We present new preliminary results for the leptonic decay constants f{sub B} and f{sub D+} determined in 2+1 flavor lattice QCD at lattice spacings a = 0.09, 0.12 and 0.15 fm. Results are obtained using the MILC Collaboration gauge configuration ensembles, clover heavy quarks in the Fermilab interpretation and improved staggered light quarks. Decay constants, computed at partially quenched combinations of the valence and sea light quark masses, are used to determine the low-energy parameters of staggered chiral perturbation theory. The physical decay constants are found in an extrapolation using the parameterized chiral formula

Bernard, C.

DeTar, C.

Di Pierro, M.

El-Khadra, A. X.

Evans, R. T.

Freeland, E.

Gamiz, E.

Gottlieb, Steven

Heller, U. M.

Hetrick, J.E.

Jain, R.

Hetrick, James E.

Gottlieb, S. A.

Pacific McGeorge School of Law

The decay constants fB and fD+ from three-flavor lattice QCD

UNT Digital Library

PoS(LATTICE 2007)370The decay constants fB and fD+ from three-flavorlattice QCDC. Bernarda, C. DeTarb, M. Di Pierroc, A.X. El-Khadrad , R.T. Evansd , E. Freelande,i,E. Gamizd , Steven Gottlieb f , U.M. Hellerg, J.E. Hetrickh, R. Jaind , A.S. Kronfeldi,J. Laihoia, L. Levkovab, P.B. Mackenziei, D. Renner j, J.N. Simone∗i, R. Sugark,D. Toussaint j, and R.S. Van de WateriaDepartment of Physics, Washington University, St. Louis, Missouri, USAbPhysics Department, University of Utah, Salt Lake City, Utah, USAcSchool of Computer Sci., Telecom. and Info. Systems, DePaul University, Chicago, Illinois, USAdPhysics Department, University of Illinois, Urbana, Illinois, USAeLiberal Arts Department, The School of the Art Institute of Chicago, Chicago, Illinois, USAf Department of Physics, Indiana University, Bloomington, Indiana, USAgAmerican Physical Society, One Research Road, Box 9000, Ridge, New York, USAhPhysics Department, University of the Pacific, Stockton, California, USAiFermi National Accelerator Laboratory, Batavia, Illinois, USAjDepartment of Physics, University of Arizona, Tucson, Arizona, USAkDepartment of Physics, University of California, Santa Barbara, California, USAE-mail: simone@fnal.govFermilab Lattice and MILC CollaborationsWe present new preliminary results for the leptonic decay constants fB and fD+ determined in2+ 1 flavor lattice QCD at lattice spacings a = 0.09, 0.12 and 0.15 fm. Results are obtainedusing the MILC Collaboration gauge configuration ensembles, clover heavy quarks in the Fermi-lab interpretation and improved staggered light quarks. Decay constants, computed at partiallyquenched combinations of the valence and sea light quark masses, are used to determine the low-energy parameters of staggered chiral perturbation theory. The physical decay constants are foundin an extrapolation using the parameterized chiral formula.The XXV International Symposium on Lattice Field TheoryJuly 30-4 August 2007Regensburg, Germany∗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-07-742-TPoS(LATTICE 2007)370The decay constants fB and fD from three-flavor lattice QCD J.N. Simone1. IntroductionThe D meson decay constants, when compared to precise experimental results, are a criticalcheck of the lattice methods needed for fB. In Ref. [1] we predicted fD+ = 201± 3± 17 MeV ingood agreement with the CLEO-c measurement fD+ = 223±17±3 MeV revealed days later [2].In this work we present new results for the D and B meson decay constants. Precise determi-nations of fB, fBs and the ratio fBs/ fB are needed to study the Standard Model picture of B- ¯B andBs- ¯Bs mixing. A progress report for the mixing matrix element study is presented in Ref. [3].2. Simulation detailsWe use the MILC Collaboration three-flavor asqtad ensembles [4]. Details are tabulated inTable 1. For these ensembles, ml denotes the mass of the two degenerate lighter sea quarks. Asingle heavier sea quark has a mass mh near the strange quark mass. Upsilon spectroscopy tells usthe heavy quark potential scale r1 = 0.318(7) fm [5]. The number of valence quark masses, #mq,used in this study is listed in the last column of the table.The leptonic decay constant fHq for a meson Hq is defined by〈0 | Aµ | Hq(p)〉= i fHq pµ . (2.1)The combination φHq = fHq√mHq emerges from a combined fit to lattice 2-pt functions:CO(t) =〈O†Hq(t) OHq(0)〉(2.2)CA4(t) =〈A4(t) OHq(0)〉, (2.3)where OHq can be either a smeared or local operator.The axial current renormalization is taken to beZQqA4 = ρQqA4√ZQQV4 ZqqV4 . (2.4)a [fm] amh aml β r1/a configs # mq0.09 0.031 0.0031 7.08 3.69 435 110.0062 7.09 3.70 557 100.0124 7.11 3.72 518 80.12 0.05 0.005 6.76 2.64 529 120.007 6.76 2.63 833 120.01 6.76 2.62 592 120.02 6.79 2.65 460 120.03 6.81 2.66 549 120.15 0.0484 0.0097 6.572 2.13 631 90.0194 6.586 2.13 631 90.029 6.600 2.13 440 9Table 1: MILC three-flavor lattice parameters. The last column lists the number of valence light quarksused in this study.2PoS(LATTICE 2007)370The decay constants fB and fD from three-flavor lattice QCD J.N. SimoneFactors Z f fV4 are fixed nonperturbatively from scattering 3-pt functions and the known normalizationof the vector current. Factors ρQqA4 are known to one-loop order and are close to unity [6].3. Staggered Chiral Perturbation Theory (SχPT)With staggered quarks the (squared) taste-nonsinglet pseudoscalar meson masses are split:M2ab,ξ = (ma +mb)µ+a2∆ξ , (3.1)where ma, mb are quark masses and the (sixteen) mesons are labeled by their taste representationξ = P,A,T,V, I with ∆P = 0.At next-to-leading order (NLO) in χPT the expression for the decay constants isφHq = ΦH [1+∆ fH(mq,ml,mh)+ pH(mq,ml,mh)] (3.2)where ∆ fH denotes the “chiral logs” and pH denotes terms analytic in the meson masses.With staggered quarks∆ fH =−1+3g2H∗Hpi2(4pi fpi)2[¯hq +hIq +a2(δ ′AhAq +δ ′V hVq)]. (3.3)Taste-breaking effects arise at finite a from the meson mass splittings and the δ ′A and δ ′V hair-pinterms [7]. Finite a effects reduce the chiral logarithm curvature, however, the expected QCD chirallogarithm is recovered in the continuum limit.The NLO analytic terms arepH =12(4pi fpi)2 [p1(ml,mh)+ p2(mq)] (3.4)p1 = f1(Λχ)[119 µ(2ml +mh)+a2(32¯∆+ 13∆I)](3.5)p2 = f2(Λχ)[53µmq +a2(32¯∆− 23∆I)], (3.6)where ¯∆ is the weighted average of taste splittings. The O(a2) terms ensure that dependence uponthe chiral logarithm scale, Λχ , in f1 and f2 cancels that of ∆ fq.Equation (3.2) with the addition of four NNLO analytic terms parameterizes our chiral ex-trapolations. We fit φHq to determine the parameters. Constraints (value and width) for µ , ∆ξ ,fpi , δ ′A and δ ′V come from χPT for lattice pions and kaons [8]. The coupling g2D∗Dpi = 0.35± 0.14is likewise constrained by the CLEO measurement [9]. From heavy quark symmetry we expectg2B∗Bpi ≈ g2D∗Dpi . The remaining parameters ΦH , f1 and f2 and the NNLO analytic parameters aredetermined in the fit.In order to extrapolate to the physical results we set ∆ξ = δ ′A,V = 0, mh → ms and ml →(mu +md)/2. Then φHd (φHs) is found in the limit mq → md (ms).3PoS(LATTICE 2007)370The decay constants fB and fD from three-flavor lattice QCD J.N. SimoneFigure 1: Chiral fits for the D (left) and B (right) mesons. Each fit is viewed along the direction in whichmq = ml . Each fit is shown as a set of solid curves with the 68% confidence limits denoted by broken curves.Only statistical errors are shown. The a = 0.09 fm curve and data points are shown in blue, 0.12 fm in greenand 0.15 fm in magenta. The a2 → 0 extrapolation curve and the φHd,s points at physical values of mu, mdand ms are shown in red. The statistical errors on the D and B physical points are of comparable size.4. The Fit and Extrapolation for D and BWe determine both φD+ and φDs from a single fit of φDq simulation results using the expressionin Eqn. (3.2), adding the four NNLO analytic terms and allowing for an explicit O(a2) term. Wecombine simulation results from 11 gauge ensembles at lattice spacings of a = 0.09, 0.12 and0.15 fm in the fit. A total of 116 points are included in the fit. A bootstrap procedure propagateserrors and correlations among the simulated results through to the statistical errors on our results.An analogous fit procedure for the B meson simulation results yields φBd and φBs .The D and B meson fits combining the three lattice spacings are shown in Figs. 1 and 2.Figure 1 shows each fit and the data points along the mq = ml direction while Fig. 2 shows thevalence mass dependence of the fit at fixed values of the sea quark mass. All of the fit points arevisible in Fig. 2 while only the subset of points with mq = ml is visible in Fig. 1.Figure 1 shows the D system on the left and the B system on the right. In each plot, the solidblue, green and magenta curves are the fit to the lattice data for lattice spacings 0.09, 0.12 and0.15 respectively. These curves include the a2 effects described by the chiral fit function. The 68%confidence limits for each curve are indicated by dotted contours of the corresponding color.In Fig. 2 we show the valence mass dependence the D and B systems. In each plot the D system(blue points and curves) is shown together with B system (green points and curves). Each plot inthe figure corresponds to a single combination of lattice spacing and ml from Table 1. Together,the points represent all of the simulation results for φHq used in this study. The fit curves includethe a2 effects described by the chiral fit function. Each curve is shown with its 68% confidencecontours. The expression in Eqn. (3.2) predicts a divergent logarithmic rise as mq → 0 with mlfixed. Our fits detect these logarithms even though taste breaking effects obscure them, so they arenot immediately obvious in the plots.The extrapolation, a → 0, mh → ms and ml = mq is shown in Fig. 1 as a solid red curve foreach of the D and B systems. Our result for φD+ (φBd ) is found from the extrapolation by settingml = mˆ = (mu+md)/2 and mq =md . Likewise, φDd (φBs) is found by setting mq =ms. The physical4PoS(LATTICE 2007)370The decay constants fB and fD from three-flavor lattice QCD J.N. SimoneFigure 2: The mq dependence of the single D fit (in blue) and the single B fit (in green) at fixed ml . Thefigures ordered from left-to-right and top-to-bottom, have aml equal to 0.0031, 0.0062, 0.0124 (a= 0.09 fm),0.005, 0.007, 0.01, 0.02, 0.03 (0.12 fm), 0.0097, 0.0194 and 0.029 (0.15 fm) respectively. On the y-axis isr3/21 φHq and on the x-axis is r21m2pi . All 116 points in each fit are shown. The χ2 = 98.6 for the D fit and 48.5for the B fit.5PoS(LATTICE 2007)370The decay constants fB and fD from three-flavor lattice QCD J.N. Simonequantity valueφDs 0.356(11) GeV3/2φDd 0.293(11) GeV3/2RDd/s 0.824(8)φBs 0.556(12) GeV3/2φBd 0.453(13) GeV3/2RBd/s 0.815(15)Table 2: The main results of this preliminary study.results are indicated with the red burst symbols. Since these points are projected into the mq = mlplane of each figure, the central values do not lie on the red curve. The statistical errors for thephysical φHq values are shown in each figure. We find statistical errors comparable in magnitudefor the physical φDq and φBq results.5. Results and OutlookOur preliminary results for the physical φHq values and ratios with their statistical errors are inTable 2. We tabulate the major sources of uncertainty in Table 3. We omit listing uncertainties aris-ing from terms of order 1/mH in the chiral extrapolations since adding such terms changes the finalresults by less than the statistical errors. Such effects are still under investigation. Uncertaintiesfrom the input parameters r1 and the light quark masses are found by propagating the uncertaintiesfound in the MILC fpi and fK determinations [5]. We estimate a 3.8% uncertainty in the bare charmmass and a 6.8% uncertainty in the bare bottom mass from variations in tuning procedures for the0.09 fm lattice. Using simulation results for two heavy-quark masses near both charm and bottom,we estimate the uncertainties in φHq . The uncertainties in Zf fV are statistical. Errors from unknownhigher orders in ρA4 are estimated by considering higher orders effects to be as large as the 1-loop terms. Heavy quark discretization effects are estimated by power counting arguments. Thedominant uncertainty in φHq comes from effects of order αsΛa×h(am) and a2Λ2, where h(am) issome mild function of the heavy quark mass. The uncertainties in the ratios are smaller by a factorof ms/Λ. Light quark discretization effects are estimated by varying the extrapolation precedure.Finite volume effects are estimated by comparing theories at finite volume to the continuum.From Table 2 and the experimental D+, Ds, B0 and Bs masses we compute the decay constants:fDs = 254±8±11 MeV (5.1)fD+ = 215±8±11 MeV (5.2)fBs = 240±5±11 MeV (5.3)fBd = 197±6±12 MeV (5.4)where each of the first errors is statistical. The second error is the systematic error combined inquadrature from Table 3.We also consider ratios of B to D decay constants where statistical and systematic errors areexpected to be reduced due to cancellations. Statistical errors in the ratios are from a bootstrap6PoS(LATTICE 2007)370The decay constants fB and fD from three-flavor lattice QCD J.N. Simonesource φDs φDd Rd/s φBs φBd Rd/sstatistics 3.1 3.8 1.0 2.1 3.1 1.8inputs r1, ms, md and mu 1.4 2.0 0.5 3.1 3.8 0.6input mc or mb 2.7 2.7 <0.1 1.1 1.1 <0.1ZQQV and ZqqV 1.4 1.4 0 1.4 1.4 0higher-order ρA4 0.3 0.3 <0.2 1.3 1.1 <0.2heavy quark discretization 2.7 2.7 0.3 1.9 1.9 0.2light quark discretization 1.0 2.7 1.8 2.0 3.8 1.8finite volume 0.2 0.6 0.6 0.2 0.6 0.6total systematic 4.4 5.3 2.0 4.7 6.1 2.0Table 3: The error budget for the decay constants and their ratios. Uncertainties are quoted as a percentage.The total combines systematic errors in quadrature.procedure in order to preserve statistical correlations.fD+/ fDs = 0.845±0.008±0.017 (5.5)fBd/ fBs = 0.821±0.015±0.017 (5.6)fBd/ fD+ = 0.919±0.051±0.056 (5.7)fBs/ fDs = 0.945±0.043±0.043 (5.8)The overall systematic errors for the first two ratios come from Table 3. Systematic errors for thelast two ratios also come from combining errors in quadrature. These errors may be overestimatessince we have not studied possible correlations at present.We will extend this study to include a lattice spacing of a = 0.06 fm. We will improve statisticsat a = 0.09 and 0.12 fm and add another ensemble (sea quark mass combination) at 0.09 fm. Afiner lattice spacing, more sea quark combinations and better statistics will help control light- andheavy-quark discretization effects and improve statistical errors. The new gauge configurationswill be used by MILC to refine r1 and the light quark masses inputs used in this study.References[1] C. Aubin et al., Phys. Rev. Lett. 95, 122002 (2005) [arXiv:hep-lat/0506030].[2] M. Artuso et al. [CLEO Collaboration], Phys. Rev. Lett. 95, 251801 (2005) [arXiv:hep-ex/0508057].[3] R. T. Evans, E. Gamiz, A. X. El-Khadra and M. Di Pierro, arXiv:0710.2880 [hep-lat].[4] C. Aubin et al., Phys. Rev. D 70, 094505 (2004) [arXiv:hep-lat/0402030].[5] C. Bernard et al. [MILC Collaboration], PoS LAT2005, 025 (2006) [arXiv:hep-lat/0509137].[6] A. X. El-Khadra, E. Gamiz, A. S. Kronfeld and M. A. Nobes, arXiv:0710.1437 [hep-lat].[7] C. Aubin and C. Bernard, Phys. Rev. D 73, 014515 (2006) [arXiv:hep-lat/0510088].[8] C. Aubin et al. [MILC Collaboration], Phys. Rev. D 70, 114501 (2004) [arXiv:hep-lat/0407028].[9] A. Anastassov et al. [CLEO Collaboration], Phys. Rev. D 65, 032003 (2002) [arXiv:hep-ex/0108043].7

The decay constants f(B) and f(D+) from three-flavor lattice QCD

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The decay constants f(B) and f(D+) from three-flavor lattice QCD

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