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FUB-HEP/93-21 HLRZ-93-93 1 December 1993 (cid:3) The Nambu-Jona-Lasinio model with staggered fermions a a a;b c b;a c A. Ali Khan M. Go(cid:127)ckeler R.Horsley P.E.L.Rakow G. Schierholz and H. Stu(cid:127)ben a Ho(cid:127)chstleistungsrechenzentrum HLRZ, c/o Forschungszentrum Ju(cid:127)lich, D-52425Ju(cid:127)lich, Germany b Deutsches Elektronen-Synchrotron DESY, Notkestra(cid:25)e 85, D-22603Hamburg,Germany c Institut fu(cid:127)r Theoretische Physik, Freie Universita(cid:127)t Berlin, Arnimallee14, D-14195 Berlin, Germany We investigate the neighbourhood ofthe chiral phase transition in alattice Nambu{Jona-Lasinio model, using both Monte Carlo methods and lattice Schwinger-Dyson equations. 1. INTRODUCTION We report the results ofan investigationintoa lattice Nambu{Jona-Lasinio model. For further details and references the reader is referred to [1, (a) 2]. TheNambu{Jona-Lasiniomodelisapurelyfer- mionicmodel. The interaction of the particles is (b) givenbyachirallyinvariantfourfermioninterac- tion [3] Z (c) (cid:8) S = d4x (cid:22)(x)(cid:13)(cid:22)@(cid:22) (x) h(cid:0) (cid:1)2 (cid:0) (cid:1)2io (cid:0) g0 (cid:22)(x) (x) (cid:0) (cid:22)(x)(cid:13)5 (x) :(1) Figure1. TheSchwinger-Dysonequationsforthe Our investigationwas motivatedby the fact that fermionpropagator. four fermion interactions have become popular again. They appear for example in technicolour Hybrid-Monte Carlo method. Additionally nu- models,top-quark condensate models [4]and ex- merical Schwinger-Dyson techniques were app- tensions of QED. lied. To enable a direct comparison we studied In their original Hartree-Fock calculation [3] the same lattice version of (1) in both cases. Be- NambuandJona-Lasiniofoundthat forlargeva- causechiralsymmetryisimportantintheoriginal lues of the coupling g0 the fermion mass (cid:22)R is model we used staggered fermions in our lattice generated dynamically and that there exist two action, since they allowa continuous chiral sym- bound states of a fermion and an antifermion. metry on the lattice. The lattice action reads There is scalar state with mass 2(cid:22)R and a mas- 1X sless pseudoscalar state which is the Goldstone- S = (cid:17)(cid:22)(x)[(cid:31)(cid:22)(x)(cid:31)(x+(cid:22)^)(cid:0)(cid:31)(cid:22)(x+(cid:22)^)(cid:31)(x)] 2 boson associated with the spontaneous break- x;(cid:22) X down of chiral symmetry. + m0 (cid:31)(cid:22)(x)(cid:31)(x) Inordertoachieveanunderstandingofthechi- x X ralphasetransitionthatgoesbeyondtheHartree- (cid:0) g0 (cid:31)(cid:22)(x)(cid:31)(x)(cid:31)(cid:22)(x+(cid:22)^)(cid:31)(x+(cid:22)^) (2) Fockapproximationwe studied the modelby the x;(cid:22) (cid:3)CombinedtalksofP.RakowandH.Stu(cid:127)ben where (cid:17)(cid:22)(x) = ((cid:0)1)x1+(cid:1)(cid:1)(cid:1)+x(cid:22)(cid:0)1; (cid:17)1(x) = 1. For 2 pagator. Theequationsaspresented thereareex- act, but must be truncated to give a numerically tractable system of equations. The (cid:12)rst approxi- mation (known as the gap equation) keeps only the termlabelled (a). This is the leadingtermin the usual 1=N expansion. The next approxima- tion is to keep (a) and (b). We refer to this system as the O(g0) equations. Finally we keep all three terms but approximate the full four-point function by the bare four-point vertex. This ap- 2 proximationisreferred toastheO(g0)equations. The two methods have di(cid:11)erent strengths and weaknesses. The Monte Carlo yields results wi- Figure 2. The Monte Carlo data for the chiral thout any approximations,but can only be used condensate and our (cid:12)t. Symbol shape denotes on relatively small lattices. Comparing the re- bare mass (diamonds 0.09, triangles 0.04, squa- sults of the methods we were able to check that res 0.02 and circles 0.01). 124 lattices are black theapproximationsmadeintheSchwinger-Dyson points, 83(cid:2)16 white. equations are reasonable. Knowing this we could con(cid:12)dentlyusetheSchwinger-Dysontechniqueto go to large and even in(cid:12)nite lattices. the MonteCarlotreatmentthefourfermionterm We measured the chiral condensate h(cid:31)(cid:22)(cid:31)i, was rewritten in a quadraticformbyintroducing an auxiliary(cid:12)eld (cid:18)(cid:22)(x)2[0;2(cid:25)): p X h i(cid:18)(cid:22)(x) Sint = g0 (cid:17)(cid:22)(x) (cid:31)(cid:22)(x)e (cid:31)(x+(cid:22)^) x;(cid:22) i (cid:0)i(cid:18)(cid:22)(x) (cid:0) (cid:31)(cid:22)(x+(cid:22)^)e (cid:31)(x) : (3) In Fig. 1 we represent graphically the Schwinger-Dyson equations for the fermion pro- Figure3. AcomparisonbetweentheMonteCarlo dataforthechiralcondensate andthethree trun- Figure 4. A plot of the chiral condensate (cid:27) cations of the Schwinger-Dyson equations. The against the renormalised fermion mass. Symbols 3 lattice size is 8 (cid:2)16,m0 =0:02. as in Fig.2, lines as in Fig.3. 3 the renormalised fermion mass (cid:22)R, energies of fermion-antifermion composite states and the pion decay constant f(cid:25). With these results at hand we considered renormalisationgroup (cid:13)ows. P M 2. THE CHIRAL CONDENSATE AND CRITICAL COUPLING First we looked at the chiral condensate in or- dertomapoutthephasediagram. Usinga(cid:12)tan- satzthatwasbasedonamodi(cid:12)edgapequationwe determinedthecriticalcouplingtobegc (cid:25)0:278. M K P The data and (cid:12)t are shown in Fig. 2. In Fig. 3 we compare Monte Carlo results for the chiral condensate with the predictions of three di(cid:11)erent truncations of the Schwinger- Figure6. TheSchwinger-Dysonequationsforthe Dyson equations. The agreement improves as meson propagator P. more terms are included, and the (cid:12)nal curve is only slightlyshifted fromthe true results. shows that the data lie on a nearly universal The fermion mass (cid:22)R behaves similarlyto the curve. The curves are the predictions of the gap chiral condensate. A plot (Fig. 4) of h(cid:31)(cid:22)(cid:31)i vs. (cid:22)R 2 equationandoftheO(g0)Schwinger-Dysonequa- tions. We see that the predictions lie in a very narrow band. The di(cid:11)erences between the curves are caused by (cid:12)nite size e(cid:11)ects and a small residual dependence on bare mass. As can be seen by comparing the white and black squares in Fig. 2 (cid:12)nite size e(cid:11)ects are rat- her severe in the Nambu{Jona-Lasinio model. It is therefore fortunate that by analysing the Schwinger-Dyson equations we can (cid:12)nd a usea- ble description of the (cid:12)nite size e(cid:11)ects. For large lattice size Ls Figure 5. Finite size e(cid:11)ects for the chiral con- 4 3 densate on Ls (solid points) and Ls (cid:2)2Ls (open Figure 7. Energy levels for the (cid:25) channel. The points) lattices. dotted line is the threshold E =2(cid:22)R. 4 3. SPECTROSCOPY In the meson channels we measured propaga- tors for the local bilinear operators and for wall sources. We compared Monte Carlo results with the solution of the Schwinger-Dyson equations sketched in Fig. 6. The extra information from the wall sources allowed us to (cid:12)nd not only the ground state energies but also the (cid:12)rst excited state. In Figs. 7 and 8 we compare our Monte Carlo results with the levels predicted (with no adjustable parameters) in the Schwinger-Dyson approach. 0 To understand the meson sector properly we Figure8. Energylevelsinthe(cid:25) =(cid:27)channel. Solid 0 need to consider the in(cid:12)nite volume limit. (This curves are the (cid:25) levels (the s-wave channel) and is particularly important for the levels above the the dashed curves are the (cid:27) levels (p-wave). threshold E =2(cid:22)R.) Therefore in Fig.9 we have 3 plotted the (cid:25) ground state energies on increasin- h(cid:31)(cid:22)(cid:31)i(cid:0)h(cid:31)(cid:22)(cid:31)i(1)/((cid:22)R=Ls)2 exp((cid:0)(cid:22)RLs)(cid:17)(cid:1):(4) gly large lattices (83 (cid:2) 1, 123 (cid:2) 1, 203 (cid:2) 1 3 and 52 (cid:2)1). For the largest lattice (thin so- Thisequationcanbetested byplottingthechiral lid lines) we have also shown the excited states. condensate from di(cid:11)erently sized lattices against 4 Also shown are Monte Carlo data from 12 and (cid:1), as in Fig. 5, (if the equation holds the data 3 8 (cid:2) 16 lattices. On an in(cid:12)nite lattice the ex- should lie on straight lines). Figure 9. Pion energy levels in increasing volu- Figure 10. The energy levels of the scalar meson. 3 mes. (20 (cid:2)1 lattice, m0 =0:02). 5 Figure 12. The spectral function for the scalar 0 particle (dashed curve) and the (cid:25) (solid curve). The scalar is a narrow resonance near the threshold. (Note that the two curves are shown at di(cid:11)erent scales.) state. In the whole broken phase the (cid:27) mass is about 2(cid:22)R. Using the Schwinger-Dyson equations we can also investigate the limit of in(cid:12)nite lattice volume,wherethecloselyspacedenergylevelslying abovethe threshold goover intoacontinuum. In this case we can de(cid:12)ne a spectral function (cid:26). In Figure11. Thepionspectralfunction. Inthe(cid:12)rst Figs.11and12weshowspectralfunctionsforthe picture, fromthe broken phase, the pionis atrue pion and scalar channels. We see delta functions boundstate andthe spectral functionhas adelta when a true bound state exists, and resonances function at E = m(cid:25). In the second diagram the when a state is heavy enough to decay into two pionis a resonance, able to decay into afermion- fermions. Wefoundthe massesandwidthsofthe antifermionpair. resonances by locating poles on the second Rie- mannsheet in complexenergy [5]. cited states form a continuum. In the symme- The vector meson ((cid:26)) also turned out to be tric phase there isa resonance inthis continuum, a resonance. In contrast to the (cid:25) and (cid:27) which which we have plotted as a thick line. On the become massless at the critical point the (cid:26) does 3 52 (cid:2) 1 lattice there is a sequence of avoided not become light. Its mass scales with the cuto(cid:11) level crossings These are signatures of the reso- when the critical point is approached. nance on a (cid:12)nite lattice. For the pseudoscalar meson ((cid:25)) we found a light bound state in the p 4. RENORMALISATION GROUP TRA- broken phase. This mass vanishes like m0 as JECTORIES expected from chiral perturbation theory. In the symmetric phase the (cid:25) turned out to be a reso- Using the results from spectroscopy we ob- nance. The energy levels of the scalar meson ((cid:27)) tained renormalisation group (cid:13)ows, i.e., contour obtainedbytheSchwinger-Dysonmethodarealso plots of dimensionless ratios of physical quanti- showninFig.10. The(cid:27)isaresonance (boldline) ties in the plane of the bare parameters m0 and if g0 (cid:25) gc and in the symmetric phase. Deep in g0. In particular we looked at the ratios m(cid:25)=(cid:22)R thebrokenphaseitbecomesaveryweaklybound and (cid:22)R=f(cid:25), Fig. 13. We found that the lines 6 Figure 13. A comparison between the (cid:13)ows of Figure14. Them(cid:25)=(cid:22)R (cid:13)owonanin(cid:12)nitelattice. constantm(cid:25)=(cid:22)R (solidlines)andconstant(cid:22)R=f(cid:25) (Dotted line resonance, solid bound state.) 3 (dotted lines) on 8 (cid:2)16 lattices. of constant m(cid:25)=(cid:22)R (cid:13)ow into the critical point while the lines of constant (cid:22)R=f(cid:25) end on the m0 axis. The two sets of lines cross everywhere in the area around the critical point. Hence there cannot be lines of constant physics in this re- gion which implies the absence of renormalisa- bility. Some caution has to be exercised in in- terpreting a (cid:13)ow diagram measured on a (cid:12)nite lattice. As we have seen the ground state energy in the pion channel is a true bound state in the broken phase but in the symmetric phase it cor- responds toanunboundfermionantifermionpair Figure 15. The (cid:22)R=f(cid:25) (cid:13)ow on an in(cid:12)nite lattice. with an energy strongly in(cid:13)uenced by the lattice volume. Wehavethereforerepeatedtherenorma- REFERENCES lisationgroup trajectory onin(cid:12)nitelattices using the Schwinger-Dyson equations. The resulting 1. A.AliKhan, M. Go(cid:127)ckeler, R.Horsley,P.E.L. (cid:13)ow diagramsare shown in Figs. 14 and 15. The Rakow,G.SchierholzandH.Stu(cid:127)ben,Spectro- (cid:13)ow lines cross, so there are no lines of constant scopy and Renormalisation Group Flow of a physics. Lattice Nambu{Jona-Lasinio Model, Preprint FUB-HEP/93-14, HLRZ-93-51(1993). 2. H. Stu(cid:127)ben, Ph.D. Thesis, Freie Universita(cid:127)t Berlin (1993). 5. ACKNOWLEDGEMENTS 3. Y. Nambu and G. Jona-Lasinio, Phys. Rev. This work was supported in part by the Deut- 122 (1961) 345. sche Forschungsgemeinschaft. The numerical 4. W. Bardeen, C. Hill and M. Lindner, Phys. calculations were performed on the Fujitsu VP Rev. D41 (1990) 1647. 2400/40 at the RRZN Hannover and the Crays 5. R.J. Eden, P.V. Landsho(cid:11), D.I. Olive and at the ZIB Berlin and the RZ Kiel. We thank all J.C. 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