We determine the quark masses and the chiral condensate in the MSbar scheme
at NNLO from Lattice QCD in the quenched approximation at beta=6.0, beta=6.2
and beta=6.4 using both the Wilson and the tree-level improved SW-Clover
fermion action. We extract these quantities using the Vector and the Axial Ward
Identities and non-perturbative values of the renormalization constants. We
compare the results obtained with the two methods and we study the O(a)
dependence of the quark masses for both actions.Comment: LATTICE98(spectrum), 3 pages, 1 figure, Edinburgh 98/1

Gimenez, V.

Giusti, L.

Rapuano, F.

Talevi, M.

Vladikas, A.

English

arXiv

We determine the quark masses and the chiral condensate in the  scheme at NNLO from Lattice QCD in the quenched approximation at beta = 6.0, beta = 6.2 and beta = 6.4 using both the Wilson and the tree-level improved SW-Clover fermion action. We extract these quantities using the Vector and the Axial Ward Identities and nonperturbative value of the renormalization constants. We compare the results obtained vr th the two methods and we study the O(a) dependence of the quark masses for both actions

Giménez Gómez, Vicente

Talevi, Mauro

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

arXiv:hep-lat/9809037v1  8 Sep 19981Quark masses and the chiral condensate with a non-perturbativerenormalization procedureV. Gime´neza, L. Giustib∗, F. Rapuanoc, M. Talevid, A. Vladikase.aUniv. de Vale`ncia and IFIC, Dr. Moliner 50, E-46100, Burjassot, Vale`ncia, SpainbScuola Normale Superiore and INFN Sezione di Pisa, P.zza dei Cavalieri 7 - I-56100 Pisa ItalycUniversita` di Roma ‘La Sapienza’ and INFN, Sezione di Roma, P.le A. Moro 2, I-00185 Roma, ItalydDepartment of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, UKeINFN-Sezione di Roma II, Universita` di Tor Vergata, Via della Ricerca Scientifica 1 Roma, ItalyWe determine the quark masses and the chiral condensate in the MS scheme at NNLO from Lattice QCD inthe quenched approximation at β = 6.0, β = 6.2 and β = 6.4 using both the Wilson and the tree-level improvedSW-Clover fermion action. We extract these quantities using the Vector and the Axial Ward Identities and non-perturbative values of the renormalization constants. We compare the results obtained with the two methods andwe study the O(a) dependence of the quark masses for both actions.1. INTRODUCTIONTo describe the light hadron spectrum one con-siders massless QCD a good approximation tothis world and the vacuum not symmetric un-der chiral transformations. The chiral condensateis the order parameter which governs the spon-taneous chiral symmetry breaking. The pseu-doscalar mesons are identified with the goldstonebosons and their physical masses are attributedto the mass term in the QCD Hamiltonian. Sincefree quarks are not physical states, quark massescannot be measured directly in the experimentsand can be determined from the meson spectrumusing non-perturbative techniques. On the latticeone can compute the quark masses and the chiralcondensate from first principles and it is the onlyprocedure that can be systematically improved.The methods and the symbols we have used andall the results we have obtained are fully describedin [1,2].2. QUARK MASSESThe usual on-shell mass definition cannot beused for quark masses and their values depend onthe theoretical definition adopted. In the follow-∗Talk presented by L. Giustiing we will give our final results for the runningquark masses defined in the MS scheme. Quarkmasses can be defined from the Vector Ward Iden-tity (VWI). Neglecting terms of O(a), the VWIbetween on-shell hadronic states can be writtenas [3]〈α|∂µV˜µ|β〉 =12( 1k2−1k1)〈α|S|β〉 . (1)Eq. (1) fixes the relation between the lattice barequark mass in lattice units and the hopping pa-rameter, i.e. for the Wilson actionm = 1/2(1/k−1/kc). Quark masses can also be extracted fromthe Axial Ward Identity (AWI). Neglecting termsof O(a), the AWI can be written as [3]ZA〈α|∂µAaµ|β〉 = 2(m0 −m)〈α|Pa|β〉 , (2)where m is defined in [3]. The light and strangequark masses are determined by fixing to theirexperimental values the masses of the pi and Kmesons. The standard perturbative approach[1,4] uses lattice and the continuum perturba-tion theory to connect the “bare” lattice quarkmass to the renormalized MS one. The scale1/a, where a is the lattice spacing, of our simula-tions is a−1 ≃ 2 − 4 GeV. At these scales we ex-pect small non-perturbative effects. However the2Table 1Summary of the parameters of the runs analyzed in this work.Matrix ElementsC60a C60b W60 C62 W62 C64 W64β 6.0 6.0 6.0 6.2 6.2 6.4 6.4Action SW SW Wil SW Wil SW Wil# Confs 490 600 320 250 250 400 400Volume 183 × 64 243 × 40 183 × 64 243 × 64 243 × 64 243 × 64 243 × 64Renormalization ConstantsC60Z W60Z C62Z W62Z C64Z W64Zβ 6.0 6.0 6.2 6.2 6.4 6.4Action SW Wil SW Wil SW Wil# Confs 100 100 180 100 60 60Volume 163 × 32 163 × 32 163 × 32 163 × 32 243 × 32 243 × 32Table 2Quark Masses an the Chiral Condensate from theVWI in MeV. MS values are at µ = 2 GeV.Run mMSl mMSs −1Nf〈ψ¯ψ〉1W60 5.8(2)(1) 130(3)(2) (247± 2± 1)3W62 5.4(2)(1) 124(4)(2) (250± 3± 1)3W64 4.9(2)(1) 112(5)(2) (258± 4± 1)3“tadpole” diagrams, which are present in latticeperturbation theory, can give rise to large pertur-bative corrections and hence to large uncertain-ties in the 1-loop matching procedure at valuesof β = 6/g2L = 6.0 − 6.4. A non-perturbativerenormalization (NP) technique eliminate theseuncertainties [5,6]. ZA, ZP and ZS can be cal-culated by imposing the renormalization condi-tions, proposed in [5], on the quark states of mo-mentum p2 = µ2 and in the Landau gauge [2].This procedure works if µ satisfies the conditionΛQCD ≪ µ ≪ 1/a to avoid chiral symmetrybreaking effects, large higher-order perturbativecorrections and discretization errors. In figure 1we show the scalar renormalization constant di-vided by its Renormalization Group (RG) evolu-tion (ZRGIS ). The data show that discretizationerrors are within statistical errors in the range0.5 < µa < 2 where the NP renormalization isapplied to compute the quark masses. The NPmethod allows a fully non perturbative defini-tion of the renormalized quark masses in the RIschememRI(µ) =1ZRIS (µa)ma−1mRI(µ) =ZRIAZRIP (µa)ρa−1 (3)where for large time separationsρ(a) =12sinh(MPS)〈A0(τ)P (0)〉〈P (τ)P (0)〉. (4)The MS definition of the renormalized quarkmasses is intrinsically perturbative and can berelated to the RI one through continuum per-turbation theory only:mMS(µ) = UMSm (µ, µ′)ZMSm (µ′)ZRIm (µ′)mRI(µ′) , (5)where UMSm (µ, µ′) is the RG evolution of thequark mass. ZMSm /ZRIm is the matching factorcomputed in perturbation theory at scales µ ≃2−4 GeV large enough to avoid non-perturbativeeffects and/or higher order corrections.3. THE CHIRAL CONDENSATEThe chiral condensate can be defined using theAWI arising from the variation of the non-singletpseudoscalar density. The integrated AWI be-comes [2,3]1Nf〈ψ¯ψ〉 = limm0→mC2(m0 −m)∫d4x〈P (x)P (0)〉 .3Table 3Quark Masses and the Chiral Condensate fromthe AWI in MeV. MS values are at µ = 2 GeV.Run mMSl mMSs −1Nf〈ψ¯ψ〉2C60a 6.0(2)(9) 136(4)(20) (242± 3± 12)3C60b 5.7(2)(8) 132(4)(19) (244± 2± 12)3W60 5.7(2)(8) 127(4)(17) (248± 2± 11)3C62 5.8(5)(6) 131(7)(14) (245± 4± 9)3W62 5.4(2)(5) 122(4)(12) (251± 3± 8)3C64 4.4(3)(3) 104(5)(6) (265± 4± 5)3W64 4.7(2)(3) 108(5)(8) (262± 4± 6)3In the chiral limit one can show that the aboveexpression is equivalent to1Nf〈ψ¯ψ〉 = − limm0→mCf2PM2P4(m0 −m), (6)which is the familiar Gell-Mann–Oakes–Renner(GMOR) relation. We write the relation (6) forthe renormalized condensate as1Nf〈ψ¯ψ〉1 = −12a−1f2χZSCHS1Nf〈ψ¯ψ〉2 = −12a−1f2χZPZACAWI , (7)where fχ = 0.1282 GeV is the “experimental”value in physical units [2] andM2P = CHS(1κ−1κC)2aρ =1CAWIM2P . (8)The main advantages of the above formulas areto avoid the error amplification of the standardmethod because we are left with only one powerof the UV cutoff a−1 and to determine the slopeCHS (CAWI) without extrapolation to the chirallimit. Note that since we work at κ values typ-ical of the strange quark mass, we are implicitlyassuming that the slope will not change in thechiral region.4. RESULTSWe have computed the renormalization con-stants of bilinear quark operators with the NPFigure 1. ZRGIS vs µ for the run W64.method proposed in [5]. The µ-dependence ofZA and ZS are in excellent agreement with theRG predictions. ZP is in good agreement but thechiral symmetry breaking effects deserve a moreaccurate analysis. The parameters and the ac-tion used in each simulation are listed in Table1 and the main results we have obtained are re-ported in Tables 2 and 3. The data at β = 6.4have to be consider for an exploratory study only,since the physical volume and the time extensionof the lattice are too small to be considered re-liable. The MS values of the quark masses andthe chiral condensate reported in Tables 2 and 3show a very good agreement between the valuesextracted from the VWI and from the AWI usingthe non perturbative determinations of the renor-malization constants, while the same comparisonusing the perturbative values of ZA, ZP and ZSfails giving inconsistent results for the two meth-ods [1]. This pattern is also found by [7,8] fordifferent fermion actions. In the β range studied,there is no statistical evidence for an ”a” depen-dence of the quark masses and the chiral conden-sate. We believe that the best estimates for thelight and strange quark masses and the chiral con-densate aremMSl (2 GeV ) = (5.7±5±8±8)MeV ,mMSs (2 GeV ) = (130 ± 8 ± 15 ± 15) MeV and−1Nf〈ψ¯ψ〉MS(2 GeV ) = (245± 4± 9± 7MeV )3,where the first error is statistical, the second isdue to the non-perturbative renormalization andthe third is an estimate of the overall systematicerrors on the bare quantities [1].REFERENCES41. V. Gime´nez et al. hep-lat/9801028 and hep-lat/9806006.2. L. Giusti et al. hep-lat/9807014.3. M.Bochicchio etal.Nucl.Phys.B262(1985)331.4. R.D. Kenway, these proceedings.5. G. Martinelli et al. Nucl. Phys.B445(1995)81.6. S. Capitani et al., Nucl. Phys. B (Proc.Suppl.) 63 (1998) 153.7. D. Becirevic et al. hep-lat/9807046.8. N. Ishizuka, these proceedings.

Quark masses and the chiral condensate with a non-perturbative renormalization procedure

Quark masses and the chiral condensate with a non-perturbative
  renormalization procedure

Quark masses and the chiral condensate with a non-perturbative renormalization procedure

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