We study the four-dimensional SU(3) gauge model with a fundamental and an
adjoint plaquette term in the action. We investigate whether corrections to
scaling can be reduced by using a negative value of the adjoint coupling. To
this end, we have studied the finite temperature phase transition, the static
potential and the mass of the 0^{++} glueball. In order to compute these
quantities we have implemented variance reduced estimators that have been
proposed recently. Corrections to scaling are analysed in dimensionless
combinations such as T_c/\sqrt{\sigma} and m_{0^{++}}/T_c. We find that indeed
the lattice artefacts in e.g. m_{0^{++}}/T_c can be reduced considerably
compared with the pure Wilson (fundamental) gauge action at the same lattice
spacing.Comment: 36 pages, 12 figure

A. Vaccarino

B. Beinlich

C. Borgs

C. Liu

C. Morningstar

C.J. Morningstar

CP-PACS collaboration

D.W. Bliss

H. Wittig

H.B. Meyer

K. Dietz

L. Susskind

M Hasenbusch

M. Laine

M. Lüscher

M. Minami

M.J. Teper

R. Gupta

S Necco

UKQCD collaboration

Y. Iwasaki

English

arXiv

Martin Hasenbusch

Silvia Necco

Blum

Heller

Hasenbusch

Borgs

Alves

Iwasaki

Boyd

Beinlich

Lüscher

Necco

Meyer

Teper

Vaccarino

Crossref

Lattice artefacts in SU(3) lattice gauge theory with a mixed fundamental and adjoint plaquette action

SU(3) lattice gauge theory with a mixed fundamental and adjoint plaquette action: Lattice artefacts

Talk presented at Lattice2004 (improved) Fermilab, June 21-26, 2004We investigated SU(3) lattice gauge theory with a fundamental and adjoint plaquette term in the action. The purpose is to test whether the choice of a negative adjoint coupling can reduce lattice artefacts and improve the scaling b ehaviour. To this end, we have studied the finite temperature phase transition, the static potential and the mass of the 0^{++} glueball. We found that indeed the lattice artefacts in e.g. m_{0^{++}}/T_c can be reduced considerably compared with the pure Wilson (fundamental) gauge action at the same lattice spacing

Hasenbusch, Martin

Necco, Silvia

HAL AMU

Lattice artefacts in SU(3) lattice gauge theory with amixed fundamental and adjoint plaquette action.Martin Hasenbusch, Silvia NeccoTo cite this version:Martin Hasenbusch, Silvia Necco. Lattice artefacts in SU(3) lattice gauge theory with a mixedfundamental and adjoint plaquette action.. CPT-2004/P.052, SHEP-0428. Talk presented atLattice2004 (improved) Fermilab, June 21-26, 2004. 2004. <hal-00002834>HAL Id: hal-00002834https://hal.archives-ouvertes.fr/hal-00002834Submitted on 14 Sep 2004HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.ccsd-00002834, version 1 - 14 Sep 20041CPT-2004/P.052SHEP-0428Lattice artefacts in SU(3) lattice gauge theory with a mixed fundamentaland adjoint plaquette actionMartin Hasenbuscha, Silvia Neccob∗aDepartment of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UKbCentre de Physique The´orique, CNRS Luminy, F-13288 Marseille – Cedex 9, FranceWe investigated SU(3) lattice gauge theory with a fundamental and adjoint plaquette term in the action. Thepurpose is to test whether the choice of a negative adjoint coupling can reduce lattice artefacts and improve thescaling behaviour. To this end, we have studied the finite temperature phase transition, the static potential and themass of the 0++ glueball. We found that indeed the lattice artefacts in e.g. m0++/Tc can be reduced considerablycompared with the pure Wilson (fundamental) gauge action at the same lattice spacing.1. INTRODUCTIONThe choice of the gauge SU(3) action plays animportant role in QCD lattice simulations, andmany efforts have been spent in searching for aformulation where discretisation errors are reducedand/or topological dislocations are suppressed. Inthis work we investigate a gauge action containingplaquettes in both fundamental and adjoint repre-sentation. The motivation for this choice is thatin the (βf , βa) plane, the pure SU(3) gauge theoryin four dimensions has a line of first order phasetransitions with an endpoint at [ 1](βf , βa) = (4.00(7), 2.06(8)) . (1)The presence of this transition line causes largelattice artefacts; in particular, the mass m0++ ofthe lightest glueball 0++ is expected to go to zeroas the end-point is approached [ 2]. As a relicof this behaviour, the estimate of m0++ in somephysical unit at βa = 0, for 5.5 < βf < 6.0 is muchsmaller than the continuum result. The purposeof this work is to study whether the discretisationerrors can be reduced by choosing a negative valueof βa, i.e. by moving away from the transition line.∗Supported by TMR, EC-Contract No. HPRNCT-2002-00311 (EURIDICE)2. THE ACTIONWe consider lattice SU(N) gauge theory with amixed fundamental-adjoint action,S = βf∑P[1− 1NReTrfUP]+ (2)βa∑P[1− 1N2TrfU†PTrfUP],where βf , βa are the fundamental and adjoint cou-plings and UP is the elementary plaquette. Weadopted 4-dimensional lattices with spatial exten-sions aNs and temporal extension aNt, with pe-riodic boundary conditions in all directions. TheYang-Mills theory is recovered in the naive con-tinuum limit if the bare coupling g0 is defined by6/g20 = βf + 2βa.Details on the simulation algorithm and its per-formance are given in [ 3].3. THE FINITE TEMPERATUREPHASE TRANSITIONFor a lattice withNt points in the time direction,in the limit Ns →∞, the deconfinement tempera-ture is given by1Tc= Nta({βf , βa}c) , (3)where {βf , βa}c indicates the critical coupling. Inour study we kept fixed the adjoint coupling to2Table 1Numerical results for the finite temperature phasetransition obtained for the fundamental-adjointgauge action.Nt; βa 0.0 –2.0 –4.02 5.0948(6) 6.4475(6) 7.8477(6)3 5.5420(3) 7.1603(3) 8.8357(4)4 5.6926(2) 7.4433(3) 9.2552(6)6 - 7.8056(5) 9.7748(11)βa = 0,−2.0,−4.0 and determined βf,c for Nt =2, 3, 4, 6 by adopting the method discussed in [ 4].Our results for βf,c are summarised in Table 1;for βa = 0 we find a substantial agreement withother results present in the literature [ 5, 6, 7, 8].For βa = 0, Nt = 6 we performed no own simu-lation, but used in the following the result βf,c =5.89405(51) of ref. [ 6].4. THE STATIC POTENTIALIn order to fix the scale we computed the staticpotential (at zero temperature) and extracted thestring tension a2σ. The static potential has beencomputed through the Polyakov loop correlationfunction:aV (r) = − 1Nt[log〈P (x)∗P (y)〉+ ǫ] , (4)where y = x+ r1ˆ and ǫ is the correction due to ex-cited states in the string spectrum. Here we useda large temporal extension aNt >> 1/Tc to ensurethat finite-Nt corrections are negligible; in particu-lar we adopted Nt = 6/(aTc) for all computations.The string tension has then been evaluated fromthe ansatzV (r) = σr + µ− π12r(1 +br), (5)where b has been recently shown from theoreticalprinciples to be zero [ 9].2 Moreover, we adoptedthe tree-level improved distance rI , defined suchthat the force at tree-level has no lattice artefacts.In order to compute the static potential up to largedistances by keeping the statistical uncertainties2In our evaluation we considered b = 0.04fm, which wasevaluated numerically in [ 10]. Nevertheless the effect ofhaving b 6= 0 is only minor and not crucial for our compu-tation of σ.under control, we adopted a variant of the algo-rithm proposed by Lu¨scher and Weisz [ 10]. Themain difference consists in using factorisation inthe spatial directions, in addition to the one in thetemporal direction. The full details of the proce-dure are given in [ 3]. For 1/(aTc) = 4 we were ableto extract a2σ from the force at r/a = 7 (βa = 0);in the other cases our final values were taken fromr/a = 4 and r/a = 6. In all computations wewhere confident the quoted error also covers pos-sible systematic uncertainties. The numerical re-sults for a2σ are reported in [ 3].The dimensionless quantity Tc/√σ as functionof the lattice spacing is plotted in fig. 1, togetherwith other values obtained for the Wilson action[ 8]. We notice that for βa = −2 and −4 theestimate for Tc/√σ is closer to the continuum limitthan for βa = 0. However, the difference between1/(aTc) = 3 and 1/(aTc) = 4 is larger than thatfor the different values of βa at fixed 1/(aTc).5. THE 0++ GLUEBALL MASSThe mass of the lightest glueball is expected tobe most sensitive to the choice of the action inthe (βf , βa) plane. The 0++ glueball mass hasbeen computed through the connected correlationfunction between spatial Wilson loops, by adopt-ing substantially the same method used in [ 11].Also for the glueball 2-point function we made useof an error reduction procedure based on the ideaproposed in [ 10], and already applied for the com-putation of glueball masses [ 12]. We extracted theglueball masses at t/a = 2, 3 for 1/(aTc) = 2, 3 andt/a = 3, 4 for 1/(aTc) = 4, 6. Fig. 2 shows our fi-nal results for m0++/Tc as function of (aTc)2; herethe errors are dominated by the uncertainties onm0++ . By averaging several results in the litera-ture [ 13, 14, 15, 16] for m0++r0 and then usingthe continuum limit relation [ 11]Tcr0 = 0.7498(50), (6)we obtainm0++/Tc|a=0 = 5.73(9) (7)as estimation of the continuum result. At a ≃0.11fm we do observe a moderate reduction of thelattice artefacts by using βa < 0 with respect to3Figure 1. Results for Tc/√σ as a function of(aTc)2. In addition, we report results from [ 8]for the Wilson action at smaller lattice spacings.the usual Wilson action (βa = 0). For βa = 0,the deviation from the continuum result of eq. (7)amounts to ∼ 18%, while for βa = −2,−4 itslightly decreases to ∼ 12%.At a ≃ 0.17fm one observes discretisation errors of∼ 40% for the Wilson action, while for the mixedaction they amount to ∼ 25% for βa = −2 and∼ 20% for βa = −4.6. CONCLUSIONSWe investigated scaling properties of a SU(3)lattice gauge action with plaquette terms in thefundamental and in the adjoint representation,with negative adjoint coupling βa. By studyingthe scaling behaviour of the quantity Tc/√σ forthe different βa at our disposal, we did not observea significant improvement at negative adjoint cou-plings in comparison to the Wilson case βa = 0.The values obtained with negative βa are a lit-tle closer to the continuum limit. As expected,the mass m0++ of the lightest glueball is moresensitive to the variation of βa. We investigatedthe scaling behaviour of the dimensionless quan-tity m0++/Tc. Here indeed, we observed a signifi-cant reduction of the lattice artefacts for negativeβa. At a ≃ 0.17 fm, the lattice artefacts for βa = 0Figure 2. m0++/Tc for βa = 0 (filled squares)βa = −2 (open squares) and βa = −4 (triangles)as function of (aTc)2. The circle gives the contin-uum results extracted from the literature.are 40%, while for βa = −4 they decrease to 20%.REFERENCES1. T. Blum et al, Nucl.Phys. B 442 (1995) 3012. U. M. Heller, Phys.Lett. B 362 (1995) 123.3. M. Hasenbusch and S. Necco, JHEP 0408(2004) 005.4. C. Borgs, Int.J.Mod.Phys. C 3 (1992) 897 andrefs. therein.5. N. A. Alves et al, Nucl.Phys. B 376 (1992) 218.6. Y. Iwasaki et al, Phys.Rev. D 46 (1992) 4657.7. G. Boyd et al , Nucl.Phys. B (1996) 469.8. B. Beinlich et al Eur.Phys.J. C 6 (1999) 133.9. M. Lu¨scher and P. Weisz, JHEP 0407 (2004)014.10. M. Lu¨scher and P. Weisz, JHEP 0109 (2001)010, JHEP 0207 (2002) 049.11. Silvia Necco, Nucl.Phys. B 683 (2004) 137.12. H. B. Meyer, JHEP 301 (2003) 48, JHEP 0401(2004) 030.13. M. J. Teper, hep-th/9812187.14. A. Vaccarino and D. Weingarten,Phys. Rev. D60 (1999) 114501.15. C. J. Morningstar and M. J. Peardon,Phys.Rev. D 60.16. C. Liu, Commun.Theor.Phys. 35 (2001) 288.

Lattice artefacts in SU(3) lattice gauge theory with a mixed fundamental and adjoint plaquette action.

SU(3) lattice gauge theory with a mixed fundamental and adjoint
  plaquette action: Lattice artefacts

SU(3) lattice gauge theory with a mixed fundamental and adjoint plaquette action: Lattice artefacts

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