I review our qualitative understanding of the increase in the value of the B
meson decay constant (fB), when dynamical fermions are included in lattice QCD
calculations.Comment: 4 pages. Talk at UK Phenomenology Workshop on Heavy Flavour and CP
  Violation, Durham, 17 - 22 September 2000 (Minor typo fixed

Aoki S (CP-PACS)

Bali G S

Bernard C

Bernard C W

Booth M J

Bowler K C (UKQCD)

Butler F

Collins S

CP-PACS

Craig McNeile

Draper T

Eicker N (SESAM)

El-Khadra A X

Groom D E

Gupta R

Horsley R (UKQCD)

Ishikawa K-I (JLQCD)

Nir Y

Sharpe S R

English

arXiv

I review our qualitative understanding of the increase in the value of the B
meson decay constant (fB), when dynamical fermions are included in lattice QCD
calculations

McNeile, Craig

University of Liverpool Repository

arXiv:hep-lat/0012008v2  12 Dec 2000Do we understand the unquenched value of fB?Craig McNeileTheoretical Physics Division, Dept. of Mathematical Sciences, University ofLiverpool, Liverpool L69 3BX, UKE-mail: mcneile@amtp.liv.ac.ukAbstract. I review our qualitative understanding of the increase in the value of theB meson decay constant (fB), when dynamical fermions are included in lattice QCDcalculations.1. IntroductionThe over determination of the CKM matrix, using the new data from the B factoriesat SLAC and KEK, is a sensitive test of the standard model [1]. The determination ofthe CKM matrix elements from experiment depends critically on hadronic parameters,such as fB, BB, BBs and BK , most of which are inaccessible to experiment, but can becalculated from lattice QCD.In a recent survey of the latest lattice QCD results for the fB (the decay constantof the B meson) Bernard [2] quotes: f quenchedB = 175(20) MeV from quenched QCD andfB = 200(30) MeV for the value of the B decay constant in full QCD. Although all latticeQCD calculations [2] have seen an increase in fB between quenched and unquenchedQCD, the effect in the world “average” is only at the one σ level. The increase in fBbetween quenched and full QCD is more significant for an individual collaboration’sresults, for example CP-PACS, obtain [3]fnf=2Bfnf=0B= 1.11(6). As unquenched simulationsare so computationally demanding, it seems useful to review the additional argumentsthat support the increase in decay constants due to the inclusion of dynamical fermions.2. What are unquenching effects?Lattice QCD is a “clever” finite difference approximation to continuum QCD [4]. LatticeQCD calculations involve computing the partition functionZ =∫ ∏xdU(x) exp(−SG)(det(M))nf (1)where U describe the gauge fields, SG is the lattice representation of the gauge action(14FµνFµν) and M is a lattice representation of the Dirac operator for quarks. Thequark fields have been integrated out. The dynamics of the gluon fields depends onDo we understand the unquenched value of fB? 2the determinant of the Dirac operator. The determinant describes the dynamics of thesea quarks and is very expensive to compute. Until recently, most phenomenologicallattice calculations did not include the determinant in the dynamics (quenched QCD).Surprisingly, quenched QCD calculations describe experiment reasonably well. Thebiggest recent study of quenched QCD by the CP-PACS collaboration [5] found thatthe spectrum of light hadrons disagreed with experiment at the 10% level.The results from an individual calculation depend on the lattice spacing a andphysical volume. These errors may be removed by repeating the calculation at differentlattice spacings and volumes and then extrapolating the results to the continuum andinfinite volume limit. For example, in the recent CP-PACS calculation [3] they obtainfB = 287(7) MeV, 234(8) MeV, and 208(10) MeV, at lattice spacings: 0.22 fm, 0.16fm, and 0.11 fm respectively, from a calculation of nf = 2 QCD. CP-PACS prefer toquote fB = 208(10)(11) MeV as their continuum result, however Bernard [2] prefers toextrapolate the CP-PACS data to the continuum assuming a quadratic dependence onthe lattice spacing (there are good, but not totally rigorous arguments for this type ofextrapolation) and obtains fB = 190(12)(26) MeV. This kind of ambiguity in the finalanalysis of lattice results is the cause of the large systematic errors in the final resultsof lattice calculations. The cost of lattice QCD calculations that include dynamicalfermions goes (something) like [6] 1a6.5where a is the lattice spacing, so halving thelattice spacing is very computationally expensive.3. How to understand unquenchingThere is a simple model of the effect of unquenching that is based on the quarkmodel [7, 8, 9]. Consider the Richardson heavy quark potential [10].V (q) ∼4pi(11− 2nf3)1q2 ln(1 + q2/Λ2)(2)where nf is the number of flavours. Equation 2 or the potential extracted from a latticecalculation is used in Schro¨dinger’s equation to calculate the wave function of mesons.The decay constant is computed using the Van Royen-Weisskopf formulaef ∝| ψ(0) | (3)In position space the potential in equation 2 at small radial separations is deeper inthe full theory (nf=3) relative to the quenched theory (nf=0). So the decay constant(computed from equation 3) is higher in the full theory than in quenched QCD. Thereis evidence from many lattice QCD calculations of the heavy quark potential, thatdynamical fermions produce a similar effect to the nf dependence of the Richardsonpotential [11].The MILC collaboration [9] have systematically studied this model in anunquenched simulation using staggered fermions. From the graphs in the paper [9],at mPS/mV ∼ 0.58, the effect of unquenching is 3% for fpi and 7% for fB. The MILCcollaboration found that unquenching was smaller on the ratio of decay constants thatDo we understand the unquenched value of fB? 3on individual decay constants in this model. A key qualitative prediction of this modelis that the decay constant in unquenched QCD is greater than the quenched value. Alsothe unquenching effect should be larger for fB than fpi.Another way to understand the effect of unquenching is to use a quenchedLagrangian [12, 13]. The idea is similar to Chiral Lagrangian’s, where a Lagrangianis written in terms of hadron fields with the same symmetries as QCD. A number ofunknown parameters enter the Chiral Lagrangian, that must be fixed from experiment(or from lattice calculations). Once the parameters are fixed, the Lagrangian can beused to make predictions. For the case of quenched QCD, ghost fields are introduced tocancel the quark determinants. The resulting theory has a different symmetry to QCD,but the idea is basically the same as for continuum chiral perturbation theory.The formalism for heavy-light mesons was developed by Booth [14] and Sharpe andZhang [15]. The quenching errors are estimated by comparing the chiral logs (some ofthe loop effects) in quenched and full QCD. This is thought to be an upper bound on themagnitude of the quenching effects [15]. Unfortunately, the estimate of the quenchingerrors involves 7 parameters that are hard to determine accurately. The message fromthis analysis was that fB in full QCD could be either greater than or less than thequenched value. Also, it was possible that the ratio fBs/fB had large quenching errors(this has not been found in simulations [2]).4. Lattice results for known decay constantsThe results of the first systematic studies of decay constants in quenched QCD havebeen reviewed by Sharpe [6]. Sharpe concluded that fpimρwas lower than experiment,after the continuum limit had been taken. The recent large CP-PACS collaboration [5]study of quenched QCD found that in quenched QCD f quenchedpi (fquenchedK ) = 120.0±5.7(138.8± 4.4) MeV, that are smaller than experiment by 2σ (5σ).After the continuum extrapolation, the light decay constants from quenched QCDare lower than experiment, consistent with the quark model in section 3. The SESAMcollaboration [16] found at one lattice spacing that the pion decay constant wasapproximately one sigma larger with dynamical fermions, than from an equivalentquenched simulation. The opposite trend was seen in the QCDSF/UKQCD [17] data. Itwill be difficult to determine the effect of dynamical fermions on light decay constants,until a continuum extrapolation is done.Including heavy quarks (Charm and Bottom) in lattice QCD calculations requiresthe introduction of new techniques, such as effective field theories for the heavyquarks [2]. As a test of the new heavy quark methods, the fDs decay constant iscomputed and compared against experiment. The value for fDs quoted in the particledata table [18] is 280± 19± 28± 34 MeV. In table 1, I have collected some results forfDs from lattice gauge theory calculations (all the various errors have been added inquadrature). All the lattice results for quenched QCD are lower than the experimentalvalue. This is consistent with the picture that unquenching raises the value of a decayDo we understand the unquenched value of fB? 4Group Comments fDs MeV σAverage (lattice) [19] Review in 1998 220(25) 2.2JLQCD [20] Continuum limit 224(20) 2.8MILC [21] Continuum limit 210(27) 2.6Collins et al. [22] a = 0.18 fm 223(54) 1.1UKQCD [23] a = 0.068 fm 241(30) 1.3Table 1. Summary of recent quenched lattice results for fDs . The σ column is thenumber of lattice errorbars below the central experimental value of 280 MeV.constant. The latest results [2] from lattice calculations show a 3-8% increase in fDs,when dynamical fermions are included.5. ConclusionsA consistent picture does seem to emerge from lattice QCD calculations, thatunquenching does raise the value of decay constants. However, the errors on thecalculations need to be reduced. Ideally, a calculation of fB similar to the recent CP-PACS calculation is required [3], but at lighter sea masses and smaller lattice spacings.Improvements in lattice techniques and faster computers will reduce the errors on fB.References[1] Y. Nir, 2000, Preprint hep-ph/0008226.[2] C. Bernard, 2000, Preprint hep-lat/0011064.[3] CP-PACS, A. A. Khan et al., 2000, Preprint hep-lat/0010009.[4] R. Gupta, 1999, Preprint hep-lat/9905027.[5] CP-PACS, S. Aoki et al., Phys. Rev. Lett. 84, 238 (2000), Preprint hep-lat/9904012.[6] S. R. Sharpe, 1998, Preprint hep-lat/9811006.[7] A. X. El-Khadra, Nucl. Phys. Proc. Suppl. 30, 449 (1993), Preprint hep-lat/9211046.[8] F. Butler, H. Chen, J. Sexton, A. Vaccarino, and D. Weingarten, Nucl. Phys. B421, 217 (1994),Preprint hep-lat/9310009.[9] C. Bernard et al., Phys. Rev. D62, 034503 (2000), Preprint hep-lat/0002028.[10] J. L. Richardson, Phys. Lett. B82, 272 (1979).[11] G. S. Bali, 2000, Preprint hep-ph/0001312.[12] S. R. Sharpe, Phys. Rev. D46, 3146 (1992), Preprint hep-lat/9205020.[13] C. W. Bernard and M. F. L. Golterman, Phys. Rev. D46, 853 (1992), Preprint hep-lat/9204007.[14] M. J. Booth, Phys. Rev. D51, 2338 (1995), Preprint hep-ph/9411433.[15] S. R. Sharpe and Y. Zhang, Phys. Rev. D53, 5125 (1996), Preprint hep-lat/9510037.[16] SESAM, N. Eicker et al., Phys. Rev. D59, 014509 (1999), Preprint hep-lat/9806027.[17] UKQCD, R. Horsley, 2000, Preprint hep-lat/0010059.[18] D. E. Groom et al., Eur. Phys. J. C15, 1 (2000).[19] T. Draper, Nucl. Phys. Proc. Suppl. 73, 43 (1999), Preprint hep-lat/9810065.[20] JLQCD, K.-I. Ishikawa et al., Phys. Rev. D61, 074501 (2000), Preprint hep-lat/9905036.[21] C. Bernard et al., Phys. Rev. Lett. 81, 4812 (1998), Preprint hep-ph/9806412.[22] S. Collins et al., 2000, Preprint hep-lat/0007016.[23] UKQCD, K. C. Bowler et al., 2000, Preprint hep-lat/0007020.

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