Table Of ContentFUB-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 sy-
stem 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 cur-
ves 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 thres-
hold. (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 vo-
lume,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
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(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
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