Lambda-bar, lambda_1 and m_b in three-flavor (lattice) QCD

a r X i v :h e p -l a t /0610108v 1 19 O c t 2006ˉΛ

,λ1and m b in three-?avor (lattice)QCD ?Speaker.

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192π30.3689 1+1.54

αs π2?1.648ˉΛπ)?0.946ˉΛ2M 2B +0.02λ2

M 3B

) ,(1.1)where |V cb |is the CKM matrix element of interest;M B is the B -meson mass;and,ˉΛ

,λ1,and λ2are scheme-dependent hadronic matrix elements de?ned in HQET.Currently,the HQET ma-trix elements are determined by ?tting measurements of various moments of heavy-meson decay distributions to corresponding HQET expressions [5,6,4,7].

In addition to using experimental measurements,one would like to calculate the HQET matrix elements from ?rst principles.In fact,because the same non-perturbative quantities appear in the HQET expression for the meson mass,there exists a rather direct method for calculating them using lattice QCD.This method was ?rst proposed in Ref.[9],where the corresponding quenched calculation was reported.

The HQET expression for the mass of a heavy-light meson is [2,4,8]

M =m +ˉΛ?λ12m

+O (1/m 2),

(1.2)where J is the total meson angular momentum,and d 0=3and d 1=?1for the pseudoscalar and vector mesons respectively.The mass of the heavy-light meson is M and that of the heavy quark is m .Working with the spin-averaged meson mass,

M ?m =ˉΛ?λ1M 1?m 1=ˉΛ

(a )?λ1(a )

M1is the spin-averaged rest mass on the lattice.

To arrive atˉΛandλ1for B±and B d we would need an expression for the(spin-averaged) meson mass from chiral perturbation theory.Continuum HQET expressions exist in the litera-ture[13].The full expression including effects from staggered quarks and HQET is being derived now.Because we can do simulations with light valence quarks near or at the strange-quark mass, and because we?nd that the effect of sea quarks is mild,we obtain preliminary results for the bottom quark mass,m b,usingˉΛandλ1from the B s meson.

In the following,we?rst discuss the lattice calculation of meson and quark masses.We then discuss how

M,m1and m2

We use the MILC unquenched gauge con?gurations[14]with2+1?avors of sea quarks and a Symanzik-improved gluon action.We use three lattice spacings:a=0.18,0.15,and0.12fm.Both sea and light valence quarks use the“asqtad”staggered-fermion action[15].Light valence quarks have masses ranging from m q=1.1m s to0.1m s,where m s is the(physical)strange quark mass. Masses of the two light sea quarks range from approximately0.05m s to0.1m s.For heavy quarks, we use the Fermilab action[16].In anticipation of the full calculation ofˉΛandλ1,we use seven or more heavy-quark masses at each lattice spacing.They range in mass from heavier-than-bottom to lighter-than-charm.

Pseudoscalar and vector meson masses are obtained from two-point correlation function?ts done using multi-state,constrained curve?tting[17].Bothχ2and?t stability are used to determine the goodness of?t.The results are spin-averaged to obtain

MS mass does not run correctly for renormalization scales below the heavy-quark mass scale,it is not appropriate.Several other short-distance mass de?nitions are available in the literature.Here,we use the potential-subtracted mass,m PS[20],which is based on the static quark potential and introduces a separation scale,μf, whereΛQCD<μf 2GeV.

Forαs,we use the V-scheme;scale setting,q?,is done via the Brodsky-Lepage-Mackenzie (BLM)prescription[21].The value ofαs(q?)is obtained from the average value of the plaquette and the four-loopβ-function as described in[22].

3.Sea Quark,Valence Quark and Lattice Spacing Dependencies

The value of the meson binding energy,

M1?m1,PS vs1/2m2,PS for three values of light sea quark masses;the valence quark mass m q=m s and a=0.125fm.(b)

M1?m1,PS vs1/2m2,PS for three values of sea-quark mass ratios m u,d/m s.For each graph,ˉΛis the intercept andλ1is the slope,while the curvature is related to a combination of HQET matrix elements at O(1/m2)[9]. Figure1(a)allows one to view the dependence of these quantities on the sea quarks.One can see that varying the light sea-quark mass has only a small effect on

M1?m1,PS for B s we used,at each value of a,the ensemble with the lightest available m u,d sea quarks,and used the variation from different ensembles in our estimate of the systematic error.

Figure1(b)is a plot of

M1?m1,PS for B s,we use the m q=1.06m s result and allow an uncertainty based on the value at m q=0.77m s.

Figure2is a plot of

M1?m1,PS in a2and a4were used to assign

4

M1?m1,PS vs1/2m2,PS for three lattice spacings:0.125,0.15and0.18fm.For all cases m q=m s. Error bars are statistical only.Bottom and charm quarks have values of1/2m2,PS=0.13and0.58GeV?1 respectively.

a systematic error for the discretization of light quarks and gluons.Our?nal error budget includes an uncertainty in the determination of a itself.

4.Result for the b-quark mass

Focusing on m b,we calculate the binding energy of a(spin-averaged)B s meson,by averaging the results at the a=0.125fm and0.15fm spacings.Our result is0.99(18)GeV in the potential-subtracted scheme with a factorization scale ofμf=https://www.sodocs.net/doc/d45351475.html,ing this value of the binding energy,we can make a preliminary estimate of the value of the bottom-quark mass.

m b=M exp?(

MS scheme at this time.For com-parison,a QCD sum rule calculation[24]obtains,m b,PS=4.52(6)GeV atμf=2.0GeV and m b,

statistical0.005

inputs(a,κb,κcrit,u0)0.041

sea-quark mass dependence0.04

strange-quark mass tuning0.025

perturbation theory(heavy quark discretization)0.10

light quark and gluon discretization0.14

M1?m1,PS for a spin-averaged B s meson.

ofμf=2.0GeV.The dominant uncertainties in this calculation can be reduced by the inclusion of2-loop effects in the perturbative expansions for m1and m2,and with improved understanding of light quark and gluon discretization effects.Future work will include the calculation of HQET matrix elementsˉΛandλ1for B±and B d,which can be used in the determination of CKM matrix elements from their inclusive,semileptonic decays.

Acknowledgments

E.D.

F.would like to thank the Fermilab Theoretical Physics Department for their hospitality while this work was being done,and Don Holmgren and Amitoj Singh for their help with comput-ing issues.E.D.F.is supported in part by the American Association of University Women.Fermilab is operated by Universities Research Association Inc.,under contract with the DOE.

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