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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