Nonperturbative approximations in the Walecka model beyond the mean field approximation

Nuclear

Physics A486 (198X) 623-633

North-Holland.

Amsterdam

NONPERTURBATIVE BEYOND

APPROXIMATIONS THE

MEAN

Thomas

LIPPERT

Received (Reviwd Abstract:

On the basis

derive

nucleons. dercribe

of formal

nonperturbati\e

wlutionh

IN THE

FIELD

and Werner

IX March

1988

16 May 1988)

for the nucleon field in Walecka’s

approximations

MODEL

SCHEID

neglecting

free mesonic

In lowest order the method yields the Hartree-Fock the propagation

WALECKA

APPROXIMATION*

of particles and hole5 correlated

cliective meson theory we

degrees of freedom

equations.

to particle-hole

and anti-

Higher approximation\ excitations.

1. Introduction

Walecka’s effective meson field theory ‘-‘) describes the nuclear many-body system as an interacting field system of nucleons and mesons, where the meson parameters, i.e. coupling constants and masses, are adjusted to the properties of nuclear matter and nuclei. Within the framework of relativistic quantum field theory one has - at least in principle - a well-defined basis for developing certain approximations. The nonperturbative approximation of lowest order is the mean field approximation (MFA) ‘.‘). It is obtained from Dyson’s equation “) for the one-nucleon propagator, which sums up selected perturbative contributions to the nucleonic proper selfenergy by restriction to tadpole contributions ‘). The inclusion of exchange graphs yields the Hartree-Fock approximation ‘.‘). In the present study we examine an alternative access to the problem, which has no correspondence to known classes of Feynman graphs and maintains the full time dependence at all steps. We start with a formal solution of the field equations similar to those used by Kallin ‘) who carried out perturbation expansions of fields in QED. In order to obtain nonperturbative approximations it is necessary to take certain perturbative field parts of all orders into account. We accomplish this by considering a special approach which neglects free mesonic degrees of freedom and approximates the nucleon field by a normal ordering prescription in the field equations. The normal ordered nucleon field can be reordered into nonperturbative spinor components, which are finally determined by a set of coupled integrodifferential equations. To lowest order this procedure yields the Hartree-Fock equations. Higher approximations describe the propagation of particles and holes correlated to particle-hole excitations. They have no counterpart in the usual Feynman graph expansions. l

Work supported

0375-9474/8X/$03.50 (North-Holland

by BMFT

(0661774).

@ Elsevier Science Publishers Physics Publishing

I)i\ision)

R.V.

T. Lippert,

624

W. Scheid / Non-perturharive

approximations

2. Field equations and basic approximations We restrict nucleons

our considerations

interacting

self-interactions

with

to the original

isoscalar,

scalar

‘,‘). The field equations

Walecka

and

vector

of this model

model ‘,“) which assumes mesons

without

mesonic

are given in the notation

of

ref. ‘) by (r,a,+

M)@(x) ==&L{gb(x), cc,(x)I+:~&{V,(x),

Y,+(x)]

I

c-0 + mfM(x) =Mglr(x), 4,(x)1, C-0 + 4

(1) (2)

V,(x) = $‘%[~clcx,,r,4(x)l .

(3)

The -y-matrices are hermitean, y = icup, y,, = p. The four-vectors have an imaginary fourth component, e.g., a, = (a, u4) = (a, ia,) with a,a, = a2 - ai. Furthermore, natural units are used: h = c = 1. The parentheses [ ] and { } denote commutators and anticommutators, respectively. G(x) is the fermion field of mass R/r, (cl(x) = $‘(x) yo; 4(x) and V,(x) are hermitean scalar and vector fields of mass m, and m, with dimensionless coupling constants g, and g, to the fermion field. The commutators are introduced to preserve charge conjugation invariance I”). The anticommutators pay regard to the fact, that V, is not an independent field and therefore does not commute with 9 [ref. I’)]. The mean field approximation (MFA) to the field equations (l)-(3) consists in replacing the meson fields with c-number fields which are determined as expectation values of the corresponding operators. This yields the following set of coupled equations (r,a, (-0 (-0

+ M -&a(x) + m%(x) + mt)u,(x)

- is,r,u,(x))lClo(x) = $g,(xl[&(x),

= 4igv(xl[&(x),

= 0,

(4)

rLob)llx),

(5)

r,(cldx)1lx) .

(6)

Here, 4,,(x) denotes the nucleon field in MFA and o(x) and v,(x) the classical meson fields. The MFA describes the dynamics of nucleons represented by the state 1,~). The nucleons move in the collective mean meson fields with sources generated by the current and scalar density of the nucleons. Written in the form of eqs. (4)-(6) the MFA equations also incorporate corrections due to the vacuum polarization. In order to derive an effective theory including correlations, we first give the formal solutions to the field equations (l)-(3). With the retarded Green function D:(x) (4 = s, v) of the Klein-Gordon equation, (-0

+ mt)DY,(x)

D”,(x) = 0 the solutions

= -c~‘~‘(x),

for xo
of eqs. (2) and (3) for the meson

4(x) = 4”(x) -

(7)

fields may be written

d“y G(x -y)ig,[i(y),

~lr(y)l,

as ‘):

(8)

V,(x) =

Vi(x) -

d”v Dk(x -yfhJ&y),

Here, 4” and VE are free fields, obeying

homogeneous

r,vK~)l. Klein-Gordon

(9) equations.

The expressions (8) and (9) can be interpreted by means of the adiabatic hypothesis I>). This hypothesis assumes that the interactions are adiabatically switched off in the remote past and future. We achieve this by damping the coupling constants, F>O,

g<,+ gC,exp (-4?‘ii/~

(10)

and taking the limit E -+ 0 at the end of all calculations. Due to this hypothesis the meson fields reduce to the free fields 4” and VF in the limit x0+ ---CC.Therefore, c$” and V(: describe free incoming mesons, which will be neglected in the following considerations. Let us introduce the retarded Green function S,(.X, y) for the Dirac equation (r,i),+M+g,U,(x)-cig,.U,(X))S,(X,yy)=-6’4’(X-y).

Then the formal

solution

for the nucleon

(11)

field can be written

+hL.IVJyh, + U(Y), $f(.Yfl). Here, again the coupling constants g, have to be replaced The incoming nucleon field J/,) obeys the Dirac equation (r,a,+M+g,Li,(x)+ig,U,.(x))~,,(x)=O.

112) by the convention

(IO).

(131

The quantities U, and U, are c-number fields which in general are matrices in spin space and may also contain nonlocal parts, not denoted explicitly. They are introduced to define the dynamics of the incoming nucleons and substitute the true interaction in the remote past. Therefore, they can be chosen proportional to the mean fields w(x) and U,(X) of the true interaction in order to obtain realistic initial conditions for the nucleon field. This will be clarified later on. The initial nucleon field can be expanded in a complete set of time-dependent spinor functions: $0(x) -II

(F,,(-x)b,, + k(xK)

(14)

with (15)

626

T.

The creation

Lipperi, W

Sdwid

and annihilation respectively.

operators for nucleons and antinucleons are denoted The nuclear ground state is described by

by b,, b:,, d,,, d:,

i.e. it is represented

/ ~(~i~-periurb~rii~e Rpprl),~imuti~~.~

by a Slater determinant

for independent

particles

in the remote

past. In order to obtain an approximate expression for the nucleon field we introduce the basic approximation replacing the anticommutators in (12) by normal ordered products: I{6,(x), 1WJ.4,

J/(x))+

:+(x)J/(-u): ,

r,Q(X)}--,:V,(X)Y~~(X):

(17)

*

The normal order prescription is to be performed with respect to the basis operators h,, , d<, defined in eq. (14). This replacement allows one to derive a non-perturbative expression for the nucleon field by an algebraic rearrangement of its perturbative parts. The normal order prescription necessitates a further approximation, namely the neglection of the antinucleonic parts of the expansion (14). Denoting the approximate nucleon field with G(x), we obtain the integral equation

with the abbreviations

3. Structure of the nucleon field and new order of perturbative parts Eq. (18) can be solved operators:

with an ansatz

for $,(x) in form of particle

and hole

where e.g.,

The spinors in eq. (20) are of perturbative nature, because, e.g., s<,(x) obeys a (20) free Dirac equation for times (x01< F-‘. The essential feature of the expansion is that it contains number operators n,, = bzb,,. They can be factorized out onto the right-hand side of any given operator term by changing at most the sign of this

T Lippcrt,

term. Looking number

W. Scheid / Non-perturhatioe

appro*_imution.s

627

at the effect of Ifi on the state Ix) (eq. (16)), we may eliminate

operators

the

in (20) using the identity

%1X)=

( >

i L5, Ix>.

(22)

,:I

This suggests to rearrange the expansion form without any number operators: 9(X) =C.f;,(.X)h,,
+

(20) with respect

C’ .f;“,,,~,,,(x)b~,h,,~h,“,+. o,‘“~c”l

to the state Ix) in a

...

(23)

The prime indicates that the sum does not contain any number operators. $_ is obtained from 9 by replacing n,, + Cf_, S,r,T,and reordering the corresponding terms into the form (23). These operations will be denoted by

R[$(-~)l= 4(x) _ Each spinor,f;,,,,,

in the expression

(23) includes

(24) contributions

A,(x) = 3
of all orders,

e.g. (25)

Of course the rearrangements indicated by eq. (25) (and similar equations for higher spinors) are impossible to carry out in praxis, because one has to know the infinite set of spinors 9Ce,C,, of the expansion (20). Therefore, we need an equation which determines the spinors J;,,,.,.. in a closed form without regression to the operations implied by R[ 1. From eq. (1X) we get

Due to the normal order prescription as 13) (for an outline see the appendix)

R[:~(4fk,~(dlf/Wl=

the last term on the r.h.s. can be written

W@4~,~b)$(y):l.

(27)

This relation implies that eq. (26) only depends on the spinors f;V,,,.... Therefore, eq. (26) can be used to obtain the equations of motion for the various components of the field 9. the expansion (23) of 4(x) into the r.h.s. of (27) yields _ Introducing _ an expression of the form

7I Lippert, W.

628

Scheid / ~~~~-per~l~rh~ii~~ R~p~ox~~ff~i~~s

The quantities Rfl,lI(~j, z) are functions of the spinors of eq. (23). Inserting eqs. (23) and (28) into eq. (26) and equating the c-number coefficients of both sides we find: f;, ,...
F,,,(x)hm-

y-r,”

d4yS,(x,y)(gsU,(yf+ig,U,(y)lf, ,,._
J

Here, 9; (i = 1,2) denote permutations. In the functions RY we have separated the indices into the groups N,, . _ . , CY,and CX,,+~,. . . , a,,,+, referring to the creation and annihilation operators connected with these indices, respectively. In order to obtain eq. (29), the following symmetry property has been used:

4. Approximations The approximation

of zeroth

order will be the first component ,(%)

In order to determiners from eq. (29):

-

C gi

y=s,v

of lowest order of 6(x),

(31)

=C,,L(x)&~ 0

we obtain

the following

i.e.

integro-differential

d4y%Xx-Yf exp(--EM) Rocc C

equations

.fd~)~,M~l.L(x)

(32) These equations

have to be solved with the initial lim L(x) I;,,- -W

conditions

= F,,(x).

Eqs. (32) are Hat-tree-Fock equations for the spinors limit F + 0, the arbitrary potentials US and UV disappear,

(33) J;,(x). If we perform the leaving the Hartree-Fock

T Lippert,

potentials

Ur, and

W. Scheid / Non-perturhatior

629

approximations

Up alone: (y*iJ, + A4 + U,(x)

+ U’;(x))f,,(x)

= 0

(34)

In the case that the potential g\U.,(x)+ ig, U,.(x) is chosen equal to U,(x) + U:(x), we get: ir.(x) = F,,(x). Then the initial condition (33) is trivially satisfied. The Hat-tree-Fock equations (34) are identical to those obtained earlier by Brockof mann 14) and similar to th ose of Miller 15), who neglected the state-dependence U;‘, which arises due to retardation. The first-order

approximation

is given by

$J”‘(x)=C.f;,(x)h, + C’ .f;~,,~,,,,(x)b,l,h,,,6,,,
(35)

<“,
Within this approximation the nucleon held describes the correlated propagation of a particle or hole and a particle-hole pair. The coupled integro-differential equations for the spinors,f;,(x) and.f;,,,,,,,,(x) are obtained from eq. (29). The first components of the field, i.e.,ff(x), become modified by contributions of the second components, and these in turn depend also on the functions Lt. Therefore, we get a rather complicated self-consistency problem to this order of approximation. The complete coupled equations for these functions can be found in ref. “) and will not be given here. We rather like to present a stationary version of these equations, which is obtained by linearization with respect to the spit-m-s f;K,cr,ce,(x). In the stationary case we have the following

separation

A,,,$, ..(.x) =.f;“,,?, (1) exp (-i&,,,,, This yields a time-independent nucleon Linearizing eq. (29) we obtain J%,f;“(x)= Yo(Y. v-t Iv+

density

U,,(x)+

of the time coordinate: (36)

.x,,) .

(xl$‘(x)@(x)lx).

u;(x)+

(37)

~P(x))L(x),

E
(38)

where we set E,,,,_ = E_+ EC,,- E,,, in order that the potentials entering eqs. (37) and (38) are time-independent. U,(x) and U:(x) are the stationary Hartree-Fock potentials: U,(X)=

-

1 1 g; ‘I_ 5,,’;r<,uc

d’y

exp (-m,lx-yl) 47+

-yl

($(Y)f,f;,(Y))T,,

>

(39)

(40)

630

T. Lippert,

The other potentials

W. Scheid / Non-pertttrhnriw

are found

ffpprl~xi?~fftiofl.~

to be:

(41)

+[“2-%1

1,

(42)

These equations may be further specialized for the example of homogeneously distributed, infinitely extended nuclear matter. The first-order approximation (35) can be extended by including higher field components. The form of the higher functions R:l,,,,,, needed in eqs. (29) can be straightforwardly derived by using eq. (28). This extension describes the effects of further particle-hole pairs correlated to the propagation of particles. Of course the corresponding self-consistency problems for the various components of the nucleon field increase rapidly in complexity.

T. Lipprrr,

M’. Scheid / NrJn-perrtcrharioe 5.

~pf)ro.yin?ation.s

631

Summary and eonciusions

The suggested method for treating the Walecka model rests on three assumptions: Firstly, we introduced the adiabatic hypothesis for the formal solutions; secondly, we neglected the contributions of the free mesons and antinucleons; and thirdly, we substituted the nucleon field by a normal ordered field. These approximations allowed us to derive nonperturbative expressions for the components of the approximate nucleon field. The components describe particle or hole propagation correlated with particle-hole pairs. In lowest order our method reproduces the time-dependent HF-theory. In the next order there is no correspondence to approximations obtained from usual Feynman graph expansions. Therefore, detailed investigations will be necessary to test the range of validity of this order of approximation. As test for the method nuclear matter calculations should be performed. It will be very interesting to clarify the effect of the particle-hole components of the field on the binding energy per nucleon. The approach can be generalized to include electromagnetic interactions and other mesons, as long as the field equations remain linear in the bosonic fields. E.g. the refined Walecka-model “‘“), which incorporates the exchange of photons and isovector p-mesons, can be treated quite similary. Systems with meson-meson interactions or mesonic self-interactions necessitate further investigations, because there exist no formal solutions of the meson field equations to our knowledge. The presented approach is inRexible in comparison to methods using Feynman diagrams. The normal order prescription determines the approximation scheme. It is by no means obvious how systematic improvements can be performed. Especially, an appropriate method should be developed in order to incorporate the effects of antinucleons which are excluded by the present approximation outlined in this paper. This remains a task for future work. Appendix

In the following

we present

an outline

of the proof of eq. (27), i.e.

~[:~(~)~~~~(~)~(~):I = ~~:~(~)~~,~(z)~~~):l.

(A.11

The fields r/f and $ have expansions given in eqs. (20) and (23): respectively, and are related to” each-other by eq. (24): R[$]_ = -+!J.The operation R[. . .] means the elimination of number operators defined in eq. (22) from products of field operators enclosed by the parentheses. Let us first consider the effect of the operation R on the field IL alone. A general term of the expansion of If, may be written (see eq. (20)): I ~(“‘(X-) = {;) % I,__
the abbreviations

* * * a;%+,

hz = ai,

. * . a>,,+1 ,

h,, = (Y.

(AZ)

T. Lippert,

632

W. Scheid / Non-perturbative

approximarion.s

Theeliminationofanumberoperatoryields(i=l,...,n; @“‘(x) =

k=l,...,n+l):

c %,...
. . . Q2n+l

x(Y;...(Y:-,‘?:+I . . . a;a,,+, . . . (Y,+h_,(Y,+h+,

. . .

(A.3)

Qy?n+1 .

The second sum of this equation includes a summation over the states 6 occupied in the state Ix). Note, that this summation is restricted by the condition S # {a,} which arises from the anticommutator rules for the fermion operators. Successive use of equations like (A.3) leads to the expression R[I/J’“‘]. For the proof of (A.l) we first show that the restriction 6 # {a,} in eq. (A.3) can be disregarded. Of course, relation (A.3) with this restriction omitted is no longer an identity. Therefore, we write an arrow instead of an equality sign in the subsequent relations in which the restriction 8 # {cr,} has been disregarded. We prove that the successive use of such a modified relation yields the exact expression R[+‘“‘] for an arbitrary component I/J”” or, more generally, for any normal ordered operator. Let us consider eq. (A.>). Extracting from this sum consecutively all parts, which contain a number operator built with the operators (Y,,, , , CY,,+~,. . , aZn+, , we arrive at:

XC?:.

Here, 1’ denotes

. .

cr:-,a:,, . . . cYT:(Y,,, , . . . an+h-lan+L+l..

a sum without

any number

operator

a2ni

I.

(A.4)

and the second term contains

those parts of $‘n) with at least one number operator. The restriction in the summation appearing in eq. (A.4), namely {(Y,, . . , a,,} f CY,,+~-,}, gains importance, if further number operators are split off from {a nt,,“‘, the second term. It prohibits multiple counting of field parts containing more than one number operator. Therefore, we introduce the substitution M

1 %“,f

...

(A.51

(i) t

for

where M is an operator, which means a multiplication exactly p number operators by a factor (p!) ‘. With

of a field part containing (AS) inserted into relation

T. Lipperr, W. Scheid / N(~n-Fe~l~rbaii~e appr~1~jmflti~)n.s

(A.4), it is easy to realize

that (A.4) becomes

coinciding

with one of the {a,} drop

restrictions

S f {m,} are finally

the intermediate of the functions

an identity,

although

with 6

R[$/““] the + they have been disregarded in

out. Therefore,

guaranteed

since all terms

633

in calculating

steps. Notice, that our proof does not make any use of the properties For this reason it applies equally well to an arbitrary .9~l_.,CzL,,+,.

normal ordered operator. Using the modified relation (A.3) (without 8 # {cu,}) we are now in a position to verify eq. (A.l). We have to pay attention to number operators, which occur ~+thin the expansion of one of the tilded fiefds on the left-hand side of eq. (A.I), because just these number operators represent the difference between both sides. Writing a general term of the left-hand side of (A.l), namely :ft;(h)L2nqJ/(“J’+i’z’:,and applying the modified relation (A.3) to an arbitrary number operator originating from one of the three field components, we verify that the same result is obtained, as if the modified relation (A.3) were employed to the corresponding isolated field component. The just outlined considerations are sufficient to prove eq. (A.l), since the successive use of the modified relation (A.3) yields the operation R. Performing the explicit calculation is tedious but straightforward.

References I) J.D. Walecka, Ann. of Phys. 83 (1974) 491 2) B.D. Serot and J.D. Walecka, Phys. Len. R87 (1979) 172 3) B.D. Serot and J.D. Walecka, The relativistic nuclear many-body problem, in: Adv. in Nucl. Phys., ed. J.W. Negele and E. Vogt, 16 (1986) 1 4) S.A. Chin, Ann. of Phys. 108 (1977) 301 5) A.F. Bielajew and B.D. Serot, Ann. of Phys. 156 (1984) 215 6) J. Roguta and A.R. Rodmer, Nucl. Phys. A292 (1977) 413 7) P.G. Reinhard, M. Rufa, J. Maruhn, W. Greiner and J. Friedrich, Z. Phys. A323 ( 1986) 13 8) A.L. Fetter and J.D. Walecka, Quantum theory of many-particle systems (McGraw-Hill, New York, 1971) 9) G. Klllin, Handbuch d. Physik, Vol. V.l (Springer, Berlin, 1958) p. 169 IO) S.S. Schweber, Introduction to relativistic quantum tield theory (Row and Peterson, Evanston, 1961) 11) G. Wentzel, Einfiihrung in die Theorie der Wellenfelder (F. Deuticke, Wien, 1943) 12) G. KiiIICn, Physica 19 (1953) BSO 13) T. Lippert, Diploma thesis 1987, University of Giessen 14) R. Brockmann, Phys. Rev. Cl8 (1977) 1510 15) L.D. Miller and A.E.S. Green, Phys. Rev. CS (1972) 241; L.D. Miller, Phys. Rev. C9 (1974) 537; Phys. Rev. Cl2 (1975) 710 16) C.J. Horowitz and B.D. Serot, Nucl. Phys. A368 (1981) SO3