graphs
  update

DeTar, Carleton

Du, Daping

Kronfeld, Andreas

Laiho, Jack

Meurice, Yannick

Qiu, Si-wei

English

arXiv

We present a (2+1)-flavor lattice QCD calculation of the form factor ratio between the semileptonic decays {bar B}{sub s}{sup 0} {yields} D{sub s}{sup +}l{sup -}{bar {nu}} and {bar B}{sup 0} {yields} D{sup +}l{sup -}{bar {nu}}. This ratio is an important theoretical input to the hadronic determination of the B meson fragmentation fraction ratio f{sub s}/f{sub d} which enters in the measurement of BR(B{sub s}{sup 0} {yields} {mu}{sup +}{mu}{sup -}). Small lattice spacings and high statistics enable us to simulate the decays with a dynamic final D meson of small momentum and reliably extract the hadronic matrix elements at nonzero recoil. We report our preliminary result for the form factor ratio at the corresponding momentum transfer of the two decays f{sub 0}{sup (s)} (M{sub {pi}}{sup 2})/f{sub 0}{sup (d)} (M{sub K}{sup 2})

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arXiv:1111.1796v1  [hep-lat]  8 Nov 2011 Semileptonic form factor ratio Bs → Ds/B→ D and itsapplication to BR(B0s → µ+µ−)Daping Du∗,a,b,c, Carleton DeTard, Andreas Kronfeldb, Jack Laihoe,Yannick Meuricea, and Si-wei QiudaDepartment of Physics and Astronomy, University of Iowa, Iowa City, IA 52240, USAbFermi National Accelerator Laboratory, Batavia, IL 60510, USAcPhysics Department, University of Illinois, Urbana, IL 61801, USAdDepartment of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USAeSUPA, Department of Physics and Astronomy, University of Glasgow, Glasgow, Scotland, UKE-mail: ddu@illinois.eduFermilab Lattice and MILC CollaborationsWe present a (2+1)-flavor lattice QCD calculation of the form factor ratio between the semilep-tonic decaysB̄0s → D+s l−ν̄ andB̄0 → D+l−ν̄. This ratio is an important theoretical input to thehadronic determination of theB meson fragmentation fraction ratiofs/ fd which enters in the mea-surement of BR(B0s → µ+µ−). Small lattice spacings and high statistics enable us to simulate thedecays with a dynamic finalD meson of small momentum and reliably extract the hadronic ma-trix elements at nonzero recoil. We report our preliminary result for the form factor ratio at thecorresponding momentum transfer of the two decaysf (s)0 (M2π)/ f(d)0 (M2K).XXIXth 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-11-610-TSemileptonic form factor ratio Bs → Ds/B→ D Daping Du1. IntroductionThe rare decayB0s → µ+µ− is a process that is potentially sensitive to physics beyondthestandard model (SM). In the SM, the decay can go only through penguin or box topologies at theloop level. Thus, a small branching fraction has been predict , with the aid of lattice QCD, to be3.2(2)×10−9 [1, 2]. Recently, LHCb [3] and CDF [4] reported bounds on the branching fraction,to be followed by upcoming results from CMS. It is likely thata 5σ measurement will be made,even at the SM branching ratio, in the near future.At LHCb, the extraction of the branching fraction relies on the normalization channelsB+u →J/ψK+, B0d → K+π− andB0s → J/ψφ [5], through the following relationBR(B0s → µ+µ−) = BR(Bq → X)fqfsεXεµµNµµNX, (1.1)whereε andN are the detector efficiencies and number of events. The fragmentation fractions,fq (q = u,d,s or Λ), denote the probability of ab quark hadronizing into aBq meson or ab-flavored (e.g.,Λb) baryon. The fragmentation fraction ratiofs/ fd is crucial in the extraction ofBR(B0s → µ+µ−). Currently, the uncertainty infs/ fd is the major source of uncertainty. Tra-ditionally, fs/ fd was measured using the ratio of the corresponding semileptonic decays. Fleis-cher, Serra and Tuning proposed [6] that the ratio can also bemeasured using the non-leptonicdecaysB̄0s → D+s π− andB̄0d → D+K−, which has the advantages of a cleaner background, similarreconstruction of final states,etc. The approach is based on factorization of the nonleptonic am-plitudes into fπ or fK and corresponding semileptonic form factors. The ratiofs/ fd is related toBR(B̄s → Dπ)/BR(B̄→ DK) in a way similar to Eq. (1.1). With the efficiencies and event countscombined with the factorization approximation, we havefsfd=|Vus|2 f 2K|Vud|2 f 2π× τB0τB0s×[εDKεDsπNDsπNDK]× 1NaNF, (1.2)whereτ is the lifetime andNa ≈ 1 with corrections of a few percent due to nonfactorizable ef-fects [6]. The semileptonic form factor ratioNF = [ f(s)0 (M2π)/ f(d)0 (M2K)]2 is currently the decisivecontributor to the theoretical error. The value currently used at LHCb is an estimate from QCDsum rules,NF = 1.24(8) [7, 6]. However, this theoretical input and the size of its error need tobe validated by a nonperturbative method such as lattice QCD. This paper is devoted to such acalculation.The matrix elements of theB → D semileptonic decay (and similarly forBs → Ds) can bewritten as〈D(p′)|V µ |B(p)〉= f+(q2)[(p+ p′)µ − M2B−M2Dq2qµ]+ f0(q2)M2B−M2Dq2qµ . (1.3)However, for heavy quarks it is convenient to use the variablesh±, defined by〈D(p′)|V µ |B(p)〉√MBMD= h+(w) (v+v′)µ +h−(w) (v−v′)µ , (1.4)wherev= p/M and the recoil variable isw= v·v′ . We will use the form factorsh± in our entireanalysis and convert them in the end tof+, f0 using Eqs. (1.3) and (1.4).In these proceedings, we report a preliminary result of the form factor ratiof (s)0 (M2π)/ f(d)0 (M2K)by analyzing the semileptonic decaysB̄0s → D+s l−ν̄ and B̄0 → D+l−ν̄ on the lattice. We use an2Semileptonic form factor ratio Bs → Ds/B→ D Daping Duidentical subset of the MILC gauge configurations for both oftheBs → Ds andB→ D processes.To reduce the statistical errors effectively, we constructa set of ratios at small recoil, from whichwe extract the lattice form factorsh±. The extrapolation to physical light quark masses and to thecontinuum is performed using root staggered chiral perturbation theory (rSχPT). The results areextrapolated to maximum recoil by employing a model-independent parametrization. In Sec.4, wereport our lattice result.2. Numerical details2.1 Data setupOur calculation uses four ensembles of the MILC’s (2+1)-flavor gauge configurations [8], twoat each of the lattice spacingsa≈ 0.12 fm and≈ 0.09 fm. The ensembles as well as the parametersused are summarized in Table 1. The strange and light sea quarks we e simulated using the asqtad-improved staggered action [9]. The action is also used in ourstrange and light valence quarks. Theheavy quarks (charm and bottom) are simulated using the Sheikhol slami-Wohlert (SW) cloveraction with the Fermilab interpretation [10]. For theB → D decay, the spectator light quark isdegenerate with the light sea quark (full QCD). While for theBs → Ds decay, the strange quark isset close to its physical value. The charm and bottom quarks in our calculation are tuned to theirphysical values up to a tuning uncertainty. The corresponding bare hopping parameterκb(c), aswell as the coefficient for the clover termcSW are given explicitly in Table 1.2.2 Lattice extractionIn this work, we are interested only in the vector current operator. On the lattice we defineVµ =√ZccV4ZbbV4 ΨciγµΨb, whereZhh are normalization factors. The vector current in the continuumis V µ = ρVµVµ , whereρ2Vµ = ZbcVµ ZcbVµ/ZbbV4ZccV4. The factorρVµ can be calculated perturbatively andhas been found to be very close to one [11, 12]. We expect theρVs to largely cancel in the ratio ofthe form factors. Hence this correction is negligible compared with other systematic errors and wetakeρV = 1 for this analysis.We employ the following three-point functions in our analysis,CDVµ B3pt (0, t,T ;pD) = ∑x,y〈0|OD(0,0)ΨciγµΨb(t,y)O+B (0,x)|0〉 eipD·y, (2.1)CDVµ D3pt (0, t,T ;pD) = ∑x,y〈0|OD(0,0)ΨciγµΨc(t,y)O+D (0,x)|0〉 eipD·y, (2.2)CBV4B3pt (0, t,T ;0) = ∑x,y〈0|OB(0,0)Ψbiγ4Ψb(t,y)O+B (0,x)|0〉. (2.3)TheB meson is at rest. To obtain the dependence of the form factorsa small recoilw, we simulatethe final stateD meson at a few small momenta,i.e., p= 2π(1,0,0)/L , 2π(1,1,0)/L, 2π(1,1,1)/La (fm) aml/ams Nconfs cSW κc κb amx(B→ D) amx(Bs → Ds)≈0.12 0.020/0.050 2052 1.525 0.1259 0.0918 0.020 0.0349≈0.12 0.007/0.050 2110 1.530 0.1254 0.0901 0.007 0.0349≈0.09 0.0124/0.031 1996 1.473 0.1277 0.0982 0.0124 0.0261≈0.09 0.0062/0.031 1931 1.476 0.1276 0.0979 0.0062 0.0261Table 1: MILC ensembles of configurations used in this analysis.3Semileptonic form factor ratio Bs → Ds/B→ D Daping Duand 2π(2,0,0)/L. The correlation functions forD → D andB → B serve as normalization. TheD → D correlation function with a non-zero final state momentum isused to extract the recoilw toalleviate the need of renormalizing the four velocity.From these correlation functions we construct three different atios, fits to which include con-tributions from the lowest-order excited states. Explicitly,CDViD3pt (0, t,T;p)CDV4D3pt (0, t,T ;p)= di (1+D02 e−∆E(0)(T−t)+D20 e−∆E(p)t), (2.4)CDViB3pt (0, t,T;p)CDV4B3pt (0, t,T ;p)= bi (1+B02 e−∆m(T−t)+B20 e−∆E(p)t), (2.5)CDViB3pt (0, t,T ;p)CDV4B3pt (0, t,T;0)×[Z0(0)Z0(p)√E0(p)E0(0)e(E0(p)−E0(0))t]= ai (1+A02 e−∆m(T−t)+A20 e−∆E(p)t +A ′20 e−∆E2(0)t)eδ t . (2.6)The factor in the square brackets of Eq. (2.6) cancels the timdependence of the ratio, stemmingfrom the fact that the numerator and denominator have final stateD mesons with different momenta.∆E and∆mdenote the lowest splittings andδ is a parameter that accounts for the impreciseE(p)−E(0) in the bracket in Eq. (2.6). In the fits the lowest-lying energy splittings∆E, ∆m are treatedas fit parameters. The splittings can be extracted from the two-point functions. So, we employ amulti-channel fitting procedure, combining the two-point functions and the ratios of the three-pointfunctions. We find that such a treatment results in more robust fits and more precise splittings.Fromdi ,bi andai we can easily recover the form factorsh± at small recoilw,w =1+d ·d1−d ·d , (2.7)h+(w) = h+(1)(ai/bi −a ·d), (2.8)h−(w) = h+(1)(ai/bi −ai/di). (2.9)3. ResultsThe extrapolation of our lattice results to the physical quark masses and the continuum isguided by rSχPT [13, 14]. However, in the case ofh+ the light quark mass dependence is accom-panied by a small recoilw dependence. Such a dependence was included in the continuumchiralperturbation theory in Ref. [15] and was extended to the NLO rSχPT in Ref. [16]. Forh−, theNLO correction is simply a constant which is inversely proportional to the charm quark mass. Wefollow the same setup, adding NNLO analytic terms and including a2 dependence. The remainingrecoil dependence of the form factors is fitted to a simple quadratic expansion at zero recoil.The results of the chiral/continuum extrapolation are shown in Fig. 1. The form factorh+ forboth of theB→ D andBs → Ds decays shows a small dependence on the light quark masses andlattice spacings. The extrapolated physical values are very close to the lattice data points. Thissuggests thath+ is insensitive to the light degrees of freedom. However, sizable light quark massand lattice spacing dependence appears in the case ofh−, as indicated by the variation due to thesea quark masses and the differences betweenhB→D− andhBs→Ds− (spectator mass). Note that thedifference betweenhB→D+ andhBs→Ds+ is minor. Considering the subleading role that− plays incontributing tof+, f0, we expect the U-spin symmetry breaking effect to be smallerthan what was4Semileptonic form factor ratio Bs → Ds/B→ D Daping Duexpected in [6, 7]. Such an observation is bolstered by the rec nt lattice calculations onf+, f0 oftheD(s) → π(K) decays [17].With the physical values ofh±, we can easily calculatef0, f+ using the physical masses ofthe D andB mesons. However, to evaluate the form factors at a small momentu transfer (q2 =M2π ,M2K), we need to extrapolate the results near maximum recoil. Weuse the model-independentz-parametrization [18] with the constraintf0(0) = f+(0). We take five synthetic points in therecoil range where we have lattice data points. We take the values of f+, f0 by evaluating ourchiral/continuum extrapolation result at these five recoilp nts and perform the z-parametrization.The result is shown in Figure 2. We study the effect of a pole ata vectorBc meson in the Blasckefactor of thez-expansion off+. We find that the shapes of the form factors are only weakly affectedby the inclusion of such a pole.By expanding the form factors at the respective momentum transfers, we finally arrive atf (s)0 (M2π)/ f(d)0 (M2K) = 1.035(39)(20). (3.1)The first error is from statistics. The second error is the systematical error due only to the un-certainty ongDD∗π and to the variation of fits in thez-parametrization. We are in the process ofbuilding a full systematic error budget. 0.75 0.8 0.85 0.9 0.95 1 1.05 1  1.05  1.1  1.15  1.2h+w B->D Chiral/Continuum:  χ2/dof=15/20, p-value=0.78 physical007/050020/0500062/0310124/031 0.75 0.8 0.85 0.9 0.95 1 1.05 1  1.05  1.1  1.15  1.2h+w Bs->Ds Chiral/Continuum:  χ2/dof=8/20, p-value=0.99 physical007/050020/0500062/0310124/031-0.16-0.14-0.12-0.1-0.08-0.06-0.04-0.02 0 1  1.05  1.1  1.15  1.2h-wB->D Chiral/Continuum:  χ2/dof=9/16, p-value=0.89 Physical007/050020/0500062/0310124/031-0.16-0.14-0.12-0.1-0.08-0.06-0.04 1  1.05  1.1  1.15  1.2h-wBs->Ds Chiral/Continuum:  χ2/dof=18/16, p-value=0.31 Physical007/050020/0500062/0310124/031Figure 1: Chiral/continuum extrapolation ofh±(w) for theB→ D (left) andBs → Ds decays (right).5Semileptonic form factor ratio Bs → Ds/B→ D Daping Du 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1  1.1  1.2  1.3  1.4  1.5  1.6w B->D z-expansion:  χ2/dof=1/7, p-value=1 f0(MK2) no polef0(MK2) f+ polef0(fit)f+(fit)f0(data)f+(data) 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1  1.1  1.2  1.3  1.4  1.5  1.6w Bs->Ds z expansion:  χ2/dof=1/7, p-value=0.99f0(Mπ2) no polef0(Mπ2) f+ polef0(fit)f+(fit)f0(data)f+(data)Figure 2: The z-expansion of form factorsf0, f+. The points that we include in the z-expansion fits areshown explicitly. The dashed curves indicate the result of chiral/continuum extrapolation.4. ConclusionsIn summary, we present a (2+1)-flavor lattice QCD calculation of the form factor ratiof (s)0 (M2π)/ f(d)0 (M2K), which is a major theoretical input for the extraction of thefragmentationfraction ratio fs/ fd. The essential part of our calculation is to extract theB → D andBs → Dssemileptonic form factors at non-zero recoil. We reduce thesystematic uncertainty by fitting thelowest-order excited states, and we employ a simultaneous multi-channel fit procedure to addresscorrelations and reduce the statistical uncertainty. Our chiral/continuum results show that the cor-rections to the finite lattice spacings and finite light quarkmasses are small. Our preliminary resultis f (s)0 (M2π)/ f(d)0 (M2K) = 1.035(39)(20), with a partial systematic error budget. As a consequence,we obtainNF = 1.071(78)(40) which implies a smaller U-spin breaking effect than that suggestedin [7], NF = 1.24(8). A more comprehensive analysis with a detailed error budgetis s ill inprogress and will be reported in a forthcoming paper.D.D. thanks Aida El-Khadra for several helpful discussionsthe perturbation theory. D.D.has also received considerable help from Chris Bouchard, Elizabeth Freeland, James Simone, JonBailey, Elvira Gamiz and Ran Zhou, who shared their numerical techniques or codes. Compu-tations for this work were carried out with resources provided by the USQCD Collaboration, theArgonne Leadership Computing Facility, the National Energy Research Scientific Computing Cen-ter, and the Los Alamos National Laboratory, which are funded by the Office of Science of the U.S.Department of Energy; and with resources provided by the Nation l Center for Supercomputer Ap-plications, the National Institute for Computational Scien , the Pittsburgh Supercomputer Center,the San Diego Supercomputer Center,and the Texas Advanced Computing Center, which are fundedthrough the National Science Foundation Teragrid/XSEDE Program. D.D. was supported in partby the URA Visiting Scholars’ program at Fermilab. This workwas supported in part by the U.S.Department of Energy under Grants No. DE-FG02-91ER40664 (Y.M., D.D.), DE-FE06-ER41446(C.D.), No. DE-FG02-91ER40677 (D.D) and in part by the U.S. National Science Foundation un-der Grants PHY0757333 (C.D.) and PHY0903571 (S.-W.Q.). J.Lis supported by the STFC andby the Scottish Universities Physics Alliance. Fermilab isoperated by Fermi Research Alliance,LLC, under Contract No. DE-AC02-07CH11359 with the United States Department of Energy.6Semileptonic form factor ratio Bs → Ds/B→ D Daping DuReferences[1] HPQCD Collaboration, E. Gámiz, C. T. H. Davies, G. P. Lepage, J. Shigemitsu, and M. Wingate,Neutral B meson mixing in unquenched lattice QCD, Phys. Rev.D80 (2009) 014503[arXiv:0902.1815 [hep-lat]][2] A. J. Buras,Relations between∆Ms,d and Bs,d → µµ̄ in models with minimal flavor violation, Phys.Lett.B566 (2003) 115–119 [hep-ph/0303060][3] LHCb Collaboration, R. Aaijet al., Search for the rare decays Bs → µµ and Bd → µµ , Phys. Lett.B699 (2011) 330–340 [arXiv:1103.2465 [hep-ex]][4] CDF Collaboration, T. Aaltonenet al., Search for Bs→ µ+µ− and Bd → µ+µ− Decays with CDF II,arXiv:1107.2304 [hep-ex][5] LHCb Collaboration, B. Adevaet al., Roadmap for selected key measurements of LHCb,arXiv:0912.4179 [hep-ex][6] R. 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Semileptonic form factor ratio B_s->D_s/B->D and its application to BR(B^0_s->\mu^+\mu^-)

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