Ntrclaar Phyiltx A276 (1977) 433-44,4 ; © Nortlt-Hopattd PttbIWilttp Co., Atn~terdanr

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AN APPROACH TO THE NUCLEAR MANY-BODY PROBLEMINCLUDING VIRTUAL MESONS

(n. HF approximation io finite nadei

F. CATARA, M. CONSOLI, A . INSOLIA, L . PAPPALARDO and S . PAPPALARDOIstituto di FYsica Teoriaa dell'Universltd di Catania,

Istituto Nazionale ä FErica Nucleare-Sezione di Catania,Centro Skl/lano di Fttlca Nucleare e di Struttura della Materfa, Catania,

Corso Italia, 57-95129 Catania

Received 25 March 1976(Revised 27 July 1976)

A6àract : A method is presented allowing a self-consistent treatment of meson dynamics in finite nuclei .The single-nucleon sdf-energy is derived in the framework ofthe HFapproximation . The diffet~enceawith respect to the usual (static) HF approach are the energy dependence ofthe self-energy and theappearance df a new term, and both are due to the time delay in meson propagation. The first effectintroduces a double self-consistency condition on both the single-particle energies aqd wavefunctions . The reported results on the ~~C and ' 60 nuclei show the importance of the correcttreatment ofmeson degrees of freedom.

1. IntrodactionIt is well established that nuclear forces are mediated by mesons t) . This statement,

if completely accepted, implies that the nuclear system has to be described as anaggregate of nucleons in the presence of virtual mesons .

Usually the mesonic degrees of freedom are not treated on the same footing asthe nucleonic ones. Indeed, a first steptowards a many-body theory includingmesoniceffects is to build up a nuclear potential able to reproduce the two-nucleon data ;the potential so constructed is then introduced in the many-body Problem 2). Thisprocedure is clearly inconsistent since the behaviour oftwo free nucleons is differentfrom that ofa pair ofnucleons embedded in the nuclear system, and since the mesondynamics are affected by the presence of nucleons . This method is usually appliedbecause of its computational advantages ; indeed it makes it possible to take intoaccount the exchange ofvarious kinds of mesons and to use well-known techniquesof many-body theory .The problem of a self-consistent many-body theory including mesons has been

approached from two different points of view .Schûtte ~) constructs ascheme which allows one to introduce the meson dynamics

in the framework ofBrueekner theory . Alimitation of this method is that the oorreo-tions to mesons propagation due to the presence of nucleons are ignored. On the

433

434

F. CATARA et al.

other hand, by construction, it is suitable only for a perturbative approach to themany-body problem.Another way to take into account the meson degrees of freedom is to solve the

equations of motion for the coupled fields 4_e). Writing the equation of motionfor the single-nucleon Green function, with aHamiltonian includinga meson-nucleoncoupling term, and putting into the expression obtained the formal solution of themeson field equation of motion, one gets an equation in which the two-nucleonGreen function appears.

In this way explicit reference to the mesonic degrees offreedom has been eliminatedin favour of the nucleonic ones ; their presence is reflected in retardation effectsdue to the time delay between emission and absorption.Dover and Lemmer `) developed and applied this method to nuclear matter .

We will apply it to finite nuclear systems by introducing a Hartree-Fock approxima-tion, i.e . by factorizing the two-body Green function as the antisymmetrizéd productof two one-body Green functions.

In this approximation we calculate the nucleon self-energy which turns out to beenergy dependent ; another difference with respect to static case is the appearanceof a new term which turns out to be very important.

Since our aim was just to display the importance ofa self-consistent treatment ofmesonic degrees of freedom in finite nuclei, we consider in this paper the simplest(even if not the most realistic) case of a neutral scalar exchanged meson of mass~ = 135 MeV. In this way we describe the nucleon-nucleon interaction only by itslong range part.

In sect . 2 we give an outline ofthe formalism involved, while in sect . 3 we presentsome results and comments on the single-hole energies in 1ZC and ' 60 nuclei .

2. Description of the methodLet us consider a finite system ofA nucleons ofmass m interacting by the exchange

of neutral scalar mesons of mass ~.The Hamiltonian of such a system is

H(t) = Hn~-lt)+H~e~t)+hi<,~~(t) =Jd3kEkl~r (t)Y'~(r)+J d3k~k"t (i)Bt(t)+~int~t~

where ajr; (Bk) create and annihilate a nucleon (meson) with kinetic energy

Instead of the B-operators let us introduce the corresponding canonical coordi-nates

VIRTUAL MESONS

435

in terms of which the Hamiltonian for the free mesonic field becomes

The coupling term can be written as

H~ôt~(t) = ZoJd3X~ +(x, t~(x, t)J~(x, t)

where

°~(t) _ ~Jd3k~pt(t~k(t)+~Qk (tXZk(t)]~

- (2n)~ Jdak fd3YWk+,(tK,(t)~k(t) _ ~

Jd39P,(t~,,(t),

and analogously for the nucleon operators ~+ and ~ ; ~, _ ~,ol(2n)~.We remark that our convention for the Fourier transform is the symmetric one, i.e .

Let us define the one-/nucleon 3reen function

where ~A~ is the enact ground state ofour A-nucleon system and Tis the usual time-ordering operator for fermions. Note that, since we consider a finite system ; trans-lational invariance does not hold and G depends on both p and p' .The equation of motion for G, with the Hamiltonian H(t), is

Ct at -s°)Gr.r(t-~ _ -b(Y-~(p-t~+it~d3k<A~T{Q_k(t)~k+,(t)~;(~}IA)

_ -a(t-~(p-P~-ii~ ~d3k Jdt,D,~(t-tJ<A~T{Pk(t,~l~k+P(t~P(~}IAi,

where we have introduced the formal solution of the equation of motion for themeson field

a2(ate +ak) Q_~(r) _ -~Pk(t~

by means of the free meson propagator Dk(t-t~ :

436

F. CATARA et ol.

(,

Since in the nuclear system only virtual mesons are present

_a __at -ePI GP.P

.(t-t')

-~t-~P-P~

_ -b(t-t')S(p-p~+iJd3k

Jdtl

Jd3k"V,~(t-t,)

lim D,~(t) = 0.

(10)r~t~

The Hartree-Fork approximation is introduced in eq . (~ by assuming that thetwo-body Green function appearing in it can be factorized into the antisymmetrizedproduct of two single-particle Green functions . Eq . (~ then becomes

-~Jdsk

J dr1D~t-ti) Jd3k"[<"41T{~t+k"(ti~k'dti)}IAi

X <AIT{fit+p(t)~p(r~}~Ai - <AIT{~k+k"(ti)~k+p(t)}IA~

x (AIT{~~"(t,)~p(t')}IA>7

Gk.k'(t -~) _ -t6(t'-t)~A~~k'1~)Wk (t)~Ai "

~;

1

2mJd3k

Jd3k"D` 0~~-~

0

w(~ -EP''P.P'(~)+

2a ~-~

k( 1"k+k",k "( )~k+p.P'( )

Introducing in eq . (11) the Fourier transforms ~ and f~ofthe G- and V-functions,we get

-iJdak fd3k�

JdCD'Vk(CU-CD)Vk+P.k+k.'(CD'Nk

. . .P.(CO),

from which, remembering the definition') of the self-energy operator E, we have

-i rd3k�~~,Yk..(CU-CU')ij~.,+P.k+k"(~')'

(15)

X [Gkt~t ".k"(~l"k+p . P'(t- t7 -Gk+P . t+k.'(t- t1)Gic".p. (t 1 -~~, (1 1)

whereVk(t-tl) _ -~,2Dk(i-r1h (12)

with

(14)

VIRTUAL MESONS

437

We note that in the second term of the r.h.s . of eq . (15) the complete G-functionappears ; the term containing G~ + ~ gives rise to divergences due to the contributionfrom processes in which the nucleon goes into an intermediate scattering state.Anyway it seems to us not completely correct to eliminate these terms by using thearguments reported in ref. ~) . Indeed, if we arc considering the propagation of aparticle (k > kF) state, this contribution is analogous to the divergent part of theself-energy of a free nucleon (see fig. 1) but the 1?auli principle prevents us fromsaying that the meson is emitted and absorbed by the "same" nucleon.

Fig. 1 . Self-energy part for partide propagation containing G~+ ~ in the intermediate state.

Fig. 2 . Self-energy part for hole propagation containing G~+ ~ in the intermediate state.

Viceversa, ifweareconsideringthe propagation ofahole state, the termcontainingGt+ ~ corresponds to a process in which a meson is absorbed by a nucleon of the"core", creating a particle-hole excitation which then annihilates by emission of ameson (sce fig. 2).Thus it is an interaction term and cannot be neglected.This implies that the propagation of a hole in the system is affected also by the

particle states and vice versa. We will see that this contribution is important.This process is present in our approach since we include directly the meson

dynamics in the many-body system . Indeed, if one makes the intermediate step ofconstructing the nucleon-nucleon interaction by considering the two free nucleonsystem such a process is not incorporated, simply because it cannot occur in such asystem .

Inthe absence ofa consistent renormalization procedure, the divergences appearingin the term eontaing Gt+l are eliminated by introducing "form fedora" taking intoaccount the extended structure of the meson-nucleon vertex s . ~. The expressionsfor these form fedora are usually fated in a more or less phenomenological wayandthe wtoff parameter is chosen by fitting the experimental data.

VKe have chosen a simple form factor, i.e . a sharp cutoff function, limiting themomentum of the exchanged virtual mesons to a maximum value. To fix this value,let us remark that, in our approach, the presence of mesons in the nucleus is dueonly to their emission from bound nucleons.

This implies that the momentum of the mesons cannot be larger than twice therms momentum ofa nucleon inside the nucleus. In the case of an infinite system thestatement can be made clearer by noting that the moments of the nucleon before

438

F. CATARA et al.

and after the emission process must be less or equal to the Fermi momentum IPFIso that their difference cannot exceed 2IPF1~

Obviously this is just an order of magnitude estimate of the cutoff momentumwhose precise value could be fitted in order to reproduce, for example, the experi-mental pion-nucleus scattering cross section 6).

Let us now consider a set of bound states {b} and the corresponding creation andannihilation operators ab and ab :

ab = fd3k<klbi~k =Jd3kUb(k~k~

(16)

The set of bound state wave functions is truncated because it obviously does notcontain single-particle scattering states . We then complete it by introducing lo)

si = ~t - ~ an Ub(k),

(1~6

where a,~ creates a nucleon in a continuum state Ik) with momentum k. Aconditionto be satisfied if IAA contains only single-particle bound states is

a~lA) = 0.

(18)Denoting by P the projection operator onto the subspace of the single-particle

normalized states, the set {Iki} defines its complementQ = 1_P,

where I is the identity operator in the space of single-particle states .In calculating the term in eq . (15) containing G~+~, we have to consider in addition

to a sum over the bound particle states, an integral over the scattering states. Theminimal condition these have to satisfy is to be orthogonal to the bound states{Ib~} . This condition defines the "orthogonality scattering" states i°) which differfrom planewavesjust because theyhave to lie in the subspace onto which the operatorQ Projects .

Obviously this approximate treatment of the continuum states makes their contri-bution dependent on the bound state wave functions .

Let us nowchoose the Ub wave functions as those diagonalizing the single-particleGreen function ; by using eqs. (16) and (1~ and denoting by {19i} and {IB')} theparticle and hole bound states respectively, we get

Eb(Eb) =Jd3PUb(P) J ~.d'kUb(k~(P, kr E6) _ ~6°~Eb)+~61~E6)+~b ~Eb)+~3~Eb~

(19)

wlth

~° lEb) _ ~°~ =

2nJd3pUé(P) fd3kUb(k)~r_,(0)

~61~E6)+~b `Eb) =

2n Jd3PUb(P)

Jd3kUb(k)

VIRTUAL MESONS

439

xJd3k" fdW>~t..(sb-w) j~

Ua.(P+k)Ua'(k+k'7 +~ Ua(P+k)i1â(k+k")J

le, W - sa,- iq e W-8e+bl,

L63- 1E6)

~Jd3PU6(P) J

d3kUb(k)Jd3k"

JdW'Vk'(Eb_

-W')

where U,~(k') are the orthogonality scattering wave functions given by '°)

U,~(k') = 83(k-k~- ~ Ub(k)U6(k~.

(211

The single-particle energies eb are solutions of the equation

J W'-k'Z/2m+iq ,

Eq . (22) has the usual form ofthe Hart~ee-Fock equations, apart from the appearanceof the energies in the "HF potential" Eb. This energy dependence comes out fromthe time delay in the meson propagation. This implies that, in order to solve eq. (22)we have to get a "double self-consistency" condition on both the wave functions{Ub} and the energies eb.This energy dependence of the "Harttee-Fork potential" can make the {fib)}

states with different angular quantum numbers non-orthogonal .The situation in analogous to that encountered in constructing the effective

Hamiltonian in shell-model calculations ' 1) . Anyway we stress that in our casesuch troubles do not occur, because we have only one single-particle state for eachsymmetry of the Hamiltonian. In any case the slow energy dependence of E seemsnot to affect practical calculations .Due to the continuum intermediate states contribution, E is non-hermitian so

that the eigenvalues of eq . (22) are complex.The solution of eq . (22) can be found from 's)

<6~tJb)+E~(Re eb) = Re E~Im eb = Im E~(Re sb).

(20b)

(20c)

(22)

This procedure is known to be correct when Imeb is small, as in our case. Indeedwe found that Im Eb is ofthe order of 10 - a of its real part and therefore negligible .This smallness is due to the Harttee-Fock approximation in which high-ôrdercontributions are not taken into account.The usual (static) Hart~ee-Fork equations are obtained by fast constructing the

nucleon-nucleon potential arising from the exchanged mesons and then factorizing,

440

F. CATARA et d.

as in the present case, the two-body Green function . In this way the equation ofmotion for G corresponding to eq . (11) is

+iJd3k~~(0)

Jd3k

�~Gk+~k

�.k1~)GP+k

.P(t-tr)-G~+k.k+k"(D~k".P~(r-t~)h (11~)

A comparison of eq. (11) with eq . (11~ and eqs. (19) and (20) with (19') and (20')shows that the correct treatment of the mesonic degrees of freedom leads to newterms, Ecs~ and ~c3~, in the expression forE. The appearance of these terms, togetherwith the energy dependence of E, are the characteristic features which make ourresults dit%rent from those obtained in the static approximation .

3. Results and die~ion

In this section we report some results obtained on the basis of the approach de-scribed in the previous one. Acomplete calculation involves the problem of doubleconsistency which can be tackled, in a truncated basis, by using amethod for solvingnon-linear equations, as for example, the Newton-Kantorovic one ' 3).The results obtained in the static HF approximation show that, in the expansion

of the b-states in a truncated harmonic oscillator basis, there is a leading coefficientwhile the others are quite small. Since we wantjust to make an order-of-magnitudeestimate of the corrections induced by the exact treatment of meson dynamics, wewill assume that the single-particle states are well described by harmonic oscillatorwave functions . With this choice and with the values for the "range parameter"reported below, the cutoff momentum, i.e. the rms momentum, turns out to be1.79 and 1 .704 fm- ' for 1 ZC and ' 60, respectively. Obviously this approximationalso reflects on the calculation of our orthogonal scattering states which can beaf%ctod by the rapid decrease of these wave functions for large ~r~ . Anyway we thinkthat even with this limitation, our results can give some indication on howthings go .We put then

U~(x) = U�t.,(x) = Y,�,(~)RA~(x),

(23)

from which we get~.~~

b_

b~(o.a+~1 .,~

6 (19~)where

~b'~~ _ -2x d3PUâ(P) d3kU~(k)J J J

~d3k,~~k,~(0)~ Ue'(P+k)Ué.(k+K").e'

(20b')

VIRTUAL MESONS

441

where a is the "range parameter" and the polynomials1,

n = 1

. . . . . . . . . . . . . . .

n >2.

We do not report here any explicit development of the expression for ï' . Sometechnical details on the calculation of the integral appearing in them can be found`in ref. 10~.

We halve applied the method previously described to the 160 and 1zC nuclei .

Tüe mrcleus 160. The set of hole states for 160 is 1s~, 1p~ and 1 pt, while we haveconsidered the 1 d~, 2sß and 1 d~ particle states . The "range" parameter has beenchosen 1~ to be

and the meson mass to be

p = 135 MeV = 0.74fm-1 .

(27)

TneLE 1

Single-hole energies of'60 in fm- ' giving the results obtained in the static approximation

TABLE 2The experimental single-particle energies (' 60) in fm - '

ls,~ z 1P3n

1P~n

ldsn

~in ldan

-0.20 -0.11 -0.08 -0.02 -0.016 +0.005

TABLE 3Relative variations with reaped to the static calculation in '60

1P~n

dE,(ea 0.98 0.74 0.74

ls, n 1s,r~(st) 1Pan 1Pan(~) 1Pcn 1Pu~(~)

Eco) -0.0187 -0.0187 -0.0156 -0.0156 -0.0157 -0.0157Ecu 0.0097 0.0102 0.0014 0.0014 0.0014 0.0014Ec :) -0.0038 -0.0004 -0.0004EI3) -0.0121 -0.0163 -0.0165

i' -0.0249 -0.0085 -0.0311 -0.0143 -0.0311 -0.0143

442

F. CATAItA et al.

In table 1 we compare the values of E for the hole states of 160 obtained in the staticapproximation and in the complete calculation, by using for the single-particleenergies the experimental data reported in table 2.

In order to givea better idea ofthecorrections toEinduced by the correct treatmentof the meson dynamics, we calculated the relative variations

ls,n 1Pan

dEy(E6) = I~b(E6)-~6`11~2~~6(E6)+~61~~,

and these values are given in table 3.The nucleus 12C. The hole states in 1ZC are 1sßand 1pt while the particle ones are:

1p.~, 1d.~ and 2s}.The "range" parameter has been chosen 1~ as

a = 0.609 ftn-1 .The single-particle and single-hole experimental energies in 1 ZC are given in

table 4.TABLE 4

Single-hole energies in "C in fm- '

TABLE 5The experimental single-particle energies ("C) in fm' 1

1P~n

ldsn

2s,r,

-0.0175 -0.093

-0.0025 -0.010

TABLE 6As in table 3 but for'~C

ls, r ~ 1Par~

dE,(s~

0.66 0.88

(26a)

-0.006

The results obtained in the same manner as for 160 are reported in tables 5and 6.Notethat the relative Wariations dE areindependent ofthe nucleon-meson coupling

constant ~.. As is evident from tables l, 3, 4 and 6, the corrections are quite importantand give larger binding energies.

ls, r = ls, r2(st) 1Pan 1Par =(st)

i"°~ -0.0169 -0.0169 -0.0115 -0.0115i'" `~ 0.0087 0.0043 0.0010 0.0008Eu~ -0.0068 -0.0011ira~ -0.0101 -0.0159ï' -0.0251 -0.0126 -0.0275 -0.0107

VIRTUAL MESONS

443

Other calculations with different sb values show the weak dependence of theresults on energy. Indeed, the largest correction to E is not due to its energy depen-dence but to thepresence ofthe terms Et 21andEt3l (see tables 1 and 4) . In a completelyself-consistent calculation this dependence could turn out not to be negligible.As one can see from tables 1 and 4, the contribution from the continuum states

is quite large ; this seems to be physically unsatisfying . It is not clear how much thisis due to the "bad" normalized wave functions we used or to the orthogonalityscattering approximation . Perhaps it is necessary to introduce some "vertex correo-tions" .Anyway, we want to stress that the sign of the contribution due to the continuum

is to increase the binding energy and that even in the case where some improvementin the treatment of this contribution makes it negligible, the relative variations dueonly to the energy dependence and to its ~ should be appreciable (see tables 7 and 8) .

TAHLS 7

Relative variations with respect to the static calculation in ' 60, but without the contribution from .thecontinuum intermediate states

ls,~ z1P3n

lPii:

dEb(e,)

0.40

0.02

0.02

TABLE 8

As in table 7 but for ' zC

dE~(e~

0.17

0.08

The calculations reported here can be improved by fulfilling the double self-consistency condition on both the energies and the single-particle wave functions.This involves only calculational difficulties which can be surmounted by using, forexample, the Newton-Kantorovic method to solve non-linear equations t').

This~will be worthwhile in amore realistic approach to the problem ofthe nucleon-nucleon interaction by the exchange of pseudoscalar and/or other mesons.An extension ofthe method should allow the evaluation ofthe one-nucleon poten-

tial due to the exchange mesons ; this requires the calculation of the E-operatorbetween scattering states .

It should be also interesting to apply the present procedure to a nuclear system inwhich pairing correlations are present. To this end the most suitable approach seemsto be the Gorkov formalism') which has been successfully applied to the study ofpairing correlations in finite nuclei . In this method the presence of pairing correla-tions is introduced by factorizing the irreducible part of the two-body correlation

444

F. CATARA et al.

functions connecting the ground state of the A-nucleus to those of the (A-2}and (A+2}nuclei.By introducing in eq . (~ the expression so obtained and by writing down the equa-

tion of motion for the correlation function in the same approximation, we get a setof two côupled equations which differ from those of the standard Gorkov methodfor the presence of time delay in, for example, an energy dependence of the gapparameter analogous to that occurring for the self-energy part .

It is then also possible to study in a self-consistent framework the consequencesof pairing correlations on the pion-nucleus optical potential .

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2) T. Obinata and M. Wad~, Prog . Theor. Phys. 53 (1975) 732;T. Obinata, M. Wad~ and T. Kasahara Prog . Theor. Phys . 53 (1975) 1406 ;K. Erkelenz, Phys. Reports 13 (1974) 291, and references therein

3) D. Schütte, Nucl . Phys. A220 (1974) 4504) C. B. Dover and R. H. Lemmer, Phys . Rev. 165 (1968) 1 l055) C. H. Dover, Ann. of Phys . 50 (1968) 449 ;C. B. Dover and R. H. Lemmer, Phys . Rev. 183 (1969) 908 ;W. D. Brown, R. D. Puff and L. Whets, Phys. Rev. C2 (1970) 331

6) C. B. Dover, J. Hüfner and R. H. Lcmmer, Ann. of Phys . 66 (1971) 248 ;C. B. Dover and R. H. Lemmer, Phys. Rev. C7 (1973) 2312

7) A. A. Abrikosov, L. P. Gorkov and I . E. Dryaloshinaki, Methods of quantum field theory instatistical physics (Prentia-Hall, NY, 1%3) ;A. B. Migdal, Theory of finite Fermi systems and applications to atomic nuclei (Whey, NY, 1%7)

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9) A. D. Jackson, D. O. Riska and B. Wervest, Nucl . Phys. A2~9 (1975) 397 ;H. Holinde and R. Machleidt, Nud. Phys . A256 (1976) 479 and 497

10) A. Agodi, F. Catara and M. Di Toro, Ann. ofPhya.10 (1966) 1 ; 49 (1968) 44511) M. H. MacFarlane, Proc . Iat . School of Physics "E. Fermi", Course XL, ed. M. Jean (Academic

Press, NY, 1969) ;C. Block and J. Horowitz, Nucl . PhyB . 8 (1958) 9l

12) H. Feshbacb, Am. of Phys. S (1958) 357; 19 (1%2) 287;L. Fonda and R. G. Newton, Ann. of Phys. 10 (1960) 490

13) F. Catara, M. Di Toro, E. Pace and G. Schiffrer, Nuovo Cim. All (1972) 733;G. Schifïrer, The nuclear many-body problem, ed . F. Calogero and C. Ciofi degli Atti (EditriceCompoaitori, 1973)