QCD at finite baryon density

322

Nuclear Physics B (Proc. Suppl.) 4 (1988) 322-326 North-Holland, Amsterdam

QCD AT FINITE BARYON DENSITY

A. VLADIKAS *Department

of Physics and Astronomy,

The University,

Glasgow G12 8QQ, U.K.

Direct attempts to measure the chlral condensate in QCD wlth dynamical fermlons and non-zero chemical potential fall to predict the chlral symmetry restoration transition point. We propose an alternative method and present some early results.

1.

INTRODUCTION

2. NC SIHULATIONS WITH DYNAMICAL FERNIONS

The work reported here i s being done in

In o r d e r to r e s o l v e

c o l l a b o r a t i o n with lan Barbour of Glasgow U n i v e r s i t y and Clive B a i l l l e of Caltech.

the above d i s c r e p a n c y ,

we i n c l u d e Kogut S u s s k l n d f e r m i o n s i n t h e The

theory.

One t y p i c a l

MC ( m u l t i ) s w e e p c o n s i s t s

computers used are an IBM In Glasgow and the

of visiting

Crays XMP in Rutherford and San Diego.

l a p p i n g a r o u n d e a c h o f them 4 t i m e s ,

We

half

the lattice

have studied the c h l r a l symmetry r e s t o r a t i o n

each of i t s

links

t r a n s i t i o n a t (almost) zero temperature and

we u s e t h e

(exact)

non-zero chemical p o t e n t i a l in SU(3) with

calculate

dynamlcal fermlons.

and t h e d i f f e r e n c e

We have been constrained

hypercubes,

10 t i m e s .

hitting

While we u p d a t e ,

Lanczos a l g o r i t h m

the ratio

of f e r m i o n i c of the traces

[sJ to

determinants of the i n v e r s e

to very small l a t t i c e s and very low s t a t i s t i c s

o f t h e f e r m t o n m a t r i x M.

which, we hope, w i l l reveal the b a s i c mechanism

M-1 and M-2 f o r t h e h y p e r c u b e i n q u e s t i o n .

c h a r a c t e r l s l n g the t r a n s i t i o n , although they

Details

w i l l not give p h y s i c a l l y r e l i a b l e answers.

action

The chemical p o t e n t i a l u (in l a t t i c e u n i t s ) i s

fashion.

included by m u l t i p l y i n g l a t t i c e l i n k s in the

15 m i n u t e s on t h e Cray.

positive e~(e-U)

(negative) [1,2].

The objective is to evaluate

the chiral condensate

as an order parameter

can be f o u n d in [ 6 ] .

For SU(3),

for

Metropolis

such a multisweep takes

problem is that

f o r SU(3),

detM i s c o m p l e x when ~ 0 .

Since the

Metropolis

on c o n f i g u r a t i o n s

algorithm

relies

the restoration

of chlral symmetry with

being generated

increasing u.

This transition

we use W=ldetM{ exp(-Sg).

is expected to

end, we need

The W i l s o n

Sg i s u p d a t e d i n s t a n d a r d

The m o s t s e r i o u s

direction by a factor

To t h i s

via a real

positive

w e i g h t W,

W i t ht h i s weight,

occur at a critical value Uc=(mp/Nc)

(mp:= the

the condensate becomes a r a t i o of two

proton mass, Nc:= number of colours)

[2] but

observables

quenched approximation

( m . : = the pion mass)[3].

values

coincide

for

SU(3)

this

for Is

has been attributed approximation

w

results suggest a value

uc=m./2

SU(2) a

The two

(Nc=2, mp=mw) but

discrepancy,

which

to the quenched

[4].

=

(1)

w

w h e r e ~ i s t h e p h a s e o f detM. imaginary parts

denominator of the above r a t i o identically

[7].

Note t h a t

vanish

We p e r f o r m e d MC m e a s u r e m e n t s

o f t h e n u m e r a t o r and d e n o m i n a t o r o f eqn. b a r e q u a r k mass mq-0.1 and i n v e r s e * I.N.F.N. F r a s c a t l , C.P. 13, 00044 F r a s c a t i , Italy

0920-5632/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

the

o f b o t h t h e n u m e r a t o r and

6-1.5 on a 44 l a t t i c e .

(1) a t

coupling

We work in strong

coupling so as to be able to compare with

A. Vladikas / QCD at finite baryon density similar

results

[8] and a l s o

decrease

correlations.

updating

each hypercube

measurements/sweep) values

of u=0.3,

uncorrelated) present

distribution

(i.e.

for

0.9,

language of refs.[ 3,4]

1.0.

the massless

5

Because

function

fluctuating,

p of the numerator

In t e r m s o f p ( x ) ,

any observable

x c a n be w r i t t e n

the results

is

0/0 answer.

and

the average

of

for

X

xi

i=1

= I

the case where of eiffenvalues In Figure

with

the

a nd

In thls

as

the

indeterminate

The same b e h a v i o u r

u = 0 . 4 a nd 0 . 9 .

of

2 we

Since the distributions

with being symmetric,

occurs

region,

at

t h e mass i s

fCOS(~) 16

max

N 1

is

In the

the denominator

compatible

X

:=

this

the strip

at ,=0.5.

condensate

expectations.

fermion matrix.

are compatible

we

the normalised

denominator.

plot

numerator

( w h i c h w e r e f o u n d t o be

through

with naive

t h e ma s s i s o u t s i d e

32

5 s w e e p s and f o r

0.5,

were wildly

results

which agrees

to

We m e a s u r e a f t e r

0.4,

our measurements

in order

323

(2)

x p(x) dx

Xmin

1./.

for a large number of measurements

N.

In Figure I we show p(cos¢) for .=0.3.

Its

1"2

asymmetry resulted in our being able to obtain a condensate of about 200 (in lattice units),

1.0

5'0

0-8

e(cosm) 0"6 ~0

tto[, 3.0

cos(C) -1-0 Figure

20

-0"6 2a:

-0'2

0-6

Same a s i n F I g .

~(Re[ei# Tz M" ])

{

0-2

10

1 for .=0 5

,0"01 ,0008

10

1ooo6t

}

t

t

I i I

-.-oa ~-06' t-oL, , -oz-O~

I

I

01 0-2

I I

cos(m) 0/,,

06

08

1"0

f

0.001 Re[el# Tz Mm ]

Figure I: The normallsed distribution of measurements for the denominator of the condensate at u=0.3

-114 F i g u r e 2b: Same a s i n F i g . 2 a of the condensate

112 for

the numerator

A. Vladikas / QCD at jqnite baryon density

324

inside the strip of eigenvalues [3,4],

of M(mq=O)

and MC appears to be failing.

behaviour,

j~(Re[ei~Tz M-i]}

This

1.0

a result of a rapidly fluctuating

phase of detff, is in disagreement analysis of [7].

wlth the

But at u=l.0, Figure 3

suggests that the numerator has a symmetric distribution denominator

(i.e. it vanishes) does not.

0.6

but the

Thus the condensate

either very small or compatible with O.

Is

O.t,

In

this case the mass sits inside the cavity formed by the elgenvalues

[3,4].

The above results suggest that in the region , ½,

u¢(0.3,1.0),

the condensate

cannot be measured

-1.o

-o.~

~4

o

25 Figure 3b: by MC;

.p(cos ~ )

~. Re,ei#,z . ~ 1- H") 1.45

Same as in Fig.2b for u-l.0

this behaviour may signal that we are

near the critical

region.

At any rate, our

results differ from the quenched result

2.0

~c-0.3.

This is mildly encouraging.

3. THE DETERMINANT

EXPANSION

In order to deal wlth the above problems,

we

use the fact that detM can be expanded in terms of Wilson loops and Polyakov lines.

,1.5

We group

these expansion terms according to the number of time-like Polyakov lines they contain. Each such Polyakov

llne contributes

exp(Nt# ) or exp(-Ntu)

a factor

and thus we have

M

1.0

detM=Co+ ~ [c m=l

exp(mNt~)

+ c: exp(-mNt~)]

where M:= Ns3Nc in obvious notation. coefficients space-llke tlme-llke

O.S

contain contributions

loops and lines, as well as from

+t as in the -t direction.

coefficients eqn.(3).

-1.0

-~6

-0-2

0.2

0.6

because

This explains

of the +~ and -~ exponentials

The last terms'

Same as in Flg.2a for ~-1.0

coefficient

it consists of the determinant

product of all the time-like Polyakov

interpreted

as representing

the

of the

Each of the terms in the expansion Figure 3a:

The

from

loops which have as many links in the

complex conjugate relationship

cos @

(3)

m

in

is I, of the lines.

can be

a contribution

from

A. Vladikas / QCD at finite baryon density

quarks

325

(anti-quarks) winding around in the

TABLE 1

forward (backward) time direction m times. Now recall that under Z 3 transformations of the t-like gauge links, we obtain a valid SU(3) configuration which leaves the path integral measure and the pure gauge action invariant but changes d e t N .

The real part of the terms (C3m exp(3mNtu)} of the determinant expansion (eqn.(6)), for typical configurations at different values of u. The lattice size is 23x4. m 0

~--0.2 5.25 0.74E-1 0.32E-2 -0.48E-4 -0 11E-6 -0 32E-9 -0.16E-12 0.42E-15 0.14E-19

We may thus write the partition

function as follows I Z = ~ ~ [DU] exp(-Sg)

3 ~ det(Mi) i=1

(4)

where the 3 determinants differ by such a Z 3 transformation.

2 3 4 5 6 7 8

u=0.4 12 8 0.81E-1 0.37E-1 0.12E-1 0.20E-3 0.11E-5 0.55E-7 0.34E-9 0.25E-12

u=O.6S

u=0.5 0.08 0.30 0.81 -0.97 -0,17 -0 27E-2 -0,49E-4 0.75E-6 0 11E~7

0.75E-4 -0.40E-2 -0.14E-1 0.88 -0.86 -0.67 -0.20E-1 0.59E-2 0.53E-3

Thus, the problematic exp(u) takes over and more terms come into

denominator of eqn.(1) can now be obtained as 3

play.

3

These terms correspond to the lattice

being gradually filled up by quarks that wind i[DU]exp(_Sg) idetMll~, det M i - I d e t M11

=

<

~ - det M i > w I d e t M11

(5)

around 3,6,9 etc times in the standard picture of ref.[ 2], until at large enough u, the whole

with the same MC weight as before.

The

lattice is completely filled up (i.e. the last

important fact is that the sum of 3

term with exp(NtM~) dominates).

determinants which appears in eqns.(4) and (5), when evaluated using eqn.(3),

becomes:

values u~0.6 are characterlsed by cancelling

3 M ~detMi=3co+~. [C3m exp(3mNt~)+C;m exp(-3mNt~)](6) i =1 m=l

"noise" which is

If, however, the noise is

similar

u=0.4 v a l u e

lattice.

It

again

difficult.

To assess how relevant these considerations

even

at

when'dealing

with

a 44

may thus

be necessary

strength of each contribution from the positive (6).

Table I we show such contributions

from

clear

directions.

Then, instead of dealing with the

sum of 3 determinants,

our is given by Early trial

runs have shown that a few results obtained in

very different from those obtained in Section 2

expectations of [2].

Work which is currently

in progress along these lines will hopefully Note

clarify the situation.

t h a t f o r small u the z e r o t h o r d e r term c0 dominates.

Thus,

f o r t h i s small l a t t i c e we

and found it to agree with its ~=0 value. the strength of the powers of

ACKNOWLEDGEMENTS I would llke to thank my collaborators,

were a b l e to measure the condensate a t u=0.4

increases,

up

and possibly in accordance with the theoretical

In

equilibrium configurations at different f o r a v e r y small 23x4 l a t t i c e .

to

the

this way at a=0.4 on a 4 ~ lattice are certainly

are in practice, we examine the relative

exponential terms in the expansion

We

difficulties

the sum of 34 determinants.

cancelled expllcitly.

values,

is

trick of Z 3 group rotations in all four

statistics to cancel and may be hiding our

we evaluate using eqn.(6),

results

the signal from further noise by using the same

present in eqn.(1) and will require enormous

signal when evaluating eqn.(2).

reliable

smaller

The cancelling

terms of eqn.(3) are, however,

dominant terms, which means that obtaining

encountered

i.e. only loops that wind around the lattice 3,6,9 etc times contribute.

Unfortunately,

we see from Table 1 that the intermediate

As

Clive Baillie and Ian Barbour for their contributions to the above.

I am also

A. Vladikas / QCD at finite baryon density

326

grateful to Phil Gibbs, Christine Davies and Peter Hasenfratz for useful discussions.

5.

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I.M. B a r b o u r e t . (1986) 296.

4. P.E. Gibbs, "Understanding Finite Baryonic Density Simulations in Lattice QCD', Glasgow preprlnt, April 1986 (unpublished) and Phys. Lett. B182 (1986) 369.

al.,

Nucl.

P h y s . B275[FS17]

I.M. B a r b o u r e t . (1987) 227.

al.,

J.

Comput. P h y s . 68

6. C. B a i l l i e e t . 8 1 . , "The C h J r a i C o n d e n s a t e i n SU(2) QCD a t F l n l t e D e n s i t y " , G l a s g o w preprlnt, to appear in Physics Letters. 7.

B. Berg e t .

al.,

8.

E. D a g o t t o e t . (1986) 1292.

Z. P h y s . al.,

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