We have developed two methods for handling b quarks in lattice QCD. One uses NRQCD (now improved to include radiative corrections) and the other uses Highly Improved Staggered Quarks (HISQ), extrapolating to the b quark from lighter masses and using multiple lattice spacings to control discretisation errors. Comparison of results for the two different methods gives confidence in estimates of lattice QCD systematic errors, since they are very different in these two cases. Here we show results for heavyonium hyperfine splittings and vector current-current correlator moments using HISQ quarks, to add to earlier results testing the heavy HISQ method with pseudoscalar mesons. We also show the form factor for B -&gt; pi l nu decay at zero recoil using NRQCD b quarks and u/d quarks with physical masses. This allows us to test the soft pion theorem relation (f_0(q2max)=f_B/fπ) accurately and we find good agreement as M_pi -&gt; 0

Colquhoun, Brian

Davies, Christine

Donald, Gordon

Dowdall, Rachel

Follana, Eduardo

Galloway, Ben

Horgan, Ron

Koponen, Jonna

Lepage, G. Peter

McNeile, Craig

Enlighten

     Davies, Christine, Colquhoun, Brian, Galloway, Ben, Donald, Gordon, Koponen, Jonna, Dowdall, Rachel, Horgan, Ron, Follana, Eduardo, Lepage, G. Peter, and McNeile, Craig (2013) Bottomonium and B results from full lattice QCD. In: 31st International Symposium on Lattice Field Theory LATTICE 2013 , 29 Jul – 3 Aug 2013, Mainz, Germany.  Copyright © 2013 The Authors    http://eprints.gla.ac.uk/90854/     Deposited on: 26 June 2014                         Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk PoS(LATTICE 2013)438Bottomonium and B results from full lattice QCDHPQCD CollaborationC. T. H. Davies∗, B. Colquhoun, B. Galloway, G. C. Donald, J. KoponenSUPA, School of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, UKE-mail: Christine.Davies@glasgow.ac.ukR. J. Dowdall, R. HorganDAMTP, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UKE. FollanaDepartamento de Física Teórica, Universidad de Zaragoza, E-50009 Zaragoza, SpainG. P. LepageLabortory for Elementary-Particle Physics, Cornell University, Ithaca, NY 14853, USAC. McNeileSchool of Computing and Mathematics, Plymouth University, Drake Circus, Plymouth, PL48AA, UKWe have developed two methods for handling b quarks in lattice QCD. One uses NRQCD (nowimproved to include radiative corrections) and the other uses Highly Improved Staggered Quarks(HISQ), extrapolating to the b quark from lighter masses and using multiple lattice spacings tocontrol discretisation errors. Comparison of results for the two different methods gives confi-dence in estimates of lattice QCD systematic errors, since they are very different in these twocases. Here we show results for heavyonium hyperfine splittings and vector current-current cor-relator moments using HISQ quarks, to add to earlier results testing the heavy HISQ methodwith pseudoscalar mesons. We also show the form factor for B→ pilν decay at zero recoil us-ing NRQCD b quarks and u/d quarks with physical masses. This allows us to test the soft piontheorem relation ( f0(q2max) = fB/ fpi ) accurately and we find good agreement as Mpi → 0.31st International Symposium on Lattice Field Theory - LATTICE 2013July 29 - August 3, 2013Mainz, Germany∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/PoS(LATTICE 2013)438Bottomonium and B results from full lattice QCD C. T. H. Davies0.10 0.15 0.20 0.25 0.30 0.35M−1ηh (GeV−1)20406080100120Mφh−Mηh(MeV)MηcMηbFigure 1: The heavyonium hyperfine splitting as a function of the inverse heavyonium mass. Resultsare given in different colours for coarse (a=0.12fm), fine (a=0.09fm), superfine (a=0.06fm) and ultrafine(a=0.045fm) lattices, including 2+1 flavours of sea quarks. The shaded band shows the fit described in thetext and the black points at the b and c the current experimental averages [4]. Note that ηb and ηc annihilationeffects are not included in the lattice results or the shaded fit.1. Vector Heavyonium - hyperfine splittingFig. 1 shows the heavyonium vector-pseudoscalar (hyperfine) mass splitting as a function ofinverse heavyonium mass for HISQ quarks [1] for a range of masses from charm upwards. We haveused MILC gluon field configurations that include 2+1 flavours of asqtad sea quarks with latticespacings ranging from 0.12 fm to 0.045 fm [2]. The finer lattice spacings allow a much larger reachin meson mass because a given meson mass corresponds to a smaller quark mass in lattice units.We have used quark masses in lattice units up to 0.8. Having so many values of the lattice spacingallows the discretisation errors to be well determined from the fit. The grey band shows the resultof the physical heavy quark mass dependence determined at zero lattice spacing resulting from afit function of form [3]:F(M,a) = A(MM0)b 7∑i=03∑j=0ci j(M0M)i(am)2 j (1.1)where M is the pseudoscalar meson mass and m the quark mass and we also include terms allowingfor sea quark mass dependence (which are then extrapolated to the physical point). Priors on thecoefficients are generally taken as 0(1) and M0 is taken as 1 GeV.The result at the b, after an increase of 3(3) MeV for ηb annihilation effects not included inour calculation (and also NOT included in the plot) isMϒ−Mηb = 53(5)MeV. (1.2)This is in reasonable agreement with the current experimental average of 62.3(3.2) MeV [4]. Theresult we obtain at c is 116.5(3.2) MeV [6], in good agreement with the experimental average of113.2(7) MeV [4]. We are currently extending these calculations to the MILC ‘second-generation’configurations that include 2+1+1 flavours of HISQ sea quarks [5].2PoS(LATTICE 2013)438Bottomonium and B results from full lattice QCD C. T. H. Davies2 3 4 5 6 7 8 9 10Mηh  (GeV)0.000.050.100.150.200.250.300.35(4thmoment)1/2 (GeV−1)Mηc Mηb2 3 4 5 6 7 8 9 10Mηh  (GeV)0.00.20.40.60.81.01.2(8thmoment)1/6 (GeV−1)Mηc MηbFigure 2: The 4th and 8th moments of the heavyonium vector current-current correlator as a functionof heavyonium mass. Results are given for coarse (a=0.12fm), fine (a=0.09fm), superfine (a=0.06fm) andultrafine (a=0.045fm) lattices, including 2+1 flavours of sea quarks. The shaded band shows the fit describedin the text and the black points at the b and c are the results derived from experiment [8].We have also calculated the bottomonium hyperfine splitting using NRQCD for the b quark [7].We have included spin-dependent relativistic corrections throughO(v6), radiative corrections to theleading spin-chromomagnetic coupling (at O(v4)) and, for the first time, non-perturbative 4-quarkinteractions. We use the MILC configurations that include 2+1+1 flavours of HISQ sea quarks. Weobtain a splitting of 62.8(6.7) MeV [7], again in good agreement with experiment.2. Vector Heavyonium - current-current correlator momentsTime moments of the vector heavyonium correlator are defined by:GVn = Z2 ∑˜tt˜nCV (t˜) (2.1)where t˜ is the lattice time variable, symmetrised around the centre of the lattice and Z is the currentrenormalisation factor, derived from continuum QCD perturbation theory for the 6th moment [6].3PoS(LATTICE 2013)438Bottomonium and B results from full lattice QCD C. T. H. Daviesf 0(q2m ax) /( f B/ f /)  M/ (GeV)a = 0.15 fma = 0.12 fma = 0.09 fm 0.5 0.6 0.7 0.8 0.9 1 0  0.05  0.1  0.15  0.2  0.25  0.3  0.35  0.4Figure 3: The ratio of f0(q2max) for B→ pilν decay to fB/ fpi as a function of Mpi . The points are latticeresults, at 3 values of the lattice spacing, with the leftmost points being at the physical value of Mpi . Theshaded band shows a simple extrapolation through the two most chiral sets of points to Mpi = 0 and a = 0.The lattice time-moments can be compared to q2-derivative moments of the heavy quark vac-uum polarisation. Values for these can be extracted [8] from experimental results in the c and bregions for Re+e− whereRe+e− =σ(e+e−→ hadrons)σpoint. (2.2)The two plots in Fig. 2 show our results for the 4th and 8th moments, calculated with HISQquarks, as a function of heavyonium mass. Again we have results for a wide range of lattice spacingvalues, from 0.12 fm to 0.045 fm, on the MILC gluon field configurations that include 2+1 flavoursof asqtad sea quarks.The shaded bands show the results of fits to the form given above for the hyperfine splitting(Eq. 1.1) but with leading power, b=-1. The results show excellent agreement with experiment,both at c [6] and at b. The experimental values [8] are shown as the black points. This is a stringenttest of lattice QCD since experimental errors are small.3. B→ pilν decay and soft pion theoremsThe matrix element of the temporal vector current for B→ pi decay at zero recoil (both mesonsat rest) is given by:〈pi|V 4|B〉= f0(q2max)(MB +Mpi). (3.1)Soft pion theorems relate this to decay constants. At leading order, for Mpi → 0,f0(q2max) =fBfpi. (3.2)This result seemed not to hold well in the quenched approximation (see [9] for a review), butlarge uncertainties arose from large pion masses (along with the absence of significant pion massdependence), quenching and current renormalisation.4PoS(LATTICE 2013)438Bottomonium and B results from full lattice QCD C. T. H. DaviesHere we have small uncertainties because we are using improved NRQCD b quarks combinedwith HISQ light quarks. We go down to physical pion masses on the MILC configurations thatinclude 2+1+1 flavours of HISQ sea quarks. Z factors cancel between f0 and fB from staggeredchiral symmetry. We include O(Λ/mb) relativistic corrections to axial and vector currents.The plot in Fig. 3 shows the ratio of f0 to the decay constant ratio as a function of Mpi . Thedecay constants are the current lattice state-of-the-art results obtained in [10] and [11]. We seethe ratio is around 0.75, quite far from 1, for physical pi masses. However, there is clearly strongdependence on Mpi . Using an extrapolation form that includes linear terms in Mpi (coming from thedependence of q2max on Mpi [12]) readily gives a result in agreement with 1 in the Mpi → 0 limit.We are currently calculating form factors for B/Bs semileptonic decays using HISQ quarks andextrapolating to the b as described above. This will give us another opportunity to test systematicerrors by comparing the two formalisms.References[1] E. Follana et al, HPQCD Collaboration, Highly Improved Staggered Quarks on the lattice, withApplications to Charm Physics, Phys. Rev. D75:054502 (2007), [hep-lat/0610092].[2] A. Bazavov et al, Nonperturbative QCD simulations with 2+1 flavors of improved staggered quarks,Rev. Mod. Phys. 82:1349 (2010), [arXiv:0903.3599[hep-lat]].[3] C. McNeile et al, HPQCD Collaboration, Heavy meson masses and decay constants from relativisticheavy quarks in full lattice QCD, Phys. Rev. D86:074503 (2012),[arXiv:1207.0994[hep-lat]].[4] J. Beringer et al, Particle Data Group, Phys. Rev. D86:010001 (2012).[5] A. Bazavov et al, MILC Collaboration, Lattice QCD ensembles with four flavors of highly improvedstaggered quarks, Phys. Rev. D87:054505 (2013), [arXiv:1212.4768[hep-lat]].[6] G. C. Donald et al, HPQCD Collaboration, Precision tests of the J/ψ from full lattice QCD: mass,leptonic width and radiative decay to ηc, Phys. Rev. D86:094501 (2012),[arxiv:1208.2855[hep-lat]].[7] R. J. Dowdall et al, HPQCD Collaboration, Bottomonium hyperfine splittings from lattice NRQCDincluding radiative and relativistic corrections, [arXiv:1309.5797[hep-lat]].[8] K. G. Chetyrkin et al, Charm and Bottom Quark Masses: an Update, Phys. Rev. D80:074010 (2009),[arXiv:0907.2110[hep-ph]].[9] S. Hashimoto, B decays on the lattice, Nucl. Phys. Proc. Suppl. 83:3 (2000), [hep-lat/9909136].[10] R. J. Dowdall et al, HPQCD Collaboration, B-meson decay constants from improved lattice NRQCDand physical u, d, s and c quarks, Phys. Rev. Lett.110:222003 (2013),[arXiv:1302.2644[hep-lat]].[11] R. J. Dowdall et al, HPQCD Collaboration, Vus from pi and K decay constants in full lattice QCD withphysical u, d, s and c quarks, Phys. Rev. D88:074504 (2013), [arXiv:1303.1670[hep-lat]].[12] K. C. Bowler et al, UKQCD Collaboration, An Investigation of the soft pion relation in quenchedlattice QCD, Nucl. Phys. Proc. Suppl. 83:313 (2000), [hep-lat/9910020].5

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Bottomonium and B results from full lattice QCD

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Bottomonium and B results from full lattice QCD

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