Statistical theory of pion-nucleus reactions in the (3, 3) resonance energy region

ANNALS

OF PHYSICS

140, l-28 (1982)

Statistical Theory of Pion-Nucleus Reactions in the (3, 3) Resonance Energy Region* D. AGASSI

AND D. S. KOLTUN

Department of Physics and Astronomy, University of Rochester, Rochester, New York

I4627

Received March 26, 1981; revised October 23, 1981

A statistical theory of pion-induced nuclear reactions is presented, and developed for application in the energy region of the A( 1232) rcN resonance. The dynamical picture is based on three types of channels: z-scattering channels, A-channels, and absorption channels. The statistical assumptions lead to a transport equation for the A in the nuclear target, from which the inclusive cross sections for inelastic scattering and absorption of pions may be obtained. The theory is more general than various semiclassical approaches, but can be shown to go over to a Boltzmann-like equation in the appropriate limits.

1. INTRODUCTION Pion-induced nuclear reactions at several hundred MeV show features of very strong interaction. The elastic and inelastic differential cross sections to low excited states have diffractive angular distributions [ 1, 21. The angle-integrated elastic, inelastic, and true absorption cross sections are large, from hundreds of mb for light targets to barns for heavy nuclei (31. For energies close to that of the A(1232) nN resonance (3,3), the nucleus becomesopaque to pions, and cross sections approach their geometrical limits. From the point of view of multiple scattering theory, this behavior is expected, since the mean free path for a pion becomesless than a fermi (using A-’ = p,,u, with p,, the density of nuclear matter, and u the average XIV total cross section), which is short compared to a nuclear radius R (see [l, Chap. 4.11). This means that coherent waves should not penetrate far inside the target, leading to diffractive elastic and almost-elastic scattering. But a further consequenceis that the nonelastic cross sections should involve multiple scattering: an energetic pion which traverses the target should collide inelastically many times (-R/A) leaving the target in a multiply excited state. Since true absorption of the meson is a significant competing channel, some fraction of deeply penetrating pions should be absorbed, after multiple scattering. The purpose of this paper is to give a statistical theory appropriate to nonelastic pion-nuclear reactions in this energy domain. The statistical aspects will be suitable * Work supported in part by the U. S. Department of Energy. 0003-49 I6/82/05OOO I-28$05.00/O Copyright 0 1982 by Academic Press, Inc. All rights of reproduction in any form reserved

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for dealing with average features of those reactions which involve many interactions of the pion with target nucleons, exciting the target in many steps. Since details of the nuclear target are necessarily simplified and phase information is neglected in the statistical averaging procedure, such an approach can only be expected to work well for the more general features of the reactions, for example, the inclusive differential cross actions for inelastic scattering, or for pion absorption with fast nucleon emission. Our approach is to derive a transport equation for the meson-nucleus system, following a general method used in previous studies of statistical theory applied to nucleon induced reactions [4] and heavy-ion scattering [5]. The general approach is similar to that of [5], which we refer to as AKW. The nuclear degrees of freedom are treated statistically. Transitions induced by scattering interactions are averaged, leading to a transport (or master) equation, expressed in terms of correlation functions, for a density matrix for the system. From the solution to this equation, various inclusive cross sections are obtained by integration. In order to apply such methods to the present problem, we choose to describe 71. nucleus scattering in this energy regime in terms of a A-resonance propagating in the nuclear system. The A is excited by the incoming pion, interacts within the nuclear target, causing nuclear excitation. If the A ultimately decays, releasing a pion which leaves the target, the inelastic pion scattering is induced. If the A converts to a nucleon by collision with other nucleons, without emission of a pion from the target, then the reaction involves pure pion absorption. In either case, one or more nucleons may have been ejected from the nucleus during the propagation of the A (see Fig. 1). The description of pion-nucleus reactions as mediated by A-propagation in the nuclear target has been the basis of several models [6] of elastic scattering (and total cross sections) in recent years. Here the A-propagation is coherent, and the nuclear NUCLEON

EMISSION

.. .

.-* .-- . . .

.. . c NUCLEON

EMISSION

FIG. 1. Schematic representation of the physical processes included in the statistical work. Channels and transition operators are defined in Section 3. Horizontal transitions shown here) are neglected.

model of this (only V” is

STATISTICALTHEORYOF7h‘f

REACTIONS

3

excitation is limited to the first level of complexity: the d-hole states. The feeding of more complicated channels, including the absorption channels is usually treated parametricalty, as a medium modification of the position and width of the d. It is precisely what happens to the d after the initial excitation of the target that is our principal interest in this work. Models of inelastic scattering and absorption of pions, based on a d-propagation, have also been developed following semi-classical lines using statistical methods. Ginocchio [7] has developed a cascade code which treats the entire reaction as a series of elastic collisions in a Fermi gas. Hiifner and Thies [S] have introduced a Boltzmann equation, based on the samesemi-classicalpicture, as a simpler method of calculating the inclusive inelastic scattering and total absorption cross section, than the cascade model. These methods treat the pion propagation as entirely on-energyshell between elastic collisions with the nuclear target treated as a Fermi gas. Our statistical theory differs from these in its generality: we are not restricted to a gas model, and off-shell pion propagation is possible. If we make the appropriate approximations within our formulation, we can in fact recover the Boltzmann equation as a low-density and local limiting case, as we show below. Other authors have recently discussedgeneral statistical methods which lead to transport equations for high energy nuclear reactions, which are similar in general spirit, but not at all in detail, to our present work [9, lo]. Within a general statistical theory, the treatment of multi-step direct reactions by Feshbach et al., [ 111 bears a closer similarity to our method, although it is not developed with the same kind of reaction in mind as in the present work. The organization of the paper is as follows. Section 2 presentsthe derivation of a transport equation for scattering, following closely the methods of AKW, and serves as a summary of their results. In Section 3 the isobar model (A) for pion-induced reactions is introduced, and the appropriate transport equation derived, generalizing the theory of Section 2 to include three kinds of channel. In Section 4, a set of approximations is introduced which makes the schememore compact and amenable to calculation by reducing the problem to a one-channel equation for the d. Specific model forms for the one-channel transport equation are introduced in Section 5, and equations suitable for calculation are derived. In Section 6 we discussthe relation of our theory to other statistical formulations: specifically, to the Boltzmann equation method of [8]. Conclusions follow.

2. TRANSPORT

EQUATION

AND INCLUSIVE

CROSS SECTIONS

We review the statistical theory of AKW [5], applied to the scattering of a particle from a complex target. We reduce the scattering problem to a transport equation, from the solution of which the inclusive differential cross section for inelastic scattering may be directly obtained. We work in a channel representation. For the present we consider only a pion scattering from a nuclear target, so each channel is representedby a state of the nuclear target, denoted the s, and a variable for the pion,

4 which could be the we shall generalize the target, and pion The target states

AGASSI

AND

KOLTUN

position or momentum, as well as the isospin. In the next section the definition of channels to include those with d-resonances in absorption channels. s are eigenstates of the nuclear Hamiltonion HN H!v Is> = 6 Is>*

The complete Hamiltonian

(2.1)

for the pion plus nucleus is given by H=h,+H,+

I’,

(2.2)

where h, is the kinetic energy operator for the pion, with eigenvalues ok for plane wave pions of momentum k, and V is the pion-nuclear interaction. We consider the partial matrix elements of this interaction ~st=(SlVO

(2.3)

between nuclear states, which are still operators on the pion variables (e.g., Vst= VJr,) for a local n-nucleus interaction, in the pion coordinate r,). The basic statistical assumption, following AKW, is that the matrix elements (2.3) may be treated as Gaussian random variables, such that the first and second moments are given by (2.4)

where the bar denotes an average, which may be considered to be taken over an ensemble of Hamiltonians V, for a given set of states s, t, s’. We have assumed a vanishing mean (first moment); the second moment is specified by the quantity VStVl,, in which the line connects the two correlated matrices, which are still an implicit operator on the pion variables. The statistical assumption is based on the complexity of the target states s, for excitation gS sufficiently above the ground state that the density of states is very large. This becomes the case when the pion has given up some tens of MeV to the target, which can happen within very few collisions. For a particular Hamiltonian (2.2) the scattering equation for the pion-nuclear system can be written (co, + go - H) ) I,Y:“) = 0,

(2.5)

where &+’ is the complete scattering wave for a pion of initial momentum k,, impinging on the target ground state, of energy &,. It is useful to introduce the density matrix corresponding to this scattering solution, which we write as a matrix in the nuclear (channel) labels:

P,,=(sIplt)=(slW:+‘)(r/l~+‘It). Again, pSl is an operator with respect to the pion variables.

Q-6)

STATISTICAL

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5

For a statistical theory, we require the ensemble average of (2.6) denoted by pSl. One of the main results of AKW is that the average density matrix is diagonal: pS,I=plGsr, and th at it satisfies the following linear equation:

P;= &,p’,+c G,V%i+?s;,, G:7

(2.7)

(see [5, Eq. (6.2)]). We use a Green operator G,, which is defined in terms of a unperturbed Green operator Gy and an optical potential P’,, as follows: G, = G; + G;F;G,, G~=(w,+~~---ho+ir)-‘, q=c

(2.8)

vmkt,

where h, is the pion kinetic energy. Note that the equations for G, and q are coupled, and involve coupling of all channels, s. The operator p” in the inhomogeneous term of (2.7) is given by fi,, = 1&,+‘)#,+‘[, an operator in the elastic channel, in which 46” is the elastic optical wave, corresponding to the optical potential 7;: (0, - h, - 7;) 1fg”)

= 0.

(2.9)

We also introduce an optical Moller operator Q, for channel t; 0&G;)-‘G,=(o,+&-gt-h,)G,=(l+Y
(2.10)

It is now possible to rewrite Eq. (2.7) in the form of a linear transport equation for pt. Beginning with the commutator [(Gy)-‘,p;] (for t # 0) and using Eqs. (2.7) and (2.10) repeatedly, we obtain

In this form of the dynamical equation for P;, the right-hand side has three terms. The first, which is inhomogeneous, contains the initial conditions, through p’o. The second is a gain term, giving transitions r--, t, while the third is a loss term, giving transitions from the t-channel, through the optical potential “t’; of Eq. (2.8). In aer to turn Eq. (2.11) into a workable equation the dynamical ingredients (h,, Vt, Vst, G,) must be specified: this is the subject of the following sections. The result is a coupled integral equation for the density matrices P;, in terms of their kinetic variables (e.g., r,, r; for the pion). The information of interest is in the solutions P; of the integral transport equation, One quantity of immediate interest that may be obtained directly from the P; is the

6

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inclusive cross section for inelastic section may be written

AND

KOLTUN

scattering

of the pion. This differential

d’o k’ wow’ dn, 6(~%-~ + w’ - & - q,> I T,, I’, df2’ do’ = k, (2792 I

cross

(2.12)

where dn, is the phase-space volume for the target final states of energy of gS’,,and the final pion has momentum k’ = (k’, a’) and energy 0’. The f-matrix r,, in (2.12) may be written (2.13)

r,, = (k’, s I VI wb”),

where (k’, ~1 is a plane wave state in the final channel s, and vi+’ was defined in Eq. (2.5). (We normalize (k’, s’ ] k, s) = (27~)’ 6’(k’ - k) 6,,,.) We combine (2.6), (2.12), and (2.13) to write d’a k’ w,,o’ an, (k’,sl df2’ dw’ = k, (2792 ( ac%==, )

V,Vlk’,

s),

where the bar denotes an average over the degenerate target states s with energy gS = ?YO+ o0 - co’. The corresponding total cross section for pion absorption is given below in Eq. (3.17). The last factor in Eq. (2.14) can be connected to the solution P; of the transport equation, Eq. (2.1 I), if we make an ergodic assumption, namely, that the average over target final states is equivalent to the average over ensembles of Hamiltonians V. The ensemble average was used in our basic statistical assumption in Eq. (2.4). This equivalence may be justifies for a sufficiently dense and complex set of final target states, s. With this additional assumption, and following the approach of AKW, we find that (k’,s]

VpV(k’,s)=(k’1f2,~

i’,,p;Sj/k’),

(2.15)

where R, is defined in Eq. (2.10). Thus we can calculate the inclusive inelastic cross section from the solution of the transport equation.

3. TRANSPORT EQUATION FOR AN ISOBAR MODEL We now apply the statistical method of the previous section to a more specific dynamical model of n-nucleus scattering, which is appropriate to the energy region of one hundred to several hundred MeV z kinetic energy. The model is based on the formulation of nucleon isobars, in particular the d( 1232) resonance of the ~cN system, which then propagates in the target nucleus. This model, which is closely related to the d-hole models of r-nucleus elastic scattering [6], has several advantages for the present case. It has a simple structure which builds in the strong energy-dependence

STATISTICAL

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I

REACTIONS

of scattering in this energy domain, and allows for the inclusion of pion absorption in a simple way. We construct our model by introducing three kinds of channels: n-nucleus scattering channels, as in Eq. (2.1), which will be denoted by rc; d-nucleus channels, which will be treated as closed (no free d) and denoted by A; and n-absorption channels, in which nucleons are emitted from the residual target, which has no pions or A remaining, and denoted by ab. As in most such models, the A channels serve as a “doorway” between the pion elastic channels, and all other reaction channels; both inelastic scattering and absorption are coupled through the A. It is convenient to introduce a matrix notation for operators in the three-channel space (rr, A, ab). We write H = H,, + V, with H, diagonal in the channel matrix space:

Ho =

h,+H, 0 0

0

0 h,+H; 0

0 h,, + H;

(3.la)

and

v= (F

T

ibj.

(3.lb)

The zeros in (3.lb) correspond to the “doorway” assumptions. The interaction h, + HN in (3.la) corresponds to that given in Eq. (2.2) for the n-nucleus scattering channels, along with I’” of (3.lb). We now label the eigenstates of H,v by IA) for the target system (of A nucleons), instead of 1s). (See Fig. 1.) The interaction in the A channel space has been divided into h,, the one-body Hamiltonian of a A bound in the target, Hh, the interaction of the residual nucleus (of A - 1 nucleons), and VA, the interaction of the A with the residual target. The eigenstates of Hh are denoted by la). Similarly, in the absorption channel we take the interaction to consist of habr the kinetic energy of emitted nucleons, HG, the residual target interaction (with eigenstates la)), and I@ the coupling between specific absorption channels. Since our model of absorption is based on the two-body reaction A + N+ N + N, we shall assume that there are two nucleons: h,, = h, + h,, and la) is a state of (A - 2) nucleons. The operators J and Jt couple the 71and A spaces, while j and jt connect the A and ab spaces. We shall use the partial matrix elements of these, between nuclear states, as in Eq. (2.3) J:,=(A/J+Ia), J‘A, = (a 1j+ 1a). These objects are still operators

(3.2)

on the remaining degrees of freedom in the channel-

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state spaces: for A, in the position or momentum

of the rc, for a, the variables of the

A, and for a, the variables of two nucleons (and also, spins and isospins). In this and

the following section we use the position representation, denoting the n-position by 5, the A-position by x, and the two-nucleon-positions (ab) by y,, yz. Thus the full matrix elements of Jt andjt in this representation are written

(SlJf;aIx>

(3.3a)

and (3.3b) The operator Jt is a one-baryon operator, transforming a single A into a single nucleon plus a pion. If we introduce a complete set of single-nucleon states, labelled by S, we can write the amplitude for the elementary transition in which a A at point x emits a pion at point r, leaving a nucleon in state 6,

u,*G x>= (t, 6IJ+ Ix>, where we have use the position representation. Then the full many-body element (3.3a) of Jt may be expressed as a sum of factors

(Sl Jk Ix> = 5 @ I a, 6 4% 4.

(3.4)

matrix

(3.5)

The coefficients (A I a, S) are overlap integrals for nuclear states, whose absolute squares give the conventional spectroscopic factors S$. The j are two-baryon operators which transform a A and a nucleon into two nucleons. If we start with a A at point x and a nucleon in the single particle state 6, the transition to two nucleons at positions yi, y2 may be written

~,(~,,~,;x)=(~,,y,Ijlx,~).

(3.6)

The full matrix element (3.3b) of jt may be expressed

Wd, Iyl, Y*>= C (a I Q,6) GYY~,y24. Again (a ) a, S) is a nuclear overlap amplitude for removing one nucleon 6 from state a to reach state cr. Note that one nucleon (in the ab channel) comes from the A, and one from the nucleon in state 6. The functions u*, u* represent the elementary dynamics of our model, and will be discussed further in the following section. The statistical assumptions are made with respect to the overlap amplitudes in Eqs. (3.5), (3.7), based on the complexity of the relevant nuclear states. We treat these amplitudes as gaussian random variables, so that the averages (over ensembles of nuclear interactions) have the form

STATISTlCAL

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9

REACTIONS

(A 1a, 6) = (a ) a, 6) = (A 1a, @(a’ 1a’, 8’) = 0, (A I a, Wa’~

I A’) = ~A,AsJa,at&,st

I@ I a, S>l’,

(3-Q

(a I a, @(a’, 6’ I a’> = &,al~a,nl~6,6~ I@ I a, +I’.

This random variable assumption then leads to the following ensemble averages for

Jk,jk

and also -~V”AA’ = VAaa’ = pbmu’- - 0 *

The quantities Et andz+ are the correlation functions which determine the statistical transitions of the system, following the method outlined in Section 2. Note that these are still operators in the sense mentioned after Eq. (3.2). We introduce the averaged density matrix p, which is diagonal in channel space (see discussion below Eq. (2.6)) and therefore takes the form p=

p” 0 i 0

0 p” 0

0 0

.

(3.10)

pa” 1

It is now a straightforward matter to obtained a linear transport equation for the density matrix P; following the procedure given in Eqs. (2.7) to (2.11). In terms of the three channel spaces: 71,A, and ab, the resulting equations have the form of three coupled equations in p”, PA, and pb. It turns out to be possible to eliminate p” and /Fb from the equations, leaving a transport equation for the p” in the d-channels only. This is also a convenient choice from the point of view of later approximations, sincee the A can be considered relatively localized, being heavy (compared to the pion). The resulting linear equation for pA may be written in a form similar to Eq. (2.11):

- hermitian conjugate.

(3.11)

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The Green operators in Eq. (3.11) are obtained from Eq. (2.8) by defining an optical potential for each channel (x = z, A, ab): G(X) = Go(X) + @(X)7-(X,G(X) I t 7 I I I

(3.12)

where GF’“’ are defined as in (2.8), using the definition of Ho in Eq. (3.la). Note that the first right-hand term of Eq. (3.11) is determined by the initial conditions, through the density matrix p; for the incoming pion in the elastic scattering channel. The second and third terms of Eq. (3.11) are gain terms, by which probability is fed into channel a, while the fourth and fifth terms give loss from channel a to the pion and absorption channels, respectively. The total probability Tr p is conserved, although the pion probability will be depleted in part by the absorption channels. The linear equation (3.11) becomes a coupled integral equation for the density matrix jr:, when we express this matrix in terms of variables for the A, say the position, pf(x, x’) = (xl DA Ix’). It will be more convenient later to introduce the Wigner transform of this matrix, defined by f’,(X,K)=(2n)-3J’d3re-iK’~;

(X++-+).

(3.13)

This quantity is real, and represents an average distribution in position X and momentum K. It is a quantum mechanical analog of the classical probability distribution function in the phase-space of a particle. Carrying the Fourier transform of (3.13) through Eq. (3.1 l), we obtain a transport equation for P,, of the form +V

- K - VU(X)

. V,

P,(X, K)

A

= Z,(X, K) + j dx’ dK’ - WcL’(XK. 3X’K’) a

I

c W$‘(XK;

X’K’)

P,(X’,

K’)

b

P fl(X’ 9K’)

I

.

For the left-hand side of this equation, we have assumed that + U(X)

(3.14)

STATISTICAL

THEORY

OF

r-A

11

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with a real, local single particle potential of the size of the target. Terms of higher order than VU(X) are neglected. The terms on the right-hand side of Eq. (3.14) are the inhomogeneous terms I,, gain and loss terms, with kernels WCC’ and l@), respectively. These quantities are given by: I,(X,K)=2(2z)-31m W$‘(XK;

I

drePiK’I

X+$ (

JmiOG,dtIX--$ I

, i

(3.15a)

X’K’)

= 2(27~)-~ Im 1 dr dr’ exp i(K’ . r’ - K . r)

(3.15b) WF’(XK;

X’K’)

= -24(27c)-3

x 7‘;

(

Im 1 dr exp(-zK . r) exp(iK’ . [2X’ - 2X + rl)

X+?,2X’-x++

3 )

(3.15c)

where 7~~(x,x’)=(x(iyIx’).

(3.15d)

Explicit forms for the quantities in Eq. (3.15) require not only information on the interaction correlation functions m,y+, but also on the Green operators and optical potentials given in Eqs. (3.12), which are really a set of dynamically coupled equations for the GCX)and Y’^(“). Short of complete solution of the dynamics, it will be necessary to make approximations which lead to more tractable expressions for the functions in Eq. (3.15) which define the transport equation of Eq. (3.14). Solution of the latter as an inhomogeneous linear integral equation yields the Wigner density function P,(X, K) for the A in the target, and by inversion of Eq. (3.13), the density matrix p$ itself. From this matrix, and using Eqs. (2.14) and (2.15), we may obtain expressions for the averaged cross sections for scattering and absorption of pions. The inclusive differential cross section for inelastic scattering takes the form

Note that because of the doorway assumption of our model, expressed in (3. lb), we require only the density matrix for the A to find the piun differential cross sections.

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Similarly, partial or integrated cross sections for pion absorption expressed in terms of pA; we give the integrated cross section:

may also be

(A-2) + E(Pi) + &(P2) - ag”’ - wO)

and s(p) = p2/2M. As mentioned above, the transport equation conserves probability, so that the cross sections calculated from PA will obey the usual unitary relations. If we add the integrated scattering cross section crScobtained from (3.16) and crab from (3.17), we obtain the total compound (or fluctuation) cross section (TV= (fc + Gab,

(3.18)

UT = 0” + UC,

(3.19)

The total cross section is given by

where uop is the cross section for elastic scattering given by the optical potential in the initial channel, in this case by ‘F-t of Eq. (3.12). (In a statistical theory, some of the elastic scattering is included in us’; this is the compound elastic or fluctuation scattering.) Now, the total compound cross section can be calculated from the antihermitian part of the optical potential, through the relation

where the velocity u. = k/o,. In a unitary theory, the total compound cross section calculated by (3.20) will agree with that of (3.18), calculated from Eqs. (3.16) and (3.17). The unitarity of the present statistical theory is shown in Appendix A. We shall insist on the constraints of unitarity in formulating approximations in the following sections. 4. APPROXIMATE

ONE-CHANNEL

EQUATION

FOR THE ISOBAR

The behavior of the transport equation (3.14) is controlled by the kernels and inhomogeneous term defined in Eq. (3.15). Two of the basic dynamical quantities on which these depend are the correlation functions .?? andJy+, given in Eq. (3.9). We shall specify these in terms of simple models in the following section. However, the correlation functions determine the form of the transport equation, only through a set of coupled dynamical equations (3.12), from which we may obtain the Green operators and optical potentials that are also required for the kernels.

STATISTICAL

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13

In the present section we show that it is possible to circumvent the problem of coupling, and simplify the determination of the transport equation, if certain quantities in the theory are of sufficiently short range. The approximations lead to a transport equation which only involves the variables of the d; the influence of the II and ab channels enter through independently specified functions. The method of approximation is designed to preserve the unitary structure of the equations, so that the expressions for the scattering and absorption cross sections, given in Eq. (3.16) and (3.17), remain valid under simplification, and flux is conserved throughout (see Appendix A). First we remove the explicit dependence of our equations on the variables of the pion or absorption channels. We write the matrix elements of the interaction operator fl of Eqs. (3.2) and (3.3a) in the following form: =JLwf*(L

xl.

(4. la)

The factor f(& x) gives the dependence of the pion coordinate relative to the position x of the d, and is assumed to be independent of the nuclear states A, a, as would be the case for any one-nucleon vertex operator Jt, as in Eq. (3.5). The function f could also depend on derivatives of either variable, representing momentum-dependent coupling. This will be treated explicitly in Section 5. Ignoring this possibility here, translational invariance gives f = f(c - x); the fourier transform of this function is the normal vertex-function or form factor in three-momentum transfer. The range of the function f is given in various theories of the d-resonance, and is presumably no larger than 0.7 fm. In a zero-range limit,f(c - x) + S3(c - x). A similar factoring of the interaction operatorj+ of (3.2) and (3.3b) takes the form (4.lb)

(XIjj;uIY1,Y2)=jota(x)h*(x,Yl,Yz).

The interaction operators appear in the kernels (3.15) of the transport equation in the combinations (4.2)

and their hermitian conjugates. By using the factored forms (4. l), we are now able to integrate over the variables of the pion, or the two nucleons of the absorption channel, so that the combinations (4.2) can be written in terms of the d-variables only:

(x’ )JaAG,“J,tb1x) = JaA(x’) g;(x’, X) Jib(X),

(4.3a)

cx’

(4.3b)

idaGzbjab

1x>

=jLm(x)

g:b(x’9

x> jab(X)T

where

g:(x’, x) = f 8’ d5SW, 5’) (t’ I G:, IS>f *(x, 5)

(4.4a)

14

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and qb(x’,

x) =

dy; dy; dy, dy, h*(x’,

I

Y;I Gib 1~1, ~2 4x, ~1, YZ).

y;y;)(y;,

(4.4’~)

Now the kernels of Eq. (3.15) may be expressed entirely in terms of the d-coordinate. (See Appendix B.) Second we simplify the expressions for the terms of Eq. (3.15) by approximation, as follows. Consider first the inhomogeneous term I, ; Eq. (3.15a); we write the integrand, using the notation of (4.3a) and omitting the exponential factor

I dzJ,,(x)&Xx, 4Jot,W(zI G:+lx’), I ’ where x = X + fr and x’ = X - fr. If we move the operator J,,(x) (4.5), we obtain the following combination of factors in (4.5):

(4.5) to the right end of

(4.6) A combination

very similar to (4.6) appears in the expression I U,“+(x, x’) = c J,Jx)(x a

1 1G: + (xl) &(x’),

(4.7)

which is defined (in its adjoint form) as the pion optical potential in d-coordinates, and which is related to 7-z of Eq. (3.12) (with A = 0 here) in pion coordinates by the transformation (Cl 7;

It’) = j d3x d3x’ f*(&

x) U$(x, x’) f(e’,

The similarity of (4.6) to (4.7) can be exploited obtain: first, that x = x’ for the combination (4.6), (4.6) on the state a be very weakly dependent on these approximations for (4.6) into one expression, I

.$,(z)(z)

x’).

(4.8)

to simplify (4.5), if two conditions and second, that the dependence of the position variables. Combining we write:

1

G;+ lx’) JaA(x) ‘v d,(x - x’) RAa U;+(z, x’),

(4.9)

where d,(x - x’) is a short-range function normalized to unity, and R,, is a positionindependent matrix, with C, R,, = 1. (We have given the approximation for any value of A; A = 0 applies to (4.5).) Applying (4.9) to (4.5) we obtain a simplified expression for the homogeneous term (3.15a): Z,(X, K) = 2(27r)-3 R,, Im 1 dr epiK.‘dJ(r) X++,y)

U$(y,X-+).

(4.10)

STATISTICAL

THEORY

Z-A

OF

15

REACTIONS

How do we justify the approximations leading to the relation (4.9)? First, it is clear that we can approximation x N x’ in (4.6), if both of the following short-range relations obtain:

mz>&A(4+0

and

(z[G:+lx’)#O

for

Iz-x/
(4.11a)

for

Iz-x’I
(4.1 lb)

where CJ and I are length parameters, both of which are small compared to the dimensions of the nucleus. The parameter u denotes the correlation length for be expected to be of the order excitation of the nucleus, which would 0.7 fm. The length A is the mean propagation distance of the A, which can aLk;‘0.5 fm. Therefore we expect that be estimated from the decay width to be L which is indeed a short distance in the nucleus. The lx - x’ 1< 1 fm in Eq. (4.6), separation of dependence on A from space-dependence follows from assumptions of our specific model, as will be shown in Section 5: see particularly Eq. (5.7). The Gi will depend more weakly on a than will the correlation function; see, e.g., (5.12). We may use (4.9) to make a similar simplification to the expression (3.15b) for the kernel Wzb, in which the combination (4.5) also appears (with A # 0). Additional simplifications obtain by using the optical Msller operators (2.10) for the z-channel, transformed to the A-coordinates (see Appendix B): FA” = g;wA”

and

.p+U+=w”.4 A

A

13

(4.12)

where g; = f Gpf ‘. We now use (4.9) and (4.12) to rewrite the part of the integrand (4.6) of the gain term kernel WL:‘, which pertains to the coupling to the z-channels. Similar forms to (4.9) and (4.12) may be obtained for the part of the integrand of Whz’ which relates to the absorption channels, based on a short-range approximation for 17’ analogous to (4.9) and (4.11). Using these approximate forms in (3.15b), we obtain W;;‘(XK;

X’K’)

= q~~j-3 lrn

d,, drt ,i(K’.r’-K,r) .I

16

AGASSI

AND

KOLTUN

Note that in the zero range limit (d(r) + S”(r)) FVG) becomes independent of K. The loss term IV) which is expressed in terms of the optical potential Yt in the Achannel in Eq. (3.15c), can also be written in the form w;‘(XK, =-24(2n)-3

X’K’) Im

dre-[K.I+iK’.(2X’-2Xtr) I

(4.14) which is not further simplified by the short-range approximations. The result of the short-range assumptions is a reduction in the complexity of the kernels (3.15) of the transport equation (3.14). The various integrals are now to be calculated in terms of the following independent quantities: 1

dbtx)

I

JbA(X’),

(xl (xl

LTA” g”,”

lY>9

@xx9

Y),

lY>,

w:bb(x,

Y),

(4.15)

which are given as functions of the A-variables only. The correlation functions fi and]?+ may be given by a model of the medium; we do this specifically in the next section. The free Green functions g”, gab may be calculated directly fromA2.8). The Moller matrices c9, UP* do depend implicitly on the correlation functions JtJ and]?+, through the optical potential. However, they are also determined by the interactions within the 7c-and &channels (V”, Vab) as well. It is therefore reasonable to treat We and uPb as independent quantities in the set (4.15). What makes this possible is the fact that the form of approximation given in Eqs. (4. lo), (4.13) and (4.14) maintains the unitarity of the transport theory, even with the w chosen independently of the FJ and yj (see Appendix A). This means that the cross sections for inelastic scattering and for pion absorption (3.16) and (3.17) calculated from the transport equation using the approximations just given throughout, will be consistent with unitarity, such that the loss of flux from the elastic channel is exactly compensated by the flux out. It is for this reason that it is necessary to calculate the loss term in the form given in (4.14), rather than in terms of an arbitrary optical potential Ut . (Note that iZJg in (4.15) is determined by g$ and 0:. In fact, the constraint of unitarity only requires that the flux loss be properly related to Im U{, as in Eq. (A.3).) Thus we have accomplished the goal of this section, to reduce the coupled-channel transport equation of Section 3 to a one-channel problem, with simpler kernels given in terms of the six independent quantities of (4.15) but maintaining the unitarity of the theory.

STATBTICALTHEORY 5. A MODEL

FOR THE ONE-CHANNEL

OF R-A REACTIONS ISOBAR TRANSPORT

17 EQUATION

The approximations of the previous section provide us with a transport equation for the d, given in Eq. (3.14), where the kernels of the integral equation are given in Eqs. (4.13) and (4.14), and the inhomogeneous term in (4.10). These, in turn, are to be calculated in terms of the correlation functions, Green functions, and Msller matrices of Eq. (4.15), which consequently control the dynamics of the transport equation, and, in turn, the reaction cross sections of Eqs. (3.16) and (3.17). For example, the width of the d propagating in the nuclear medium is controlled, in our formulation, by the Msller matrices w” and cPb, which give the damping of the Achannel into the pion and absorption channels, respectively. The p-wave aspect of pion scattering in the A energy region should appear in the form of the correlation function .!0. In this section we take specific forms for the various quantities required, e.g., in Eq. (4.15). We follow the general physical features that we know for z interactions in this energy domain, but simplifying all algebraic or functional complexity that we believe is washed out by the statistical aspect of the transport process. The resulting quantities are assumed to have simple functional form, with a small number of adjustable parameters on which the model depends. In Section 3 we gave the matrix elements of the transition operators J, j in position representation. Here it is more convenient to use a mixed representation with position x for the d, but the momenta k for the pion and p,, p2 for the nucleons in the absorption channels. We rewrite (3.5) and (3.7) in the form

(k I Jk, lx> = c (A I a, d>$Yk, xl, (xlj:.,p~~~)=~jbl~,~)~~(x,~lh).

(5. la) (5.lb)

For our model we take the function J8 to have the simple factored form (5.2)

appropriate to a static nNo A transition, where #s(x) is the orbital state for the nucleon, S is the spin transition operator (see [ 121) which takes the nucleon spin (l/2) into the A spin (3/2). The operator Q3,* projects isospin 3/2 for the ZN system, and is given by Q3,* = 2/3 + l/31, mr. The strength of the IrN et A transition is g, and the form factor F(k*) gives the momentum dependence of 51 which we expect to be slowly varying for (kl less than some value, of the order of the nucleon mass, and to drop of rapidly above that value. The S . k factor represents the p-wave rrN interaction; the linear k dependence in (5.3) leads to a vertex function f(& x) of Eq. (4.1) with a gradient in 5. For the absorption function& we take as our model form .&(x9 pIp2) = CQ,Wlp,

- p21)h(x) ei(pl+h).x,

(5.3)

18

AGASSI

AND

KOLTUN

where C is the strength, and L3 a form-factor for the relative momentum of the nucleons emitted in the absorption process, which we would expect to be a slowly varying function of ] pi - pZ ], reflecting short range or high momentum transfer in the AN c) NN transition. (We normalize D to unit integral.) Again we assume the xdependence is given by the functional form of the nucleon orbital $s(x) (and the exponential), as in any impulse approximation model. We have treated the interaction as S-wave, averaging out any angular dependence. The projection operator Q,limits both AN and NN to S = 1, T = 1 states, as required for AN H NN, with spin and isospin invariance. We need the free Green functions g;(x, x’) and gz!‘(x, x’), transformed to the Acoordinates, as in Eqs. (4.3) and (4.4). We use a relativistic (modified Kleinand a nonrelativistic form for the A. Gordon) form for the pion propagator, Integration over the pion or absorption-nucleon momenta is then easily performed using the factored forms of the transition (vertex) function (5.2) and (5.3): ik.(x-x’)kZF2(kZ) ga”(”

“)

=

j

dk

(E

TEA)2

x’) =

j

dp, d3p,

_

k2

_

mT, +

ei(P,+Pz).(~--x’)D2(lp,

g:%,

E -E,

(5.4a)

jr’

- (A2/2M)(pf

_

p21)

+ pi) + iv ’

(5.4b)

The correlation functions for the A-channel are obtained from the transition operators (5.7), using the statistical assumptionsgiven in Eqs. (3.8) and (3.9):

(5.5b) where the numerical factors come from spin-isospin sums over states, assuming N = 2 for simplicity here. Except for these factors, and for the coupling strengths g2 and C*, the correlation functions are characterized by the structure of the nuclear medium, which we approximate by some simple considerations To specify the spectroscopic factors in (5.5) we adopt an exciton model of the nucleus [ 131. Each nuclear state is first characterized by the number of particles and holes relative to the ground state of the target Fermi gas. The exciton number is the sum: particles plus holes. There are a large number of states of any given exciton number. After averaging, only the energy of the state remains as a relevant quantum number (and the charge, which we shall not specify). We therefore label states in the various channels only by exciton number and energy, as follows:

A: a: a:

M, E,, m, E, P? Em

(5.6)

STATISTICAL

THEORY

OF

x-A

19

REACTIONS

with exciton numbers M, m, ,u, and energies E, , E,, E,. We then assume that the statistically averaged spectroscopic factors take simple forms given by the densities of states, conditions on the exciton numbers and energies. After some analysis and approximation within the exciton model, which we do not give here in detail [ 141, we obtain the following forms for the correlation functions J,,(x>jatA(X’) =P,‘(&A) 1 Lb> ~aa(x’> = PW’W

4&m-*4&.4

- 53) WV x’),

(5.7a)

4n,,- 1&J

- CJ CCG x’),

(5.7b)

where U(E) is a smooth function giving the spread of values of E = E, -8, around (Ed), and where (Ed) is an average value of the orbital energy below E,. Based on the Fermi energy E, and corresponding momentum k,, we assume a gaussian form for U(E): -(fa)Plm

e-” U(E)=

(5.8)

m

with (E*) - 30 MeV and r- 30 MeV. The function C(x, x’) should be only weakly dependent on energy E; we neglect the dependence, and take the simple form (5.9) where the first factor is characterized by a correlation length [T, with we would expect to be of order k;‘. The second factor Q simply restricts the average position to within the nuclear target, and should have the functional form of the nuclear density, and be normalized to give the number of target particles; J” dx Q(x) = A. The coefficient N is approximately given by N - AE/E~ with AE - 5-10 MeV. The function Pi is the average density of states with energy E, and exciton number M, for the target. The remaining factors we must specify for our model are the Msller matrices w,; and ozb of Section 4, which are defined in terms of the channel optical potentials Vi and U”,“, in Appendix B, Eqs. (B.4) and (B.5). We assume that the nuclear medium is uniform with density p,, neglecting density variation, and particularly surface effects, for the evaluation of these terms. Then we may assume the potentials to be independent of position, given by complex functions of energy U”(E) and Uab(e); the energy E is that carried by the n, E = E - EA, ic the n-channel, and that carried by the nucleons E = E - E,, in the absorption channel. For the pion potential we may put Im U=(E) =

hw%) 2.

,

(5.10)

k

where Go is the RN total cross section in the medium. It is here that the A-resonant behavior of the ZN system enters: the cross section can be specified by a resonant

20

AGASSI

AND

KOLTUN

function of energy, with parameters to give the position and width. These parameters may be taken as depending on the nuclear medium, e.g., through the local density, as in [6, 81, and therefore on the central density po. With a potential independent of position, the Msller matrices in the pion channel take the form oY(X, x’) = S3(x - x’) + UZ(&) gT(l x - x’ I)

(5.11)

with the Green function given by

g:w-‘l)=-~

wk exp &z(e) 1x - x’ 1 lx--x’1

(5.12a)

and ICY = k2 + 2~0, U(E), e=wK-rnx,

oi=k2+m2,.

(5.12b)

The form given in (5.11) will obey the extinction theorem of scattering theory, replacing a plane wave of momentum k by one of complex momentum K(E). The treatment of the Moller matrices for the absorption channel is somewhat more problematical. The form of the Green function given in (5.12a) is not valid for the propagation of two nucleons, without a fixed mass in the two-nucleon c. of m. A better treatment would involve using Eq. (5.4b) in Eq. (5.11). However, since the nucleons have a longer mean-free path than the pions and with less energy dependence, the details of propagation in the channel may not be so important to the results. This concludes our outline of a physically reasonable model for the one-channel transport equation. The dynamical input is given in terms of a small number of parameters which fix interaction strengths and ranges, average optical properties of the pion and absorption channels, and average properties of the target nucleus. These dynamical properties of the target are included in the energy spreading function (5.8), and correlation function (5.9). We have not yet applied this model to calculation of the kernels of the transport equation, although this would be a straightforward matter. Numerical solution of the full transport equation in three dimensions is a nontrivial problem of calculation, however, and may well require further approximation.

6. OTHER MODELS FOR PION REACTIONS The statistical formulation developed in Section 2 is rather general and can be applied somewhat differently to the the problem of pion induced reactions than in the isobar transport model developed in the preceeding sections. In particular, we could have decided to follow only the pion in the target, rather than the A, as in Section 4.

STATISTICAL

THEORY

7c-A

OF

21

REACTIONS

In our three-channel model this is quite cumbersome since it is impossible to eliminate analytically both the A and ab channels. However, for higher pion energies for which no particular resonance dominates it is reasonable to treat together all nonpionic channels as a single set. This leads us to consider a two-channel model in the present section. We shall show that this kind of model underlies the recent work of Hilfner and Thies [8); we obtain their Boltzmann Equation as a limiting case of the two-channel model. The two-channel model is formulated along the same lines given in Section 3; in analogy to Eq. (3.1) we write the hamiltonian in (2 x 2) matrix form: (6. la) (6. lb)

We have lumped the A and ab channels of (3.1) into one channel-space, containing a D-particle or “pseudo-A” plus (A - 1) nucleons. The eigenstates of H,, in the Dchannel (labeled i, j...) have overlaps with the A and ab channels, which provides a decay or absorption mechanism. The statistical assumptions of Eq. (3.9) are now replaced by new assumptions of the same form, in the new channels: JAi = 0,

Jia

= 6,6,, >JTJIi

(6.2)

and yx7i AR!= v,=o

(6.3)

(although we could also assume nonzero averages for (6.3) with little change). Note that these assumptions are not equivalent to (3.8). Now we write Eq. (2.11) for p: in this representation:

with 7-i and GP as in Eq. (3.12) modified to two channels. We then rewrite (6.4) in the form of a transport equation for the Wigner function P,(X, K), using (3.13) for p;, +K

. VP,(X,

K) =i

dx’

dK’

n

]x

.W$(XK;

X’K’)

PB(X’,

K’)

B - W’,L’(XK;

X’K’)

PA(X’,

K’)

. I

(6.5)

The kernels WCC) and IV’ are integrals of the matrix elements of the operators on the right-hand side of (6.4), as in Eq. (3.15). There is no inhomogeneous term; the initial conditions are in the elastic density P,,(X, K).

22

AGASSI

AND

Let us write the gain term kernel W”’ elements W’,C,‘(XK; X’K’)

X IF

KOLTUN

as an integral over momentum-space

matrix

= 2(27~)-~ Im ( dq dq’ eics’x-s’.x’)

(K+~~):iG~~i~

lK’+~)(K’-fl~~iGP’3i,,G;‘,K-~)i.

(6.6)

We now consider a number of simplifying approximations, which lead us to the form of transport equation proposed by Hiifner and Thies. First we assume the target can be treated as a cold uniform Fermi gas of nucleons. In this case G;+ is diagonal in momentum: G,“t(K - q). Now consider the matrix element

in (6.6) which act as transition amplitudes for scattering a pion of momentum (K’ f fq’) to (K f fq) while the target goes from B to A. For Fermi gas states, the nuclear transition involves one nucleon in momentum state pe recoiling into state pA . Momentum conservation requires K + $9 + pA = K’ + fq’ + pB+ Similarly, for the adjoint matrix element in (6.6), we have K - iq + pa = K’ - fq’ + pe, so that q=q’.

Now we assume that the matrix elements (6.7) depend only on momentum so that (KfqlJ~iGf’J,IK

fq)=(KIJJiGf’J,IK’),

transfer, (6.8)

where we have used q = q’. This property is equivalent to locality in x-space, i.e., w h’rch in turn would obtain if the J were zero range (no (x~~GDJlx’)~63(x-x’), form factor: see Eq. (4.la)) and GD were also of zero range: e.g., a static A-no recoil. This could alternatively be considered as a high energy assumption, for K %-q, with q -R-l for R the target radius. In the same spirit, we will take Git(K - q/2) N Gzt(K). Combining the Fermi gas and high energy (or static A) approximations, we may now rewrite Eq. (6.6) as WiT(XK;

X’K’)

= Im

I

dq eiq’(x-x’)

c ((K, pa 1JtGyJ i

IK’, pB)12

x [ 1- @,)I n&J @WI d3W+ pa -K’

- P&

(6.94

where we have factored momentum conserving &functions from the amplitudes (6.7) and integrated q’. The absolute square in Eq. (6.9a) represents the quantity r

1% Pa I JtGiJ IK’ The q-integral

PB)I* z

,

I

1

WI JzA,iGIJi,p,IK’)(K’ IJiB,iGTJi,pAIK)* (6*9b)

will give (27r)3 6(X - X’).

The factors n(p,),

n(p,)

are the cold

STATISTICAL

THEORY

OF

n-A

23

REACTIONS

Fermi-gas occupation numbers for the initial and final states: partial depletion is neglected. We shall also assume that the pion propagates with free kinematics, that is, GA” N_ G$ in Eq. (3.9); this is a low density approximation. The contribution to (6.9a) will then take the form Im G;‘(K)

N &(w,

(6.10)

+ e0 - E, - o(K)),

which guarantees conservation of energy in the gain term of (6.5). It is consistent with this assumption to keep energy conservation in the Wigner probability function for the pion f’,(X

K) = f(X

K 4,) 4%

- E, - dK))/W,

(6.1 I)

>

where E, = w,, + e0 is the initial energy of the system and W, is the density of states aNA/&, . This gives the semi-classical form of transport theory, for which total energy is conserved at every step. The more general approach used in our method does allow for off-energy-shell propagation. The assumption (6.8) that the scattering amplitudes are local implies that the optical potential 7-2 should also be taken to be local: T‘,“(x, x’) = 7 ‘A” X + +, X - +)

= “/ ‘“(X)

d3(r).

(6.12)

We also neglect the state dependence (this is similar to the assumption in (4.10)). Inserting this result into (3.15~) leads to a simple expression for the kernel WcL’ of the loss term in (6.5), Wy’(X,

K; X’, K’) = -2 Im Y-“(X)

S’(X - X’) 6’(K - K’),

(6.13)

which is independent of K, and therefore also of the state A. The transport equation (6.5) can now be written in terms off(X, K, E,), using Eqs. (6.9), (6. lo), (6.11) and (6.13) integrating i dp, and summing spin projections, to obtain ~K.V~(X,K,E,)=!.~K’~(X,K,K~)/(X,K’,E,) R + (2) Im y-“(X)

f(X,

K, E,),

+ OA)

-UK,

(6.14)

where

x a3W + ~a - K’ - P,) J(w, x c I(K, P,,,I J+G,J IK’, PB)I*.

- Qs)) (6.15)

24

AGASSI

AND

KOLTUN

The first summation is over nucleon spin projections. The factor of p(X)/p,, with p(X) the nucleon density, and p,, its average value, is introduced here as a local density approximation to the Fermi gas; it supplies the only dependence on X. It is consistent with (6.15) to set Im Y,(x) = -(Ku/2w,) p(x) with the pion-nucleon total cross section (as in Eq. (5.20)). The results we have obtained in Eqs. (6.14) and (6.15) are essentially the same as those of Hiifner and Thies (see Section II of [S]), if the last factor in (6.15) is reexpressed as a barycentric cross section, except that we have used a steady state formulation, with fixed energy E, (6.1 l), rather than the time-dependent form of their work. (See [lo] for a different derivation of the Boltzmann equation method.) To summarize, the Hiifner-Thies Boltzmann equation corresponds to a twochannel model (pions, and other channels), in the limit of: a cold Fermi-gas model of the nuclear target, on-shell propagation of the pion between collisions, no recoil of the d-intermediate state, and local optical potentials. As mentioned above, the approach of following the original projectile @ion channel) outlined here would be more appropriate for higher energy reactions in which many N* or A* resonances could enter as doorway channels, and no one dominates. The Eq. (6.5) is a suitable starting point for developing a model transport equation. It is not necessary to make all the simplifying assumptions that lead to the Boltzmann equation form (Eqs. (5.14-(6.15)) given by Hiifner and Thies. 7. CONCLUSIONS

We have presented a statistical theory of pion-induced nuclear reactions, in two stages. First, we have applied a general statistical method in which the scattering theory leads to a transport equation. This integral equation must be solved for the density matrix of Wigner density, from which one calculates certain inclusive cross sections for scattering or absorption. Second, we have given a multichannel model, appropriate to the A( 1232) energy region, that is, for pions of 1001100 MeV. The model is given in a form such that one need solve only the A-channel transport equation. We have given details of the structure of such a model. It is intended to develop the present method for application to calculation of the general features of pion-induced reactions, particularly those involving absorption. Given the statistical assumptions and further simplifying approximations employed, under what conditions would we expect the present methods to be justified? For the general method, we expect the basic statistical assumptions to be adequate when we are dealing with transitions between dense sets of complex nuclear states, which presumably happens after two or three inelastic collisions in the target. Reactions which involve penetration by the pion of considerable nuclear matter should be suitable: forward inelastic scattering with energy loss of at least 20-40 MeV, and most pion absorption reactions, both on all but the lightest (A < 20) targets. For the specific model we have employed, we depend on further assumptions. We invoke a short-range approximation for propagation in the pion and absorption channels; this depends on strong coupling between the channels, which seems

OF R-A REACTIONS

STATISTICALTHEORY

25

appropriate for the strong EN ++ LI interaction, and probably also for the AN cf NN interaction. The exciton model is invoked as a simple way of transferring energy between the active channel particle(s) and the target. This implies fast mixing of the excitation energy within the nuclear volume: there must be no localized storage of energy (“hot spot”). This picture is perhaps oversimplified, but would be difficult to improve dynamically. Similarly, we have no nuclear correlation effects in our stepwise picture of single-target-particle interactions-the doorway picture. Several features recommend the present approach. The transport equation provides a relatively simple and direct way to translate the phenomenology of meson-nucleon and resonance-nucleon interactions into a fairly compact integral equation (which does, however, still have to be solved numerically). It is possible to include the nonlocality of the interactions, and some features of off-energy-shell propagation in the nucleus, which are left out of the more classical gas methods: the intranuclear cascade [7] and the Boltzmann equation [8]. We are able to maintain the unitarity of the theory in its approximate forms. We are also able to include the absorption channels on a similar footing to the scattering channels. It is not clear whether the nonclassical aspects of our theory will prove to be important in understanding the gross features of pion absorption, which are at present only crudely reproduced by the semiclassical calculations [7, 81. The present method is rather flexible, in that one can adapt the description of channels to single out features of particular reactions. This was illustrated in Section 6, where we modified the A and absorption channels to emphasize the inelastic scattering. A rather direct extension of the present method could be used to calculate particular aspects of absorption, such as the inclusive cross section for emission of protons: (n, p), which has been studied in a number of recent experiments [ 151. For this problem, it would be more useful to treat the absorption channels as one nucleon + nuclear states (a), and follow only single nucleons after absorption. This will be the subject of a future work. APPENDIX

A:

UNITARITY

We show that the structure of the dynamical equations (3.11) or (3.14) guarantees unitarity, so that (3.18) and (3.20) lead to the same value for the total compound cross section. We are interested in the detailed form the unitary relations take, since we wish to maintain these relations in making approximations to the general theory. We begin with Eq. (3.18) for the compound cross section (f = (yc + rp, where we integrate Eq. (3.16) to obtain

and uob is given in Eq. (3.17).

‘(A.1)

26

AGASSI

AND

KOLTUN

Since the cross sections are calculated from pA, we consider the dynamical equation (3.11) for that quantity. Treating this equation as a matrix equation in (x 1M, 1x’), we take the trace (tr M = C, j dx (x ( M lx)) of both sides of (3.11). The trace of the commutator on the 1.h.s. gives zero. The trace of the first term on the r.h.s. yields tr{JmJ,Gt+

- h.c.} = tr$$V”^;+ - Y;) = iu,(uT - fP),

(A-3)

where we use the cyclic property of the trace (here on the pion variable: Id<) and Eq. (3.12), in the second line, and Eq. (3.20) in the third line. The trace of the second term on the r.h.s. of (3.11) may be written: I I tr (JG”JtpAJG”+JtGA - h.c.} = tr{ G”JtpAJGntJtGAJ - h.c.} = tr{G”JtpAJ(Qnt - 1) - h.c.}, (A-4) where we have used Eq. (3.12) in the last line. We also calculate tr {JmtpA

- h.c. } = tr { G,J,“j+

- h.c. }

(A-5)

from the last line of Eq. (3.11). Adding (A.4) and (A.5), and using G” = Go’%“, we obtain tr (GnJtpAJWt - h.c. I= tr{(GO” - Go”+) PJ+pAJQn+J. (A*61 For the free Green function we have Go” - Gent = -2xi

d(h, + HN - E,)

(A.7)

so that the sum of traces may be compared to (A.2a) to find (A.6) = (A.4) + (A.5) = -izIousc.

G4.8)

The trace of the remaining terms on the r.h.s. of (3.11) can be calculated completely in analogy to (A.4)-(A.7), and compared with Eq. (3.17) to produce tr (j-terms ) = -iv,

aa*.

(A.9)

Thus the trace of Eq. (3.11) leads to the unitary relation (oT - oa*) = use + ga*

(A. 10)

so that Eqs. (3.18) and (3.20) must agree. The approximations of Section 4 lead to the expressions given in (4.13), (4.14), (4.10) for the kernels of the transport equation (3.14). The gain term in Eq. (4,13) is expressed in terms of the same combination of matrices as appears in the brackets of the last line of (A.4), with G” = g”R”, and an analogous term inj, jt. The loss term

STATISTICALTHEORY

OF 7k-,4

21

REACTIONS

(4.14) has the same form as (A.5), with an analogous j, jt term. And the inhomogeneous term (4.10) is an integral of the operator product in (A.3). It follows that the unitary relation (A.lO) still holds for the results of the solutions of the approximate equations, and that cross sections calculated from (3.16) and (3.17) are consistent with the optical flux loss given in (3.20).

APPENDIX

B:

TRANSFORMATION

TO ~-COORDINATES

In Section 4 we introduced structure functions f(& x) and h(x, y,, yJ which relate the pion coordinate 5 or absorption channel coordinates y, , y2, to the A-coordinate x. This allowed us to transform functions of the rr- or &-coordinates to A-coordinates, as in Eqs. (4.4) and (4.7). We can treat the transformation in linear operator form, for compactness, writing F?A”=fcA”j-

+

(B.1)

for (4.4a) and s-,; = f +u;j-

(B-2)

for the definition of lJ; in (4.8). Note that (B.2) implies a transformation to obtain a well-defined U:, this requires that f have an inverse f -I:

I‘f %

X)fW, x) dx = S3(C- 5’).

Similarly, we may introduce Msller matrices w,“(x, x’) Moller operators L2; of Eq. (2.10) (in n-space), by

w;=f and it follows

from x to g:

+-‘n;f +,

by transformation

of the

(B.3)

that 0; = 1 + u,n.LF;.

Similarly, for the ab-channel to A-channel transformations, with (4.4b):

(B.4) we may define, starting

Fib = h+Gzbh, Uab n

=

h-‘P”“bht-’ LT

ozb = h-‘Rzbh+ The unitary quantities.

properties

demonstrated

3

(B-5)

= 1 + u;“gtb.

in Appendix A still apply to the transformed

28 For the calculation binations

AGASSI AND KOLTUN

of cross sections in Eqs. (3.16), (3.17) we require the comi2; J,t, = f+oA” Jl,(x)

and

(B-6)

.n”,“j,, No inverse transformations f -‘, h-l,

= hw~bj,,(x). need be calculated.

REFERENCES 1. J. M. EISENBERG AND D. S. KOLTUN, “Theory of Meson Interactions with Nuclei,” Chap. 5, WileyInterscience, New York, 1980. 2. C. H. Q. INGRAM, in “Meson-Nuclear Physics--1979” (E. V. Hungerford III, Ed.), American Institue of Physics, New York, 1979. 3. I. NAVON et al., Phys. Rev. Lett. 42 (1979), 1465; D. ASHERY, Nucl. Phys. A 335 (1980), 385; D. ASHERY et al., TAUP 836-80, to be published. 4. D. AGAS’SI AND H. A. WEIDENM~LLER, Phys. Lett. B 56 (1975), 305; D. AGASSI, H. A. WEIDENMUELLER, AND G. MONTZOURANIS, Phys. Rep. C 22 (1975), 147. 5. D. AGASSI, C. M. Ko, AND H. A. WEIDENM~LLER, Ann. Phys. 107 (1977), 140. 6. L. S. KISSLINGER AND W. L. WANG, Phys. Rev. Len. 30 (1973), 1071; Ann. Phys. 99 (1976), 374; A. N. SAHARIA, R. M. WOLOSHYN, AND L. S. KISSLINGER, Phys. Rev., in press; F. LENZ, Ann. Phys. 75 (1975), 348; M. HIRATA, J. H. KOCH, AND E. J. MONIZ, Ann. Phys. 120 (1979), 205. 7. J. N. GINOCCHIO, Phys. Rev. C 17 (1978), 195; J. N. GINOCCHIO AND M. B. JOHNSON, Phys. Rev. C 21 (1980), 1056. 8. J. H~~FNER AND M. THIES, Phys. Rev. C 20 (1979), 273. 9. E. A. REMLER, Ann. Phys. 75 (1975), 455. 10. M. THIES, Ann. Phys. 123 (1979), 411. 11. H. FESHBACH, A. KERMAN, AND S. KOONIN, Ann. Phys. 125 (1980), 429. 12. G. E. BROWN AND W. WEISE, Phys. Rep. C 22 (1975), 281. 13. J. J. GRIFFIN, Phys. Rev. Lett. 17 (1966), 478; Phys. Lett. B 24 (1967), 5; see also M. BLAIN, Ann. Revs. Nucl. Sci. 25 (1975), 123, and references therein. 14. See, however: B. R. BARREN, S. SHLOMO, AND H. A..WEIDENM~~LLER, Phys. Rev. C 17 (1978). 544, and S. SHLOMO, B., R. BARREST, AND H. A. WEIDENM~~LLER, Phys. Rev. C 20 (1979), 1. 15. H. E. JACKSON et al., Phys. Rev. Lett. 39 (1977), 1601; Phys. Rev. C 16 (1977) 730; R. D. MCKEOWN et al., Phys. Rev. Lett. 44 (1980), 1033.