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Pion scattering from a bound nucleon at medium energies
ANNALS
OF
PHYSICS
103, 141-168 (1977)
Pion Scattering
from a Bound Nucleon
at Medium Energies*
K. K. BAJAJ AND Y. NOGAMI Department of Physics, McMaster
University, Hamilton, Ontario, Canada LAS 4Ml
Received May 5, 1976
In order to examine the validity of the impulse approximation for pion-nucleus scattering in the 33-resonance energy region, we consider pion-scattering from a “nucleus” which consists of a single nucleon bound in a harmonic oscillator potential. A separable ?rN interaction is assumed. The oscillator parameter is chosen such that the nuclear sizes are fitted for 4He - I60 . The binding effect is found to result in a downward shift of the resonance energy (by about 20 MeV), and an increase (by 50 - 70 %) of the total cross section near the resonance. The angular distribution is also strongly modified. In connection with the binding effect, the importance of a careful treatment of nucleon recoil is emphasized. It is pointed out that the closure approximation which is often used to sum over intermediate nuclear states leads to very misleading results. The effect of the Pauli principle is also examined by excluding some intermediate states.
1. INTRODUCTION A considerable amount of information has been accumulated for pion-nucleus scattering at intermediate energies.For example, the total and elastic cross sections for several nuclei ranging from 4He to 32Sare known [I]. On the theoretical side, it has been found that these cross sections can be reproduced quite well by means of the optical model [2], and the Glauber approximation [3]. In these calculations, the pion-nucleus (z-A) scattering amplitude is obtained in terms of the pion-nucleon (-rrN) scattering amplitude. This nN scattering amplitude should in principle be related to the t-matrix from a bound nucleon, but in practice the free ?TNt-matrix is usually used. This is the impulse approximation (IA). It is generally expected that the IA is valid if the energy of the projectile is much larger than the binding energy of the nucleon in the target. More precisely, it can be shown [4] that the t-matrix for 7TNbscattering1is related to that for niVp scattering by
* Work supported by the National Research Council of Canada. 1 We denote a free nucleon by Nr and a bound nucleon by Nb .
141 Copyright All rights
0 1977 by Academic Press, Inc. of reproduction in any form reserved.
ISSN
0003-4916
142
BAJAJ
AND
NOGAMI
wherefis the nNr scattering amplitude, h = h/p, T is the collision time, and SE is a “level shift” which is of the order of the average binding energy of the nucleon in the target. As was pointed out a long time ago by Goldberger and Watson [4], 1(f/h)(T@/fi)I = 0.5 around the 33-resonance, and hence the validity of the IA is questionable. This is because the relatively long collision time at the resonance enhances the importance of the interaction between the struck nucleon and surrounding nuclear medium. Some of the typical processesin TA scattering are depicted in Fig. 1. The shaded blob stands for tb , which is replaced by tf in the IA. The methods such as the Glauber approximation and the optical model take care of thesemultiple scattering processesdue to different nucleons, and the accuracies of these methods have been examined by many authors. On the other hand, relatively little attention has been paid to the validity of the IA [5]. If we assumethe Yukawa or Chew-Low interaction for the basic mechanismof TN scattering, the blob consistsof those processes shown in Fig. 2, where the double line connecting two nucleon lines stands for an N-N interaction. The processesof the type shown in the first row will give tf , while those in the secondrow will causethe difference between tb and tf . This is the kind of correction that we are going to examine in this paper. There are two corrections to the IA which we will refer to as the “binding effect” and the “Pauli principle effect”. The former is due to the fact that Nb is described by a bound state wavefunction, whereas Nr is described by a plane wave, while the
FIG. 1. Typical processes in ?rA scattering. The shaded blob stands for
1 +
\/ ,‘,I‘
/
\ +
I
tb .
/
){11;;’ f.0.
FIG. 2. The structure of interaction.
fb
. The double line connecting two nucleon lines stands for an N-N
PION
SCATTERING
FROM
A BOUND
NUCLEON
143
latter is due to the Pauli principle which prevents Nb from being excited into states which are occupied by other nucleons. Kohmura [6] and Schmit [7] estimated the binding effect for a nucleon bound in a harmonic oscillator (HO) potential. The only significant effect they found is that the 33-resonance energy is shifted; upward in [6] but downward in [7]. For the deuteron, Julius [8] and Myhrer and Koltun [9] have estimated the binding effect in the resonance energy region. The former finds an increase in the elastic cross section of up to 40 % for backward angles, while the latter obtain an upward shift of the 33-resonance by about 15 MeV. Some other authors have examined the Pauli principle effect by considering the Fermi gas model of nuclear matter [IO]. More recently, Julius and Rogers [ll] have estimated the binding effect on the optical potential by including the first-order correction to the IA in Eq. (1.1). They find that the resulting correction to the potential is of the order of 50 - 100 yi near the 33-resonance, with an increase in the TNI, total cross section by the same amount. In the present paper we examine the validity of the IA for the rrN interaction in the 33-resonance region for a simple model. We consider pion scattering from a “nucleus” which consists of only one nucleon bound in an HO potential. For the TN interaction, we assume a separable-type one, which reproduces rNr scattering at intermediate energies including the 33-resonance. In a real nucleus, of course, there are many nucleons, and the nA scattering amplitude can be constructed in terms of tb , which satisfies tb = L‘ + VGtb .
(1.2)
Here c is TN interaction and G an appropriate many-body propagator. The most important intermediate states that contribute to tb in Eq. (1.2) are probably those in which the struck nucleon is excited while all other nucleons remain undisturbed. Then the other nucleons can be regarded as spectators, and, so far as the binding effect on tb is concerned, they simply generate the potential which binds the struck nucleon. This is the idea that underlies our model calculation. Since we have only one nucleon in our “nucleus,” we do not consider multiple scattering due to different nucleons. We are aware of the ambiguity as to the distinction between our binding effect and that of the last diagram of Fig. 1. However, since the latter effect is not included in the Glauber approximation or in the usual optical model, tr in these methods can be replaced by fb which we obtain, without danger of double counting. Unlike some previous calculations [6, 71, we do not use the closure approximation (CA) but do the summation over nuclear intermediate states explicitly. Our results are very different from those of [6, 71, but are similar to those of Julius and Rogers [ll]. We find that the 33-resonance energy is shifted downward (in terms of the pion energy in the TA laboratory system) by the binding effect. Crucial differences between our analysis and the previous ones [6, 71 lie in the treatment of 595/103/I-10
144
BAJA1
AND
NOGAMI
nucleon recoil as well as in the summation for intermediate nuclear states. The main result of our calculation is that the total cross section for rrNb scattering increases by about 50 N 70 % compared with the IA2 near the resonance, and the TNb elastic cross section is more strongly forward-peaked. We also examine the Pauli principle effect by excluding appropriate intermediate states in the summation. In Section 2 we present our model, examine TNf scattering and thereby determine the parameters of the model. The TNb scattering with the nucleon bound in the ground state of the HO potential is considered in Section 3. In Section 4 we describe the IA for our model and in Section 5 we examine the validity of the closure approximation. We consider the Pauli principle effect in Section 6. Results are summarized and discussed in the last section. In Appendix A we show that, so long as the binding effect is concerned, the Chew-Low interaction and the separable Z-N interaction will lead to very similar results. Some details of the calculation in the text are given in Appendixes B and C.
2. THE MODEL
AND TNf
SCATTERING
We consider a “nucleus” which consists of a single nucleon bound in a “shellmodel potential” V(r). The nucleon also interacts with a pion field through a seperableinteraction. The Hamiltonian for the TN system is given by H=H,v+
H,f
HI,
(2.1)
where HN and H, are given in standard notations by HN = p2/GW + W,
(2.2)
fL = 4 c 1 dr (~2 + (VbJ2 + ~‘9~~). a
(2.3)
The subscript 01refers to the charge of the pion. We useunits such that c For the interaction HI let us first consider HI
=
-@Id
s drl
dr2
4
r -
rl
I> 4
r -
r2 I) c&J
4(r2),
?i = 1.
(2.4)
where r is the nucleon coordinate, h is a (dimensionless) coupling constant, and J drv(r) = 1. We have suppressedisospin indices in HI , but it is understood that HI or h and v are determined for a given isospinchannel. e As emphasized in Section 4, however, the IA is beset with ambiguities.
PION
SCATTERING
FROM
A BOUND
145
NUCLEON
For the pion it is convenient to work in the momentum +(r) in plane waves as usual;
space. Hence we expand
r$(r) = (257-3/Z j dk (~uJ)-~/~ (ukeik’r + ak+e-ik.r),
(2.5)
where a, and ak+ are the annihilation and creation operators, respectively, and w = (JL~+ k2)li2. Then H, and H, become H, =
s
dk uaL+ak ,
HI = -@/~)(27r-~ X (akeik.r
(2.6) s dk dk’ (4ow’)-112
+
ak+e-ik.r)(ak,eik’.r
= -(h/~)(271)-~
+
1 dk dk’ (ww’)-~/~
g(k) g(k’) a;,,-ik’.r)
(2.7)
g(k) g(k’) ak+akei(k’-k).r
+ a..,
(2.8)
where w’ = (LL~+ k’2)1/2, and g(k) = s dr U(T) eik.r,
g(0) = 1.
(2.9)
In Eq. (2.8) we have omitted terms of the form of a,~,’ and uk+akt+ . This is because we are going to drop intermediate states containing more than one meson. The interaction HI of Eq. (2.8) causes rrNf scattering [for I’ = 0] for the s-wave only. Therefore we will modify H, by introducing the projection operator P& k’) defined by I’&
k’) = (3k . k’ - a . ko . k’)/p2
(2.10)
which satisfies
I Ai? P(k,
k”) P(k”, k’) = 47~(k”/p)~ P(k, k’).
(2.11)
&
(2.12)
We replace HI of Eq. (2.8) by --A
HI
=
(2743
p
I
mc,
g(k)
&w (ww,)l,2
P(k, k’) ak+alcjei(k’-k).r,
which causes VTN~ scattering in the p3/2 state only. Also it should be understood that this HI acts only for the ?TN isospin I = 312. We will determine the coupling constant X and form factor g(k) so that the
146
BAJAJ
AND
NOGAMI
TNf scattering parameters are reproduced. First note that when V(r) = 0 in Eq. (2.2) the total TN momentum defined by P = p + J dk k ak+ak is a constant of motion. Tn the center of mass (CM) system, we put P = 0, and hence p = - Jdk k a,‘-a, . Then HIV and H, become H,, + HN =
[ dk ija,+ak t I dk dk’ k . k’a,~‘a:,aka~,/(2nz), *
(2.13)
where ii3 = w + (k’/h).
(2.14)
Since we are going to restrict ourselves to the one-meson state, the last term in Eq. (2.13) doesnot contribute. The Schrodinger equation for the pion wavefunction then becomes
in the CM system. Equation (2.15) can be solved exactly and the -rrNr scattering amplitude is given by .m, 0) =
eib sin S P(k’, k) = k Wp.)”
177
177j- w
Xg2(k) P(k’, k) 2npD(k) ’
(2.16)
where S is the 33-phase shift, k and k’ are the initial and final momenta in the TN CM system, 0 is the scattering angle, and D(k) _ 1 - h 2gp.3
j= &' _ k'4&?k') ~'(6' -- L; - ie) . o
(2.17)
Becauseof the projection operator in H, , phase shifts in partial waves other than the 33-state vanish. The tota! cross section for &p scattering from an unpolarized proton target is given by uf,tot = (&r/k”) sin3S = (471./k)Imf(k,
0),
(2.18)
where the subscript f refers to n7f scattering. Note that P(k, k) = 2(k/p)2. The angular distribution is of the form of (I + 3 cos28). The on-shell t-matrix is related to the scattering amplitude f through
f = -457%Jt{m/(m + co)}.
(2.19)
We now want to determine X and g(k) by fitting the energy and width of the
PION
SCATTERING
FROM
A BOUND
NUCLEON
33-resonance. The resonance energy & is determined by Re D(G) width ij is determined by expanding Re D(G) around & ;
147 = 0. The
(2.20) where D’(G) = dD(G)/& and r = 2 Im D(ij,)/Re D’(G,). Then the scattering amplitude takes the Breit-Wigner form. For the energy and width of the resonance we take the experimental values, &= 1233MeV-m=2.13p and r= 116 MeV = 0.84~. For the masses we take p = 138 MeV and m = 6.80~. We choose the square cutoff g(k) = &kc - k) for the form factor. We tried other forms such as the gaussianbut the results are quite insensitive to the details of the form factor. We also evaluate the z-N scattering length a which is given by 1 -
a
z
k3
cot
6 IkzO
_
%k!?
m
2y3
D(O).
(2.21)
The resonanceenergy Gir and width I’ are fitted by taking the following values for h and wc = (kc2 + pL2)lj2. h = 0.1536,
WC= 7.21 p.
(2.22)
With these parameters the scattering length is found to be a = 0.10 p-3. The experimental value [12] is a = 0.21 p-3. Note that this scattering length is for the 33-state. It is possible to improve the fit for the scattering length by choosing an appropriate form of g(k), but since we are mainly interested in the 33-resonance energy region the scattering amplitude obtained above is satisfactory. Unlike the Chew-Low model [13], for example, the separable model of the p-wave TN interaction has no theoretical foundation and is used only becauseof its simplicity. We believe, however, that for the problem under consideration the separable model is in fact a good simulation of the Chew-Low model. Let us compare some typical diagrams for nN scattering in the two models shown in Fig. 3, where diagrams A’ and B’ represent the first- and second-order Z-N scattering in the Chew-Low theory, while diagrams A and B are the corresponding ones for the separable interaction. The binding effect for TNb scattering is due to the change in the nucleon propagator in the intermediate states. Then diagram A’ in Fig. 3 is modified but A is not. We show in Appendix A, however, that the binding effect on A’, or, more generally, on a similar part (with two-pion intermediate state) of a larger diagram, is quite negligible and the main contribution stemsfrom the one-pion intermediate state in which the energy denominator can vanish. Hence it is a good approximation to replace A’ by A, and the circled part of B’ by the corresponding part of B in Fig. 3.
148
BAJAJ AND NOGAMI
B’ 3. Diagrams A’ and B’ represent the first- and second-order TN scattering in the ChewLow theory. Diagrams A and B are the corresponding ones for separable interaction. FIG.
3. rril$, SCATTERING
Unlike rNr scattering described in Section 2, the Schriidinger equation for TNb scattering cannot be solved analytically. This is because, although the 7dv interaction is separable, the nA interaction is not. Therefore we obtain TNb scattering amplitude up to the second order with respect to h and then unitarize it by means of the Pad6 approximant. Note that the Padt approximant gives the exact result for ?rNr scattering. We introduce a “single particle wavefunction” & for the nucleon by H.&v = Ev#v,
(3.1)
where v stands for the set of quantum numbers (FZ,1,m), and also we set E, = 0 for the ground state. In order to evaluate the matrix element (v’ 1HI 1v), it is convenient to write & as W)
= fn*w y&9,
F = (e, $).
(3.2)
Then we obtain (3.3) where I-k, V) standsfor the state such that the pion momentum is k and the nucleon state is v. Also, q = k’ - k and (3.4)
PION
SCATTERING
FROM
A
BOUND
149
NUCLEON
We consider the elastic scattering process such that the nucleon is in the ground state before and after the collision. The t-matrix elements in the first and second order with respect to X are given, respectively, by (3.5)
t12’ = - C j- dk” 1’ --
W, 0 I HI I k”, VW’, v I HI I k 0) E + Wn _ w _ ic ”
X (” dk’) g(k) I (277)3 p ) (w’w)1/2 x c 1 dk” Y
g2(k”)
P(k’, k”) P(k”, k) F,,(k” - k’) F,,(k JyW” - w + E, - ie)
-
k”) ,
(3 6)
So far we have not specified the “shell model potential” V(r), which we now assumeto be a harmonic oscillator potential
V(r) = &m7j2r2.
(3.7)
Then E,, &, F,,, are all known explicitly. In particular, E, = E, = tzq, and F,,(q) = exp(--b2q2/4), where b2 = l/(mT). The summations with respect to m and I for tt2’ can be done [seeAppendix B], and we obtain t2 for k = k’ as
x
b2(k” - k’) . (k” - k) [
2
n I
exp -
b2{( k” - k’ I2 + 1 k” - k 1”)
[
4
1. (3.8)
Although only the paj2state contributes to nNr scattering, all partial waves enter for nNb scattering. Before applying the Pad6 approximant we have to decompose t(l) and t (2) into partial waves. This can be done as follows. Since the nucleon is bound in an s-state our “nucleus” has spin 4. The TNb scattering amplitude (and hence the on-shell t-matrix) takes the form [14] F(e) = f(e) + io * ng(Q
(3.9)
where 0 is the scattering angle and the unit vector n is defined by n = & x &. It
150
BAJAJ
AND
NOGAMI
is understood that F, f, and g depend on k also. The spin-nonflip and spin-flip amplitudesfand g are related to partial wave amplitudesf,* as follows.
de) = f {h- - .h+>P,‘(cos0).
(3.11)
Z=l
Here I& correspond to J = I f 3. Equations (3.10) and (3.11) can be inverted to give PE(cos e) -
.fL* = t j;l 4cos af(e)
In order to obtain the spin-nonflip for elastic scattering k = k’,
[COS ~P,(COS
e)
-
P~*,(co~
e)l g(e)>.
(3.12)
(NF) and spin-flip (F) parts of t(l), note that
P(k’, k) = (k/#
(3.13) (3.14)
(2 COS e - io . n),
4’ = (k’ - k)’ = 2k2(1 - COS d).
Using these, we obtain from Eq. (3.5) h
t(l) NF--oqz
Cl)tl-2 ---
h
(2743
p”
!ft?@ w
2
cos
e exp{ -Pk2(1
k2gL(k) exp{ -b2k2( 1 w
cos
-
cos
Q/2),
(3.15)
@/a}.
We substitute Eqs. (3.15) and (3.16) into Eq. (3.12) to obtain the first order t-matrix t(l) for the Zth partial wave. For the second-order t-matrix element t(2), using F$. (C.5) and (C.6) for the product P(k’, k”) P(k”, k’) in Eq. (3.Q we obtain the spin-nonflip and spin-flip parts as follows.
x
3 cos2 i co? cy-- sin2 i sin2 a: cos2 8) + cos e 1 ( I
x
b2(k”2 - 2k”k [
x exp -
COS(e/z)
COS
2
a + k2 COS d) n
I
b2(kn2- 2k”k cos(oj2) COS a! f k2)
2
I,
(3.17)
PION
x
SCATTERING
b2(k”” - 2k”k 1
x exp -
FROM
A BOUND
NUCLEON
151
cos(0/2) cos a + k” cos 6) ‘L 2
- 2k”k b”(k””
cos(0/2) cos a + 2
1 k”) I.
(3.18)
The angles a and /3 specify the orientation of vector k” as defined in Appendix C. Again using Eq. (3.12) together with t:J and ti”, we obtain t,(z). We then construct the partial wave t-matrix t,* by means of the Pad& approximant I,* = t$‘/[l
-
t$/t,‘:‘].
(3.19)
Knowing t,+ we can obtain the total %-Nb t-matrix from Eq. (3.9 - 11). The TNb elastic scattering amplitude F is related to the on-shell f-matrix element by F(k, 0) = -4n%t(k, The total and elastic (differential) Ob,tot ‘Jb,el(o)
d).
(3.20)
cross sections (for r+p) are given by =
(47/k)
h
=
j F(k
e)l”,
W,
01,
(3.21) (3.22)
where the subscript b refers to the 77Nb scattering. As we shall see the rfNb total cross section Qb,tot as a function of energy shows a broad peak. This peak is often referred to as the “33-resonance,” although, unlike in nNr scattering, partial waves other than p3,2 are also contributing in TNb scattering. Since the peak is broad the resonance energy becomes quite ambiguous. We define the resonance energy by the condition Re F(k, 0) = 0. Then the energy for the peak of the cross section will be considerably lower than the resonance energy. Since the center of the “nucleus” is fixed, there is no distinction between the rrA CM system and nA laboratory (lab) system in our model. It is, however, more natural to interpret our system in which the nucleus is fixed as the TTTACM system because for elastic scattering such that k = k’ the initial and final energies of the pion are the same, and this is the case in the nA CM system. We therefore interpret that all the above formulas for nNt, scattering are for the nA CM system. When we compare various results in the following, however, we give the cross section, etc., as a function of the pion lab energy. For example, for the case which simulates 160, first we obtain the cross section in the nA CM system and then transform the CM energy to the VA lab energy using the “nuclear mass” 16m.
152
BAJAJ AND NOGAMI
In Figs. 4 and 5 we have plotted Re F(k, 0) vs the pion lab energy for two cases corresponding to Nb in 160 and 4He, respectively. The solid curves show the results of ‘rrNb calculation, whereas the other curves are for different forms of the IA described in the next section. The resonance energy is found to be 174 MeV and 186 MeV for 160 and 4He, respectively. There is thus a downward shift in the resonance energy as compared with the nNr resonance energy of 192 MeV in the TNf lab system. Note that the F here contains all partial waves, significant contributionscoming from 5 to 10 partial waves depending upon the incident pion energy. The p-wave alone in the zNb scattering exhibits a resonance at an energy lower than that of total F. For example, for I60 it is at 163MeV. Next we have shown the total cross section Ub(r+p) from a nucleon bound in the Is state of the HO potential, against the pion lab energy. The solid curves in Figs. 6 and 7 correspond to the TNb scattering for 160 and 4He, respectively. We see that the TNb cross section is about 50 % higher than that of z-Nr near the 33resonance. The cross section for the IA will be discussedin the next section. The significant contributions to TNb scattering come from the first 5 to 10 partial waves as pointed out above, whereas rrNr scattering takes place only through p-waves. The p-wave scattering in rrNb is suppressedby binding to about half that in the TABLE Partial Wave Contributions
I
(nr6) to the Total Cross Section at Various Energies” Ecb (MeV)
50
I
IA (B) ?TN~
0
0.2 0.4 1.9 4.4 0.4 0.7 0.1
1 2 3 4 5 6 7 8 9 10
IA(B)
200
150
100
aNb
2.9 9.0 12.6 37.0 5.8 16.1 1.3 3.3 0.2 0.3
IA (B) ?rNb 15.7 50.8 33.2 11.0 2.4 0.4
39.4 99.6 88.8 40.5 10.7 1.8 0.2
250
IA(B) nh% 25.4 70.2 55.8 24.1 7.0 1.5 0.3
21.8 54.7 61.2 45.9 25.7 11.1 3.7 0.9 0.1
IA(B) 16.7 43.0 38.5 20.0 7.2 1.9 0.5 0.1
rrNb 11.8 30.1 36.7 29.9 19.5 10.4 4.8 1.9 0.7 0.2 0.1
DThe rrNb calculation and the IA (B) are shown for a comparison. This is for the nucleon bound in an HO potential well appropriate to raO.
0
50 150 El,ob( M EN
2 50
FIG. 4. The real part of the “Nb forward scattering amplitude, Re(F) in units of cc-l, as a function of pion lab energy (MeV). This is for the nucleon bound in an HO potential well appropriate to leO. The solid curve represents the rrNb calculation, the other curves correspond to various forms of the IA. The dotted curve shows the IA(B), the dashed IA(C), and the dash-dot-dot shows the IA(A). The meaning of the IA’s is explained in Section 4.
-10
-. 5
-‘3. Lj 2
20
FIG.
Er
(MEV)
250 5. The same as in Fig. 4, but for Nb in “He.
150
154
BAJAJ
AND
NOGAMI
t; 0 0
m
(W)D
0 N
0 0
Q. iz
0
’
’
60
180 cross section divided by F&(q)
120
FIG. 8. The differential for ?rh% scattering (mb) as a function of rrA CM scattering angle. This is for the nucleon bound in an HO potential well appropriate to 160. Pion lab energy = 150 MeV. The curves are explained in Fig. 4, except for the dash-dot curve which corresponds to IA(D).
0’
60
0
; 60
120
180
FIG. 9. The same as in Fig. S, for pion lab energy -: 200 MeV.
0
20
40
60
80
2
E
156
BAJAJ
AND
NOGAMI
free case. The overall increase in rrNb total cross section is the effect of binding, with contributions from a large number of partial waves. The contributions of various partial waves are shown in Table I. The TNb differential cross section [divided by F&(q)], for 160, at three different pion lab energies, 150,200, and 250 MeV is shown in Figs. 8,9, and 10, respectively. The curves other than the solid ones show the results of the IA. Notice that the angular dependence is very different from (1 + 3 cos2 0) for rrNr scattering. The .rrNb forward cross section is much greater than that in the IA at all energies considered. The backward cross section, on the other hand, is much smaller than that in the IA at and above the resonance energy.
I/ I I
60
I
I
/
I
I
I’
I’
:
-._
0 60 120 180 FIG. 10. The same as in Fig. 8, for pion lab energy = 250 MeV. 4. IMPULSE APPROXIMATION In this section we describe the IA for our model and compare its results with those obtained in Section 3. The IA, as we have noted earlier, consists in replacing the ?TNb t-matrix tb by the 7rNr t-matrix tf , i.e., lb = tt . Therefore the pion interacts with Nb as if it were free. Unlike Np , however, the nucleon inside the potential well
PION
SCATTERING
FROM
A BOUND
NUCLEON
157
does not have a fixed momentum. This can be taken account of by averaging the TNr t-matrix with appropriate nucleon momentum distribution [ 151. We need the TNr t-matrix t(wo) in the nA CM system. This is related to the t-matrix in the TN CM system f( IV) by [ 161 (k’, P’ I h,)l
k, P; = y Xx’ I f(W
x:,,
(4.1)
where w,, is the TN energy, k, p and k’, p’ are the initial and final momenta of the pion and the nucleon, respectively, in the nA CM system, while H and H’ are the initial and final momenta and W is the rrN energy in the Z-N CM system. The factor y is given by
y =
-6,(K) Ev(K’> -b(K) -htK’> I” E,(k) E,(k’) E,(k) E,(k’) .
1
(4.2)
The pion momentum x and energy C(X) = (x2 + p2)1’2in the VN CM system are related to k, p by a Lorentz transformation, which implies the following reIation for C(K) E,E, + p2 - pk cos ep ‘(x) = (2E,E, + m2 - @ - 2pk cos 6Dj1/2’
(4.3)
where E, , EIv are the pion and nucleon energies, and 8P is the angle of nucleon momentum with respect to the incoming pion. With this we obtain t as a function of (k, p, cos 0,). This is averaged over the nucleon momentum distribution to obtain f(k), the nNb t-matrix in the IA in the 7rA CM system W
=
j- d3p P(P)
W,
P, ~0s
e,>,
(4.4)
where p(p) is the nucleon momentum distribution. For the nucleon bound in the s-state of the HO potential it is given by p(p) = (n/b3)-‘f@exp(--b2p2).
(4.5)
Finally, we obtain the scattering amplitude in the IA from the on-shell ?TNbtmatrix f(k) using Eq. (3.16). We will refer to the IA of this form as IA (A). Note that we have obtained the forward scattering amplitude only. For any finite momentum transfer, the initial and final momentum space wavefunctions are different. In such a case the nucleon momentum distribution with which the t-matrix should be averaged is not uniquely known. So, we do not present the results for angular distributions in the IA (A).
158
BAJAJ
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NOGAMI
There is another form of the IA for a fixed nucleus, which takes account of the nucleon momentum distribution. This is due to Chew [17]. We will call this IA(B). According to this, the scattered amplitude is given by
f = {(P+ m>/m: f& , WJ’(q)
(4.6)
where f(ki , kr) is the rrNr scattering amplitude and F(q) is the nuclear form factor; ki = mk,/(m + pL), kr = ki + q, and q = k, - k, is the momentum transfer, k, , k,, being the initial and final pion momenta. In deriving this, a local potential has been assumed for the interaction between the two particles, whereas our TN interaction is nonlocal. We nevertheless assume the above relation for our model with an additional replacement of p by w, the incoming pion energy in the VA CM system. It is not clear how the factor y should be incorporated in the relativistic version of the above form of the IA, and hence we do not include this factor in the IA(B). For the sake of completenesswe would like to discussyet another form of the IA which has been used by several authors [18, 191. Tn this case the motion of the nucleus as a whole is considered with nucleons “frozen” in it, which gives a fixed value of the nucleon momentum. Clearly, this is different from our model which assumesa fixed nucleus with the moving nucleons in it, and hence the results of this IA cannot be compared directly with those of Section 3. Nevertheless, it is an interesting comparison. The matrix element of the -rrNr t-matrix (k’, p’ / t(w)i k, pi depends on an energy variable w in addition to the initial and final pion and nucleon momenta. We choose the value of w, called wO, such that it represents the collision energy of the TN collision as seenin the nA CM system. For a given value of the nucleon momentum p, wOis given by (q)
Ezz (k”2
$
pyi:!
+
(p”
+
pp)lP.
(4.7)
It is hoped [18] that with this choice of w the correction to the IA will be small. Various authors make different choices for p. We will consider two such cases, one due to Tabakin et al. [I81 and the other due to Landau [19] with the corresponding form of the IA referred to as IA(C) and IA(D), respectively. According to the former, the nucleons are “frozen” initially and finally in the moving target nucleus, so that each nucleon has a momentum p = -k,/A and p’ = -k,‘/A, where k, , k,’ are the initial and final on-shell VA CM momenta. Note that the momentum is not conserved for ~TNcollisions, even for energy conserving n-A scattering (k, = ko’), since the initial and final total 7rN momenta are k,(l - A-l) and k,‘(l - A-l), respectively. With this choice of the nucleon momentum p, the energy W of the TN system
PION
SCATTERING
FROM
A
BOUND
159
NUCLEON
in its CM frame, which is different from that in the nA CM system, can be obtained by using the Lorentz invariance of s = (p, + pN)2 w*
=
S =
[E,,(K)
+
= [E&J where u is the related to r(w) The relation be obtained by
EN(K)]’
+ Eh&hq12
- k(12(1 - A-Y,
xN CM momentum. The t-matrix in the TN CM system according to between the scattering angle tiZA in using the invariance of t = ( ppitia’
cos t&.v =
{E,(K)
&(k’)
-
E,,(k)
I?,#‘)
(4.8)
t(w,,) in the TA CM system is relation (4.1). nA CM and 8,, in TN CM can - ~f”~‘)~, +
kk’
cos
B,A}/~~‘.
(4.9)
This angle transformation leads to unphysical angles On,vfor large angle scattering in 7-rA CM, even for energy conserving scattering, i.e., k = k’ in rA CM system. As we have noted above the 7rN momentum is not conserved, although the rrN energy is conserved for on-shell n-A collisions. This is the origin of the difficulty of unphysical angles B,,.y . Landau [I91 has proposed to take a different nucleon momentum in the above “frozen” nucleon approximation, which conserves ZTN momentum for on-shell Z-A collisions. His choice is P = i--k,
+ &(A -
I) q}/A,
(4.10)
where q is the momentum transfer in the TA CM system. This implies the following relation instead of Eq. (4.8). [E,,(K)
$
E,,,(K)]’
=
[E,(k)
+ a&(p)]’
- ;:-k2(1 - A-1)2 (1 + cos e,,).
(4.11)
This implies that the momenta X, K’ depend upon the rrA scattering angle enA . This is a result of the q-dependent choice of p for energy conserving QTAscattering. Only physical values of cos OWNoccur in Eq. (4.9). The results of these IA’s are shown in the Figs. 4-10 along with the nNb results of the last section. We note that the IA results differ from one another to some degree. For example, the total cross section for 160 is 176 mb in IA(A) and 182 mb in (IA(B) at the resonance peak. Similarly, the resonance energy is 194 MeV and 208 MeV for the above example. For the total cross section, the IA results differ from one another by 10 N 20 “/ But the .rrNb total cross section is about 50 N 70 % greater than any of the IA cross sections. So, the ambiguity in the IA does not in any way obscure the binding correction. The results for the differential cross section show a similar binding effect. The rrNb differential cross section near the resonance
160
BAJAJ AND NOGAMI
is strongly forward-peaked compared to any of the IA differential cross section. Also, the magnitude of the TNb cross section in the forward direction is much larger than that in the IA. 5. VALIDITY
OF THE CLOSURE APPROXIMATION
In Section 3 we have done the summation over intermediate nuclear states explicitly. In order to examine the validity of the CA and also the relation between our rNr-, results and those of Kohmura [6] and Schmit [7] who used the CA, we perform closure on the intermediate nuclear states, i.e., use the relation 1 F&k"
- k') F,,(k - k") = F&k - k').
Y
With this we get the following expression for tt2’. g2(k")P(k', k")P(k",k) WH(W" - w + n7j - iE) ' (5.2)
where Q is the mean excitation energy of the nucleon. This term is usually set equal to zero in the CA. It is argued that only low lying states will contribute significantly. With fin = 0 and Eqs. (C.5) and ((2.6) we obtain the spin-nonflip and spin-flip parts of tL2A
x (8~ cos 0) /= dk’ u,(u:‘4g2E”
ic) ,
0
k’4g2(k’) x (-477) 6 dk’ w,(w, _ w - k) *
The angular dependences of t&2 and t(l) are the same, so that their ratio is independent of angles. The use of the Padt approximant to obtain the .rrNb scattering amplitude in the CA then leads to a denominator that has no l-dependence. The TNb scattering amplitude in the CA becomes FCA =
k2W
27~4
W, k) &dk) ’
(5.5)
PION SCATTERING FROM A BOUND NUCLEON
161
where D,,(k)
h k14g2(k’) * dk’ = 1- w’(w’ - W - k) . 2372p3I0
(5.6)
We note immediately that DC- is the same as D for rrNr except that there is no recoil term in Dca . Thus the rrNt, scattering amplitude in the CA reduces to the nNr amplitude in which the nucleon is fixed. With the parameters h, wc of the z-N interaction given by Eq. (2.22), it turns out that the .rrNb scattering amplitude (5.5) exhibits no resonance at any incident pion energy. Instead there is a bound state with the binding energy of 69 MeV. The CA in our model of nNb scattering, therefore, leads to results which are qualitatively different from those obtained in Section 3. We conclude this section by discussingthe limiting casesof nucleon binding. In the limit of infinite binding 7 ---f co, only the II = 0 term contributes in Eqs. (3.17) and (3.18), and hence the TNt, scattering amplitude of Section 3 is reduced to F,, of Eq. (5.5). As we have pointed out above, it is very different from the nNr amplitude. In the weak binding limit 77-+ 0, on the other hand, we would expect that the 7iruf scattering amplitude is recovered. This can be verified for zero energy, but we have not been able to do so for finite energy.
6. THE PAULI PRINCIPLE EFFECT So far we have considered a “nucleus” which consists of only one nucleon, and have allowed the nucleon to be excited to all levels in the intermediate state. Now let us suppose that a few lowest levels are occupied by other spectator nucleons. According to the Pauli principle, the nucleon which is struck by the incident pion cannot go to the levels alredy occupied. For example, in n-l60 scattering the IZ= 1 level is forbidden for the s-state nucleon as an intermediate state. Examination of Eqs. (3.17) and (3.18) for tc2) together with the PadC approximant (3.19) reveals that this should push the resonanceenergy upwards. In the caseof n-l60 scattering, the TNb resonanceis found to occur at pion lab energy of 200 MeV. This is to be compared with 194 MeV in the IA(A) and 174 MeV in z-Ni, scattering with binding correction alone. When the Pauli principle effect is combined with the binding effect, the resonanceenergy becomesclose to that found in the IA(A). For the cross section, its effect is to decreasethe mNa total cross section below the resonance energy, whereas there is a significant increase above the resonance. For example, at pion lab energiesof 150 and 200 MeV, the TNb total cross sectionswithout and with the Pauli principle effect are 281, 192mb and 225, 322 mb, respectively. We noted in Section 2 that the separableTN interaction is similar to the more basic CL interaction so far as the binding effect is concerned. However, the results
162
BAJAJ
AND
NOGAMI
of the two interactions may be quite different with respect to the Pauli principle effect. The lowest order diagram in the CL interaction is modified by the Pauli principle, whereas the corresponding diagram for the separable interaction is not. Hence the effect is probably more appreciable for the CL interaction.
7. DISCUSSION It is clear from our analysis that the IA is a rather poor approximation in the neighborhood of the 33-resonance. The binding correction to the IA is indeed quite significant. The binding effect pushes the resonance energy downwards compared with that in the IA. The 7TNb total cross section near the resonance is about 50 - 70 “/o greater than the corresponding IA cross section and the elastic cross section shows a much stronger forward peak than that of IA. For instance, for Nb corresponding to 160, the nNb resonance energy is found to be 174 MeV with the total cross section at the resonance peak of about 290 mb, where as the corresponding figures in the IA(A) are 194 MeV and 176 mb, respectively. The basic mechanism for the downward shift of the resonance energy is that the nucleon mass is effectively increased by the binding effect. When the strength of the potential which binds the nucleon is increased, the nucleon becomes less and less mobile and finally in the tight binding limit the Z-N scattering amplitude reduces to that in the static limit in which m --, co. As we noted at the end of Section 5, the resonancedisappearsin this limit; instead, a TN bound state emerges.For the CL interaction the effect of the increasing mass is somewhat less dramatic. The resonanceremains in the tight binding limit, but at 17 MeV [20]. Of course, for a realistic strength of the HO potential, the effect is much lessdrastic. In addition to the binding effect, we simulated the effect of the Pauli principle by excluding some lower intermediate states. This effectively weakens the TNb interaction and hence the resonanceenergy is shifted upwards, for example, from 174 MeV to 200 MeV in the 160 case. Note that the effective strength of the z-N interaction increases with increase in pion energy (because of the momentum factor in the interaction). In discussingthe Pauli principle effects in nA scattering Landau and McMillan [21] found that the ETAtotal cross section decreasesaround the resonance energy (corresponding to .rrNr scattering), but increasesabove the resonance.They did not emphasizethis point. In their model they have also used a separableTN interaction. We now want to comment on some related works and compare our results with them, Kohmura [6] examined the binding correction to the IA for a model similar to ours, but he assumedthe n-A interaction to be separable, used the CA to sum over the intermediate nuclear states, and neglected the nucleon recoil in fitting the mNf scattering. We have pointed out that the n-A interaction is not separable, the
PION
SCATTERING
FROM
A BOUND
NUCLEON
163
CA is misleading, and the nucleon recoil is important. Schmit [7] considered a model much the same as Kohmura’s. He also used the CA and neglected the nucleon recoil in nNf scattering. His treatment of the binding effect is in the socalled “quasiclassical” approximation, which amounts to shifting the energy argument of the nN t-matrix from E to E’ = E - ZJ, U (< 0), being the binding potential of the target nucleon. Since E’ - E = 20 - 30 MeV, the resonance energy is lowered by the corresponding amount. However, the magnitude of the cross section is not modified substantially. The Pauli principle effect cannot be taken account of in this treatment. Note that our rNh calculation does not use the quasiclassical approximation. Myhrer and Koltun [9] have studied nd scattering as a three body problem in the 33-resonance energy region. They examined the effect of dynamical excitations of the target as well as of multiple scattering and nucleon motion. They find that the resonance occurs at p = 2 15 MeV/c in the nd CM system, whereas the 33-resonance momentum in the IA(C) with nucleons frozen in the deuteron is 205 MeV/c in the z-d CM. Thus the binding effect results in an upward shift of the resonance energy. The multiple scattering effect. which contributes a downward shift, is small for the deuteron and hence the net effect is an upward shift. They explain this result in the following way. The energy argument of the TN scattering amplitude is assumed to be shifted due to binding from E to E’, E’ = E + UN,,
(7.1)
where UNA is the average potential in the initial state which binds the nucleon. In the intermediate states the binding potential is small and is neglected, so that the replacement (7.1) implies an upward shift of the resonance by -U,“, MeV in nuclear medium. For the deuteron this shift is about 15 MeV. (Incidentally Schmit’s choice of energy replacement is E’ = E - U and differs from Eq. (7.1). The difference in the sign of the potential term changes even the qualitative nature of the binding effect. There is some confusion on this point in the literature.) In our calculation the nucleon is bound in the intermediate states as well as in the initial state, and it is not possible to interpret our results by means of Eq. (7.1) or the like. Hiifner’s [5] analysis or conjecture of the binding effect is similar to our basic idea which we have described earlier, i.e., the pion scatters repeatedly from one bound nucleon, which is excited while all other nucleons are spectator nucleons. But for the binding correction in the intermediate state, he argues that it is unimportant. It is not obvious whether the neglect of binding in the intermediate states is a priori justified. Moreover, how well can the binding effect be simulated simply by replacing E by an effective energy E’ as the argument of the nN scattering amplitude is also not clear. Julius and Rogers [I I] included the first-order binding correction to the TNr b-matrix by retaining the second term in Eq. (I .l). The resulting corrections to the
164
BAJAJ
AND
NOGAMI
optical potential are 50 - 100 % near the resonance and fall to negligible values at energies less than 120 MeV and greater than 450 MeV. Their Fig. 5 in which they plot the binding correction to the optical potential for ,rr-Y (Re V,,,(r)) indicates that Re V,,,,(r) vanishes at an energy slightly lower than that for the optical potential without the binding effect. Unlike their calculation in which binding effect is taken account of only up to second order, we do not make an expansion with respect to the binding potential. We have used the Pad6 approximant to obtain the 7TNbamplitude: the accuracy of the Pad6 approximant for various scattering problems is known to be generally good [22]. This is the only approximation in our calculation. We are aware of the unrealistic nature of the HO potential, but we believe that for the problem under consideration it provides us with a reasonable starting point. At least it has been used quite extensively in literature for similar problems. At this point we would like to note that the 50 - 70 % increasein the nNb total cross section as compared with that in the IA, near the resonance, will not necessarily increase the nA cross section by the same amount. Julius and Rogers [ll] have shown by meansof a simple model that the n-A cross section at the resonance peak is infact quite insensitive to u,,~ . For example, doubling the ~TNcross section increasesthe n-A cross section by ( 10 “/, at the resonanceenergy. But the effect of corrections will be much larger at energiesaway from the resonance peak. As we have seen above the binding corrections to u,,,, are negligible at energies 2 100 MeV so that the TA total cross section is unaffected at such energies. At energies >, 200 MeV the binding effect increases u,~ by 20 - 25 ‘A and then decreasesto negligible effect at energies >, 400 MeV. The n-A cross section will thus increase by about 20 2, at energies above 200 MeV, i.e., on the tail of the peak. Tt is in this energy region that the experimental values of n-A cross section are greater than what most VA calculations using the IA prediction. Our binding correction will thus increase these TTAcross sections resulting in better agreement with experimental data. Finally, let us briefly summarize the results of various calculations which have examined the effect of the Pauli principle by considering nuclear matter [lo]. In such calculations the initial and final wavefunctions are taken to be plane waves. Unlike our calculation, where the potential binding the nucleon modifies the bound state wavefunction, there is no binding effect in nuclear matter. Only the Pauli principle modifies the Z-N scattering amplitude in nuclear matter. Kimura and Nagashima’s [lo] calculation exhibits a very large upward shift of the resonance energy, by about 200 MeV for the CL amplitude. Bethe’s [lo] result, on the other hand, shows a downward shift of the resonance to w,. = 1.15 p. There are some differences in the two approaches, but they both consider the CL 7ri? amplitude. Dover and Lemmer [lo] predict an upward shift of the resonance energy due to the Pauli principle effect which reducesthe effective TN coupling constant. In their
PION
SCATTERING
FROM
A
BOUND
NUCLEON
165
most recent calculation, Weber and Eisenberg [lo] claim that the Pauli blocking in 7~,4 scattering has little effect at higher energies; the main blocking effect occurs below the resonance energy. It is obvious from these diverse results that the situation is far from having been resolved about the Pauli principle effects for nuclear matter. It is difficult to compare these results with our model calculation since the former completeIy ignore the binding effects.
APPENDIX
A
We will evaluate the lowest-order diagram for nN scattering in the Chew-Low theory when the nucleon is bound in a harmonic oscillator potential. The binding effect manifests itself through the modification of the propagator, and the vertices have extra factors coming from the bound state wavefunctions. Apart from the vertex factors which are common to free and bound cases, we have I=C
” OJ -
Fodk) Mk’) (w + w’ + q)
(A-1)
’
where F’,(k) is defined by Eq. (3.4) and the energy in the intermediate state is w + w’ + nv. After manipulations similar to those shown in Appendix B, Eq. (A.1) can be reduced to
The summation
for IZin Eq. (A.2) can be converted into an integration using
z.A wTnl =lrn4 e+exp(xe-9. Hence we obtain I=
Ia d[ exp[+b2k * k’(epen -
1) - &J].
0
(A-4)
In the tight binding limit 77-+ co, Eq. (A.4) reduces to (l/w) which is the propagator for the Chew-Low interaction with a static nucleon. In the weak binding limit 3 + 0, retaining terms up to c in (A.4), we obtain I(7) ---f 0) =
s0m de ev-Ha
+ K’)]/(u
+ r),
(A.5)
166
BAJAJ
where K’ = k * k’/(2m). order in f, we obtain
AND
NOGAMI
For finite but weak binding, retaining terms up to second
I = Jrn df (1 f (l/2) S”K’v) exp[-(Q2)(w
+ K’)]
0
=
CA.6)
1 w+K’
[ 1+
K’rl (w + K’)2 I .
The second term in the bracket is the binding correction
which satisfies (4.7)
B
APPENDIX
For the HO potential, the function -Fyo(q) takes the form = (4~Y2 +Gd
Ejyo6-d
(B.1)
K%(4),
where G,(q) = c, exp(--b2q2/Wq)n.
(B-2)
The coefficient c, is known, but we do not need its explicit expression. To do the m and 1 summations in Eq. (3.6), we start with the form factor for the Is-state F,,(q). Foe(q) = (0 j eiq” 10) = exp( -b2q2/4) = exp
-
;
(k2
f
p)]
[
f
$
( bZk2.
k’
)y
(B.3)
7X=0
Using Eqs. (B.l) and (B.2) this can be rewritten
as
F,,(q) = C (0 [ eik”r j V)(V 1 e-ik.r 10)
(B-4)
= 4~7 C (21 + 1) P,(cos 8) G,(k’) G,(k) ?L,Z = exp[-(b2/4)(k2
+ kf2)] f TZ=O
(b”k . k’)”
c (21 + 1) &P,(cos 2
0).
167
PION SCATTERING FROM A BOUND NUCLEON
Here and also in Eq. (3.6), I takes all values that are allowed for a given value of n. In the third line of Eq. (B.4) we have made use of the following relation for msummation. c Y,T,,(k’) Y,,,(i-i) r (21 + 1) P&k). 1,L Comparison
(B.5)
of Eqs. (B.3) and (B.4) yields
cI (21 + 1) &P,(cos
0) = (2” . n!)-’
COSR0.
u3.6)
APPENDIX C Consider the product
of projection
P(k’, k”) P(k”, k) = [9(k’ . k”)(k” . k”)(k”
- 3(k’ . k”)(k” Let us choose the coordinates
P(k’, k”) P(k”, k).
. k) - 3(k’ . k”)(o
- 3(ci . k’)(o = [(3(k’
operators
. k”)(k”
. k”)(o
. k)
. k) + (c . k’) /P(o
. k)]/p*
. k) + (k’ . k) ,Y2} + ia . (V2(k’ x k) - 3(k’ x k”)(k”
Cl) x k)
. k):/p4.
(C.2)
as follows.
k = k(sin (e/2), 0, cos (e/2)), k’ = k(-sin
(e/2), 0, cos (e/2)),
(C.3)
k” = k”(sin cycos /3, sin ci sin /3, cos a). Then we obtain k’ ’ k = k2 cos 8, k” * k = k”k(sin
(8/2) sin a: cos /3 + cos (e/2) cos a),
k” * k’ = k”k(-sin With these, the spin-nonflip p-4(k”k)2
(C.4)
(d/2) sin cycos /3 + cos (e/2) cos a).
and spin-flip
terms in Eq. (c.2) become
[3(cos2 (O/2) cos2 a - sin2 (e/Z) sin* a: ~0~2 /3) +
ia * (f’ x ic>p-4(k”k)2
cos
[l - 3{(sin 01cos /3)” + cos2 a}).
01,
(C.5) (C.6)
ACKNOWLEDGMENT We would like to thank Dr. II-Tong Cheon for helpful discussions on his calculation [15].
168
BAJAJ
AND
NOGAMI
REFERENCES la.
J. P. STROOT, “Proceedings of the Fourth International Conference on High Energy Physics and Nuclear Structure” (V. P. Dzhelepov, Ed.), p. 221, Dubna, 1972. lb. F. BINON et al., Nacl. Phys. B17 (1970), 168. lc. F. BINON et al., Nucl. Phys. B 33 (1971), 42. Id. A. S. CLOUGH et al., Phys. Lett. B43 (1973), 476. 2a. R. H. LANDAU, S. C. PHATAK, AND F. TABAKIN, Ann. Physics 78 (1973), 299. 2b. S. C. PHATAK, R. H. LANDAU, AND F. TABAKIN, Phys. Rev. C7 (1973), 180. 2c. L. S. KISSLINGER, Phys. Rev. 98 (1955), 761. 2d. L. CELENZA, L. C. Lru, AND C. M. SHAKIN, Phys. Rev. Cl1 (1975), 1593. 3a. C. SCHMIT, Lett. Nuovo Cim. 4 (1970), 454. 3b. C. WILKIN, Lett. Nuovo Cim. 4 (1970), 451. 3c. C. WILKIN, “Proceedings of the Fifth International Conference on High Energy Physics and Nuclear Structure” (G. Tibell, Ed.), p. 157, North-Holland, Amsterdam, 1973. 3d. M. E. BEST, Canad. J. Phys. 50 (1972), 1609. 3e. R. J. GLAUBER, “High Energy Physics and Nuclear Structure,” p. 207, North-Holland, Amsterdam, 1967. 4. K. M. WATSON AND M. L. GOLDBERGER, “Collision Theory,” Sect. 11.1, Wiley, New York, 1964. 5. J. H~~FNER, Phys. Reports C 21 (1975), 1. 6. T. KOHMURA, Nucl. Phys. B 36 (1972), 228. 7a. C. SCHMIT, Nucl. Phys. A 197 (1972), 449. 7b. C. SCHMIT, Orsay preprint IPNO/TH 73-17 (June, 1973). 8. D. I. JUL~JS, Ann. Physics 87 (1974), 17. 9. F. MYHRER AND D. KOLTUN, Phys. Lett. B 46 (1973), 322; Nucl. Phys. B 86 (1975), 441. 10a. H. A. BETHE, Phys. Rev. Lett. 30 (1973), 105. lob. C. B. DOVER AND R. H. LEMMER, Phys. Reu. C7 (1973). 2312. 1Oc. Y. KIMURA AND Y. NAGASHIMA, Prog. Theoret. Phys. 33 (1965), 43. 10d. J. M. EISENBERG AND H. J. WEBER, Phys. Lett. B45 (1973), 110. 10e. J. M. EISENBERG, Phys. Left. B 49 (1974), 224. 1Of. H. J. WEBER AND J. M. EISENBERG, Phys. Rev. Cl0 (1974), 925. 11. D. I. JULIUS AND C. ROGERS, Phys. Rev. C 12 (1975), 206. 12. J. HAMILTON AND W. S. WOOLCOCK, Rev. Mod. Phys. 35 (1963), 137. 13. G. F. CHEW AND F. E. Low, Phys. Rev. 101(1956), 1571. 14. S. GASIOROWICZ, “Elementary Particle Physics,” Chap. 23, Wiley, New York, 1966. 15. I. T. CHEON, Phys. Lett. B 55 (1975), 463. 16. C. MBLLLER, Kgl. Danske Videnskab. Selskab., Mat-.fys. Medd. 23 (1945), 1. 17a. G. F. CHEW, Phys. Rev. 80 (1950), 196. 17b. N. F. MOTT AND H. S. W MASSEY, “The Theory of Atomic Collisions,” Chap. 12, Oxford Univ. Press, London, 1965. 18. R. H. LANDAU, S. C. PHATAK, AND F. TABAKIN, Ann. Physics 78 (1973), 299. 19. R. H. LANDAU, Scattering of pions from 4He and 4He calculated with realistic form factors, preprint. 20. K. K. BAJAJ AND Y. NOGAMI, Phys. Rev. Lett. 34 (1975), 701. 21. R. H. LANDAU AND M. MCMILLAN, Phys. Rev. C 8 (1973), 2094. 22. J. Z~NN-JUSTIN, Phys. Reports Cl (1971), 55.
OF
PHYSICS
103, 141-168 (1977)
Pion Scattering
from a Bound Nucleon
at Medium Energies*
K. K. BAJAJ AND Y. NOGAMI Department of Physics, McMaster
University, Hamilton, Ontario, Canada LAS 4Ml
Received May 5, 1976
In order to examine the validity of the impulse approximation for pion-nucleus scattering in the 33-resonance energy region, we consider pion-scattering from a “nucleus” which consists of a single nucleon bound in a harmonic oscillator potential. A separable ?rN interaction is assumed. The oscillator parameter is chosen such that the nuclear sizes are fitted for 4He - I60 . The binding effect is found to result in a downward shift of the resonance energy (by about 20 MeV), and an increase (by 50 - 70 %) of the total cross section near the resonance. The angular distribution is also strongly modified. In connection with the binding effect, the importance of a careful treatment of nucleon recoil is emphasized. It is pointed out that the closure approximation which is often used to sum over intermediate nuclear states leads to very misleading results. The effect of the Pauli principle is also examined by excluding some intermediate states.
1. INTRODUCTION A considerable amount of information has been accumulated for pion-nucleus scattering at intermediate energies.For example, the total and elastic cross sections for several nuclei ranging from 4He to 32Sare known [I]. On the theoretical side, it has been found that these cross sections can be reproduced quite well by means of the optical model [2], and the Glauber approximation [3]. In these calculations, the pion-nucleus (z-A) scattering amplitude is obtained in terms of the pion-nucleon (-rrN) scattering amplitude. This nN scattering amplitude should in principle be related to the t-matrix from a bound nucleon, but in practice the free ?TNt-matrix is usually used. This is the impulse approximation (IA). It is generally expected that the IA is valid if the energy of the projectile is much larger than the binding energy of the nucleon in the target. More precisely, it can be shown [4] that the t-matrix for 7TNbscattering1is related to that for niVp scattering by
* Work supported by the National Research Council of Canada. 1 We denote a free nucleon by Nr and a bound nucleon by Nb .
141 Copyright All rights
0 1977 by Academic Press, Inc. of reproduction in any form reserved.
ISSN
0003-4916
142
BAJAJ
AND
NOGAMI
wherefis the nNr scattering amplitude, h = h/p, T is the collision time, and SE is a “level shift” which is of the order of the average binding energy of the nucleon in the target. As was pointed out a long time ago by Goldberger and Watson [4], 1(f/h)(T@/fi)I = 0.5 around the 33-resonance, and hence the validity of the IA is questionable. This is because the relatively long collision time at the resonance enhances the importance of the interaction between the struck nucleon and surrounding nuclear medium. Some of the typical processesin TA scattering are depicted in Fig. 1. The shaded blob stands for tb , which is replaced by tf in the IA. The methods such as the Glauber approximation and the optical model take care of thesemultiple scattering processesdue to different nucleons, and the accuracies of these methods have been examined by many authors. On the other hand, relatively little attention has been paid to the validity of the IA [5]. If we assumethe Yukawa or Chew-Low interaction for the basic mechanismof TN scattering, the blob consistsof those processes shown in Fig. 2, where the double line connecting two nucleon lines stands for an N-N interaction. The processesof the type shown in the first row will give tf , while those in the secondrow will causethe difference between tb and tf . This is the kind of correction that we are going to examine in this paper. There are two corrections to the IA which we will refer to as the “binding effect” and the “Pauli principle effect”. The former is due to the fact that Nb is described by a bound state wavefunction, whereas Nr is described by a plane wave, while the
FIG. 1. Typical processes in ?rA scattering. The shaded blob stands for
1 +
\/ ,‘,I‘
/
\ +
I
tb .
/
){11;;’ f.0.
FIG. 2. The structure of interaction.
fb
. The double line connecting two nucleon lines stands for an N-N
PION
SCATTERING
FROM
A BOUND
NUCLEON
143
latter is due to the Pauli principle which prevents Nb from being excited into states which are occupied by other nucleons. Kohmura [6] and Schmit [7] estimated the binding effect for a nucleon bound in a harmonic oscillator (HO) potential. The only significant effect they found is that the 33-resonance energy is shifted; upward in [6] but downward in [7]. For the deuteron, Julius [8] and Myhrer and Koltun [9] have estimated the binding effect in the resonance energy region. The former finds an increase in the elastic cross section of up to 40 % for backward angles, while the latter obtain an upward shift of the 33-resonance by about 15 MeV. Some other authors have examined the Pauli principle effect by considering the Fermi gas model of nuclear matter [IO]. More recently, Julius and Rogers [ll] have estimated the binding effect on the optical potential by including the first-order correction to the IA in Eq. (1.1). They find that the resulting correction to the potential is of the order of 50 - 100 yi near the 33-resonance, with an increase in the TNI, total cross section by the same amount. In the present paper we examine the validity of the IA for the rrN interaction in the 33-resonance region for a simple model. We consider pion scattering from a “nucleus” which consists of only one nucleon bound in an HO potential. For the TN interaction, we assume a separable-type one, which reproduces rNr scattering at intermediate energies including the 33-resonance. In a real nucleus, of course, there are many nucleons, and the nA scattering amplitude can be constructed in terms of tb , which satisfies tb = L‘ + VGtb .
(1.2)
Here c is TN interaction and G an appropriate many-body propagator. The most important intermediate states that contribute to tb in Eq. (1.2) are probably those in which the struck nucleon is excited while all other nucleons remain undisturbed. Then the other nucleons can be regarded as spectators, and, so far as the binding effect on tb is concerned, they simply generate the potential which binds the struck nucleon. This is the idea that underlies our model calculation. Since we have only one nucleon in our “nucleus,” we do not consider multiple scattering due to different nucleons. We are aware of the ambiguity as to the distinction between our binding effect and that of the last diagram of Fig. 1. However, since the latter effect is not included in the Glauber approximation or in the usual optical model, tr in these methods can be replaced by fb which we obtain, without danger of double counting. Unlike some previous calculations [6, 71, we do not use the closure approximation (CA) but do the summation over nuclear intermediate states explicitly. Our results are very different from those of [6, 71, but are similar to those of Julius and Rogers [ll]. We find that the 33-resonance energy is shifted downward (in terms of the pion energy in the TA laboratory system) by the binding effect. Crucial differences between our analysis and the previous ones [6, 71 lie in the treatment of 595/103/I-10
144
BAJA1
AND
NOGAMI
nucleon recoil as well as in the summation for intermediate nuclear states. The main result of our calculation is that the total cross section for rrNb scattering increases by about 50 N 70 % compared with the IA2 near the resonance, and the TNb elastic cross section is more strongly forward-peaked. We also examine the Pauli principle effect by excluding appropriate intermediate states in the summation. In Section 2 we present our model, examine TNf scattering and thereby determine the parameters of the model. The TNb scattering with the nucleon bound in the ground state of the HO potential is considered in Section 3. In Section 4 we describe the IA for our model and in Section 5 we examine the validity of the closure approximation. We consider the Pauli principle effect in Section 6. Results are summarized and discussed in the last section. In Appendix A we show that, so long as the binding effect is concerned, the Chew-Low interaction and the separable Z-N interaction will lead to very similar results. Some details of the calculation in the text are given in Appendixes B and C.
2. THE MODEL
AND TNf
SCATTERING
We consider a “nucleus” which consists of a single nucleon bound in a “shellmodel potential” V(r). The nucleon also interacts with a pion field through a seperableinteraction. The Hamiltonian for the TN system is given by H=H,v+
H,f
HI,
(2.1)
where HN and H, are given in standard notations by HN = p2/GW + W,
(2.2)
fL = 4 c 1 dr (~2 + (VbJ2 + ~‘9~~). a
(2.3)
The subscript 01refers to the charge of the pion. We useunits such that c For the interaction HI let us first consider HI
=
-@Id
s drl
dr2
4
r -
rl
I> 4
r -
r2 I) c&J
4(r2),
?i = 1.
(2.4)
where r is the nucleon coordinate, h is a (dimensionless) coupling constant, and J drv(r) = 1. We have suppressedisospin indices in HI , but it is understood that HI or h and v are determined for a given isospinchannel. e As emphasized in Section 4, however, the IA is beset with ambiguities.
PION
SCATTERING
FROM
A BOUND
145
NUCLEON
For the pion it is convenient to work in the momentum +(r) in plane waves as usual;
space. Hence we expand
r$(r) = (257-3/Z j dk (~uJ)-~/~ (ukeik’r + ak+e-ik.r),
(2.5)
where a, and ak+ are the annihilation and creation operators, respectively, and w = (JL~+ k2)li2. Then H, and H, become H, =
s
dk uaL+ak ,
HI = -@/~)(27r-~ X (akeik.r
(2.6) s dk dk’ (4ow’)-112
+
ak+e-ik.r)(ak,eik’.r
= -(h/~)(271)-~
+
1 dk dk’ (ww’)-~/~
g(k) g(k’) a;,,-ik’.r)
(2.7)
g(k) g(k’) ak+akei(k’-k).r
+ a..,
(2.8)
where w’ = (LL~+ k’2)1/2, and g(k) = s dr U(T) eik.r,
g(0) = 1.
(2.9)
In Eq. (2.8) we have omitted terms of the form of a,~,’ and uk+akt+ . This is because we are going to drop intermediate states containing more than one meson. The interaction HI of Eq. (2.8) causes rrNf scattering [for I’ = 0] for the s-wave only. Therefore we will modify H, by introducing the projection operator P& k’) defined by I’&
k’) = (3k . k’ - a . ko . k’)/p2
(2.10)
which satisfies
I Ai? P(k,
k”) P(k”, k’) = 47~(k”/p)~ P(k, k’).
(2.11)
&
(2.12)
We replace HI of Eq. (2.8) by --A
HI
=
(2743
p
I
mc,
g(k)
&w (ww,)l,2
P(k, k’) ak+alcjei(k’-k).r,
which causes VTN~ scattering in the p3/2 state only. Also it should be understood that this HI acts only for the ?TN isospin I = 312. We will determine the coupling constant X and form factor g(k) so that the
146
BAJAJ
AND
NOGAMI
TNf scattering parameters are reproduced. First note that when V(r) = 0 in Eq. (2.2) the total TN momentum defined by P = p + J dk k ak+ak is a constant of motion. Tn the center of mass (CM) system, we put P = 0, and hence p = - Jdk k a,‘-a, . Then HIV and H, become H,, + HN =
[ dk ija,+ak t I dk dk’ k . k’a,~‘a:,aka~,/(2nz), *
(2.13)
where ii3 = w + (k’/h).
(2.14)
Since we are going to restrict ourselves to the one-meson state, the last term in Eq. (2.13) doesnot contribute. The Schrodinger equation for the pion wavefunction then becomes
in the CM system. Equation (2.15) can be solved exactly and the -rrNr scattering amplitude is given by .m, 0) =
eib sin S P(k’, k) = k Wp.)”
177
177j- w
Xg2(k) P(k’, k) 2npD(k) ’
(2.16)
where S is the 33-phase shift, k and k’ are the initial and final momenta in the TN CM system, 0 is the scattering angle, and D(k) _ 1 - h 2gp.3
j= &' _ k'4&?k') ~'(6' -- L; - ie) . o
(2.17)
Becauseof the projection operator in H, , phase shifts in partial waves other than the 33-state vanish. The tota! cross section for &p scattering from an unpolarized proton target is given by uf,tot = (&r/k”) sin3S = (471./k)Imf(k,
0),
(2.18)
where the subscript f refers to n7f scattering. Note that P(k, k) = 2(k/p)2. The angular distribution is of the form of (I + 3 cos28). The on-shell t-matrix is related to the scattering amplitude f through
f = -457%Jt{m/(m + co)}.
(2.19)
We now want to determine X and g(k) by fitting the energy and width of the
PION
SCATTERING
FROM
A BOUND
NUCLEON
33-resonance. The resonance energy & is determined by Re D(G) width ij is determined by expanding Re D(G) around & ;
147 = 0. The
(2.20) where D’(G) = dD(G)/& and r = 2 Im D(ij,)/Re D’(G,). Then the scattering amplitude takes the Breit-Wigner form. For the energy and width of the resonance we take the experimental values, &= 1233MeV-m=2.13p and r= 116 MeV = 0.84~. For the masses we take p = 138 MeV and m = 6.80~. We choose the square cutoff g(k) = &kc - k) for the form factor. We tried other forms such as the gaussianbut the results are quite insensitive to the details of the form factor. We also evaluate the z-N scattering length a which is given by 1 -
a
z
k3
cot
6 IkzO
_
%k!?
m
2y3
D(O).
(2.21)
The resonanceenergy Gir and width I’ are fitted by taking the following values for h and wc = (kc2 + pL2)lj2. h = 0.1536,
WC= 7.21 p.
(2.22)
With these parameters the scattering length is found to be a = 0.10 p-3. The experimental value [12] is a = 0.21 p-3. Note that this scattering length is for the 33-state. It is possible to improve the fit for the scattering length by choosing an appropriate form of g(k), but since we are mainly interested in the 33-resonance energy region the scattering amplitude obtained above is satisfactory. Unlike the Chew-Low model [13], for example, the separable model of the p-wave TN interaction has no theoretical foundation and is used only becauseof its simplicity. We believe, however, that for the problem under consideration the separable model is in fact a good simulation of the Chew-Low model. Let us compare some typical diagrams for nN scattering in the two models shown in Fig. 3, where diagrams A’ and B’ represent the first- and second-order Z-N scattering in the Chew-Low theory, while diagrams A and B are the corresponding ones for the separable interaction. The binding effect for TNb scattering is due to the change in the nucleon propagator in the intermediate states. Then diagram A’ in Fig. 3 is modified but A is not. We show in Appendix A, however, that the binding effect on A’, or, more generally, on a similar part (with two-pion intermediate state) of a larger diagram, is quite negligible and the main contribution stemsfrom the one-pion intermediate state in which the energy denominator can vanish. Hence it is a good approximation to replace A’ by A, and the circled part of B’ by the corresponding part of B in Fig. 3.
148
BAJAJ AND NOGAMI
B’ 3. Diagrams A’ and B’ represent the first- and second-order TN scattering in the ChewLow theory. Diagrams A and B are the corresponding ones for separable interaction. FIG.
3. rril$, SCATTERING
Unlike rNr scattering described in Section 2, the Schriidinger equation for TNb scattering cannot be solved analytically. This is because, although the 7dv interaction is separable, the nA interaction is not. Therefore we obtain TNb scattering amplitude up to the second order with respect to h and then unitarize it by means of the Pad6 approximant. Note that the Padt approximant gives the exact result for ?rNr scattering. We introduce a “single particle wavefunction” & for the nucleon by H.&v = Ev#v,
(3.1)
where v stands for the set of quantum numbers (FZ,1,m), and also we set E, = 0 for the ground state. In order to evaluate the matrix element (v’ 1HI 1v), it is convenient to write & as W)
= fn*w y&9,
F = (e, $).
(3.2)
Then we obtain (3.3) where I-k, V) standsfor the state such that the pion momentum is k and the nucleon state is v. Also, q = k’ - k and (3.4)
PION
SCATTERING
FROM
A
BOUND
149
NUCLEON
We consider the elastic scattering process such that the nucleon is in the ground state before and after the collision. The t-matrix elements in the first and second order with respect to X are given, respectively, by (3.5)
t12’ = - C j- dk” 1’ --
W, 0 I HI I k”, VW’, v I HI I k 0) E + Wn _ w _ ic ”
X (” dk’) g(k) I (277)3 p ) (w’w)1/2 x c 1 dk” Y
g2(k”)
P(k’, k”) P(k”, k) F,,(k” - k’) F,,(k JyW” - w + E, - ie)
-
k”) ,
(3 6)
So far we have not specified the “shell model potential” V(r), which we now assumeto be a harmonic oscillator potential
V(r) = &m7j2r2.
(3.7)
Then E,, &, F,,, are all known explicitly. In particular, E, = E, = tzq, and F,,(q) = exp(--b2q2/4), where b2 = l/(mT). The summations with respect to m and I for tt2’ can be done [seeAppendix B], and we obtain t2 for k = k’ as
x
b2(k” - k’) . (k” - k) [
2
n I
exp -
b2{( k” - k’ I2 + 1 k” - k 1”)
[
4
1. (3.8)
Although only the paj2state contributes to nNr scattering, all partial waves enter for nNb scattering. Before applying the Pad6 approximant we have to decompose t(l) and t (2) into partial waves. This can be done as follows. Since the nucleon is bound in an s-state our “nucleus” has spin 4. The TNb scattering amplitude (and hence the on-shell t-matrix) takes the form [14] F(e) = f(e) + io * ng(Q
(3.9)
where 0 is the scattering angle and the unit vector n is defined by n = & x &. It
150
BAJAJ
AND
NOGAMI
is understood that F, f, and g depend on k also. The spin-nonflip and spin-flip amplitudesfand g are related to partial wave amplitudesf,* as follows.
de) = f {h- - .h+>P,‘(cos0).
(3.11)
Z=l
Here I& correspond to J = I f 3. Equations (3.10) and (3.11) can be inverted to give PE(cos e) -
.fL* = t j;l 4cos af(e)
In order to obtain the spin-nonflip for elastic scattering k = k’,
[COS ~P,(COS
e)
-
P~*,(co~
e)l g(e)>.
(3.12)
(NF) and spin-flip (F) parts of t(l), note that
P(k’, k) = (k/#
(3.13) (3.14)
(2 COS e - io . n),
4’ = (k’ - k)’ = 2k2(1 - COS d).
Using these, we obtain from Eq. (3.5) h
t(l) NF--oqz
Cl)tl-2 ---
h
(2743
p”
!ft?@ w
2
cos
e exp{ -Pk2(1
k2gL(k) exp{ -b2k2( 1 w
cos
-
cos
Q/2),
(3.15)
@/a}.
We substitute Eqs. (3.15) and (3.16) into Eq. (3.12) to obtain the first order t-matrix t(l) for the Zth partial wave. For the second-order t-matrix element t(2), using F$. (C.5) and (C.6) for the product P(k’, k”) P(k”, k’) in Eq. (3.Q we obtain the spin-nonflip and spin-flip parts as follows.
x
3 cos2 i co? cy-- sin2 i sin2 a: cos2 8) + cos e 1 ( I
x
b2(k”2 - 2k”k [
x exp -
COS(e/z)
COS
2
a + k2 COS d) n
I
b2(kn2- 2k”k cos(oj2) COS a! f k2)
2
I,
(3.17)
PION
x
SCATTERING
b2(k”” - 2k”k 1
x exp -
FROM
A BOUND
NUCLEON
151
cos(0/2) cos a + k” cos 6) ‘L 2
- 2k”k b”(k””
cos(0/2) cos a + 2
1 k”) I.
(3.18)
The angles a and /3 specify the orientation of vector k” as defined in Appendix C. Again using Eq. (3.12) together with t:J and ti”, we obtain t,(z). We then construct the partial wave t-matrix t,* by means of the Pad& approximant I,* = t$‘/[l
-
t$/t,‘:‘].
(3.19)
Knowing t,+ we can obtain the total %-Nb t-matrix from Eq. (3.9 - 11). The TNb elastic scattering amplitude F is related to the on-shell f-matrix element by F(k, 0) = -4n%t(k, The total and elastic (differential) Ob,tot ‘Jb,el(o)
d).
(3.20)
cross sections (for r+p) are given by =
(47/k)
h
=
j F(k
e)l”,
W,
01,
(3.21) (3.22)
where the subscript b refers to the 77Nb scattering. As we shall see the rfNb total cross section Qb,tot as a function of energy shows a broad peak. This peak is often referred to as the “33-resonance,” although, unlike in nNr scattering, partial waves other than p3,2 are also contributing in TNb scattering. Since the peak is broad the resonance energy becomes quite ambiguous. We define the resonance energy by the condition Re F(k, 0) = 0. Then the energy for the peak of the cross section will be considerably lower than the resonance energy. Since the center of the “nucleus” is fixed, there is no distinction between the rrA CM system and nA laboratory (lab) system in our model. It is, however, more natural to interpret our system in which the nucleus is fixed as the TTTACM system because for elastic scattering such that k = k’ the initial and final energies of the pion are the same, and this is the case in the nA CM system. We therefore interpret that all the above formulas for nNt, scattering are for the nA CM system. When we compare various results in the following, however, we give the cross section, etc., as a function of the pion lab energy. For example, for the case which simulates 160, first we obtain the cross section in the nA CM system and then transform the CM energy to the VA lab energy using the “nuclear mass” 16m.
152
BAJAJ AND NOGAMI
In Figs. 4 and 5 we have plotted Re F(k, 0) vs the pion lab energy for two cases corresponding to Nb in 160 and 4He, respectively. The solid curves show the results of ‘rrNb calculation, whereas the other curves are for different forms of the IA described in the next section. The resonance energy is found to be 174 MeV and 186 MeV for 160 and 4He, respectively. There is thus a downward shift in the resonance energy as compared with the nNr resonance energy of 192 MeV in the TNf lab system. Note that the F here contains all partial waves, significant contributionscoming from 5 to 10 partial waves depending upon the incident pion energy. The p-wave alone in the zNb scattering exhibits a resonance at an energy lower than that of total F. For example, for I60 it is at 163MeV. Next we have shown the total cross section Ub(r+p) from a nucleon bound in the Is state of the HO potential, against the pion lab energy. The solid curves in Figs. 6 and 7 correspond to the TNb scattering for 160 and 4He, respectively. We see that the TNb cross section is about 50 % higher than that of z-Nr near the 33resonance. The cross section for the IA will be discussedin the next section. The significant contributions to TNb scattering come from the first 5 to 10 partial waves as pointed out above, whereas rrNr scattering takes place only through p-waves. The p-wave scattering in rrNb is suppressedby binding to about half that in the TABLE Partial Wave Contributions
I
(nr6) to the Total Cross Section at Various Energies” Ecb (MeV)
50
I
IA (B) ?TN~
0
0.2 0.4 1.9 4.4 0.4 0.7 0.1
1 2 3 4 5 6 7 8 9 10
IA(B)
200
150
100
aNb
2.9 9.0 12.6 37.0 5.8 16.1 1.3 3.3 0.2 0.3
IA (B) ?rNb 15.7 50.8 33.2 11.0 2.4 0.4
39.4 99.6 88.8 40.5 10.7 1.8 0.2
250
IA(B) nh% 25.4 70.2 55.8 24.1 7.0 1.5 0.3
21.8 54.7 61.2 45.9 25.7 11.1 3.7 0.9 0.1
IA(B) 16.7 43.0 38.5 20.0 7.2 1.9 0.5 0.1
rrNb 11.8 30.1 36.7 29.9 19.5 10.4 4.8 1.9 0.7 0.2 0.1
DThe rrNb calculation and the IA (B) are shown for a comparison. This is for the nucleon bound in an HO potential well appropriate to raO.
0
50 150 El,ob( M EN
2 50
FIG. 4. The real part of the “Nb forward scattering amplitude, Re(F) in units of cc-l, as a function of pion lab energy (MeV). This is for the nucleon bound in an HO potential well appropriate to leO. The solid curve represents the rrNb calculation, the other curves correspond to various forms of the IA. The dotted curve shows the IA(B), the dashed IA(C), and the dash-dot-dot shows the IA(A). The meaning of the IA’s is explained in Section 4.
-10
-. 5
-‘3. Lj 2
20
FIG.
Er
(MEV)
250 5. The same as in Fig. 4, but for Nb in “He.
150
154
BAJAJ
AND
NOGAMI
t; 0 0
m
(W)D
0 N
0 0
Q. iz
0
’
’
60
180 cross section divided by F&(q)
120
FIG. 8. The differential for ?rh% scattering (mb) as a function of rrA CM scattering angle. This is for the nucleon bound in an HO potential well appropriate to 160. Pion lab energy = 150 MeV. The curves are explained in Fig. 4, except for the dash-dot curve which corresponds to IA(D).
0’
60
0
; 60
120
180
FIG. 9. The same as in Fig. S, for pion lab energy -: 200 MeV.
0
20
40
60
80
2
E
156
BAJAJ
AND
NOGAMI
free case. The overall increase in rrNb total cross section is the effect of binding, with contributions from a large number of partial waves. The contributions of various partial waves are shown in Table I. The TNb differential cross section [divided by F&(q)], for 160, at three different pion lab energies, 150,200, and 250 MeV is shown in Figs. 8,9, and 10, respectively. The curves other than the solid ones show the results of the IA. Notice that the angular dependence is very different from (1 + 3 cos2 0) for rrNr scattering. The .rrNb forward cross section is much greater than that in the IA at all energies considered. The backward cross section, on the other hand, is much smaller than that in the IA at and above the resonance energy.
I/ I I
60
I
I
/
I
I
I’
I’
:
-._
0 60 120 180 FIG. 10. The same as in Fig. 8, for pion lab energy = 250 MeV. 4. IMPULSE APPROXIMATION In this section we describe the IA for our model and compare its results with those obtained in Section 3. The IA, as we have noted earlier, consists in replacing the ?TNb t-matrix tb by the 7rNr t-matrix tf , i.e., lb = tt . Therefore the pion interacts with Nb as if it were free. Unlike Np , however, the nucleon inside the potential well
PION
SCATTERING
FROM
A BOUND
NUCLEON
157
does not have a fixed momentum. This can be taken account of by averaging the TNr t-matrix with appropriate nucleon momentum distribution [ 151. We need the TNr t-matrix t(wo) in the nA CM system. This is related to the t-matrix in the TN CM system f( IV) by [ 161 (k’, P’ I h,)l
k, P; = y Xx’ I f(W
x:,,
(4.1)
where w,, is the TN energy, k, p and k’, p’ are the initial and final momenta of the pion and the nucleon, respectively, in the nA CM system, while H and H’ are the initial and final momenta and W is the rrN energy in the Z-N CM system. The factor y is given by
y =
-6,(K) Ev(K’> -b(K) -htK’> I” E,(k) E,(k’) E,(k) E,(k’) .
1
(4.2)
The pion momentum x and energy C(X) = (x2 + p2)1’2in the VN CM system are related to k, p by a Lorentz transformation, which implies the following reIation for C(K) E,E, + p2 - pk cos ep ‘(x) = (2E,E, + m2 - @ - 2pk cos 6Dj1/2’
(4.3)
where E, , EIv are the pion and nucleon energies, and 8P is the angle of nucleon momentum with respect to the incoming pion. With this we obtain t as a function of (k, p, cos 0,). This is averaged over the nucleon momentum distribution to obtain f(k), the nNb t-matrix in the IA in the 7rA CM system W
=
j- d3p P(P)
W,
P, ~0s
e,>,
(4.4)
where p(p) is the nucleon momentum distribution. For the nucleon bound in the s-state of the HO potential it is given by p(p) = (n/b3)-‘f@exp(--b2p2).
(4.5)
Finally, we obtain the scattering amplitude in the IA from the on-shell ?TNbtmatrix f(k) using Eq. (3.16). We will refer to the IA of this form as IA (A). Note that we have obtained the forward scattering amplitude only. For any finite momentum transfer, the initial and final momentum space wavefunctions are different. In such a case the nucleon momentum distribution with which the t-matrix should be averaged is not uniquely known. So, we do not present the results for angular distributions in the IA (A).
158
BAJAJ
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NOGAMI
There is another form of the IA for a fixed nucleus, which takes account of the nucleon momentum distribution. This is due to Chew [17]. We will call this IA(B). According to this, the scattered amplitude is given by
f = {(P+ m>/m: f& , WJ’(q)
(4.6)
where f(ki , kr) is the rrNr scattering amplitude and F(q) is the nuclear form factor; ki = mk,/(m + pL), kr = ki + q, and q = k, - k, is the momentum transfer, k, , k,, being the initial and final pion momenta. In deriving this, a local potential has been assumed for the interaction between the two particles, whereas our TN interaction is nonlocal. We nevertheless assume the above relation for our model with an additional replacement of p by w, the incoming pion energy in the VA CM system. It is not clear how the factor y should be incorporated in the relativistic version of the above form of the IA, and hence we do not include this factor in the IA(B). For the sake of completenesswe would like to discussyet another form of the IA which has been used by several authors [18, 191. Tn this case the motion of the nucleus as a whole is considered with nucleons “frozen” in it, which gives a fixed value of the nucleon momentum. Clearly, this is different from our model which assumesa fixed nucleus with the moving nucleons in it, and hence the results of this IA cannot be compared directly with those of Section 3. Nevertheless, it is an interesting comparison. The matrix element of the -rrNr t-matrix (k’, p’ / t(w)i k, pi depends on an energy variable w in addition to the initial and final pion and nucleon momenta. We choose the value of w, called wO, such that it represents the collision energy of the TN collision as seenin the nA CM system. For a given value of the nucleon momentum p, wOis given by (q)
Ezz (k”2
$
pyi:!
+
(p”
+
pp)lP.
(4.7)
It is hoped [18] that with this choice of w the correction to the IA will be small. Various authors make different choices for p. We will consider two such cases, one due to Tabakin et al. [I81 and the other due to Landau [19] with the corresponding form of the IA referred to as IA(C) and IA(D), respectively. According to the former, the nucleons are “frozen” initially and finally in the moving target nucleus, so that each nucleon has a momentum p = -k,/A and p’ = -k,‘/A, where k, , k,’ are the initial and final on-shell VA CM momenta. Note that the momentum is not conserved for ~TNcollisions, even for energy conserving n-A scattering (k, = ko’), since the initial and final total 7rN momenta are k,(l - A-l) and k,‘(l - A-l), respectively. With this choice of the nucleon momentum p, the energy W of the TN system
PION
SCATTERING
FROM
A
BOUND
159
NUCLEON
in its CM frame, which is different from that in the nA CM system, can be obtained by using the Lorentz invariance of s = (p, + pN)2 w*
=
S =
[E,,(K)
+
= [E&J where u is the related to r(w) The relation be obtained by
EN(K)]’
+ Eh&hq12
- k(12(1 - A-Y,
xN CM momentum. The t-matrix in the TN CM system according to between the scattering angle tiZA in using the invariance of t = ( ppitia’
cos t&.v =
{E,(K)
&(k’)
-
E,,(k)
I?,#‘)
(4.8)
t(w,,) in the TA CM system is relation (4.1). nA CM and 8,, in TN CM can - ~f”~‘)~, +
kk’
cos
B,A}/~~‘.
(4.9)
This angle transformation leads to unphysical angles On,vfor large angle scattering in 7-rA CM, even for energy conserving scattering, i.e., k = k’ in rA CM system. As we have noted above the 7rN momentum is not conserved, although the rrN energy is conserved for on-shell n-A collisions. This is the origin of the difficulty of unphysical angles B,,.y . Landau [I91 has proposed to take a different nucleon momentum in the above “frozen” nucleon approximation, which conserves ZTN momentum for on-shell Z-A collisions. His choice is P = i--k,
+ &(A -
I) q}/A,
(4.10)
where q is the momentum transfer in the TA CM system. This implies the following relation instead of Eq. (4.8). [E,,(K)
$
E,,,(K)]’
=
[E,(k)
+ a&(p)]’
- ;:-k2(1 - A-1)2 (1 + cos e,,).
(4.11)
This implies that the momenta X, K’ depend upon the rrA scattering angle enA . This is a result of the q-dependent choice of p for energy conserving QTAscattering. Only physical values of cos OWNoccur in Eq. (4.9). The results of these IA’s are shown in the Figs. 4-10 along with the nNb results of the last section. We note that the IA results differ from one another to some degree. For example, the total cross section for 160 is 176 mb in IA(A) and 182 mb in (IA(B) at the resonance peak. Similarly, the resonance energy is 194 MeV and 208 MeV for the above example. For the total cross section, the IA results differ from one another by 10 N 20 “/ But the .rrNb total cross section is about 50 N 70 % greater than any of the IA cross sections. So, the ambiguity in the IA does not in any way obscure the binding correction. The results for the differential cross section show a similar binding effect. The rrNb differential cross section near the resonance
160
BAJAJ AND NOGAMI
is strongly forward-peaked compared to any of the IA differential cross section. Also, the magnitude of the TNb cross section in the forward direction is much larger than that in the IA. 5. VALIDITY
OF THE CLOSURE APPROXIMATION
In Section 3 we have done the summation over intermediate nuclear states explicitly. In order to examine the validity of the CA and also the relation between our rNr-, results and those of Kohmura [6] and Schmit [7] who used the CA, we perform closure on the intermediate nuclear states, i.e., use the relation 1 F&k"
- k') F,,(k - k") = F&k - k').
Y
With this we get the following expression for tt2’. g2(k")P(k', k")P(k",k) WH(W" - w + n7j - iE) ' (5.2)
where Q is the mean excitation energy of the nucleon. This term is usually set equal to zero in the CA. It is argued that only low lying states will contribute significantly. With fin = 0 and Eqs. (C.5) and ((2.6) we obtain the spin-nonflip and spin-flip parts of tL2A
x (8~ cos 0) /= dk’ u,(u:‘4g2E”
ic) ,
0
k’4g2(k’) x (-477) 6 dk’ w,(w, _ w - k) *
The angular dependences of t&2 and t(l) are the same, so that their ratio is independent of angles. The use of the Padt approximant to obtain the .rrNb scattering amplitude in the CA then leads to a denominator that has no l-dependence. The TNb scattering amplitude in the CA becomes FCA =
k2W
27~4
W, k) &dk) ’
(5.5)
PION SCATTERING FROM A BOUND NUCLEON
161
where D,,(k)
h k14g2(k’) * dk’ = 1- w’(w’ - W - k) . 2372p3I0
(5.6)
We note immediately that DC- is the same as D for rrNr except that there is no recoil term in Dca . Thus the rrNt, scattering amplitude in the CA reduces to the nNr amplitude in which the nucleon is fixed. With the parameters h, wc of the z-N interaction given by Eq. (2.22), it turns out that the .rrNb scattering amplitude (5.5) exhibits no resonance at any incident pion energy. Instead there is a bound state with the binding energy of 69 MeV. The CA in our model of nNb scattering, therefore, leads to results which are qualitatively different from those obtained in Section 3. We conclude this section by discussingthe limiting casesof nucleon binding. In the limit of infinite binding 7 ---f co, only the II = 0 term contributes in Eqs. (3.17) and (3.18), and hence the TNt, scattering amplitude of Section 3 is reduced to F,, of Eq. (5.5). As we have pointed out above, it is very different from the nNr amplitude. In the weak binding limit 77-+ 0, on the other hand, we would expect that the 7iruf scattering amplitude is recovered. This can be verified for zero energy, but we have not been able to do so for finite energy.
6. THE PAULI PRINCIPLE EFFECT So far we have considered a “nucleus” which consists of only one nucleon, and have allowed the nucleon to be excited to all levels in the intermediate state. Now let us suppose that a few lowest levels are occupied by other spectator nucleons. According to the Pauli principle, the nucleon which is struck by the incident pion cannot go to the levels alredy occupied. For example, in n-l60 scattering the IZ= 1 level is forbidden for the s-state nucleon as an intermediate state. Examination of Eqs. (3.17) and (3.18) for tc2) together with the PadC approximant (3.19) reveals that this should push the resonanceenergy upwards. In the caseof n-l60 scattering, the TNb resonanceis found to occur at pion lab energy of 200 MeV. This is to be compared with 194 MeV in the IA(A) and 174 MeV in z-Ni, scattering with binding correction alone. When the Pauli principle effect is combined with the binding effect, the resonanceenergy becomesclose to that found in the IA(A). For the cross section, its effect is to decreasethe mNa total cross section below the resonance energy, whereas there is a significant increase above the resonance. For example, at pion lab energiesof 150 and 200 MeV, the TNb total cross sectionswithout and with the Pauli principle effect are 281, 192mb and 225, 322 mb, respectively. We noted in Section 2 that the separableTN interaction is similar to the more basic CL interaction so far as the binding effect is concerned. However, the results
162
BAJAJ
AND
NOGAMI
of the two interactions may be quite different with respect to the Pauli principle effect. The lowest order diagram in the CL interaction is modified by the Pauli principle, whereas the corresponding diagram for the separable interaction is not. Hence the effect is probably more appreciable for the CL interaction.
7. DISCUSSION It is clear from our analysis that the IA is a rather poor approximation in the neighborhood of the 33-resonance. The binding correction to the IA is indeed quite significant. The binding effect pushes the resonance energy downwards compared with that in the IA. The 7TNb total cross section near the resonance is about 50 - 70 “/o greater than the corresponding IA cross section and the elastic cross section shows a much stronger forward peak than that of IA. For instance, for Nb corresponding to 160, the nNb resonance energy is found to be 174 MeV with the total cross section at the resonance peak of about 290 mb, where as the corresponding figures in the IA(A) are 194 MeV and 176 mb, respectively. The basic mechanism for the downward shift of the resonance energy is that the nucleon mass is effectively increased by the binding effect. When the strength of the potential which binds the nucleon is increased, the nucleon becomes less and less mobile and finally in the tight binding limit the Z-N scattering amplitude reduces to that in the static limit in which m --, co. As we noted at the end of Section 5, the resonancedisappearsin this limit; instead, a TN bound state emerges.For the CL interaction the effect of the increasing mass is somewhat less dramatic. The resonanceremains in the tight binding limit, but at 17 MeV [20]. Of course, for a realistic strength of the HO potential, the effect is much lessdrastic. In addition to the binding effect, we simulated the effect of the Pauli principle by excluding some lower intermediate states. This effectively weakens the TNb interaction and hence the resonanceenergy is shifted upwards, for example, from 174 MeV to 200 MeV in the 160 case. Note that the effective strength of the z-N interaction increases with increase in pion energy (because of the momentum factor in the interaction). In discussingthe Pauli principle effects in nA scattering Landau and McMillan [21] found that the ETAtotal cross section decreasesaround the resonance energy (corresponding to .rrNr scattering), but increasesabove the resonance.They did not emphasizethis point. In their model they have also used a separableTN interaction. We now want to comment on some related works and compare our results with them, Kohmura [6] examined the binding correction to the IA for a model similar to ours, but he assumedthe n-A interaction to be separable, used the CA to sum over the intermediate nuclear states, and neglected the nucleon recoil in fitting the mNf scattering. We have pointed out that the n-A interaction is not separable, the
PION
SCATTERING
FROM
A BOUND
NUCLEON
163
CA is misleading, and the nucleon recoil is important. Schmit [7] considered a model much the same as Kohmura’s. He also used the CA and neglected the nucleon recoil in nNf scattering. His treatment of the binding effect is in the socalled “quasiclassical” approximation, which amounts to shifting the energy argument of the nN t-matrix from E to E’ = E - ZJ, U (< 0), being the binding potential of the target nucleon. Since E’ - E = 20 - 30 MeV, the resonance energy is lowered by the corresponding amount. However, the magnitude of the cross section is not modified substantially. The Pauli principle effect cannot be taken account of in this treatment. Note that our rNh calculation does not use the quasiclassical approximation. Myhrer and Koltun [9] have studied nd scattering as a three body problem in the 33-resonance energy region. They examined the effect of dynamical excitations of the target as well as of multiple scattering and nucleon motion. They find that the resonance occurs at p = 2 15 MeV/c in the nd CM system, whereas the 33-resonance momentum in the IA(C) with nucleons frozen in the deuteron is 205 MeV/c in the z-d CM. Thus the binding effect results in an upward shift of the resonance energy. The multiple scattering effect. which contributes a downward shift, is small for the deuteron and hence the net effect is an upward shift. They explain this result in the following way. The energy argument of the TN scattering amplitude is assumed to be shifted due to binding from E to E’, E’ = E + UN,,
(7.1)
where UNA is the average potential in the initial state which binds the nucleon. In the intermediate states the binding potential is small and is neglected, so that the replacement (7.1) implies an upward shift of the resonance by -U,“, MeV in nuclear medium. For the deuteron this shift is about 15 MeV. (Incidentally Schmit’s choice of energy replacement is E’ = E - U and differs from Eq. (7.1). The difference in the sign of the potential term changes even the qualitative nature of the binding effect. There is some confusion on this point in the literature.) In our calculation the nucleon is bound in the intermediate states as well as in the initial state, and it is not possible to interpret our results by means of Eq. (7.1) or the like. Hiifner’s [5] analysis or conjecture of the binding effect is similar to our basic idea which we have described earlier, i.e., the pion scatters repeatedly from one bound nucleon, which is excited while all other nucleons are spectator nucleons. But for the binding correction in the intermediate state, he argues that it is unimportant. It is not obvious whether the neglect of binding in the intermediate states is a priori justified. Moreover, how well can the binding effect be simulated simply by replacing E by an effective energy E’ as the argument of the nN scattering amplitude is also not clear. Julius and Rogers [I I] included the first-order binding correction to the TNr b-matrix by retaining the second term in Eq. (I .l). The resulting corrections to the
164
BAJAJ
AND
NOGAMI
optical potential are 50 - 100 % near the resonance and fall to negligible values at energies less than 120 MeV and greater than 450 MeV. Their Fig. 5 in which they plot the binding correction to the optical potential for ,rr-Y (Re V,,,(r)) indicates that Re V,,,,(r) vanishes at an energy slightly lower than that for the optical potential without the binding effect. Unlike their calculation in which binding effect is taken account of only up to second order, we do not make an expansion with respect to the binding potential. We have used the Pad6 approximant to obtain the 7TNbamplitude: the accuracy of the Pad6 approximant for various scattering problems is known to be generally good [22]. This is the only approximation in our calculation. We are aware of the unrealistic nature of the HO potential, but we believe that for the problem under consideration it provides us with a reasonable starting point. At least it has been used quite extensively in literature for similar problems. At this point we would like to note that the 50 - 70 % increasein the nNb total cross section as compared with that in the IA, near the resonance, will not necessarily increase the nA cross section by the same amount. Julius and Rogers [ll] have shown by meansof a simple model that the n-A cross section at the resonance peak is infact quite insensitive to u,,~ . For example, doubling the ~TNcross section increasesthe n-A cross section by ( 10 “/, at the resonanceenergy. But the effect of corrections will be much larger at energiesaway from the resonance peak. As we have seen above the binding corrections to u,,,, are negligible at energies 2 100 MeV so that the TA total cross section is unaffected at such energies. At energies >, 200 MeV the binding effect increases u,~ by 20 - 25 ‘A and then decreasesto negligible effect at energies >, 400 MeV. The n-A cross section will thus increase by about 20 2, at energies above 200 MeV, i.e., on the tail of the peak. Tt is in this energy region that the experimental values of n-A cross section are greater than what most VA calculations using the IA prediction. Our binding correction will thus increase these TTAcross sections resulting in better agreement with experimental data. Finally, let us briefly summarize the results of various calculations which have examined the effect of the Pauli principle by considering nuclear matter [lo]. In such calculations the initial and final wavefunctions are taken to be plane waves. Unlike our calculation, where the potential binding the nucleon modifies the bound state wavefunction, there is no binding effect in nuclear matter. Only the Pauli principle modifies the Z-N scattering amplitude in nuclear matter. Kimura and Nagashima’s [lo] calculation exhibits a very large upward shift of the resonance energy, by about 200 MeV for the CL amplitude. Bethe’s [lo] result, on the other hand, shows a downward shift of the resonance to w,. = 1.15 p. There are some differences in the two approaches, but they both consider the CL 7ri? amplitude. Dover and Lemmer [lo] predict an upward shift of the resonance energy due to the Pauli principle effect which reducesthe effective TN coupling constant. In their
PION
SCATTERING
FROM
A
BOUND
NUCLEON
165
most recent calculation, Weber and Eisenberg [lo] claim that the Pauli blocking in 7~,4 scattering has little effect at higher energies; the main blocking effect occurs below the resonance energy. It is obvious from these diverse results that the situation is far from having been resolved about the Pauli principle effects for nuclear matter. It is difficult to compare these results with our model calculation since the former completeIy ignore the binding effects.
APPENDIX
A
We will evaluate the lowest-order diagram for nN scattering in the Chew-Low theory when the nucleon is bound in a harmonic oscillator potential. The binding effect manifests itself through the modification of the propagator, and the vertices have extra factors coming from the bound state wavefunctions. Apart from the vertex factors which are common to free and bound cases, we have I=C
” OJ -
Fodk) Mk’) (w + w’ + q)
(A-1)
’
where F’,(k) is defined by Eq. (3.4) and the energy in the intermediate state is w + w’ + nv. After manipulations similar to those shown in Appendix B, Eq. (A.1) can be reduced to
The summation
for IZin Eq. (A.2) can be converted into an integration using
z.A wTnl =lrn4 e+exp(xe-9. Hence we obtain I=
Ia d[ exp[+b2k * k’(epen -
1) - &J].
0
(A-4)
In the tight binding limit 77-+ co, Eq. (A.4) reduces to (l/w) which is the propagator for the Chew-Low interaction with a static nucleon. In the weak binding limit 3 + 0, retaining terms up to c in (A.4), we obtain I(7) ---f 0) =
s0m de ev-Ha
+ K’)]/(u
+ r),
(A.5)
166
BAJAJ
where K’ = k * k’/(2m). order in f, we obtain
AND
NOGAMI
For finite but weak binding, retaining terms up to second
I = Jrn df (1 f (l/2) S”K’v) exp[-(Q2)(w
+ K’)]
0
=
CA.6)
1 w+K’
[ 1+
K’rl (w + K’)2 I .
The second term in the bracket is the binding correction
which satisfies (4.7)
B
APPENDIX
For the HO potential, the function -Fyo(q) takes the form = (4~Y2 +Gd
Ejyo6-d
(B.1)
K%(4),
where G,(q) = c, exp(--b2q2/Wq)n.
(B-2)
The coefficient c, is known, but we do not need its explicit expression. To do the m and 1 summations in Eq. (3.6), we start with the form factor for the Is-state F,,(q). Foe(q) = (0 j eiq” 10) = exp( -b2q2/4) = exp
-
;
(k2
f
p)]
[
f
$
( bZk2.
k’
)y
(B.3)
7X=0
Using Eqs. (B.l) and (B.2) this can be rewritten
as
F,,(q) = C (0 [ eik”r j V)(V 1 e-ik.r 10)
(B-4)
= 4~7 C (21 + 1) P,(cos 8) G,(k’) G,(k) ?L,Z = exp[-(b2/4)(k2
+ kf2)] f TZ=O
(b”k . k’)”
c (21 + 1) &P,(cos 2
0).
167
PION SCATTERING FROM A BOUND NUCLEON
Here and also in Eq. (3.6), I takes all values that are allowed for a given value of n. In the third line of Eq. (B.4) we have made use of the following relation for msummation. c Y,T,,(k’) Y,,,(i-i) r (21 + 1) P&k). 1,L Comparison
(B.5)
of Eqs. (B.3) and (B.4) yields
cI (21 + 1) &P,(cos
0) = (2” . n!)-’
COSR0.
u3.6)
APPENDIX C Consider the product
of projection
P(k’, k”) P(k”, k) = [9(k’ . k”)(k” . k”)(k”
- 3(k’ . k”)(k” Let us choose the coordinates
P(k’, k”) P(k”, k).
. k) - 3(k’ . k”)(o
- 3(ci . k’)(o = [(3(k’
operators
. k”)(k”
. k”)(o
. k)
. k) + (c . k’) /P(o
. k)]/p*
. k) + (k’ . k) ,Y2} + ia . (V2(k’ x k) - 3(k’ x k”)(k”
Cl) x k)
. k):/p4.
(C.2)
as follows.
k = k(sin (e/2), 0, cos (e/2)), k’ = k(-sin
(e/2), 0, cos (e/2)),
(C.3)
k” = k”(sin cycos /3, sin ci sin /3, cos a). Then we obtain k’ ’ k = k2 cos 8, k” * k = k”k(sin
(8/2) sin a: cos /3 + cos (e/2) cos a),
k” * k’ = k”k(-sin With these, the spin-nonflip p-4(k”k)2
(C.4)
(d/2) sin cycos /3 + cos (e/2) cos a).
and spin-flip
terms in Eq. (c.2) become
[3(cos2 (O/2) cos2 a - sin2 (e/Z) sin* a: ~0~2 /3) +
ia * (f’ x ic>p-4(k”k)2
cos
[l - 3{(sin 01cos /3)” + cos2 a}).
01,
(C.5) (C.6)
ACKNOWLEDGMENT We would like to thank Dr. II-Tong Cheon for helpful discussions on his calculation [15].
168
BAJAJ
AND
NOGAMI
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