One-dimensional QCD with finite chemical potential

Volume 212, number 1

PHYSICS LETTERS B

15 September 1988

O N E - D I M E N S I O N A L QCD W I T H F I N I T E C H E M I C A L P O T E N T I A L Neven BILI(~ and Kregimir DEMETERFI Ruder BoYkovi~ Institute, P.O.B. 1016, YU-41001 Zagreb, Croatia, Yugoslavia Received 2 June 1988

We consider a one-dimensional S U ( N ) model with Kogut-Susskind fermions in order to investigate the effects of finite chemical potential. We calculate the chiral condensate exactly and compare it with the quenched approximation.

In lattice calculations of the chiral condensate ( ~ / / ) at finite chemical potential one encounters difficulties associated with a complex fermion determinant [ 1,2 ]. In this case, a straightforward numerical simulation of an S U ( N ) theory with N > 2 becomes impossible since an ordinary Monte Carlo procedure requires a positive statistical weight. The complex Langevin simulation which seemed promising in the study of effective spinsystem-type theories [2,3] is not yet well founded, especially when applied to the full SU (N) theory [ 4 ]. Numerical calculations at # ¢ 0, therefore, have been performed mostly in the quenched approximation [ 5,6 ]. The results obtained in this way do not fit in with expectations. For SU (3) one would expect to find a threshold value/to= roB~3, with mB the mass of the lightest baryonic state. However, a comparison of the "measured" values for #o with known hadron masses in the strong-coupling limit [ 7 ] shows that instead of being related to the nucleon mass, #o is determined by #o = m J 2 , with rn~ being the lowest meson mass [6 ]. The critical value seems to approach zero as the quark mass is lowered. Gibbs has recently argued [ 8 ] that the reason for such an unexpected behavior can be the use of the quenched approximation. The argument is based on the study of a one-dimensional U ( 1 ) model. Gocksch has drawn a similar conclusion by studying an eight-flavor QCD in which the absolute square of the determinant enables numerical simulation [ 9 ]. In this paper we wish to address the problem of chemical potential along the lines of ref. [8 ]. However, as the model we consider one-dimensional SU (N) theory instead of U ( 1 ). This model can be integrated exactly in both quenched and unquenched cases and is physically more interesting for the following reasons. Firstly, unlike the U( 1 ) model, the S U ( N ) model contains baryonic excitation, and secondly, this model can be related to the mean-field approximation of a realistic d-dimensional theory at infinite coupling. Consider an S U ( N ) gauge theory on one-dimensional lattice with n sites described by the Kogut-Susskind action:

S = ½ ~ 2(i) (e u Ui,j(~i,j--I --e-U u~idi,j+ ~+ 2mdii)z(j) .

(1)

i,j= 1

The integration over one-component fermionic fields yields a determinant of the (n.N) × (n.N) matrix:

H,,=

mI - e-UuT2 \-½eUu~,l

½ e u u12 mI ½ e-U u~3 0

0 ½e uu23 mI 0

0 0

... ...

e

u,,,l

0 -½e-Uu~_l,~

(2) mI

/

The partition function can be brought into the form in which only one group integration remains: 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

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PHYSICS LETTERSB

15 September 1988

(3)

Zn= ~ dUdetDn(U), with D~(U) being an N × N m a t r i x : D~(U) = 2 ch(nm' ) + e -~u Ut+e~U U,

(4)

with m' = s h - l m .

(5)

The matrix U can be put into the diagonal form Ujk =ei°J 8jk.

(6)

In this case the normalized integration measure is given by 2n ~ I dO~) iA(eiO,, .... e i°~) 12 8(81 +...+ON) d U = ~.. =1 2zrJ

(7)

where A (x,..., XN) is the Vandermonde determinant [ 10 ]. After replacing the 8 function in (7) by 1

oo

8(0) = ~ k__~_o° e ik0 ,

(8)

we can easily perform the final integration by making use of the orthogonal-polynomial method [ 11 ], in the same way as was done for the U (N) group [ 12]. The result for SU (N) reads Z, = 2 ch(Nn#) +sh[ (N+ 1 )nm']/sh(nm'),

(9)

whereas for U ( N ) the first term is absent. The vacuum expectation of Z~ is then given by 1

~mln Zn = 1 { N c h [ ( N + 1 )rim'

In the limit n ~ ,

]/sh(nm' )-sh(Nnm' )/sh2(nm ' ) } / c h m ' .

(1o)

we obtain a threshold behavior:

(Z~) = 0 ,

/~>sh-~m,

=N/(l+m2)

1/2 ,

g
(11)

in contrast to the U( 1 ) as well as to the U ( N ) case, where there is n o / t dependence. In the quenched theory the chiral condensate is given by (Z~)q =

1

(12)

n f dU d-~ln[detD~(U) ] "

With the help of eq. (8) this reduces to a single integral, 2~

sh(nm' ) r dO N - ( - 1 )N.2 cos(N0) (77Z)q- - -chin - - - S - J 2rtch(nm')+cos(O-inlz)'

(13)

0

which can he easily evaluated using contour integration. The result is (Z~)q = (e Nn~m'-u)-e-Nn~m' +u) ) / c h m ' ,

/z> m'

=(N--eN'~(u-m')--e-Nn(u+m'))/chm ' , ~
(14)

Volume 212, number 1

PHYSICS LETTERS B

15 September 1988

Obviously, the quenching provides a threshold behavior even for finite n. In the limit n~oo, we again recover the full-theory result ( 11 ). To get a feeling of the n dependence, we plot (ZZ) for SU (3) as a function of g for different values of the product n m ' (fig. 1 ). We see that for n r n ' >~3, the quenched theory and the full theory essentially agree. The situation would be different for U (N); there we would find the threshold only in the quenched theory since in the full theory the/t dependence integrates out as in the U ( 1 ) example [ 8 ]. The above analysis could also be carried out for the one-dimensional eight-flavor theory [ 9 ]. In this theory we have to integrate the absolute square of the determinant of the matrix D, given by (4). The application of the orthogonal-polynomial method [ 11,12 ] involves slightly more algebraic manipulations. However, for large n, the result is very simple: 2N e 2Nnm' m ' e zNnrn" + e 2Nn# + e ( n )

(ZZ) = ch

,

(15)

where e(n) denotes the terms vanishing exponentially with n. It is remarkable that in this model eq. ( 15 ) holds for both SU(N) and U ( N ) gauge groups. Again, in the limit n~oo, we observe the same threshold at/Zo=m' for U ( N ) and SU(N) in both quenched and unquenched cases, thus confirming the conclusions resulting from the Monte Carlo simulation [ 9 ].

~

nmql

0.~ 1

2

®

z, LL

0.2 ^ 3

nm~ 3

u

1

1 ,

I

p 2

1

®

2.~ A x /

v

nm/_-10

I

/ /

1,2

/ / /

/

/ I

I

I

I

J

i

I

~t 2

Fig. 1. Chiral condensate as a function o f / t for three different values of nm' for the full theory (full line) and the quenched approximation (dashed line ). kt is given in units of m' = s h - Jm.

r

/

I

0.4

i

~°

I

0.8

i

~.

Fig. 2. Mean-field results for m = 0 and n=4, (a) Free energy as a function of the mean field 2 for the critical value/to=0.55. (b) Chiral condensate as a function of/~. The dashed line represents the instability region.

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It is not surprising that U ( N ) and S U ( N ) give the same threshold behavior in this m o d e l The two kinds of flavor, Zl and X2, described by Gocksch [ 9 ] transform according to the 3 and 3* representations of U (3) and consequently baryons can be built up of XI flavors and Z2 antiflavors. For example, the combination XIX122 can be made color singlet in both SU (3) and U (3). The eight-flavor model [9], therefore, contains baryons even in the case when the gauge group is U (3). The one-dimensional S U ( N ) model is related to the mean-field approximation of a d-dimensional S U ( N ) gauge theory at infinite coupling [ 13 ]. In this case we expect the theory to have a non-trivial chiral limit, i.e. the chiral condensate
F=NnAE/d - 1 - l n Zn,

(16)

where Z, is given by (9) and ( 5 ), with m replaced by m + 2 and 2 is an auxiliary mean field. The solution to the mean-field equation d F / d 2 = 0 gives the point of the chiral transition ;to which is related to the chiral condensate = 2 N 2 o / ( d - 1 ) .

(17)

The solution is particularly simple in the limit n--, oo. In this limit, the derivative of In Z is proportional to a step function O(x) and we have to solve the mean-field equation 2N2 ( d - 1 ) - N 0 [ s h - 1( m + 2 ) - g ] / [ 1

+ ( m +,~) 2 ] 1/2=0o

(18)

For m = 0, we find a second minimum at

2o={ [ ( d - 1 ) E + l ]l/2-1} l/2/x/~.

(19)

for all/t ~ sh - 12o. The corresponding threshold value of the chemical potential/to is obtained from the requirement F(0) =F(2o),

(20)

which yields /*o = s h - L~o-A2/d - 1.

(21)

In the large-dlimit, we recover the result ofBarbour et al. [6 ], ao_~ln(2d)~/2- ½,

(22)

as well as the result ao=0.55 for d = 4 . The results are not much altered at finite n. In fig. 2. we show the ~t dependence of obtained numerically for the full theory with n = 4, d = 4 and N = 3. The critical value/~o is given by the Maxwell construction corresponding to eq. (20). From our analysis we may draw the following conclusions: (a) The quenched approximation works well for SU (N) theory for reasonably large lattices (nm ~>3 ). Therefore the Monte Carlo simulation of the full theory is likely to give results similar to those obtained with the quenched approximation [ 6 ]. (b) The quenched approximation is invalid for U ( N ) gauge groups, similarly to the U ( 1 ) example [8 ]. The exception is the eight-flavor model with a U ( N ) gauge group [9 ] in which baryons can be built up of flavors and antiflavors. (c) The strange behavior of baryonic matter in the chiral limit observed in Monte Carlo simulations remains a puzzle. The naive expectation that the threshold value of the chemical potential should be at roB~3 is probably incorrect. The work of N.B. was supported in part by the US National Science Foundation under grant no. YOR 82/ 051. 86

Volume 212, number 1

PHYSICS LETTERS B

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References [ 1 ] J. Engels and H. Satz, Phys. Lett. B 159 ( 1985 ) 151. [2] F. Karsch and H.W. Wyld, Phys. Rev. Lett. 55 (1985) 2242. [ 3 ] N. Bili~, H. Gausterer and S. Sanielevici, Phys. Lett. B 198 ( 1987 ) 235; Brookhaven preprint BNL-40296 (1987), Phys. Rev. D, to be published. [4] F. Karsch, Nucl. Phys. A 461 (1987) 305. [ 5 ] J. Kogut, H. Matsuoka, M. Stone, H.W. Wyld, S. Shenker, J. Shigemitsu and D.K. Sinclair, Nucl. Phys. B 225 ( 1983 ) 93. [6] I. Barbour, N.-E. Behilil, E. Dagotto, F. Karsch, A. Moreo, M. Stone and H.W. Wyld, Nucl. Phys. B 275 (1986) 296. [ 7 ] H. Kluberg-Stern, A. Morel and B. Petersson, Nucl. Phys. B 215 ( 1983 ) 527. [8] P.E. Gibbs, Phys. Lett. B 182 (1986) 369. [9] A. Gocksch, Phys. Rev. D 37 (1988) 1014. [ 10 ] H. Weyl, The classical groups, their invariants and representations (Princeton U.P., Princeton, 1946 ). [ 11 ] Y.Y. Goldschmidt, J. Math. Phys. 21 (1980) 1842. [ 12 ] P.H. Damgard, N. Kawamoto and K. Shigemoto, Nucl. Phys. B 264 (1986) 1. [ 13] E.M. Ilgenfritz and J. Kripfganz, Z. Phys. C 29 (1985) 79.

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