We present a scaling study of lattice QCD with O(a) improved Wilson fermions
and a chirally twisted mass term. In order to get precise results with a
moderate computational effort, we have considered a system of physical size of
0.75^3 * 1.5 fm^4 with Schroedinger functional boundary conditions in the
quenched approximation. Looking at meson observables in the pseudoscalar and
vector channels, we find that O(a) improvement is effective and residual cutoff
effects are fairly small.Comment: 5 pages, 4 figures, Lattice 2000 (Improvement and Renormalization),
  two misprints correcte

Della Morte, M.

Frezzotti, R.

Heitger, J.

Sint, S.

English

arXiv

We present a scaling study of lattice QCD with O(a) improved Wilson fermions and a chirally twisted mass term. In order to get precise results with a moderate computational effort, we have considered a system of physical size of 0.75^3 * 1.5 fm^4 with Schroedinger functional boundary conditions in the quenched approximation. Looking at meson observables in the pseudoscalar and vector channels, we find that O(a) improvement is effective and residual cutoff effects are fairly small.We present a scaling study of lattice QCD with O(a) improved Wilson fermions and a chirally twisted mass term. In order to get precise results with a moderate computational effort, we have considered a system of physical size of 0.75^3 * 1.5 fm^4 with Schroedinger functional boundary conditions in the quenched approximation. Looking at meson observables in the pseudoscalar and vector channels, we find that O(a) improvement is effective and residual cutoff effects are fairly small

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arXiv:hep-lat/0010091v2  4 Sep 20011Bicocca-FT-00-20Non–perturbative scaling tests of twisted mass QCD ∗M. Della Mortea, R. Frezzottia, J. Heitgerb and S. Sint caDipartimento di Fisica, Universita` di Milano–Bicocca and INFN Sezione di Milano, Milano, ItalybWestfa¨lische Wilhelms-Universita¨t Mu¨nster, Institut fu¨r Theoretische Physik, Mu¨nster, GermanycCERN, Theory Division, Geneva, SwitzerlandWe present a scaling study of lattice QCD with O(a) improved Wilson fermions and a chirally twisted massterm. In order to get precise results with a moderate computational effort, we have considered a system of physicalsize of 0.753 × 1.5 fm4 with Schro¨dinger functional boundary conditions in the quenched approximation. Lookingat meson observables in the pseudoscalar and vector channels, we find that O(a) improvement is effective andresidual cutoff effects are fairly small.1. IntroductionTwisted mass QCD (tmQCD) was introducedin [ 1] to solve the problem of unphysical fermionzero modes in lattice QCD with two degenerateflavours of Wilson fermions. The fermion latticetmQCD action reads:SF = a4∑xψ(x)(D +m0 + iµqγ5τ3)ψ(x) , (1)where D is the O(a) improved Wilson lattice reg-ularization1 of D/ and m0 ≡ 1/2κ− 4 and µq aretwo real quark mass parameters. The lattice sym-metries and power counting determine the coun-terterm structure for renormalization and O(a)improvement of the infinite volume theory. Thephysical equivalence of tmQCD to QCD in thecontinuum limit has been argued in [ 1] and willbe further detailed in [ 3].We study tmQCD in a four dimensional boxwith Schro¨dinger functional (SF) boundary con-ditions. We wish to check that after on-shell O(a)improvement a` la Symanzik [ 4] a few renormal-ized quantities related to meson physics approachthe continuum limit with residual scaling viola-tions that are small and compatible with beingO(a2). Following ref. [ 7], we keep the spatial size∗Based on a poster by M. Della Morte presented at theInternational Symposium on Lattice Field Theory, August2000, Bangalore, India.1See ref. [ 2] for notation and details.of the box moderately small (L ≃ 0.75 fm) andchoose T = 2L.The transfer matrix of lattice tmQCD with ac-tion (1) and cSW = 0 can be constructed in closeanalogy to ref. [ 10] and turns out to be self-adjoint and strictly positive for |κ| < 1/6 [ 3].Following refs. [ 8, 11], the Schro¨dinger functionalin tmQCD can be conveniently defined as the in-tegral kernel of an integer power T/a of the trans-fer matrix [ 3]. It has an Euclidean representationgiven byZ[ρ′, ρ′, C′; ρ, ρ, C] =∫fieldse−S[U,ψ,ψ¯] (2)and can hence be considered as a functional ofthe fields at the Euclidean times 0 and T . Thestructure of the transfer matrix implies that theboundary conditions for gauge and quark fieldsare the same as in the standard framework (µq =0). The action S[U, ψ¯, ψ] in eq. (2) is the sumof the usual plaquette pure gauge action and thequark action, which takes the same form as ineq. (1), provided one adopts the notational con-ventions of Subsect. 4.2 of [ 2].The tmQCD Schro¨dinger functional is expectedto be finite after the standard couplings andboundary renormalization [ 3]. At order anew boundary counterterms proportional to aµqarise [ 3]. Their effect beyond tree–level will beneglected in this study.22. Correlation functionsIn order to motivate our choice of observ-ables, we recall some properties of renormalizedtmQCD. For simplicity, we first consider the the-ory with boundary conditions that are periodicin space and as specified in [ 10] in time direc-tion. The renormalization scheme (R) of tmQCDcan be chosen [ 3] to be consistent with the Wardidentities of the flavour chiral symmetries:∂µ(AR)aµ = 2mR(PR)a + δa3iµR(SR)0 (3)∂µ(VR)aµ = −2µRεab3(PR)b (4)where the above relations are to be understoodas operator insertions2 in the correlation func-tions and ε is fully antisymmetric with ε123 = 1.Note that our chirally twisted parameterization ofQCD involves two renormalized mass parameters,mR and µR.It can be shown [ 1, 3] that, within a certainsubset of the renormalization schemes that pre-serve the above Ward identities, the on-shell cor-relation functions of tmQCD can be mapped ontothe ones of standard QCD in a related renormal-ization scheme. In the latter scheme the flavourchiral Ward identities (at the operator level) read:∂µ(AR)aµ = 2m′R(PR)a , ∂µ(VR)aµ = 0 (5)for a = 1, 2, 3 with the mass m′R satisfying:(m′R)2 = m2R + µ2R , tan(α) = µR/mR . (6)Here α is an unphysical angle which just specifiesthe quark mass parameterization. An example ofthis mapping is given by:〈(AR)20(x)(PR)2(y)〉c{gR,m′R, 0} =〈(A′R)20(x)(P′R)2(y)〉c{gR,mR, µR} , (7)where x 6= y and by definition:(A′R)aµ = cos(α)(AR)aµ + sin(α)εab3(VR)bµ ,(P ′R)a = (PR)a a = 1, 2 . (8)It is understood that in eq. (7) the l.h.s.must be evaluated at renormalized couplings2The quark bilinears are defined in the standard way, withflavour matrices τa/2 (for a = 1, 2, 3) or τ0 = 1.{gR,m′R, 0} and the r.h.s at renormalized cou-plings {gR,mR, µR}, with the quark mass cou-plings related via eq. (6). Note that the abovemapping takes the same form as at the classicallevel, where it is obtained via [ 1]ψ′cl = eiαγ5τ3/2ψcl , ψ¯′cl = ψ¯cleiαγ5τ3/2 .In the following we will also need to consider,again for a = 1, 2 only, the primed fields:(V ′R)aµ = cos(α)(VR)aµ + sin(α)εab3(AR)bµ ,(T ′R)aµν = (TR)aµν a = 1, 2 . (9)Coming back to our scaling tests of tmQCDwith SF boundary conditions, we focus on a fewrenormalized quantities that can be extractedfrom the following correlators:f22R,A′(x0) = −〈(A′R)20(x)O25〉/√f221 ,f22R,P′(x0) = −〈(P′R)2(x)O25〉/√f221 (10)and (with sum over k = 1, 2, 3 understood)k22R,V′(x0) = −13〈(V ′R)2k(x)Q2k〉/√f221 ,k22R,T′(x0) = −13〈(T ′R)2k0(x)Q2k〉/√f221 , (11)where Oa5 and Qak are the SF boundary fields de-fined in ref. [ 7] at zero Euclidean time3. In theabove correlators the division by the square rootof the quantityf221 = −1L6〈O′25O25〉 , (12)takes care of the boundary field renormalization.In the quantum mechanical representation ofthe renormalized correlation functions that areobtained from eq. (10) and eq. (11), the onlynon–vanishing contributions (up to cutoff effects)arise when inserting states with vacuum quantumnumbers at Euclidean times between x0 and T ,and states with pion and ρ–meson quantum num-bers, respectively, at Euclidean times between 0and x0. Let us consider for instance the SF cor-relator f22R,A′ : the matrix elements between inter-mediate states |k〉 and |n〉〈k| cos(α)(AR)20 − sin(α)(VR)10|n〉 , (13)3The analogous fields localized on the other SF time-boundary are labelled with a prime.3which enter its quantum mechanical representa-tion, are independent of the SF boundary condi-tions and also appear in the quantum mechanicalrepresentation of the r.h.s. of eq. (7). The equal-ity of the correlators in eq. (7) implies the equalityof their quantum mechanical representations forarbitrary values of (x0 − y0). From this equalityit follows that in tmQCD the matrix elements ofthe form (13) with the states |k〉 and |n〉 carry-ing vacuum and pion quantum numbers, respec-tively, are non-zero. Analogous arguments holdfor the remaining SF correlators in eqs. (10)–(11).The fields Oa5 and Qak are chosen so that the cor-responding SF boundary states have a non–zerooverlap with the pion and the ρ–meson states.3. Definition of the observablesIn this study we consider as observables someratios of the previously introduced SF finite vol-ume correlators that are expected to approach awell defined continuum limit4:mPS =∂˜0f22R,P′(x0)f22R,P′(x0), x0 = T/2 , (14)m˜PS =∂˜0f22R,A′(x0)f22R,A′(x0), x0 = T/2 , (15)mV =∂˜0k22R,V′(x0)k22R,V′(x0), x0 = T/2 , (16)m˜V =∂˜0k22R,T′(x0)k22R,T′(x0), x0 = T/2 , (17)ηPS = CPSf22R,A′(x0) , x0 = T/2 , (18)η˜PS = C˜PSf22R,A′(x0) , x0 = T/2 , (19)ηV = CVk22R,V′(x0) , x0 = T/2 , (20)η˜V = C˜Vk22R,V′(x0) , x0 = T/2 . (21)As T = 2L → ∞, mPS and mV yield estima-tors of the pion and ρ-meson mass, respectively.The constants CPS and CV are defined as in [7] in terms of mPS and mV. CPS is such thatηPS → Fpi as T = 2L → ∞. The quantity ηV isnot related to the decay of the ρ–meson because ofits unphysical normalization. Analogous consid-4 The symbol ∂˜µ denotes the symmetric lattice derivative.erations hold for the alternative scaling quantitieslabelled with a ”tilde”.The fR- and kR correlators entering our observ-ables are defined in eqs. (10)–(12) via eqs. (8)–(9)and(AR)aµ = ZA(1 + bAamq)(AI)aµ ,(VR)aµ = ZV(1 + bVamq)(VI)aµ ,(PR)a = ZP(1 + bPamq)(PI)a ,(TR)aµν = ZT(1 + bTamq)(TI)aµν , (22)where (restricting attention to a = 1, 2)(AI)aµ = Aaµ + cAa∂˜µPa + aµqb˜Aεab3V bµ ,(VI)aµ = Vaµ + cVa∂˜νTaµν + aµqb˜Vεab3Abµ ,(PI)a = P a ,(TI)aµν = Taµν + cTa(∂˜µVaν − ∂˜νVaµ ) . (23)For the definition of the bare lattice fields Aaµ,P a, V aµ and Taµν we follow Sect. 2 of ref. [ 9].The suffix I refers to the O(a) improvement ofthese fields, which together with O(a) improve-ment of the action, eq. (1), implies that -in thelimit T → ∞- the observables in eqs. (14)–(21)deviate from their continuum limit by O(a2) cut-off effects. The b˜ coefficients multiply countert-erms that are needed to subtract (bulk) cutoffeffects of order aµq.4. ResultsWe choose non–perturbative mass–independentrenormalization conditions for the couplings andthe observables of interest. As L/a is increased,the relation between β = 6/g20 and a/r0 [ 12] isemployed in order to keep the physical size of thebox fixed at L = 1.49r0, with r0 being a hadronscale of order 0.5 fm. The mass parameter κ istuned so to keep fixed the PCAC renormalizedmass in units of L:LmRdef=La{ZAZP∂˜0f22A2f22P}= 0.020 , (24)where:ZAf22A = −〈(AR)20(x)O25〉 ,ZPf22P = −〈(PR)2(x)O25〉 . (25)4The mass parameter µq is chosen so to fulfill thecondition:LµRdef= Z−1P Lµq = 0.153 . (26)This choice is justified by the exact lattice Wardidentity ∂∗µV˜2µ = 2µqP1, where V˜ aµ is the one–point split vector current that is conserved atµq = 0 and ∂∗µ denotes the backward latticederivative. The angle α, given by tan(α) =µR/mR, is close to pi/2.β µq κ L/r0 LmR6.0 0.01 0.134952 1.490(6) 0.0228(23)6.14 0.00794 0.135614 1.486(7) 0.0203(30)6.26 0.00659 0.135742 1.495(7) 0.0201(23)6.47 0.00493 0.135611 1.488(7) 0.0180(24)Table 1The bare and renormalized parameters in our sim-ulations. As for L/a and LµR see the text.The renormalization factors ZA, ZV and ZPwere determined in [ 6] and [ 13], where ZP isgiven as a function of β at the scale L0 = 1.436r0.As for the improvement coefficients, we employnon–perturbative estimates of cSW, cA [ 5] andbV [ 9]. Moreover, we use 1–loop estimates of cV,cT, bA, bP, bT [ 9] as well as of the SF-boundarycoefficients ct and c˜t [ 11]. The coefficients b˜A andb˜V vanish at the tree level and are of order 10−2at 1–loop level [ 3]. We have hence varied theirvalue in the analysis in the range −0.2 ÷ 0.2 forb˜A and −0.1÷ 0.1 for b˜V, without observing anystatistically significant variation in our results.Our simulation points are listed in table 1: thefour values of β correspond to L/a = 8, 10, 12, 16.The quoted error on LmR is purely statistical,due to the negligible uncertainty on ZA/ZP. AsZP ≡ ZP(L0) is known with a relative uncer-tainty of about 0.5%, the condition (26) is imple-mented with this precision by adjusting µq. Wecorrect for small mismatches with the conditioneq. (24) using numerical estimates of the LmR–dependence of our observables obtained via someextra simulations at β = 6. We finally extrap-olate our data to the continuum limit assumingconvergence with a rate ∝ a2. Our fits are shownin figures 1- 4, where only statistical errors are0.000 0.005 0.010 0.015 0.020(a/L)21.601.651.701.751.801.85mPSLmPSLmPSL~Figure 1. Scaling of mPSL and m˜PSL.0.000 0.005 0.010 0.015 0.020(a/L)22.202.302.402.502.602.70mVLmVLmVL~Figure 2. Scaling of mVL and m˜VL.displayed. The results are compatible with O(a2)deviations from the continuum limit.Because of the finite value of T ≃ 1.5 fm,the observables (14)–(21) still depend on the SFboundary action and fields. This induces residualO(a) effects due to the imperfect knowledge of ctand c˜t and a further coefficient5 that is associatedwith O(aµq) effects [ 3]. Following [ 7], we haveperformed a few extra simulations at β = 6 andβ = 6.26 with values of ct − 1 and c˜t − 1 that areabout 2 and 10 times, respectively, larger thanthe 1–loop values. The discrepancies with theprevious results are at most about two standarddeviations (at β = 6) and can be considered neg-5Due to LµR = 0.153 ≪ 1, this coefficient was set to itstree level value and never varied.5ligible for the purposes of this study.0.000 0.005 0.010 0.015 0.020(a/L)20.5200.5300.5400.5500.5600.5700.5800.590η PSLηPSLηPSL~Figure 3. Scaling of ηPSL and η˜PSL.0.000 0.005 0.010 0.015 0.020(a/L)20.1400.1500.1600.1700.1800.1900.2000.210η VηVηV~Figure 4. Scaling of ηV and η˜V.In table 2 we compare the estimated continuumlimit value of our observables with the value atβ = 6. The deviations from the continuum limitappear to be fairly small and not larger than theones found in an analogous study at µq = 0 [ 7].5. ConclusionsIn the parameter region specified by β ≥ 6,LµR = 0.153 ≫ LmR = 0.020 and T =2L ≃ 1.5 fm the O(a) improvement programme oftmQCD has been successfully implemented andmPSL mVL ηPSL ηV1.80(3) 2.62(4) 0.559(7)[1] 0.164(5)[1]3.7% 2.8% 2.6% 4.8%m˜PSL m˜VL η˜PSL η˜V1.66(1) 2.29(3) 0.581(5)[1] 0.200(6)[1]2.7% 4.4% 2.5% 7.7%Table 2Continuum limits and deviations from β = 6. Er-rors due to ZX–uncertainties in square brackets.tested for a few meson observables. The renor-malization of the twisted mass parameter is easyin practice, as it can be traced back to the renor-malization of the non-singlet pseudoscalar densityin the massless theory.This work is part of the ALPHA collabora-tion research programme. M. Della Morte andR. Frezzotti thank MURST for support. S. Sintacknowledges support by the European Commis-sion under grant No. FMBICT972442. We thankDESY and INFN for allocating computer time onAPE100 machines to this project.REFERENCES1. R. Frezzotti, P. A. Grassi, S. Sint and P.Weisz, Nucl. Phys. B (Proc. Suppl.) 83-84(2000) 941-946.2. M. Lu¨scher, S. Sint, R. Sommer and P. Weisz,Nucl. Phys. B478 (1996) 365.3. R. Frezzotti, P. A. Grassi, S. Sint and P.Weisz, to appear.4. K. Symanzik, Nucl. Phys. B266 (1983) 187and 205.5. M. Lu¨scher et al., Nucl. Phys. B491 (1997)323.6. M. Lu¨scher, S. Sint, R. Sommer and H. Wit-tig, Nucl. Phys. B491 (1997) 344.7. J. Heitger, Nucl. Phys. B557 (1999) 309.8. S. Sint, Nucl. Phys. B421 (1994) 135.9. S. Sint and P. Weisz, Nucl. Phys. B502 (1997)251.10. M. Lu¨scher, Commun. Math. Phys. 54, 283(1977).11. M. Lu¨scher, R. Narayanan, P. Weisz and U.Wolff, Nucl. Phys. B384 (1992) 168.12. M. Guagnelli, R. Sommer and H. Wittig,Nucl. Phys. B535 (1998) 398.13. S. Capitani, M. Lu¨scher, R. Sommer and H.Wittig Nucl. Phys. B544 (1999) 669.

Non-perturbative scaling tests of twisted mass QCD

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