A lattice determination of the form factor and decay constants for the
semileptonic decay of heavy pseudoscalar (PS) mesons at zero recoil is
presented from which the soft pion relation is satisfied. Chiral extrapolation
of the form factor is performed at constant $q^2$. Pole dominance is used to
extrapolate the form factor in heavy quark mass. At the B mass, the form factor
at zero recoil lies somewhat below the ratio of decay constants; the relation
remains satisfied within error.Comment: 3 pages, 2 figure

Aoki

Brown

Burdman

C.M. Maynard

Cabibbo

D.G. Richards

J. Nieves

J.M. Flynn

K.C. Bowler

L. Del Debbio

L. Lellouch

Luscher

Matsufuru

Sheikoleslami

Sint

V.I. Lesk

English

arXiv

Crossref

An investigation of the soft pion relation in quenched lattice QCD

A lattice determination of the form factor and decay constants for the semileptonic decay of heavy pseudoscalar (PS) mesons at zero recoil is presented from which the soft pion relation is satisfied. Chiral extrapolation of the form factor is performed at constant $q^2$. Pole dominance is used to extrapolate the form factor in heavy quark mass. At the B mass, the form factor at zero recoil lies somewhat below the ratio of decay constants; the relation remains satisfied within error

Collaboration, UKQCD

bowler, K.

maynard, C.

Lellouch, L.

Nieves, J.

richards, D.

Del Debbio, L.

flynn, J.

lesk, V.

Edinburgh Research Explorer

     Edinburgh Research Explorer                                      An Investigation of the Soft Pion Relation in Quenched LatticeQCDCitation for published version:Collaboration, UKQCD, bowler, K, maynard, C, Lellouch, L, Nieves, J, richards, D, Del Debbio, L, flynn, J &lesk, V 1999, 'An Investigation of the Soft Pion Relation in Quenched Lattice QCD', Nuclear Physics B -Proceedings Supplements, vol. 83-84, pp. 313–315. https://doi.org/10.1016/S0920-5632(00)91658-XDigital Object Identifier (DOI):10.1016/S0920-5632(00)91658-XLink:Link to publication record in Edinburgh Research ExplorerPublished In:Nuclear Physics B - Proceedings SupplementsGeneral rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 04. Jan. 2021arXiv:hep-lat/9910020v1  8 Oct 19991An Investigation of the Soft Pion Relation in Quenched Lattice QCDUKQCD CollaborationK.C. Bowlera, L. Del Debbiob, J.M. Flynnb, L. Lellouchc, V.I. Leskb∗, J. Nievesd, C.M. Maynarda andD.G. Richardsa†.aDepartment of Physics and Astronomy, University of Edinburgh EH9 3JZ, UKbDepartment of Physics and Astronomy, University of Southampton, SO17 1BJ, UKcTheory Division, CERN, CH 1211, Geneva 23, SwitzerlanddDepartamento de Fisica Moderna, Universidad de Granada, Granada 18071, SpainA lattice determination of the form factor and decay constants for the semileptonic decay of heavy pseudoscalar(PS) mesons at zero recoil is presented from which the soft pion relation is satisfied. Chiral extrapolation of theform factor is performed at constant q2. Pole dominance is used to extrapolate the form factor in heavy quarkmass. At the B mass, the form factor at zero recoil lies somewhat below the ratio of decay constants; the relationremains satisfied within error.1. IntroductionHeavy-quark and chiral symmetries combine topredict the following relation for a heavy-lightmeson H the mass of whose heavy quark is largerthan some arbitrary hadronic scale, and for a softpion:f0(q2max)H→π = fH/fπ (1)where f0(q2) is the form factor in the 0+ channelfor the vector current hadronic matrix element〈H(p)|V µ|π(p − q)〉. The two quantities on theright-hand side are the corresponding PS mesondecay constants. Equation 1 is the soft pion rela-tion for heavy-light PS mesons [1].Results are given for f0(q2max)H→π and fH/fπfor heavy-light PS mesons and are extrapolatedto the B mass. Errors are quoted asfB = 190(5)stat.(10)sys.MeV. (2)2. Extracting the form factorsThe form factors for the hadronic matrix ele-ment of semileptonic heavy pseudoscalar decay toa pion are extracted from mass extrapolations of∗Presenter†Jefferson Laboratory, Newport News, VA 23606, USAthe appropriate matrix elements of quenched lat-tice QCD with SW action [2] with a−1=2.6(1)GeV, β=6.2 on a 243×48 lattice using 216 gaugeconfigurations generated with a combination ofthe Cabbibo-Marinari algorithm [3] and an over-relaxed algorithm [4]. The matrix elements arecalculated using the method of extended quarkpropagators with all combinations of the follow-ing hopping parameters:κheavy ∈ {0.1200, 0.1233, 0.1266, 0.1299}κlight ∈ {0.1346, 0.1351, 0.1353}κspectator ∈ {0.1346, 0.1351}.From the light PS meson massesκcrit=0.13582(1)(2) is found.The action and operators are improved to O(a)and renormalized to a continuum scheme, usingcoefficients determined non-perturbatively wherepossible [5,6]. The following values are used:ZV = 0.792 ZA = 0.807bV = 1.41 bA = 1.11cV = −1.58 × 10−2 cA = −3.71 × 10−2cSW = 1.61 bM = 0.583Of these, only cV , bA and bM are determined froma perturbative scheme [7].2In the continuum, the form factors f+ and f0are defined from hadronic matrix elements as fol-lows:〈H(p)|V µ|π(p − q)〉 =f+(q2)(2 pµ −[(M2H − M2π)/q2 + 1]qµ)+f0(q2)([M2H − M2π]/q2)qµ3. Pion physics on the latticeLattice correlators in this study are directly rel-evant to the physics of heavy-light mesons withmass 1300− 2200 MeV, and of light-light mesonswith mass 350 − 850 MeV. Form factors at thesame value of q2 for decays to a pion can be es-timated by assuming the following dependence ofa form factor f ∈ {f+, f0} on the meson masses:f(q2, M, m) = f(q2, Mchiral, 0)+a×∆M+b×m2(3)and fitting the data to determine f(0) and a. Sim-ilarly light-light meson and heavy-light meson de-cay contants fL and fH are extrapolated in themeson mass according to the following:fL(m) = fL(0) + a′ × m2 (4)fH(M) = fH(Mchiral) + a′′ × ∆M (5)where ∆M ≡ M − Mchiral.The q2 dependence of the form factors can bemodelled using the following parameterization [8].f+(q2) =cB(1 − α)(1 − q2/M2∗ )(1 − αq2/M2∗ )(6)f0(q2) =cB(1 − α)(1 − q2/βM2∗ )(7)where M∗ is the heavy-light vector meson mass.f0(q2max) is extracted by extrapolating f0 upwardsin q2 using the best fit curve from a fit of 7(fig. 1). One finds the soft pion relation to bewell satisfied at simulated values of heavy quarkmass.4. The soft pion relation for B → πHeavy quark effective theory predicts the fol-lowing form for the dependence of f0 on the heavyquark mass, at constant recoil variable v · (p− q):f0(M, ω)Θ(M)√M = a+b/M+c/M2+O(1M3)(8)−0.5 −0.3 −0.1 0.1 0.3 0.5 0.7 0.9q2/M*20.00.51.01.52.0f(q2 )Figure 1. Example fit to f+, f0(q2) for decay to apion, κheavy = 0.1200.Table 1The soft pion relation for a massless pion.κheavy f0(q2max) fPS/fπ0.1200 1.5(2)(3) 1.57(6)(9)0.1233 1.5(2)(2) 1.53(6)(9)0.1266 1.5(2)(2) 1.48(6)(9)0.1299 1.4(1)(1) 1.41(5)(9)and for a heavy-light pseudoscalar decay con-stant:fH(M)Θ(M)√M = a′+b′/M+c′/M2+O(1M3)(9)where Θ(M) is a perturbative matching coeffi-cient function whose value is very close to 1.These prescriptions are used as fit ansatz to de-termine a best fit quadratic in 1/M to the data intable 1. The curve is shown in (fig. 2). Values forf0(q2max) and fH extrapolated to the b mass arepresented in table 2. It is instructive to repeatTable 2The soft pion relation for B decay to a masslesspion.f0(q2max) fB/fπMB 1.1(2)(2) 1.45(6)(9)the whole procedure, extrapolating form factorsfor a pion which is not massless but has its physi-cal mass. Results are presented in table 3 and arepractically unchanged from the massless case.30.0 0.5 1.0 1.5 2.0 2.5 3.01/MH0.0400.0600.0800.1000.120fΘMH0.5Figure 2. Best fit fB vs. heavy mass accordingto eqn. 9.Table 3The soft pion relation for B decay to a 140 MeVpion.f0(q2max) fB/fπMB 1.1(2)(2) 1.44(6)(9)5. Systematic errorSystematic error in this work arises from:1. discretization errors of O(a2)2. quenched approximation3. estimation of correlation matrix for fits4. interpolation of lattice form factors in q25. corrections to heavy quark effective theory6. choice in setting the scale.The quoted systematic error is an estimate ofthe variation arising from the last sources 3 − 6,achieved by repeating the analysis with differ-ent numbers of bootstrap sets, varying the modelfunction used to interpolate the form factor inq2, changing the degree of the polynomial in 1/Mused as an ansatz for the extrapolation in heavymass, and setting the scale alternatively againstMρ and the gluonic scale r0. Quenching error isnot quantified here. Residual discretization er-rors also may be significant, particularly in theheavy extrapolation where the fitted curve usedmay be diverted substantially by a correction tothe lattice form factor at the largest simulatedheavy quark mass, for which (Ma)2 ≃ 0.65.6. ConclusionsFor a simulated heavy-light decay whose massis within 20% of MD, the soft pion relation isreproduced in this study. On extrapolating theform factor and the heavy-light decay constant tothe B mass, systematic errors become O(30%),to within which precision the soft pion relationholds for B meson decays. Other approaches todate [9,10] have generally violated this relation,with the form factor lying significantly below theratio of decay constants. Work is ongoing withthe prospect of controlling further the systematicerror, and a simulation at β = 6.0 including astatic heavy quark will address a large source ofsystematic uncertainty as well as providing scal-ing information [11].REFERENCES1. G. Burdman et al. Phys. Rev. D49 2331,1994.2. B. Sheikoleslami and R. Wohlert. Nucl. Phys.B259 572, 1985.3. N.Cabibbo and E. Marinari. Phys. Lett. B199387, 1982.4. F. Brown and T. Woch. Phys. Rev. Lett. 582493, 1987.5. M. Luscher et al. preprint CERN-TII/96-218hep-lat/9609035, 1996.6. M. Luscher et al. Nucl. Phys. B491 344, 1997.7. S. Sint and P. Weisz. Nucl. Phys. B502 251,1997.8. D. Becirevic and A. Kaidalov. hep-ph/9904490, 1999.9. H. Matsufuru et al. JLQCD Collaboration.Nucl. Phys. B(Proc.Suppl.)63A-C 368, 1998.10. S. Aoki et al. JLQCD Collaboration. Nucl.Phys. B(Proc.Suppl.)63A-C 380, 1998.11. K. Bowler et al. UKQCD Collaboration. inpreparation.

An Investigation of the Soft Pion Relation in Quenched Lattice QCD

A lattice determination of the form factor and decay constants for the semileptonic decay of heavy pseudoscalar (PS) mesons at zero recoil is presented from which the soft pion relation is satisfied. Chiral extrapolation of the form factor is performed at constant $q^2$. Pole dominance is used to extrapolate the form factor in heavy quark mass. At the B mass, the form factor at zero recoil lies somewhat below the ratio of decay constants; the relation remains satisfied within error.A lattice determination of the form factor and decay constants for the semileptonic decay of heavy pseudoscalar (PS) mesons at zero recoil is presented from which the soft pion relation is satisfied. Chiral extrapolation of the form factor is performed at constant $q^2$. Pole dominance is used to extrapolate the form factor in heavy quark mass. At the B mass, the form factor at zero recoil lies somewhat below the ratio of decay constants; the relation remains satisfied within error

Bowler, K.C.

Flynn, J.M.

Lesk, V.I.

Maynard, C.M.

Richards, D.G.

CERN Document Server

1An Investigation of the Soft Pion Relation in Quenched Lattice QCDUKQCD CollaborationK.C. Bowlera, L. Del Debbiob, J.M. Flynnb, L. Lellouchc, V.I. Leskb, J. Nievesd, C.M. Maynarda andD.G. Richardsay.aDepartment of Physics and Astronomy, University of Edinburgh EH9 3JZ, UKbDepartment of Physics and Astronomy, University of Southampton, SO17 1BJ, UKcTheory Division, CERN, CH 1211, Geneva 23, SwitzerlanddDepartamento de Fisica Moderna, Universidad de Granada, Granada 18071, SpainA lattice determination of the form factor and decay constants for the semileptonic decay of heavy pseudoscalar(PS) mesons at zero recoil is presented from which the soft pion relation is satisfied. Chiral extrapolation of theform factor is performed at constant q2. Pole dominance is used to extrapolate the form factor in heavy quarkmass. At the B mass, the form factor at zero recoil lies somewhat below the ratio of decay constants; the relationremains satisfied within error.1. IntroductionHeavy-quark and chiral symmetries combine topredict the following relation for a heavy-lightmeson H the mass of whose heavy quark is largerthan some arbitrary hadronic scale, and for a softpion:f0(q2max)H!pi = fH/fpi (1)where f0(q2) is the form factor in the 0+ channelfor the vector current hadronic matrix elementhH(p)jV µjpi(p − q)i. The two quantities on theright-hand side are the corresponding PS mesondecay constants. Equation 1 is the soft pion rela-tion for heavy-light PS mesons [?].Results are given for f0(q2max)H!pi and fH/fpifor heavy-light PS mesons and are extrapolatedto the B mass. Errors are quoted asfB = 190(5)stat.(10)sys.MeV. (2)2. Extracting the form factorsThe form factors for the hadronic matrix ele-ment of semileptonic heavy pseudoscalar decay toa pion are extracted from mass extrapolations of∗Presenter†Jefferson Laboratory, Newport News, VA 23606, USAthe appropriate matrix elements of quenched lat-tice QCD with SW action [?] with a−1=2.6(1)GeV, β=6.2 on a 24348 lattice using 216 gaugeconfigurations generated with a combination ofthe Cabbibo-Marinari algorithm [?] and an over-relaxed algorithm [?]. The matrix elements arecalculated using the method of extended quarkpropagators with all combinations of the follow-ing hopping parameters:κheavy 2 f0.1200, 0.1233, 0.1266, 0.1299gκlight 2 f0.1346, 0.1351, 0.1353gκspectator 2 f0.1346, 0.1351g.From the light PS meson massesκcrit=0.13582(1)(2) is found.The action and operators are improved to O(a)and renormalized to a continuum scheme, usingcoefficients determined non-perturbatively wherepossible [?, ?]. The following values are used:ZV = 0.792 ZA = 0.807bV = 1.41 bA = 1.11cV = −1.58 10−2 cA = −3.71 10−2cSW = 1.61 bM = 0.583Of these, only cV , bA and bM are determined froma perturbative scheme [?].2In the continuum, the form factors f+ and f0are defined from hadronic matrix elements as fol-lows:hH(p)jV µjpi(p− q)i =f+(q2)(2 pµ − (M2H −M2pi /q2 + 1 qµ+f0(q2)(M2H −M2pi/q2qµ3. Pion physics on the latticeLattice correlators in this study are directly rel-evant to the physics of heavy-light mesons withmass 1300− 2200 MeV, and of light-light mesonswith mass 350 − 850 MeV. Form factors at thesame value of q2 for decays to a pion can be es-timated by assuming the following dependence ofa form factor f 2 ff+, f0g on the meson masses:f(q2, M, m) = f(q2, Mchiral, 0)+a∆M+bm2(3)and fitting the data to determine f(0) and a. Sim-ilarly light-light meson and heavy-light meson de-cay contants fL and fH are extrapolated in themeson mass according to the following:fL(m) = fL(0) + a0 m2 (4)fH(M) = fH(Mchiral) + a00 ∆M (5)where ∆M  M −Mchiral.The q2 dependence of the form factors canbe modelled using the following parameteriza-tion [?].f+(q2) =cB(1− α)(1− q2/M2 )(1 − αq2/M2 )(6)f0(q2) =cB(1− α)(1− q2/βM2 )(7)where M is the heavy-light vector meson mass.f0(q2max) is extracted by extrapolating f0 upwardsin q2 using the best fit curve from a fit of 7(fig. 1). One finds the soft pion relation to bewell satisfied at simulated values of heavy quarkmass.4. The soft pion relation for B ! piHeavy quark effective theory predicts the fol-lowing form for the dependence of f0 on the heavy−0.5 −0.3 −0.1 0.1 0.3 0.5 0.7 0.9q2/M*20.00.51.01.52.0f(q2 )Figure 1. Example fit to f+, f0(q2) for decay to apion, κheavy = 0.1200.Table 1The soft pion relation for a massless pion.κheavy f0(q2max) fPS/fpi0.1200 1.5(2)(3) 1.57(6)(9)0.1233 1.5(2)(2) 1.53(6)(9)0.1266 1.5(2)(2) 1.48(6)(9)0.1299 1.4(1)(1) 1.41(5)(9)quark mass, at constant recoil variable v  (p− q):f0(M, ω)Θ(M)pM = a+b/M+c/M2+O(1M3)(8)and for a heavy-light pseudoscalar decay con-stant:fH(M)Θ(M)pM = a0+b0/M+c0/M2+O(1M3)(9)where Θ(M) is a perturbative matching coeffi-cient function whose value is very close to 1.These prescriptions are used as fit ansatz to de-termine a best fit quadratic in 1/M to the data intable 1. The curve is shown in (fig. 2). Values forf0(q2max) and fH extrapolated to the b mass arepresented in table 2. It is instructive to repeatTable 2The soft pion relation for B decay to a masslesspion.f0(q2max) fB/fpiMB 1.1(2)(2) 1.45(6)(9)30.0 0.5 1.0 1.5 2.0 2.5 3.01/MH0.0400.0600.0800.1000.120fΘMH0.5Figure 2. Best fit fB vs. heavy mass accordingto eqn. 9.the whole procedure, extrapolating form factorsfor a pion which is not massless but has its physi-cal mass. Results are presented in table 3 and arepractically unchanged from the massless case.Table 3The soft pion relation for B decay to a 140 MeVpion.f0(q2max) fB/fpiMB 1.1(2)(2) 1.44(6)(9)5. Systematic errorSystematic error in this work arises from:1. discretization errors of O(a2)2. quenched approximation3. estimation of correlation matrix for fits4. interpolation of lattice form factors in q25. corrections to heavy quark effective theory6. choice in setting the scale.The quoted systematic error is an estimate ofthe variation arising from the last sources 3 − 6,achieved by repeating the analysis with differ-ent numbers of bootstrap sets, varying the modelfunction used to interpolate the form factor inq2, changing the degree of the polynomial in 1/Mused as an ansatz for the extrapolation in heavymass, and setting the scale alternatively againstMρ and the gluonic scale r0. Quenching error isnot quantified here. Residual discretization er-rors also may be significant, particularly in theheavy extrapolation where the fitted curve usedmay be diverted substantially by a correction tothe lattice form factor at the largest simulatedheavy quark mass, for which (Ma)2 ’ 0.65.6. ConclusionsFor a simulated heavy-light decay whose massis within 20% of MD, the soft pion relation isreproduced in this study. On extrapolating theform factor and the heavy-light decay constant tothe B mass, systematic errors become O(30%),to within which precision the soft pion relationholds for B meson decays. Other approaches todate [?, ?] have generally violated this relation,with the form factor lying significantly below theratio of decay constants. Work is ongoing withthe prospect of controlling further the systematicerror, and a simulation at β = 6.0 including astatic heavy quark will address a large source ofsystematic uncertainty as well as providing scal-ing information [?].

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An Investigation of the Soft Pion Relation in Quenched Lattice QCD

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