Chiral Symmetry in Nuclear Matter

ANNALS OF PHYSICS 188, 61-l 19 (1988)

Chiral Symmetry in Nuclear Matter

W. BENTZ, L. G. LIU, AND A. ARIMA

Department of Physics, Faculty of Science, University of Tokyo.

Hongo, Bunkyo-ku, Tokvo 113. Japan

Received May 20, 1988

We use the chiral c--w model at zero temperature to study the role of chiral symmetry in nuclear matter. The central role played by the pion propagator in connection with the energy density of nuclear matter and the problem of chiral phase transitions is pointed out. Results obtained previously are reexamined from the standpoint of chiral symmetry. Consequences of baryon current conservation and the renormalization of the neutral vector boson field in connection with many-body problems are also treated in detail. .?I 1988 Academic Press, Inc.

1. INTRODUCTION

Since the works of Lee and collaborators [ 1 ] on abnormal nuclear states there has been much interest in chiral models for nuclear matter and nuclei (see, for example, Refs. [2-g]). In particular, recent works [S-7] stressed that chiral models provide insights into the mechanism of saturation in nuclear matter. From the theoretical viewpoint, chiral symmetry provides important constraints on strong interaction processes not only in free space [lo] but also, in a similar way, in the nuclear medium, and it is important to explore these constraints further and to study their consequences. Among chiral models, the linear CJ model [ 11, 123 is the simplest one, and therefore it has been widely used as a means to demonstrate chiral symmetry constraints both in free space [ 1 l-131 and in the nuclear medium [5, 6, 8, 281.

In this work, we will start from the (T model lagrangian and add a neutral vector boson (w meson) in order to account for the short range repulsion between nucleons. In such a model, which we will call the chiral C-O model, nucleons interact via exchange of scalar (a), vector (o), and pseudoscalar (rr) mesons. Besides its inclusion of pions, it differs from the more popular non-chiral g-o model [ 14, 151 (called QHD I in Ref. [ 161) in nonlinear meson interaction terms (like a3, a4) which are required by chiral symmetry. The quantity we are mainly interested in is the energy density of nuclear matter. The aim of our investigation is threefold: First, we want to set up the general formalism consistent with symmetry constraints, which in principle enables one to calculate the energy density, including the effects of quantum fluctuations. We will use the results of Refs. [ 11, 121 to discuss the renormalization of the c model in nuclear matter. Second, we wish to

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62 BENTZ, LIU, AND ARIMA

elucidate some general properties of the energy density which mainly follow from chiral invariance. In particular, we will generally demonstrate and explore the fact that the energy minimization condition is equivalent to the Goldstone theorem in nuclear matter. That is, if the energy density is minimized with respect to the nuclear matter expectation value of the scalar field, either the pion propagator has a pole at q=O (Goldstone mode) or the nuclear matter expectation value of the scalar field vanishes (Wigner mode). The stability conditions for the two modes can also be formulated in terms of the pion propagator. These features will enable us to discuss the chiral phase transition to the abnormal state studied previously [ 1, 9,281 more transparently from the standpoint of chiral symmetry. Our third purpose is to study the form of the energy density in specific approximations. It is well known that the simple-minded Hartree approximation is, strictly speaking, not applicable in the present model due to the appearance of “tachyon poles” in the meson propagators [l, 3, 5, 17,281. For this reason we attempt to discuss at least the formulation of the problem beyond the Hartree approximation. The numerical calculations based on an approximation to the Hartree-Fock scheme will be discussed in a different paper [18]. In order to obtain numerical estimates in the framework of the Hartree approximation, it is common to add higher order loop terms to the Hartree result in a somewhat ad hoc manner. We will discuss results obtained by other authors and give some numerical estimates for the case of adding the pions to the scenario of Refs. [ 1, 5, 61. We emphasize, however, that it is not the main purpose of this paper to present numerical results. Instead, it is to explain general and rather formal aspects of the chiral g--w model in nuclear matter and to illustrate these general considerations using simple examples. In this sense, the present paper should be considered to provide a basis for quantitative numerical works.

The paper is organized as follows: The general formalism is set up in Section 2, where special emphasis is placed on the renormalization procedure consistent with chiral symmetry and baryon current conservation. In Section 3 we discuss some general properties of the energy density, in particular its relation to the Goldstone theorem. There we also present some considerations on the Landau-Migdal force in the present model. Section 4 gives the form of the energy density when particular approximations are introduced. In Section 5 we use the Hartree approximation to illustrate some of our general results and to present some numerical estimates. A summary is given in Section 6.

2. RENORMALIZATION OF THE CHIRAL ~-0 MODEL IN THE PRESENCE OF NUCLEAR MATTER

In this section we explain the formalism needed to discuss renormalizations in the presence of the nuclear medium. We emphasize the role of the hypothesis of the partially conserved axial vector current (PCAC) and of the conservation of the baryon current. As for the treatment in free space we follow Refs. [ 11, 123.

CHIRAL SYMMETRY IN NUCLEAR MATTER 63

2.1. Lagrangian and Renormalization Constants

If we add the o meson interacting with the nucleon to the chiral D model lagrangian [ 111, we have in terms of unrenormalized quantities (characterized by a subscript (0))

- 2 = yco,Ci v - go(dco, + in,,, . fY5) - ETWOY,, yo,1 y’,o,

+; C@,d,oJ2 + GQ,o,)~I -f Gto,,,$?,ib”,

-~(i~o)+x:,,,~-~(~:o)+~:u))2

+m+ VO),, yo, + cobto,. (2.1)

Here Y, 4, n, and VP are the fields of the nucleon, the B meson, the pion, and the o meson, respectively, and G@‘” = LJp v” - aV VP. The model contains the three coupling constants g, g,, and ,I. The last term in (2.1) explicitly breaks the chiral symmetry, which leads to PCAC,

with

a,& = -cOfqo) (2.2a)

The invariance of the lagrangian under global phase transformations of the nucleon field leads to baryon current conservation

aj;=o (2.3a)

with

.G = ~co)Y~(oj. (2.3b)

The assumption PL:, < 0 leads to spontaneous symmetry breaking. The c and the w meson fields have (constant) expectation values in nuclear matter, which we denote by 17 and V, respectively. (Here the tilde indicates that these quantities are density dependent. As the density goes to zero we have 9’ + 0 (see Eq. (2.36) below) and fi + u.) Accordingly we translate the fields as follows,

ho, = O(O) + 60 (2.4a)

P 10) = qo, + %. (2.4b)

595/188/l-5


Furthermore, we introduce the renormalized fields, coupling constants, and mass parameters according to

Y(O) = Jz YY; (G(O)> v”,, R(O)) = & (a, 6, XL

Pi= p2 + 6p2 z,;

m2 - rnt + srnt,

wo - ZW

z, Z

go = ‘N & g’ g~o=zN~gwl

Co=&.

(2Sb)

(2%)

Here we introduced the wave function renormalization constants Z,, Z,, and Z,; the mass counterterms 6~’ and 6rni; and the vertex renormalization constants z,, &IN and Z,. Putting Eqs. (2.4) and (2.5) into (2.1) we obtain up to density independent c-number terms

with

2& = -; 12Z,(iT2 - u2)2 - 4 +-p; (62 _ u2)

+c(v”-u)+ mt, + bm2

2 w w3, (2.7a)

L$ = qiz, !u - tiN - 662, - gZ,(a + in. tys)

- &oz,,Y,(~p + WI y

+; [zM(d,o)2 - (6; + hi?;) 0’1

+~[z,(a,n)‘-(a:+6A:)nZ]--aZ,G,,~~

+ m2, + drnt,

2 c!Jp~ -; A2Z,(02 + lc2)2 - A2Z~kT(fJ2 + x2)

.%- = a(c - iY(rC~ + Sri?~)) + co,W(m$ + dmt).

(2.7b)

(2.7~)

In Eqs. (2.7) we delined the density dependent mass parameters and their counterterms as follows:


riii, + &iii, = gz,lT (2.8a)

62; + s6i; = p2 + 6p2 + 312Zj,C2 (2.8b)

62; + s@ls, = /.i2 + is/l2 + L2Z, 2. (2.8~)

Equations (2.8), without tildes, define the masses mN, m,, and mrr of the nucleon, the sigma meson, and the pion together with their counterterms for zero density. Following Ref. [ 121, we consider these masses as free parameters and define the coupling constants g and I’ by the relations

m,=gv

m2 - m2 = 2A202. ” n

(2.9a)

(2.9b)

Then Eqs. (2.8) for zero density determine the vertex renormalization constants Z, and Z, in terms of the mass counterterms as

Z,,l+dm, mN

cSrn~-drn3 Z,=l+ m2-m2

n L

(2.10a)

(2.10b)

Equation (2.8a) can be written in the form gN + c%, = mN + dm, + gZ,(C - v). This relation, together with similar relations for the cr meson and the pion, can be used to derive separate relations for the mass parameters and the counterterms:

fi,=gfi=m,! V

c-l; = nz; + 317 f? - v2)

FE’, = m’, + L2(iT2 - v2)

(2.11a)

(2.1 lb)

(2.1 lc)

and

b+zN=g(ZR-1)3=6mNj V

(2.12a)

SEz~ = drni + 3L2(Z, - 1 )( C* - v2) (2.12b)

662~ = Srnt + A2(Z, - I )(I? - v2). (2.12c)

Summarizing up to this point, if we impose renormalization conditions to determine the mass counterterms dm,, drnz, and am: in free space, the vertex renormalization constants and the density dependent mass parameters and counterterms are obtained from Eqs. (2.10) to (2.12).

In Eq. (2.6) we split the lagrangian into three parts: YM, is a constant depending


on the “mean fields” 0” and t?. The lagrangian YL consists of terms with at least two field operators. This part will generate the loop terms in the energy density. The last piece &., is linear in the fields 0 and oP and, as we will discuss below, generates counterterms in order to cancel contributions from loop terms to the nuclear matter expectation values of CT and op.

Let us now discuss the renormalization conditions. For this we consider the self- energies z,, C,, z,, and ,YE of the nucleon, the CJ meson, the pion, and the o meson in the nuclear medium as calculated from 6pL of Eq. (2.7b). To exhibit the contributions due to the mass and wave function renormalization counterterms, we split them as

Z,(k) = C,(k) + c%zi, - f(Z, - 1) (2.13a)

Z,(k) = C,(k) + S&~ - k2(Z, - 1) (2.13b)

C,(k) = C,(k) + S&i;: - k2(Z, - 1) (2.13~)

C:(k) = Z:(k) - gp” bmt, - (kpkY - k’g”‘)(Z, - l), (2.13d)

thereby introducing the “unrenormalized” self-energies z‘. In Eq. (2.13a) we introduced /%” = kp - gc,,Zg,/ZN G@. The renormalization conditions are imposed on the zero density values (denoted by a subscript f) of the self-energies as follows:

2,,(;9 = PN) = O,

(2.14~)

Here the renormalization points are denoted as pI (a = N, (T, 7c, 0). In the case where the above expressions are complex (which happens, e.g., for ,u: > 2mz), the real part must be taken. Cj,j is the transverse part of the w meson self-energy in free space:

(2.15)

The longitudinal part of the self-energy (2.15) is not an independent dynamical quantity: Baryon current conservation (2.3) requires that (see Ref. [19] or Appen- dix C of Ref. [20])

k,D”“(k) = m2 :lrn2 0, u

(2.16a)


for the renormalized o meson propagator. Due to the Dyson equation (2.20d) given below, Eq. (2.16a) is equivalent to

k,C$+‘(k) = -k” drn$, k&f,“(k) = 0, (2.16b)

and hence Clfj= -hrni, independent of k. The quantity dm2, is finite [19,20], but the bare o meson mass is infinite, as can be seen from the second equation in (2Sb). (Note that Z;’ diverges.)

The seven conditions (2.14) determine the mass counterterms and the wave function renormalization counterterms of Eqs. (2.13) in free space. Let M, (a = c, o, n, N) be the pole positions of the propagators in free space. Taking the pion as an example, we see that Mz is related to the parameter rn3 in the lagrangian by

m2 = M2 - ZJM’). I[ n n (2.17)

If we choose the renormalization points as p1 = m,, then the propagators in free space have poles at k2 = mf = Mz with residues equal to one (except for the 0 meson, which can decay into two pions). In this case the m, are the physical masses.

It is still necessary to determine the renormalization constant Z,, for the oNN vertex. Here we follow Refs. [ 19,201 and define the coupling constant g, such that the oNN source vertex r;,(q) in free space (f) satisfies

cd(q) -+ -&WY” as q+O,#+mN, (2.18)

where q = p‘ - p is the momentum transferred by the o meson to the nucleon. Then the Ward-Takahashi identity for baryon current conservation determines Z,, in terms of 8mZ, and Z,:

z,,=z, l+g$ . ( > w

(2.19)

Equation (2.19) is derived in Appendix C of Ref. [20] and is equivalent to Eq. (3.11) in Ref. [19].

Now all renormalization constants are fixed, and in the next section we discuss the determination of the mean fields i? and K+‘. Before that, however, we write the Dyson equations for the single particle Green functions for later use: In terms of the unrenormalized and renormalized self-energies, the single particle Green functions satisfy the Dyson equations

- - s=s()+&~,s=&+s,c,s

A,=~,,+6,,~,A,=A,,+A,o~:,A.

A,=a,,+a,,c,A,=A,,+A,,C,A,

Dp” = &” + &,~~~,l, D”’ = DoMv + D&&,,f, D”‘.

(2.20a)

(2.20b)

(2.2Oc)

(2.20d)


The lowest order unrenormalized propagators in these equations are

a,, = 1

Z,k2-r%-i%; (c( = ?r, a)

&y(k) = 1 kpk’ 1

Z,k2-mt,-6mZ, +-

k2 rn: + drnz,’

In Eq. (2.21a) we introduced [16]

(2.21a)

(2.21b)

(2.21c)

(2.22)

where n(k) is the Fermi distribution function. The lowest order renormalized propagators So, dmo, and Dg” are obtained from Eqs. (2.21) by setting the wave function renormalization constants equal to one and the mass counterterms equal to zero.

Since the unrenormalized o meson self-energy satisfies Eq. (2.16b), we see that the terms proportional to kpkY in D;’ do not contribute in Eq. (2.20d). In the actual calculation we can therefore replace

as long as we consider products with Z’,. Equation (2.20d) then becomes

DP=AP’Y- w

(2.23)

(2.24)

where A, satisfies A: = g”” a,, + a,oL?,p, A;. (2.20d’)

The second term in (2.24), although necessary to make Dpy finite, will not contribute in our subseqeunt discussions, since eventually either it drops in the energy density after subtracting the vacuum expectation value, or the product with C, is involved. In the nuclear medium there can occur mixing of the 0 and w meson degrees of freedom; i.e., the self-energies C, or 2, contain intermediate states with one o or one (T meson, respectively. It is sometimes convenient to define a new self- energy [ 151 which is irreducible with respect to both (T and o meson lines, and to combine Eqs. (2.20~) and (2.20d’) to

dab = A-ab + A-act, Adb 0 0 cd . (2.25)


Here, as in the following,’ latin indices take the values a = - 1, 0, 1, 2, 3, while Greek indices run from 0 to 3. We define

A-l-1 =A 0 (2.26a)

A”” = A”” (0 (2.26b)

and due to Eqs. (2.21b) and (2.23)

A-ab = 0 (2.27)

In the following developments we will make use of the field theoretical definition of r, for the field c1 (CI = CJ, TI, o, N) in terms of the renormalized single particle Green function A,

(01 %%(x) =x(.x’)) lo) = 5 d4$,(x - y) A,(y - x’), (2.28)

where zZ is the operator for the field a and (-&,) is equal to the “unrenormalized source” (see Section 2.2).

2.2. Determination of the Mean Fields

The quantities i? and V are the nuclear matter expectation values of the scalar and vector fields 4 and VP, and therefore they are determined by the conditions (see Eqs. (2.4))

(01 CJ IO) =o (01 cop IO) = 0.

(2.29a)

(2.29b)

The equations of motion for (T and w derived from the lagrangian (2.6) are

(Z, 0 + 622 + ST%;) = j, = J, - is, (2.30)

with

J, = c - qfii5, f Sfi2,)

is, = gZ, W + A’Z,(3ija2 + fin2 + o3 + an*),

(2.31a)

(2.31b)

and

(Z, 0 + mt, + drn$) 0.P = j$ = -.I; + is; (2.32)

’ The inverse of a matrix M is defined by (M m1)uh (M)br = g”, with g”, = (A $,). Indices - 1 may be written above or below.

70

with

BENT& LIU, AND ARIMA

J; = aqmt, + sm;) (2.33a)

8; = g, Z,, !Fyp IF/. (2.33b)

In Eqs. (2.30) and (2.32) we split the “unrenormalized sources”j, and j$ (which do not include the mass and wave function renormalization counterterms) into constant terms (from P& of Eq. (2.7~)) and terms involving the quantized fields. The conditions (2.29) imply

j, = J, - 8, = 0 (2.34a)

-j; = J$ - is:, = 0, (2.34b)

where the quantities j, and S, without hats in these expressions are defined as the nuclear matter expectation values of the corresponding operators. Since the quantities S, and S,,, themselves depend on the mean fields, Eqs. (2.34) can be considered as the eigenvalue equations [11] to determine v” and CV. In terms of Feynman diagrams, S, and S; are given by the sum of all loop diagrams with one external line (a or o, respectively; see Fig. 1 for S,). From (2.34) we see that the role of JZ&., Eq. (2.7c), is just to provide the necessary counterterms in order to cancel these loop terms. Let us first consider the simpler case of Eq. (2.34b). Using Eqs. (2.33) and (2.3b), we can write this equation as

- j$ = iV(m% + &rz~) - g,z Ajg=O,

Zl-4

(2.35)

where jg = (p, j,) is the baryon current in the nuclear matter ground state. Baryon current conservation implies Eq. (2.19), and therefore

(2.36)

Thus, the mean field produced by the o meson is completely determined by the baryon current, as expected. Now consider Eq. (2.34a) and compare it with the low energy theorem obtained from the axial Ward-Takahashi relation [ 11, 211 for the pionic decay vertex P(q) in the medium:

qpW(q) = i(c + iY d;‘(q)). (2.37)

FIG. 1. Graphical representation of Eq. (2.41). The dashed-iiotted line denotes the CJ meson, the dashed line the pion, and the full line the nucleon. The hatched circles represent the T matrices. The expressions obtained from the Feynman rules must be multiplied by the numbers shown in parentheses.


Here d; l(q) = q2 - fii’, - L?,(q) is the inverse renormalized pion propagator in the nuclear medium. Setting q=O in the above relation we obtain the low energy theorem

-ad,‘(0)=~(rE;+C~(O))=C. (2.38)

Equation (2.38) expresses the Goldstone theorem in the nuclear medium [ 11, 213: For exact chiral symmetry (c = 0) we have either d;-‘(O) = 0 (Goldstone mode; the pion is a Goldstone boson), or fi=O (Wigner mode; the nucleon effective mass is zero). Instead of Eq. (2.34a) we could equally well use the Goldstone theorem (2.38) to determine the physical value 6,. (Note that C, is a function of 6). In fact, we will show that the relation

jb=L7d,1(o)+c (2.39)

holds for any value of 6. If iY= v”,, both sides of Eq. (2.39) vanish identically. Using the methods of Ref. [ 111, we derive Eq. (2.39) in Appendix A. In the following, however, we demonstrate Eq. (2.39) in a more explicit way, which will be more convenient for later use. Namely, from Eqs. (2.30) and (2.31a) we see that (2.39) implies the following relation between the source function S, and the pion self- energy:

i+ C,(O). (2.40)

To demonstrate this relation explicitly, we take the expectation value of (2.31b) between the nuclear matter ground state to obtain

+ 3A2ZzC s & (d,(q) + d,(q))

(do(q,) do(q2) d,(q, + 42) T(q,, q29 -41-41;)

(2.41)

This expression is shown graphically in Fig. 1. For the mesonic T matrices we follow largely the conventions of Ref. [ 111: T(p, , . . . . pn; q, , . . . . qm) is obtained from the connected part of the full n sigma-m pion Green function by amputating the external legs. The arguments of T denote the incoming momenta, and the phase convention is such that T equals the expression for the Feynman diagram calculated by using the Feynman rules given in Ref. [ 111. (It is equal to i times the T matrix of Ref. [ 111.) For the three-meson vertices, which appear in Eq. (2.41), the T matrix is equivalent to the 6, rr irreducible vertex. In Eq. (2.41) we used the fact that for the case m = 2 the isospin dependence is simply

Wp,, .--3 pn; ql,qz) = ~“T(P,, . . . . P,,; q1,qr). (2.42)


On the other hand, the expression for the unrenormalized pion self-energy ci = Sg,!?, is (see Fig. 2)

C!(k) = igz, (2n)” I -f!k Tr(y,r’S(q) Q(q, k + q)S(k + q)) (2.43a)

+ cS”iA*Z, (2n)4 5 2.L (d,(q) + 5d,(q)

+2iv”d,(q)d,(k+q)T(-q;k+q, -k))

-i;l*Z, j$$ j$$( 6” d,(q,) d,(q,) d,(q, + 42 -k)

(2.43b)

x zlq,, 42; -ql-qz+k -k)+fd,(q,)d,(y,)d,(y,+q,-k)

x (,‘a #Jr + 6’P pL’ + pjb sd) FL% (; ql, q2, -ql-q2+k, -k) >

. (2.43~)

For the NN-meson irreducible vertices r, (o! = rc, 0, o) we follow the convention that T,(p’, p) equals the expression for the amputated Feynman diagram with one outgoing (momentum p’) and one incoming (momentum p) nucleon line, and one external meson line. Equation (2.43) can be derived straightforwardly from the definition of the unrenormalized pion self-energy given in Eq. (2.28). The connection between (2.41) and (2.43) for k=O is provided by the low energy theorems derived from the axial Ward-Takahashi relations for the T-matrices, given by Eq. (6) in Section 5b of Ref. [ 111 for the purely mesonic case and reproduced in Appendix A, Eqs. (A. 12). There we also demonstrate (see Eq. (A.13)) that by simply inserting these relations into (2.43) one is led directly to (2.40).

Before concluding this section, we add the following remarks. First, the symmetry breaking parameter c is expressed in terms of physical quantities by evaluating Eq. (2.37) in free space at the pole position of the pion propagator (q* = Mz) and using the definition D;(q) = iqpFn(q2) with F,JMz) = F,, the physical pion decay constant. One obtains

c= F&fI$ (2.44)

FIG. 2. Graphical representation of the pion self-energy (2.43). For an explanation of the symbols see the caption to Fig. 1.


Equation (2.38) for zero density determines the value of v. Using (2.44) and (2.17) we can write this equation as

(2.45)

Explicit one-loop calculations [12] of the pion self-energy, renormalizing at p’, = mf, show that C,,(O) < MZ,, and therefore from Eq. (2.45), v z Fn. Note that for exact symmetry (M, = 0) the relation u = F, holds exactly, since then the renormalization condition (2.14b) implies a vanishing derivative of Z,, at q2 = 0. The same conclusion holds with the renormalization prescription usually adopted in many-body calculations [ 1, 161, namely pi = 0 and m, taken as the experimental pion mass. In this case, v = F,Mz/rni z F,. Therefore, if we use the experimental values for the nucleon mass mN and the pion decay constant F,, Eq. (2.9a) shows that the parameter g has the value g% 10. The Goldberger-Treiman relation in the present framework is [ 12,211

MN g, = W,miv 9 (2.46)

where g, and gnNN are defined via the irreducible NN axial vector and NN7c vertex, respectively, in the q + 0 limit. If we renormalize the nucleon propagator at fi(N = mN, such that mN = M,, this relation can be written as

g-“N-gnNN V gA .

(2.47 )

It is possible to reproduce the experimental value gnNN z 13.5 in the (T model in the one-loop approximation using g= 10 as an input parameter [ 121. On the other hand, in many works [4-6, 131, g is considered as a free parameter and Eq. (2.9a) is used to determine the corresponding value of v. If then one uses Eq. (2.45) to assign a theoretical value to F,, it is possible to carry out consistent calculations also in such a scheme.

Second, Eq. (2.38) implies the following behaviour of the quantity (-d; l(O)), which can be used to define an “effective pion mass”: For c # 0, it is enhanced in the medium by a factor v/i?= m,/rFz,. (We assume fiN < mN.) For c = 0 (exact symmetry), it is zero as long as the system remains in the Goldstone mode but changes discontinuously to a non-zero (positive) value if at some density the system goes into the Wigner mode. (As we will discuss in the next section, the stability of the Wigner mode requires that -A;‘(O) > 0.) We also note that it is necessary to take into account the self-energies in the definition of “effective meson masses,” since the mass parameters defined in (2.11 b) and (2.1 lc) are not positive definite (“Tachyon poles” in the Hartree approximation) Cl, 3, 5, 17,281. We will come back to this point later.

Our third comment concerns the role of the counterterm lagrangian (2.7~) [ 111. In actual calculations one can ignore it, treat fi as a free parameter, and leave out


all 0 tadpole diagrams in the calculation of Green functions. (Note that the role of L&- is just to cancel these diagrams.) At the end of the calculation the physical value of i? is determined from Eq. (2.34a) or (2.38), or by minimizing the energy density (see Section 3). Thus, unless the opposite is stated explicitly, all Green functions in this work are considered not to contain 0 tadpoles and to depend on v” as a free parameter. Consider, for example, the quantity S, itself: If we discard Y&, all propagator lines in the loops of Fig. 1 contain in principle also self-energy corrections due to q tadpoles. These a-reducible contributions can be separated schematically as S, = S,; + B(SPi + iJ,) with a u-reducible block B, whose exact definition is not important for the argument. By definition, the quantity Soi is u irreducible and is thus given by the set of diagrams (Fig. 1) without 0 tadpoles on the internal propagator lines. From this it follows that the condition J, - 8, = 0 (Eq. (2.34a)) is equivalent to J, - is,, = 0. Thus, the eigenvalue equation formally does not change if we apply the prescription of omitting all (r tadpoles, as discussed above. An analogous discussion holds for the treatment of o tadpoles as well.

3. GENERAL PROPERTIES OF THE ENERGY DENSITY

The hamiltonian density Y?’ = Cj piQi - 9, where qi and pi are the fields and their conjugate momenta, can be split, analogously to Eq. (2.6), as &? = zMF + SL + %&. We therefore write for the energy density E = (01 2 IO)

E=E,,+E,

with the mean field dependent part

(3.1)

E,, = f L2Z,(iT2 - u2)2 + 4 ym: (52 _ u2)

- c(a - u) - rn$ + 6m2

2 o KG w, (3.2)

and the loop part E,. It is convenient to separate the dependence of EM, on the renormalization points in the following way: Using Eqs. (2.10b), (2.9b) to express Z, by the mass counterterms, and Eqs. (2.13) to replace the latter by the self- energies at zero momenta, we get

with E MF= u+du (3.3a)

UC _ d,‘(O) - d,‘(O) 8V2

(e2-"2)$!?q!4~2~u2)

-c(C-V)- mt, + 6m2

2 w ic,lv

dU= /Lw-zrAO) 8V2

(az_v2)2-y(~2~u2),

(3.3b)

(3.3c)


The dependence on the renormalization points is contained in the finite “potential” U, which as a function of fi has the familiar double minimum shape. If we choose the renormalization points as ~2 = 0 (c( = rr, rr, o), U reduces to

(3.3b’)

The divergencies, which eventually cancel against those in the loop part E,, are contained in the term 6U, which is independent of the choice of the renormalization points. Explicit forms for E, will be given in later sections. Here we wish to study some properties of E = E(iT, W, n,) as a function of fi, iP’, and {n,}, where nj are the occupation numbers for nucleons.

3.1. Variations with Respect to the Mean Fields

The following relations hold’:

(3.4a)

a2E as’= -d,‘(O); a2E - = - (A, ‘(O))? aGp ak,, (3.4b)

Consider first the mean field term E,, of Eq. (3.2). Using Eqs. (2.8b) and (2.8~) we have

a&4,, -= -J,; aa

a2-h, - = - gYmZ, + 6mi), aclp a$,,

where J, and 5; are given by Eqs. (2.31a) and (2.33a). Thus, in order to prove Eqs. (3.4) we must show that for the loop terms

(3.6a)

a’E, - = E;‘,(O); a3 g&=QYO,. (3.6b)

Thus, by differentiating the loop terms in the energy density we generate the amputated n-point meson Green functions with external momenta equal to zero.

z In this work, the limit q + 0 of any quantity A(q) is delined as A(0) = lim,“+, lim,,, .4(q). If the order of the limits is reversed, the corresponding quantities carry an index (St), i.e.,


(Here we treat the cases n = 1, 2.) Although this is intuitively clear and can be shown more elegantly using the concept of the effective action [22], we wish to indicate here a proof of relations (3.6) for completeness: One splits LZL of Eq. (2.7b) according to _4pL = 6p0 + $f, where Yt contains all the terms with explicit dependence on fi and VP’. For example, one possible choice is to consider & as the free lagrangian depending on the free mass parameters. Use 2, to define an interaction picture (ip) and write for any operator product &x,, .x1, . ..) = A(x,)B(x~)...

where (0) is is the ground state in the absence of Yr and

Using the following property of U,

and substituting aH;P/Sz = - j d3x aP’~p/&, we obtain from Eq. (3.7)

+ijdxbJd’x’{(Ol T(P(x,,s,,...)F) IO)

- (01 IQ,, x2, . ..) IO) (01 F IO)1 (2 = 6, V). (3.9)

The order of integrations in (3.9) is important, since in a many-body system the operator (a/&) YI can give rise to particle-hole excitations at the Fermi surface which vanish when integrated over x’ first but survive when the order of integration is reversed [23]. The second term in { } of (3.9) subtracts the disconnected parts from the first term. For the case P= XL(~) the expression (3.9) is independent of x. Since IO) is an eigenstate of the hamiltonian and we can replace sL by the full hamiltonian density yi” + YMF in the last two terms, these cancel each other. Noting that (a/&)&’ = - (a/&) L& one arrives at Eqs. (3.6a). To show the first equation of (3.6b), we take p = is,, to obtain from (3.9) and (2.31b)

i%= A2z, (01 3a2 + 7t2 lo) + i j” d4X’ (01 ~($,(sb(x’)) to>..... o-xred.5 (3.10)

where the order of integrations is defined as in Eq. (3.9). Remember that by S, we


mean only the o-irreducible pieces, although in the formal expression also the a-reducible pieces enter (see the remarks at the end of Section 2). Since this irreducibility is not affected by the differentiation, we must take only the a-irreducible pieces of the expression on the right-hand side of Eq. (3.10). If the equation of motion for c is used, the last term in (3.10) becomes

- s d4X’ (01 T(S,(x)[Z, q .y + Pi?;+ Sfi~] 0(x’)) IO)

= -i2z, (01 3o(x)2 + n(x)2 IO) - 1 d4x’

x [Z, Cl,. + cl; + Sri?;] (01 T&(x) S,(xy IO). (3.11)

In Eq. (3.11), first, we used the fact that according to our discussions at the end of Section 2 the source term of the quantized CJ field now does not include the term J, of Eq. (2.31a) (since we discarded the counterterm Yc- and treat t? as a free parameter); second, we used the canonical commutation relation between ti and (T to take the d’Alembert operator out of the T product. The first term on the right- hand side of (3.11) cancels the first one in (3.10) and for the second term we use the definition (2.28) of the unrenormalized self-energy. Taking the o-irreducible part means that we should replace A,, -+ a,, in that expression, which leads to

i % = 1 d4x’ C,(x - x’) = z&(q = 0). (3.12)

The proof of the second equation in (3.6b) follows the same procedure and is not reproduced here.

The relations (3.4a) imply that the minimization conditions for the energy density with respect to the mean fields are equivalent to Eqs. (2.29).

3.2. Minimization of the Energy Density and the Goldstone Theorem

The considerations of Section 2 based on baryon current conservation and PCAC allow us to rewrite Eqs. (3.4a) as (see Eqs. (2.35), (2.19) and (2.39))

(3.13)

From Eq. (3.13) it follows that the minimization condition of the energy density with respect to v” is equivalent to the Goldstone theorem (2.38). That is, for exact chiral symmetry (c = 0), the minimization of the energy density leads either to a


pole in the pion propagator at q = 0 (Goldstone mode) or to a vanishing expectation value fi= (01 4 IO) (Wigner mode):

dE z= -i7A,'(O)=O+ -A,‘(O) = @ri + C,(O) = 0 (Goldstone mode) (3.15a)

iT=o (Wigner mode). (3.15b)

The stability criterion

a2E -= -A,'(O)=k;+c,(o)>o ai7 (3.16)

decides which of the two modes is actually realized. Due to the relation -A;'(O)= -A;l(0)-v"(a/av")A;l(O), which follows from Eqs. (3.15) and (3.16), or from the identities in Appendix A, the stability criterion can also be written in terms of the pion propagator as

d'E z=-A,l(0)>O

a-$(-A;'(0))=8-$A'+Z.,(O))>O (Goldstone mode) (3.17a)

-A,'(O)=fi;+Z,(O)>O Pikwr mode). (3.17b)

These considerations are illustrated schematically in Figs. 3 and 4. Let us first consider the upper parts of these figures, which illustrate the self-consistency relation (3.15). The dashed line represents the quantity -fif, which is equal to m2/2(1 - (C/a)‘) for exact symmetry (see Eq. (2.1 lc); we assume that the renormalization point is chosen such that m, --+ 0 if M, 4 0, see Eq. (2.17)), and the full line represents C,(O). The situation is shown for three densities, increasing from left to right. The intersections of the dashed and the full line are solutions to Eq. (3.15a). Due to Eq. (3.17b), the Wigner mode solution (i?= 0) is stable if the full line lies above the dashed one at 6 = 0, while the Goldstone mode solution (0” # 0) is stable if for 6 > 0 (6 < 0) the derivative of the full line is larger (smaller) than that of the dashed line at the intersection. (By “stable” we mean stability with respect to variations in u” at fixed density.) In the lower parts of the figures we show the corresponding curves for the binding energy per nucleon (E,/A = E/p - mN) as a function of 6.

Figure 3 illustrates a continuous (second order) chiral phase transition: The physical value of 6, which is a solution to A;'(O) = 0, decreases continuously to zero. At the critical density pc = p2, the value of the energy density at C=O turns from a maximum into a minimum. For densities higher than pc, only the Wigner mode solutions exists.

Figure 4 represents an example of a discontinuous (first order) chiral phase transition: At the first density pl, both the Goldstone and the Wigner mode


595/188/l-6

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solutions are stable and separated by an energy barrier. At the second density p: = pZ, the local minima and local maxima of the energy density for both 6 > 0 and 6 < 0 have coalesced. At this density the two equations

A,‘(O)=A,‘(O)=O (fi#O) (3.18)

are satisfied simultaneously. These two equations determine the density p; and the corresponding value of 6 As we approach this density from below, the quantity (-A.(O)) and hence the attraction due to the rr meson exchange in the Landau-Migdal force (see Eq. (3.24) below) goes to infinity. For higher densities there is no longer a Goldstone mode solution. We use the notation pk to distinguish this “critical density” from pC, above which the energy density of the Wigner mode is an absolute minimum (p, < p: in Fig. 4). The case shown in Fig. 4 is actually the “quasiclassical approximation” discussed in Section 5.

If the chiral symmetry is explicitly broken (c #O), a term (c/a-- mi) should be added to the dashed lines in Figs. 3 and 4 according to Eq. (3.13). In the example shown in Fig. 4 there then still exists a discontinuous transition to a very small value of 6. Equation (3.17a) is formally unchanged if c # 0. We remark that in the schematic plots of Figs. 3 and 4 we assumed that for any baryon density the energy density is real for all values of 6. In general this is not the case, since, for example, at low densities E may become complex at small 6 (see Ref. [3] or the examples in Section 5).

3.3. The Landau-Migdal Interaction

Let us now study some aspects of the dependence of E(v”, W, ni) on the occupation numbers {ai}. For this, let us assume that the conditions

CUE dE -=-= aa as,

0 (3.19)

are already satisfied, and that accordingly iJ= C(n,) and V = +‘(n,). In the Landau-Migdal theory [24] the first and second variations of E with respect to the ni determine the quasiparticle energy si and the quasiparticle interaction fii at the Fermi surface:

dE aE aE ai? aE aiv aE &i=-=-+y-+--=- dn, ani av ani hv ani dn,

(3.20a)

- ‘- lIL--+ds,d3” d&. a&. aE &?

dn, dn, dn, an, 6% anj &V an, ’ (3.20b)

It does not seem possible to derive the general forms of &i and f, using only these prescriptions and the underlying field theoretical model. However, we can specify those terms in fi/ which arise from the density dependence of the mean fields, i.e., the last two terms in Eq. (3.20b), thereby generalizing some of the results in Ref. [25]. (Note that in the Hartree approximation used in Ref. [25] the first term on


the right-hand side of Eq. (3.20b) is zero; see Section 5.) To determine the quantities Z/lanj and &P’/lh, we use the following conditions, which follow from (3.19)

d aE a2E d’E c% a+ aiv --=- --= dnj aa anjaii+dazdn,+aV aa anj

0 (3.21a)

d 8E a2E a% as a92 aa --= dnj i3w anj &v +-- asaw an.

+ --zoo. J aaaapan. J

(3.21b)

In the limit jB + 0 (the Landau-Migdal interaction is defined in this limit) the last terms in Eqs. (3.21) vanish: For p = i this is obvious since i32E/lhCi % must be proportional to jBi. For p = 0 we can use the form of aE/&G” as given by Eq. (3.14) with j”, = Cj ni. Noting that Go and v’ are independent variables, we get a2E/No % = 0. Therefore, using Eqs. (3.4b) and (3.20a) we obtain from (3.21)

(3.22b)

In Appendix B we prove the relations

f$= if(P) T&2 &,=J(P) = 4,(P) (3.23a)

(3.23b)

where the r, (c( = IT, CD) are the irreducible clNN vertices and f(p) is the correctly normalized quasiparticle spinor. Using Eqs. (3.22) and (3.23) in (3.20b) we obtain

Lj=z - Ycr(Pi) dc(o) Yo(Pj) - Ycop(Pi) dZ:‘(o) Ycov(Pj) I

(3.24)

with y,, and y; defined in Eqs. (3.23). We thus see that the {nil dependence of the mean fields gives rise to the direct interaction terms due to cr and o meson exchange, i.e., the last two terms in Eq. (3.24). Equations (3.21), (3.22), and (3.24) generalize Eqs. (8), (9), and (10) of Ref. [25], respectively. This correspondence will be made more explicit later when we discuss the Hartree approximation in Section 5.

Baryon current conservation can be applied to further specify the o meson exchange part, i.e., the last term in Eq. (3.24). The condition (2.16a) gives for the o meson propagator at k = 0 (see also Eq. (2.24))

A;‘(O) = 1

2; mZ, + 6m, AZ(O) = 0; A;““(O) = m2 f;m2 . (3.25)

w 0


(See the footnote in Section 3.1 concerning our definitions of the limit q + 0. Note also that we set j, =O.) The last equation in (3.25) means that the w meson self- energy i?:(q) vanishes for q + 0 if we first let q,, + 0 and then q + 0: Setting Ci(O) = gvC,(0) we have

cjy(o) = 0, (3.26)

as follows also from Eq. (2.16b). In Eq. (3.24), however, we need the self-energy Z,(O), which is obtained by setting q -+ 0 first. The difference between these two limits [23] of the self-energy is due to particle-hole (ph) excitation processes at the Fermi surface, which contribute to cEt)(0) but not to C,(O). Denoting the contribution of these ph excitations by Clph’, we have shematically L?‘p”(O) = c,(O) + C:“)(O), and due to Eq. (3.26), c,(O) = -Zzh)(0). Therefore the self-energy in the expression

A!(O) = g”

mZ, + 6rn2, + E-,(O) (3.27)

vanishes for zero density and, due to gauge invariance, can be related to ph excitation processes at the Fermi surface. (For the Hartree approximation, see Eq. (4.41d).) Next let us discuss the oNN vertices in Eq. (3.24). They are defined in Eq. (3.23b) as the matrix elements of the irreducible oNN vertex r;. The relation to the wNN source vertex l-z,, on which we imposed the renormalization condition (2.18), is [20] (using q = p’- p)

Wq) ~C,“(P’? PI = W(q) ~“>S,JP’Y PI. (3.28)

The matrix element of the wNN source vertex

YZs(P)=.f(P) CJPY P)I,,=,,f(P) (3.29)

at the Fermi surface has the same form as the isoscalar electromagnetic current (times a factor -2ig,), which was determined in Refs. [23,26] from gauge invariance:

Y&(P)= -is, 1,: . ( 1 P Using Eq. (3.28) and the propagators as given in Eqs. (3.25) and (3.27) we obtain for the matrix elements of the irreducible wNN vertex

drni y”,(P)= -ig, 1 t---p ( > w

y,(p)= -igcop C,(O) . P (

1+2+7;;i- w w >

(3.31b)


Therefore we obtain finally for the o meson exchange part (fj,!“‘) of Eq. (3.24)

“f-y= -Y,,(PJ 430) Y,“(Pj)

(3.32)

The o meson exchange part in (3.24) and the first term in (3.32) due to CO’ exchange contribute to the Landau-Migdal parameter Fo, while the second term in (3.32), which comes from CO exchange, contributes to F,.

The first derivative of the binding energy per nucleon E,/A = E/p - mN with respect to the density is given by

(EF - E,/A - m,), (3.33)

where I+ is the Fermi energy, i.e., the value of E; at the Fermi surface. Hence at the saturation point the binding energy per nucleon is E,/A = cF - mN. Due to our above considerations, the second derivative is given by

with

d*(EJA) 1 dc, 2 d(E,/A)

aP* =-----

P dp P dp (3.34a)

2=&l +Nofo)+l +Fo), 0 0

1 X2 aE -z-L No 2Pz, aP &-,=Jq;

Fo = NoJo 1

(3.34b)

(3.34c)

(3.34d)

At the saturation point the second term on the right-hand side of (3.34a) is zero, and one obtains for the compressibility

K= 9p’ d*(E,IA 1

4’ =3p,3 ap = (l+FO),

P PF

4. FORM OF THE ENERGY DENSITY

(3.35)

In this section we derive the form of the loop part of the energy density (3.1). As usual [15, 161, we first express it generally in terms of single particle Green functions and then consider specific models for the latter. The pion self-energy has already been represented graphically by Fig. 2. In this section we will also make use


of the self-energies for the nucleon, the (T meson, and the w meson, which are shown in Fig. 5. It should be noted that some of the results derived in this section are essentially already known [ 1, 3-61. However, as stated in the Introduction, we wish to present the general method of calculating the energy density beyond the Hartree approximation in accordance with the renormalization scheme discussed earlier, and to illustrate some of the more general considerations of Sections 2 and 3 in simple models. To achieve these aims in a self-contained manner, we rederive some results obtained previously.

4.1. General Form of the Energy Density

The loop part E, in Eq. (3.1) is the expectation value of

(4.1)

where qi and pi are the fields and their conjugate momenta and LZL is given by Eq. (2.7b). The kinetic energy part X0 is understood to include the contributions due to the mass and wave function renormalization counterterms. In order to express the expectation value of the interaction part & in terms of single particle Green functions, we use the connection between the unrenormalized source -is, and the unrenormalized self-energy c, for the field u = g, rr, w, N as given in Eq. (2.28). In this way we obtain

E,=CEo,+E, (a = N, c, co, n) (4.2) 1

.-. -i;N’ i’ ‘& + :‘--‘& +

FIG. 5. Graphical representation of the self-energies of the nucleon, the o meson, and the o meson. The o meson is denoted by a wavy line. For an explanation of the other symbols, see the caption to Fig. 1.

86 BENT& LIU, AND ARIMA

with the “kinetic energies”

EON= -i (2n)4 s d4k - Tr S(Z,y’k, - s; ‘)

Eo, = 3Eo,(a -+ ~1,

and the “potential energy”

E,= -i 5 $$ Tr(~,W + f [ $ (co3 A, + 3Z,, A,)

+fr,, d4k (c,, A, + 3zin4 A,).

(4.3a)

(4.3b)

(4.3c)

(4.3d)

(4.4)

Here we split the meson self-energies according to

2, = -CL nut + L + En4 I (4.5)

where the first and second terms refer to the first and second diagrams in Fig. 2, respectively, and zr4 stands for the remaining four diagrams. A similar notation is used for the CJ meson; i.e., z:,,.UC refers to the first diagram in Fig. 5b, z03 to the next two diagrams, and zg4 to the rest. We note that due to the Dyson equation (2.20a) the term involving SC’ in (4.3a) could be cancelled against the first term in (4.4) [ 163.

The kinetic terms due to the mesons cancel parts of the potential energy, as can be seen as follows: Due to the Dyson equations (2.20b), (2.2Oc), and (2.20d’), we can replace

A, 2,’ + c, A, (a = 0, 0, ?T)

in Eqs. (4.3b) to (4.3d). Therefore the terms involving 1~~ in those equations give a contribution

(4.6)


The following relation was used to derive Eq. (4.6),

s o,nuc A, + 3~,,.., A, + WC, A,))

which is verified by noting that the same types of vertex functions enter into the nucleon self-energy of Fig. 5a and the nucleon loop terms of the meson self-energies in Figs. 2, 5b, and 5c. Adding the contribution (4.6) to E, of Eq. (4.4) we see that the meson kinetic energies cancel half of the potential energy due to the nucleon loops. We obtain

d4k

with

E,=Eo,+i ~k~CZH(A,+3A.)-Z,A,,‘,l+E; s (4.8a)

E;= -ij$Tr(,NS)-ij$(,, A,+3Z’,, A,)

(c,, A, + 3L?,, A,). (4.8b)

The terms involving ki in Eq. (4.8a), which are the remainders of the meson kinetic energies, are usually discarded in actual Hartree-Fock calculations done so far.

The method for expressing the energy density explained above does not directly make contact with the two-line irreducible graphical expansion [ 1, 3, 22, 271 of the energy density in terms of full propagators but bare vertices. In the latter method, the two-line reducible diagrams (“ring diagrams”) are contained in a compact mathematical expression, which is sometimes more convenient in actual calculations. The connection to the expressions given above is provided by the relation

(4.9)

and analogous relations for the meson propagators. Using these equations, we can rewrite the kinetic energy terms (4.3a) to (4.3d) as

I d4k EoN=Ek-i mkoTr

Eo,+Eo,,=E,,,+; j&$, k,, Tr A- ( it)

3i d4k - Eo,=Ei+T jpj$oA.~

(4.10a)

(4.10b)

(4.1Oc)


with the “ring energies”

EL= -i (Zn)4 s d4k -Tr(ln S-s;‘S)

’ d4k E;,,=L j7

2 (2x) Tr(lnd-a;‘d)

E&21 d4k “2(271)4 s

(In d, - J;l d,).

(4.11a)

(4.1 lb)

(4.1 lc)

Here we combined the (T and CO meson contributions by using the propagator (2.25). The total loop part then becomes

with

and

E,=E,+E, (4.12)

E,=x E; (a = N, ND, TC) (4.13) rr

4=4-i jfik {Tr(S~)-~Tr(~~)--~,4.~}, (.&q, o (4.14) 0

where E, is given by Eq. (4.4). Although it is not obvious from the expression (4.14), E, is equal to the sum of the two-line irreducible diagrams with two or more loops. We will demonstrate this explicitly in the next subsection for the Hartree- Fock approximation, which leads to the two-loop contribution to E,.

4.2. The Hartree-Fock (HF) Approximation

The HF approximation consists in taking only the two-loop part of E, (Eq. (4.14)), or, equivalently, calculating the single particle propagators self- consistently in the one-loop approximation. The one-loop expressions for the unrenormalized self-energies are as follows (we use the same symbols as before for the HF self-energies and propagators):

Z‘,(k)= -ij& r,W + 4) rb dab(q)

(4.15)

(4.16a)


with

C$,(k) = i j g‘$ Tr(T”S(k + q) rbS(q)) (4.16b)

Z‘,,(k) = 6i(13’Z,)* i7* i d4q (d,(q) d,(k + 4) + 3d,(q) A,@ + 4)) (2n)4 (4.16~)

Z,,(k) = 3iA2Z, j -g$ (d,(q) + ‘4,(q)); (4.16d)

and

z',(k) = &.., + zixx + En4 (4.17a)

with

C,. n.,(k) = GsQ2 j $$, Tr(y5W + 4) y5S(q)) (4.17b)

z‘,,(k) = 4i(;i2ZJ2 fi2 j $ d,(q) d,(k + q) (4.17c)

E’,,(k) = il’Z, { 3 (d,(q) + 5d,(q)). (4.17d)

In Eqs. (4.15) and (4.16b) we used the notation

r”=(-igZ,, -ig,Z,,Y’). (4.18)

Equations (4.15) to (4.17) are shown graphically in Fig. 6. Inserting the above forms of the self-energies into Eq. (4.14) we obtain (see Appendix C)

(a)

*P, /-\

0 ‘-‘\ 1 . 1 I

-iF,= - - + -4.’ ;L- + -iA- + -‘a a/-- ‘--’ (l/2) flfil

Id FIG. 6. Graphical representation of the HF self-energies (4.15) to (4.17). The wavy line with dots

denotes the combined propagator for o and o. In the combined self-energy Z, mixing occurs only through the lirst diagram (the nucleon loop diagram). For the other symbols, see the caption to Fig. 1.


EW), -f I . ~Tr(S~~)+~f~(T~1A,+32,,A.) s 2 (2n)

+fSpp d4k (zc?,, A, + 3C,, A,),

where the first term can alternatively be written as

i d4k - Tr(Sz,)

-z (2n)4 s

=if$ Urkf~,,,)+ 34,~,,,,,).

@“‘) is shown graphically in Fig. 7. The total HF energy density is then

E’HF’ = ~2;) + ,?T~HF’= ET;) + EW=‘+ E(HF’, r

(4.19)

(4.20)

(4.21)

The three terms on the right-hand side of (4.21) are given by Eqs. (3.3), (4.13), (4.1 l), and (4.19), where now all single particle Green functions are solutions to Eqs. (4.15) to (4.17). Note that the ring energies (4.11) contain counterterm contributions through the unrenormalized propagators (see Eqs. (2.21)). As was noted in Ref. [IS], those counterterm contributions in Eqs. (4.11b) and (4.11~) which are due to the nucleon loop self-energies can be combined with Eq. (4.20). The resulting expression then again has the form (4.20) with the unrenormalized meson self-energies replaced by the renormalized ones.

If we consider the HF energy density (4.21) as a functional of the free variables S, A, and A,, and require stability with respect to variations in them, we get back the

./‘f-. ’ i

\ /‘P i

; i . + 1 ! :

‘\.i.’ ‘NJ,/

FIG. 7. Graphical representation of the two-loop (HF) contribution to E,, given by Eq. (4.19). The three terms in that equation correspond to (a), (b), and (c) in the figure.


Dyson equations (2.20a), (2.20b), and (2.25): From Eqs. (4.11) and 4.19) we have, using the form of the self-energies as given by Eqs. (4.15) to (4.17),

aE’HF’ -= +-‘-~,‘+&J=() as(k)

aE’HF’ i

ad(k)=Z(d-‘-6,‘+f)=0

(4.22a)

(4.22b)

(4.22~)

The self-energies in these equations arise from the variation of EjHF) of Eq. (4.19), while the other terms come from the variation of the ring energies (4.11). Due to the relations (4.22) it is very simple to calculate the derivatives of the HF energy density with respect to the mean fields and thereby to derive Eqs. (3.6a) (or, as a consequence, Eqs. (3.4a)) in the present model: We need only take into account the mean field dependence of the various unrenormalized lowest order propagators in EC”“’ of Eq. (4.11) as well as the explicit 6 dependence due to I03 and C,, in Eq. (4.19). Using the forms of the lowest order propagators given in Section 2 we have

as, 1 -= -guJZ,,Yp aiv (4.23a)

a(a, 1 )ob = -da- 1 db- I(j12z -. sag

a6 AU, -= -2A22,ii a6

(4.23b)

In deriving these relations we made use of Eqs. (2.8). We therefore obtain

aEy’ -= -igZ, ~TrS(k)+3ii2Zj,iI~(d.(k)+d.(k))

aa s

-6(n’Z,)‘i~~~~d.(k,)

x (I, A,@, + 4 + d,(k,) A,@, + k,)) = i,‘$HF), (4.24)

which is the same as (2.41) with the full vertices replaced by the lowest order ones ( -6iA2ZAv” for the o3 vertex and -2iA2Z,i7 for the arc2 vertex). Variation with respect to GP gives

aEy’ -= -igwZRw

s d4k

a+,, 4 Tr(S(k) y”) = is;, %I

(4.25)


as is clear from the definition of Sz, Eq. (2.33b). In Sections 2 and 3 we discussed the equivalence between the minimization condition for the energy density and the Goldstone theorem, which is expressed by Eq. (2.40). It is clear, however, that this relation does not hold if we simply replace S, and 2, by their HF expressions given above, since S’bHF’ contains two-loop terms (since it is derived from the two-loop approximation ECHF)), while c, of Eq. (4.17) does not. In general, if we calculate E in the I-loop approximation, the pion self-energy satisfying (2.40) contains 1 loops also and is different from the (1- 1)-loop expression used as an “input” to calculate E. Therefore, in the HF case, the relevant pion self-energy has the form (2.43), where the single particle Green functions are given by their HF expressions, while the vertex functions r, and T are determined by the one-loop approximation and satisfy the low energy theorems (A.12) together with the HF propagators. As discussed earlier, this guarantees the validity of (2.40).

4.2. The Hartree Approximation

The HF approximation discussed in the previous subsection is a self-consistent two-loop approximation to the energy density. The Hartree approximation is obtained by neglecting all self-energies in the calculation of the loop part E,, i.e., by neglecting E, of Eq. (4.14) as well as the counterterm contributions in Eqs. (4.11). It is a self-consistent one-loop approximation to the energy density. The variations (4.22) then give S=&, A= A,, and A,= A,,. The energy density thus becomes

E(H)=E(H)+EW) MF r.

( 4.26)

Here EE$ is to be calculated from Eq. (3.3) with the propagators in free space determined (non-self-consistently) in the one-loop approximation, and EjH’ is the sum of the three terms

s d4k E!yd= -i ~TrinS,

E;H,1, = E;H,’ = 4 s

d4k yin A,

2 (271)

3i E(H),- d4k In A r,n 2 (2x)4 s no. (4.27~)

Both terms in Eq. (4.26) individually are divergent, but their sum is finite. Let us first consider Eg.j of Eq. (3.3). In order to calculate the finite potential UCH) we need the free space self-energies for (T and 7~. These are given in Appendix D for arbitrary choice of the renormalization points. If the mesonic renormalization points are taken to be zero, UCH’ reduces to (3.3b’). The divergent term 6lJ is obtained from the unrenormalized 0 and R self-energies in free space given in Appendix D. The result is


Here T(x) is consider the convenient to

with

~(~(2-~)--lnm~) (2g4(~2-02)2+4g2m~(~2-a2))

+(,(2-i)--hrrn;)(i A4(iT2 - u2)2 + 2 A2mz(iT2 - u2) >

+(l-(2-~)-lnm~)(~ i4(82-07)‘f~12m:(a2-c’) . >

(4.28)

the gamma function and n is the number of dimensions. Next we loop terms (4.27). For the nucleonic contribution (4.27a) it is split the propagator S, according to (see Eqs. (2.21a) and (2.22))

So(k) = E _ 1, isk = so, + SOD (4.29a)

s”‘=~e;+id .@I

so, = (E + cl) Xl 7 q/T, - E,) 4

(4.29b)

(4.29~)

with E, = $&%?. By a partial integration we can separate the contribution due to SODI

Tr In SOF

Z Er n(k) + g, --$! @‘Op + 4i

d4k m ln(k2 - 6i2 + id). (4.30)

Here the first two terms on the right-hand side are due to the integral over SOD. Since Z, = 1 in the Hartree approximation, we will omit it from now. With (4.30) used, the ring energy (4.27) becomes, after the vacuum expectation value is subtracted,

(4.31)


Here we introduced (a = N, 0, rc)

with d,=tii~-rn;, and

(4.32)

(4.33)

In going from (4.27) to (4.31) we essentially only subtracted the term 6E and added it again in order to separate the divergent term. Note that the function F(y) defined by Eq. (4.32) is finite. An elementary calculation of 6E shows that

6E= -6U, (4.34)

where SU, the divergent part of E #, has been given in Eq. (4.28). This shows that the energy density (4.26) is finite. The function F(y) is calculated to be [l, 3, 5, 63

F(y)=y21ny-+(y2-1)+2(y-1). (4.35)

The energy density in the Hartree approximation is obtained by adding Eqs. (4.31) and (3.3),

EcH) = UcH’ + g,Z,,pfi’ + 4 l$$ n(k) E,

4 -@F(YN)+& F(Y,), (4.36)

with UCH’ given by Eq. (3.3b) (see also Appendix D) or, for the choice ~2 = 0, by (3.3b’). Using Eqs. (2.36) and (2.19) with ZN = 1, the terms due to the w meson in (4.36) can be combined to

(4.37)


If we set the mesonic renormalization points equal to zero and omit the pionic contributions, Eq. (4.36) reduces to the expression used in Refs. [S, 61. Let us calculate the derivatives of the energy density (4.36) with respect to the mean fields. We obtain (without changing the notations for the Green functions)

dE'"' -= -v"Ll,'(O)-c

a6

with

+4g2 I

3m212 +~GCY,,+~G(P,,

(4.38a)

(4.38b)

G(y)=Illnv-y+l. (4.38d)

The term Ci’, in Eq. (4.38b) as well as the first two terms in (4.38~) involving the free space self-energies at q = 0 has been derived by differentiating 17’“’ of Eq. (3.3b), while the last four terms in (4.36) give rise to the last four terms in (4.38~). It is not diflicult to convince oneself that C,(O) of Eq. (4.38~) is indeed the one-loop expression for the pion self-energy at q = 0: If we split the renormalized free space self-energies in (4.38) according to C,(O) = c,-(O) + hmf (c( = cr, n), we see that the counterterm contribution contained in (4.38~) is just

SmZ,+(0"2-uUZ)~2(ZA- 1)=&2,

due to Eqs. (2.10b) and (2.12~). Therefore Eq. (4.38~) takes the form Z,(O) = Z;,(O) + sq, where r,(O) is obtained by replacing C,,(O) --) C,,(O) in (4.38~). A simple calculation, not reproduced here, then shows that this expression for Z,(O) is equivalent to the expression obtained from Eqs. (2.43a) and (2.43b) or the graphs in Fig. 6c calculated with the lowest order (Hartree) propagators and bare vertices. If we choose the renormalization points as 11: =&=O, the first two terms in (4.38~) vanish. The next two terms are due to the nucleon loop diagram of Fig. 6c and the last two terms are due to the meson loop diagrams. Using Eqs. (4.29), the nucleon loop diagram in Fig. 6c can be split into a “density part,” which involves at least one propagator Soi, and is given by the third term in (4.38c), and a “Feynman part,” which is given by the fourth term in (4.38~). Since we are considering the limit k + 0, there are only particle-antinucleon (pm) and no particle-hole (ph) contributions to the nucleon loop term.


Equation (4.38a) is a special case of the Goldstone theorem (3.13). In particular we note that in the so-called “quasiclassical approximation,” which is obtained by neglecting the last three terms in (4.36) and choosing the form (3.3b’) for U, only the third term in the pion self-energy (4.38c), i.e., the “density part” of the pN loop diagram, remains. It is repulsive. The attractive contribution due to the interaction of the pion with the condensed scalar field is contained in the mass parameter fiis, = rni + A2(C2 - v*). The cancellation between these two terms is analogous to the cancellation which leads to the small s-wave scattering length ah+’ in pion-nucleon scattering. In the framework of the quasiclassical approximation, the assumption of exact chiral symmetry then requires a complete cancellation of these two terms also in the nuclear medium (as long as we stay in the Goldstone mode), since then A; l(O) = 0 becomes the energy minimization condition to determine the physical value of fi. The Wigner mode (v”= 0) becomes stable in the quasiclassical approximation if the density is large enough that the third term in (4.38~) exceeds the value mz/2 (which is equal to -fiz at C = 0; compare Eq. (3.17b) and Figs. 3,4).

Let us briefly also discuss the forms of the c and o meson self-energies as derived from the expression (4.36), since these determine the Landau-Migdal interaction (3.24) in the Hartree approximation. Differentiating (4.38a) once more we obtain

with

,jZEfH) -= -d,‘(O) ai7

4,‘(0)=rG~+.&(0),

(4.39a)

(4.39b)

3m2A2

+16n2

3m2A2 ff,(Y,) + * HAYA

H,(g)=(y+F)lny-y+I,

H,(y)=(y+E$)l” y-y+l.

(4.39c)

C,(O) of Eq. (4.39~) agrees with the expression obtained from Fig. 6b in the Hartree approximation. The third term in (4.39~) is the “density part” of the nucleon loop contribution in the limit lq( + 0 first followed by q. + 0. Only the pN contributions survive in this limit. The first two terms in (4.39~) vanish for the choice & = pi = 0. In the Wigner mode (5 = 0) the expression (4.39b) is equivalent to (4.38b).


Using (3.3b), differentiation of (4.36) with respect to W gives

with j”, = p and

aEcH’ -= j$ = -mt,Z,,W + g,Z,,jk a3,

(4.40a)

Setting the right-hand side of Eq. (4.40a) equal to zero then gives Eq. (2.36). Differentiating (4.40a) once more we obtain

a2E’H’ aj; -= -(~,‘(O)Y”= -gg”“m;zx,+g,z,,-.-. asp a+,, G

(4.41a)

Using (4.40b) together with (2.19) for Z, = 1 this becomes

(A, ‘(0))” = mt, + hrnt,

(dJ'(O))"= g”(mZ, + 6mt) + Z;(O)

(4.41b)

(4.41c)

with

(4.41d)

where i?, =&m. Equation (4.41b) has been derived more generally in Section 3 (see Eq. (3.25)), and c,(O) of (4.41d) is the Hartree expression of the w meson self-energy at q = 0 (compare Eq. (3.27)). In accordance with the discussion in Section 3, X,(O) vanishes if the density goes to zero. The propagators given in (4.39) and (4.41) should be inserted into the general expression (3.24) to determine the Landau-Migdal interaction. The first term in (3.24) is zero in the Hartree approximation since the quasiparticle spectrum

&,=Ep+g,Z,,iGO (4.42)

does not depend explicitly on the distribution function. The vertex r”, in the Hartree approximation can be derived from the definition (3.23b) or from the more explicit form (3.31b) using (4.42) and the expression (4.41d) for c,(O). One obtains

y;(p)= -Ig,zg..,$ (p” = Gp9 P)). P

As expected, this is just the matrix element of the free ?/NN vertex, since in the Hartree approximation the nucleon self-energy ,JC, and therefore, due to gauge


invariance, also the correction to the irreducible oNN vertex is zero. (This is in contrast to the source vertex which is renormalized by the RPA-type vertex corrections [20, 263.) Equation (3.23a) gives for y,

y,(p)= -kS. EP

(4.43b)

We therefore obtain for the Landau-Migdal interaction in the Hartree approximation from (3.24)

(4.44)

with z,(O) given by (4.39~) and ,X:,,(O) = r;,(O) + drnt, with z;,(O) given by (4.41d). In the quasiclassical approximation, where Z,, = 1 and only the third term in (4.39~) contributes, Eq. (4.44) differs from the result derived in Ref. [25] only by the appearance of fit instead of the free mass parameter rnz.

5. SIMPLE EXAMPLES

In this section we use the form of the energy density in the Hartree approximation, Eq. (4.36), to discuss various simple cases, including those studied previously by other authors [l, 4-6,9]. Our aim is to illustrate the connection to the more general results derived in Sections 2 and 3. In the numerical calculations discussed below, we will fix the coupling constant g = 10 due to reasons discussed in Section 2. (Note, however, that for the cases discussed in Sections 5.1 and 5.2 only the ratio g2/m2 enters.) Thus, fixing also m, = 783 MeV, the free parameters are m, and g,. Concerning m,, we will refer either to the case m, =0 or to the case rnz = 140 MeV.

5.1. The Quasiclassical Approximation

The quasiclassical approximation consists in neglecting all loop terms in (4.36) except the loop term which gives the energy of the filled Fermi sea. In the frame where the total three-momentum of the system is zero, we have

(5.1)

where U, is given by (3.3b’), and Z,, = 1. By differentiating this expression with respect to r? we obtain the corresponding expression for the pion self-energy (see Eq. (4.38~))

p=4g2 s PF d3k 1 03%’ (5.2)


As discussed earlier, this is the density part of the pN loop diagram in the pion self- energy. For simplicity, let us discuss the case of exact chiral symmetry (c = m, = 0). The solution of the self-consistency equation

-A;1(0)=rkiZ,+C~1'(0)~m,*2=0 (5.3)

has already been illustrated graphically in Fig. 4. Here, for convenience of comparison, we use parameters identical to those in Ref. [9], namely m, = 0.8 GeV and g, = 9.1. (The choice of parameters has no influence on the qualitative features discussed below.) The stability condition is

-A;'(O)=GI~+~~,')(O)~~~'>O (5.4a)

with (see Eq. (4.39~))

(5.4b)

which is the density part of the pN loop contribution to the g self energy. As is clear from Fig. 4, due to the large curvature of the pion self-energy at fi = 0 there exist both the Goldstone mode and the Wigner mode solutions up to a density ph = pz. For p > p: only the solution fi = 0 remains. In Fig. 8 we show the effective nucleon mass mN = gfi and the binding energy per nucleon. As noted by other authors [9], independent of the choice of parameters, the normal state (Goldstone mode) does not show saturation with respect to variations in the density. On the contrary, for g, # 0, the abnormal state saturates. (Note that since in the abnormal state the attractive g meson exchange part of the Landau-Migdal force (4.44) is ineffective, an extremum of E,/A(p) is necessarily a minimum.)

a b

0 0.1 0.2 density [fmw3]

03 04

FIG. 8. The nucleon effective mass (a) and the binding energy per nucleon (b) in the quasiclassical approximation. Beyond a certain density p: the normal state disappears and only the abnormal state, where &., = 0, exists. At the critical density pc <p: the system goes into the Wigner mode and the

effective mass becomes zero, as indicated by the dashed line in (a). The parameters used here are g = 10, m, = 0.8 GeV, m, = 0, and g,, = 9.1. The phase transition shown here is analyzed in more detail in Fig. 4.


Let us look at these results more closely: We have seen that the discontinuous phase transition is caused by the large curvature of C;‘(O) at v”=O. In fact,

a2 T(“(O, 0; 0,O) =- Zk’)(O)cCln C -+ - cc

aiT as v”+ 0. (5.5)

The reason for this singularity is the following: The quantity (5.5) is that part of the nucleon loop contribution to the a*~* vertex (with all external momenta equal to zero) which depends explicitly on the density. The full nucleon loop contribution is graphically represented by Fig. 9. For 6 -+ 0 the loop integral behaves like 1 d4k/k4. However, since due to the time ordering at least one of the internal nucleon lines must be a hole line, the Pauli principle forbids (kj < J+ and thus excludes the infrared singularity. However, since the quantity (5.5) is only one part (the Pauli correction term) of the full nucleon loop contribution, the infrared singularity is present. In other words, once we include also the “Feynman part” Xi*’ in the pion self-energy, its curvature at v’= 0 will be finite. Thus, one is naturally led to include the nucleon vacuum fluctuation term, which will be discussed in the next subsection.

At the density where Zh”(O) = mz/2 at v” = 0, the abnormal state becomes stable with respect to variations in fi. This occurs at

1 mn /+=-L,

vf 2 g (5.6)

which, with the present parameters, corresponds to 0.3 p0 (p0=0.17 fmm3). The “critical density” p; (Fermi momentum pFc) and the corresponding value x, = CC/u are determined by the two equations (3.18); i.e., in the present case

m,*’ = m,*’ = 0, (5.7)

I FIG. 9. Nucleon loop contribution to the &r* vertex with zero external momenta.


which can be brought into the form

C-z3+z=Cz z-(z2-l)ln- (

z+l z-1 >

(5.8b)

with

c=2(%)‘: z=2(=)‘. (The trivial solution z = 1, I, = 0 corresponds to the case (5.6), where the abnormal state turns from an energy maximum into an energy minimum.) With the present parameters we obtain p; z 0.9 p0 and x c z 0.55 (see Fig. 4). The critical density where the Wigner mode solution becomes an absolute minimum lies slightly below PL (P F z 0.8 pO). The instability of the normal state can therefore also be viewed in the following way: In the Goldstone phase we can use the equation mX’= 0 to bring m,*’ into the form

L+cp: rn*’

mZ mkE, (Goldstone phase)

with E, = dmN. The quantity (5.9) determines the cr meson exchange part of the Landau-Migdal parameter f,, (obtained by setting -d;‘(0)=m,*2 in Eq. (3.34d)). The attraction increases monotonously as the density increases, until a critical density is reached where m,*’ vanishes. This behavior is due to the attractive contributions from the nonlinear terms, contained in the mass parameter fii’,. Note that if we had replaced fiz -+ rnz, the quantity ( -A; ‘(0)) would be positive definite.

5.2. The Role of the Nucleon Vacuum Fluctuation Term

Including the nucleon vacuum fluctuation term in (4.36), we obtain the energy density

44 E”’ = E”’ - 8712 F(J+,),

where EC” is given by Eq. (5.1). For simplicity we assume the renormalization points pL, = pL, = pn = 0. This is the model investigated in Ref. [4]. In the calculations below we use m, = 0.6 GeV and g, = 4.8, which are very close to the values used in Ref. [4]. (Note that we fixed g = 10.) From Eq. (4.38~) we obtain the corresponding expression for the pion self-energy

ii ,?y(O) = q/‘(O) -& Lhlnh-.h+l).


CF) is the full pN loop contribution. For simplicity we again refer to the case of exact chiral symmetry (C = m, = 0). The second term in (5.11), which is the vacuum fluctuation term, is negative definite. It vanishes for v”= o and takes the value -~hg*/(2x*) at o”= 0. The solution of the self-consistency equation

-d,‘(o)=rs~+cy’(o)=o (5.12)

is illustrated in Fig. 10 for p = 0.2 fm P3. The vacuum fluctuation term is small for C= u, but it changes the form of the pion self-energy drastically for smaller values of 0”. The curvature at i?=O is now finite. The stability criterion (3.17a) is obviously satisfied for the Goldstone mode solution. As the density increases, the full line in Fig. 10 moves upwards without changing its shape appreciably. If the full line touches the dashed one at o’= 0, the Wigner mode becomes stable (see Eq. (3.17b)) and beyond this density the “effective pion mass” (-d;‘(O)) is positive; i.e., Eq. (5.12) no longer has a solution. The Fermi momentum corresponding to this critical density is, using L’,(O) = g2/rr2(& -m&/2),

44 7t2m2 ,>m; p:,=y+ zg2 =y2' (5.13)

and therefore the critical density is roughly 15 times the normal nuclear matter density independent of the choice of parameters. There is certainly little meaning in using the present schematic model at such high densities, but the conclusion we can draw is that due to the large attractive contributions of the vacuum fluctuations to the pion self-energy the chiral phase transition becomes a continuous (second order) one and the critical density is shifted to very high values. This change of the

FIG. 10. Illustration of the self-consistency equation (5.12) for p = 0.2 fm m3. The dashed line shows -fi5, = 1’(u2 - 15’) and the full line shows the pion self-energy (5.11). As the density increases, the full

line moves upwards and, at the critical density (5.13), touches the dashed line at G/L> = 0. Compare this with the fictitious case shown in Fig. 3.


FIG. 11. The nucleon elfective mass (a) and the binding energy per nucleon (b) when the nucleon vacuum flutuation term is included. The parameters are g = 10, nt, = 0.6 MeV, NI, = 0. and g,,, = 4.86.

phase transition from a discontinuous to a continuous one due to the inclusion of the nucleon vacuum fluctuations has also been observed in Ref. [28].

In Fig. 11 we show the effective nucleon mass and the binding energy per nucleon. We see that for densities larger than the critical density (5.13) there is no minimum in the binding energy curve. One can easily show from Eq. (3.33) that for 6 = 0 the binding energy per nucleon actually has a minimum at a density which is, however, slightly below the critical density, and hence the abnormal state is still unstable there with respect to variations in 6. The results are repeated, on a smaller scale, by the dashed lines in Fig. 12.

a b 10 ,,,, ,,,, ,,,, ,,,, ,,,, ,,,,

-20 ” ” ” “‘1. ” ” ” ” ” ” I”” 0 a* 0.2 0.3

density [frn-‘j 0.5 0.6

FIG. 12. The full lines show the u meson mass parameter of Eq. (5.4) and the nucleon effective mass (a), and the binding energy per nucleon (b) when the vacuum fluctuation term due to the sigma is

included in addition to that due to the nucleon. The parameters are the same as those used in Fig. 11. The dashed lines reproduce the results shown in Fig. 11 for comparison.


The 0 meson self-energy in the present model is given by the sum of the third and fourth terms in Eq. (4.39~). Once the self-consistency equation (5.12) is satisfied we have

(Goldstone mode), (5.14)

where C was introduced in Eq. (5.8~). The second term on the right-hand side of (5.14) is due to (but not identical with) the repulsive contribution due to the nucleon vacuum fluctuation term. This sizable term excludes the pole in the c meson propagator at 4 = 0 and thereby also the discontinuous phase transition observed in the quasiclassical approximation.

Some results are shown in more detail in Table I in the column “QCL + N.” There we split the binding energy per nucleon at the saturation point into the four contributions E,, EN, E,, and E,, which are due to the first three terms, the fourth, fifth, and sixth terms in Eq. (4.36) respectively. Further we give the numerical value of (5.14), which, however, seems extremely large. The next three rows give the three contributions to the Landau-Migdal parameter F, shown in Eq. (3.34d), i.e., Fg’, Fg’, and Fhw) corresponding to the first, second, and third terms in (3.34d), respectively. Finally we show the total F. and the resulting compressibility.

The above discussion has shown the necessity of including the nucleon vacuum fluctuation term in order to exclude the discontinuous phase transition at normal densities. The actual size of its contribution to various physical quantities is, however, extremely large, as can be seen from the third and sixth rows of Table I or from Fig. 10.

TABLE I

Nuclear Matter Properties in the Hartree Approximation”

Case QCL+N QCL+N+u QCL+N+a+n

(PO, -WA ) 4 WV) EN WV) .% (MeV) E, WW -d,‘(O)/7g FX’, Fb”’ F’“’ 0 FO K (MeV)

(0.16, -15.9) (0.17, - 17.4) (0.18, - 15.8)

-42.5 -45.8 -39.5

26.6 30.0 36.1

0 - 1.6 - 16.8 0 0 4.4 5.1 5.0 2.2

0 - 0.01 7.63 - 2.33 - 2.29 - 7.42

1.81 1.81 0.62 - 0.52 - 0.49 0.82 115 131 482

y The results shown in the three columns are based on the expressions (5.10), (5.17). and (5.18) of the energy density. With g= 10 and M, =0.783 GeV in all three cases, the other parameters are m, = 0.6 GeV, M, =O, and 8, =4.86 in the first two cases, and m, = 1 GeV, m, =0.14 GeV, and g, = 2.81 in the third case. The first row shows the resulting saturation density p0 in fm-’ and the corresponding binding energy per nucleon E,/A in MeV. For further explanation, see the text.


5.3. The Role of the Meson Loop Terms

We now discuss the role of the last two terms in the Hartree energy density (4.36), which represent the “zero point energy” contributions of the boson fields. As we discussed before, these terms are ill defined since yg and y, are not positive definite, which is especially serious for the pionic contribution. The sigma meson contribution has been treated in many works [ 1, 5, 61 by replacing the Hartree mass parameter &i by the quantity m, *’ defined in (5.4), which is the self-consistent rr meson mass parameter in the quasiclassical approximation. In a similar way one can replace fiz by m, *’ of Eq (5 3) which is the self-consistent pion mass parameter . . , in the quasiclassical approximation. (By “self-consistent mass parameters” we mean the quantities (-d;‘(O)) and (-A; ‘(0)) satisfying Eq. (3.16) and the Goldstone theorem (3.13) for a given model energy density E.) Thus, in order to obtain some insight into the role of the meson loop terms, we replace

(5.15)

in the last two terms of the energy density (4.36). As discussed earlier, these mass parameters m,*’ and m,*’ . include the density dependent part of the pN polarization term at q = 0. We must note, however, that there are many problems associated with such an ad hoc replacement: First, although it seems quite natural, it is not unique. This is clear from the fact that the replacement (5.15) cannot be done from the start, since then the cancellation of the divergences, expressed by Eq. (4.34) would no longer be valid. If one includes the “density parts” of the polarizations consistently in the ring energies (4.1 lb) and (4.1 lc) from the beginning, the finiteness of the result is due to the l/k’ behaviour of the polarizations for large meson momenta [15]. Therefore it is not possible to include the polarizations uniquely by replacing them by their values at k = 0. Second, the prescription (5.15) destroys chiral symmetry. In particular, the Goldstone theorem (4.38a) no longer holds: If we differentiate the energy density k(H) modified due to (5.15) with respect to 6 and write the result in the form (4.38a), thereby defining a modified quantity -d^; ‘(0) = fif, + f’,(O), the quantity f,(O) is obtained from (4.38~) by replacing

WY,) + Cl+ c,PW,) (ct = 0,7c)

with Zf) and Zr) given by Eqs. (5.4b) and (5.2). The presence of the factors c, makes it impossible to interpret the result of the differentiation as a pion self- energy, and, moreover, gives rise to the following further problem: For fi -+ 0 these factors behave like In i7 (compare the discussion around Eq. (5.5)). This has the


consequence that the second derivative of the energy density with respect to i? (i.e., the quantity --a;‘(O)) goes to minus infinity as fi+ 0. (The first derivative is finite due to the factor fi in (4.38a).) In other words, due to the ad hoc replacements (5.15) the point v”= 0 is always a maximum of the energy density, and the transition to the abnormal state cannot take place no matter how high the density is. On the other hand, for the cases discussed below, the factors c, are quite small compared to 1 for densities around the normal nuclear matter density and iJ/v 2 0.7. Then the nucleonic term (5.11) is dominant in g‘,(O) and we get a behaviour similar to that shown in Fig. 10. For values c//v 2 0.5, however, the c, become large and the “depth” of f’,(O) at fi = 0 becomes infinite, which prevents the Wigner mode from becoming stable. Thus, while the method (5.15) of including the effect of the polarizations might be reasonable for normal densities, where we expect fi & u, it certainly gives incorrect results for high densities. A more consistent treatment of the self-energy corrections to the meson propagators by starting from the HF framework discussed in Section 4 is now in progress [ 181. One can show that if the polarizations L’:)(q) and C:‘(q) are taken into account consistently from the start, the three diseases discussed above are cured. Such an attempt, using approximate forms for the polarizations, has been made in Ref. [3].

With these limitations in mind, let us discuss the role of the [T meson loop term; i.e., we consider

where E(‘) is given by Eq. (5.10). As noted in Refs. [S, 61, the size of the g meson loop term depends sensitively on the assumed value for m,. If we use parameters identical to those in the previous subsection (g = 10, m, = 0.6 GeV), its contribution is negligibly small, even for higher densities. This can be seen from Fig. 12 by comparing the results for the binding energy per nucleon and the effective nucleon mass including the g meson loop term (full lines) with those including only the nucleon loop term (dashed lines). Figure 12 also shows the mass parameter m,*’ which is used in the calculation of the meson loop.

Comparison of the values in the second and third columns of Table I shows again the small influence of the 0 meson loop term. Concerning the parameter FO, we must note that due to the replacements (5.15) in the energy density the quasiparticle spectrum also changes due to Eq. (3.20a) and in fact now depends explicitly on the density. Therefore also the first term in Eq. (3.34d) is non-zero, and the rrNN vertex (3.23a) is changed from its Hartree value (4.42b). For the present case, however, these modifications are negligibly small, as can be seen from Table I.

Finally we consider

(5.18)


where EC3’ is given by Eq. (5.17) and the last term represents the effect of the pion loop. Use of the same parameters (g = 10, m, = 0.6 GeV) leads, however, to an unreasonably high value for m,*' and therefore also for 9, even at low densities. This can be foreseen, since using simply the values of 0” and p at the saturation point shown by the full lines in Fig. 12, we obtain mx/m, = 2.6. This somewhat artificial problem, which has also been discussed in Ref. [ 171, arises because the mass parameter m,*' of Eq. (5.3) is a “self-consistent” mass parameter (in the sense defined above) only in the quasiclassical approximation. Only in that case does it agree with the inverse pion propagator (-A; ‘(0)) which satisfies the Goldstone theorem, Eq. (4.38a), and vanishes for exact symmetry once the energy density is minimized. When we go beyond the quasiclassical approximation, no such con- straint on rnx' exists. If one insists on using m,**, one must raise the value of m, up to 1 GeV or higher, while keeping g = 10 fixed, in order to counterbalance the strong repulsion due to ,ZZ I’) by the attractive contributions due to the nonlinear meson interaction terms contained in the parameter fizi. For g = 10, m, = 1 GeV, M, = 0.14 GeV, and g,, = 2.8, we get the results shown in Fig. 13. Even in this case the repulsion due to the pion loop is substantial for higher densities, as is clear from the increase of m,*' with increasing density, and accordingly the value of g, must be reduced in order to get the right saturation point. The results using this parameter set are collected in the last column of Table I. Due to the larger value of m,, the CJ ring energy now cancels about half of the nucleonic ring energy. The repulsive pionic contribution is still moderate at the normal nuclear matter density. The value of (-A;‘(O)) is changed little, compared to its value in the two cases discussed above, with the largest contribution still due to the nucleon vacuum fluctuation term. The modifications of the quasiparticle spectrum due to the replacement of 5; by m,*' m the energy density lead to enhancement of the aNN vertex 7, of Eq. (3.23a), thus increasing the contribution to F, due to the second

a

FIG. 13. The meson mass parameters of Eqs. (5.3) and (5.4) and the nucleon effective mass (a), and

the binding energy per nucleon (b) when the vacuum fluctuation term due to the pion is included in addition to those due to the sigma and the nucleon. The parameters are R= 10, M, = I GeV, m,=O.l4GeV, and g,,,=2.8.


term in (3.34d). This additional attraction, however, is cancelled largely by the contribution due to the first term in (3.34d), and the result is a positive total value F0 = 0.82 and a compressibility of K = 482 MeV. We also wish to mention that the behaviour of the effective meson masses with increasing density shown in Fig. 13 is qualitatively consistent with recent results obtained in effective quark theories [34].

Altogether, the assessment of the vacuum fluctuation energies due to the meson fields poses many problems, especially concerning the pionic contribution. Based on the present approximate treatment we cannot draw firm conclusions. In our estimate, the repulsive quantum fluctuation contribution due to the nucleon dominates over the mesonic contributions at normal densities. It is partially cancelled by the attractive 0 meson loop term, the degree of cancellation depending on the assumed value for m,. The pionic contribution is repulsive and increases with increasing density. A more reliable treatment of the meson loop contributions based on the formalism of Section 4 will be discussed in a future paper [ 181.

6. CONCLUSION

In this paper we used the chiral g--w model to study the role of chiral invariance in the nuclear medium. We discussed the formalism needed to carry out calculations in nuclear matter, particularly emphasizing a renormalization procedure consistent with symmetry requirements. Our main object was to elucidate certain constraints imposed by chiral symmetry and baryon current conservation. In particular, we focussed our attention on the Goldstone theorem in nuclear matter and its connection to the energy density and the problem of chiral phase transitions. These general considerations have been illustrated in the Hartree approximation, where we have shown how one can conveniently discuss the phase transition in terms of the pion propagator. A reliable treatment of the meson vacuum fluctuation terms requires at least the inclusion of particle-antinucleon and particle-hole polarization insertions in the meson propagators, and for this reason we also discussed some formal aspects of the energy density in the HartreeeFock approximation. Details of the procedure for taking into account these polarizations consistently as well as the numerical work required will be discussed in a separate paper [ 181.

At present there is much interest in the possibility of chiral phase transitions on the quark level in connection with the existence of the quark-gluon plasma at finite temperature [29, 303 and finite temperature and density [31-343. The developments of the present paper, utilizing the equivalence of the self-consistency condition and the Goldstone theorem in order to obtain information on the nature of the phase transition, might turn out to be useful in discussing these topics, too. For example, it is interesting that in the framework of the Nambu-Jona-Lasinio model in the mean field approximation a relation analogous to (3.15) has been derived in Ref. [33] for finite density and temperature.

Apart from chiral symmetry, some developments contained in the present paper might be of general interest for recently developed and widely used many-body


theories. For example, the constraints imposed by the baryon current conservation on the omega-nucleon interaction in the medium and the renormalization of the omega meson field have been investigated systematically. The results of Ref. [25] on the part of the Landau-Migdal interaction which arises from the density dependence of the mean fields produced by the CJ and o mesons have been generalized. For the case of the Hartree-Fock approximation we have demonstrated explicitly the connection between more conventional approaches, in which the energy density is viewed as the sum of kinetic and potential energy terms, and the diagramatic expansion method, thereby putting some of the formal developments in Ref. [ 151 on a somewhat broader basis.

In conclusion, the chiral (T--W model provides us with a useful tool for obtaining insight into the way in which chiral symmetry governs strong interaction processes in the nuclear medium. We attempted to clarify some of the consequences of these constraints and to illustrate them by simple examples. Further attempts in this direction are necessary in order to decide whether the model is sufficiently powerful to study many-body problems for purposes going beyond a formal exploration of symmetry constraints.

APPENDIX A: LOW-ENERGY THEOREMS IN THE o MODEL

1. Proof of Eq. (2.39)

In the following we will discuss two possible ways to derive Eq. (2.39).

(a) The axial Ward-Takahashi identities in the limit q + 0, where q is the momentum of the external axial vector field, impose so-called “consistency relations” or “low-energy theorems” on the strong interaction vertices. These identities are given in Section 5b of Ref. [ 111, Eq. (6) (referred to as Eq. (L5b.6)), for the mesonic T matrices (T), and in Eq. (11) for the 0, n-irreducible parts (r). The latter quantities are related to the T matrices by

T(P 1 7 -**> pn ; q 1 , **., qm) = r,, ,(p,, . . . . pn; ql, . . . . q,,,) + 0, n-reducible part.

Here the arguments denote the incoming meson momenta. The identities for the T’s are

-~~:_.;~A+,(PI, . . . . d,, . . . . j.‘,,; 41, . . . . qm, p,)-ic6,,6,, 6”‘. (A.l)

The notation GS means that the variable qs must be omitted. The isospin indices i, and j are written in the same order as the pion momentum variables. For n = 0 the second term on the right-hand side of (A.l) is zero by definition. We have

~,.,(P, -Pi) = id,‘(p); &(; p, -p)=id;‘(p)@; ~*,o(O;) = ii,, (A-2)


where j, is the a-irreducible part of the c source as used in the main text. At first sight it therefore seems that Eq. (2.39) is simply the n = 0, m = 1 case of Eq. (A.1 ). One must note, however, that in Ref. [ll, Eq. (L5b.3)], T,,O is assumed to vanish identically, in which case (A.l) for n = 0, m = 1 reduces to the Goldstone theorem (2.38) which holds only at the physical point 6,. One can, however, show quite easily that Eq. (2.39) is a necessary consequence of the relations (A.l) for (n, m) # (0, 1). (A direct proof of Eq. (2.39) will be given below.) For this, define a function (we leave out isospin indices)

F,,,(E)=dT,,,(;p, -p)-T,,o(O;)+ic. (A.3)

We know, due to Eqs. (2.34a) and (2.38), that FO,, vanishes at the physical point: FO,i(&,) = 0. One can easily show that also all derivatives with respect to 6 vanish at the physical point: F$(i&,) = 0 for k = 0, 1, 2, . . . . It then follows that F,,i(C) = 0 for all 6. To show that all derivatives vanish at v”,, we note that by generalizing the method discussed in Section 3.1 one can show that

f f2:iL(P 1, . . . . P,,; 41, . . . . qm)=~,+,~o,,,(p,, . . . . pn, O,, . . . . O,;q,, . . . . q,,,); (A.4)

i.e., taking the kth derivative with respect to fi generates a Green function with k additional (amputated) cr meson legs with zero momenta, indicated by the symbol kc”. Using (A.4) one finds that the-kth derivative of Eq. (A.3) is given by

Fhk;( 17) = F 0 + !@, 1(3,

where for general integers n,m the function F,,,, is defined by

F,.,(~)=v”T,,,+,(p,,..., ~n;q~>..., q,,,O)

-c r,+ I.??- ,(PI2 . ..> Pn, qs; 41, “.3 i,, -..* qm) s

(A.5 1

(‘4.6)

Due to Eqs. (A.1 ) and (AS), all derivatives of F,,, vanish at Co, and therefore F,,, vanishes identically. This establishes Eq. (2.39) as the generalization of the Goldstone theorem (2.38). Moreover, from the above it is clear that we have generally

Fck’ (fi) = F mm n + k%n(fi),

which confirms that, if the identities (A.l) hold at the physical point, they also hold for any value of 0’.

(b) Let us now discuss a direct derivation of Eq. (2.39). For this, we note that, according to our discussions at the end of Section 2, once we regard fi as a free parameter, Eq. (2.29a) is not satisfied and therefore, a priori, all Green functions


calculated from the lagrangian (2.7b) contain also (T tadpole insertions. Let us characterize those Green functions by a hat; e.g., f,, contains also the Feynman graph shown in Fig. 14. The Goldstone theorem was derived in Ref. [ 1 l] under the assumption (a) = 0. It is given in Eq. (L5b.4), which is equivalent to Eq. (2.38). If the restriction (0) = 0 is lifted, it is changed to

i(~+(a))~~.2(;0,O)=C; (A.7)

i.e., instead of 0” we have the expectation value (4) of the original scalar field on the left-hand side, and the inverse pion propagator includes also the g tadpole insertions. Since the quantity T,,O is by definition the o-irreducible part of the one- point Green function (a), we have

r1,0= -f2,,(O,O;)<a>. (‘4.8)

The a-reducible part of f,,2 (i.e., the part mvolving c tadpole insertions) can be separated as follows,

G,,(; (40) = ro,,t; 0, 0) + (a) f,,,(O; 0, O), (A.9)

where the arc2 vertex f,,? satisfies (see Eq. (A.l))

v”f,,,(O; 0,O) = f’z,o(o, 0;) - f&, 0, 0). (A.lO)

The second term on the right-hand side of (A.9) is graphically represented by Fig. 14. Note that since the vertex f,,2 itself contains all possible (T tadpole insertions in the an2 vertex, the connected part of the second term in (A.9) includes all a tadpole insertions in the pion self-energy. Using (A.9) and (A.lO) in (A.7) we obtain

c = iu”T,,,(; 0, 0) + i(a) P,,,(O, 0;)

= iW,,,(; 0, 0) - iT,,,(O;) = -a d,‘(O) +j,, (A.ll)

FIG. 14. Graphical representation of the second term on the right-hand side of Eq. (A.9). The hatched rectangular box stands for the vertex r’,,,.

595:188/1-X


where in the second step we used (A.8) and in the third one (A.2). Equation (A.ll) is the same as Eq. (2.39).

2. Demonstration of the Identity (2.40)

The low-energy theorems for the mesonic T matrices can be derived from the identities for the irreducible vertices, Eq. (A.l), and are given in Eq. (L5b.6) of Ref. [ 111. Eq. (2.43) we need the special cases

W -4; q, 0) = i(d,‘(q) - d,‘(q)) (A.12a)

v”T(q,, 42; -41-q2~O)=d,(ql+qz)d,l(ql+qz)T(q1,q2, -qj-q2;)

-d.(q,)d,‘(q,)T(qz;q1, -41-42)

-d.(q,)d,‘(q,)T(q,;q,, -41-92) (A.12b)

~T”“%IA~, -ql-q2,O)=~~d,(q,)d,‘(q,) T’%,;q,, -ql-42)

+6~1id,(q,+q,)d,‘(q,+qz)T’y(-q1-q2;q1,q2)

+@‘d,(q,) d,‘(qz) T”“(qz; q,, -ql-42). (A.-)

If we insert these relations together with the identity for the nNN vertex [35],

into Eq. (2.43) we obtain

C,(O)= -i%f$TrS(q)

+3i*2zA s d4q

(2n)4 (d,(q) + d,(q))

(A.12d)

(A.13a)

(A.13b)

+34.(q,)d.(q,)d,(q,+qz)T(-q,-q,;q,,q,)), (A.13~)

where we also used (2.42). The terms (A.l3a), (A.l3b), and (A. 13~) correspond to (2.43a), (2.43b), and (2.43c), respectively. Comparing (A.13) with (2.41) we arrive at (2.40).

APPENDIX B: PROOF OF RELATIONS (3.23)

The spinor f(p) obeys the Dirac equation

(B.1)


and the normalization condition

i(P)(qJf(P)= 1.

Equation (B.2) is derived in Appendix B of Ref. the end of this appendix. We differentiate (B.1) then multiply from the left byJ‘(p):

a& -$ (f(P) a”;p;p)

( I > PO = Ep f(P) +f(Pl aS a’q z PO = Ep

Due to (B.2) we obtain

(B.2)

[26], and we will comment on it at with respect to the mean fields and

f(p)=0 (z=IT, V). 03.3)

aE,- a,-

- -fjp) as ;;(p) / f(P) (2 = I?, iv). PO = Ep

(B.4)

Let us calculate the derivative of the nucleon propagator using the formula (3.9). For z = 0” we obtain

(B.5)

Here, as in Eq. (3.10), we must take only the o-irreducible part, since we under- stand that the propagator S does not include 0 tadpoles, i.e., is 0 irreducible. Further evaluation of (B.5) follows closely the steps which led from Eq. (3.10) to (3.12): We use the equation of motion for c to replace -8, in (B.5) by (Z, 0 + fiz + &&:)a. Since C+ commutes with the nucleon field at equal times, the d’Alembert operator can be taken out of the T product, and we can then use the definition of the irreducible aNN vertex.

(01 WW)$(-MY)) 10) = -ijd‘u jd%jd4.YJS(X~-d)

xf,(u’-y’,u--‘)S(u-x)d,(y’-y). 03.6)

To obtain the a-irreducible piece of (BS), we again must replace d, + B,, effec- tively (see Eq. (2.21b)), and we obtain

as(~~x)=ij~4yjd4u’ j ~4uS(x’-u’)f,(u’-~y,u-y)S(u-x). (B.7)

In momentum space this becomes

as(p) . -= ZS(P) ~A& P)S(Pf ai7 (B-8)

114

or

BENT& LIU, AND ARIMA

as -l(p) aa = -irAp, p).

A completely equivalent treatment for the case z = K+’ gives

as-b) aa,

= -ir;o& p).

(B.9a)

(B.9b)

Using Eqs. (B.9) in (B.4) we arrive at Eqs. (3.23). Let us add a few comments concerning Eq. (B.2): It is equivalent to the condition

[26] that the quantity C,f,(p)x(p) be the residue of S(p) at the quasiparticle pole pO = E,. For zero density Ed --t E, = dm. If we choose the renormalization point pN = mN in Eq. (2.14a), where rnN is the nucleon mass parameter in the lagrangian, then M, =rnN, and for zero densityf(p) -+ MN/E, u(p) with Uu = 1, as follows from Eq. (B.2). For other choices of pN the spinors in free space, normalized as UIA = 1, must be multiplied by a (finite) renormalization factor, which depends on the self-energy and is determined from Eq. (B.2).

APPENDIX C: PROOF OF EQ. (4.19)

Let us start from the expression for E,, Eq. (4.14), in the HF approximation. We split it into two parts,

E,= EL... + EI,,,,~

with

(C.1)

cc.21

where we used the notations introduced in (4.16a) and (4.17a) for the meson self- energies, Let us also split the nucleon self-energy as C, = zr,,i + EN2 according to


the two terms in (4.15). Consider first the second term in (C.l ), which has the form [ 151 (see Eq. (4.16b))

i d4k 1

acb,a,, -k’A,,-

2 (2n)4 Jko

= -fj~~&$k”A~h(k)Tr(~hdS~~q)I.YS(q))

=;jg‘ijg$q” ( as(k)

Aah Tr TbS(k + q) r”x (C.3) 0

where the second equality follows from a partial integration and interchanging k and q. We combine (C.3) with the part involving TiNi in the first term of (Cl):

=-jg%j&$ A.,(q) Tr(S(k) rbW + q)r?

- j$+ j$$ ( A,,(q) Tr S(k) Tb “~Tk’ ‘) f u o )(k”+$ (C.4)

If we replace k + k - q and then q -+ -q in the last term and note that Aab( -4) = A,,(q) (this relation follows from the form of the self-energy (4.16b) and the Dyson equation (2.25)), it is seen [ 151 that the last term is equal to -A,,, and hence

Am= -; jf$, j$ A.,(q) Tr(SW) rbW + q)W

Proceeding in the same way for the pionic part A,, we obtain

E i d4k

Lnuc = 5 s - (TWO,,, + 34, &. . ..) (2n)4

i d4k --[~‘WS~N),

2 (2x) (C.5)

as in Eqs. (4.19) and (4.20). This contribution is shown graphically in Fig. 7a. Consider now the purely mesonic terms. The first term in (C.2) appears, in the same


form, in (4.19) and is shown graphically in Fig. 7c. The sum of the last two terms in (C.2) is

=B,+B,. (C.6)

We now consider the terms B, and B, separately and treat them in the same way as A,, in Eq. (B.4); i.e., first we make a partial integration in k,, which produces two terms, one involving simply a product of propagators and the other involving one derivative. In the latter term we replace k + k - q and then q + -q to obtain

B, =3(112Z,)2 v”‘j$, j$ A,(k) A,(q) A,@ + 4)

B2=3(~‘z,$ 6’ j$$, j$$

(C.7)

A,(k) A,(q) A,(k + q).

Therefore B can be written as

Using this in (C.2) we obtain

as in Eq. (4.19). The second term in Eq. (C.8) is shown graphically in Fig. 7b.

APPENDIX D: LOWEST ORDER G AND TI SELF-ENERGIES IN FREE SPACE

In the potential (3.3b) there enter the inverse propagators in free space

A,‘(k’)=k’-mf-C,,(k2) (a=cT, 71) (D.la)

with

C,,(k2) = z&k’) + drn$ - (Z, - l)k’, (D.lb)


where the unrenormalized self-energy c, is calculated from the Feynman diagrams in Figs. 6b and 6c in lowest order, and the renormalization constants are determined from Eq. (2.14b) as

(D.2a)

(D.2b)

If gz > 2mz, the real part of L?;,(/,L~) should be taken in Eq. (D.2b). The unrenormalized self-energies are (see Figs. 6b, 6c and Eqs. (4.16) (4.17))

C,(k’) = L’(3Fl(m;) + 3F,(mz) + 6E.‘u2F2(k’, mf, mZ,)

+ 18L2u’F,(k’, rnz, mz)) - 8g2F,(m&) - 4g2(kZ -4m&) F,(k’, rnh, m&),

L?‘,(k’) = A’(Q’,(m~) + F,(m%) + 4%‘u2F,(k2, rnz, mz))

- 8g2F,(mk) + 4g’k’F2(k2, m&, mZ,).

Here we introduced the functions [12]

F,(m2,_ijd’y& (2n)4q2-m2’

F,(k’, m:, m:) = i I

d4q 1

(271)4(q2-m:)((k-q)2-m~)’

which are calculated by using dimensional regularization,

with

’ f2(k2, m:, m<) = I d-u ln(m: + (m: - mf)x - k2x( 1 - x)). 0

(D.3)

(D.4)

(D.5a)

(DSb)

(D.6a)

(D.6b)

(D.6c)

From the expression (D.3) and (D.4) the counterterms (D.2) are calculated, and the results are inserted into (D.lb). For k’ = 0, which is the case required in Eq. (3.3b), we obtain


16n2Z,r(0) = 6A402(f2(0, rn$ mZ,) -fi(&, mZ,, mi))

+ 18A402(f2(0, m$ mz) -f2(p2, m3, mi))

- 16g2m&(.f2(0, mZ,, nz&)-f2(pz, mZ,, mh))

+4~4v2~~f;(~~,m~,m~)+4g2~~~~f;(~~,m~,m~)

+@4tf2(& nail m~)--f2(Lt2,, 4, mh)), (D.7)

16n2C,,(0) = 412~'(f2(0, rn;, rni) -f2(pi, rn:, mz))

+4~4u'~~f;(~~,n~~,,,~)+4g2~L4,f;(~~, mZ,,mk). (J3.8)

Using the form (D.6c) one can calculate these expressions analytically. In particular, for m, = m, the function fi takes the simple form

f2(p2, m2, m’) = In mz - A,,,

where

A = 2 - 2yiLm arctg(yji,’ ) Pm 1

if pc2m

2-y,,lnl(l+y,,)l(l-y,,)I if ,u>2m

with y,, = dm. Th e se lf- energies at finite density for arbitrary choice of the renormalization points are obtained by inserting (D.7) and (D.8) into (4.38~) and (4.39~).

In the Hartree approximation, the dependence of the energy density on the renormalization point pLt, can be absorbed into a redefinition of the coupling constant g,, as is clear from the formulae in the main text. (For the explicit form of drn$, see Ref. [20].)

ACKNOWLEDGMENTS

The authors express their thanks to K. Yazaki and M. Asakawa for many helpful discussions.

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Chiral Symmetry in Nuclear Matter