Volume 110B, number 3,4 PHYSICS LETTERS 1 April 1982

THE HARTREE APPROXIMATION AS A HIGH-DENSITY LIMIT

Enrique ALVAREZ Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA

Received 16 November 1981

By doing a formal series expansion of the parti t ion funct ion in terms of the inverse o f the chemical potential associated with baryon number conservation, ~z, it is shown that the dominan t terms for large baryon densi ty correspond to the Hartree, or mean-field, approximation.

1. Introduction. The problem of finding good approximation schemes for the study of matter at high densities is a very interesting one from the astrophysical point of view: there are at least two places in the universe (namely, the neutron star cores, and the primeval cosmological matter *~ ) where densities several times greater than the nu- clear one are predicted by the standard models.

When matter is described by an asymptotically free theory (such as QCD for quark matter), the renormalization group equations lead to the introduction of an effective coupling constant depending on the baryon density n (or the chemical potential, tt) say g(/l) [1 ], which at high densities decreases when/x grows, so that in fact, ordinary perturbation theory can be applied to those problems [2,3]. Of course this does not mean that even in those theo- ries the perturbative expansion gives all the information relevant for large/x; there could be processes similar to the tunnelling effect which do not show up at any order.

In nonrelativistic many-body theory a typical high-density approximation is the Hartree (or Hartree-Fock) one, which physically consists in assuming that the correlations can be neglected, so that if we add two particles to the system, they will propagate completely independently of each other. In fact this kind of approximations have been applied to quark matter [4,5]. The aim of this note is to show directly from the partition function, that the Hartree (mean-field) expansion always dominates at high densities for a large class of physical systems.

2. The partition function in a Yukawa model. In what follows, a model of fermions interacting via a scalar par- ticle exchange (by a Yukawa-like coupling) will be considered. None of the results, however, will crucially depend on the details of the model; only the fact that the fermionic degrees of freedom can be integrated away is essential.

The partition function can be represented as a functional integral ,2.

Z(J, rT, ?7) = f ~ ~ ~ ~ CD ~ exp ( f d4 x [L (~) + J'~ - ~ (i ~ + m + t~'~ ) ~ + g~ ~ + ~ ~ + ~rl ] ). (2.1)

In terms of Fourier-transformed variables ,3 ~0(k) = f exp (ikx)¢(x) d4x the operator which acts quadratically on

,1 There is an impor tan t difference, however: the ratio o f the localization to the thermal energy, nlla/kT (n being the baryon num- ber density) is very big in the neu t ron star case (so that the mat ter is in a q u a n t u m degenerate state), bu t not so in the second, because in terms of the Rober t son-Walker scale factor, n ~ R -3 and kT ~ R -2.

,2 This integral is expressed in euclidean space, E4, with convent ions such that (-rta , 3,v) = - 2 ~ v , x 4 ----- ixo. L(~o) is the lagrangian density for the scalar field; uP is the unit (u 2 = - I in E4) four-vector defined by the mean value o f the baryonic current. The integral symbol means f d4x = fo ~ dx 4 f d3x.

t3 When the temperature T 4: 0, only discrete values o f k 4 (~ 2~rn/#) can appear, owing to the periodicity o f ~0 in euclidean t ime ~o(x4 + ~) = ~o(x4). In order to avoid unnecessary complications, we shall work in the sequel at T = 0.

315

Volume 110B, number 3,4 PHYSICS LETTERS 1 April 1982

the Fermi fields is

- (2rc )4D(p ,q ) - (~ + m +/a.t~)f(4)(p - q) + [g/(27r) 4 ] g/tP - q), (2.2)

so that if the gaussian integral over those Fermi fields is made, the parti t ion function appears as a single functional integral.

Z(J, r l , f l ) = f C-l) tpexp ( - t r f d4p d4q logD(p - q ) - f d 4 p d4q ~(p)D-1 (p - q) r l (q )+ fd4x[L (~o )+ J'~0]) . (2.3)

The inverse operator D -1 (p - q) can be expressed in terms of two new functions (to be defined later), A and A u, a s

- D - l ( k - l) = (270 4 {[m - (/¢ + u.~k)l / [m 2 + (k + U'u) 2 ] }6(4)(k - l) + A(k - l) + A ( k - l), (2.4)

The auxiliary functions A(k - 1) and

A o ( k - 1) - [(qp + la'uo)/(q + U'u) 2 ] Z(k - l)

are to be obtained in terms of the functional integration variables ~ k ) , by means of the integral equation with two kernels ,4, K and N:

f d 4 s A ( k - s )K({@, s - q) =N({¢} , k - q), (2.5)

2 ( k - q) - m A ( k - q) + (mg/[ (q + Uou) 2 + m 2] }~p(k - q) + i g ~ f d4s A(k - s)~o(s - q). (2.6) \ ~ - - ]

By making use of this expression for the inverse quadratic operator, (2.4) and by functionally deriving the for- mula for the part i t ion function, (2.3), algebraic relationships can be obtained relating, for example, the two-point function to the mean values of the constructs A and Au:

(~ (p)~(q) ) = - ( 2 ~ ) 4 ([m - (/b + U't~)]/[m 2 + (p + p ' u ) 2 ] }6(4)(p - q) + (A(p - q)) + (4t(/9 - q)). (2.7)

3. Expansion in terms o f e - 1/la. Although this last equation is an exact one (no approximations a t all have been made up to now), it is not very useful until (2.5) is solved [then (2.6) would give automatically A o as an in- tegral] , allowing (3,) and b t u) to be expressed in terms of mean values of the physical fermionic and bosonic fields.

In fact this can be easily done to lowest order in e (corresponding to the high-density limit). The result is:

A(p - k) = eZtp(p - k) + O(e3), (3.1)

Z ( p _ q ) = _ e 2 ( 2 m g g / ( p _ q ) + g2 f d 4 s c ( p _ s ) ~ ( s _ q ) ) + O ( e 3 ) . (3.2) (21r) 4

Using those values of A and 2; in the computat ion of the two-point function in (2.7), a Hartree result is obtain- ed: the medium behaves as a noninteracting fermionic plasma composed of particles with an effective mass m e given by a self-consistent equation. Explicitly , s .

,4 The explicit expression for the kernels is: g2

K( (~}, s - q) ~ 2g ~(s - q) + f d4t ¢(s - t)~(t - q) + {[(q + u.u)2 + m2 ] /m}6(4)(s - q), (27r)4m

N({~p}, k - q) ~ (2~r)4g (q + la'u)2 - rn2 g2 ~0(k - q) f d4u ~0(k - u) ¢(u - q). m (q + ta .u)2 + m 2 (q + ta .u)2 + m 2

,5 It should be kept in mind that in the real Minkowski space, the true (grand canonical mean values) Green functions of the sys- tem are to be obtained from the auxiliary ones used all over the text by replacing k p by k p - ~auP.

316

Volume 110B, number 3,4 PHYSICS LETTERS 1 April 1982

(ff~) = [m e - (/b + O't~)]/[m2e + (p + ~ ' u ) 2 ] , m e -- m + (@. (3.3)

4. Asymptotic expansion. In this section an alternative, less convincing, method for obtaining high-density ex- pansions will be sketched. As a matter of fact, it has been noticed long time ago in a different context [6] that if the thermodynamic potential ~2 - - k T l o g Z is very big [for example, if ~2(/3,/~) = (~//J0)f2(fl,/J0)], then the cor- relations are negligible, i.e. the nonconnected four-point functions are dominated by its uncorrelated parts*6"

(¢J(z) ¢J(y) ¢J(t) ¢J (x)) : (/~ 2/t~2)[(¢J(z)~(x))~°)(¢~(y) ~(t))~ °) - (~(z)~(t))~°)(¢J(y) ~ (x))~ °)

+ (@(t) ff(x))~°)(if(z) qj(y))~0) ] + (/j//~0) (if(z) ~(y) ~(t) ¢(x)>~ °). (4.1)

Using standard arguments [7], this expression can be shown to lead to the Har t r ee -Fock approximation. A first difference thus appears already here: in the regime (4.1) direct and exchange effects are equally important, whereas in the former expression (3.3) only the Hartree term did appear.

Such a physical situation [i.e. f2 = ~//~0)f2(°)] would be expected to correspond (when/~ -+ oo) to a system with a partition function given by (2.1) if the functional integration variables were bosonic ones: we would sim- ply do a saddle-point approximation around the stationary points of the action.

There is always possible to build up another (different) partition function ,7, with the same stationary points as (2.1), by partially bosonizing the fermionic variables, in which case the expansion described above could be ap- plied.

It is clear, however, that such a procedure is plagued with ambiguities. Moreover, as has been already pointed out, the two systems (described by Z and Z) are not fully equivalent; they are so only in the classical limit. For all these reasons, we found the approach leading to (3.3) much more clean and convincing.

5. Conclusions. It has been formally shown that for a wide class of theories, the dominant terms in the two- point function at high enough density correspond to the Hartree approximation.

All this formalism applies equally well to a gauge theory, with more algebraic complications. In a situation like QCD, however, since the mean value of the gluonic field is assumed to be zero, (A~) = 0, the first nontrivial result appears only at the next order in e.

It is worth stressing, finally, that the Har t r ee -Fock approximation (not Hartree only) is exact at zero coupling [7] ; this means that when the plasma is almost noninteracting; i.e., when the Debye length - the effective range of the interaction in the medium - is much shorter than the mean interparticle distance, that approximation is necessarily good. The Debye length, )~D, is a quantity that must be computed using some approximation, so that this criterion is a self-consistent one. If the zero-temperature perturbative value for ~k D is used, then QCD is not ,8 quasi free at high density (in the sense defined above), because )~D ~ [Pg~)] -1 ~ n-l~3 (logn)l/2. In fact, this can be the origin of some problems in the perturbative treatment of a QCD plasma [8].

It is a pleasure to acknowledge very stimulating discussions with Dr. D. Gross, Dr. J.L. Lucio-Martinez and Dr. E. Witten. This work was supported by a Fulbright-MUI grant.

+6 The subscript "c" stands for connected; the superscript "0" means that ~t =/~0. ,7 For example for h~b 4 theory, a new partition function corresponding to an action which leads to the same classical equations

of motion as the usual theory is: - - + 1 2 Z(J,~,~)~fCD~oQ)tpC~CDx(Jgn~expfd4x[-~(i~+m)~ ,u tr ~X ~o(a~au- ms)~O

- x~¢ 2 + g x ¢ + ] . ~ + ~ + ~ n + ~(x - ~ ) + ½x~21. ,s We want to correct here the erroneous statement made on tiffs subject in previous papers [5].

317

Volume 110B, number 3,4 PHYSICS LETTERS 1 April 1982

References

[1] G. Baym, Les Houches lectures (1978). [2] P.D. Morley and M.B. Kislinger, Phys. Rep. 51 (1979) 64. [3] E.V. Shuryak, Phys. Rep. 61 (1980) 71. [4] E. Alvarez and R. Hakim, Phys. Rev. D19 (1979) 1696. [5] E. Alvarez, Phys. Lett. 98B (1981) 140; Phys. Rev. D23 (1981) 1715. [6] C.M. Bender, F. Cooper and G.S. Guralnik, Ann. Phys. (NY) 109 (1977) 165. [7] L.P. Kadanoff and G. Baym, Quantum statistical mechanics (Benjamin, New York, 1961). [8] D. Gross, R.D. Pisarski and L.G. Yaffe, Rev. Mod. Phys. 53 (1981) 43.

318