Isobar-hole description of pion elastic scattering and the pion-nuclear many-body problem

2.L

[

Nuclear Physics A319 (1979) 4 7 7 - 5 1 7 ; (~) North.HollandPublishino Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

ISOBAR-HOLE DESCRIPTION OF PION ELASTIC SCATTERING AND THE PION-NUCLEAR MANY-BODY P R O B L E M E. OSET t and W. WEISE

Institute ~[ Theoretical Physics, UniversiO' ~[~Re.qenshur,q, D-8400 Re,qenshur.q, W. German)' Received 7 August 1978 (Revised 12 October 1978) Abslra¢t: A calculation of the pion-nuclear T-matrix is performed within the framework of the isobarhole model. The emphasis here is on a consistent microscopic treatment of A-isobar self-energy and particle-hole interactions. This approach allows incorporating medium corrections (Pauli and binding effects together with absorption and reflection contributions) in a non-phenomenological way. The effects of various medium corrections are discussed for the case of g-~He scattering.

1. Introduction Pion-nucleus scattering in the (3, 3) resonance region has for a long time been subject to simple (and often surprisingly successful) multiple scattering treatments, frequently treating the nucleons as infinitely heavy objects. More recent investigations of the many-body aspec.~ of pion nuclear-elastic scattering both in the resonance region 1-6) and at low energy 7 9) have pointed out that the apparent success of such simple approximations is misleading. It has been emphasised that higher order medium corrections to first order optical potentials are important, although there is partial cancellation between various effects. Several authors have pointed out the relevance of so-called "true-absorption" channels. In fact, a major fraction of the reaction cross section below resonance is due to true absorption 10). The absence of such reaction channels in most of the simple fixed scatterer treatments is hard to justify. The purpose of the present paper is to incorporate what is commonly called Pauli and binding effects, together with absorption and reflection contributions, into a many body description of the scattering amplitude. This is appropriately done in a picture which describes pion-nucleus scattering in terms of an excitation and propagation of A-isobar-hole states 1.4-6). In such a model, many-body effects appear through self-energy modifications of the A-isobar Green function 2,~), or through vertex corrections ~,9), which can be understood as a generalised form of the Lorentz-Lorenz correction 11.12). Altogether, the many-body corrections contribute to the energy dependent shifts and widths of the relevant isobar-hole doorway states ~3.1, z). Up to now, absorption and reflection terms have been included into isobar-hole models on a purely phenomenological basis 1,2): The isobar is assumed to move in * On leave from University of Barcelona. 477

478

E. OSET A N D W. WEISE

a simple local complex optical potential, the (energy dependent) parameters of which have been adjusted freely to obtain an optimum fit to total and elastic cross sections. Although the diversity of effects simulated by such a treatment cannot be controlled, we note that the best-fit isobar optical potential obtained in this way generates contributions to the shifts and widths of the dominant isobar-hole states which are comparable in magnitude to other important many-body corrections (e.g. Pauli principle corrections) which can be calculated rather reliably. It is therefore somewhat unsatisfactory to relaize that a given class of many-body effects can be evaluated to a considerable precision, while other effects of similar order of magnitude are either left out our simulated by purely phenomenological methods. In a recent paper 14), the authors have presented a microscopic calculation of absorption and reflection terms together with Pauli and binding contributions for g-4He scattering. The absorption is described in terms of the coupling of the A-hole states to the two-nucleon two-hole continuum. It can be expected that such a calculation accounts for the larger part of the p-wave absorption in the (3, 3) resonance region, inasmuch as it takes into account the influence of the (g, 2N) reaction on the elastic channel. It is well known that the results of such calculations (in particular for the dispersive real parts associated with the absorptive widths) depend strongly on the off-shell properties of the effective meson-baryon Lagrangians [se.e refs. 1~, 16) in connection with s-wave absorption]. On the other hand, detailed studies of the p-wave pion absorption in the deuteron, ~+d -~ pp, have been performed which impose strong constraints on the relevant ~NA and pNA vertex functions 17). The calculations of absorption channels performed in this paper strictly fulfill the important requirement of consistency with gd absorption. The purpose of this work is to present the model of ref. 14) in detail and extend the material in various directions. The general scheme will follow that of ref. 5), where the pion-nuclear many-body problem in the isobar-hole framework is conveniently split into one part which deals with effective isobar-hole interactions including vertex corrections, and a second part which concerns the self-energy interactions of the isobar and the hole with the surrounding nucleons. The isobar-hole interactions and self-energies are then combined in a matrix inversion scheme, closely related to the response function formalism developed by Werner is), to obtain the full isobarhole Green function and the pion-nucleus T-matrix. Our emphasis will be on a detailed discussion of the various ingredients which determine the dynamics of isobar propagation: The reduction of the isobar width due to Pauli correlations, the broadening due to absorption channels; the shifts in the energies of isobar-hole states due to Hartree potentials (binding effects); dispersion effects related to true absorption and reflection terms; the role of p-meson exchange in the isobar-hole interaction. Since our investigations are in many respects related to the work of ref. 1), we would like to comment here in more detail upon the degree to which we regard our approach as an extension of ref. 1) in various directions.

ISOBAR-HOLE MODEL

479

The important development of ref. 1) was to point out that an understanding of n-nuclear scattering requires a careful consideration of absorption and quasi-elastic channels. The treatment of elastic broadening and lowest order Pauli corrections in ref. 1) has been on a microscopic level, while all other parts of the pion-nuclear many-body problem, like absorption terms and reflection contributions to quasifree scattering, p-meson exchange and other short-range baryon-baryon correlations, are supposed to be summarized in the phenomenological complex spreading potential already mentioned. Our aim is to replace the spreading potential phenomenology by a microscopic treatment of many-body effects based on effective meson-baryon Lagrangians as input. The question is then : To what extent are we able to describe absorption and other many body degrees of freedom in pion-nuclear scattering on a microscopic level using meson-baryon interactions constrained by nd absorption? Applications will be presented for 4He. They indicate that part of the success of simple multiple scattering pictures is accidental due to cancellations in the various higher order many-body corrections. We shall not include Coulomb and s-wave interactions in our investigation, so that the comparison with experimental data should be limited to the vicinity of the (3, 3) resonance. Nevertheless, some exploratory statements will be made about the relationship between the dispersive part of the p-wave absorption and the LorentzLorenz correction, which is of interest in the analysis of low energy scattering.

2. Elementary input: A-isobars and effective Lagrangians As in earlier work 4, 5), the A-isobar is introduced as a particle to represent the resonant part of the nN scattering amplitude. The non-relativistic effective Lagrangian which couples a pion of charge 2, four-momentum q = (~o, q) and mass kt to the NA system is 5) 6H(q) = i f,(q2) S . q T~[n~(q) + n~ ( - q)] + h.c.,

(2.1)

where nz(q) annihilates a pion; S and T~ are transition spin and isospin operators, respectively. The form factor is taken to be of the monopole form, f,(q2) = f

* ( A2 -/~2 ) AZ-~ ,

(2.2)

as in ref. 17). The free A-isobar GrEen function is G a(p, E) = I x ~ s - M n + ½ir(s)]- 1,

(2.3)

where ~/s = [ ( M + E) z _pZ]~, the total c.m. energy of the nN system. Here M is the nucleon mass, and M~ = 1232 MeV that of the isobar. In the non-relativistic limit,

p2 x/s~m+E

2(M+E)"

(2.4)

480

E. OSET AND W. WE1SE

The coupling strength f * and the width F are then prepared such as to yield a good description of the resonant part of the nN amplitude, as outlined in appendix A. The adjustment is made in the nN c.m. system, where we choose for the effective n N A coupling strength: f*2 2M a 4n - 0.32 Ma +~/s ' A = 1.2 GeV,

(2.5a) (2.5b)

as in ref. 17). The additional factor in eq. (2.5a) takes into account relativistic corrections (see appendix A). The isobar width is then obtained as 2 f.2

~

1" - 3 4nkt 2 q3

O(E-#),

(2.6)

where again relativistic recoil corrections are included. The resonant amplitude of appendix A, if combined with the background from crossed nucleon Born and crossed isobar terms, reproduces the 63a phase shift between T, = 100 and 250 MeV within 3 %, without additional cutoff factors. Note that the pion momentum q in the preceeding equations is in the nN c.m. system, while the n N A effective Lagrangian will finally be applied in the n-nucleus c.m. frame. If an average over the internal nucleon momenta is performed (see appendix A), the relative n-nucleon momentum q in the n-nucleus c.m. frame, written in terms of the pion momentum k and the frequency ~oin that frame, becomes q = ~k,

~

--

M + ~o/A

M+~o

,

(2.7)

to be inserted in eq. (2.1). Thus calculations in the n-nucleus frame can be carried out keeping the form $. k for the effective Lagrangian, and replacing f * of eq. (2.5a) by f * ~ ~f*. 3. Isobar-hole description of the ~-nuclear T-matrix

We proceed now to formulate the n-nuclear response problem in the isobar-hole framework, continuing stepwise from the simplest multiple scattering approach to higher degrees of complexity in the pion-nuclear many-body problem. We assume isobar dominance, i.e. we omit direct and crossed nucleon Born terms, the sum of which provide only a small background for nuclei with zero total spin and isospin 5), which we shall consider throughout in this paper. The incoming pion excites A-isobar nucleon-hole states, Is) = I(A~)J", T),

(3.1)

coupled to angular momentum J~ = 0-, 1+, 2-, 3+, etc. and isospin T = 1. The

ISOBAR-HOLE MODEL

48 I

nucleon hole N is described by a single particle shell model wave function *. The scattering process is then determined by the propagation of these AN states in terms of the full A-hole Green function. In this first step, we shall develop the formalism in the Tamm-Dancoff approximation (TDA), which is equivalent to taking into account only "forward-going" isobar-hole states 5). The resonant part of the pion-nuclear T-matrix within TDA is written as 1

(k'l T(09)lk)

= ~

~ (k'

IfiH Is')G~,~(09)(slfHIk),

(3.2)

between in- and outgoing pion states ]k) and Ik'),1, where fiH is the rcNA vertex operator of eq. (2.7), including kinematic factors discussed in sect. 2. The quantity of central interest is the isobar-hole Green function 1

Gs,~(09) = ( s' 1-09-- .OF Is)'

(3.3)

where off is the total Hamiltonian of the coupl.ed system of nucleons and A-isobars. The scattering amplitude is ~(0, 09) = - 2-~ (k'l T(09)lk>,

(3.4)

normalised such that d~

d--~ = 1.7(0,09)1z. The Tamm-Dancoff approximation considers G~,s(09) only for positive frequencies 09. Inclusion of the negative frequency part is equivalent to passing from TDA to the full RPA scheme t s, 5), which we shall do later. Since G~,~ is diagonal in the A-hole spin J~, eq. (3.2) is equivalent to an expansion in partial waves ~(0, 09) = ~ (2J + 1)Aj(09)Pj(cos 0).

(3.5)

d

In the absence of A-hole interactions and for freely propagating isobars, the A-hole Green function is written G~,0~(09) _

6~,~

(3.6)

09_,% '

where e s = M A - M - ½iF(09 + er~) + e~ - e N ~ In the specific application to rt-4He, we shall choose harmonic oscillator wave functions. ~t The pion charge index will be dropped for simplicity.

(3.7)

482

E. OSET AND W. WEISE

is the unperturbed A-hole energy. Here eN is the single particle energy of the bound nucleon and ea is, in this limit, the kinetic energy of the A-isobar as it appears in eq. (2.4). The free isobar width F [see eq. (2.6)] describes the coupling to the free ~znucleon continuum. Note that the energy variable in F is shifted by the single particle energy of the hole from which the isobar is generated. Next, we add the one-pion exchange interaction,

f.Z(qZ) V n ( ( . o , q ) ~-

]2 2

S 1 • qS~ . q ~o2_qz_]22+io(Tl"

T~-),

(3.8)

to the Hamiltonian of eq. (3.3). This generates matrix elements of the isobar-hole interaction in the direct channel (see fig. 1) n [ R~,s = (s i I V,,s).

(3.9)

S'

$ Fig. 1. Direct part of the isobar-hole interaction. Clearly, this particle-hole interaction is strongly energy dependent and develops an imaginary part for ~o > ]2. The isobar-hole Green function now becomes G~,~(co) -- [(o9- e~)6~,~- R~,~(og)]- t

(3.10)

When inserted into the T-matrix of eq. (3.2), the one-pion exchange in G,, s generates multiple scattering. To make the connection clear, we mention that - except for crossed isobar terms to be included later - the fixed scatterer approximation (without any medium corrections) is obtained if the nucleon single particle energies eN are neglected in es. The imaginary part of R~,S combines with the free isobar width and leads to the elastic broadenin~ t) o f the relevant isobar-hole states, while the real part of R" causes a downward shift of the peak in the total cross section. Medium corrections enter in two ways to modify eq. (3.10): (a) The particle-hole interaction matrix Rs,~ is generalised to include p-meson exchange, spin-isospin independent short-range correlations, and other types of vertex corrections. These have been discussed in ref. ~), so that only a brief chapter will be reserved for such corrections in the following section. (b) The isobar experiences various self-energy interactions. The most important ones among these are considered to be the Hartree potential, Foek terms (Pauli corrections), and the coupling to more complicated continuum configurations of

ISOBAR-HOLE MODEL

483

the highly excited many-body system, including true absorption. As in ref. 14), the self-energies will be summed up in a matrix Zs,~(o9) which is non-diagonal only in the quantum numbers of the isobar:

Xs, ~(o3) = ( A'JX(~o)IA )6r~,N.

(3.11)

The calculation of various contributions to the A self-energy will be presented in detail in sect. 5. With inclusion of the self-energy and particle-hole matrices S~,~ and R~,~, the particle-hole Green function becomes

~,s(o)) : [(~o - ~ ) , ~ , ~ - s~,s(~o)- ~,~(~o)3 -~

(3.12)

Calculation of the T-matrix of eq. (3.2) requires a combined inversion of ~ and R. Finally, we add crossed terms in going from TDA to the full RPA scheme which includes "backward going" A-hole excitations as shown in fig. 2. Let us illustrate this

~ ~.~.. ~ I ~

~,k Fig. 2. Schematic representation of the pion-nuclear T-matrix in the isobar-hole RPA approach. The full isobar-hole interaction is denoted by R, self-energy renormalisations of the isobar propagator are denoted by /2. Not shown here are binding corrections on the nucleon-hole lines.

first in the lowest order pion self-energy, employing the free A-hole Green functions G (°) ofeq. (3.6). Adding the crossed term, we obtain the lowest order optical potential, with binding effects for the nucleon holes included, but omitting any other medium corrections: 1


(k'l//~(~o)lk>

1

- 2~o ~ {(k'l~SHls')G~9~(°o)(slfiHlk) + (016nlk; s')G~9~(-o~)(s; k'16H[0)} a'$

1

- 2~9 ~ (k'lfHls'){G~9](~o) + G~°~)(- o~)}(s[6Hlk), ~d'S

(3.13)

484

E. OSET AND W. WEISE

where the last equality follows for a spin-zero N = Z nucleus from the crossing property of the vertex functions and the symmetry under s ~ s'. The full T-matrix follows then from eq. (3.2) upon replacing Gs,~ by the full RPA Green function fgs,s(e)) = {[(~o6~,~- Ms,~(e))) 1 -(e~fi~,~+ ~s,s(e))- 1]-1 _ R~,~(~o)}- 1. (3.14) We have introduced here the mass matrix M~,~(co) = ~(co)6~,~+ Ss,~(~o),

(3.15)

~s,s(co) = Mss,(-co).

(3.16)

and its crossed analogue

Detailed formulae will be presented in appendix B. 4. Isobar-hole interactions The model which we shall use forthe isobar-hole interaction to generate the matrix Rs,s will be essentially the same as in ref, 5). The driving term in R is given by (nonstatic) one-pion exchange (OPE), but vertex correction due to mechanisms other than OPE have to be taken into account. Among these, p-meson exchange in combination with short-range correlations is regarded to be most important as a generator of the Lorentz-Lorenz correction ~2), while other types of vertex corrections, for example through exchange terms of the isobar-hole inter.action, are estimated to be comparatively small 5). Thus we choose the following model for the effective interaction in the NA -~ AN channel, the one which is responsible for the direct isobar-hole interaction, fig. 1 : f d3k

17(o9,q) = J ( ~ ) 3 f2(q - k ) [ V,(co, k)+ Vp(~o,k)],

(4.1)

where V~ is the OPE interaction of eq. (3.8), while V, is the p-meson exchange force,

Vp(m,k) - f~Z(k2) (S, x k)" ( S ] x k) 2

mp

~ 2 _ k 2 -- mp2

T~.T~.

(4.2)

The NAp vertex is not very we~ determined; we choose 2

f~(q2) = f f A ~ - m o A2 o ~ q~"

(4.3)

and use the quark model relation.~/f, = ~ 7 2 / 2 5 , with the pNN coupling constant fp = g, (1 +K)mp/2M. Values commonly used are g~/4~ = 0.5 and K between 3.7 and 6.6. A value o f K = 4.5 to 5 together with Ap around 2 GeV is consistent with an analysis of ~-absorption in the deuteron ~v). Furthermore, ~(q) is the Fourier transform of the static baryon-baryon correlation function which receives contributions primarily from multiple exchange of ~-

ISOBAR-HOLE

MODEL

485

mesons. We write 27~ 2

q~- C(q).

(2(q) = (2~)363(q)

(4.4)

A simple schematic form of the correlation function is obtained by choosing for C(q) the dominant Fourier component 5),

C(q) = ,5(q-qc),

with qc ~ rn,o,

the o~-meson mass. Then

~'(~o,q) = V~(o~,q)+ Vo(~o,q)- [ V2(~o,q)+ V;(~o, q)],

(4.5)

where 2~2 f d3k V;(~o, q) = ~ j ( 2 ~ ) 3 b(Iq - k l - qc)V~(co,k),

(4.6)

and an analogous expression for V2. An appropriate angular average yields V~(m, ff)= ( f f (; q ) +q~) z S," qS~+ "q+~q~S~'S~ ' z + ~ - _ ~ ~ T~. T ; , m --~ --q¢ --~ V;(~, q) = [~ } ~pk

)

xq)'(S~+ ~q)+~q¢S~" S,+ -?~~.._z T~" Tz+, ~ - q -q~ -mo

(4.V)

(4.8)

In the limit mz, qZ << q~ ~ m~, z eq. (4.5) reduces to the form

P(o,q) ~ V~(~,q)+

g'S~. S~T~. T~,

(4.9)

where g' plays the role of a Landau parameter transcribed to the NA force, or equivalently, the Lorentz-Lorenz correction ~). In our model, 2

2

*

2

2

2

2

g, ~ I ~ ( A ~ - , ~ ~ q~ + 2p~(ff~ * (A;-mg~a ~ q~ -~ ~ [ ~ ] q~+,a-ma mo ~ f ] ~A~+q~] q~+m~J

(4.10)

For the role of exchange terms, we refer to the discussion in ref. ~) and to ref. ~9). Clearly, the larger part of eq. (4.10) comes from correlated p-exchange. The effective interaction of eq. (4.5) together with (4.7) and (4.8) will be used to calculate matrix elements of the direct isobar-hole interaction: R,,~(o) = (s'l ff(~)~s). Formal details will again be presented in appendix B.

(4.11)

486

E. OSET AND W. WEISE 5. Isobar- self-energies

The isobar propagating through the nuclear environment experiences all kinds of self-energy interactions. We wish to include all terms which would, in infinite nuclear matter, correspond to corrections of first order in density p in the self-energy ~, i,e. of order p2 in the pion optical potential. We distinguish between those parts of _r which can be absorbed in a real, local and energy independent (Hartree) potential and others (Fock terms, true absorption and reflection contributions) which develop imaginary parts and show a characteristic energy dependence. 5.1. HARTREE POTENTIAL FOR ISOBARS The isobar moves in a Hartree-type background potential, as shown in fig. 3, This potential receives contributions mainly from two-pion exchange and combined np exchange. The process fig. 3a involves mostly the second order tensor force and provides attraction comparable to the analogous piece in the nucleon single particle potential, fig. 3a', if the quark model is used to relate coupling constants 4). The twopion or ~tp exchange contribution of fig. 3b has its counterpart in a corresponding, piece of the 2re or r~p exchange nucleon-nucleon interaction, fig. 3b'. On the other hand, figs. 3c and d, which also contribute to the attraction (through 2n exchange) and short range repulsion (through ~tp exchange) in the NN force, have no analogue in the NA interaction. On the basis of the work of Durso et al. 20), we expect that the diagram fig. 3b', with all combinations of ~trt and ~tp exchange, provides about half of the combined effect of what would be described as "tr" and "co" exchange in oneboson exchange potentials. Using quark symmetry relations, the summed contributions of figs. 3a and b are then expected to lead to a moderately attractive local and

÷

Crossed ferms

(a)

(b)

(a') (b') (c) (d) Fig. 3. Main contributionsto isobar Hartree potential (a and b), as compared to mechanismsgenerating the Hartree potential for nucleons (a', b', c and d).

ISOBAR-HOLE MODEL

487

energy independent Hartree potential. The precise depth of it is not very well under control. We choose a parametrisation of the form vAHarlree(/") =

--

Vof(l'),

(5.1)

where the depth is assumed to range between 20 < Vo < 50 MeV and a harmonic oscillator form is used for f(r). This Hartree potential serves to provide a single particle basis of A-isobar states and corresponding single particle energies ez which replace the kinetic energy in the free isobar-hole propagator, eqs. (3.6) and (3.7). In the discussion of our results, we shall comment on uncertainties in the n-nuclear T-matrix owing to lack of information about Vo. 5.2. P A U L I A N D B I N D I N G C O R R E C T I O N S

Here we shall study the Fock term shown in fig. 4. This involves the decay of the isobar into nucleon and pion. If the nucleon and pion were both free, non-interacting particles, the imaginary part of this self-energy contribution would just generate the free isobar width, eq. (2.6), and the real part would be absorbed in the renormalised free mass of the A. We are interested here in modifications of the free mass and width from the following sources: (a) The intermediate nucleon states in fig. 4 are partly blocked for the isobar decay because some of them are occupied (Pauli blocking). This leads to an energy dependent reduction of the decay width. (b) The intermediate nucleon feels a single particle potential. We refer to this as a binding effect. ~e

Fig. 4. Fock term of the AN interaction. The cut nucleon line indicates the corrections due to Pauli and additional binding effects. The wavy line corresponds to n- and p-meson exchange plus additional shortrange correlations.

The combined Pauli and binding corrections lead to an energy dependent complex shift of the isobar position which we would like to study in detail. We shall call the corresponding self-energy matrix elements (A'ISp(E)IA), to be taken between A-isobar basis states. Suppose first that only a single pion appears in the intermediate state of the Fock diagram, fig. 4. The corresponding isobar self-energy operator is

488

E. OSET AND W. WEISE

then

= ,f dtzn~

qo--q --~ +~0

where the integration runs over the pion four-momentum (qo, q)- Here 6 H is the vertex operator of eq. (2.1). The G~°) is the free nucleon propagator,

1

1

G~)(E) - E - T + i 6 - ~ IN> (N'I, ~, E - T ~ , - i6

(5.3)

with T the kinetic energy operator and {IN)} an arbitrary basis set of nucleon states. Furthermore, Pv 1 - P~ G~(E) - E - Ho(E ) - i6 + E - Ho(E ) + i6

is the propagator for a nucleon inside the nucleus, where Ho = T + V(E) is the single particle Hamiltonian describing the interaction of this nucleon with the residual nucleus. The energy dependent single particle potential V(E) is taken to be of the form (5.5)

V ( E) = - Vo(E) f (r),

where Vo is equal to 50 MeV at E _<_ 0, and the extrapolation to high energy is done in a power series in E following the real part of the optical potential according to ref. 21). For)'(r), a harmonic oscillator form is chosen, since the asymptotic behaviour of the nucleon wave functions is not relevant for the calculation of (AIZvlA'). Finally, Pv in eq. (5.4) is the projector onto the filled Fermi sea of bound nucleons, i.e. we write

PF = ~ JN)(NI

(5.6)

N~F

in terms o f the bound spectrum of H0. Note that the free propagator vN~(°) has to be subtracted in eq. (5.2) because its contribution is already absorbed in the free space A mass and width. We can rewrite: GN(E)- G~)(E) = 2niPv6(H o - E) +

[1

- E - H o + i6

1]

E - T + i6 "

(5.7)

The term 2 n i P v 6 ( H o - E) is referred to as Pauli correction, while additional nucleon binding corrections appear through the term in square brackets. Note that 6Ulq)(q[6H

qo~ _ q 2 _ ~2 + i(f

= V,~(qo, q)

(5.8)

ISOBAR-HOLE M O D E L

489

is simply the one-pion exchange interaction, analogous to eq. (3.8). While the imaginary part of Sp, which reduces the free decay width, is determined by OPE alone, the real part of Sp receives contributions also from p-meson exchange and additional short range correlations. In order to incorporate them, V~ is replaced by the effective re- and p-exchange NA interaction P, as in eqs. (4.5), (4.7) and (4.8). Detailed formulae are again given in the appendix. We shall discuss here a few M~V 20

~,..~"

10

~ -10

~

~... -

p.a ~ ~

~IZp ~~ ~

~

"~ I~>

l& >= Ils3/~ >, 1.0

~

,

~.~

",--~_ __j

1.8

2.2

MeV Im /"~

20

P÷B(TI:)

" ~ - ~ :

~

-20 -30

~

~

.

.~

_ ~

Irn _._ ._

~

E/~

P ~

//~/ ,,,,"

10

P÷B

/ z/z/ ?

~ ~,.¢/, ~ ' ~ " •

\

-10

I

\

IA> = Ilp~n>

P, ~,,~

-'° ,i0 Fig, 5. Examples of diagonal matrix elements o~ Fock self-energy Z~,(E) between harmonic oscillator states of the A-isobar. P: Pauli corrections only (no binding potential acting on intermediate nucleon line). P + B: Bin~ding correction i0cluded in addition, as explained in text. Input parameters for p-exchange areJ~*/x/4n = 4.1, Ap = 2.5 GeV. Correlation parameter qc = 5.5p. Pion exchange as in eq. (2.5). Upper curves show imaginary parts, lower curves show real parts of Zp. (a) For isobar in Is3~z orbit (oscillator parameter: h~oo~~ = 20 MeV). Curve P+B(~) shows real part with one-pion exchange only; to be compared with P + B (including p-exchange). (b) Same for isobar in lp,/2 orbit.

490

E. OSET AND W. WE1SE

instructive examples of important diagonal matrix elements (dl~'pIZI), see figs. 5a and b. Due to the frequency dependence of the OPE interaction, the Fock terms are strongly energy dependent, as expected. Again, Pauli corrections (P) refer to the 2rCiPF6(Ho - E) terms in eq. (5.7), whereas the inclusion of binding effects leads t6 the curves denoted by (P + B). Obviously, binding corrections are quite important in some channels. Also shown is the effect of p-exchange, in combination with other short range correlations, on (AIRe S~,IA); because of the short range of this part of the interaction, the corresponding (attractive) contribution is almost independent of energy in our region of interest. In general, the combined action of 2p in various states of the A leads to a positive imaginary part (a sizeable reduction of the free isobar width) and to an attractive real part (a downward shift of the resonance position), once p-exchange is included. We repear that, whereas Im Sp is solely determined by the exchange of an on-shell pion in the Fock term, Re ~p depends on cutoff mechanisms in the off-shell extrapolation of the various exchange interactions. The freedom of such extrapolations is reduced by the requirement of consistency with nd absorption. Nevertheless we would like to emphasise that Re ~p is less well determined than Im L'p although the overall tendency of an attractive (though energy dependent) Re ~p persists if cutoff parameters and the correlation length q~l are varied within the range of possibilities. 5.3. ABSORPTION AND REFLECTION CONTRIBUTIONS

Experimental spectra of protons produced by pions at various energies covering the (3, 3) resonance region 22) indicate that the dominant process for r~-absorption in 4He is a two-body mechanism in which the absorption proceeds through rc+ N --, A followed by A + N ~ 2N, very much in the same way as in the microscopic description of the ~td ~ pp reaction 17). Thus at least for '~He one can hope to incorporate the main contribution of absorption to pion elastic scattering by coupling the A-hole states to the two-nucleon two-hole continuum. This coupling is described by the self-energy contribution shown in fig. 6a.

Fig. 6a. Self-energy coupling of isobar to intermediate two-nucleon one-hole states (denoted by z~r~ in text), including crossed piece. This incorporates two-nucleon emission channels

Fig. 6b, Same for intermediate,~N hole states (denoted by Z'~a~ in text), incorporating resonant reflection terms.

ISOBAR-HOLE

MODEL

491

At the same time we wish to include so-called reflection contributions to quasi-free scattering, where a real pion appears together with an on-shell nucleon, the pion being rescattered by the optical potential. All these processes can be combined in the isobar self-energy operator

f

d3q ["d3q ' fdqo G~(E- qo)(qllI(qo)lq') (q'16H, XA(E) = i.~ (2z03,~ (2~z)3J 2n 6HIq) (q~ _ q2 _ 1~2+ i,~)(q2o_ q,2 _ #2 + i6)

(5.9)

where 6H is again the gNA vertex operator and GN is the nucleon Green function describing the propagation of one of the two nucleons in the intermediate state. Furthermore, H is the lowest order pion self-energy which receives contributions from the formation of intermediate nucleon-hole (ph) and isobar-hole states: //=//r~+//~,

(5.10)

where

Flr~(qo)= ~ 6H~lph) {~ ph

1

--e~--qo

-[- ~P - ~h - [ ' -

(phl6Hr~,

(5.11)

for a spin-zero N = Z nucleus. Here ep and eh are single particle energies. The summation over the intermediate particle states (p) covers an integration over the full continuous spectrum of unbound single nucleon states. The ~rNN vertex operator is denoted by 6H N and has the usual structure analogous to eq. (2.1) where f * S " qT~ is to be replaced by f a . qz~ with f z / 4 ~ = 0.08. The resonant p a r t / / n of eq. (5.-10) is given by eq. (3.13). In accordance with our attempt to keep terms in the isobar self-energy which correspond in nuclear matter to terms linear in density, we replace G~ in (5.9) by the free nucleon Green function G(O) N and take the same approximation for the particle spectrum in HN of eq. (5.11). The replacement of the intermediate two-particle states by plane waves is not expected to be critical (at least for the imaginary part of ZA), since the two nucleons have large momenta once the pion is absorbed. The OPE interaction in the NA -~ NN or NA -* AN channels appears twice in eq. (5.9). In order to arrive at a more realistic approach, p-meson exchange and additional short range correlations are once again added by replacements analogous to eqs. (4.5), (4.7) and (4.8). Clearly, the energy q0 in eq. (5.9) can be distributed into two types of physical decay channels: (i) The non-mesonic decay (AN -~ NN) of the A leading to the emission of two nucleons into continuum states corresponds to combined poles in G N ( E - q o ) and Hs(qo). The pions are virtual in this case, since large momenta q and q' are required to meet the on-shell conditions for the two nucleons. This is the "true absorption" contribution arising from the (~r, 2N) channel, as mentioned before. (ii) The rescattering of a real pion in the presence of an on-shell nucleon corresponds

492

E. OSET AND W. WE1SE

to combined poles in the pion propagator and in G~°)(E- qo). Most of this reflection contribution to the quasi-free channel comes from the resonant rescattering, fig. 6b; the pion rescattering through virtual nucleon-hole excitations, included in fig. 6a, provides only a small background. The imaginary part of '~a coming from the reflection terms, can be combined with the lowest order diagram, i.e. the imaginary part of Sp according to eq. (5.2), and the sum of both is to be interpreted, to this order, as the modification of the free width due to quasifree nN collisions. The principal value integrations associated with each of the absorption or reflection poles yield corresponding dispersive real parts of S A. These give additional energy dependent shifts of the isobar-hole states. Calculation of the real and imaginary parts of (A'IEAIA) have been carried out in a harmonic oscillator basis of A-isobar states, for the case of 4He. We refer to appendix D for details. The calculation requires a partial wave decomposition of the intermediate two particle states in 2;A(E), where angular momenta up to ! ~ 10 have to be taken into account in order to obtain sufficient convergence at energies as large as E = 2.6/~. We present a few instructive examples of diagonal matrix elements (A IL'AIA) in figs. 7a and b; the contributions from figs. 6a and b are discussed separately and are denoted by X~ ) and X~ ~, referring to either H N or H~ in the intermediate state. As mentioned before, Z~A N~ involves primarily the two-nucleon true absorption channel, while S~ ~carries most of the contribution from reflection corrections to the quasifree channel. Clearly, at low energy, Im S A is completely determined by the two-nucleon absorption, Im Z ~ ). The matrix elements of both real and imaginary parts of XtA N) are relatively smooth functions of energy. Of course, (AlIm 2;AIA) is always negative and effectively enhances the isobar decay width. The resonant rescattering contribution 2;~ ) generally shows a strong variation with energy, due to the resonant behaviour of the pion self-energy Ha, and also a strong dependence on the particular state of the A-isobar. (The example shown in fig. 7b can be regarded as typical, whereas the imaginary part of (ls~12;~)lls~), not shown in fig. 7a, happens to be accidentially small.) The imaginary part of the rescattering term receives contributions from the propagation of real pions, hence Im S~ ) goes to zero at low energy. It is also worthwhile noting that at low energy, matrix elements of Re S A provide attraction of roughly the same magnitude as Im ~A" A large part of this attraction comes from virtual rt- and p-exchange in fig. 6b which involves the NA -~ AN interaction in second order. There is a clear distinction, in terms of energy dependence, between these terms and the (energy independent) Hartree pieces discussed in subsect. 5.1 t The matrix elements of 2~A, in particular the real parts, are sensitive to the cutoffs in the meson-baryon form factors and to the p-baryon coupling strength, and to the * In addition,we wouldlike to mentionthat (lm~An~) at sufficientlyhigh energyreceivescontributions from the open NA ~ zlN channel.

ISOBAR-HOLE M O D E L

493

correlation length l / q c. Fortunately, the dependence on details of the correlation function is only moderate: a 20 % change in qc induces changes on the 10 % level in the matrix elements of2~A. Concerning cutoff factors and p-meson coupling constants, the situation is similar to ~td absorption, although not as extreme. In the ~d --, pp case, the tensor part of the interaction mediated by the rescattering of either g or p is of foremost importance, and the role of p-exchange lies in cutting down the abnormally large tensor force from OPE ~v). In the calculation of %A, the summation over many partial waves makes the relative importance of the tensor force with MeV 10

IA>=

Ils3/2>

5

-5

......

~~" ~ ~

_R~_

~

-~0

"'. -15

i

I0

~

I~

l

I

18

22

~

EI~

MeV

~)

IA> = Ild~/2>

lO

R~ ;__-/

/

/! -

-4,

-5---~-..~., ~ -----~... -10

-15

~

\

\\

//

/ ~

//

~"~'~Irn~AIY~IIA> i I //

\\

/ -. \,,." - ~ x~\/ \

/

\\

(~) \ \ Im

/

/

/ / I

/! .,~.~/

-20

& Fig. 7. (a) Examples of diagonal matrix elements of --Y'~AN)(E)and Z~Aa)(E) according to figs. 6a and b, respectively, for isobar in I s3/z orbit. (Not shown here is (A ]Im ~ I A ) which is small in this channel.) Input parameters: pion exchange as in eq. (2.5); p-exchange:J~*/~/4n = 4.1, Ap = 2.5 GeV; correlation parameter: qc = 5.5#. (b) As fig. 7a, for isobar in ld~/2 orbit.

E. OSET AND W. WEISE

494

respect to the spin-spin part of the interactions less pronounced. As in the nd case, the dependence on the p-baryon cutoff mass Ao [see eq. (4.3)] is quite pronounced in individual partial waves, but the overall result is stable with respect to variations in the range Ao ~ 2 GeV ~. We emphasise once again that the consistency of the meson-baryon effective Lagrangians used here with those appearing in the r~d ~ pp calculation is a crucial constraint to minimize the dependence of S A on the off-shell properties of these Lagrangians. /x'

A

(a)

(b)

Fig. 8. Direct a and exchange term b of the absorptive isobar self-energy,describing the coupling of the isobar to the two-particle one-hole continuum. Finally, we comment on exchange pieces, fig. 8b, to the direct absorption term fig. 8a. Both of these couple the isobar to the two-nucleon one-hole continuum. A calculation of the imaginary parts of the self-energies, figs. 8a and b, performed in nuclear matter (employing pion and p-meson exchange, form factors and additional short range correlations) reveals that the magnitude of the exchange diagram is always less than 30 % of the magnitude of the direct one, with opposite sign. Since the uncertainties in the direct absorption term cannot be claimed to be less than 20 ~o we have neglected the exchange diagram. We have no explanation for the large exchange term found in ref. 23), where short-range correlations are, however, omitted. 6. Results and discussion In this section we would like to present results for the separate influence of various medium corrections discussed in the preceeding sections. We calculate the pionnuclear T-matrix, eq. (3.2), for the case of n-4He scattering, with the R P A isobar-hole Green function, eq. (3.14), and include stepwise corrections in the particle-hole interaction matrix R~,~ and in the mass (or self-energy) matrix M~,~. Included in the calculation are all partial waves (or isobar-hole state) with J " = 0 - , 1 +, 2 - , etc. up to 6 - . A harmonic oscillator basis has been used for the bound nucleons as well as for the A-particle states, with a c o m m o n oscillator parameter ho9o = 20 MeV appropriate for 4He. F o r the construction of the matrix elements Rs, ~ and M~,~ between isobart For example, a change from Ap = 2 GeV to Ap = 2.5 GeV induces changes in the matrix element (A(Isa/2)IZAIA(Is3/2)~ of about 10 ~o in the imaginary part at threshold and less than 5 % in the 3, 3 region. For the real part, the change is somewhat larger at threshold (an increase in magnitude of about 20 %). If Re ~A develops a zero, the location of it is also slightly shifted (generally downward) by increasing A~.

ISOBAR-HOLE MODEL

495

hole states Is) = [(Ah)J"), A-particle states up to principal quantum number n = 3 have been taken into account, together with all possible angular momenta la required at a given J". Inclusion of higher principal quantum numbers would have changed the T-matrix by about 1% in the 3, 3 resonance region. In the first step we would like to study the effect of p-meson exchange (combined with short range correlations) in the isobar-hole interaction R, on the total cross section 4/~

Otot(Co) = -~- ~ (2./+ I) Im As(co) "3-

(6.1)

[see eq. (3.5)]. For reference, we show in fig. 9 a calculation of O-tot with the following ingredients: (a) Only the one-pion exchange part of the effective interaction V of eq. (4.1) is included in Rs, s. (b) The isobar-hole Green function is of the "free A" type, G ~°) of eqs. (3.6)-(3.7), i.e. omitting any medium self-energy corrections, but including nucleon binding in terms of the single particle energy ~s(~s = - 2 0 MeV for 4He). This case is roughly comparable with a standard lowest order optical potential calculation, except for nucleon binding and a (small) Lorentz-Lorenz type effect coming from the action of the correlation function in eq. (4.1) in combination with Otot [mb]

T~He ,,~--~

350 ~ ~q

~ x~ "~

,~/

'",

300

250

/ ....... -----

free A, f~ =0 free A ÷p exchtange

~

Hartree+p exthnnge

/

I

I

I

~50

200

250

T~[HeV] Fig. 9. Effects of p-exchange in the isobar-hole interaction, and of isobar Hartree potential, on total cross section for n-4He scattering.

496

01.

[rob]

i

~

~

~

~

------"free" A - - - - - Pauti ÷ Binding ........

150

Absorpfion. Refiechon / ~ ,

100

50

/ I0

I

5

I

100

150

I

I

200

250

Tv [HeV] i

0 2-

[mb]

i

free"A -----Pauli+ Binding .

.

.

.

.

........ 150

Abs orpfi on + Reflection fofal

100

/// \

50

S0

100

1~0

2~o

2'50

T~ (HeV] Fig, 10. Effects of various medium corrections on partial cross sections of dominant ~-4He isobar-hole states: J " = 1 + (a) and J " = 2- (b). "Free A": Free isobar (no self-energy interactions), but binding of nucleon-hole included. "'Pauli+ binding": Combined effect of Hartree and Fock self-energies 27~ added to "free A". (Hartree potential depth Vo = 50 MeV ; for input parameters of Fock terms see caption fig. 5.) "'Absorption + reflection": Combined effect of self-energies Z'~N~and , ~ (see fig. 6) added to "free A". For input parameters see text and caption of fig. 7. "'Total": Summed effects of "'Pauli+ binding" and "absorption + reflection".

ISOBAR-HOLE MODEL

497

the rtNA form factor. [Had we omitted these effects (by putting er~ = 0 and f2(q) = (270333(q)), the peak position in atot would be moved down in energy by about 20 MeV and raised beyond 350 mb.] The large elastic broadening is included here because of the propagation of on-shell pions in R~,~. Next, we incorporate p-meson exchange [i.e. we take the full V of eq. (4.1)], but still keep the free-A Green function. Since p-exchange acts repulsively, the peak position of atot is moved upward, with a corresponding decrease of the cross section. The value of the pNA coupling used here is fp*/.,f4~ = 4.1, together with a cutoff Ap = 2.5 GeV. With a correlation parameter qc ~ 5.5/z, this would give a LorentzLorenz correction of 9' "~ 0.5. Note that the effect of p-exchange is only moderate at energies above 200 MeV, but large at low energies. We then show the effect of a phenomenological isobar Hartree potential, parametrised in the form of eq. (5.1), with a potential depth V0 = 50 MeV, the same as in the nucleon single particle potential. The isobar energies e~ in eq. (3.7) are then replaced by the single particle energies in this potential well. As can be seen from fig. 9, the attraction from such a Hartree potential, although large individually, roughly cancels the repulsion from p-exchange in Rs,s; however, this may be accidental for *He. Of course, a variation of Vo or of the p-meson coupling constant would have partly destroyed this balance. We proceed now to discuss the influence of various self-energy corrections. In fig. 10 we show selectively these effects on the most important pion-nuclear partial waves, which turn out to be J~ = 1 +, 2- for 4He. We have plotted there the partial cross sections, 47~

~rs(co) = ~ ( 2 J + 1) Im As(m),.

(6.2)

where A s is the partial wave amplitude defined in eq. (3.5). The curve denoted by "free A" is calculated with plane-wave isobars; it includes, however, the binding of the nucleon-holes in the 1s~ shell. It also uses the full ~ of eq. (4.5) (re- and p-exchange plus correlations) in the isobar-hole interaction R(to). The curve showing Pauli and binding effects incorporates in addition an isobar Hartree potential as in fig. 9 together with the Fock terms 2~p discussed in subsect. 5.2 (including binding corrections on intermediate nucleon lines). The effect of the Fock terms is a narrowing of the width (6F ~ - 3 0 MeV at resonance in the 1 + partial wave) together with a downward shift of the peak position due to attraction in both the Hartree potential and in Re 2;p, once p-exchange and additional short range correlations are included in the Fock term. Also shown in fig. 10 is the separate effect of the absorption and reflection terms, i.e. the self-energy 2~A of subsect. 5.3 (combined with the isobar Hartree potential, but omitting the Fock terms 2;p). The width of the isobar-hole states is increased by this agency, as expected. In all these curves, the p-meson coupling parameters and the short-range

E. OSET A N D W. WE1SE

498

correlation parameter have been chosen to be identical (f~p/x/~ = 4.1, Ap = 2.5 GeV, qc = m,o = 783 MeV) with the parameters in the isobar-hole interaction and consistent with the nd absorption. The combined effect of all self-energies (Hartree together with Sp and 2~A) is presented in the "total" curves of fig. 10 and shows a partial cancellation between Pauli and absorption plus reflection terms, as far as the widths are concerned. The downward shift of the peaks in "total" with respect to "free A" would have been about 20 MeV less, if the isobar Hartree potential had been set to zero. ~

do/d~ [rob/st]

10 2

r

r

r

T~ ~He (Trt= 160 MeV)

~~x,

10 ~

/ //

10 o

i0 -~ ----

--

--

"--

-

~°-~

-

'o

2

"free"A Pouli .Binding fofol t~O '

60 '

8' 0

t~

100 '

120 '

lt.O '

160

[deg]

Fig, 11. Effects of m e d i u m corrections on differential cross section for ~t-*He at 160 MeV. For explanation of curves see fig. 10.

Fig. 11 presents an example of how various medium corrections affect the differential cross section. All partial waves up to 6- have been summed up in the scattering amplitude. These corrections become increasingly important under backward angles. On the other hand, because of the sensitivity with respect to input parameters (cutoffs, coupling constants) especially in the real parts of self-energies, it would be premature to draw firm conclusions.

ISOBAR-HOLE MODEL

499

o

imb]

T~He 150

IO0

50

3" /,,0-

5o

~oo

15o

200

250

Trt iMeV]

Fig. 12. Partial wave cross sections for ~-4He in each isobar-hole channel of given J~. This calculation includes: Pauli+ binding, absorption + reflection, as in fig. 10.

Fig. 12 shows the partial wave cross sections, eq. (6.2), including all self-energy interactions together with the full isobar-hole interaction. The dominance o f J ~ = 1 + and 2 - partial is demonstrated and agrees with the conclusions ofref, i). In the energy region of interest, there is rapid convergence beyond J~ = 4 - . As mentioned before, we include partial waves up to 6-. In the isobar-doorway picture 13), each one of these resonant A-hole partial waves would be identified with an isobar-doorway state, of given J, the width of which is considerably larger than the width of the free isobar. Because of their short lifetime, we avoid terms like "collectivity" 6) in connection with these states. The matrix inversion scheme which we use in order to calculate the T-matrix (independent o f the particular set o f A-isobar basis states) does not necessarily suggest a collective interpretation. Furthermore, we show in fig. 13 the various effects discussed before on the real part of the forward scattering amplitude f(0) = ~-(o~, 0 = 0) and on the total cross section, summed over all partial waves (up to 6-). The total cross section shows tendencies similar to those observed for the dominant partial waves. Clearly, medium corrections are by no means negligible in the total cross section. But as we have pointed out, there are partial cancellations between various such corrections, although they are large individually. Fig. 14 shows once again separately the effect of p-exchange in the isobar-hole

500

E OSET A N D W. WEISE

Re f(o) [

[fro]

1

1

/

.--~ T

/

/

~"

F

-~

free"A

.....

/~"~\

2

T - - 1 - - - -

--~,,%1

,,~', ' ~ \

........ ~o~.0!,o~.

n; ~He -2 I

L

50

_

L

100

150

_

2OO

250

300

350

T~ tMeV]

O f or

[mb]

~00

/\ '\ TC- ~'He

. 350

L\

"~

300 ~

250

2~ •

150

.

.

.

.

~" q /~/~//,'~/

........ Absorp,ion+ Reflection --,o,o[

50 I

I

50

100

Pauli+ Binding

I

150

I

~0

I

250

l

~00

T=IMeV] Fig. 13. (a) Real part of/~-4Hc forward scattering amplitude showing various medium corrections, as explained in fig. 10. All partial waves from 0- to 6- are included. (b) As in (a) for total n-4He cross section. Experimental data are from ref. 29) (dots) and ref. 3o) (triangles).

ISOBAR-HOLE MODEL ~

Of of

I

1

--

501

I

I

r--

[mb]

T~He

t, O0

~50

300

250

2OO

150

IO0

~0

5

I

I

I

I

I

100

150

200

250

300

T~ [MEV] Fig. 14. Effect of p-exchange in isobar-hole interaction on total =-*He cross section; (a) no p-exchange, but all self-energy interactions incorporated; (b) p-exchange included.

interaction, if all self-energies are included. The relative importance of p-exchange is quite pronounced at low energies, but becomes small above 180 MeV. The upward shift of the peak in 6tot is in accordance with the repulsive nature of p-exchange in the relevant isobar-hole channels. Fig. 15 presents differential cross sections calculated at various energies. On the whole, comparison with experimental data is quite satisfactory at energies around the (3, 3) resonance and above. Below T, = 150 MeV there seems to be still lack of absorption, and the real part of the forward scattering amplitude is not very well reproduced. On the other hand, we have left out a number of ingredients, like s-wave interaction and s-wave absorption, direct and crossed nucleon Born terms, which become increasingly important at low energy. One might ask whether it is sufficient to include only the lowest order self-energies in the calculation of absorption and reflection terms in Z A. Especially the reflection terms which involve the pion rescattering through the resonant part H a of the optical potential would tend to be enhanced in the resonant region and should require the consideration of multiple rescattering. Formally, the ultimately correct procedure would be to replace the pion self-energy (qlFl(qo)lq') in eq. (5.9) by the n-nuclear T-matrix, 2q0 (qlT(qo)[q'), and perform a fully self-consistent calculation. While

do/d~ [mb/srl

~He (Trt=150MeV)

lt;

10 z

,

~He (T~ =180 MeV)

i0 ~

',\ ,~,,'

10 o

100

104

i0 -~

0

2

i

i

i

i

t

I

t,O

60

80

100

120

1~0

10 -2

I

I

20

160

I

~0

60

flO

i

i

i

100

120

I/+0

16C

(9~.~Ideg]

8~r" [deg] do/d~

do/dQ [mblsr]

[mbbsrl

10;

10 ~

r~He

~: ~He

(Tr~=220 M e V )

10 ~

10 o

lO-~

I0 ~

1o-~

10 `2

20

t,O

60

80 0 ~ [deg]

100

120

lt, O

(Trt: 26,0MeV)

10 ~

i0 ~

10-~ ~

i

10 2

10 ~

10-z

i

dold~ [mblsr]

160

10 3

20

aO

60

80

(9~

100

120

lt.O

toe9]

Fig. 15. Differential re-'tHe cross section calculated at various energies with full isobar-hole interaction and all self-energy corrections. Experimental data are from ref. 29).

160

ISOBAR-HOLE M O D E L

503

such an effort is beyond the range of possibilities of our microscopic approach, we can at least estimate the importance of higher order rescatterings in XA by comparing the on-shell quantities (q IH(~o)tq') and 2o9(q IT(o~)Iq'). In the (3, 3) resonance region, higher order rescatterings would reduce the imaginary part coming from H alone by about 25 ~o. Thus the imaginary part of S~ ~tends to be overestimated by considering only the lowest order in H~. On the other hand, the absorption part in ~ is expected to be much less affected by such considerations, since it involves the propagation of highly virtual pions of large momentum, so that the short range of the interaction which couples the A to the two-nucleon continuum prevents higher orders in H N from becoming important. It is also worthwhile noting that the difference between RPA and TDA calculations of O-tot is only about 5 ~ at resonance, while at lower energies the background from crossed isobar-hole terms becomes more important (about 20 ~o at T~ = 100 MeV). We summarize our discussion for n-4He so far: (a) We have analysed the influence of p-meson exchange and short range correlations as vertex corrections to the standard OPE isobar-hole interaction. In the interpretation of Baym and Brown 12), p-exchange is the major source of the LorentzLorenz correction, while the influence of short-range repulsive correlations in addition to OPE is cut down by finite range nNA vertex factors. The effect of pexchange is repulsive and depends on the pNA coupling strength f~*. For values of f~* currently used, the result is an upward shift of the peak position in 6tot b y 10-20 MeV. (b) A Hartree potential for isobars is believed to provide net attraction, the precise amount of which is hard to estimate. For a Hartree poiential of depth - 5 0 MeV, this binding alone would give a downward shift of the resonance position by about 10 MeV, and a 10 ~o increase of O-tot at the peak * (c) Fock-type self-energies introduce Pauli effects and additional binding corrections. They give an attractive shift of the isobar mass with a relatively strong energy dependence coming from the long range OPE interaction. This shift depends on gNA cutoffs and p-exchange coupling parameters. The energy dependent imaginary part is determined by on-shell properties of OPE and gives a reduction of the isobar width, typically by about 30-40 MeV at resonance. (d) The coupling of the isobar to the 2 N continuum effectively increases the width of the important isobar-hole states by 10-20 MeV. This imaginary part is a slowly decreasing function of energy. The real (dispersive) part provides moderate repulsion in the (3, 3) resonance region. At low energy, it is small and appears with varying sign in different channels. (e) Reflection terms increase the isobar width and provide a mass shift, in a strongly energy dependent way because of resonant rescattering. This mass shift is attractive at t If the Hartree potential depth Vo is allowed to vary between 20 and 50 MeV, the peak position is uncertain by about 5 MeV, while a,ot in the neighbourhood of the maximum varies within only 5 ~ .

504

E. OSET AND W. WE1SE

low energy (where the corresponding imaginary part is zero) and may change sign at higher energy, depending on the magnitude of the imaginary part. Both absorption and reflection contributions depend quite strongly on off-shell properties of the g- and p-baryon Lagrangians. A crucial point is to restrict the off-shell freedom by adjusting the effective Lagrangians to ~d absorption.

7 . Conclusions

Our results show that absorption and reflection contributions, described by. the coupling of A-isobar-hole states to more complex two-nucleon two-hole or rtA-hole continuum configurations, provide sizeable corrections to pion-nuclear elastic scattering. In the (3, 3) resonance region the broadening of various isobar-hole states due to absorption and reflection mechanisms is comparable in magnitude with the reduction of the width due to Pauli effects. The mass shifts of isobar-hole states receive contributions from at least four agencies (besides the downward shift due to multiple scattering in the elastic channel): Hartree- and Fock-type self-energies give net attraction; dispersive real parts associated with absorption and reflection terms also provide net attraction, if summed over all channels. Repulsive contributions come from p-exchange in the isobar-hole interaction. The results are sensitive to the off-shell behaviour of the effective meson-baryon Lagrangians. In order to minimize this sensitivity, the input parameters have been adjusted to the p-wave pion absorption in the deuteron, a necessary (but not sufficient) condition. In that respect, imaginary parts are determined more reliably than real parts. Although these corrections are individually large, there is partial cancellation between them. This is believed to be part of the reason why simple optical potential calculations omitting such effects are reasonably successful 2,, 25). The absorption channels generally lead to non-local isobar self-energies. An appropriate framework to deal with such effects is the combined treatment of isobarhole interactions and self-energies in a generalised RPA matrix inversion scheme. Comparison with a simple phenomenological local "spreading" potential used in ref. t) to simulate medium corrections turns out to be reasonable for the imaginary parts and shows that a large part of the absorption needed is indeed reproduced by our microscopic approach, although there seems to be still too little broadening at low energies. For the real parts, a comparison is hardly possible, because of the uncontrolled diversity of effects covered by the local background potential. In our model, it seems that a large part of the repulsion in that potential would be associated with the simulation of p-meson exchange (i.e. the Lorentz-Lorenz correction), while the strong energy dependence might be associated with reflection terms. [Note that an isobar Hartree potential with a depth - 7 5 MeV has been used in

ISOBAR-HOLE MODEL

505

ref. 1). It may well be that part of the repulsive background has to be introduced in order to balance this overly strong Hartree potential.] Our investigation is not comparable with that of ref. 26), because of the unrealistic model for the short-range part of the two-body absorption mechanism used there. It is somewhat more closely related to the work of refs. 27.31) which deal with pwave absorption at low energy using similar microscopic input, but neglect any dispersive real parts. The total cross section otot in the resonance region is better reproduced than the elastic cross section by our calculations. This deficiency can be traced back to the behaviour of Re f(0) in the J~ = 0- partial wave, which is not correctly accounted for by our model because of the lack of an appropriate repulsive s-wave background. The detailed discussion in ref. 1) related to this problem applies to our case as well. Direct comparison of our calculated absorptive damping widths with (poorly) measured nuclear absorption cross sections is complicated by the fact that absorption and quasielastic channels always appear together in Im S A. We can however extrapolate down to threshold, where two-nucleon absorption remains as the only absorptive channel, and extract an equivalent imaginary part in the optical potential. Our discussion reopens the question about the determination of the parameters C o and 2 in the p-w.ave optical potential:

4n v(r) =

Cop(r) q- C0p2(r) v. 1

Cop

v,

which is commonly used in calculations of low energy n-nucleus scattering 9). Clearly, C o is related, in our model, to the isobar self-energy interactions studied earlier, whereas 32, the Lorentz-Lorenz correction, is identified in our model with the Landau parameter g' of eq. (4.10) (up to small exchange terms). By direct comparison of the absorptive part of our second order pion self-energy with the Im CoV. p2V term of the optical potential we find, using a Gaussian density distribution for ~He, that our model at 09 = p gives the value Im C O = 0.06/~- 6. Although 4He cannot be regarded as representative for a larger class of nuclei, this value is quite close to the one usually required by the analysis of pionic atoms. In addition, we find that the dispersive real parts of absorption and reflection terms are attractive at low energy and comparable in magnitude to the imaginary parts. This would not be in accordance with the usual (arbitrary) choice Re C O = - I m C O [Stricker et al. 9)]; that is, Re C O would probably have to be compensated by an increase in 2 over the value 2 = 1.2 used in refs. 7 -9). [Note that in refs. 7,28), Pauli correlations and p-exchange in the Fock terms are absorbed in the Lorentz-Lorenz parameter 2.] This discussion suffers somewhat from the uncertainty in the isobar

506

E. OSET A N D W. WElSE

Hartree potential V0, but any attractive Vo would maintain this tendency. The balance between Re C O and ;t is clearly an interesting point which will be discussed in connection with similar results for 160, to be reported in a forthcoming paper. One of us (E.O.) would like to thank the Ministerio de Educaci6n y Ciencia, Madrid, for financial support. Helpful support from the CERN Computer Center, where parts of the calculations were performed, is also gratefully acknowledged. We thank G. E. Brown, J. Hiifner and E. Werner for discussions and comments.

Appendix A d - I S O B A R K I N E M A T I C S A N D RELATIVISTIC C O R R E C T I O N S

The relativistic form of a Breit-Wigner resonance of angular momentum l is given by the partial wave amplitude

At(s ) =

2 7 q 2~ S~ - - S - - 2 i ~ q 2t+

~/x/S ,

(A.1)

with x/s the c.m. energy, and q the c.m. momentum. We can write the usual amplitude ~(s), satisfying the unitarity relation Im f~(s) = q[f~(s)[~,

(A.2)

f~(s) = At(s)/x/s.

(A.3)

as

For the particular case of the A-resonance, we would get from our non-relativistic Lagrangian model the amplitude (with inclusion of recoil correction) 1 1 f.2 f~(s) - 3 4~ p2

q2

1

Ma

M~_x/~_½irx/s

,

(A.4)

2~,q2/(Ma + ~s) 2iyqa/~(Ma + ~s)"

(A.5)

whereas the equivalent relativistic amplitude would be

1 A(s) = ~

M a-~-

Clearly some energy dependent factors are missing in (A.4), but they can be easily taken into account by redefining an energy dependent coupling constant. We can establish the equival~ce of (A.4) ~ d (A.5) at ~ = Ma, the resonance energy, ~ d this gives the following relationship:

1 1 f,2 ~ 3 4~ #2 - M~"

(h.6)

ISOBAR-HOLE MODEL

507

The factor 1/~/s multiplying A~ should be kept explicitly, since it is the kinematical factor that multiplies the invariant amplitude to give the c.m. cross section. If we now want to include relativistic effects in our model, it can be simply accomplished by taking an effective coupling constant

\Ma+x~s } "

(A.7)

The width will then be given by

1 1 f~qaM,~ x/s"

½F = 3 4n g~

(A.8)

TRANSFORMATION FROM n-NUCLEON TO n-NUCLEUS C.M. FRAME

The effective Lagrangian (2.1) is to be understood as a nonrelativistic reduction of a more general relativistic invariant Lagrangian and holds only in the n-nucleon c.m. system, i.e. it has then lost its invariant properties. The vertex factor is S . q where q is the pion-nucleon c.m. momentum. To make the reduction from the pion-nucleus c.m. to the n-nucleon c.m. we can use a simple transformation that considers the nucleons non-relativistically while the pion is considered relativistic. If k and PN are the pion and nucleon momenta in any given frame of reference, the n-nucleon c.m. momentum is

q = (Mk - OgPN)/(M + o9).

(n.9)

where ~o is the pion energy and M the nucleon mass. In the n-nucleus c.m. system, if we take into account the recoil of the nucleus, we have

Pr~ = Pi,,t- k/A,

(A.10)

where Pint is the internal nucleon momentum, the distribution of which is given by the nuclear wave function. If we then assume a spherically symmetric distribution o f the nucleons and average over the internal nucleon momenta, we get q -

(M +og/a)k , M+o9

(A.I I)

which is the expression used in eq. (2.7). We have estimated the effects of going beyond this simple averaging procedure over the nucleon momenta and found them to be small, modifying the pion optical potential by about 3 ~o in the resonance region, but becoming progressively important at smaller energies. In fact, this error is state dependent. The quoted 3 ~o refer to the sum over all states which dominantly contribute to the construction of the optical potential.

508

E. OSET AND W. WE1SE Appendix B

DETAILS ON THE PION-NUCLEUS /'-MATRIX WITHIN RPA In order to carry out the angular momentum algebra we will separate the particles and holes, denoted by A and ~, respectively. We can then write for the pion selfenergy, corresponding to a forward propagating bubble, as in fig. 2,

(q'lHa(e~)lq) = ~

~ (~,q'16HIA')[eo6aa,-Maa,]-~(Al6Hl~,q),

(B.1)

~eF d, d'

where Maa., given by ~ . (3.15), is diagonal in j, M~ and M,, the total spin, spin Nojection and isospin third component of the A-state. The vertex functions are given in t e ~ s of the nucleon and isobar wave functions by

(~, q'lfglA') -

xo~(x)e- ~"'S" q'r~'oa,(x),

(B2)

f.(q2) P - ~ j~0 ~ye~ty~ *" "e" ~S +'qT+xe~(y),

where 2, 2' are pion charge indices. We can sum over the isospin component of the occupied nucleon states and the A-isospin and obtain ~ ~ a'3 3 + ~l~m,)= ~ ( ~ ~m,lfixa,-~a,rxl~m,) ~ ~ ,,~,(~mlT I~M,)(~M,IT

~

~6 3 22'~

(B.3)

~l

which gives us the conservation of the pion charge. Analogously we can sum over the spin component of the occupied nucleons and have ~

1 3 ~ 3 + 1 (~m~lS" q ~I~M~>(~M~[S • ql~m,)

ms

= ~ n q ' q ' ~ (lv "~msl~M,~ 3 , l ~m~l~M,) ~ 3 , Y~.,,, Y~~,(q) ~{q ).

(B.4)

~s~v'

We then decompose the exponential factors of (B.2) in partial waves:

e-'~"* = 4~ ~ ( - i ) ~ ( q ' x ) ~ Ya,z,(#') Yff~,(1), 2"

(B.5)

~'

e '~" = 4n ~ iaj~(qy) ~ Y~(~)Yau(~); 2

(B.6)

~

furthermore, = ~ (~M~- M ~ M~IjM~)R.,(7) ~, ~ ~_.,(P)l~Ms),

(B.7)

Ms

(xlA') = ~ (l'M~- M~M'~IjM~)R.,r(x)~,, ~ _ u;(g)l~M;),

(B.8)

M~

(xl~) = (for the special case of aHe),

R~o~X)~ool~m,)

(a.9)

ISOBAR-HOLE

MODEL

509

The angular integrations in (B.2) can then be carried out immediately. The next step is to couple the spherical harmonics of the pion momentum variables as *

^

*

r~,.~_~(q)~.~

^ ~(~)

[- 3(2•+ 1) -~

= E .~/4-'--~-+~)/ L ,~-~o (1M~-mflMj-M~IdM~IOIOISO)YTu(~)" EM

(B.10)

"

Using symmetries and orthogonality relations of the Clebsch-Gordan coefficients finally allows us to write the pion self-energy as



= ~ ~ q' Y~M(~'~=I~ (q')l£(n'l'j'))[~&zz, -- Mz,zJ- ~(A(nlj)iQ~(q)i~)qY~(~), z~' ~

(BAD

where

O(nlj)l@~(q)~) =2( -

1 J){J

i)'f*(q2)[~n(2j+l)(2/+l)]'(-)s(; ~

0 0

I j}

~ ~

nT"~""(q)'

(~.]2)

and (for the special case of ~He)

K~,~"O(q) = ~r~drR~o(r)j~(qr)R.~(r).

(B.13)

The sum over 3, 3' runs now over the quantum numbers n, l,j. Eq. (B.I 1) gives us a partial wave decomposition of the pion s~lf-energy by rec~lling that

~ r,.(~')r£.(~) u

23+ 1

e~(cos 0),

~.~4)

4g

where ~ is the angle between q and q'. The pion self-energy (B.11) can be iterated, through the exchange of correlated n- and p-mesons (4.5), to give the full response function. We can illustrate the iteration to second order through one pion exchange:

fd~q '

= / ~

,,

,

Do(~, q ),

Jtz~) where Do(~, q') is the free pion propagator

1

Do(~, q') = wz_q,~_#: +i5.

(B.15)

(B.16)

510

E. OSET AND W. WEISE

By using the orthogonality of the spherical harmonics we can write _- ~ ~

~ q"YjM(~"}(otlQf(q")[d'l(n'~l'lj'l))

J M d d ' dtd't

x [~o6,~t~.~-M,~;at]- 1(~, A l(nl l~jx.)lRj(og)[~, A'(n'lT')) x [e~6az,- M~,z] - l(A(nlj)lQj(q)la)q Yj*M(~),

(B.I 7)

where (~, AIRj(~)I~, A') = R~s,

=

fo~ q ,2 d q ' q ,2 t2+i ~ (a]Q~(q')ld') (~)3(AIQs(q)J~)ogZ_q,T~_t, ,

(B.18)

is the matrix for the particle-hole interaction through one pion exchange ofeq. (3.9). The iteration can be continued to all orders and we can sum the terms of the series in the following formula:

j~(q), = ~ ~ q' YjM(O'~(o~IQf(q')lZ')(ct, A'IGj(m)I ct, A)(A IQj(q)l~)q Y* "

(B.19)

J M AA'

where G~ is the particle-hole Green function given by (ct, A'IGj(w)I~, A) -= G~,d~o) = [mfis,~-M~,,-R~,~]- 1,

(B.20)

and s, s' refer to the particle quantum numbers n, l, j of the A coupled with the nucleon hole ~ to the total A-hole spin J. Only one state ~ enters for ~He. The evaluation of the Green function now requires the selection of all the states of the A that can couple with the hole state to a given partial wave J and then perform for each J a matrix inversion within that space of particle-holes. Inclusion of the remaining parts of the particle-hole interaction, eq. (4.5), in the matrix R,,~ proceeds in an analogous way. The response f u n c t i o n / / ~ represents, up to a factor, the T-matrix in the TDA scheme: 1

( q'lT(~o)lq > = ~ ( q'lll~(~)]q>.

(S.21)

To generate the full RPA series we would then include the "backward going" A-hole excitation according to formulae (3.12)-(3.16). Finally, to account for the c.m. motion of the nucleus, we multiply the T-matrix by the Tassie-Barker factor exp [(q --q')2/4A~], where ~2 = Mto/h is the harmonic oscillator parameter.

ISOBAR-HOLE MODEL

511

Appendix C

PAULI AND BINDING SELF-ENERGIES We present here details on the calculation o f Pauli and binding effects in the A-selfenergy, given by eq. (5.2). An exact calculation of this formula with the nucleon propagator eq. (5.4) would require the full knowledge of the spectrum of Ho, or alternatively an inversion scheme similar to the one developed in appendix B, if an arbitrary basis o f nucleon states is chosen. But one can take advantage of the fact that one has to subtract an equivalent piece with the free nucleon propagator. For high-lying states of the spectrum the effect of the potential will be very small and we can expect large cancellations with the terms that have no binding at all. In order to carry out the calculation it is useful to split the nucleon propagator (5.7) in the following way: Pt~ GN(E)- G~)(E) = E - Ho(E ) - ib -

f 1t P~E-T+ib

+ E-tto(E)+i~

s

-

(I-P~)E-T+i~

s'

(C.I)

where the symbol S stands for the symmetrized product of the projection operator PF and the free propagator. The difference between the last two terms can be interpreted as binding corrections acting on the unoccupied nucleon states. The effects of the potential are mainly felt by the occupied states and, according to our arguments, the summed contribution from these last two terms to Zp should be small. We calculate this contribution in an approximate way. We choose a complete set of discrete states which approximate the true eigenstates of T in a region of space which overlaps with the relevant A wave functions. In that region we have the following approximate formula: •

~

Rnt(ot, r) ,~ N nlJl(Tntotr),

(C.2)

where Rnl(O~,r) is a harmonic oscillator radial wave function, )'nl = 4 n + 2 l - 1 and a2 = Mogo/h (~0 being the oscillator frequency), which becomes progressively better for larger values of n. For the restricted r-space we can thus consider the harmonic oscillator wave functions as eigenstates of T with eigenvalue Tnl = (4n + 2 l - 1)~t2/2M.

(C.3)

The potential V(E) is then treated as a perturbation and to first order the eigenvalues of H 0 will be E~ = T~ + (nllV(E~t)}nl). (C.4) We_then evaluate eq. (52.) with the last two terms of the nucleon propagator of (C.1) by summing over the unoccupied states of the harmonic oscillator and using the spectra (C.3) and (C.4) respectively for the two terms. The combined contri-

512

E. OSET AND W. WE1SE

bution to _~p from these two terms accounts for about 20 % or less of_~p and justifies such approximation. The contribution from the first two terms can be carried out in a straightforward way since they only involve the occupied states. The explicit calculations can be performed by making a partial wave decomposition o f the vertex functions and using similar angular momentum algebra as in appendix B. We shall only present the final formula here giving the contribution to Xp from every piece of(C. 1). Also we will only write the contribution from one-pion exchange. With minor modifications one can include the other terms of the full interaction of eq. (4.5), to take into account p-meson exchange, and additional correlations.

6G~(E) =

Pv . E - H o - i(~

(A'I6,Y,~I)(E)IA) = 6 ~ , ~ 6 ~ 6 s s ,

~~ [~,,._~0,~,+ ~,~,+ ~,.,.-,,_,~.. (; ~0 0~ "~' ~, ~'}] x

; ~q~q(~p; ~

'

,

~qZK~"a'(q)K~'"(q)2~(q) E + ~ ( q ) - e , '

,~.s,

where ~(q) = [q~ +~z]~, with ~, the binding energy of the occupied states and ~ , a(q) given by (B. 13). The two t e ~ s in the curled bracket correspond to the spinspin part and tensor part of the one-pion exchange. This piece would contribute only to the real part of the self-energy, while all the other terms would contribute to both real and imaginary parts.

(ii)

~6~[E)= -

P~E-T+i

s:

(A'{dZ~2)(E)[A) = - (4K) 3OM,~6M~j6 jj ,

~ [~,,.-.~'+'~'+ ~,~"-',-~+~(~ ~0 0~.~~ ~, ~'}] l~o~,~a, V ca, (s.~e~V(k ~ ,/ E )- ~ ( q ) - ~ 1 ,2 M ) + i 6 7~Jo ~ k - - ~ )

x~

~ [t~',,~, ~ ) # ~ ) ~ ' %) + ~,~, q)#~)nl' qq)], where

I ~t(p, q) = ~ ; rZdrR~.,(r)jo(pr)j,(clr), (C.7)

..¢t~(p) = f ; r2drR ~°(r)j°(pr)"

ISOBAR-HOLE

(iii)

6Gc’ =

513

MODEL

_‘-p,__ E-H,+i6

’

(d’&Y’3’(E)ld > = 47~13

hftM;dM,M;djj'

C

C

11’ 6 ib

(ZT+

1)(22+1)(2X+

1)

f(1.0)

and a similar piece foI; the last term in (C.l) by suppressing V6i in (C.8). We also write the contribution that we call Pauli correction, coming from (iv)

SGg’(E) = 2d(E (d’(Gz~4’(E)lLl)

- H,)P,:

= -4nc5Mt‘bf;S~jhf;~jj~

(C.9)

The binding corrections corresponding to the bracket of.eq. (5.7) will then be evaluated as the contribution C, calculated from the four terms of (C.l) minus the Pauli correction of (C.9). Appendix D ABSORPTION

AND

REFLECTION

TERMS

In this appendix we present details on the d-self-energies from the absorption and reflection channels corresponding to eq. (5.9). Let us concentrate on the piece

Fig. 16 Absorption

term of d-isobar

self-energy

514

E. OSETAND W. WEISE

containing the pion absorption into two-nucleon continuum states, fig. 16, with only pion exchange. As argued in subsect. 5.3 we will consider plane waves for the intermediate nucleon lines. We can then write an explicit exprbssion for eq. (5.9): <~'lS~'(~)l~>

=i E f d3p f d3p' I ~ ~,d

fd-~ -p0

f~

dtzrcl d(zrc I j z~

d

1 1 x ( E - ~o) ~ - ~'~ - i~ ~ + i f (=' q'l,~HIp'> (E - po) _ ~(p,) + ~,, + i,~

×


1

1

( E - pO)2_ q2 _ #~ + ib (p' qlbnlA )pO_ ¢(p) + i6'

(D.1)

with the plane wave states normalized to a ~-function. The matrix elements of the interaction are given by expressions analogous to (B.2). If the sums over intermediate spin-isospin states are p e r f o ~ e d , (D. 1) becomes

' fdZq fdaq ' <~,lsk~,(e)~>= ~4i jf d~p ° fj .dap~ fd3p j ~ j ~ j ( ~ x f*(e~)y*(~'=)y(~)y(~'~)s,s~{~

• ~ ' ~ },

(~.2)

~

where S~ is the product of all the propagators in (D.l), and S ~ the product S~

(D3)

= l f l f l fl~,

where

11 = Jd 3xq~],(x)ei*'' Xei~'"x~ ~2

~

f d Sxq~*~(x)e- i~, . ~ein, . ~~ (D.4)

13 =

jd~xe-'~"*e"*~,(x),

I, = _[d3xe- i~ "*e- iq' x~A(X). By explicitly carrying out the sum over the spin proj~tions of the nucleon state IP) we obtain the useful expression

q" q' ~, (~m'~ls+ " q'1½m~)(½mslS" ql~m~> ms

= qZq,2 ~ ~ 16r~(2~+ 1) z~

× (-~mk~-#l~#+

1 0 0

0

o

oJu

~

2)U

mAX~mAZ#I## + m~) Y#,,. ' 3 , +.,#q ^,)Y#,~+.~(@),

# (D.5)

ISOBAR-HOLE

515

MODEL

where ~ and/3 can have the values 0 or 2 as they account for the spin-spin or tensor part of the one-pion exchange. The p-meson exchange can be introduced then in a straightforward way since, apart from the different coupling constant and propagator, the spin-spin piece is twice the one in one-pion exchange and the tensor piece has opposite sign. One can work out the angular momentum algebra in (D.1) and (D.2) and perform the angular integrations analytically with the following result:

F(p, p' ; q, q') ~- f dOq f dOc f dOp f dOp,SM~q " q'( A'iS+ " qS " q'lA ) -- ~ ~ ~ ~ 4i"-~'(4~z)6(22+ 1)(22'+ 1)(2q+ 1)(2t/'+ l)(2~+ 1)(2p+ l) a# 2~' ~o ~/~'

(2~ +1)(2/3 +1)[(21+ l )(21' + l )] ~

1

~

1

1

/3~[r

o

o

o

o

o)\o

('0)(

--

~ o

~

D(' :) o

o

o

(~,, ~)(., ~-~-~ ,~-~-~ ,~,, ~ <~,~ :,} 0

0

~' j

×

0

0)(1

0

l )~'

~

&J(l

/3 z J ( ¢

~

2'1(~

/3

ft q2q'2Uc¢(P'~

q )I¢¢(p. q)lo,.(p. q )I°"(P' q)'

(D.6)

where l~¢(p, q) =

C r~drR~(r)j¢(pr)j¢(qr)' ~,J

(D.7)

I J,(p, q) = j~r~ drR ~(r)j~(pr)j~(qr.). On the other hand one has to evaluate the pO integration of the set of propagators. We can carry out the integration in the complex pO plane and after some rearrangements one gets the following formula: 1 1 G(p, p' ; q, q' ; E) =- i -I"dp° .J ~ (E--p°)2--qZ--122 +i~ ( E - p ° ) 2 - q ' 2 - ~ 2 +i¢~

×

1

1 -

1

pO _ e(p) + i6 E - pO _ e(p') + ~ + i,~ 1

1

1

1

1

1

-

2 09o9' 09+~o' 0 9 + ~ - ~ 0 9 ' + e - ~ E - ~ - 0 9 + i 6 E - ~ - 0 9 ' + i 6 i

X

E - e - e ' + e~+ i6 {(09+09'- E + eXe'- e~)~+(e'-e~)[(09+09')(09 +09'- E + e)

+ (09 -- E + aXed - E + e)] + 0909'(09+ 09' - E + ~) + (09 + 09')(09- E + eX09'- E + e)}, (D.8)

516

E. OSET AND W. WEISE

where ~ -- e(p),

~ ~ ,~o(q) = [ q : +

~2]~,

~:' -= ~(p'),

o9' -= e)(q') = [ q , Z + g / ] ~ .

(D.9)

The A self-energy can then be written as

(A'IZ~'(E)IA) = ~ 6~,~;,6~,~,6~, (P~P (p'Zdp' (qZdq (q'2dq' ~

' .

j(2g)3j(2g)3 J(2~)3j(2~) 3

x f*(q2)f*(q'2)f (q:)f (q'Z)F(p, p'; q, q')G(p, p'; q, q' ; E).

(D.10)

T h e fourfold i n t e g r a t i o n with respect to the m a g n i t u d e o f the m o m e n t a is carried out numerically. (A G a u s s i n t e g r a t i o n with 12 points, where good convergence was f o u n d , was actually used for the integral evaluation.) T h e f u n c t i o n (D.8) exhibits the p r o p e r analytical structure o f the F e y n m a n d i a g r a m o f fig. 16, with cuts c o r r e s p o n d i n g to h a v i n g the lines p, q or p, q' o n shell (quasielastic channel), a n d also the iinesp, ~, p ' o n shell which c o r r e s p o n d s to the p i o n a b s o r p t i o n channel. I n c l u s i o n of the i n t e r m e d i a t e A-hole excitation the exchange o f p - m e s o n a n d the r e m a i n d e r of the i n t e r a c t i o n eq. (4.5) can be carried out in a straightforward m a n n e r from the f o r m u l a e presented here. T h e same can be d o n e with the crossed d i a g r a m s with m i n o r modifications, b u t we will o m i t the f o r m u l a e here.

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ISOBAR-HOLE MODEL 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31)

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