In the past year, we calculated with lattice QCD three quantities that were
unknown or poorly known. They are the $q^2$ dependence of the form factor in
semileptonic $D\to Kl\nu$ decay, the decay constant of the $D$ meson, and the
mass of the $B_c$ meson. In this talk, we summarize these calculations, with
emphasis on their (subsequent) confirmation by experiments.Comment: v1: talk given at the International Conference on QCD and Hadronic
  Physics, Beijing, June 16-20, 2005; v2: poster presented at the XXIIIrd
  International Symposium on Lattice Field Theory, Dublin, July 25-3

Allison, IF

Aubin, C

Bernard, C

Davies, CTH

Detar, C

Di Pierro, M

El-Khadra, AX

Fermilab Lattice Collaboration,

Freeland, ED

Gottlieb, S

Gray, A

Gregory, E

Hellerk, UM

Hetrick, JE

HPQCD Collaboration,

Kronfeld, AS

Levkova, L

MacKenzie, PB

Maresca, F

Menscher, D

MILC Collaboration,

Nobes, M

Okamoto, M

Oktay, MB

Osborn, J

Renner, D

Simone, JN

Sucar, R

Toussaint, D

Trottier, HD

English

arXiv

Fermilab Lattice Collaboration

MILC Collaboration

HPQCD Collaboration

Enlighten

Predictive lattice QCD

In the past year, we calculated with lattice QCD three quantities that were unknown or poorly known. They are the $q^2$ dependence of the form factor in semileptonic $D\to Kl\nu$ decay, the decay constant of the $D$ meson, and the mass of the $B_c$ meson. In this talk, we summarize these calculations, with emphasis on their (subsequent) confirmation by experiments

S. Kronfeld, Andreas

F. Allison, I.

Aubin, C.

Bernard, C.

T. H. Davies, C.

DeTar, C.

Di Pierro, M.

D. Freeland, E.

Gottlieb, Steven

Gray, A.

Gregory, E.

M. Heller, U.

E. Hetrick, J.

X. El-Khadra, A.

Levkova, L.

B. Mackenzie, P.

Maresca, F.

Menscher, D.

Nobes, M.

Okamoto, M.

B. Oktay, M.

Osborn, J.

Renner, D.

N. Simone, J.

Sugar, R.

Toussaint, D.

D. Trottier, H.

Edinburgh Research Explorer

     Edinburgh Research Explorer                                      Predictive Lattice QCDCitation for published version:S. Kronfeld, A, F. Allison, I, Aubin, C, Bernard, C, T. H. Davies, C, DeTar, C, Di Pierro, M, D. Freeland, E,Gottlieb, S, Gray, A, Gregory, E, M. Heller, U, E. Hetrick, J, X. El-Khadra, A, Levkova, L, B. Mackenzie, P,Maresca, F, Menscher, D, Nobes, M, Okamoto, M, B. Oktay, M, Osborn, J, Renner, D, N. Simone, J, Sugar,R, Toussaint, D & D. Trottier, H 2005, 'Predictive Lattice QCD', International Journal of Modern Physics A,vol. 21, no. 04, pp. 713-719. https://doi.org/10.1142/S0217751X06031934Digital Object Identifier (DOI):10.1142/S0217751X06031934Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Publisher's PDF, also known as Version of recordPublished In:International Journal of Modern Physics AGeneral 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/0509169v2  30 Sep 2005Predictions from Lattice QCD FERMILAB-CONF-05-428-TA.S. Kronfeld∗,a, I.F. Allisonb, C. Aubinc,d, C. Bernardd, C.T.H. Daviesb, C. DeTare,M. Di Pierro f , E.D. Freelandg, Steven Gottliebh, A. Grayi, E. Gregory j , U.M. Hellerk,J.E. Hetrickl , A.X. El-Khadram, L. Levkovah, P.B. Mackenziea, F. Marescae,D. Menscherm, M. Nobesn,o, M. Okamotoa, M.B. Oktaym, J. Osborne, D. Renner j,J.N. Simonea, R. Sugarp, D. Toussaint j, H.D. TrottieroE-mail: ask@fnal.govaFermi National Accelerator Laboratory, Batavia, Illinois60510, USAbDepartment of Physics and Astronomy, Glasgow University, Glasgow, Scotland G12 8QQ, UKcPhysics Department, Columbia University, New York, New York 10027, USAdDepartment of Physics, Washington University, St. Louis, Msouri 63130, USAePhysics Department, University of Utah, Salt Lake City, Utah 84112, USAf School of Computer Science, Telecommunications and Information Systems, DePaul University,Chicago, Illinois 60604, USAgLiberal Arts Department, School of the Art Institute, Chicago, Illinois 60603, USAhDepartment of Physics, Indiana University, Bloomington, Indiana 47405, USAiDepartment of Physics, The Ohio State University, Columbus, Ohio 43210, USAjDepartment of Physics, University of Arizona, Tucson, Arizona 85721, USAkAmerican Physical Society, Ridge, New York 11961, USAl Physics Department, University of the Pacific, Stockton, California 95211, USAmPhysics Department, University of Illinois, Urbana, Illinois 61801, USAnLaboratory of Elementary-Particle Physics, Cornell University, Ithaca, New York 14853, USAoPhysics Department, Simon Fraser University, Burnaby, British Columbia V5A 1S6, CanadapDepartment of Physics, University of California, Santa Barbara, California 93106, USAFermilab Lattice, MILC, and HPQCD CollaborationsIn the past year, we calculated with lattice QCD three quantities that were unknown or poorlyknown. They are theq2 dependence of the form factor in semileptonicD → Klν decay, thedecay constant of theD meson, and the mass of theBc meson. In this talk, we summarize thesecalculations, with emphasis on their (subsequent) confirmation by experiments.XXIIIrd International Symposium on Lattice Field Theory25-30 July 2005Trinity College, Dublin, Ireland∗Speaker.PoS(LAT2005)206c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licen. http://pos.sissa.it/Predictions from Lattice QCD A.S. Kronfeld1. Introduction and BackgroundIn recent years, lattice QCD has reached the stage where manycalculations of hadron masses,mass splittings, and operator matrix elements agree with experimental measurements. The keyhas been the inclusion of sea quarks. The progress has been especially striking [1] when the lightquarks (sea and valence) are implemented as staggered quarks, with an improved action.Some of the ingredients of these calculations are controversial. Staggered quarks come in fourtastes, three of which must be removed to obtain each individual flavor. For sea quarks, this isdone by taking the fourth root of the fermion determinant; for valence quarks, by projecting ontothe desired taste sector. Furthermore chiral perturbationtheory must be modified [2]. Althoughevidence for the validity of these “tricks” is slowly accumulating, a proof remains at large [3]. Inaddition, much of the success of Ref. [1] comes from hadrons with heavy quarks. Although debateon heavy quarks in lattice QCD seems to have subsided, checksare still useful.In this paper, we discuss three calculations, with emphasison their subsequent experimentalconfirmation. They are the normalization andq2-dependence of theD → Klν form factor; thedecay constants of theD+ andDs mesons; and the mass of theBc meson. Each tests a somewhatdifferent combination of the ingredients, and the following table gives an informal guide:calculation light sea light valence heavysemileptonicf+(q2) ⋆⋆ ⋆⋆ ⋆⋆leptonic fD ⋆⋆ ⋆⋆⋆ ⋆⋆Bc mass ⋆⋆ — ⋆⋆⋆The chiral extrapolation, which is more sensitive to valence quarks than sea quarks, turned out tobe more important for the decay constant than the form factor. The Bc meson has no light va-lence quarks at all, but one should expect an accurate calculation only if heavy-quark discretizationeffects are under control.Successful predictions are, of course, not a substitute fora proof. They are still useful. Evenif the experts are confident of all the elements of their numerical calculations, non-experts areinterested in an end-to-end check [4]. The quantities discussed here are ideal candidates: they arestraightforward to compute; the first “good” experimental measurements were not expected untilthis year; and new physics is unlikely to contribute significantly.2. Semileptonic D DecaysSemileptonic decays such asD → Klν are mediated by electroweak vector currents. Thematrix element〈K|Vµ |D〉 is parametrized by form factors. For a vector current there are two, butexperimentally only the one calledf+(q2) is accessible; the rate from the other one,f0(q2), issuppressed bym2l . Hereq2 is the momentum transferred to the lepton-neutrino system,falling inthe range 0≤ q2 ≤ q2max = (mD −mK)2. In lattice QCD, discretization effects are smallest whenthe momentump of the kaon is small, and thenq2 is not too far fromq2max.Experiments usually measure the branching fraction and quote the normalizationf+(0), aftermaking assumptions about theq2 dependence. While our results were still preliminary [5], exper-imental results came out for the normalization ofD → Klν [6] andD → π lν [7]. The agreementPoS(LAT2005)206206 / 2Predictions from Lattice QCD A.S. Kronfeldwith our final results [8] is excellent. For example, we findf D→K+ (0) = 0.73(3)(7) [8] while BESmeasuresf D→K+ (0) = 0.78(5) [6]. Our calculations of the normalization are also consistent withthe soft pion theorem, which statesf0(q2max) = fD/ fπ .In principle, the shape of the form factors can be computed directly in lattice QCD. In practice,we calculated at a few values ofp and used the Bécirević-Kaidalov (BK) form [9] to fix the fullq2dependence off+ and f0. Then the normalization off+ comes mainly fromf0 through a kinematicconstraint f+(0) = f0(0). The BK Ansatz and calculations nearq2max determine the shape. It wasimportant, therefore, to measure theq2 dependence experimentally. In photoproduction of charmoff fixed nuclear targets, the FOCUS Collaboration was able to collect high enough statistics totrace out theq2 distribution of the decay [10]. This setup does not yield an absolutely normalizedbranching ratio, so one is left to comparef+(q2)/ f+(0).In Fig. 1 we plot our result forf+(q2)/ f+(0) vs. q2/m2D∗s . The errors fromf+(0) must bepropagated to non-zeroq2, so for f+(q2)/ f+(0) the errors grow withq2. Figure 1 shows 1-σ bandsof statistical (orange) and all uncertainties (yellow) adde in quadrature [11]. As one can see, theq2 dependence of lattice QCD (curve and error band) and experiment (points) agree excellently,although the uncertainties are still several per cent.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45q2/mDs*200.511.522.5f +(q2 )/f +(0)q2max/mDs*2experiment [FOCUS, hep-ex/0410037]lattice QCD [Fermilab/MILC, hep-ph/0408306]1σ (statistical)1σ (stat + syst)D → KlνFigure 1: Shape of form factorf+(q2)/ f+(0) vs.q2/m2D∗s , compared with experiment [10].3. Leptonic D DecaysWe also computed the hadronic matrix element for the leptonic decay of charmed mesons,fD+and fDs. The first (experimental) measurements offD+ appeared in 2004, with three events fromBES [12] and eight from CLEO [13]. Neither provides a stringet test of QCD, but CLEO-c wasjust starting its run and promised 5–8 times higher statistics by the Summer 2005 Lepton-PhotonSymposium [4]. At Lattice 2004 [14], we presented preliminary results for fD+ , based on onelattice spacing,a ≈ 0.125 fm. Our aim was to extend the running to two other lattice spacingsand, of course, to improve our understanding of other aspectof the calculation, such as the chiralextrapolation. Details are given in the ensuing publication [15]. We findfD+ = 201±3±17 MeV, (3.1)where the first error is from finite Monte Carlo statistics, the second is a sum in quadrature ofseveral systematics. A conservative (but not naïve) estimate of heavy-quark discretizations effects,PoS(LAT2005)206206 / 3Predictions from Lattice QCD A.S. Kronfeldas discussed in Ref. [16], is the second largest (largest) syematic onfD+ ( fDs). A few days afterour paper was posted on the arXiv, CLEO-c announced its new measurement [17]fD+ = 223±17±3 MeV, (3.2)based on 47±8 events. At the 1-σ level, the agreement between Eqs. (3.1) and (3.2) is fine. Oneshould keep in mind that the experiment actually determines|Vcd| fD+ . CLEO-c [17] assumes that|Vcd| = |Vus| and uses a recent average of|Vus| from semileptonicK decay.It is interesting to look at thenf dependence ofDs, shown in Fig. 2(a). Of course, quenchedresults vary widely, but we show one [18] carried out with similar choices for heavy quarks, renor-malization factors, etc. One sees a trend offDs to increase withnf . A similar comparison offD+ ,in Fig. 2(b), is less instructive, because the chiral extrapol tions in Refs. [18, 19] started at largequark masses and are, hence, less reliable than in the present work.(a)0 1 2 3nf160180200220240260280300320f Ds (MeV)hep-ph/9711426 [Fermilab]hep-lat/0206016 [MILC]hep-lat/0506030 [Fermilab + MILC]PDG average(b)0 1 2 3nf160180200220240260280300320f D+ (MeV)hep-ph/9711426 [Fermilab]hep-lat/0206016 [MILC]hep-lat/0506030 [Fermilab + MILC]hep-ex/0508057 [CLEO-c]Figure 2: Dependence of (a)fDs and (b) fD+ on the numbernf of sea flavors. Quenched (nf = 0) [18];nf = 2 [19]; nf = 3 [15]. Solid (dashed) error bars are statistical (statistical+systematic).4. Mass of the Bc MesonThe pseudoscalarB+c meson is the lowest-lying bound state of a charmed quark and ab qu rk.CDF [20] first observed it during Run I of the Tevatron in the semil ptonic decayB+c → J/ψ l+ν .During Run II, DØ has confirmed the discovery in the same mode [21]. Because the neutrino isundetected, the mass resolution in semileptonic modes is poor, ±(300–400) MeV. Now, however,the upgraded detectors are able to reconstruct hadronic modes, such asB+c → J/ψπ+, which givemuch much better precision onmBc [22].At Lattice 2004 we presented results in nearly final form [23], and posted the final results onthe arXiv in mid-November [24]:mBc = 6304±12+18− 0 MeV, (4.1)where the last error is a rough estimate of residual heavy-quark discretization effects. Soon after-wards, CDF announced a precise mass measurement. They find [25]mBc = 6287±5 MeV, (4.2)which agrees with Eq. (4.1) at slightly more than 1-σ .Two comments are in order. First, the agreement at the gross level of the calculation withexperiment shows that discretization effects are well under control with lattice NRQCD [27] and thePoS(LAT2005)206206 / 4Predictions from Lattice QCD A.S. KronfeldFermilab method [28]. Of course, this follows from the careful application of effective field theoriesfor heavy quarks [29, 30]. Indeed, as seen in Fig. 3(a), almost no lattice spacing dependence is seenin the splitting∆ψϒ = mBc − (m̄ψ +mϒ)/2 that is at the crux of the calculation [26]. Moreover, it isstriking how much the splitting∆ψϒ changes when sea quarks are included. Figure 3(b) comparesEq. (4.1) with an old quenched calculation [26] (and the measurement [25]). The solid error barshows the non-quenching errors, and the dashed includes thestimate of the quenching error. Theinclusion of sea quarks has reduced the splitting by a factorof three or four, bringing an essentiallydiscrepant result into agreement.(a)0.00 0.05 0.10 0.15 0.20a (fm)4050∆ ψΥ (MeV)(b)6300640065006600mBc (MeV/c2 )lattice QCD, Feb. 1999lattice QCD, Nov. 2004CDF, Dec. 2004Figure 3: (a) Dependence of the splitting∆ψϒ on the lattice spacinga. (b) Comparison of the quenched [26],nf = 2+1 [24], and experimental [25] values ofmBc; the dashed line denotes the baseline(m̄ψ +mϒ)/2.5. ConclusionsIn the past year, three lattice-QCD calculations have been confirmed by experiment. FOCUS[10] confirmed theq2-dependence of theD → Klν form factor [8]; CLEO-c [17] confirmed theD-meson decay constant [15]; and CDF [25] confirmed the mass of theBc meson [24]. To obtain theseresults it is essential to have heavy-quark discretizationeffects under control, as one expects fromtheoretical foundations [27, 28, 29, 30]. Furthermore, thecomparison of quenched QCD, QCDwith 2+1 staggered flavors, and experiment shows that sea quarks are needed to obtain agreement,and that staggered quarks (in these cases) capture the needed effect.This work has been supported in part by the U.S. National Science Foundation, the Office ofScience of the U.S. Department of Energy (DOE), and the U.K. Particle Physics and AstronomyResearch Council. Fermilab is operated by Universities Research Association Inc., under contractwith the DOE.References[1] C. T. H. Davieset al. [HPQCD, MILC, and Fermilab Lattice Collaborations], Phys.Rev. Lett.92,022001 (2004) [hep-lat/0304004].[2] C. Aubin and C. Bernard, Phys. Rev. D68, 034014 (2003) [hep-lat/0304014];ibid., 074011 (2003)[hep-lat/0306026].[3] S. Dürr, “Theoretical issues with staggered fermion simulations,” hep-lat/0509026.[4] I. Shipsey, Nucl. Phys. Proc. Suppl.140, 58 (2005) [hep-lat/0411009].[5] M. Okamotoet al. [Fermilab Lattice, MILC, and HPQCD Collaborations], Nucl.Phys. Proc. Suppl.129, 334 (2004) [hep-lat/0309107].PoS(LAT2005)206206 / 5Predictions from Lattice QCD A.S. Kronfeld[6] M. Ablikim et al. [BES Collaboration], Phys. Lett. B597, 39 (2004) [hep-ex/0406028].[7] G. S. Huanget al. [CLEO Collaboration], Phys. Rev. Lett.94, 011802 (2005) [hep-ex/0407035].[8] C. Aubin et al. [Fermilab Lattice, MILC, and HPQCD Collaborations], Phys.Rev. Lett.94, 011601(2005) [hep-ph/0408306].[9] D. Bećirević and A. B. Kaidalov, Phys. Lett. B478, 417 (2000) [hep-ph/9904490].[10] J. M. Link et al. [FOCUS Collaboration], Phys. Lett. B607, 233 (2005) [hep-ex/0410037].[11] P.B. Mackenzieet al. [Fermilab Lattice, MILC, and HPQCD Collaborations], work in progress.[12] M. Ablikim et al. [BES Collaboration], Phys. Lett. B610, 183 (2005) [hep-ex/0410050].[13] G. Bonviciniet al. [CLEO Collaboration], Phys. Rev. D70, 112004 (2004) [hep-ex/0411050].[14] J. N. Simoneet al. [Fermilab Lattice, MILC, and HPQCD Collaborations], Nucl.Phys. Proc. Suppl.140, 443 (2005) [hep-lat/0410030].[15] C. Aubinet al., [Fermilab Lattice, MILC, and HPQCD Collaborations], Phys. Rev. Lett.95, 122002(2005) [hep-lat/0506030];J.N. Simoneet al., [Fermilab Lattice, MILC, and HPQCD Collaborations], these proceedings.[16] A. S. Kronfeld, Nucl. Phys. Proc. Suppl.129, 46 (2004) [hep-lat/0310063].[17] M. Artuso, talk at the XXII International Symposium on Lepton-Photon Interactions at High Energy,Uppsala, Sweden,http://lp2005.tsl.uu.se/˜lp2005/ (June 30–July 5, 2005);M. Artusoet al. [CLEO Collaboration], “Improved measurement ofB(D+ → µ+ν) and thepseudoscalar decay constantfD+ ,” hep-ex/0508057.[18] A. X. El-Khadra, A. S. Kronfeld, P. B. Mackenzie, S. M. Ryan, and J. N. Simone, Phys. Rev. D58,014506 (1998) [hep-ph/9711426].[19] C. Bernardet al. [MILC Collaboration], Phys. Rev. D66, 094501 (2002) [hep-lat/0206016].[20] F. Abeet al. [CDF Collaboration], Phys. Rev. Lett.81, 2432 (1998) [hep-ex/9805034].[21] DØ Collaboration, DØ Note 4539-CONF (August 14, 2004);E. Cheu [DØ Collaboration], Int. J. Mod. Phys. A20, 3664 (2005).[22] K. Anikeevet al., “B physics at the Tevatron: Run II and beyond,” hep-ph/0201071.[23] I. F. Allison et al. [HPQCD and Fermilab Lattice Collaborations], Nucl. Phys. Proc. Suppl.140, 440(2005) [hep-lat/0409090].[24] I. F. Allison et al. [HPQCD and Fermilab Lattice Collaborations], Phys. Rev. Lett. 94, 172001 (2005)[hep-lat/0411027].[25] D. Acostaet al. [CDF Collaboration], “Evidence for the exclusive decayB±c → J/ψπ± andmeasurement of the mass of theBc meson,” hep-ex/0505076.[26] H. P. Shanahan, P. Boyle, C. T. H. Davies, and H. Newton [UKQCD Collaboration], Phys. Lett. B453, 289 (1999) [hep-lat/9902025].[27] G. P. Lepage and B. A. Thacker, Nucl. Phys. Proc. Suppl.4, 199 (1987);B. A. Thacker and G. P. Lepage, Phys. Rev. D43, 196 (1991).[28] A. X. El-Khadra, A. S. Kronfeld, and P. B. Mackenzie, Phys. Rev. D55, 3933 (1997)[hep-lat/9604004].[29] G. P. Lepage t al., Phys. Rev. D46, 4052 (1992) [hep-lat/9205007].[30] A. S. Kronfeld, Phys. Rev. D62, 014505 (2000) [hep-lat/0002008].PoS(LAT2005)206206 / 6

Predictive Lattice QCD

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Predictions from Lattice QCD

Predictions from Lattice QCD

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