many-body approach to low-energy pion-nucleus scattering

Nuclear Physics A554 (1993) 554-579 North-Holland

NUCLEAR PHYSICS A

Many-body approach to low-energy pion-nucleus scattering

J. Nieves and E, Oset Departament# de Fisk-a Teorica and IFIC, Centro Mixto, ~ni~rs~dad de Valenc~a CSIC, 46100

Burjassot, Valencia, Spain

C. Garcia-Recio Departamento de Fisica Moderna, Universidad de Granada, 18071 Granada, Spain

Received 30 October 1991 (Revised 12 August 1992)

Abstract: We have extrapolated for low-energy pions (T, = O-50 MeV) the results for the pion-nucleus optical potential previously developed for pionic atoms. The evaluation is done using microscopic many-body techniques which allow us to separate the different contributions to the imaginary part of the potential and relate them to the different reaction channels: quasielastic and absorption. Elastic differential, reaction, absorption and quasielastic cross sections are evaluated for different nuclei and energies and contrasted with experiments. The agreement with data for the different channels, energies and nuclei is rather good with some isolated discrepancies.

1. Introduction

Pion-nucleus elastic scattering at low energies has been the subject of considerable experimental ‘-‘O) and theoretical ‘1-15) efforts. In conjunction with the data of pionic atoms, it was soon realized “) that the pion-nucleon interaction is appre~ably modified in the nucleus since the lowest-order density terms of the optical potential (RY = 2wV,,, = tp, with t the mN scattering f-matrix) were insufficient to describe the data. Genuine many-body corrections, such as Pauli or short-range correlations, or pion absorption were soon identified as important ingredients in the pion-nucleus interaction and subsequent work has confirmed it.

These many-body corrections have usually been parametrized, and fits to the data have been carried out in order to obtain optical potentials which describe the present data with high accuracy ‘3-‘5)_ However, although some eftorts have been made to understand parts of the optical potential from a microscopical many-body point of view *6-22), a thorough and detailed many-body study had not been done until recently in the case of pionic atoms 23). In this paper a systematic study of all the second-order terms in a density expansion of the pion selfenergy is done for the

0375-9474/93/$06.00 @ 1993 - Elsevier Science Publishers B.V. All rights reserved

J. Nieces et al. / zany-~dy approach to low-energy 555

p-wave part of the optical potential, which, in addition to the results of a similar study of the s-wave part in ref. **), leads to a theoretical potential which provides a reasonable description of the pionic-atom data including the so-called anomalous states 24), The agreement with data is improved, with changes in the widths and shifts at the level of 15%, when some small phenomenological part is added to the theoretical potential and a fit is carried out to the data in order to determine these additional parts. The most important of these is an extra repulsion in the s-wave part which the theoretical approach does not provide.

Consistency of the theoretical approach requires that the extrapolation of the results for low-energy pions be able to describe the data of low-energy pion scattering. In this work we have extended the theoretical model of ref. 23) for energies above threshold and have introduced the contribution from the quasielastic channels which are now open. This microscopical approach not only allows the determination of the optical potential but it also allows for the separation of the channels and hence we can calculate the reaction cross section and separate from it the absorption and quasielastic channels. This is obviously a more stringent test of the theory than the reproduction of the elastic scattering alone, or the data in pionic atoms.

The theoretical approach is done in a way that at resonant energies, where the delta excitation dominates the reaction, it matches the work done in ref. *‘), where pion-nucleus elastic scattering around resonance was fairly well described, together with the different reaction channels: quasielastic, single-charge exchange, double- charge exchange and absorption *“).

We show in this paper the results obtained by extrapolating the theoretical approach used in pionic atoms. Therefore, we would like to stress that we have not fitted any parameter to the scattering data as traditionally has been done.

In sect. 2 we present our approach. We describe the different parts of the optical potential: absorption and quasielastic parts of the imaginary part of the p- and s-waves plus the corresponding real parts. We include the technical details involved in the introduction of the quasielastic channel in appendix B. The corresponding ones related to absorption can be found in refs. 22*23V27). In sect. 3 we study the splitting of the reaction cross section in quasielastic and absorption channels. In sect. 4 we discuss the results obtained and finally in sect. 5 we present our conclusions.

2. The model

The model is exactly the same as that used in ref. *3) except that the calculations are done at finite pion kinetic energies and one has to introduce the quasielastic channel. We will perform the calculations for a m-. The results for a rTTf are obtained by changing pn 9 pp in all our formulae, where pn, pP are the neutron and proton densities. The results for a TO are obtained by symmetrizing our potential with respect to P,, , pp, except for the s-wave absorption part, where we obtain a structure

of the type (P,P&

556 J. Nieves et al. / Many-body approach to low-energy

Fig. 1. Diagrams used in the evaluation of the pion-nucleus optical potential. (a) lowest-order s-wave

part; (b) lowest-order p-wave parts; (c), (d) second-order s-wave parts; (e), (f) second-order p-wave parts. The full dot stands for the WN s-wave t-matrix and the square for the p-wave t-matrix.

Our model is depicted in fig. 1 by using standard diagrammatic many-body techniques where the full dot indicates the ~TN s-wave amplitude and the square the TN p-wave amplitude. In addition, the ph or Ah excitation by the pion is iterated to all orders in the RPA sense, giving rise to an induced interaction *‘), and the pion exchange in the p-wave parts is replaced by an effective spin-isospin interaction (in our approach the latter is generated from n- and p-exchange plus the additional effect of short-range correlations).

The s-wave amplitude is taken from the experimental values of the scattering lengths and phase shifts, but for the p-wave part an explicit model is used, which is depicted in fig. 2 and consists of the nucleon and delta direct and crossed pole terms 29). The delta properties are substantially modified in the nuclear medium 29-34) and this model allows us to consider the delta degrees of freedom explicitly.

The isospin dependence is taken care of in all the diagrams this allows us to obtain an expression for the optical potential in

and pp (r).

evaluated and terms of p,(r)

Fig. 2. Model used for the p-wave t-matrix containing nucleon and delta direct and crossed pole terms.

J. P&eves et al. j Mary-buy ap~~ac~ to low-energy 557

After this brief exposition of the model used in ref. 23) we write here the results for the optical potential above threshold. The optical potential has two different parts: s-wave and p-wave. Let us start with the p-wave,

v P(r) 1+4?rg’P(r)

V+A P(r)

1 f4?rg’P(r) )I * (1)

The factor 1 + 4Tg’P(r) in the denominator of eq. (1) implements the Lorentz-Lorenz correction.

We have split P(r) in three different terms,

P(r)=P,(r)+P*(r)+SP,(r), (2)

where P,(r) is the extrapolation to finite energies of the theoretical potential [eqs. (24) and following of ref. ““)I used in pionic atoms. Pz( r) is imaginary and accounts for the lowest-order quasielastic contribution (this channel is closed at threshold) and finally 6P3(r) is a small phenomenological term obtained from the study of pionic atoms.

* 2

P,(r)=+; 5 ( )

* EfYd(r)+ff,(r)l+CYN(r)+“r(r). (3)

P,(r) has a term Q(P), which comes from the direct Ah excitation (see fig. 3), another one, a,(r), which comes from the crossed Ah excitation, a term LYE, which comes from both the direct and crossed nucleon-hole excitation, and finally a non-resonant term, a,(r), which comes from the combination of many second-order diagrams.

The resonant and non-resonant parts are explicitly given by

&)=&2 : ( ) 2

77 (1+fE)2$(Pn-PJ.

ad(r) = pn 1

~-Md-~~(2pp/~o)+i[5~A-ImZ:BA3(~I~O)I

+4p,{J;:-M~-[fZb(2p~/~o)+f~~(2p,/~o)l

+ i[$” - Im Z~A3(p/pO)]}-’ ,

(4)

(5)

Fig. 3. Feynman diagrams for the direct Ah excitation term including A-selfenergy corrections.

558 J. Nieves et al. / zany-&ody approae~ to low-energy

4r) = 4rbP * h, &--MA-,-&S-M,-w,, ImI;,=~A=Im2~A3=0), (6)

a,(r) = i Im C,““(T) PO -!- {%p( T)& Im zA(2pp/ib) Im & ( T) anor( T)

+ %( Th Im ZA (2Pp/PO) + %n( T)pp Im 2~ &%/PO)}

+ Re CoNRp2, (7)

W,=A&-i&, g’ = 0.63 , E=%rfn/fN, ~,=m,i.T,

f*2/47i- = 0.36, s/4?? = 0.08 , Pp” P,W , Pn = hl(r> ,

P = P,(r)+dr) , .s=Mh+mt+2w, MN+- (

: g), k,=(Wp)“‘.

(8)

d% is an average over the Fermi sea of the total energy of the TN system in the c.m. frame, kI: is the local Fermi momentum and T is the pion kinetic energy in the lab system. f* is the Pauli corrected A free width. Now all the coefficients depend on T, their explicit expression can be found in appendix A. On the other hand the A-selfenergy contains the diagrams depicted in fig. 4 and is given by L; + i Im EyA3, where

L(P) = Re -L(P/Po)+ i Im -&~/PO), (9) with

Re xA b/i%) = Re 5A P/PO I Re $., = -53 MeV, (10)

h SA tdfd = Im $A( T, --& arc@ (4 ~~(~/~~~) , (11)

Im ~~A3f~/~O) = -GJ T)(~/A”Q’~)- CA T)(p/po)aA3,fT’ . (12)

~

- g:: + g, + ~:::::, +...

Fig. 4. Different A-selfenergy diagrams considered in our approach including quasielastic, two- and three-body absorption contributions to the imaginary part of the A-selfenergy.

f. P&eves et al. / zany-body app~aeh to ~ow-e~e~ 559

Since Re $4 is a smooth function of the energy 35) we have taken it and Re C,“” as independent of the kinetic energy of the pion in the range T = O-50 MeV considered here.

The term Im & in eq. (11) accounts for two-body absorption and we have kept the parametrization of ref. *“). At T = 0 (as can be seen in appendix A) the results are the same as in ref. 23) for pionic atoms. The first and second terms in Im X;8” of eq. (12) account for the contribution of the quasielastic and three-body absorption 35) channels of the four last diagrams of fig. 4. These two terms are rather small in the range of 0 to 50 MeV where we make the calculations compared to the main sources of quasielastic cont~bution which we evaluate later. Therefore, for simplicity we have not considered the isospin dependence of these two terms. As we have mentioned before, fA is the Pauli corrected A width. This is given in the appendix of ref. 3s) in an invariant form for the scalar and tensor parts, We take here an alternative method which takes into account the weight that the scalar and tensor parts have in pion scattering. We take

FA = I $$ P,(lq - (1’1) dO(q') , (13)

’ (14)

where q and q’ are the incoming and outgoing pion momentam, k, is the Fermi momentum and FF (lk() is the Pauli blocking function obtained after integrating over the Fermi motion of the nucleus **).

Since q - q' is a galilean-invariant quantity and we are dealing with low-energetic pions, we have evaluated eq. (13) in the cm. of the A where dT,/d0 is proportional to [3(q^* $)*+ l] when the A has been excited by a pion and we find the analytical expression

fAGI) = GN?,M4(&+~2)1 , (15)

The expression of I, and I2 can be found in appendix B in eqs. (B.6) and (B.7), r,,, is the free A width, q the pion-nucleon c.m. momentum and f* the coupling constant TNA. The imaginary part of PI(r) contains the absorption contribution from the 32 diagrams implicit in figs. le and If. Because of the way the A-part has been treated it also includes the quasielastic terms related to the Ah excitation. We still have to introduce the quasielastic terms related to other parts of the interaction. However, before implementing this we describe the s-wave part of the potential. This is given by

560 J: Nieces et al / Many-body approach zo low-energy

where the meaning of the three different terms is similar to those which appear in eq. (2). The first term reads 23,27)

2&‘,“‘(r)= -4p[(l+&)(b,+Ab,(r))f(T)p+(l+~)b,(p,-p,)

+ i(Im B. (l+~&)Z(~~+~~~“)+rrn B~(~){l+~&)~2)~, (18)

with Ab, the Pauli corrected second-order rescattering term “*‘*)

AbO=-3l+s)[b;+2b;] (19)

and the values of the other parameters can be found in appendix A. Im B0 2(&+ pppn) comes from two-body absorption. This theoretical isospin

dependence provides a correction of (p, -pJp with respect to the traditional (p,+ pp)2 form (it is ofthe order of 20% in the Pb region). This correction is extremely important in the improvement achieved in the problem of the anomalies in pionic atoms 23).

We have also included in V, the quasielastic contribution to the imaginary part (im BT) related with the diagrams lc and Id. This small contribution was evaluated in ref. *‘).

Now we are going to evaluate the terms P2 and Vff’ in eqs. (2) and (17). These two parts, as we have mentioned before, are related with the lowest-order quasielastic contributions.

The quasielastic contributions are calculated from the diagrams of fig. 5 which are the lowest order in the ph excitation. The contribution of these diagrams is readily evaluated by means of

-~fl(q, PI = J d4q' -z (-if)(-i~)~~~(q’)~~~~(q-q’), (2r)4 S,t

with t the rrN scattering matrix including s- and p-waves, D,, the pion propagator and &,(q - q’) the Lindhard function. The most important sources of the real part associated with the diagrams of fig. 5 are accounted for by means of the A excitation

S..$“’ ..=

Fig. 5. Feynman diagrams which provide the quasielastic contributions in the lowest order in the number

of ph excitations. The square stands for the TN t-matrix.

J. Nieves et al. / Many-body approach to low-energy 561

terms in eqs. (4) and (5) and the s-wave rescattering term Abe [see the appendix of ref. *‘) for the interpretation of Ab, in diagrammatic terms and the proper subtraction of terms which are already accounted for in the empirical first-order potential]. We thus neglect the remaining real parts associated with these diagrams. This would be a small part of the total real part of the optical potential. These and other missing contributions will be accounted for by a phenomenological correction to the potential fitted to pionic atoms (6P, and SVJcs)), as we shall see later. The A excitation contribution implicit in the diagrams of fig. 5 has to be excluded, as we indicated before, because it is treated as a special term.

By using Cutkosky rules we readily evaluate the imaginary part of eq. (20) and find

5 3 I

Imn(q,P)= ~~(4°-“(9’))20tq,)“Im Wq-4)

(21)

We give in appendix B the details on how to evaluate these terms from the experimental TN phase shifts. We summarize the results of the appendix as follows:

(i) s-wave:

(ii) p-wave: &oVf’(r)=-4r(b$p+b?(p,-p,)), (22)

P*(r) = c?p + c?(P, - p,) . (23)

The information given above concludes the input of our theoretical potential. The p-wave is accounted for by eqs. (l)-(7), (23) and the s-wave part by eqs. (17),

(lg), (22). In ref. *‘) on top of this we added a small phenomenological term to the potential

and carried out a best fit to the existing data of pionic atoms. The fitted part turned out to be a small fraction of the theoretical potential and modified the results on the widths and shifts at a level of less than 15%. We assume these missing terms to be rather energy independent and add them to the theoretical potential with the values at pionic atoms. We also carry out calculations with our theoretical potential alone. As we shall see the differences between the results with the two potentials are rather small and the quality of the agreement with the experimental data is the same in both cases.

The extra terms of this phenomenological part are given by [in ref. 23) we called these terms cr, and SV!,$]

W,(r) = is Im Co *{P, Im ~AC+,~PO)+P~ Im ~A(%J,/Po)~ A

+(l+&){ScoP+Scl(p”-p,)}, (24)

562 J. Nieues et al. / Many-body approach to low-energy

and for the s-wave part we take

2w6V:“‘(r) = -4rr[(l+&)6b,p+(1+&)66,(p”-p,)

+ i8 Im B, (1 +~E)~(P~+P~P~)]. (25)

The values for these parameters from ref. 23) can be found in appendix A.

One of the interesting things of the many-body approach is that it not only

provides the optical potential, but the imaginary part is automatically split into two

parts, related to the absorption and quasielastic channels, respectively. How to

obtain the actual cross sections related to these channels is the subject of the next

section.

3. Elastic and reaction cross sections

The calculations of the absorption and quasielastic potential have been done

assuming that a pion with a certain kinetic energy reaches the nucleus, and this

kinetic energy plays some role in the evaluation of the phase space available. Here

we must take into account that, because of the Coulomb interaction, the rr+, Y

behave differently and will have different available kinetic energies when reaching

the nucleus if they start with the same energy asymptotically. For an energy w, of

the pion we evaluate the strong potential in the point r at an energy

T,, = w, - m, - V,-(r) , (26)

with V,(r) the finite-size Coulomb potential, and then solve the Klein-Gordon

equation with the physical energy of the pion, o,,

(-V2+P2+2%Kpt)+ = (% - K)*+, (27)

with p the reduced pion-nucleus mass. We solve numerically the Klein-Gordon

equation by means of an accurate method to solve the Schrodinger equation3’)

adapted for complex potentials 38).

At large values of r, eq. (27) is equivalent to a Schrodinger equation where w,

plays the role of the mass and (w’, - ~*)/2w, plays the role of the non-relativistic

energy. We know then that the wave function t,b(r) behaves asymptotically as

with I(r) and S(r) the standard Coulomb wave functions 39,40) for Coulomb scatter-

ing from a charge Z. The scattering amplitude can be split in partial waves and we

have

(29)


where q is the pion momentum in the TA c.m. frame, a[ the standard Coulomb phase shift 39*40), 6, the additional phase shift due to the strong interaction and 7, the inelasticity. We have then

me=:; (2I+l)ll-771 exp (2i(01+&))Iz,

atot =:7(21+1)[1-rl,Reexp(2i(c~,+S,))],

(30)

While u=, and a,,, are infinite, a,,,, is finite because the short-range nuclear interaction does not have any effect at large 1 and thus qr = 1 for large 1.

Thus, following a standard procedure we can calculate the elastic differential cross section and the reaction cross section. Next we show how we separate the reaction cross section into the absorption and quasielastic parts.

The imaginary part of the pion selfenergy, 17 = 2oV,,,, splits into two parts, the quasielastic, which we call Im DQ and the absorption, which we call Im flA.

Assuming little distortion of the pion waves we would have

VA=-- : I

d3r Im ITA(p(r)) ,

’ aQ=-4 I

d3r Im nQ(p(r)) . (31)

We expect these equations to give only a rough approximation since they neglect screening effects. However, the ratio of these two magnitudes should be a better approximation for the exact magnitude. Another alternative method is to take 4’742)

uA=jd’b(l-eXp(-j-l--iImn”(p(),z))dz)),

uQ=Id2bdzeXp(-l;m-; ) Im l7(p(b, 2’)) dz’ (-> i Im UQ(p(b, z))

Im nA(p(b, z’)) dz’ ,

which assumes a straight propagation of the pions. The interpretation of eq. (32) is easy: uA is given by the integral over the impact parameter of unity minus the probability that a pion crosses the nucleus without absorption. Then uQ is obtained as the probability that a pion reaches the point z without any reaction, times the

564 J. Nieves et al. ,/ Many-body approach to low-energy

probability that there is a quasielastic reaction in dz, the pion is not absorbed on its way out, integrated over z and the impact parameter.

Eqs. (32) offer a better approach to reality particularly at energies around resonance and above 42). Once again we assume that the ratios of cro/ua will be given better by the ratio of the approximate formulae in eq. (32), than by the individual quantities.

Hence we define

with cQ, aA defined by eqs. (32). We also evaluate the ratio with eqs. (31) and find differences at a level of lo-20%, which gives us an idea of the errors that we can have with this ratio.

Since the ratio R is better determined than the individual magnitudes from eqs. (32) we use this ratio R and the exact value of a,,,,, from eq. (30) and determine (TA and @Q aS

(34)

At higher energies the determination of R is more accurate. Even assuming 20% errors in R at low energies, since rQ is around a factor 4-10 smaller than (TA, the errors induced in the evaluation of oA are at the level of 2-5%. Thus our evaluation of c+A should be accurate and that of oo somewhat less, particularly at very low energies.

An alternative way of calculating (TA would be to use a distorted-wave Born approximation using pion waves, as discussed in ref. 13). However, as noted there, it has the inconvenience that it removes from the flux the pions that undergo quasielastic collisions, as well as those that undergo absorption. But, while the latter ones should be removed to avoid double counting, the pions which are simply scattered are still there and can be absorbed.

Finally, we should stress that the consideration of Coulomb interaction has some influence on the cross sections, particularly in the quasielastic channel, which however do not have to be understood as isospin effects which are much smaller.

4. Results and discussion

In fig. 6 we show the results for the differential elastic cross section for V+ scattering from “C at several energies. The results are computed with the full optical potential which contains the small phenomenological part from pionic atoms. The agreement with the data at different energies and in the whole range of angles is


T,= 20 MeV

1

lo-’ 4 0 O,, (degrees)

Fig. 6. Center-of-mass differential cross section for elastic scattering of 20,30.3,40 and 50 MeV P+ from “C. At 50 MeV: Black squares from ref. lo); crosses from ref. ‘); stars from ref. ‘); white squares from ref. 3); pentagons from ref. ‘). The data at 40,30.3 and 20 MeV are from refs. 4*5) and ref. 6), respectively.

rather good. In fig. 7 we show the results for T- scattering on ‘*C. The agreement with the data is rather good on a global scale but at the diffraction minima there is some disagreement with the data. In fig. 8 we show the results for T+ scattering on 40Ca at several energies. The calculations are performed there with the purely theoretical potential and with the theoretical plus small phenomenological parts (SP, and SVf’). The differences between the two calculations are at the level of or below lo-15%. The agreement with the data is equally good for both potentials, with none of them preferable to the other.

In fig. 9 we show the results for a- scattering on 40Ca. The agreement is also rather good but there are some small discrepancies close to the diffraction minima. In fig. 10 the results are shown for 7~+ scattering on *“Pb and the agreement is fairly good with again some discrepancies around the diffraction minimum at T, = 50 MeV. Finally in fig. 11 we show the results for T- scattering on “*Pb where the agreement is also fair but the discrepancies around the minima become more apparent. The calculations have been done by using different neutron and proton radia, as deduced from an analysis of the pionic-atom data43). We take two- parameter Fermi distributions for 40Ca, *“Pb and a harmonic oscillator for ‘*C. We


50 100

O,, (degrees)

Fig. 7. Center-of-mass differential cross section for elastic scattering of 19.5, 30 and 50 MeV T from

‘*C. At 50 MeV the black squares are from ref. ‘) and the white squares are from ref. lo). At 30 MeV the

white squares are from ref. ‘) and the black ones from ref. “). The data at 19.5 MeV are from ref. ‘).

subtract the proton or neutron size as shown in detail in ref. 23). The parameters

used are those of table I of ref. 23).

We have checked that the results for r- at T, = 50 MeV in 208Pb improve a bit

if we use the same proton and neutron radia but the differences are not significant

in our opinion to make a claim in favour of a certain neutron radius. Our study of

pionic atoms 43), which includes about 23 nuclei and makes a fit to the data to fix

the contributions needed in the potential and not accounted for by our theoretical

potential, is more reliable in order to determine neutron radia.

We should recall that although the quality of both the theoretical potential alone

and the theoretical plus the small phenomenological parts fitted to pionic atoms is

rather good, the assumption of energy independence for these phenomenological

terms (which we have done here) is not necessarily correct. Changes at the level of

10% in some terms could produce sizeable effects around the diffraction minima

where the largest discrepancies of our results with the data appear.

This of course should not divert us from the main point which is that one can

obtain a reasonable agreement with the data for different nuclei from light to heavy,

different energies and angles by means of a theoretical potential where everything

J. iVieoes et al. / zany-body approach to low-energy 567

B,, (degrees)

Fig. 8. Center-of-mass di~erential cross section for elastic scattering of 20,30,40.2 and 50 MeV T+ from %a. The data at 50 and 40.2 MeV are from ref. ‘) and ref. 4), respectively. At 30 MeV the white squares are from ref. 5, and the black ones from ref. 9). At 20 MeV the black squares are from ref. ‘) and the white ones from ref. ‘). The solid line is the result of using all terms for the optical potential. The results that we obtain by using the theoretical potential alone (excluding the semiphenomenological terms of

eqs. (24), (25)) are represented by the dashed line.

is fixed beforehand, either from elementary reactions or from previous phenomenology, like the value of the Landau-Migdal parameter.

The shift in the minimum at 50 MeV is not just a peculiar feature of our potential. It also appears in ref. “) where a fit to the data of several nuclei is done to obtain the potential.

With respect to to the reaction channels we show in fig. 12 the total reaction cross section from T- and rr+ on ‘*C. The agreement with the data is good for T- and fair for IY+ (about lo-15%). In fig. 13 we show the results for T” and 7r4 on 58Ni and we can observe that the agreement is rather good.

There are also data on the absorption cross section. In fig. 14 we compare our results with data for T- on 27Al where the agreement is good and with m+ on *‘Al where at low energies we overestimate the data by about 20%. We should also recall that the authors of the experimental work have warned about their experiment underestimating the cross section for low-energy T+ [ref. “)I_ In figs. 15 and 16 we show our results for the absorption cross section on 63Cu and r9’Au. The agreement

568 J. Nieces et al. / Many-body approach to low-energy

IO’

50 100

O,, (degrees)

Fig. 9. Center-of-mass differential cross section for elastic scattering of 50, 30 and 19.5 MeV ?r- from %a. The data at 50 and 19.5 MeV are from ref. lo) and ref. 9), respectively. At 30 MeV the black squares

are from ref. “) and the white ones from ref. ‘).

with the data is good for both nuclei and rrf and rr-. Finally in figs. 17 and 18 we show our results for rY_ and -rr+ on 4o Ca and *08Pb where there are no data.

In all figures from fig. 12 to fig. 18 we plot the three cross sections, ao, CA and creact. We found no data for mo with which to compare our data. Since the predictions vary so much from one nucleus to another and from the Y to the m+ it would be interesting to carry out systematic measurements of this magnitude which would put important constraints on theories.

We now discuss the basic facts on these reactions. We observe that the cross sections are larger for 6 than for 7~+, as a consequence of the Coulomb force which brings more negative pions close to the nucleus and repels the rr+. This feature is repeated in all the figures for the different nuclei. When we go to intermediate nuclei the differences in the cross sections for nTT+ and 6 become bigger and if we go to the 208Pb nucleus the reaction cross section for 7r- is about one order of magnitude bigger than for v+ at low energies. The differences in ao are much bigger and at T,, = 20 MeV there is nearly three orders of magnitude difference between uo for +rrTTf and rTT- in heavy nuclei. We also observe that as the energy increases o;, becomes more similar for & and -z--.


50 100

O,, (degrees)

Fig. 10. Center-of-mass differential cross section for elastic scattering of 50, 40.4, 30.7 and 20 MeV rr+

from ““Pb. At 50 MeV the black squares are from ref. I”) and the white ones from ref. ‘). The data at

40.4, 30.7 and 20 MeV are from ref. 4S5) and ref. 6), respectively.

5. Conclusions

The present work has brought together many results obtained in the last years

on a microscopic many-body description of the interaction of pions with nuclei.

The many-body approach turns out to be rewarding and one can do an expansion

in terms of ph excitations in the absorption channels (some of them are induced

by previous Ah excitations which are also considered in the scheme) finding good

convergence at the level of 3p3h excitation, although at low energies the 3p3h

excitation is not very important. Another interesting aspect of the many-body

approach is that it allows the separation of the imaginary part of the potential from

different sources and one can relate the different parts to different reaction channels.

Thus, in an approximate but rather reliable way, we can evaluate the absorption

and quasielastic cross sections for different nuclei and energies.

The structure of our potential, with its density, isospin and energy dependence,

is more involved than in usual potentials coming from fits to the data. However,

there is no freedom in the present potential. This does not mean that all the input

is absolutely determined and no free parameters are used. It simply means that each

parameter in the theory has been fixed previously to some other reactions. To be

570 J. Nieoes et of. / puny-body upp~ooch to low-energy

Fig. 11. Center-of-mass

earn (degrees)

differential cross section for elastic scattering of 50 and 30MeV T- The data are from ref. lo).

$j b

101111111111111 20 40 60

T(MeV)

:::_&; ; ,’

50 - ,*’ ,/’

1’ I’

20 - ,/I I’

( %-ni) - ,’

10 ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ 20 40 60

T(MeV)

‘OsPb.

Fig. 12. Reaction (Rf, absorption (A) and quasielastic (Q) cross sections for P” fram “C as a function of the kinetic energy of the incoming pion. The reaction cross-section data (black squares) are from

ref. 4’). The absorption datum (cross) is from ref. 48).

J. Nieoes et al. / Many-body approach to low-energy 571

.

lo- T(MeV)

2 b

I ” ’ I ” ‘I” 1

1000 -

R

:/

A

300

I Q 100 - ,’

/’ I’

I’ 30 - ,J’

- I’ (“Ni-n+) - /’

10 """""'_ 20 40 60

T(MeV)

Fig. 13. Same as fig. 12 from ‘sNi.

more precise, form factors are used (of the monopole type) and the cutoff masses

are taken from fits of the NN data4’). The results for the imaginary parts of the

pion or delta selfenergies are not strongly dependent on the cutoffs because the

momenta involved are fixed at values smaller than the cutoff masses by means of

energy and momentum conservation laws. The real parts of the pion or delta

selfenergies are not so much under control as the imaginary parts. They are more

dependent on form factors, and some parts have been neglected which are estimated

to be small. On the other hand, there is the assumption that the A feels a Hartree

potential similar to that felt by the nucleon, on top of other sources of real part

which are evaluated. The assumption is reasonable but since no calculation is done

500

200

100

r

R

A

0

:&eV) 60

z b

loo0 F-l-7 :::q-- -

r - -----!: ,I’

c /’

ii

A I’ 100 -

/’ /’

I’

50 - Q /' /'

I' (27Ahl+)-

,'

20 "1""11'11 20 40 60

T(MeV)

Fig. 14. Same as fig. 12 from 27Al. The absorption data are from ref. u, (white squares).


I ” ’ I ” ‘I”

1000

500

2

s, 200

b

100

50 t L c

R

A

Q

20 IIII(I(IIIIJI 20 ,4;Mev) 60

$ b

I ” ’ I ” ‘I”

1000

500: #ii+ 200 -

,’ , ,’

/’

100 r ,f’ 0,’

I

50 - I'

I'

I'

I'

(%-7rf) :

20 """""'- 20

,4~IddJ~

60

Fig. 15. Same as fig. 14 from 63Cu.

for it, its support has to be found in the fact that the resonance peak in pionic

reactions appears well placed in our calculations 2”).

When this theoretical approach was used to study pionic atoms and the problem

of the anomalies we found a reasonable description of the experimental data.

However, with a model like the present one, where so many diagrams are involved,

one necessarily has to admit uncertainties. Comparison with the data of pionic

atoms reveal that globally the uncertainties were at the level of lo-15%, although

in some magnitudes like the isoscalar s-wave real part of the potential the discrepan-

cies are bigger and the theoretical approach still lacks some repulsion.

b

1000 ______-----

g

b

I ” ’ I ” ’ I ‘I-

looo, fl: **- cc

/’ , I’

100 : , Q ,R’

I

I’ )I/’ (197*u_n+) :

10: ,

- ,' 7 III III II 20 40 60

T(MeV) T(MeV)

Fig. 16. Same as fig. 14 from 19’Au.

J. Nieves et al. / Many-body approach to low-energy

30 -

(40Ca-n-) -

10 ” ” ” ” ’ ” 20 40 60

T(MeV)

1000 -’ 11, #II,,,

. ..i/. I

-/

I’ 100 -_ /’

I’

,I’ I’

30 - ,/’

_ I’ I’

(40cwT+) -

10 ” ” ” ” 1” 20 40 60

T(MeV)

573

Fig. 17. Same as fig. 14 from ?a.

In the present work we extended these results to finite energies. As we have seen, the changes not only involve a change in the energy in the previous calculations but one has to introduce now the quasielastic channels which are absent in pionic atoms. Some of the work here was devoted to include consistently these channels.

In pionic atoms 23) in order to improve the agreement with data, an extra phenomenological (though small) term was added to the potential and fitted to the data. This resulted obviously in a better agreement with the data. We assumed that for continuity such a phenomenological term could be extrapolated at low pion energies without change. The term however is rather small and we observed that the results with the two potentials, theoretical or theoretical plus phenomenological part, were the same with differences below lo-15%.

c , , , , I I ’ I I-(

R

A 1000 7 _- __--- Q - __--

2

100

_s

5 7 3

b - b

10 r

(20aPb-n-) ;

I ” ’ I ” ’ I ’ ”

loo0 wfR ,I’ 0

100 r I’ I’

/’ I’

I’ 10: ,

i ' (20apb_n+) i _:

-:

1 f~l~'lnl'~l- 20 40 20

T(MeV)

Fig. 18. Same as fig. 14 from *08Pb.


The agreement found with data with both potentials is rather good for elastic

scattering, different energies, different nuclei and the whole angular range. Some

isolated discrepancies appear in some cases around the diffraction minima.

As mentioned before, one of the strengths of the present approach is that it allows

for the separation of channels. Thus, we evaluated uR, uo and a, and found also

quite good agreement with experiment, where data were available, both for rr+ and

K, which have rather different cross sections as a consequence of the Coulomb

interaction.

We found no data for cro to compare with our results. The approximate way in

which uo and uA were calculated provides more reliable results for uA than for uo.

The predictions for UQ are very much energy dependent and vary much from one

nucleus to another or for rr+ and 7~~. It would be very interesting to carry out

experiments to measure ‘+Q and thus put extra constraints on the present or other

theoretical approaches.

The conclusion that one draws from the present study is that the microscopic

description of the pion-nuclear interaction is possible and that the present work

has reached a certain level of convergence and stability where the description of

the pionic processes in the low-energy domain is quite fair at the quantitative level.

The same model used around resonance, where the background parts studied here

become very small, also provided a fair description of the different pionic re-

actions 25*26). The use of such microscopic pictures also allows one to tackle many

other reactions at intermediate energies involving pions directly or indirectly, like

photonuclear reactions, weak interactions, antiproton annihilation in nuclei, etc.

These pictures allow one to establish connections and relationships between different

magnitudes in different processes and are very useful tools in order to provide a

unified picture of different physical reactions at intermediate energies.

Finally, although the derivation of the potential was not trivial and the structure

is a little more complicated than in the ordinary potentials coming from fits to the

data, the consistency of the present potential with the data of pionic atoms and the

cross sections for the different channels at finite energies make of this potential a

useful and reliable tool to be used in the investigation of other properties or reactions

involving pionic atoms or low-energy pions.

This work is supported in part by the CICYT, grant no. AEN 90-0049. One of

us, J. Nieves, wishes to acknowledge a fellowship from the Ministerio de Educacidn

y Ciencia.

Appendix A

PARAMETERS OF THE OPTICAL POTENTIAL

The parametrizations, as function of the pion kinetic energy, T, of the different

coefficients appearing in the optical potential have been omitted in sect. 2 and are

collected in this appendix for sake of completeness.


PI part of the p-wave: a,(r) is the non-resonant part of PI(r), which appears in eq. (7). The coefficients involved in a,(r) are the following:

Im C,““(T) =[0.360-0.497T/m,+0.52T2/m~]m~6, (A-1)

Re C,“” = 0.498mG6, (A.2)

u&T) = 1.30-3.22T/m,+2.59T2/mt, (A.3)

a,,(T)=1.735-4.07T/m,+3.11T2/mt, (A-4)

upn( T) = 0.295 - 0.75 T/ mW + + 0.613 T2/ rnt , (A.5)

u,,(T)=0.57-1.617T/m,+1.436T2/m~. (A.6)

The explicit expressions for the coefficients of Im .X4 of eq. (11) and of Im EzA3 of eq. (12) are

Imid(T38.3 1-0.85X+0.54TZ MeV , m, 4

u(T)=2.72-4.07L+3.072 m, m2,’

20.2-8.58L+0.702z MeV , m, d

CA,(T)=~ -7.O8+27.4z-9.492 ( m, m,

=63.7 MeV,

T~85 MeV,

Ts85 MeV,

0.984-0.512$+0.1f . m 4

(A-7)

(A-8)

(A.9)

(A.lO)

(A.1 1)

(A.12)

VP’ part of the s-wave: The parameters appearing in the V(Is’ part of the optical potential of eq. (18) are given 23,27) by

b, = -0.013m;’ ,

b, = -0.092m;’ ,

ImB,=0.041m~4,

f(Tf=(l+&) TinMeV,

Im@(T)=$ -14.0+ 190.8 L- 182.3 T2 7r m, mZ, >

x 10-3m;4. (A.13)


P~enom~n~~ogica~ parts: S VY’ and SP,. The coefficients appearing in the phenomenological p-wave SP,, eq. (24), and s-wave SV?‘, eq. (29, are taken constant in energy and given 23) by

6bo = -0.0053m;’ ,

6b, = -O.O13m;‘,

S Im B. = 0.0064m;4,

Sc, = 0.030m,3 ,

6c, =0.10m;3,

S Im C0=0.107m;6. (A.14)

Appendix B

LOWEST-ORDER QUASIELASTIC CONTRIBUTION TO THE PION-NUCLEUS OPTICAL

POTENTIAL

We are interested in the imaginary part of the pion selfenergy diagram depicted in fig. 5, where we have shown the variables involved in the integrations. As we have seen in the text we can write

(21)

We will make use of an approximation 2’) for the Lindhard function,

Im I&(q) = -VP&l)~(lq”l - e(4)), 03.1)

where w (q’) = m and s(q) = $/2MN. The Pauli blocking factor PF is defined

in eq. (14). Then we get in symmetric nuclear matter

03.2)

For simplicity we have approximated eq. (B.2) by

Im n”(q) = -bzh p dflcYs’) dncf,,tq,) p& - s’l,.,.) . (B.3)

Since q-q’ =p’-p is Galileo invariant, then the error introduced in this part, which is only a small part of Im n, is a negligible fraction of the total Im E By using eq.

J. Nieues et al. / Many-body approach io low-energy 511

(B.3) and da/d0 in terms of rrN phase shift 46), we are able to find an analytical expression for Im nQ( q) and its first and second derivatives with respect to r which are needed to solve the Klein-Gordon equation with a non-local potential. Further details on these analytical formulae can be found in ref. 36), although a direct numerical evaluation is also possible. Since one can write da/da for n- or p-targets, we can separate the isospin dependence in this part.

On the other hand when we have included in the A-propagator of eq. (5) the effective width of the delta in the medium, we are considering some contributions to the lowest-order quasielastic optical potential. In order to avoid double counting we must subtract from eq. (B.3) the quasielastic content of the A excitation part.

Finally we obtain

and for the p-wave

2d@‘(r)lQ = -4,&p + b?&) (B.4)

P*(r) = c& + c?sp . (B.5)

bz, by, c$ and c? are purely imaginary and depend on the kinetic energy of the pion and on the nuclear density. However, they are very smooth functions of the nuclear density (they depend on p through kr). We show in table 1 these parameters as a function of the pion kinetic energy for p = 0.75p,,. The analytical expressions which we have used can be found in ref. 36). However, neglecting the p-dependence in these coefficients (these parts are small compared with the ones already considered) and using the values of table 1 is already a good approximation and simplifies the computational task.

TABLE 1

Imaginary part of the parameters b. , Q b?, CR and cp as a function of the pion kinetic energy in lab at p = 0.75p,

T (MeV) Im b$[ m;‘] Im bp[m;‘] Im c$[ mi3] Im c?[ m;“]

5.0 0.0006 -0.0002 0.0001 0.0000 10.0 0.0013 -0.0004 0.0003 0.0001 15.0 0.0021 -0.0005 0.0008 0.0003 20.0 0.0030 -0.0005 0.0014 0.0006 25.0 0.0039 -0.0005 0.0023 0.0009 30.0 0.0050 -0.0005 0.0034 0.0014 35.0 0.0062 -0.0004 0.0046 0.0020 40.0 0.0075 -0.0002 0.0061 0.0027 45.0 0.0088 0.0001 0.0078 0.0036 50.0 0.0103 0.0004 0.0097 0.0046 55.0 0.0119 0.0008 0.0118 0.0057 60.0 0.0136 0.0013 0.0141 0.0069 65.0 0.0155 0.0018 0.0166 0.0081 70.0 0.0174 0.0025 0.0192 0.0094

578 J. Nieves et at. / bang}-body approach to few-energy

The quantities I,, I2 needed in the evaluation of eq. (15) are given by

PF(~q-q’Jc.m.)=l+0(9”-1) +o(1-q”)(~-1-$j3) )

03.6)

I = dfwr-"-cos2 @

2 I 4P/3

Cm. &(l!F?‘lc.m.)

=l+@(q”_l) [ _21_4’+EL 1 5 4’ Cp $4 21 35

1) 2) 3)

4)

5)

61 7) 8)

9) 10)

11) 12)

131

14)

15) 16)

17) 18)

19) 20)

21) 22) 23) 24)

25) 26) 27)

28) 291 30)

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many-body approach to low-energy pion-nucleus scattering

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