effects of pion-fold-pion diagrams in the energy-independent nucleon-nucleon potential

Nuclear Physics A443 (1985) 601-627

@ North-Holland Publishing Company

EFFECTS OF PION-FOLD-PION DIAGRAMS

IN THE ENERGY-INDEPENDENT NUCLEON-NUCLEON POTENTIAL

G. DE GUZMAN* and T.T.S. KU0

Department of Physics, State University of New York, Stony Brook, NY 11794, USA

K. HOLINDE** and R. MACHLEIDT***

Institut fiir Theoretische Kernphysik, Universitiit Bonn, Nussallee 14-16, D-5300 Bonn, West Germany

and

A. FAESSLER and H. MOTHER

Institut fiir Theoretische Physik, Universitiit Tiibingen, Auf der Morgenstelle 14, D-7400 Tiibingen, West Germany

Received 21 November 1984

Abstract: Based on a T-matrix equivalence theory, an energy-independent or locally energy-dependent

nucleon-nucleon potential V,, derived from meson exchanges is studied. The potential, given as

a series expansion of folded diagrams, is independent of the asymptotic energy of the scattering

nucleons. It is, however, locally energy dependent in the sense that its matrix elements (alV,,lb) depend on the energies associated with its bra and ket states a and b. Our formulation makes use

of right-hand-side on-shell T-matrix equivalence of the field-theoretical and potential descriptions

when limited to the space of neutrons and protons only. This preserves not only scattering (e.g.

phase shifts, projections of wave functions) but also bound-state properties. The matrix elements

of V were calculated for two potential models, one based on one-pion exchange (OPEP) and the

other on one-boson exchange (OBEP) using {r, p, v, w, S, q}. Three types of phase-shift calculations

have been carried out to study the viability of constructing an energy-independent potential using

the folded-diagram expansion: (A) NN phase shifts for an energy-dependent OPEP and OBEP.

For the OBEP we used parameters adjusted to fit experimental data. (B) The same phase shifts

for the energy-independent case for both OPEP and OBEP. (C) Repetition of (B) with effects of

the two-pion folded diagrams included. Our results show two important points: (i) folded diagrams

are of essential importance, and (ii) the first-order folded diagrams contain the dominant effect and the neglect of terms with more than two folds can be regarded as a good approximation. The

effects of folded diagrams are large especially for low partial waves and high energies. For high

partial waves (J greater than 2) the folded terms are negligible, and the phase shifts given by (A), (B) and (C) practically coincide.

1. Introduction

In nuclear physics, it has been a long and rather successful tradition that nuclei

are treated as a system of nucleons (neutrons and protons) interacting with each

* Present address: Dept. of Physics, Bryn Mawr College, Bryn Mawr, PA 19041, USA.

** Also at Institut fiir Kemphysik KFA Jiilich, D-5170 Jiilich, West Germany.

*** Present address: TRIUMF, 4004 Wesbrook Mall, Vancouver BC, Canada V6T 2A3.

601

602 G. de Guzman et al. / Pion-fold-pion diagrams

other with a nucleon-nucleon (NN) potential V NN. But nuclei are in fact a collection

of not only nucleons but also non-nucleonic “elementary” particles such as anti-

nucleons, mesons (r, a, p, 6,. . .) and the A(3,3) isobar which corresponds to an

excited nucleon. The interaction among these “elementary” objects is represented

by appropriate field-theoretical lagrangians, such as the pion-nucleon lagrangian L sr~~. How to reconcile the above two rather different descriptions of nuclei has

been the subject of many theoretical investigations ‘). Briefly speaking, a central

problem to be resolved is the derivation of the NN potential VNN, which can be

used in conventional nuclear physics, from the underlying meson-exchange interac-

tions between the nucleons. As is well known, V,, derived in this way is expressed

in terms of irreducible meson-exchange diagrams, such as the one-pion-exchange

diagrams and so on. As we will discuss later, the V NN so derived is generally energy

dependent and it is desirable in several aspects to have an energy-independent NN

potential. General theories for energy-independent effective interactions have been

studied by many authors ‘). Their application to the derivation of an energy-

independent NN potential V,, has been investigated by Johnson 3), and more

recently by Li, Ng and Kuo (LNK) “). This type of energy-independent V,, contains

an additional type of irreducible diagram - the mesonic folded diagrams.

The purpose of the present work is to discuss some specific properties and

consequences of the above mesonic folded diagrams, and, in particular, to investigate

the effect of the pion-fold-pion (r j r) diagrams for nucleon-nucleon scatterings.

Some preliminary investigations of the mesonic folded diagrams have been per-

formed by LNK4) for neutral and spinless nucleons. Their results indicate that,

among the various mesonic folded diagrams, the rr 5 n ones are probably the most

important. In the present work, we shall calculate and investigate the rr 5 rr diagrams

for realistic nucleons, and apply them to nucleon-nucleon phase-shift calculation

for partial-wave channels ‘SO to 3PZ.

Since the method and concept of folded diagrams have been a rather specialized

subject, restricted mainly to the theory of effective interactions to be used in nuclear

shell-model calculations, we will give in sect. 2 first some general review about this

subject before discussing some specific aspects of the mesonic folded diagrams. The

LNK “) theory of V,,., is formulated in the framework of a T-matrix approach. We

shall describe, also in sect. 2, that this theory of V NN can also be obtained by using

a generalized model-space renormalization method. There are two general types of

effective interaction theories - the energy-dependent theory and the energy-indepen-

dent theory. The former may be referred to as the Brillouin-Wigner or Feshbach ‘)

type of effective interaction theory, while the latter is known as the Rayleigh-

Schroedinger or folded-diagram “) type. The NN potentials derived from meson

exchanges may also be classified into two similar general categories - the energy-

dependent NN potentials V,,(E) and the energy-independent ones VI. We shall

study in some detail the connection between these two types of potential in sects.

2 and 3. We shall examine, for example, some essential features of the variable E

G. de Guzman et al. / Pion-fold-pion diagrams 603

of V,(E) and show how it may be removed by the inclusion of the mesonic folded

diagrams. In sect. 2, we will discuss that the energy-independent NN potential

should be more appropriately referred to as being locally energy dependent. As we

will also discuss, the familiar Bethe-Brandow-Petschek theorem for constructing a

single-particle potential in nuclear matter is, in fact, equivalent to a folded-diagram

method for constructing a locally energy-dependent single-particle potential.

Physical quantities such as nuclear energies and NN phase shifts calculated with

V,(E) and those with V, should, of course, exactly agree with each other, if

calculations are carried out exactly. In practice, one usually has to make some

approximations. For example, in the calculation of V, one may have to neglect some

higher-order mesonic folded diagrams. It is important to study the accuracy of the

various approximation methods for the calculation of VI. For the usual (non-

mesonic) effective interaction problems, there have been extensive studies ‘) of the

accuracy of the approximation methods for the calculation of the energy-indepen-

dent effective interaction, usually with the help of some solvable finite-dimensional

matrix models. We would like to carry out a similar investigation for the calculation

of V,. As will be discussed in sect. 3, a simple one-pion-exchange model will be

employed from which exact NN phase shifts can be calculated. These phase shifts

will then be compared with those given by V, which is derived approximately with

the inclusion of only the lowest-order mesonic folded diagrams (i.e. r 5 v). We will

show that the effect of the r j r diagrams can be very significant and desirable in

a number of cases. An analogous study is made for a one-boson-exchange model,

considering again, however, only rr j rr folded diagrams. The model includes

(r, p, a, 6, 7, o) and the results for this model will be discussed in sect. 4. A

conclusion and some general discussions will be given in sect. 5.

2. Mesonic folded diagrams

In this section we review the origin and definition of the mesonic folded diagrams

and discuss some of their general properties. In nuclear physics, one often uses the

so-called model-space approach where the entire Hilbert space of nuclear degrees

of freedom is divided into a model space denoted by the projection operator P and

the rest denoted by the projection operator Q. Let the Schroedinger equation for

the entire Hilbert space be

H?P,,=E,IV,.

The corresponding P-space equation is given by

(2.1)

(2.2) where (Y = 1 to d and d is the dimensionality of the model space P. Clearly

only a portion of the solutions of (2.1) is reproduced by (2.2), and we obtain from

(2.2) d eigenvalues of (2.1) and the corresponding P-space projections of the


d eigenfunctions of (2.1). Formal theories for the derivation of the effective hamil-

tonian H,e from given H and P have been the subject of many theoretical investiga-

tions and are now rather well developed. [See e.g. ref. *), and references‘quoted

therein.] Note that H,e of (2.2) is of the energy-independent type; it is independent

of the nuclear energy E,. Note also that He, generally contains folded diagrams,

as given e.g. by the folded-diagram expansion of Kuo, Lee and Ratcliff “) (KLR).

The above effective interaction theories are usually used to treat nuclear problems

with only nucleonic degrees of freedom. For instance, the projection operators P

and Q are both composed of shell-model wave functions with P chosen to be a

closed I60 core plus two valence nucleons in the Odls shell, as in the case of “0.

Let us make a generalization. As mentioned earlier, we treat the nucleus as a system

of nucleons (n, p), antinucleons (n, p), the various mesons (7r, 7, w, p, . . .), and other

non-nucleonic entities such as the A(3,3) isobar which is a nucleonic excited state.

These nuclear constituents interact with each other with some field-theoretical

interaction lagrangian such as L,,, for the pion-nucleon interaction. In conven-

tional nuclear physics, the nucleus is considered as being composed of n and p

only. Thus this conventional nucleus is in fact just a model-space representation of

the physical nucleus considered as composed of not only nucleons but also non-

nucleonic “elementary” particles as mentioned above. Here, the model space P is

defined as the space composed of n and p only. Nuclear state vectors possessing

any non-nucleonic “elementary” particles, such as n, n, p and A(3,3), are classified

as belonging to the Q space (P+ Q = 1). Some examples are given in fig. 1 where

the state vector 4, belongs to P and the other three all belong to Q. With this

definition of P and Q, the conventional nuclear hamiltonian used in nuclear physics

is just the effective nuclear hamiltonian PHP.

P n pna P”

+3

‘A(3,3) pppnfin

+4 Fig. 1. Examples of P-space (0,) and Q-space (Q2 to Q4) state vectors for a nuclear system with baryon

number 2. The upward-going solid lines represent nucleons, the downward-going lines antinucleons,

and the dashed lines the mesons. The isobar is denoted by a double line.

A time-ordered irreducible diagram is defined as a diagram which must have at

least one interaction vertex and where the intermediate state between any two

successive vertices must be a Q-space state. Some examples of reducible and

irreducible diagrams are given in fig. 2. It is convenient and perhaps necessary to

use the language of the time-dependent perturbation theory and the corresponding

time-ordered diagrams, as is done in the present work. This is because our nuclear

model space P is composed of n and p only, with their antiparticles n and p excluded.


(al (b) (c) (d) (e)

Fig. 2. Irreducible (a-d) and reducible (e) time-ordered diagrams.

This exclusion is easily done when one uses the time-ordered diagrams. If one uses

the usual Feynman diagrams, then the nucleon and antinucleon intermediate states

are not separated. Consider, for example, diagram (e) of fig. 2. If it were a usual

Feynman diagram, then its intermediate states are summed over both nucleon and

antinucleon states. In this case we should not consider diagram (d) of the same

figure as it is already contained in (e). Note that in our formalism we intend to

include all the irreducible diagrams such as (a) to (d) in the nucleon-nucleon

potential VNN, while the reducible diagrams are indended to be generated by the

repeated operation of V,, within the nucleonic model space P. For example, diagram

(e) describes two consecutive V,.,, interactions with an intermediate state belonging

to the P-space. As we will discuss later, this treatment of the reducible diagrams is

closely connected to the origin of the mesonic folded diagrams.

Using the above generalized projection operators P and Q and the corresponding

definition of reducible and irreducible time-ordered diagrams, the effective nuclear

hamiltonian PHP can be readily derived. We start from a model nuclear hamiltonian.

For example, if the nuclear system is composed of nucleons and mesons only, we

may write

H = HO(N) + H,(meson) + &,eSon-nucleon+ &,eson-meson , (2.3)

where H,(N) and H,(meson) are the bare nucleon and meson hamiltonians. For

instance, we may take H,(N) as the free Dirac hamiltonian. The L’s are the

interaction lagrangians. In analogy to the Q-box of the KLR folded-diagram

method 6), we may define a X-box as composed of all irreducible diagrams such as

diagrams (a) to (d) of fig. 2. Then the KLR method is directly applicable to the

derivation of PHP from H of (2.3), giving

PHP = H,,(N) + PVP (2.4)

with

v=+++[Z-..., (2.4a)

where V is the effective nucleon-nucleon interaction and the symbol j represents a

generalized folding 6). The above result is entirely consistent with the LNK

expansion “) of the nucleon-nucleon potential which was derived using a T-matrix

method. V in general has many-body components ‘) ; its two-body part should


correspond to the usual two-body nucleon-nucleon potential V,,. As shown in

(2.4a), V,, and V in general contain mesonic folded diagrams.

In the above we have shown that the transformation of H into the nucleonic

model-space effective hamiltonian PHP gives rise to the mesonic folded diagrams

contained in V. To study the connection between V and the corresponding energy-

dependent effective NN potential and particularly the role of the folded diagrams

in this respect, it may be more convenient to use the T-matrix derivation4) of V.

We review in the following some relevant aspects of this derivation. Using the

time-ordered perturbation expansion of the time evolution operator U(0, -CO), we

can write the integral equation for the two-nucleon scattering T-matrix:

(2.5)

with n + O+. Here, A j, and i are each a two-nucleon state and j is summed over all

such states. Ei and Ej are both the bare two-nucleon energies, i.e. H,(N)Ii)= E,li)

and similarly for Ey 2 is the irreducible Z-box defined earlier. From (2.5), one can

expand T in terms of Z’s in the form of I+ Eg1+ EgEgE + * * *, where g stands

for the unperturbed Green function l/(E, -H,(N) + iv). It is important to note the

energy variable of the Zboxes. For example, the term second order in 2 of this

expansion is

(A) =c (fI~(Ei)Ij)(jI~(Ei)li) .i Ei - Ej+ iq ’

(2.6)

where the second Z-box is half-on-s%ell, i.e. Ei is the energy of the ket state i. But

the first E-box is generally off-shell, since Ei is related neither to j nor to j In fact,

f, i, and j are independent variables. We may define a potential X by defining its

matrix elements as

The second E-term of (2.6) can be replaced by (jlXli>. But then we cannot do the

same thing for the first 2 box, as (flE(Ei)lj) is by no means equal to (flxlj). We

may define a new term:

(B) =c (f~:(Ej)Ij>(jI~(Ei)I~) j Ei- Ej+ iv ’

where both S-boxes are right-hand-side on-shell. (B) is then of the form

(2.7)

The difference (A) - (B) is in fact the once-folded diagram.

Let us consider the structure of (A) and (B) in terms of time ordered diagrams.

Suppose 1 is composed of one pion exchange only. Then (A) is just the reducible

diagram shown in fig. 3, where the starting energy Ei appears in all its propagators.

G. de Guzman et at. / fan-fold-Zion diagrams 607

(A) (8) (C)

Fig. 3. Definition of the pion-fold-pion (T 5 T) diagram (C).

Note that (B) is drawn in a product form, meaning that the starting energy for its first part (fromj tof) is Ej instead of I$ In terms of time integrations, the integration limits for (B) are

where we note that tl is allowed to extend all the way down to --CO, irrespective of f2. The time integrations for (A) is, however,

Thus, in converting (A) to (B) we have introduced an incorrect contribution which is represented by diagram (C) whose time-integration limits are

Some details about the calculations of folded diagrams will be given later in sect. 3. When two nucleons interact with each other only with an effective nucleon-

nucleon potential VNN, the respective T-matrix, denoted as z is given by

(2.8)

where one should note that (_f~V,&) is independent of Ei. The matrix element (fl V,,Ij) is completely specified by its bra and ket indices f and j, without any need of knowing what El is. This is of course a reasonable property of any usual potential operator. For example, in evaluating the matrix element {f VReidlj) where VReid stands for the familiar Reid nucleon-nucleon potential ‘), we only need to know what are f and j, but not Ei- V NN is derived from (2.5), (2.8), and the requirement

(flT(Ej)Ii)-5(fJT(Ei)li>. (2.9)

The above discussion suggests clearly that, to derive VNN, we need to convert all the E-boxes of (2.5) into those of being right-hand-side on-shell, such as the


conversion of (A) of (2.6) into (B) of (2.7). The strategy is to include (B) of (2.7)

in the

<A VNNljXjl V.d) Ei - Ej + iq

(2.10)

term of (2.8), and include the difference (A)-(B) in the (flV,,li) term of (2.8).

Continuing this way, V,, is obtained “) as

t_fl~NNl~)=(fl{~(j)z- 1 ~(A+~ /z [~(i)-..*}ii), (2.11)

where the symbol j represents a generalized folding “). The once-folded term -2 j 2

is just the difference (A)-(B) mentioned earlier.

Note that (fl V,,lj) of (2.11) is in fact not energy independent, although it is

usually referred to as being so. As shown, (fl V,,(j) is dependent on the energy E,,

as every term of (11) is explicitly right-hand-side on-shell. It may be appropriate

to classify our V,, as being either right-hand-side on-shell or locally energy depen-

dent. In contrast, the usual energy-dependent potential may be more suitable denoted

as being globally energy dependent. From (2.5), we may define such an energy-

dependent potential VD(Ei)y for fixed Eiy by

(fIV,(Ei)Ij)=(fl~(Ei)Ij). (2.12)

In fact, V,,(E,) is just the operator E( Ei) itself. To illustrate the difference between

VD and VW, we consider the two typical diagrams of T and 7 shown in fig. 4. For

T, the two nucleons interact with each other by exchanging mesons, corresponding

to a Vi,-type interaction. For diagram (a), the middle Z-box has an energy

denominator given by

l/{(w-E,-E,-E,+iT)(w-Ed-E,+iT)(w-Es-Em-E7+iv))

with w = E, + Ez. o is entirely independent of indices 3,4,5 and 6 which are directly

time >

I 3 A 5

/’ \ ,

T- lT, lr,r,‘/ ‘1, ll 7r’\ /‘lr / /“\

(a) . /’ /’ ‘\ I / \

2. 4 7 6

I 3 5

=i- VNN “NN “NN (b)

2 4 6

Fig. 4. Comaprison of two typical diagrams for T (a) and F (b).


/d

6 7 I D

6 . .

(i) ( ii )

Fig. 5. Many-body diagrams with locally (i) and globally (ii) energy-dependent vertices. Note that lines

1 to 7 are. all nucleons, with 1,2, and 6 being holes and the rest being particles. The dashed lines are pions.

attached to the E-box. In contrast, because our V NN is only locally energy dependent,

the middle vertex (561 V,,134) of diagram (b) of fig. 4 is independent of E, and E2.

It depends on E,, Ed, E5, and E6 only.

For many-body calculations, it seems to be desirable and more convenient to

have the locally energy-dependent interaction V NN than the globally energy-depen-

dent interaction VD(Ei). Consider the two many-body diagrams shown in fig. 5. In

diagram (i), we have the V,,-type vertices. Thus the vertex (561 V,,146) is indepen-

dent of indices 1,2, and 3, as is in conventional many-body calculations. In diagram

(ii), we have the V,( E,)-type vertices. The composite vertex (561 VD( Ei)I46) contains

an internal energy factor (E, + E2 + E6 - E3 - E7 - Ed + iv), corresponding to

Ei=E,+EZ+E6-E3,

Thus this vertex depends not only on the external nucleon lines directly attached

to it but also on (1,2,3) which do not have any direct connection to the vertex.

This is clearly a disadvantage, as it means that this vertex cannot be replaced by

the vertex of a usual effective potential X whose matrix element (561X146) depends

only on its bra and ket indices 56 and 46. Instead, we need a potential X whose

value at a particular vertex depends on what events are going on at the other parts

of the many-body diagram.

It may be of interest to note that the idea of constructing a locally energy-dependent

effective potential by way of folded diagrams has already been used before in the

familiar Brueckner theory of nuclear matter9), although using different terminol-

ogies. The Brueckner reaction matrix G is generally off-energy-shell, or globally energy

dependent according to our classification. This makes the G-matrix not suitable to

be used as an effective interaction for constructing a Hartree-Fock-type single-

particle potential. Bethe, Brandow and Petschek (BBP) lo) have, however shown

that by summing up a certain class (the BBP class) of generalized time ordered

diagrams, the G-matrix self-energy insertions to hole lines can be made half-on-

energy-shell, i.e. locally energy dependent according to our present classification.

Then this converted G-matrix can be used to construct a single-particle potential,

610 G. de Gwzman et al. / Pion-fold-pion diagrams

(a)

I23

(b)

Fig. 6. Identification of BBP diagrams (a) as folded diagrams (b).

usually known as the Brueckner-Hartree-Fock potential, for the hole lines. The

BBP diagrams are usually drawn using a particle-hole vacuum, where “blank”

propagator represents the propagation of all the particles of the closed Fermi sea.

If we redraw the BBP diagrams using a bare vacuum, it is readily seen that the BBP

diagrams are just the folded diagrams used in the present work and the BBP series

is of the same general structure as our folded-diagram series for V,,,, given in (2.11).

In fig. 6, we give an example to explain this point. Diagram (a) is a standard diagram

contained in the BBP series, where the railed lines denote particles above the Fermi

sea. When redrawing this diagram using the bare vacuum representation, it becomes

(b) which is just a once-folded diagram. Here we have only particle lines. The

downward-going lines marked by a circle are not hole lines. Instead, they are the

folded particles lines.

3. The one-pion-exchange model

In the previous section we have discussed the advantages of having a locally

energy-dependent NN potential V NN and its derivation for meson exchanges. A

special feature of V,, is that it contains a new type of diagram - the mesonic folded

diagrams. In this section, we shall investigate the effects of the lowest-order mesonic

folded diagrams - the pion-fold-pion (r 5 rr) diagram - in the calculation of NN

phase shifts.

We first solve the T-matrix equation (2.5), with the energy-dependent X-box given

by the two time-ordered OPE diagrams of fig. 7. This approximate Z-box is denoted

by ZA. Note that here Z’A is energy dependent (or globally energy dependent, using

Pi -Pi Pi -Pi

(a) (b)

Fig. 7. Definition of the P-box terms to be used in phase-shift calculations.


the definition of the present work). Thus each diagram of .X’, is characterized by

an independent energy variable z. As an example the contribution of diagram (a)

of fig. 7 is given as

(a) = 1 d3k (-P’/Lu.~-P, k)(p’, klL,&) 6(p,+ k _p)

z-(E,~+E,+q)+i~ 9 (3.1)

where r,,, is the familiar pseudo-scalar pion-nucleon lagrangian

r rrNN = i&G g,,,?P-y5T. ?P@ .

E,, = Jp’* + m* and E = Jp*+ m* are the final and initial nucleon energies, respec-

tively, and wk = fl k + p with k = p’-p is the pion energy. For phase-shift calcula-

tions, z = +E,,,+ 2m is the asymptotic energy of the two-nucleon scattering system.

Thus, the T-matrix equation we have to solve for phase-shift calculations is in fact

(flTA(z)li)= (fJ.ZA(z)li)+Zj (f’Zf~~)~j~(z)‘i). J

(3.2)

This equation is readily solved numerically by matrix inversion in momentum

space 1’,12). From its solution, we obtain the phase shifts for the different partial-wave

channels. The coupling constant J4?r g,NN with gt,, = 14.4 has been used for all

calculations discussed in this section.

The above phase shifts can also be calculated using an energy-independent (or

locally energy-dependent) NN potential. That is, we solve the T-matrix equation

(2.8) using an appropriate VNN interaction derived from _Z according to (2.11). We

shall consider two approximations for V NN as defined in eq. (2.11). In a first step,

hereafter denoted as approximation B, we consider the one-pion-exchange terms

only:

(flvBlj)= (fl~A(Ej)lj)~ (3.3)

This means that the potential V, is simply given by the two one-pion-exchange

diagrams of fig. 7 with z = 2E,. In a second step, hereafter denoted as approximation

C, we will also take into account the contributions of the folded diagrams of lowest

order (n. 5 r):

(f Vclj) = (fl-%(j) -2.4 I um . (3.4)

From figs. 7 and 3, we see that the -XAjZ;A term of (3.4) contains the eight

pion-fold-pion diagrams of fig. 8. They are all time-ordered diagrams and they are

all right-hand-side on-shell. To illustrate the calculations of these folded diagrams,

we give the expression for diagram (iii) of fig. 8 as

(iii) = Zspins d3p, d3p, d3k, d3k2


(VI (vi) (vii) (viii)

Fig. 8. Pion-fold-pion (m j v) time-ordered diagrams. The folded nucleon lines are marked with a circle.

where each r,,, vertex contains a three-momentum conservation factor and the

summation is over the spins of the two intermediate nucleons. Clearly, (3.5) depends

only on its bra and ket energies E, and E,,, but not on the asymptotic energy z.

We will now discuss the evaluation of the OPE and the rr j 7~ diagrams in more

detail. We shall use the helicity formalism as discussed in several references [see

e.g. Brown “), Erkelenz 13) for more thorough discussion]. Invariance properties of

the NN interaction limit the set of independent helicity amplitudes to six. We take

these six independent amplitudes to be the following:

.fl =(P’++lvlP++)+w++lvlP--) 9 _I-2 = w++l VIP++> -(P’++l VIP--> 7

f =(p’+-IvlP+-)+(p’+-IUP-+) 3

(l+z) (l-z) ’ f Jp’+-IvlP+-)_(p’+-HP-+) 4 (l+z) (l-z) ’

.f5=(P’++lvlP+-YY9 f6=(P’+-lvlP-+)IY>

where (p’++~V~p++)=(p’(+$)(+~)~V(p(+~)(+~)),...,z=cos(~p,p’), and y=

Jl - z2. The singularities at z = + 1, -1 are only apparent since the numerators vanish

at the corresponding points. These amplitudes are then used in partial-wave projec-

tions for different LSJ states in the coupled and uncoupled channels [see refs. 1’*13)].

The nucleon wave function is given by

u(P~) = Np ( ) 2A;,E IAL

(3.6)

where A=F& E=E+m, E2=p2+m2, N$,= s/2E, and IA) are the Pauli helicity

spinors. The matrix conentions used were the following:


y-matrices:

%=(A :J, Yk=(_Lk ;), ,,=(:, ;); Pauli matrices:

ul=(Y ;), u*=yi ;), cQ=(:, _4). As in the previous sections, we shall work in the c.m.s. of the two nucleons. The

initial momentum p is taken to be in the +z direction and the final momentum p’

is chosen to lie on the xz plane. Thus, the helicity spinors are

IAJ=x*, ,

IA*) =x&

for the initial states of the two nucleons (1) and (2), respectively, and

IA;) = exp (-S~J)X~, ,

IA;) = exp (-&J)x-*,

(3.7a)

(3.7b)

(3.7c)

(3.7d)

are the corresponding final states. Here, 0 is angle between p’ and the +z axis and

& are the conventional Pauli spinors.

The one-pion contribution can be evaluated directly and is given by i4)

(3.8)

where w2= k2+p2, p is the pion mass, and k =p’-p. We used the form factor

F(k2) =$$

at each vertex where the cut-off mass A is typically l-2 GeV.

We note the form of the propagator {w( z - E - E’ - CO)}-‘, which is different from

the standard propagators “) used in covariant calculations of Feynman diagrams.

In the energy-independent case, z is set to 2E as prescribed in sect. 2. The calculation

of matrix elements for the helicity states is straightforward and yields the result

(A:A&(A,A,)={lA:+A,I cosi13+(A{-A,) sin;@}

X{lA;+A21cos~8-(A;-Az)sin$?}. (3.9)

Since the pion is an isovector meson, the projected partial-wave matrix elements

should be multiplied by an isospin factor T, = 2 T( T + 1) - 3 where T is the two-


TABLE 1

+amplitudes (pseudoscalar meson)

41 (E’E - m2) cos 0 - p’p

42 (FE - m’) -p’p cos e

43 (-E’E + m’+p’p) co? t9

44 (-E’E+m*-p’p)sin’fe

4s +m(E -E’) sin 0

46 fm( E - E’) sin 0

f, = -.*qz J- 4, E’E o(z-E-E’-w)

{(A2-m2)/(A2+k2)}2.

nucleon isospin quantum number. Table 1 contains the six independent reduced

amplitudes 4r, (b2,. . . , & for the exchange of a pseudoscalar particle. The f’s and

4’s are related by

+i

E’E o(z-E-E’-w)’

The calculation of the pion-fold-pion contribution V,J, however, is more in-

volved. Instead of working directly on expressions as given for an example in eq.

(3.5), we make use of the identity displayed in fig. 3 and compute (C) = (B) -(A)

in the energy-independent case (see discussion in sect. 2). After satisfying momentum

conservation at each vertex, we are left with a single free integration variable, which

we take as the intermediate nucleon momentum q.

After some simplification, we have

(pA;h;l V,&A,Az) = (2~)-~ d3q(m2/E’E)(m2/E2,)

where

XC n(-p’A:)mNNU(-qAr)ii(-qA2)r?rNNU(-pA2). h’

(3.11)

Here k = p - q, k’ = q -p’, CO: = k2+ p2, wt. = kf2+ p2, and the sums are over inter-

mediate nucleon helicity states. These sums are simplified by using the identity

Z’, ii(qA)u(qA) = (yoE, -y . q+ m)/2m. The spin functions r,* then become

r12= {A(A:A,)(A:lA,)- B(A;A,)(A:lq * &))

x{A(A~A,)(h;(A,)+B(h;A,)(-A;(q. +A,)), (3.12)

where A(AA’) = E’(E, - m)~+4AA’pp’(E~+ m) and B(AA') =2A'ps'+2Ap'&. Note


TABLE 2

Spin functions r,2 for T J P diagram

Amplitude

(++I VW,,,/++) (++I v,, JnI--)

(+-lv,J,l+-)

(+-lv,J,l-+)

(++lv,,J,l+-)

(+-/v,J,I++)

~~(x+y)cosf~-(z+w)[q+sin~O+q,cos~O~2

-pl(x-y) sin$O+(z-w)[q+co~~&q,sinfO~~

/3{(x+y)cos$O-(~+w)[q+sin~O+q,cosf~}~

p{(x-y) sin$O+(z- w)[q cosf8-q, sinjO}*

-p{(x+y) cost4 -(z+ w)[q+ sintO+q, costt9)

x{(x-y)sinfO-(z+w)lq+sin$3+q,cos~O}

P{(x+y)cos@-(z+w)[qsin$O-q, sintO}

x{(x-y)sin+tV+(z-w)[q+cos&-q, sin@}’

P =8&h, /m’(E’+m)(E+m), x=(E’+m)(E,+m)(E+m),

y=p’p(&+m), z =p(E’+ m) ,

w=p’(E+m), q = (9x2 qy> 42) 9

qT = qx T iq,

that, because of our choice of the direction of p’, the imaginary parts of r, and r,

come only from the terms (h~~q,o,,(h,) and (-A;lq,a,,l-A,). The remaining parts of

the integrand in (3.10) are all even functions of q,,. Thus, the term linear in aY in

the product r,, vanishes when the integration is performed over the q-variable.

Therefore we can neglect the imaginary part of (3.10) from the outset. The corre-

sponding isospin factor for the folded term is (2T( T-C 1) - 3)2. Consider now the

energy denominator in the integrand of (3.10). Because of momentum conservation

at each vertex the factors E’- E4 + ok, and E - E’ - ok never vanish. The third factor

2E-E’-E,-w, however, has a root in q for suitable combinations of p and p’.

The integrand is singular in that case and we opt to do a principal value integration,

dropping the imaginary part in the process. Instead of the six amplitudes fi, . . f ., 6, we give in table 2 six spin functions r,,. The corresponding J’s are obtained by

including suitable combinations of the spin functions in the integrand of (3.6).

Two independent calculations of the phase shifts given here were made and found

to agree and this suggests the correctness of our calculations. In fact we have

constructed two totally separate computer programs for calculating OPE, OBE,

rr j 7~, and the phase shifts. In dealing with the singularity of the folded term, in

one program we used the subtraction method, where the singularity is subtracted

and a principal-value integration is performed, i.e.

P I mfodx= 0 x--x0

Another method is to symmetrize a finite interval as follows:

dx [f(x + xo) -f(-x + xo)lIx +


[ PO’ ] 0!4S a=“-ld

r

9 (u a 0 6

I 9 [ PDJ ] +4!4S asVd

r

[PO’ ] 0!4S =Wd

0 CD P 0’ 0 L d

‘4 0 d d 6

[ PDJ ] i4!4S asWd

a

G. de Guzman et al. / Pion-fold-pion diagrams

[ POJ] 14!4’S =Yd

[ PDJ ] O!S =Wd

, I

618 G. de Guzman et al / Pion-fold-pion diagrams

The first integral on the right is then well behaved at x = 0. The normalization used

in the scattering equation is as given in Kothoff et al. 14). The momentum mesh

points used in the numerical integration for the interval [0, 001 was a tangent mapping

from [0, 11:

qi = C tan (isi) , Wi = CWi set’ (iti) , i=1,2 ,..., N,

where I&, Wi} are gaussian points in the interval [0, 11. We used N = 24 for all partial

waves and C was varied around C = 1 GeV to make sure the results are stable.

The main aim of the investigations discussed in this section is to test the conver-

gence of the folded-diagram expansion applied to the NN scattering problem. We

would like to demonstrate that a truncation of the expansion of the energy-indepen-

dent potential V,, in eq. (2.11) after the terms with one fold yields a good

approximation for the energy-dependent potential. For that purpose we have calcu-

lated NN phase shifts for the one-pion-exchange model of the NN phase shifts

using the three approaches discussed above. Results for the phase shifts in some

partial waves with low angular -momenta are displayed in fig. 9 as a function of the

energy of the scattering nucleons in the laboratory system. According to the nomen-

clature introduced above, the curves labeled A stand for the results of the globally

energy-dependent potential, whereas B and C represent the results of the energy-

independent calculations without and with the inclusion of the lowest-order folded

diagrams, respectively.

It is interesting to note that all phase shifts calculated in the energy-dependent

approach (A) are smaller (or more negative) than the corresponding values obtained

by solving the Lippman-Schwinger equation (2.8) for the energy-independent one-

pion-exchange potential (B). The results displayed in fig. 9 also demonstrate that

the differences between curves (A) and (B) are often rather large. As shown by the

figures, it is encouraging to observe that curves (C), which include the r 1 r diagrams,

and (A) are generally in rather good agreement with each other. Let us consider

the two specific case ‘S,, and 3S,. For ‘So, curves (B), (C) and (A) are rather close

to each other at low energies. At higher energies (-250 MeV), the difference between

curves (C) and (A) becomes appreciable. But still, (C) agrees with (A) much better

than (B). For 3S1, the difference between (B) and (A) is large. The agreement between

(C) and (A) is, however, remarkably good.

The above results indicate that the effect of the rr 5 n diagrams is generally

repulsive, as the phase shifts given by curve (C) are generally lower than those

given by (B). In addition, the good agreement between (C) and (A) suggests that

the higher-order folded diagrams, such as n j r 1 r, of V,, may be neglected, at

least as far as the calculation of the low-energy (up to -250 MeV) phase shifts is

concerned. This is an encouraging result. For partial waves with 1> 2, we note that

the curves (A), (B) and (C) do not differ from each other in a significant way. This

indicates that the potential given by the v I r diagrams is of short-range nature,

and consequently low-energy phase shifts for partial waves with high I are not

G. de Guzmnn et al. / Pion-fold-pion diagrams 619

appreciably influenced by the folded diagrams. This is in agreement with the

observation made in ref. “) for the interaction of particles without spin. There it was

found that the rr J n terms can be represented by a Yukawa potential corresponding

to a mass of twice the r-mass.

We note that the T-matrix of the energy-independent calculation (C) is identical

to the T-matrix of the energy-dependent one (A) including all terms up to second-

order one-pion exchange. Therefore the differences between curves C and A are

due to differences in the higher-order terms of V. Thus the discrepancies between

models C and A (also between models B and A) in general increase with increasing

energies, because at higher energies the nucleons feel the stronger short-range parts

of the potential and consequently the higher-order terms in T become more impor-

tant. This trend is observed in our results. On the other hand, the higher-order terms

become less important for the partial waves with higher angular momentum, since

in this case the nucleons feel mainly the long-range parts of the interaction. This

yields results for phase shifts in these higher partial waves which are essentially

identical for the three models under consideration.

All results discussed so far were obtained assuming a cut-off mass of A = 1 GeV

in the form factor of the pion nucleon vertex. This cut-off mass is relatively small

compared to those which are typically used in boson-exchange models of the NN

interaction. Therefore one may argue that the nice agreement between the energy-

dependent and the energy-independent approach with the inclusion of folded

diagrams might be obtained only when short-range components of the potential are

suppressed by a strong form factor. To investigate this point we have repeated the

calculations assuming higher values for the cut-off parameter. The example displayed

in fig. 10 are obtained using A = 2 GeV, and they clearly show that the good

convergence of the folded-diagram expansion remains.

4. Folded diagrams in one-boson exchange

While sect. 3 was devoted to an investigation of the convergence of the folded

diagram expansion by applying it to the one-pion-exchange model of the NN

interaction, it is the aim of the this section, to study the influence of the r J TT terms

on a realistic one-boson-exchange (OBE) potential. The bosons we will consider

are {r, p, w, 6, u, 7). Our 1 of eq. (2.11) is then made up of diagrams like fig. 7

summed up over all the mesons, i.e.

~*=~~+~~++~+~6+~a+~I).

Strictly speaking, the folded diagram 2 J 2 contains terms like

“JTPJP,...

(4.1)

as well as mixed terms such as

r J P, n J u, . . . , P J a, . . . ,


A=2 GeV

E,& [M&VI

- A=2GeV

:

-0. IO - A

b) -0.20

-L/y: 0 100 200 300

E,ab LMeVl

Fig. 10. Examples of NN phase shifts calculated for the one-pion-exchange model using a cut-off

parameter A = 2 GeV. For further explanations see fig. 9.

with obvious notations. In the present paper, we use a third approximation step

and restrict the investigation by assuming Z j Z = rr j GT. In terms of the familiar

Yukawa argument about the mass of the exchange meson and the corresponding

range of interaction, the 7r 1 r terms should give the contribution of longest range

among the folded diagrams.

The lagrangians used in our OBE model are defined by (for completeness, we

include the pseudoscalar case which has already been discussed in sect. 3)

scalar: L, = J’IT gstjlpP )

pseudoscalar: L,, = i&G g,,iJy,J, @(“) ,

vector: L, = 6 g,ljy5pD~‘+JZ [fJ4m]&7~““~f;~,

where g, ((Y = s, ps, v) and fV are the coupling constants. In particular, g, and f,

describe the Pauli and Dirac coupling, respectively, since they correspond to the

Dirac and anomalous (Pauli) part of the magnetic moment. a@_ and &, are given by

upV = SLY,, YJ , (4.2)

fey = a,@’ - a&’ = -j-I’,’ . (4.3)

The isospin dependence in the lagrangians has been suppressed. For isovector

mesons we should replace @(a, by 7 * @u) (a = s, ps, v) where I is the isotopic spin

operator.

To proceed in the same manner as in sect. 3, we define our energy-dependent

potential VA as

<A VA9 = (A&(z)li> , (4.4)

where ZZA is as defined above and z represents the global energy variable which is

G. de Guzmnn et al. / Ban-fold-pion diagrams 621

independent off and t We therefore need the matrix elements (fl&!i> for @ = s, psi v. The pseudoscatar case has been given previously in sect. 3. For Q = s, v the cakufations proceed similarly using heficity formalism. For the scalar case we have [cf. ref. I”)]:

x wwd3 (Ffk2))’

w(z-r-E-#) ’ (4.5)

where the symbols E’, E, w, I . . are defined exactly in the same way as the pseudoscalar case in sect. 3. Fot thr? case of a vector meson, we split the matrix element into

three parts:

(p’A~A;j~,Cz)lpA,=(p’A;A&fV,fpA,A,)S(F’A;A;I~;tipAlhz)

+(P’A:A:l K,IPA,Az) ”

The Dirac term (w) is given by

(4.6)

the E)auXi term (tt) by


and the mixed Dirac-Pauli term (vt) by

The matrix elements (A:A:lA,A,) are given in eq. (3.9). Direct calculation also gives

(A;A$, . ~&Wz)

=-{(A~+A,)sin$0+IA;-A,[ cos~13}{(A;+A,) sin$B-Ihi-A,\ cosi0)

-{lA;+A,l sin$3--(hi-h,) cosi0}{1A;+A,l sinte+(A;-A,) cos$e)

-{(A;+A,) cos;0-IA;-A,l sin$e}{(A;+A2) cos$0+lA;-A,I sini@}. (4.10)

The corresponding potential V, is given by setting z = 2Ei = 2E in

(fl V&) =(flzA(a = 2Ei)li). (4.11)

V, is then given by

(fl~~li)=(SlV~li)-(flV,~,li), (4.12)

where the folded term has been given in sect. 3.

For partial-wave calculations we need the functions fjO’ (i = 1,6; cx = s, ps, v)

corresponding to the functions f,, f2, &, f4, f 5, and f6 as defined in eqs. (3.5). These

are. given in table 1 and ref. 13). As in an updated version of ref. 14) we use an

identical form of the form factor for each of the bosons, i.e. (A’- p*)/(A’+ k*) at

each vertex, The parameters of the OBE potential like coupling constants, form

factors and the mass of the u-meson are taken from ref. 15). They have been adjusted

in order to get a reasonable fit to the experimental data 16) by using the energy-

dependent approach. The parameters which were obtained by this fit are listed in

table 3. We have used the same parameters which were then also used in the

energy-independent calculations. The results of the three calculations: energy-

TABLE 3

Parameters used in the OBE model

Meson Mass [MeVJ g2 fig Isospin A [MeVI

77 138.0 14.40 0 1 1750

;i’ 983.0 548.8 5.00 1.05 0 0 0 1 2000 1500

v 550.0 8.88 0 0 2000

P 769.0 0.90 6.1 1 1500

w 782.6 20.00 0 0 1500.


dependent, energy-independent, and energy-independent plus 7~ 5 rr terms are given

as curves (A), (B), and (C), respectively, in fig. 11.

Also in the case of the realistic OBE potential some general features of the

folded-diagram expansion of V NN persist, which were already discussed for the

one-pion-exchange model. Also in this case the differences between the energy-

dependent and energy-independent approach disappear for the higher partial waves,

and we therefore restrict the discussion and the presentation of results mainly to

partial waves with smaller angular momenta.

The one-pion-exchange model yields a very poor approximation for the NN

interaction in the ‘S, channel. This is demonstrated e.g. by the fact that the phase

shifts displayed in fig. 9 for this channel have even the opposite sign as compared

to the empirical values or the results for the realistic OBE displayed in fig. 11.

Therefore it is surprising that the n J T terms still account for roughly 1 of the

discrepancy between the energy-dependent and the energy-independent approach

for the OBE case. Since the terms with more than one fold seem to be of minor

importance only, the remaining difference may be attributed mainly to folded

diagrams of the form 7r J CT, (+ J u, etc.

The tensor component of the one-pion-exchange is a very important ingredient

of the NN interaction in the coupled 3S, and 3D1 channels. This is one reason why

the one-pion-exchange model is a fairly realistic model for the NN interaction in

this channel which leads to phase shifts which are roughly 50% of those obtained

for a realistic interaction. Consequently the inclusion of the r J r terms in the

energy-independent approach (curve C) yields phase shifts which are in a very

good agreement with the energy-dependent approach. It is interesting to note that

the rr J v terms overshoot the attractive contribution of all folded diagrams in the

OBE model for this channel. The repulsive contributions to the folded diagrams

which are still needed to obtain agreement with curve A may arise from r J p terms

since the tensor component of the p-exchange has the opposite sign as the tensor

component of the one-pion-exchange.

The results for the ‘P, are similar to those in the ‘So channel. The one-pion-

exchange yields for large energies phase shifts with the wrong sign and for these

energies the rr J n- terms only account for a small part of the difference between the

energy-dependent and energy-independent approaches. However, for the other P-

and D-channels the inclusion of the 7r J r terms alone yields results for the phase

shifts which are in a good agreement with the model A. This is trivial for channels

as e.g. ‘DZ, for which already model B is in good agreement with A, or for channels

as e.g. ‘P,, where the one-pion-exchange alone leads to quite realistic phase shifts.

However, it is interesting to see, that even in channels for which the one-pion-

exchange model is not too good, the difference between model B and A is com-

pensated to a large extent by the r J GT terms. This may indicate that for such

channels the folded-diagram terms involving other mesons than the pion have a

range which is too short to influence the phase shifts for these channels.


9 0

I

I I I

a

[ PDJ ] +4!qS =Dlld [ PDJ ] +i!IIS =D’-id

[ PDJ] +4!4S =Dlld

C. de &man et al. / fan-paid-aeon diagrams 625

3 d

[pm ] 14!4S aswd

0 0- N>

r”

s 8 w- -

0 J m (u - 0

:> d d d d

r / I I

[ piI.4 ] 84!4S asDqd


As shown in fig. 11, the phase shifts calculated from our potential model (C) are

generally in rather good agreement with the empirical phase shifts. This is certainly

encouraging that the construction of an energy-dependent (or locally energy-depen-

dent) nucleon-nucleon potential from meson exchanges appears to be quite feasible,

and in its calculation higher-order mesonic folded diagrams may be safely neglected.

5. Conclusions

In this paper, we have studied energy-independent nucleon-nucleon potentials

derived from meson exchanges, based on a T-matrix equivalence theory. The

importance of constructing an energy-independent, or more appropriately locully

energy-dependent, V,, is discussed. Briefly speaking, the energy dependence of

our V,, is restricted to the dependence of its matrix element (al V,,lb) on the

energies associated with its bra and ket states a and 6. Our V,, is obtained using

a folded-diagram expansion. Another scheme of constructing the nucleon-nucleon

potential is also discussed, and this leads to an energy-dependent, or more appropri-

ately globally energy-dependent, potential V,. The matrix elements of V,, (a( V,( E)lb) depend on a global energy variable E which is independent of the bra

and ket states a and b. Strictly speaking this global energy dependence of V, does

not allow a definition of a NN potential, and it can be difficult to use this interaction

in a nuclear structure calculation. The folded-diagram expansion of the NN interac-

tion, which is discussed in this paper, now provides a way to determine a potential

which does not depend on an external or global energy variable, and yields the

same T-matrix elements and bound-state properties as the corresponding energy-

dependent potential I’).

Our V,, contains a new type of irreducible diagram - the folded diagram. The

main purpose of this paper is to investigate how to calculate these diagrams, their

importance, and whether it is feasible in using them to construct a nucleon-nucleon

potential which can describe the nucleon-nucleon scattering data. As a first step

we consider a simple one-pion-exchange model for the NN interaction and evaluate

the phase shifts for the energy-dependent approach (model A) and the energy-

independent approach considering only terms without folds (model B) and with up

to one fold (model C). Considerable differences between the phase shifts calculated

in model A and B are obtained, indicating the importance of the folded diagrams.

We have found, however, that the pion-fold-pion (T 1 n) diagrams are important

only for low partial waves (1~ 2). This feature is desirable for actual calculations

of v,,. The fact that the inclusion of folded diagrams of first order (C) leads to a

good agreement with the energy-dependent approach indicates that the terms with

more than one fold (n J T J n etc.) are less important.

We then consider a realistic one-boson-exchange (OBE) model for the NN

interaction. In this case phase shifts are also calculated in both the energy-dependent

and energy-independent approach. The calculation of folded diagrams was also

restricted to the rr J rr terms. We have found that the phase shifts given by our


energy-independent potential (C), which includes OBE with rr j r only, reproduces

quite well the phase shifts given by the energy-dependent approach (curves A) as

well as the empirical phase shifts. As in the OPE case, we also find that the folded

diagrams are important only for partial waves with 1 s 2. The above indicates the

feasibility of constructing, from meson exchanges, an energy-independent NN

potential using the folded-diagram expansion. It also indicates that the other folded

diagrams such as rrj u, rjp and ~5 rr rr are considerably less important than j

r J T. Although this is consistent with the argument that folded diagrams involving

more and/or heavier mesons are of smaller ranges, further investigation of these

other mesonic folded diagrams is important and remains to be done. It will be of

much interest to apply the energy-independent NN potential obtained in the present

work, or to be obtained using the method described here but with more refined

calculations, to many-body nuclear structure calculations, such as binding energy

and saturation of nuclear matter. This will be an essential test of our theory that

the NN potential obtained with the present method preserves, in principle, not only

scattering but also bound-state properties of the original field-theoretical hamil-

tonian. Nuclear matter calculations using the present potential are in progress.

Many helpful discussions with G.E. Brown, R. Vinh Mau and M. Johnson are

gratefully acknowledged. This work has been supported by the US DOE under the

contract DE-AC02-76ER13001 and also by the Deutsche Forschungsgemeinschaft

(DFG, West Germany).

References

1) K. Holinde, Phys. Reports 6X (1981) 121;

R. Machleidt, Lecture Notes in Phys. 197 (1984) 352;

H. Muther, A. Faessler, M.R. Anastasio, K. Hollinde and R. Machleidt, Phys. Rev. C22

(1980) 1744; M. Lacombe, B. Loiseau, J.M. Richard, R. Vinh Mau, J. Cote, P. Pires and R. de Tourreil, Phys.

Rev. C21 (1980) 861

2) P.J. Ellis and E. Osnes, Rev. Mod. Phys. 49 (1977) 777;

T.T.S. Kuo, Topics in many body theories of effective interactions, Lecture Notes in Physics, vol.

144 (Springer, Berlin, 1981)

3) M. Johnson, Ann. of Phys. 97 (1976) 400 4) Guang-lie Li and T.T.S. Kuo, Bull. Am. Phys. Sot. 26 (1981) 607;

Guang-he Li, K.K. Ng and T.T.S. Kuo, Phys. Rev. C25 (1982) 2877

5) H. Feshbach, Ann. of Phys. 19 (1962) 286 6) T.T.S. Kuo, S.Y. Lee and K.F. Ratcliff, Nucl. Phys. Al76 (1971) 65

7) H. Miither, A. Polls and T.T.S. Kuo, Nucl. Phys. A345 (1985) 548

8) R.V. Reid, Ann. of Phys. 50 (1968) 411

9) B.D. Day, Phys. Rev. C24 (1981) 1203 10) H. Bethe, B.H. Brandow, A.G. Petschek, Phys. Rev. 129 (1963) 225

11) G. Brown and A.D. Jackson, The nucleon-nucleon interaction (North-Holland, Amsterdam, 1976) 12) M. Haftel and F. Tabakin, Nucl. Phys. A158 (1970) 1

13) K. Erkelenz, Phys. Reports 13C (1974) 191

14) K. Kotthoff, K. Holinde, R. Machleidt and D. Schiitte, Nucl. Phys. A242 (1975) 429 15) R. Machleidt, unpublished

16) M.H. McGregor, R. Arndt and R.M. Wright, Phys. Rev. 182 (1969) 1714

17) T.T.S. Kuo, H.B. Geyer and S.S. Wu, to be published

effects of pion-fold-pion diagrams in the energy-independent nucleon-nucleon potential

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