We compute the leptonic decay constants of heavy-light vector mesons in the
quenched approximation. The reliability of lattice computations for heavy
quarks is checked by comparing the ratio of vector to pseudoscalar decay
constant with the prediction of Heavy Quark Effective Theory in the limit of
infinitely heavy quark mass. Good agreement is found. We then calculate the
decay constant ratio for B mesons: $f_{B^*}/f_B= 1.01(0.01)(^{+0.04}_{-0.01})$.
We also quote quenched $f_{B^*}=177(6)(17)$ MeV.Comment: 11 pages, 3 postscript figs., revtex; two references adde

A. Abada

C. Bernard

Carleton DeTar

Claude Bernard

Craig McNeile

D. Groom

Doug Toussaint

G. P. Lepage

K. Bitar

Kostas Orginos

M. Neu-bert

M. Neubert

O. F. Hernandez

Peter Williams

R. M. Baxter

Robert Sugar

Saumen Datta

Steven Gottlieb

Urs M. Heller

Y. Kuramashi

English

arXiv

Crossref

Lattice results for the decay constant of heavy-light vector mesons

We compute the leptonic decay constants of heavy-light vector mesons in the
quenched approximation. The reliability of lattice computations for heavy
quarks is checked by comparing the ratio of vector to pseudoscalar decay
constant with the prediction of Heavy Quark Effective Theory in the limit of
infinitely heavy quark mass. Good agreement is found. We then calculate the
decay constant ratio for B mesons: $f_{B^*}/f_B= 1.01(0.01)(^{+0.04}_{-0.01})$.
We also quote quenched $f_{B^*}=177(6)(17)$ MeV

Bernard, C

Williams, P

Datta, S

Gottlieb, Steven

DeTar, C

Heller, Urs M

McNeile, C

Orginos, K

Sugar, R

Toussaint, D

University of Liverpool Repository

PHYSICAL REVIEW D, VOLUME 65, 014510Lattice results for the decay constant of heavy-light vector mesonsClaude Bernard and Peter WilliamsDepartment of Physics, Washington University, St. Louis, Missouri 63130Saumen Datta and Steven GottliebDepartment of Physics, Indiana University, Bloomington, Indiana 47405Carleton DeTarPhysics Department, University of Utah, Salt Lake City, Utah 84114Urs M. HellerCSIT, Florida State University, Tallahassee, Florida 32306-4120Craig McNeileDepartment of Mathematical Sciences, University of Liverpool, Liverpool, L69 3BX, United KingdomKostas OrginosRIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000Robert SugarDepartment of Physics, University of California, Santa Barbara, California 93106Doug ToussaintDepartment of Physics, University of Arizona, Tucson, Arizona 85721~Received 21 September 2001; published 11 December 2001!We compute the leptonic decay constants of heavy-light vector mesons in the quenched approximation. Thereliability of lattice computations for heavy quarks is checked by comparing the ratio of vector to pseudoscalardecay constant with the prediction of heavy quark effective theory in the limit of an infinitely heavy quarkmass. Good agreement is found. We then calculate the decay constant ratio for B mesons: f B* / f B51.01(0.01)(20.0110.04). We also quote quenched f B*5177(6)(17) MeV.DOI: 10.1103/PhysRevD.65.014510 PACS number~s!: 12.38.Gc, 13.20.2vThe symmetries of heavy quark effective theory ~HQET!@1# show how quantum chromodynamics ~QCD! simplifies inthe limit of an infinite quark mass. For a mesonic systemsuch as the neutral B, consisting of a heavy, but finite mass,anti-b quark and a light d quark, HQET can be applied withthe inverse b mass as a small perturbation parameter. In par-ticular, the ratio of the decay constants of the B* and B canbe calculated. Heavy quark spin symmetry implies that, inthe limit of an infinite quark mass, the spins of the quarksdecouple and the vector and pseudoscalar mesons are degen-erate, so the ratio of their decay constants is 1. Perturbativecorrections to this ratio are also calculable within the HQETframework.In this paper, we study the heavy-light vector and pseu-doscalar decay constants in quenched lattice QCD. Becausecomputational restrictions limit the range of heavy quarkmasses that are used in our simulations, the data must beextrapolated to the B mass ~or interpolated between theheavy-light data and a static-light point!. The lattice calcula-tions also inherently require extrapolations to the continuumlimit of zero lattice spacing. The comparison of the latticecalculation and the HQET calculation of the ratio of the vec-tor and pseudoscalar decay constants tests the consistency ofthe treatment of heavy quarks in lattice QCD. Calculations of0556-2821/2001/65~1!/014510~5!/$20.00 65 0145heavy-light vector decay constants on the lattice have beencarried out previously by several other groups @2#.In earlier work @3#, we computed pseudoscalar decay con-stants only. Here we extend that analysis to include vectormesons. Since our aim is to test the consistency of theselattice simulations with the results of HQET, we confine theanalysis to the quenched data sets, the details of which aresummarized in Table I. The parameters of the generation ofthese lattices, gauge fixing, and quark propagator determina-tion are found in @4,3#. We use unimproved Wilson valencequarks. ‘‘Smeared-local’’ ~SL! and ‘‘smeared-smeared’’ ~SS!vector meson propagators are calculated for the heavy-lights.TABLE I. Summary of quenched Wilson action lattices.Set b Size No. confs.A 5.7 83348 200B 5.7 163348 100E 5.85 123348 100C 6.0 163348 100CP 6.0 163348 305D 6.3 243380 100H 6.52 3233100 60©2001 The American Physical Society10-1CLAUDE BERNARD et al. PHYSICAL REVIEW D 65 014510These propagators are also calculated for the heavy-light andlight-light pseudoscalar mesons. The heavy-light pseudo-scalar decay constant f Qq is defined by^0uA0cont~0 !uP ,pW 50&52i f QqM QqwhereA0cont~0 !5Q¯ g0g5qis the 0 th component of the axial current at point x50, anduP ,pW 50& is a zero 3-momentum pseudoscalar bound state ofheavy quark Q¯ and light quark q with mass M Qq . We definethe vector decay constant in exact analogy to the pseudo-scalar decay constant ~this is standard in HQET! to simplifythe interpretation of the ratio:^0uVicont~0 !uV ,pW 50,e&5e i f Qq* M Qq*where Vicont is a spatial component of the continuum vectorcurrent,Vmcont5Q¯ gmqand uV ,pW 50,e& is the vector meson state with zero3-momentum, mass M Qq* , and polarization e .The vector propagators have the same relation to f B* asthe pseudoscalar propagators have to f B . The light-lightpseudoscalars are used to set the scale ~through f p) and thephysical value of the hopping parameter of the degenerateup/down quarks ~through mp).Each pair of SL and SS propagators for a particular masscombination is fit simultaneously and covariantly to singleexponential forms sharing the same mass; i.e., we makethree-parameter fits. The time ranges used in these fits werevaried to produce different fits ~typically 8–10 of them! thatprovided reasonable confidence levels for both vector andpseudoscalar decay constants. The alternate fit ranges werethen used to fit the ratio of the decay constants as discussedbelow. A preferred fit range was selected from the acceptablealternatives by choosing a range that provided a good blendof high confidence level and small statistical error for theratio fit. For each ratio derived from these fits, the standarddeviation of the alternate fit ranges was added in quadratureto the raw statistical error of the preferred fit to produce ameasure of the statistical uncertainty in the ratio that reflectsthe different possible plateau regions.To relate the matrix elements measured on the lattice totheir continuum counterparts we use the perturbative renor-malization factors for heavy-light currents calculated byKuramashi @5#. These renormalization factors include a de-pendence on the quark mass, which for large quark massesproduces an approximately 100% difference in the one loopcoefficients compared to those in the massless quark limit.We adjust the values calculated in @5# to correspond to ourdefinition of the mean link, u0, in terms of the critical hop-ping parameter, u051/8kc . The mass dependence of therenormalization factors is more important for the individualdecay constants, but still provides a meaningful improve-01451ment to the calculation of the decay constant ratios, whereonly the ratio of the vector and axial current renormalizationsis relevant. As in @3#, we adjust the measured meson polemass upwards by the difference of the heavy quark kinematicmass and the heavy quark pole mass. This allows us to esti-mate the kinetic mass of the meson while only looking at itszero-momentum state.A new element that is added to the previous analysis ofthis data is the choice of q*, the momentum scale that satis-fies @6#ln~q*2!5E d4q f ~q !ln~q2!E d4q f ~q !where f (q) is the integrand of a 1-loop current renormaliza-tion. Evaluating the coupling ~we use gV2 defined in terms ofthe plaquette @6,7#! at q* and using that coupling to evaluatethe renormalization should reduce higher order effects. Thevalues of q* for the heavy-light currents have not been cal-culated. In Ref. @3#, the Hernandez-Hill result @8# for thetadpole-improved static-light axial vector current scale (q*52.18/a , a value close to the tadpole-improved light-lightaxial vector current scale, q*52.32/a @9#! was used to arguethat q* was only mildly mass dependent. The light-light q*was then used for the heavy-lights. The Hernandez-Hill cal-culation has recently been repeated @10# ~see also @11#!, witha rather different result, q*;1.4/a . We believe this static-light q* is likely to be more appropriate for the heavy-lightsthan the light-light q*, and we use it here. The new calcula-tion of the static-light q* includes the continuum part ofstatic-light current, which gives rise to an amQ dependence.For the axial vector current this dependence is weak enoughthat a constant value of q* from @10# (q*’1.43/a) can beused reliably, and this has been done in @12#. However, theamQ dependence of q* for the static-light vector current ismore pronounced, so here the scale was calculated for eachheavy kappa for both the vector and pseudoscalar case. Wecompare the value of f B obtained from the mass dependentq* scheme with the results @12# for the q*>1.43/a scheme1as a consistency check.To help estimate systematic uncertainties we use threedifferent chiral fits, in which we extrapolate the results at thelight quark kappas used in the simulation to the kappa ap-propriate to physical light quarks, as determined by the pionmass. The first of these, from which the central value for theratio is taken and which will be referred to as the standardanalysis, uses quadratic fits vs am2 ~light quark kinematicmass! for mp2, and linear fits vs am2 for f p , M Qq , and f Qq .The rationale for these choices is discussed in @3#. The firstof the alternate analyses has quadratic fits for mp2 and f p1To quote f B we had to maintain the distinction between latticesets C & CP, since the static points of these lattices are calculateddifferently. This is irrelevant for the ratio of the decay constantscomputed here, but is necessary for this consistency check.0-2LATTICE RESULTS FOR THE DECAY CONSTANT OF . . . PHYSICAL REVIEW D 65 014510with all other fits linear, and the second has quadratic fits formp2, f p , and f Qq . The difference of these chiral fits is usedto assess the systematic error in the choice for the standardanalysis.For each set of lattices, the ratio of fAM for the vectorand pseudoscalar mesons at each heavy kappa is calculated.For each heavy k , we then divide out 12g2/(6p2), the lead-ing order HQET correction to the ratio @13#, using gV2 evalu-ated on the lattice at the q* appropriate to mB . The resultingdata is fit to the three parameter functionb1c/M11d/M . ~1!Since the decay constants each have a 1/M expansion inHQET, this fitting function can be viewed as the ratio of thefirst two terms from the individual expansions. If the dataproduce the correct static limit, the constant term in the nu-merator, b, should be 1. Table II shows the value of b for thestandard analysis of each set of lattices. Note that all theresults are consistent with 1. The errors are quite large on thecoarsest lattices at b55.7, but are much smaller on the finerlattices.From now on we assume consistency with HQET and usethe two parameter fitting function Eq. ~1! with b51 to ex-tract the ratio f B* / f B . The difference of these fitting meth-ods can be seen in Fig. 1. The final two-parameter fit is theninterpolated to the B mass and the leading order perturbativecorrection is reinserted. This result still includes the ratio ofthe square roots of the B* and B masses. Removing thisgives us a value of the ratio for each set of lattices, whichmust then be combined and extrapolated to zero lattice spac-ing.The fits shown in Fig. 2 are different possible lattice spac-ing extrapolations for the continuum value of the ratio. Thesedata are the result of the standard analysis on each set oflattices, but the general features of the plot are generic for allthe analyses, as can be seen in Fig. 3. We use constant fitsover different intervals in the lattice spacing: 0.2 to 0.4, 0.2to 0.5, and 0.2 to 0.75 (GeV)21. We do not include a linearfit, as analysis of the new data sets described in @12# suggeststhat the constant fits provide a good measure of the latticespacing extrapolation uncertainty. For each set of analyses,the fit to the interval containing only the values from the twoTABLE II. Agreement of the static limit of the 3 parameter ratiofits with 1.Static limit of fitting functionSet of lattices (b) ~including plateau uncertainties!H~6.52! 0.9960.05D~6.3! 0.9960.04CP~6.0! 1.0560.04C~6.0! 1.0860.08E~5.85! 0.8960.08B~5.7/16! 0.7060.42A~5.7/8! 1.0460.2501451sets with finest lattice spacing @0.2,a,0.4 (GeV)21# istaken to be the central value for that ratio.The systematic errors are obtained from various alterna-tive analyses ~see Fig. 3!. The discretization error is esti-mated by computing the difference between the average ofthe two finest lattices and the average over all lattice spac-ings. The three constant fits for the standard analysis areshown in Fig. 2, and the results of the three fits for each ofthe alternate analyses can be seen in Fig. 3. We estimate thelattice spacing extrapolation error as the largest difference ofthe three constant fits, which is >2010.02 for the standardanalysis.Higher order perturbative effects are a second source ofsystematic error. This error is estimated by taking the differ-ence of the standard analysis with the analysis performed atdifferent values of q*. In particular, we compare the standardFIG. 1. Comparison of three-parameter fits and two-parameterfits for the b56.52 lattices. Perturbative HQET corrections havebeen removed. Dotted line enforces the HQET result that f Qq* / f Qq51 at M5‘ . Solid line allows the M5‘ value to be free.FIG. 2. Three constant fits to estimate the lattice spacing ex-trapolation error in the ratio. The fit to a,0.4 ~solid line! is taken asthe central value. The data are for the standard analysis. The fitranges are 0.2 to 0.4 (GeV)21 ~solid line!, 0.2 to 0.5 (GeV)21~dotted line!, and 0.2 to 0.75 (GeV)21 ~dashed line!.0-3CLAUDE BERNARD et al. PHYSICAL REVIEW D 65 014510analysis results to the two alternate analyses 4 and 7 of Fig.3, where q* is adjusted down and up, respectively. We esti-mate the perturbative error to be >20.0110.03.The final significant contribution to the systematic errorcomes from the chiral extrapolations. Our estimate of thesystematic error involved in this extrapolation procedure isfound by taking the larger difference of the central value andthe two alternate chiral fits described above. This systematicerror can be seen in Fig. 3 by comparing fit number 1 to 3, 4to 6, or 7 to 9. We estimate the error from chiral extrapola-tion in our central value of the ratio as >2010.02. We do notinclude an analysis of the other sources of error mentioned in@3# ~difference of magnetic mass and kinetic mass, higherorder lattice extrapolation fits, and finite volume effects! be-cause they are negligible for the ratio f B* / f B .We combine the three sources of systematic error as ifthey were completely independent, because we see in Fig. 3that the results of the different changes made in the analysisare not significantly correlated. This gives us our finalquenched value of f B* / f B :1.01~0.01!~20.0110.04!.FIG. 3. Values of f B* / f B coming from the different types ofanalyses. Analysis type 1 is the standard analysis; analysis 2:change f p fits from linear to quadratic; analysis 3: change f p andf Qq fits from linear to quadratic; analyses 4–6: same as 1–3 withq* chosen so that its mean value for the heavy kappas is 1/a; andanalyses 7–9: same as 1–3 with q* chosen twice as large as thestandard analysis.01451In HQET the leading order ~in 1/M ! value of the ratio is 12g2(mB)/6p2’0.96, using L (5)50.208 GeV @14#. Notethat this is less than 1 because the perturbative correction isnegative. However the results of our simulations suggest thatthe ratio for the B is more likely to be greater than or equal to1. Neubert has calculated that the value of the ratio f B* / f Busing the subleading order terms in the 1/M expansion to be1.0760.03 @15#, which is consistent with our result.Using the same analysis, we find f B>175(7) MeV,where statistical error only is shown. This should be com-pared with the current MILC value f B5173(6)(16) MeV@12#. The latter includes improved action data and a completesystematic error analysis and, therefore, should be taken asthe most up-to-date MILC value2 for quenched f B . How-ever, the consistency of the current analysis with the previ-ous calculation is comforting and indicates, among otherthings, that the use of a scheme in which q* varies withheavy quark mass has no drastic effects.We also report a value for f B*. This quantity was notcalculated in @3# or @12#, so we perform a more detailedanalysis including estimates of systematic errors analogousto that performed in @12#. This gives f B*5177(6)(17)MeV in the quenched approximation.These results for f B* and f B* / f B are in qualitative agree-ment with what is expected. However, the main point of thispaper is not the computation of the quenched decay constantsat the B mass, but the extrapolation of our results to infiniteheavy quark mass. The agreement of f Qq* / f Qq in this limitwith the HQET prediction is an indication that the presenttreatment of the heavy quark on the lattice is consistent.Calculations for this project were performed at ORNLCCS, SDSC, Indiana University, NCSA, PSC, MHPCC,CTC, CHPC ~Utah!, and Sandia Natl. Lab. This work wassupported by the U.S. Department of Energy under grantsDE-FG02-91ER-40628, DE-FG02-91ER-40661, DE-FG02-97ER-41022 and DE-FG03-95ER-40906, and by the Na-tional Science Foundation under grants PHY99-70701 andPHY97-22022.2We note that the central value of f B in @12# is considerably higherthan the value 157(11)(29122)(20121) MeV quoted in @3#. That differ-ence is due to new data ~including improved action! and new analy-sis ~including update of q*) as explained in @11# and @12#.@1# See, for example, H. Georgi, in Perspectives in the StandardModel, Proceedings TASI 91, Boulder, CO, 1991, edited by R.K. Ellis et al. ~World Scientific, Singapore, 1992!; B. Grin-stein, in High Energy Phenomenology, Proceedings Workshopon High Energy Phenomena, Mexico City, 1991, edited by M.A. Pe´res and R. Huerta ~World Scientific, Singapore, 1992!; N.Isgur and M. B. Wise, in Heavy Flavors, edited by A. J. Burasand M. Lindner ~World Scientific, Singapore, 1992!; M. Neu-bert, Phys. Rep. 245, 259 ~1994!.@2# A. Abada et al., Nucl. Phys. B376, 172 ~1992!; R. M. Baxteret al., Phys. Rev. D 49, 1594 ~1994!; S. Collins et al., ibid. 55,1630 ~1997!; A. Ali Khan et al., ibid. 56, 7012 ~1997!; D.Becirevic et al., ibid. 60, 074501 ~1999!; S. Collins et al., ibid.60, 074504 ~1999!; K. C. Bowler et al., hep-lat/0007020.@3# C. Bernard et al., Phys. Rev. Lett. 81, 4812 ~1998!.@4# C. Bernard et al., Nucl. Phys. B ~Proc. Suppl.! 42, 388 ~1995!.0-4LATTICE RESULTS FOR THE DECAY CONSTANT OF . . . PHYSICAL REVIEW D 65 014510@5# Y. Kuramashi, Phys. Rev. D 58, 034507 ~1998!.@6# G. P. Lepage and P. B. Mackenzie, Phys. Rev. D 48, 2250~1993!.@7# K. Bitar et al., Phys. Rev. D 48, 370 ~1993!; T. Klassen, ibid.51, 5130 ~1995!.@8# O. F. Hernandez and B. R. Hill, Phys. Rev. D 50, 495 ~1994!.@9# C. Bernard, M. Golterman, and C. McNeile, Phys. Rev. D 59,074506 ~1999!.@10# C. Bernard and T. DeGrand ~in preparation!.@11# C. Bernard, Nucl. Phys. B ~Proc. Suppl.! 94, 159 ~2001!.01451@12# C. Bernard et al., Nucl. Phys. B ~Proc. Suppl.! 94, 346 ~2001!.@13# Neubert @1#.@14# Particle Data Group, D. Groom et al., Eur. Phys. J. C 15, 1~2001!, p. 91.@15# M. Neubert, Phys. Rev. D 46, 1076 ~1992!. ~The value wequote for f Qq* / f Qq at the B mass is numerically the same as thevalue of f B*AmB* / f BAmB as calculated in this reference. Theaddition of the mass dependence makes no difference at thecurrent precision.!0-5

Lattice results for the decay constant of heavy-light vector mesons

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