The Fermilab Lattice-MILC collaboration is completing a comprehensive program
of heavy-light physics on the MILC (2+1)-flavor asqtad ensembles with lattice
spacings as small as 0.045 fm and light-to-strange-quark mass ratios as low as
1/20. We use the Fermilab interpretation of the clover action for heavy valence
quarks and the asqtad action for light valence quarks. The central goal of the
program is to provide ever more exacting tests of the unitarity of the CKM
matrix. We give a progress report on one part of the program, namely the
analysis of the semileptonic decay B to D at both zero and nonzero recoil.
Although final results are not presented, we discuss improvements in the
analysis methods, the statistical errors, and the parameter coverage that we
expect will lead to a significant reduction in the final error for |V_cb| from
this decay channel.Comment: 7 pages, 3 figures, LATTICE 2011 conferenc

/Brookhaven

/Fermilab

DeTar, Carleton

Du, Daping

Kronfeld, Andreas S.

Laiho, Jack

Qiu, Si-Wei

U., /Iowa

U., /Strathclyde U. /Glasgow

U., /Utah

Van de Water, Ruth S.

English

arXiv

The Fermilab Lattice-MILC collaboration is completing a comprehensive program of heavy-light physics on the MILC (2+1)-flavor asqtad ensembles with lattice spacings as small as 0.045 fm and light-to-strange-quark mass ratios as low as 1/20. We use the Fermilab interpretation of the clover action for heavy valence quarks and the asqtad action for light valence quarks. The central goal of the program is to provide ever more exacting tests of the unitarity of the CKM matrix. We give a progress report on one part of the program, namely the analysis of the semileptonic decay B to D at both zero and nonzero recoil. Although final results are not presented, we discuss improvements in the analysis methods, the statistical errors, and the parameter coverage that we expect will lead to a significant reduction in the final error for |V{sub cb}| from this decay channel

UNT Digital Library

arXiv:1111.0677v1  [hep-lat]  2 Nov 2011Semileptonic B to D decays at nonzero recoil with2+1 flavors of improved staggered quarksSi-Wei Qiu ∗Department of Physics and Astronomy, University of Utah, Salt Lake City, UT84112, USAE-mail: siwei.qiu@utah.eduCarleton DeTarDepartment of Physics and Astronomy, University of Utah, Salt Lake City, UT84112, USAE-mail: detar@physics.utah.eduDaping DuDepartment of Physics and Astronomy, University of Iowa, Iowa City, IA 52240, USAE-mail: ddu@illinois.eduAndreas S. KronfeldTheoretical Physics Department, Fermi National Accelerator Laboratory, Batavia, IL 60510,USAE-mail: ask@fnal.govJack LaihoSUPA, Department of Physics and Astronomy, University of Glasgow, Glasgow, Scotland, UKE-mail: jlaiho@fnal.govRuth S. Van de WaterPhysics Department, Brookhaven National Laboratory, Upton, New York 11973, USAE-mail: ruthv@bnl.gov(Fermilab Lattice and MILC Collaborations)The Fermilab Lattice-MILC collaboration is completing a comprehensive program of heavy-lightphysics on the MILC (2+1)-flavor asqtad ensembles with lattice spacings as small as 0.045 fmand light-to-strange-quark mass ratios as low as 1/20. We use the Fermilab interpretation of theclover action for heavy valence quarks and the asqtad actionfor light valence quarks. The centralgoal of the program is to provide ever more exacting tests of the unitarity of the CKM matrix.We give a progress report on one part of the program, namely thanalysis of the semileptonicdecayB to D at both zero and nonzero recoil. Although final results are not presented, we discussimprovements in the analysis methods, the statistical errors, and the parameter coverage that weexpect will lead to a significant reduction in the final error f|Vcb| from this decay channel.The XXIX International Symposium on Lattice Field Theory - Lattice 2011July 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 Licen. http://pos.sissa.it/Semileptonic B → Dlν decays at nonzero recoil Si-Wei Qiu1. IntroductionPrecision tests of the standard model from flavor factory andintensity frontier experimentscan reveal new physics even if the accelerator energy isn’t sufficient to create the new particlesassociated with the new physics. Here preliminary results of a lattice-QCD calculation of thenonzero-recoil form factor for the semileptonic processB → Dℓν are presented. The principal goalis to determine|Vcb| to a high precision. The theoretical uncertainty on|Vcb| limits the precision ofthe unitarity triangle constraint from neutral kaon mixing.There are several methods for determining|Vcb|. (1) From the inclusive decayb→ cℓν , pertur-bation theory and the operator product expansion provide anstimate of the inclusive decay rate,which is then combined with the measured decay rate to obtain|Vcb| [1]. (2) From the exclusivesemileptonic decayB→D∗ℓν , lattice-QCD methods provide the hadronic contribution tothe decayrate [2]. For the special case of zero recoil, we have presentd an unquenched calculation for thisprocess three years ago [3] with a preliminary update reportd a the Lattice 2010 [4] and CKM2010 conferences [5] (3) Likewise, from the exclusive semilptonic decayB → Dℓν , lattice-QCDmethods provide the hadronic contribution [8]. Here we repot the first unquenched lattice-QCDcalculation for this process at nonzero recoil.At present there is a 1.6σ disagreement between the value of|Vcb| determined from inclusivedecays and our recent preliminary value based on the decayB → D∗ℓν at zero recoil, as illustratedin Fig. 1. Thus, further cross checks of and improvements in the theoretical calculations are needed.The value of|Vcb| can be determined from the ratio of the measured decay rate tothe calculatedhadronic form factor at any chosen recoil energy. Typically, the uncertainty in the experimentalmeasurement increases as the recoil energy vanishes [12], whereas the uncertainty in the lattice-QCD calculation decreases [6, 7, 8]. Thus, to minimize the uncertainty in the ratio, it is best tocombine results of the experimental measurement with thoseof the lattice-QCD calculation overthe full range of available recoil energies. For this reasonwe undertake an analysis over a broadrange.2. Methodology for determining |Vcb|The hadronic weak matrix element for this process is commonly parameterized as〈D(p′)|Vµ |B(p)〉=√mBmD[hB→D+ (w)(v+ v′)µ +hB→D− (w)(v− v′)µ ] . (2.1)Figure 1: The central values and errors resulting from three determinatio s of|Vcb|: (1) A “global fit” tothe unitarity triangle including all inputs except|Vcb| [9] [10], (2) inclusive measurements summarized bythe HFAG [11], and (3) ourB → D∗ lattice-QCD calculation [4, 5].2Semileptonic B → Dlν decays at nonzero recoil Si-Wei QiuHere,w is velocity transfer,w = v · v′. The differential rate for the semileptonic decayB → Dℓν isdΓ(B → Dlν)dw=G2F |Vcb|248π3(mB +mD)2m3D[w2−1]3/2|GB→D(w)|2 , (2.2)whereGB→D(w) is defined asGB→D(w) = hB→D+ (w)−mB −mDmB +mDhB→D− (w) . (2.3)Thus, once we knowhB→D+ andhB→D− , we can determine the form factorGB→D(w). Then, combin-ing it with the result of the decay rate from experiment, we will obtain |Vcb|. In order to constructhB→D+ andhB→D− , we need the following quantities:h+(w) = R+(p)[1− x f (p)R−(p)], (2.4)h−(w) = R+(p)[1−R−(p)/x f (p)], (2.5)R+(p) = 〈D(p)|V 4|B(0)〉, (2.6)R−(p) =〈D(p)|V 1|B(0)〉〈D(p)|V 4|B(0)〉 , (2.7)x f (p) =〈D(p)|V 1|D(0)〉〈D(p)|V 4|D(0)〉 , (2.8)whereV µ is the continuum hadronic weak vector current.In our calculation we use the local (nonconserved) lattice vector currentVµ(x)= Ψ̄b(x)iγµ Ψc(x)with O(a) improved heavy quark fieldsΨh(x) following [13], and we renormalize it following apartly nonperturbative method, namelyZVbc = ρV√ZVbbZV cc , (2.9)where the flavor-diagonal renormalization coefficients arecomputed nonperturbatively on the lat-tice via the conditionsZV bb〈B|V 4|B〉 = 1, (2.10)ZVcc〈D|V 4|D〉 = 1, (2.11)andρV is computed perturbatively [7]. Because of cancellations among similar loop diagrams, weexpect thatρV is nearly equal to 1 [7]. Preliminary results presented hereomit theρV factor.At zero recoil (w = 1) we use the double ratio method [2]:|h+(1)|2 = ρ2V〈D|V 4|B〉〈B|V 4|D〉〈D|V 4|D〉〈B|V 4|B〉 . (2.12)This ratio suppresses a large part of the statistical fluctuations, and it builds in the current renor-malization√ZV bbZV cc .3Semileptonic B → Dlν decays at nonzero recoil Si-Wei Qiua(fm) size ml/mh≈0.15 163×48 0.2≈0.12 203×64 0.14≈0.12 203×64 0.2≈0.12 203×64 0.4≈0.12 243×64 0.1≈0.09 283×96 0.2≈0.09 323×96 0.15a(fm) size ml/mh≈0.09 403×96 0.1≈0.09 643×96 0.05≈0.06 483×144 0.2≈0.06 483×144 0.4≈0.06 563×144 0.14≈0.06 643×144 0.1≈0.045 643×192 0.2Table 1: Summary of all ensembles to be included in the full analysis.The ensembles in bold have beenanalyzed for this report. Light(strange) sea quark masses are denoted byml(mh).3. ResultsWe report on results from an analysis of a large set of gauge field ensembles generated in thepresence of 2+1 flavors of improved staggered (asqtad) quarks [14, 15]. Further ensembles willbe included in the future. Some key ensemble parameters are listed in Table 3.On each ensemble we compute the three-point correlation functions relevant to the weak ma-trix elements and the two-point functions relevant to the propagation of theB andD mesons. Forthe heavy-light mesons we use heavy clover quarks in the Fermilab interpretation and light, im-proved staggered (asqtad) quarks [14, 15]. We use two types of interpolating operators for theB andD mesons, namely local and smeared using the 1S Richardson wave function. We set thevalence-quark masses equal to the sea-quark masses. The bare lattice charm and bottom quarkmasses are fixed by matching the kinetic masses of theDs andBs mesons, respectively, to theirexperimental values. Some small adjustments will be neededto refine this tuning but are not yetincluded in this preliminary analysis.In terms of interpolating operatorsOB andOD for theB andD mesons and the vector current,the two-point and three-point functions are given byC2pt(t;p) = ∑xexp(ip ·x)〈O(x, t)O†(0)〉= ∑nstnZn(p)[exp(−En(p)t)+exp(−En(p)(Nt − t))] (3.1)C3pt,B→DVµ (t,T ;p) = ∑x,yexp(ip ·y)〈OD(0)Vµ (y, t)O†B(x,T )〉 , (3.2)wheresn = ±1 accounts for contributions that oscillate int, andNt is the lattice extent in thetdimension. Note that in the three-point function above, we have put theD meson at the origin andtheB meson at(x,T ).The matrix elements we want from Eq. (2.6) to Eq. (2.8) are obtained by factorization andreduction of the three-point functions. The reduction is baed on the overlap coefficientsZn(p),energies and masses obtained in fits to the two-point functions. For example, theR+(p) matrixelement is obtained fromR+(t,T ;p) =C3pt,B→DV4 (t,T ;p)√C3pt,D→DV4 (t,T ;0)C3pt,B→BV4(t,T ;0)√ZD(0)EDZD(p)mDeEDt−12mDT e−mB(t−12T ) (3.3)4Semileptonic B → Dlν decays at nonzero recoil Si-Wei Qiuin the limit t → ∞ andT − t → ∞. In that limit it is a constant,R+(p), independent oft. However, atfinite t andT , complications from oscillating terms and excited states mu t be taken into account.Because they contain a light staggered quark, the meson interpolating operators excite even-as well as odd-parity channels. The unwanted even-parity sta es manifest themselves as terms thatoscillate int, corresponding to terms in the two-point and three-point fuctions withsn = −1. Tosuppress the effect of the oscillating terms in the three-point functions, we average in bothandT(after removing the dominant exponential dependence ont as in Eq. (3.3)) as follows:R ≡ 12R(0, t,T )+14R(0, t,T +1)+14R(0, t +1,T +1) . (3.4)Here,R refers to any of the quantities derived from three-point functions, such asR+, R−, andx f .Having suppressed the contributions of the oscillating state to the three-point correlators, wemust still account for contributions from excited states. For example, forD → D, the leadingcontributions areC3ptD→DV4 (p, t) =√ZD(p)e−EDt√2ED〈D|V4|D〉e−mD(T−t)√2mD√ZD(0)+√ZD′(p)e−ED′t√2ED′×〈D′|V4|D〉e−mD(T−t)√2mD√ZD(0)+√ZD(p)e−EDt√2ED〈D|V4|D′〉e−mD′ (T−t)√2mD′√ZD′(0) , (3.5)where the primes indicate the overlaps and energies of the excited states of the same parity. We haveneglected the doubly excited contribution, since the singly excited contribution is found already tobe small. Thus, at finitet andT the ratioR+(t,T ;p) in Eq. (3.3) is not constant, but has a smallcontamination from excited states. We take this into account by fitting the resulting ratio toR+(p, t,T ) = R+(p)exp(δmt)+Aexp(−∆ED t)+Bexp(−∆mB (T − t)) (3.6)as illustrated in Fig. 2. The fit parameterδm should be zero, but it is introduced to allow fora small error in determiningED and mB. The other terms involve∆ED = ED′ −ED and ∆mB =mB′ −mB. These parameters are obtained from the fits of two-point functio s. The results of two-point function fits then become prior central values and widths for fitting R+(t,T ;p). We use asimilar method to obtainR−(p) andx f (p).4. Chiral and continuum extrapolationTo complete the analysis we will extrapolate in the light quark masses to their physical valuesand take the continuum limit, using staggered chiral perturbation theory [16, 17] . At this prelim-inary stage, we don’t show physical values and chiral fit in ths report. The numerical data pointsfor h+ andh− are shown in Fig. 3. We will need these data points to finish theextrapolation ofphysical value ofh+ andh− and the chiral fit.5. Future plansTo complete the project, we need to (1) finish analyzing the avail ble ensembles, (2) makeadjustments for the tuning of the heavy quark masses, (3) compute the current renormalization (ρVfactor), and (4) combine the experimental results for the diff rential decay rate with our results forthe form factor to obtain|Vcb|, using methods employed for|Vub| to extendw over the full kinematicrange [18].5Semileptonic B → Dlν decays at nonzero recoil Si-Wei QiuFigure 2: A sample fit toR+(t,T ;p). Example three-point correlation fit for the 0.12 fm 0.14ml/mhensemble. The circles correspond to the local source, and diamonds, the smeared source. The fit result isobtained by a simultaneous fit to both correlators using Eq. (3.3). The horizontal lines represent the centralvalue and error of the result forR+(p). 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1  1.05  1.1  1.15h +wB2D h+0.12fm 0.1ms0.12fm 0.14ms0.12fm 0.2ms0.12fm 0.4ms0.09fm 0.1ms0.09fm 0.15ms0.09fm 0.2ms-0.14-0.12-0.1-0.08-0.06-0.04-0.02 1  1.05  1.1  1.15h -wB2D h-0.12fm 0.1ms0.12fm 0.14ms0.12fm 0.2ms0.12fm 0.4ms0.09fm 0.1ms0.09fm 0.15ms0.09fm 0.2msFigure 3: Form factorsh+ (left) andh− (right) vs.w, omitting the matching factorρV , on the bold ensemblesin Table 3. The color code is given in the legends.AcknowledgmentsComputations for this work were carried out with resources provided by the USQCD Col-laboration, the Argonne Leadership Computing Facility, the National Energy Research ScientificComputing Center, and the Los Alamos National Laboratory, which are funded by the Office ofScience of the U.S. Department of Energy; and with resourcesprovided by the National Center forSupercomputing Applications and the National Institute for C mputational Science, the PittsburghSupercomputer Center, the San Diego Supercomputer Center,a d the Texas Advanced ComputingCenter, which are funded through the National Science Foundation’s Teragrid/XSEDE Program.This work was supported in part by the U.S. Department of Energy under Grants No. DE-FG02-91ER40664 (D.D.), No. DE-FG02-91ER40677 (D.D), and No. DE-FC06-ER41446(C.D.); and inpart by the U.S. National Science Foundation under grants PHY0757333 (C.D.) and PHY09035716Semileptonic B → Dlν decays at nonzero recoil Si-Wei Qiu(S.-W.Q.). J.L. is supported by the STFC and by the Scottish Universities Physics Alliance. Thismanuscript has been co-authored by employees of BrookhavenScience Associates, LLC, underContract No. DE-AC02-98CH10886 with the U.S. Department ofEnergy. R.S.V. acknowledgessupport from BNL via the Goldhaber Distinguished Fellowship. D.D. was supported in part by theURA Visiting Scholars’ program at Fermilab. Fermilab is operat d by Fermi Research Alliance,LLC, under Contract No. DE-AC02-07CH11359 with the United States Department of Energy.References[1] C. G. Boyd, Z. Ligeti, I. Z. Rothstein, and M. B. Wise, Phys. Rev. D55 (1997) 3027[arXiv:hep-ph/9610518].[2] S. Hashimoto, A. S. Kronfeld, P. B. Mackenzie, S. M. Ryan,d J. N. Simone, Phys. Rev. D66(2002) 014503 [hep-ph/0110253].[3] C. Bernardet al. [Fermilab Lattice and MILC Collaborations], Phys. Rev. D79 (2009) 014506[arXiv:0808.2519 [hep-lat]].[4] J. A. Baileyet al. [Fermilab Lattice and MILC Collaborations], PoSLATTICE2010 (2010) 311[arXiv:1011.2166 [hep-lat]].[5] P. B. Mackenzie, talk at Workshop on CKM Unitarity Triangle, Warwick, UK (September 2010).[6] A. S. Kronfeld, Phys. Rev. D62 (2000) 014505 [hep-lat/0002008].[7] J. Harada, S. Hashimoto, A. S. Kronfeld, and T. Onogi, Phys. Rev. D65 (2002) 094514[hep-lat/0112045].[8] G. M. de Divitiis, E. Molinaro, R. Petronzio, N. Tantalo,Phys. Lett. B655 (2007) 45[arXiv:0707.0582 [hep-lat]].[9] J. Laiho, E. Lunghi, and R. S. Van de Water, Phys. Rev. D81 (2010) 034503 [arXiv:0910.2928[hep-ph]].[10] J. Laiho, E. Lunghi, and R. S. Van de Water,http://latticeaverages.org/, accessed July,2011.[11] Heavy Flavor Averaging Group ,http://www.slac.stanford.edu/xorg/hfag/semi/,accessed July, 2011.[12] J. E. Barteltet al. [CLEO Collaboration], Phys. Rev. Lett.82 (1999) 3746 [hep-ex/9811042].[13] A. X. El-Khadra, A. S. Kronfeld, P. B. Mackenzie, Phys. Rev. D55 (1997) 3933 [hep-lat/9604004].[14] C. W. Bernardet al. [MILC Collaboration], Phys. Rev. D61 (2000) 111502 [hep-lat/9912018].[15] A. Bazavovet al., Rev. Mod. Phys.82 (2010) 1349 [arXiv:0903.3598 [hep-lat]].[16] C. K. Chow and M. B. Wise, Phys. Rev. D48 (1993) 5202 [arXiv:hep-ph/9305229].[17] J. Laiho and R. S. Van de Water, Phys. Rev. D73 (2006) 054501 [hep-lat/0512007].[18] J. A. Baileyet al. [Fermilab Lattice and MILC Collaborations], Phys. Rev. D79 (2009) 054507[arXiv:0811.3640 [hep-lat]].7

Semileptonic B to D Decays at Nonzero Recoil with 2+1 Flavors of Improved Staggered Quarks

Van de Water, Ruth

Semileptonic B to D decays at nonzero recoil with2+1 flavors of improved staggered quarksSi-Wei QiuPhysics Department, University of Utah, Salt Lake City, UT 84112, USAE-mail: siwei.qiu@utah.eduCarleton DeTar∗†Physics Department, University of Utah, Salt Lake City, UT 84112, USAE-mail: detar@physics.utah.eduDaping DuDepartment of Physics, University of Illinois, Urbana, IL 61801, USAAndreas S. KronfeldFermi National Accelerator Laboratory, Batavia, IL 60510, USAJack LaihoSUPA, Department of Physics and Astronomy, University of Glasgow, Glasgow, Scotland, UKRuth S. Van de WaterFermi National Accelerator Laboratory, Batavia, IL 60510, USA(Fermilab Lattice and MILC Collaborations)The Fermilab Lattice and MILC collaborations are completing a comprehensive program ofheavy-light physics on MILC (2+1)-flavor asqtad ensembles with lattice spacings as small as0.045 fm and light-to-strange-quark mass ratios as low as 1/20. We use the Fermilab interpre-tation of the clover action for heavy valence quarks and the asqtad action for the light valencequarks. The central goal of the program is to provide ever more exacting tests of the unitarity ofthe CKM matrix. We present preliminary results for one part of the program, namely the analysisof the semileptonic decay B→ D`ν at nonzero recoil.The 30th International Symposium on Lattice Field TheoryJune 24 - 29, 2012Cairns, Australia∗Speaker.†Presented on behalf of Si-Wei Qiu.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/arXiv:1211.2247v1  [hep-lat]  9 Nov 2012FERMILAB-CONF-12-604-TSemileptonic B to D decays at nonzero recoil Carleton DeTar1. IntroductionThe CKM matrix element |Vcb| is a key quantity in Standard-Model tests. It normalizes the legsof the unitarity triangle. The dominant uncertainty in |Vcb| comes from theoretical determinationsof the hadronic form factors for B→ c`ν + . . . . The exclusive processes B→ D`ν and B→ D∗`ν[1, 2, 3] can be studied in lattice gauge theory. Here we report on results for B→ D`ν . Latticecalculations at zero recoil typically have the smallest errors. However, because of the phase spacesuppression near zero recoil in B→ D`ν , experimental errors are largest there. Thus, we aim towork at nonzero recoil where the combined experimental and theoretical error is minimized. Thiswork updates our previous report [4] with all ensembles now included in the analysis.The differential decay rate dΓ(B→D`ν)/dq2 is, for m2` min(M2B,q2), proportional to | f+|2for `= e,µ , where for q = pB− pD,〈D(pD)|V µ |B(pB)〉= f+(q2)[(pB + pD)µ −M2B−M2Dq2qµ]+ f0(q2)M2B−M2Dq2qµ . (1.1)Here V µ = b̄γµc is the b→ c vector current and f+ and f0 are the vector and scalar form factors,respectively. The alternative form factors h+ and h− are convenient:〈D(pD)|V µ |B(pB)〉√MBMD= h+(w)(v+ v′)µ +h−(w)(v− v′)µ , (1.2)where v = pB/MB and v′ = pD/MD. They are related to f+ and f0 throughf+(q2) =12√r[(1+ r)h+(w)− (1− r)h−(w)] ,f0(q2) =√r[w+11+ rh+(w)−w−11− rh−(w)], (1.3)where r = MD/MB and q2 = M2B +M2D−2wMBMD or w = v · v′.2. Lattice-QCD calculationWe are carrying out calculations on 14 lattice ensembles generated in the presence of 2+ 1flavors of improved (asqtad) staggered sea quarks [7] with light-quark masses and lattice spacingsshown in Fig. 1.In the B meson rest frame for any recoil D-momentum p we can obtain h+ and h− from matrixelements of the current starting from ratios of lattice matrix elements R+ and R−, and x f , whereR+(p) ≡〈D(p)|V 4|B(0)〉R−(p) ≡〈D(p)|V 1|B(0)〉〈D(p)|V 4|B(0)〉x f (p) ≡〈D(p)|V 1|D(0)〉〈D(p)|V 4|D(0)〉(2.1)w(p) = [1+ x f (p)2]/[1− x f (p)2]h+(w) = R+(p)[1− x f (p)R−(p)]h−(w) = R+(p)[1−R−(p)/x f (p)] .2Semileptonic B to D decays at nonzero recoil Carleton DeTarFigure 1: Left panel: parameters of the 14 lattice ensembles in this study. Plotted are values of the light tostrange sea quark mass ratio m`/mh vs. the approximate lattice spacing in fm. Right panel: example fromthe a = 0.06 fm, m`/mh = 0.15 ensemble with T = 24,25. Upper curve: the zero-recoil double ratio, middlecurve: the ratio R+(p, t)/R+(0, t) for the 1S smeared D-meson interpolator, and bottom curve: the localinterpolator. Red points are included in the fit (p = 0.15).At zero recoil, we can also use the double ratio of Hashimoto et al [5]:|h+(0)|2 =〈D(0)|V 1|B(0)〉〈B(0)|V 1|D(0)〉〈D(0)|V 4|D(0)〉〈B(0)|V 4|B(0)〉. (2.2)The continuum V µ and lattice V µ currents are matched through V µcb = ZV µcbVµcb. We use amostly nonperturbative method [5], writingZV µcb = ρVµcb√ZV 4ccZV 4bb , (2.3)and determine ρV µcb from one-loop lattice perturbation theory.In addition to the two-point functions, we need matrix elements 〈Y (p)|V µ |X(0)〉 for X ,Y ∈{B,D}. They are constructed from naive light spectator quark propagators and clover heavy quarkpropagators in the Fermilab interpretation [6] as shown in Fig. 2. Valence bottom and charm quarkmasses were tuned to the “kinetic” Bs and Ds masses. The mass of the naive light spectator quarkis set equal to that of the light sea quark.For interpolating operators OX , we compute two-point and three-point functionsC2pt,X(p, t) =〈O†X(0)OX(t)〉, (2.4)C3pt,X→Y,µ(p;0, t,T ) =〈O†Y (0)Vµ(t)OX(T )〉. (2.5)We use both point and 1S smeared interpolating operators for the D meson and 1S smeared inter-polating operators for the B.3Semileptonic B to D decays at nonzero recoil Carleton DeTar(t)Vt = 0 TBDµclover blight naiveclover cFigure 2: Valence quark line diagram for B→ D.2.1 Two-point and three-point correlator fitsWe obtain the lattice form factors via a two-step procedure. First, we fit the B- and D-mesontwo-point correlators to obtain the energies and overlap factors. Then we use these determinationsas constraints (with Bayesian priors) in the three-point fits. For illustration, we show the reductionof the three-point and two-point functions to obtain R+(p) =〈D(p)|V 4|B(0)〉. We include excitedB and D contributions indicated with a prime but not both together:C3pt,B→DV 4 (p, t) =√ZD(p)e−EDt√2ED〈D(p)|V 4|B(0)〉 e−mB(T−t)√2mB√ZB(0)+√ZD′(p)e−ED′ t√2ED′〈D′(p)|V 4|B(0)〉 e−mB(T−t)√2mB√ZB(0) (2.6)+√ZD(p)e−EDt√2ED〈D(p)|V 4|B′(p)〉 e−mB′ (T−t)√2mB′√ZB′(0) ,orC3pt,B→DV 4 (p, t) =C0(p)〈D(p)|V 4|B(0)〉e−EDte−mB(T−t)[1+C1(p)e−∆EDt +C2(p)e(t−T )∆mB],(2.7)where C0(p), ∆ED = ED′−ED, and ∆mB = mB′−mB come from fits to two-point correlators. Termsoscillating as (−)t (not shown) are introduced by the naive light quark. We suppress their contri-butions by averaging over T , T +1 and t, t +1, as introduced in [1].Putting information from three- and two-point functions together, we getR+(p, t) ≡C3pt,B→DV 4 (p, t)e(ED−mB)t+(mB−mD)T/2√C3pt,D→DV 4 (0, t)C3pt,B→BV 4 (0, t)√ZD(0)EDZD(p)mD≈ R+(p)[1+ s1(p)e−∆EDt + s2(p)e(t−T )∆mB]. (2.8)The zero-recoil form factor h+(0) = R+(0) can be calculated very accurately from the double ratio.A good strategy is to use it to normalize the nonzero recoil values:R+(p, t)R+(0, t)=R+(p)R+(0)exp(δmt)+A(p)exp(−∆EDt)+B(p)exp(∆mBt) , (2.9)where δm= 0, ∆ED =ED′−ED, and ∆mB =mB′−mB are constrained by fits to two-point functions.We do a simultaneous fit to three three-point functions, as illustrated in the right panel of Fig. 1and determine R+, R−, and x f for each momentum (recoil parameter w), from which we determineh+ and h−.4Semileptonic B to D decays at nonzero recoil Carleton DeTar 0.8 0.85 0.9 0.95 1 1.05 1  1.05  1.1  1.15wB2D h+physical value0.12fm 0.1ms0.12fm 0.14ms0.12fm 0.2ms0.12fm 0.4ms0.09fm 0.1ms0.09fm 0.15ms0.09fm 0.2ms0.09fm 0.4ms0.09fm 0.05ms0.06fm 0.2ms0.06fm 0.4ms0.06fm 0.14ms0.06fm 0.1ms0.045fm 0.2ms -0.16-0.14-0.12-0.1-0.08-0.06-0.04-0.02 1  1.05  1.1  1.15wB2D h-Figure 3: Form factors h+ (left) and h− (right) as a function of the recoil parameter w. The curves in eachpanel are a result of a simultaneous fit to the chiral-continuum expressions of Eq. (2.10) with p = 0.68 forthe h+ fit and p = 0.41 for the h− fit.2.2 Chiral-continuum extrapolation and q2 parameterizationThe resulting form factors h+ and h− are shown in Fig. 3. We fit them to the expressionsh+(a,m`,w) = 1−ρ2+(w−1)+ k+(w−1)2 +X+(Λχ)m2c+ c1,+m`+ ca,+a2 + ca,w,+a2(w−1)+g2D∗Dπ16π2 f 2logs1−loop(Λχ ,w,m`,a) (2.10)h−(a,m`,w) =X−mc−ρ2−(w−1)+ k−(w−1)2 + c1,−m`+ ca,−a2 + ca,w,−a2(w−1)for light spectator quark mass m`, lattice spacing a, and w = v · v′. For the one-loop chiral logs weuse a staggered fermion version of Chow and Wise [9]. Thus, these fit functions contain the correctnext-to-leading-order chiral perturbation theory expressions, including staggered discretization ef-fects [10]. As can be seen, the dependence of h+ on a and m`/mh is quite mild. We expect h−to have larger discretization effects than h+ because of different HQET power counting. This isconsistent with what we see in the data. For |Vcb|, the contribution coming from h− over the entirekinematic range is small, so the larger errors in h− don’t increase the overall error much. Thesefeatures with 14 ensembles are consistent with our previous findings with four ensembles [10].To compare the lattice and experimental form factors we need to extrapolate to larger w (equiv-alently q2). We do this using the z-expansion of Boyd, Grinstein and Lebed [11], which provides amodel-independent parameterization of the q2 dependence of f+ and f0. This expansion builds inconstraints from analyticity and unitarity. It is based on the conformal mapz(w) =√1+w−√2√1+w+√2, (2.11)5Semileptonic B to D decays at nonzero recoil Carleton DeTar 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 0  0.01  0.02  0.03  0.04  0.05  0.06  0.07form factorz valueB->D zexpansion χ2/dof=5.9/3f+f0Figure 4: Left: form factors f+ and f0 parameterized by the z expansion (p = 0.12). Right: comparisonwith experimental results from the Babar collaboration [12]. The red line gives our result. The red dottedlines show only statistical errors. The boxed region appears to have the smallest combined error.which maps the physical region w ∈ [1,1.59] to z ∈ [0,0.0644]. It pushes poles and branch cuts faraway at |z| ≈ 1. Form factors are then parameterized asfi(z) =1Pi(z)φi(z)∞∑n=0ai,nzn , (2.12)where Pi(z) are the Blaschke factors and φi are the “outer functions”. The latter are chosen tosimplify the unitarity bound:∑n|ai,n|2 ≤ 1 . (2.13)In practice, we need only the first few coefficients in the expansion. We also impose the kinematicconstraint f+ = f0 at q2 = 0 or z≈ 0.0644.To implement the z expansion, we start from the value of f+ and f0 at the physical point, asdetermined from the chiral/continuum fit. We choose four w values, w = 1.00, 1.04, 1.10, and 1.16,and use the corresponding form factor values to determine the coefficients ai,0, ai,1, and ai,2. These,then, are used to parameterize the form factors over the full kinematic range, as shown in the leftpanel of Fig. 4.We compare our result with experimental measurements from the Babar collaboration [12] inFig. 4. For present purposes we take |Vcb| from B→ D∗`ν at zero recoil [3].3. Future plansTo complete the analysis, we need to apply small corrections resulting from adjusting thecharm and bottom quark masses to their tuned values, implement the full current renomalization,and compile a complete error budget.6Semileptonic B to D decays at nonzero recoil Carleton DeTarAcknowledgementsComputations for this work were carried out with resources provided by the USQCD Collab-oration, the National Energy Research Scientific Computing Center and the Argonne LeadershipComputing Facility, which is funded by the Office of Science of the U.S. Department of Energy;and with resources provided by the National Institute for Computational Science and the TexasAdvanced Computing Center, which are funded through the National Science Foundation’s Tera-grid/XSEDE Program. This work was supported in part by the U.S. Department of Energy undergrant No. DE-FG02-91ER40677 (D.D.) and the U.S. National Science Foundation under grantsPHY0757333 and PHY1067881 (C.D.) and PHY0903571 (S.-W.Q.). J.L. is supported by the STFCand by the Scottish Universities Physics Alliance. This manuscript has been co-authored by em-ployees of Brookhaven Science Associages, LLC, under Contract No. DE-AC02-98CH10886 withthe U.S. Department of Energy. Fermilab is operated by Fermi Research Alliance, LLC, underContract No. DE-AC02-07CH11359 with the United States Department of Energy.References[1] C. Bernard et al. [Fermilab Lattice and MILC Collaborations], Phys. Rev. D79 014506 (2009).[arXiv:0808.2519 [hep-lat]].[2] J. Laiho, R. S. Van de Water, Phys. Rev. D73, 054501 (2006). [hep-lat/0512007].[3] J. A. Bailey et al. [Fermilab Lattice and MILC Collaborations], PoS(Lattice 2010)311 (2010)[arXiv:1011.2166 [hep-lat]].[4] Si-Wei Qiu et al. [Fermilab Lattice and MILC Collaborations], PoS(Lattice 2011)289 (2011)[arXiv:1111.0677 [hep-lat]].[5] S. Hashimoto, A. S. Kronfeld, P. B. Mackenzie, S. M. Ryan, and J. N. Simone, Phys. Rev. D66,014503 (2002). [hep-ph/0110253]; J. Harada, S. Hashimoto, A. S. Kronfeld, and T. Onogi, Phys.Rev. D 65, 094514 (2002) [hep-lat/0112045].[6] A. X. El-Khadra, A. S. Kronfeld, P. B. Mackenzie, Phys. Rev. D55, 3933-3957 (1997).[hep-lat/9604004].[7] T. Blum et al. [MILC Collaboration], Phys. Rev. D 55, 1133 (1997) [hep-lat/9609036].C. W. Bernard et al. [MILC Collaboration], Phys. Rev. D 58, 014503 (1998) [hep-lat/9712010].K. Orginos et al. [MILC Collaboration], Phys. Rev. D 59, 014501 (1999) [hep-lat/9805009].J. F. Lagaë and D. K. Sinclair, Phys. Rev. D 59, 014511 (1999) [hep-lat/9806014]. G. P. Lepage,Phys. Rev. D 59, 074502 (1999) [hep-lat/9809157]. K. Orginos et al. [MILC Collaboration],Phys. Rev. D 60, 054503 (1999) [hep-lat/9903032].[8] A. Bazavov et al., Rev. Mod. Phys. 82, 1349 (2010) [arXiv:0903.3598 [hep-lat]].[9] C. K. Chow and M. B. Wise, Phys. Rev. D 48 (1993) 5202 [arXiv:hep-ph/9305229].[10] J. A. Bailey et al., Phys. Rev. D 85, 114502 (2012) [Erratum-ibid. D 86, 039904 (2012)][arXiv:1202.6346 [hep-lat]].[11] C. G. Boyd, B. Grinstein and R. F. Lebed, Phys. Rev. Lett. 74, 4603 (1995) [hep-ph/9412324].[12] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. 104, 011802 (2010) [arXiv:0904.4063[hep-ex]].7

Semileptonic B to D decays at nonzero recoil with 2+1 flavors of improved staggered quarks. An update

Semileptonic B to D decays at nonzero recoil with 2+1 flavors of
  improved staggered quarks

Semileptonic B to D decays at nonzero recoil with 2+1 flavors of improved staggered quarks

Abstract

Similar works

Full text

Available Versions

UNT Digital Library

UNT Digital Library