qcd phase structure with 8 light quark flavors
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Nuclear Physics B (Proc. Suppl.) 26 (1992) 314-319 North-Holland
ith c$ht dqenerake fIavors ia &x&d on 16’ x Nt lstticcr with NC = 4, 8, 16 and 32.
mo = 0.015 WBI uucd in the simulation. In calculatin8 (j&) and hadron propagator, we of mo = o.oo4,0.01 and 0.025 in order to mtudy the cbiral property of the phma. A
t order cbiral qmlmet transition ir mead at Nt = 4 and 8 with diftcrtmt Be. However, &
mdependent of Nt when a bulk or sero tanperatw ph~e tramition with & = 4.59(2).
5.0 ir seen while in the r&on of we coupling with fi B /SC hadron propagatorn
ynunetric behavior.
reduction
11 order to understand better the physics of
it is important to dy the dependence
on number of qu ly calculations on Nr = 2, 3, 4[1] show a
creases. Work at
= 4 [2-61 and possibly bulk transition at Nt 1
6[3]. A recent calculation on M3 x 4 and 163 x 6
lattices[7] searched for the first order transition
110 5 m 5 rn, and showed that m, =
6, which is significantly larger than
= 4 case, where no first order behavior
= 0.025. On the Columbia 16 GigafIop Parallel Su-
percomputer, we have extended earlier work on
done in collaboration with Fkauk
u&r, Ho Chen, Robert Mawbin-
SW- Leo unger and Al
par%lly rugported by the Depart ment of Energy.
eight flavors to a smaller dynamical quark mass ma = 0.015. A more complete description of this
work can be found in a recent preprint[$].
Our calculation was done on lS3 x Nt lattice with Nr = 4,8, 16, and 32. We studied the tran-
sition region and determined’& for different Nt.
A larger value of p than PC was also studied for
large Nt. In order to get more information on the
chiral properties of the different phases, hadron
masses and (nx) was also calculated for valence
quark maesee m,,,,la = 0.004, 0.01, 0.025, and
0.05.
The simulation of QCD with Kogut-Susskmd
fermione was done using the R algorithm of Got-
tlieb et sL[9], with time step A7 = 0.0078125
and a molecular dynamics trajectory length 0.5
time unite. Calculations to assess the effect of finite time step at Nt = 8 euggeet we have con- trolled thie error.
The conjugate gradient method was used to
invert the Dirac operator. The stopping condi-
tion was chosen so that the reaidual, scaled by
the norm of the source, is lees than 1.13 x low5
for the inversion in the evolution of the gauge
&563 5.00 6 1992- Elsevier Science Publishers B.V. All rights reserved.
2. Lkmg et al. I QCD phase shucnuc with 8 light quark flavoJs 315
~....I....I....I....~...~ 0 100 200 300 400
7
Fig. 1. The Monte Carlo evolution of (m) at 0 = 4.60
atatting fkom an ordued gauge co&guration. The jump
ia interpreted (u tunneling from the mctastable weak cou-
pling phase to the stable strong coupling phase.
configuration, which results in 300 to 600 it-
erations. For the inversion used in calculating
hadron propagators we use 8 more stringent con-
dition, 2.21 x 10D6 , yielding 700 to 800 CC. it-
erations.
2. Strong First Order Transition and flG
Consistent with earlier calculations, we see
strong first order transition on 163 x 4 lattice.
However this strength continues for N* = 6 and
Nt = 16, where the action surprisingly shows a
discontinuity of more than 20% across the tran-
sition while (ax) changed by a factor of more
than 5 (See Table 3). In Fig. 1, we show a
strong tunneling event on 8 163 x 32 lattice at
p = 4.60, where metastable states persist for 8
few hundred time units. We 8lso did simulations
at p = 4.65 where the initially ordered and disor-
dered lattices continued for more then 300 time
units without tunnelling.(See Figure 2 of Nor-
man Christ’s contribution to this conference).
Next let UE discuss the critical value of p for
this transition. The method we used to deter-
mine PC is 8s follows. We choose a p so that
0 0 50 100 150
7
Fig. 2. Gmeration of the mixed ph+sse umfiguretion from
a dimordered rtart for N: = 4. Initially fi is +X5. We
to tune fi after 70 time units.
ordered and disordered starts give two stable
phases with different values of the action and
(Rx). We start from 8 configuration in one of
the phases, evolve it while tuning /I every 10 or
20 trajectories, so that we eventually cooked a
configuration with action and (2~) midway be-
tween the values of the two met&able phases.
Fig. 2 is the time history of (gx) obtained when
we generate a mixed phase configuration from an
Nt = 4 disordered statt.
Then from this single mixed phase configure
tion generated above, we evolve the system for
different values of 8. The (nx) time histories of
this series of evolution determines fle precisely.
Fig. 3 is the evolution of (2~) for five values
of p started from the mix configuration gen-
erated in Fig. 2. This figure shows that when
p increases the behavior of (j&) monotonically
changes from rapidly equilibrating in the strong
coupling phase to rapidly equilibrating in the
weak coupling phase. We identify PC 8s that
value of the p for which the mixed configuration
changes the slowest.
To check the consistency of our method, we
made another mixed phase from the initially OP-
dered Nt = 4 configuration. Fig. 4 shows the
(RX) evolution starting from this second mixed
2 Dang &al. lQCD phase stmchue with 8 light quarkjhvars
Fig. 3. Deruminetion of & for Nt = 4 from the mixed
co&uration generated in Fig. 2.
a d
q 0
3,
2 0 20 40
T
Fig. 4. Similar plot an Fi6. 3 but rtartiqg from a second
mixedp configuration generated from a ordered &art
lattice.
configuration. We can see that it gives the same
& as the first one. This indicates that our mixed phases have no inertia toward one phase and our
method is reliable.
With this method we determined PC for Nt =
4,8 and 16. They are respectively 4.58(l), 4.73(l) and 4.73(l).
The transition for Nt = 4 and Nr = 8 looks
very similar to the chiral symmetry restoring
transition seen for four flavor calculations. They
all have an N* dependent PC. This behavior could be easily interpreted as a finite temperature tran-
sition.
However it is very surprising to see the same PC
AT 0.002 0.005 0.0076 0.0125 1 eXact
Be 4.59(l) 4.64(l) 4.73(l) 5.29(l) 1 4.59(l)
Time rtep Ar end correspondiq & for a 165 x 8 lattice.
for Nt = 8 and Nt = 16 lattices. This indicates
that for a large enough space time volume the
transition has become a bulk or sero temperature transition with /3= independent of the lattice sise.
We conclude that for Ni = 8 there is a strong
first order bulk transition for N* 3 8.
3. Finite Time Step Effect
Since we are using an algorithm with Nl x
(a~)~ errors, and performing an unfamiliar back- wards leapfrog for Nj = 8, it is necessary to
check the finite time step effects. On the 163 x 8 lattice, we searched for PC us-
ing several different time steps. As a comparison,
PC was also determined with the exact Hybrid
Monte Carlo algorithm[lO]. The results are listed
in Table 1. For each PC search, we started from
the mixed phase configuration generated with
AT = 0.0078125. Except for the AT = 0.0125
point we can fit PC with
fjc(Ar) = 4.58(l) + 2460(25O)(Ar)’ (1)
where xa = 0.02. Fig. 5 shows that & extrap
olated to zero time step agrees with the exact
algorithm result within the error. This suggests
that we have controlled finite time step errors at
AT = 0.0078125. However when AT goes up to
0.0125 the algorithm breaks down since we can-
not get a good fit even when including higher
order terms.
Because PC depends significantly on the step
size, we decided to make a second test of the
effects of this finite time step by computing Nt =
Z. Dong et al. IQCD phase mcture with 8 light quarkjlavors 317
ml
ui
u- a
(D
4
9 *
0 5x 1 o-5 0.01 0.015
step size Ar
Fig. 5. & plotted (u a function of step rise on a l@ x 8
lattice. The triangle point ir the exact algorithm result.
8 and 16 lattices at a second time step AT =
0.005 in order to confirm the existence of a bulk
transition. The result PC = 4.62(l) for Nt = 16,
again agrees(within the error) with the Nt = 8
result at the same step size given in Table 1.
We therefore believe the finite time step doesn’t
affect the qualitative features of our calculation.
4. Chiral Properties of the Phases
In order to study the chiral properties of each
phase we calculated (RX) on both side of the
transition for Nt = 4 and 8 close to PC. At large
Nt we explored both the transition region and
p = C.O(about 0.3 above &), calculating hadron
masses as well as (RX). Table 2 shows the hadron
masses calculated from equilibrated lattices. Ta-
ble 3 contains results for (nx).
Prom the mass table we can see that, except
for the Goldstone pion, the bulk transition al-
tered the hadron masses by nearly a factor of
N 3. (j&) also dropped by a large factor from
the strong coupling to the weak coupling phase.
However this does not imply the restoration of
chiral symmetry, because we might expect (Rx)
to change by a factor 33 when masses change by
a factor of 3.
Table 2
Hsdron meaael calculated in
ond and third cohmm were &tam
weak coupling ph8ee~ with B = 4.65, while tbe forth u&
umn givea /3 =5.oInafwa.The~trowliIl~thenumber
of equilibrated time u&r wed in each a&d&on.
Table 3
RemIt of (a) for diRemxlt pheSu.
At p = 5.0 hadron masses show remark-
able parity doubling, behavior representing chi-
ral symmetry. In contrast, the Q - r doubling is
not obvious in the p = 4.65 weak coupling phase,
which suggests chiral symmetry is broken. Table
3 shows (xx) to be very small at Nr = 4 and
at /3 = 5.0 with large Nt. Note, however (Zx)
increased by almost a factor of two when Nr in-
creased from 4 to 16 at the same p.
One should study the m -+ 0 behavior of (Rx)
0
a function of m,a( for iVt = 4
6. The fits forced through origin with
x2 = 5.1 for nrt = 8 0.36 for Nt = 4.
cl
d t 0
% % - A
g” I# -! 3 0
d
0 0 0 5x10-= 0.01 0.015
mvala tted ea a function of mvol for
@ = 4.65. mara extrapolates
expected for a Goldstone pion.
. ’ .., . . . * , . . . . . . . . . 8
0 (~0)’ d
;3 2 ci d
0 0
mMla t for /3 zz 5.0, rimilar to Fig. 7. (2~) precisely
to WO indicating chir%l eynunetry.
I I I 10 0 0.02 0.04 0.06
mvala Fig. 9. A plot for the we coupling pham at /3 = 4.65,
mimilar to Fig. 7. (2~) nonlinearly extrapolatea to auo.
and the hadron spectrum to learn whether chiral
symmetry is spontaneously broken in each phase.
Instead, we have computed (2~) and the hadron
spectrum in each phase for a number of valence
quark masses, using the gauge configuration gen-
erated with fixed rrarcao = 0.015.
The valence quark calculations show that for
A$ = 4 and 8, (gx) is consistent with zero when
extrapolated to %,I = 0 (See Fig. 6). This leads
to the conclusion that for Nt 5 8 the transition
restores chiral symmetry. Fig. 7 shows (zx) and
mi as functions of m,,,,r, for the Nt = 16 strong
coupling phase at p = 4.65 . mi extrapolates to
sero as we expected for a Goldstone pion. The
fact that (xx) extrapolates to sero for p = 5.0
(Fig. 8) supports the conclusion that this sys-
tem is chirally symmetric. All these suggest our
quenched, valence quark calculation works well.
However in Fig. 9 for the p = 4.65 weak cou-
pling phase, we see (ax) curving to zero even in
the small mass region, a behavior we have never
seen before. In Fig. 10, we plot the dependence
of hadron masses on m,l in this phase. Both fig-
ures suggest that the chiral symmetry is restored
when m,l -+ 0. But compared to p = 5.0 and
smaller A& cases, the restoration is slower.
2. Dong et (II. IQCD phase smrctm with 8 Ii& qunrkjhm 319
0 0.01 0.02
mvtYio Fig. 10. Hadron maarer plotted an a function muol. The
open circlen are mv, the eolid circler mu, the open triem-
glee mp the open quarea ma,, the solid squarea mN and the rolid triangles are mNf. All parity partnera extrapo-
late to degenerate valuer as m,,l -+ 0.
20-
18-
16-
14-
g 12-
10-
8-
6-
4-
I I I/ I I I 1 I I I
24.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2
B Fig. 11. A poeaible eight flavor phase diagram. The strong
fint order tram&ion in represented by solid line. The
de+shed line indicater the continuation of the finite tan-
perature transition to iVt 2 6. Squarer are the points
where we did the simulation. Open rquares are our re-
rultr of &.
5. Conclusion
Based on the featurea doye, we conclude that
there is a finite temperature transition which re-
stores chiral symmetry when A$ 5 8 , Also a
region with chiral
fl= 5.0 on a lS3 x
chiral symmetry. Thii lattice
= 8 has a continuum lit similar to
which in consistent with a conventional picture
for the continuum lit of &flavor &CD. In this
figure, the first order transition se into a strong bulk transition (solid vertical line) and a continuation of the conventional chiial symmetry restoring finite temperature transition (dashed
lime) when Nt goes above 8. The unfamiliar chiral
behavior in the weak coupling phase at /3 = 4.85
may be caused by the relative amall spa&l size
of our lattice, i.e., l/&T, is on the order of I’, for p = 4.65. The presence of this $-flavor lattice ar-
tifact, a strong first order bulk transition, could
explain why one sees the transition strengthen- ing as AT/ increases from 2 to 8.
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PI
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