AlllDQ TflT7m

NBS MONOGRAPH 139

U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards

Interactions

of High Energy Particles

With Nuclei
NATIONAL BUREAU OF STANDARDS

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MOmi BUREAU OF STANDARDS

OCT 3 1975

^+ occ

Git 100 Interactions of High Energy Particles

With Nuclei a,

0.^

Wiesl'aw Czyz

Center for Radiation Research

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

\

U.S. DEPARTMENT OF COMMERCE, Rogers C. B. Morton, Secretary

U.S NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acfing Director

Issued September 1975
Library of Congress Cataloging in Publication Data

Czyz, Wiestaw.

Interactions of high energy particles with nuclei.

(NBS Monograph 139)

Supt. of Docs. No.: C13.44:139

1. Nuclear reactions. 2. Hadrons—Scattering. I. Title. II. Series: United States. National Bureau of Standards. Monograph 139. QC100.U556 No. 139 [QC794.8.H5] 389'.08s [539.7'6] 74/13725

National Bureau of Standards Monograph 139

Nat. Bur. Stand. (U.S.), Monogr. 139^ 73 pages (Sept. 1975) CODEN: NBSMA6

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WASHINGTON: 1975 For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C., 20402 - Price $1.35

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Preface

This monograph is based on lectures given by Dr. Wiesfew Czyz at the University of Virginia

during the spring semester of 1973. They cover selected topics in the field of high energy diffractive scattering and production processes. In addition to reviewing some well-known material there is much here that is new, both in content and form of presentation.

The material presented here is also part of a program of research and cooperation between the

author and the National Bureau of Standards that was begun in 1967 when the author was a National

Science Foundation Senior Foreign Scientist Fellow at The American University and a Guest

Worker at the National Bureau of Standards. This cooperation was continued on an informal basis

during his several visits to this country between 1968 and 1971. Since July 1972 this research has

been the subject of a grant by the National Bureau of Standards to the Institute of Nuclear Physics,

Krakow, Poland under the PL-480 program. The sponsorship of this program has greatly facilitated

this effort which has produced a series of articles written jointly by the author and NBS staff. The

result of this cooperation is reflected, however, not only in published papers, but in numerous unpublished notes as well as in these lectures themselves.

L. C. ]\Iaximon

iii

Interactions of High Energy Particles with Nuclei*

W. Czyi**

Physics Department, University of Virginia, CharlottesWlle, Va. 22904

Elastic scattering and diffractive production processes induced in nuclear targets by high

energy- projectiles are discussed in this article. Special attention is paid to the interaction of high

energi.- hadrons and photons. Interactions of high energy electrons and neutrinos are briefly mentioned. The common features of all these processes are emphasized throughout the article: The

multiple scattering and shadowing processes inside of the target nuclei. An effort is made to develop

a unified way of treating nuclear interactions of particles which are either hadrons or exhibit some

hadronic components in such interactions. This article is divided into seven sections: (1) Introduction, (2) Description of multiple scattering, (3) Elastic scattering of hadrons from nuclei, (4) Diffractive dissociation and diffractive excitation, (5) Diffractive production of hadrons in hadron-nucleon collisions, (6) Shadowing

effects in inelastic electron-nucleus scattering, (7) Shadowing effects in neutrino reactions on nuclei. Key words: Diffractive production; diffractive scattering; Glauber model ; hadronic components of photons; high energy- scattering; multiple scattering; neutrino-nucleus interactions; shadowing

effects.

1. Introduction

Let us start by gh'ing a few motives for discussing this subject:

(a) It is well kno^Mi that nuclear targets are of considerable importance in high energy ph\-sics.

Work on vector meson production on nuclei or Sw (ot) coherent diffractive production (compare

refs. [S3, S4, So]) ^ is a good example of the role of nuclear targets. One observed also excitations of

specific nuclear levels by high energy hadrons [1, 2]. This opens a possibUity [2] of selecting diffrac- tive productions -n-ith nuclear levels as their analyzers.

(b) The very high energy incident particles -may also be important for physicists working on

nuclear structure—although this point does not seem to be well established (presumably due to poor

energy resolutions of high energy beams) . Nevertheless, one still hopes to be able to learn something

new, for example about short range nucleon correlation functions in target nuclei or about the pres- ence of resonances in nuclear ground states [3, 4]—just to name two problems.

(c) One may also hope that, in the cases where the scattering from a nucleus cannot be reduced

to the "elementary amphtudes" of the incident particle—target nucleon, some new physical situa- tions may occur which stem from the complexity of the target. For example, in the case of ir-nucleus

scattering in the region of the (3, 3) resonance [5, 6] one may hope to learn something about the nature of the (3, 3) resonance because the exclusion principle (due to the many particle structure of the target) may chstort the resonance and this distortion may depend on its internal structure. In these notes the interactions of various different particles with nuclear targets are to be considered. Of course, we cannot cover all problems related to the interactions of hadrons, photons, and

electrons (virtual photons) T\'ith nuclei—we must choose a certain point of view which tmifies aU

these problems. The common denominator which we shall emphasize is the existence (or lack thereof)

of "shadow effects" (which occur mosth" for forward scattermg and production processes). Such

effects are very well established in the case of hadron scattering and photon-nucleus interactions;

they are not well kno'wn in the case of neutrino reactions and very ^drtual photons (see refs. [S3, SoJ)

* Based on a series ot lectures given at Department of Physics, University of Virginia, during the spring semester

1973.

** Guest worker of the Xational Bureau of Standards, Washington, D.C., 1973-74. On leave from Institute of Nuclear Physics, Krakow, Poland.

' Figures in brackets indicate the literature references at the end of this paper.

1
(Nonetheless there exists a motivation for behef in their existence : as in the case of photons inter- acting with nucleons—where the strongly interacting vector mesons seem to dominate—in the case

of neutrino interactions—as was pointed out by Adler [7]—vr-mesons should be important). In

other words, in all the processes we are going to discuss, strongly interacting particles appear (as

real or virtual particles) which may produce shadowing effects. So, the crucial point in our discussion is an understanding of a multiple scattering process of

strongly interacting particles inside of nuclear matter (or more generally: just a multiple scattering process with forces strong enough to insure the existence of multiple scattering). Hence we shall start with the very successful model of such processes: the Glauber model.

2. Description of Multiple Scattering

2.1. General Remarks

To construct the relevant formulae for the theory of multiple scattering one can employ various

models of potential scattering. First let me quote the well-known formulae: one particle scatters from a collection of A particles at very small angles (in the Glauber model [Si])

TARGET

CONSISTS OF A PARTICLES

This is to a very good approximation a two dimensional process. The individual amplitude

/;(5)= - / c^^fcexp (iA.b)(l-e^-x>W),

is shifted to the position of the^th nucleon:

= f

'^'^ exp (zA-b) {1- exp [%(b-S;) ]},

where k is the momentum of the incident particle in laboratory frame

A is the two-dimensional momentum transfer b is the impact parameter

X;(6) is the phase shift which characterizes the incident particle—jth nucleon elastic scat- tering amplitude.

The expression

l_eix;(ft) = ^.(6)

is called the profile of the jth. nucleon, incident particle collision. Assuming

x(b)= i:x>(b-s,)

2
and assuming that the particle goes through the target so fast that all the nucleons are 'frozen' at

certain positions, we get for the amphtude

m/i = ]- ! d^i . . . d^rA'^/*{ri . ..Ta) f

d^b exp (^A-b)

X {1- exp X;(b-jy)]}^.(ri • • • r^)

= — f

d'^b exp (^A•b) f

dhi . . . d^A'^'/

27r J J 1- n (i-7>(b-s,)) (2.1)

where and ^/ are the initial and final wave functions of the target nucleus.

One can produce many arguments which make this important formula plausible. One can use,

e.g., an optical description of attenuation of a wave penetrating a medium. One can also use some

arguments based on approximate solutions of the wave equation of the incident particle interacting

through potentials with the target particles.

For instance, in the case of the Schrodinger equation

\2m )

in the limit E-^ oo ,2 and for the incident particle moving along the z axis, we present the solution

in the form

i/'fc(x, y, z) =e*V(a;, V, z).

If the potential is smooth enough (so that second derivatives of (p can be neglected), one can show

that tp satisfies the approximate equation^

which gives

d<p{x, y, z) i

7 = - - V{x, y, z)^{x, y, z), dz V

yPk^e^'"- -

/ dz'V{x, y, z').

2 Notice that to have scattering in the limit > « we have to have "F~J?F' where V is energy independent.

Otherwise the high energy solution of the Schrodinger equation reduces to the Born approximation.

2m

=ea^ ( — + — ) ,p{x, y, -fcV*-V+2tfce'*^ — +e''-'^ ., dz dzhence, neglecting second derivatives of <p, we obtain the following equation for ip :

dip 2m ., 2iA;e'*^ — = — Fe'^v,

dz fi^

d<p i dz vn

where we have used

P p = kfi, y = — , m

3
The amplitude for the particle to scatter from k to k' is:

771/ C

9Tl(k', k) =

/ dVexp (-zV.r)F(r))A;t(r)

=^ ^ I

d^fe exp (zA-b) dze*'Y{h, z) exp ^ jdzV{h,z')^

= d-6exp (2A.b) l-exp^-^y d2'F(b, , A = k-k'.

We can see that from the additivity of the potentials

v= Zvj,

we recover additivity of phase shifts. There are many simplifications made in obtaining the fundamental formula (2.1) ; the reliability of this formula is of primary importance. The most complete analysis one can perform is presumably

to employ the Watson multiple scattering theory, b.ut we shall not present it here. In fact it is amazing that (2.1) works so well. Even in the conceptually simplest cases of relativistic potential scattering one can give examples in which it breaks down.

Examples

Example 1. Dirac particle with anomalous magnetic moment in a given electromagnetic static

field (notation from Bjorken and Drell [S7])

:

\ 4in /

1 0" 0 d~

and multiply it by 7° : y° = 0 = , a =

0 -1 d 0

where t!^=y°A°—fA, ct^,= - {y^y' — Yl") Denote K= —

2 2m

{iV-eA:+}4Ka^,Fi'^-m)xP = 0. Take

^n" +i Y

• V - e Ft" +^AKa^yF"^ - m^ 4^ = 0

to get

\^i-+ia-V-eV+^}4Ka^yF''' -^mj^ = Q.

This equation was worked out in ref. [8].

We introduce the electric and magnetic fields (E, B) in terms of which

YzKa^.F"^ = - 2Hi^ ( <Toi^. +a,2E,+ aozE,)

+2H^( <7235, +(r3l5, + .) = -Kia-E+K-L'B,

where S = ((r23, 0-31, an)

f23 —

0 o-y 0 (T, 0

, 0'31 — , ai2 —

_0 _0 ay_ _0 (7._

4
(The sign 'minus' in front of Kia-E is because

We get finally

i-^ = L-ia-V-\-eV-Kl3CL-B-ia-E)+0m^xl,.

at .(2.2)

The time dependence of \l/^e~'^' implies i >E and we have

dt

Exp = l- a- V+/3?n-A'i8(£-B-za-E)i-eF>

^ = e"iz<^= exp (iz\/E^-m-)ip^e'^'<p, (E-^cc).

Inserting this into the Dirac equation and noting

i — e'-E'v = Ee'^'<p+e'^'

dz

(f we get

^:(l-a3)^^=[-^a•V+/3w-^^:/3(E•B-^a E)+eF>. (2.3)

Hence, in the limit > oo we have to have

(l-a3).^-*0, {l+a3)<p^2<p.

This is because the right-hand side of (2.3) does not contain the energy, E. We multiply eq (2.3) from the left by i^(l+a3) and get (note that (l+as) (1- as) =0:3^ = 0)

0= -z"K(l +a3) ax-V±-tK(l+a3)a3- +M(l+a3))8m

dz

-KH(l +a3)/3(£-B-za-E)+eFH(l +a3)

But in the limit > <»

(1+0:3) (x±(p= ax{l—a3)(p—>0

(l+a3)/3v5 = /3(l-a3)s^O

(l+a3)j823^ = /323(l-a3)^0

(1 +as) /3a3<P = iSas (1 — 03) >^0,

where the transverse components (in x, y plane) are marked J. . Thus, filially, we find i— +eF-X/3(Sa..Bj.-zaj.-Ex)

dz

(2.4)

So, if the anamalous magnetic moment K= 0, we end up with an expression which is virtually the same as in the case of the Schrodinger equation

whose solution
gives

where M(k) is a four-spinor. As

5Jn(k', k) = - — / dV/7oeF(b, z)xp,

where i/'/ = ii(k') exp (z£'2;+?.'A'b), we get

3n(k', k) = — tZ(k')7oM(:(k)?y" d^h exp (/A-b) rf2 F(b, z))

So, in this case we also have additivity of phase shifts—hence the Glauber model: But when i?5^0

the principle of additivity of phase shifts breaks down. Let us consider this case in more detail.

i heF-A',3(Sx-Bx-zax-Ej.)

dz

From the equation

dependence on F by substituting

9? = Fexp^ — {e y" dz'V {x,y,z')^ ,

= 0 we can eliminate the 'trivial'

— I (j5 bz = exp £ ...)(-.£ f) -»FF exp {-u A' . . .)

and we get the following equation for F:

dz

F is a four spinor but we can reduce it to an equation for a two component spinor because F has to

satisfy the relation

{\-az)F = 0, az=

'

So, F can be taken in the form

0

i- F = K

dz i6±'E± -dx-B±

or

dz

(2.5)

which is in fact a system of first order differential equations for two unknown functions (the two

components of the spinor x) • Call

In general

(i{x, y, z) (dj.'Bx — zdi-Exo-j).

la{x, y, z), a{x, y, z')J^O.
Hence we have to use a 2-ordered product to express x in a compact form

X = jexp (^i j dz'aix, y, z') ^| XiEach infinitesimal step

x{z+Lz)-x{z) =i^zK{^i.'Bx-i63.^'E.L(J,)x{z),

x{z+^z)=e'^'^^\{z),

should be applied in order of increasing ^'s. That is what }. . . ) + means.

In any case, the additivity principle is violated : Si and (I2 generated by two sources of the elec- tromagnetic field (at two different positions) are, in general, noncommuting operators and there is no way of adding phase shifts (or, equivalently, multiplying profiles). We can also see that the

physical reason for this phenomenon is the coupling between different spin states produced by the

term K (dx-Bj. — ?dj.'Ej.(r3). So, we have to deal with a coupled channels problem. We can also make

the following remark: sometimes coupled channels can be decoupled by diagonalization.

A remark about "decoupling" channels through a diagonalization procedure

Start with a generalization (to N channels) of the eq (2.5)

_.a _^

OZ rn=l

Note that "compositeness" of the incident particle is responsible for the existence of more than one

channel. For instance, the presence of an anomalous magnetic moment can be looked upon as a mark of "compositeness." Suppose

A a«m(r, n, . . .

, r^) = G-lmir-r-).

Diagonalization should produce a diagonal matrix of the form:

Ex^^''(r-ry) 0

3=1

(r-ry)

0

0

\^^'(r-r;) 0

7
One achieves additivity if the same S diagonalizes all Q^-^ (r — ry) simultaneously. If it does not, there

is no context in which one could talk about additivity of phase shifts. In general the additivity does not occur. Take, e.g., pure Coulomb scattering (5 = 0, F = Coulomb potential) in eq (2.5):

— i—= Gix- We have to diagonalize the matrix of eq (2.5) (compare ref. [81)

:

dz

where

f 0

\i{x+iy)

iix-iy)\ /- (a;2+?/2)i/2 0 >

0 / \ 0 ix'^+y^yi'/

I {x-iyyi^ {x-iyyi^^

\ — i{x+iyyi'^ iix+iyy^i

H{x+iyy'^ -{x-iyy'-\

V^ {x+iy) {x-iyyy

1 2U/2

Even this soluble case cannot be diagonalized for more than one scattering center if the Dirac particle

has an anomalous magnetic moment, K^^O.

Without going into any details of the calculation let us quote the results. In the case when only one Coulomb potential is present (hence B = 0, but E^O) , we have

3n(A)~?x/+|y" d% exp (iA-h) l-exp^-«e j

dzV{h,z)+i j

dza{h,z)^

|

where

G(b, z) =K..F'(r) 0,

\i{x+iy),

-i{x-iyy

0

Note that since the z dependence is outside the spinor matrix this does commute at different z's:

[(J(fe, 2), GL{b, z')^ = 0. Suppose, however, we have two sources of Coulomb field at two different

points. Then

^Fi'(lr-ril) 0, -il{x-xi)-iliy-iyi)j'

r — ri

F/dr-r^l)

\il{x-xi)+iiy-yi)2, 0 /

'0, -iL{x-X2)-i{y-iy2)y\

l^-^^l \il{x-X2)+iiy-y2n 0 /

Now, however, we do not have the Glauber model any more

!

M{A)^ixf'^ Ij d'-bexp (iA-h) 1- exp(^-ie j

dz(Fi + F2)^|exp J

rf2(ai + a2)^|

where { . .

. ) + denotes the 2-ordered product.

Because

/O a\ /O c^

Kb 0/ \d Oj

^0 c\ /O a>

Kd 0/\b 0/ we have

la{x,y,z), a{x,y,z')2^0.

One could argue that the coupling to the anomalous moment is weak and hence not very relevant.

8
This is true, but one can give some other—though much more complex—exampkfs of scattering from

a classical external field in which the "principle of additivity of phase shifts" is also violated.

Let us consider a vector particle (hence a very relevant kind of particle to our further analysis)

Example 2. Scattering of a charged vector meson in a static field (we shall quote the results, for more details see refs. [9, 10])

Let us allow for our vector particle to have an arbitrary magnetic moment and define the

magnetic moment operator

M=(1 + k)-^S, (2.6) 2m

(S—spin operator) , where k determines the value of the magnetic moment. When k = 0 the equations

of motion of such a particle are the so-called Proca equations. If, however, Tct^O some additional terms

appear (as in the case of the Dirac equation with anomalous magnetic moment) . With k9^0 we have

(in the pseudo-euclidean metric, n, p = l, 2, 3, 4), (compare ref. [11]),

where

dXn dXy

We shall choose A^ = ib^iV {r) (just the static Coulomb field). The results of a long and involved

analysis [9, 10] are as follows:

1. We recover the principle of additivity of phase shifts only in the case k = 1. 2. In all the other cases (including k = 0) , there is no additivity of phase shifts. A general comment on Examples 1 and 2 is in order here. First: terminology. Many authors call

K the anomalous magnetic moment of the vector meson [9, 10]. This is presumably so because when

one starts with the free vector meson field equations

d / d d \ , ^

aXfi \dx^ dXy /

and then introduces the electromagnetic field in the standard way ( d^= ieAA one obtains the

\ dx^ /

~K = 0 case. (Similarly, if one starts with the free particle Dirac equation and introduces A^ in the same way, one obtains the k = 0 case of Example 1). So, from this point of view, the cases k = 0 of Example 1 and k = O of Example 2 are analogous, and, as we know, in the first case the spin channels

decouple in the high energy limit whereas in the second case they do not. One may ask oneself a question : is there any simple way of telling which values of k and Tc result

in decoupling of various spin states in the high energy limit? The answer seems to be: yes. It is enough to observe that in Example 1 for k = 0, the relation between the magnetic moment IVI and

spin S is M=-S (2.7) TO

where e is the charge and m the mass of the particle. Note that eq (2.6) gives the same relation

between magnetic moment and spin when /c = l! So, in both Examples, when k and k are chosen to make eq (2.7) valid, the spin states decouple in the high energy limit.

9
In order to make the condition (2.7) more plausible, let us consider a charged particle with

spin S and magnetic moment M given by (2.7) moving in an almost uniform magnetic field B. This particle follows a circular trajectory with frequency

co= - B. m

Its magnetic moment (hence its spin) precesses with frequency

M ^ e ^

o:p= —B=-B.

*S m

So eq (2.7) makes these two frequencies equal. But that means that the projection of the spin on the direction of particle velocity is a constant of the motion. Hence in this case all helicity spin states are decoupled.

Although we have considered a very special case of nonrelativistic motion in a constant magnetic

field, the condition (2.7) for the decoupling of spin channels turns out to be very general: The

relevant relativistic formulae for precession of the polarization of particles with arbitrary magnetic

moments and spins in a slowly varying (in space) electromagnetic field were given in ref. [50].

Their immediate consequence is [51] that in the high energy limit and for the gyromagnetic ratio

g = 2 (hence when (2.7) is valid because the definition of g is through the equation M= g (e/2m) S) the

projection of the polarization on the direction of motion is constant, and hence there is no coupling

between various spin channels.

To conclude this section we may say that Examples 1 and 2 warn us that if the strong inter- actions are mediated through vector fields (analogous to the electromagnetic field) one can expect

the "principle of additivity of phase shifts" to be violated.

3. Elastic Scattering of Hadrons from Nuclei

Let us go back for a moment to scattering of incident particles whose internal structure one can

neglect (in particular the internal quantum numbers can be neglected). Let us start with just one

scatterer:

TARGET

The incident wave: e'^'. The wave immediately behind the scatterer: — 7 (6)e'*^^, b={x,y).

The shape of the shadow is given by 7(6)

:

yib) = [ d^'dexp {ih'h)f{8),

2wik f

where/ (6) is the elastic scattering amplitude as shown below,

10
As long as 8h/2k'^<l, the 2-dependonco of the second term in the wave immediately behind the

scatterer is given to a good approximation by e'*^^ Otherwise one should realize that depends on o which sits in the Fourier transform of the shadow. Hence, away from the scatterer, one would guess

the following shape of the wave (compare D. R. Yennie article in [S3])

e"'- fd'Sexp {iz\/W^^) exp (z5- b)/(5)-^e'^--7 (6)e*^ (for small z's) / (3.1) ZwlK J

If the size of 7 (6) , and hence of the scatterer, is a, the representative transverse momentum transfer

is 5~a"~\ We can then estimate the "healing" length, L, of the shadow:

for a = 1 fm, A; = 10 GeV we obtain

2ka^ = 2X10 GeV X 25 GeV-^ = 100 fm.

Note that (3.1) gives, as r— , (compare D. R. Yennie article in [S3])

)/'(r) =e'*^^+[/(/cex)/r] exp {ikr), (ej. = component of r_LeJ,

with/(A;ei) correctly given by the inverse of 7 {h) . One can see this by shifting the origin of integration to kej.: 8 = ^ex+q (we introduce a new variable q)

Then

and

/ 1 Q^r zVk^-S''+^-h~kr-

^ rC

1 r ( i Q-r\ 1 - —- / d?q exp ( - - — )/(^-ej. +q) - f{kei)e'>":

Zirtk J \ 2 k / r Remark : When the incident wave already has a profile different from unity we get:

incident wave: g (x, y) e^''' transmitted wave: g{x, y)e'^^{l — y{x, y))

(this is all under the assumption z<^L)

Let us construct the "shape of the shadow" for a collection of scatterers (nucleons in a nucleus;

see e.g., fig. 1)

The incident wave : e'** after the first collision: [1 — 7i(b— si)]e'''^ after the second collision: [1 — 72(b — S2)][l — 7i(b — si)]e'''^ etc. The 'shape of the shadow' for the whole collection is then

A [l-r(b, si . . . SA)y== n [l-Ty(b-sy)]e'-*^

3=1 A r(b, Si . . . sa) =1- n [l-7;(b-S;)].

So, we get again the formulae of section 2.

11
The previous case dealt with an elementary object scattering from a composite object. We

already saw in the examples of scattering of relativistic particles from external electromagnetic fields that "internal structure" (in these cases the internal spin quantum numbers + anomalous magnetic

moment) breaks down the "ansatz" of additivity of phase shifts. We can also have a look at this problem from the point of view of a Glauber-like description of scattering of two composite objects.

The formulae given below are interesting also because they may be used to analyze high energy

nucleus-nucleus collisions (which is not an academic problem because there are experimental projects under way)

.

The geometry of the process is shown in figure 3.

B COMPONENTS

The profile describing the collision of two elements is

For the sake of simplicity let us take the wave functions of (a) and (b) in the form of products of

single particle wave functions. Let us assume also that all particles have the same single particle wave functions. The ground state wave functions are:

A B

3 I and the elastic scattering amplitude is, therefore:

ik r r f A B c^= — d% exp (zA-b) / dW^ . . . dhA^"^ / dW'>^ . . . dhe'-''^ H 11 ^o("'*(r;<'")<po<''>*(rj('") 27r J J J 3 I

X |l- n n [l-T^Kb-s^^^'+s/"))]! <po'"Kr/">)'Po<'"(n(''). (3.2)

What kind of formulae would we have if the "ansatz" of additivity of phase shifts of the composite system (b) colliding with nucleons of (a) were valid? Let us look at the profile of the jth.

nucleon:

Tj{b) = / dW^ . . . d^SijW n P'*' (si)

•' I 1- n (l-7;i(b-S,))

= l-(l-7y(fe))^

where we assumed all profiles to be identical and introduced y

7;(6)= j

dVi<^o<''^*(riW)^o(^>(n(6')7;7(b-Sz(*))

which is the profile for elastic scattering of (b) from the jth nucleon of (a) and a two-dimensional

density

/+C0

dz <po* (r) (r)

.

^

12
Then the additivity of phase shifts gives us the formula:

311= — jd}h exp (^A•b) j dW^ . . . d'sA^-^ np'^'Cs/"') |l- 11 (1 - r^Cb-s/"*) )|. (3.3)

This is different from (3.2). What is the difference? First let us note that (3.2) is a sum rule. For

instance, we can extract from (3.2) the following contribution of the second order

I

#riW^o(''*(n''')Tn(b-Si('"+SiW)72Kb-s,(^'+S2('")¥'o<'" (M

= [ (iVi('"dVi('')Vo''"*(rz)Tn(b-sj(^'+Si(°') Z'P«'"(n<''0^n<''>*(r;(^)')

= Z /

rfVi(''Vo^''^*(n<*>)7H(b-s,('" +si('")^,/'"(n('0 /"(iVi('')V„w*(nW')

XT2i(b-sjW'+S2('^))>Po"" (n'^O,

because

Z <Pn^'^ (ri^"^ ) v/" *(?•;<'"') - (n^"^ - r ' )

Hence the formula (3.2) sums over all intermediate excited states. For instance, the above

contribution gives

i

ALL POSSIBLE EXCITED STATES

Suppose we reject the intermediate excited states and take only the ground state as a possible inter- mediate state (this is the way to eliminate all channels but one) . Then each can be averaged

over rj'''': j dhi^"^ I ^o^^^in^"^ |2 7,-,(b-si('"+s/'")

= j d^Si'^'p^'"' (Si)7y((b — Si^^' +sy^"') (it does not depend on I when

all nucleons are "equivalent")

Then the formula (3.2) reduces to (3.3)

2fn= ^ Id'bexp (^•A•b) Jd • • • d^A^"^ XIp^^'Cs/"^) |i- 11 (l-7ji(b +s/''>))«|

= j d'b exp (^A•b) jd^siW . . . d'sA^"^ {l~ H (1 - ry(b+s/<") )| 11 P^"Hs/<")

(to make it identical to (3.3) we should substitute s/°>^ — Sj^"'). Hence we get a formula whicli

follows from additivity of phase shifts.

13
It would seem, therefore, that indeed "compositeness" of the incident particle is decisive in

destroying or satisfying additivity. The other "moral" is that if we know the structure of the composite body (b) we may still use a generalized Glauber model with additivity of all possible phase

shifts of the pairs of components of (a) and (6)

.

Let us consider some limiting cases of eq (3.2) (compare ref. [12]). Let the radii of the two

composite objects be Ra and Rb. The calculations of ref. [12] show that the smaller is Rb the nearer we are to the additivity of (fe)-nucleon phase shifts. But that means that this additivity improves

with increase of the binding of (6) . Of course for Rb-^0 the additivity becomes exact. One can see

this explicitly by replacing for (b) , p'-''^ (s) '^S^^) (^g) (then (6) is a point-like object) . When Rb-^0 we

in fact remove all the intermediate excited states already mentioned: In this case

ARE NEGLIGIBLE, AND

DOMINATE

Having written down the formula (3.2) this is a good place to discuss it a little further. As we

have already said, it would be very interesting to test formulae of the type (3.2) against some experimental data. There is, however, very little data in existence to analyze. To the best of my knowledge,

only deuteron-deuteron scattering data are available, but reliable calculations are very difficult because of the high spins involved. Nevertheless, there exist some calculations [Si] and there seems

to be reasonable agreement between theory and experiment. But we shall talk about comparison

with experiment at other occasions.

Some special cases of formula (3.2) were also employed to describe hadron-hadron scattering

in the high energy limit. For example, the limit when A and B become very large was considered

[13] (compare also [12])

:

lim 9Tl= — f

d'-bexp (zA-b) 1- exp(-AB [ dV"WV''Ks<'")7 (b-s('"-s('^')p(''Us<'") )

(3.4)

where the p's were defined before and we assume that all yji's are the same. One gets this formula

trivially from

3TZ^^ j

d^feexp (zA-b) 1- ^1- j

dV^W»p(«'(s"'')7(b-sW+sW)pW(s('"))

j

as AB—^'Xi. But this formula has no intermediate excited states, neither of (b) nor of (a). So, the

Chou & Yang [13] limit A, B-^ oo looses all excited state contributions and becomes (3.4) . Equation

(3.4) gives the well-known "droplet model" [13] elastic scattering amplitude of two composite objects

whose hadronic matter distributions are given by p'-"^ (s'°^) and p^*"' (s'*^)

.

If one assumes that y(b) is a very narrow function of b (hence the components of the two

hadrons are very small) we can write

J

d^sWd^s^pf^^ (s(''))7 (b-s(*) +s('^')p(''^ (§('')) j

d-spW (s)p(*> (b+s)

= j

d'qexp (-zq-b)F(„)(g)F(6)(g),

where k is a free parameter.

14
If we accept that the densities of hadronic matter are the same as charge densities, F(^a), I'\b) are the

charge form factors of the colUding hadrons. This formula was used successfully to:

(i) reproduce the proton charge form factors from elastic scattering hadron-hadron cross

sections.

(ii) predict diffractive structure (e.g., diffractive minima) of the high energy hadron-hadron

coUisions.

The very recent measurements of p-p elastic collisions confirm the existence of such a structure (CERN-Serpukhov experiment)

In the form given above, the droplet model is very crude and I do not want to go beyond this qualitative description. One should perhaps mention at this point that the amplitude (3.4) contains

the geometric shape of the colliding objects (e.g., their transverse density distributions). If these

geometric characteristics do not depend on energy, one gets the total cross section (from the optical

theorem) which is energy independent. So, it seems to be difficult to reconcile this model with the

recent evidence for the increase of the total hadron-proton cross sections at very high energies

(compare the data e.g., analyzed in ref. [14])

Selection of formulae taken from a standard partial wave expansion [86]

fei{k,e)=^ i: (2^+l)(^;-l)P:(cos0),

ZIK

(re,(A;)=TX2 J2 (2^+1) hi-l|2,

1=0

4x

<XTOT{k)= — Im/el(/c, 0), (TTOT(k) =ae\{k) +arine\{k) , k

aTorik) =27rX2 (2^+1) (1- Re vi)

1=0

<Tinel(fc)=7rX2 (2^+1) (1- \ Vl \') 1=0

Pi(cose)^/o(2(/+3^) sin H^), sm'}^d«l, (3.5)

f,,{k,d)^^ r dl{l+y2)Joi2{l+y2) sini^e) im-l),

Ilk J 0

5A; = Z+i^, kdb-=dl,

/el {k,e)^-ik r dbbJo (A&) [,7 (&) - 1],

•'0

A = 2k sin ij OA =r]iOptical theorem and. unitarity

Let us first consider the "elementary" collisions (whose scattering amplitude is determined by

the profile 7(6)). As the wave passes a scatterer it gets modified by a factor 1 — 7(6).

Hence, the probability that the particle gets removed from the incident beam is 1— | 1 —7(6) [- =

2 Re 7(6) - I 7(6) p (at the impact parameter h). Notice that here we use the same expression as

15
in the following paragraphs: we identify l — rj with 7, and 1— [ ?j |- = 1— | l — y\'\ Hence,

<Tinei= /

d'b[_2Rey{b)-\j{b)

As o-ci= j d^b \y (b) p, (see Remark below) we have

(TTOT= j

d''b\j{b) |2 + j

d'b[2Reyib)-\y{b) p]

= j

d%2Rey(b).

We have, however.

So,

/(5)= ^

/" d^feexp (z5.b)7(6)

ik C f{0)=- d%y{b).

4lt r Im/(0)=

I d^b2'Rey{b)=(7TO'r.

k

Hence we do have the optical theorem built into our model

Remark

:

da = 1/(6) |2, C7el= j

dn\f\'

= j

dd sin d dct> \ f l"" ^

"(Fy^/ I

^'^^^P (^S-b)7(fc) /

dW exp {-ih'h')y* {b')

For small 6 we have

:

d''8 = dd8d(t>^¥ddedcl>,

hence

Noiv let us go over to composite targets.

Consider the case when one "elementary" particle scatters from a "composite" nucleus. In this case the profile is {r)=(s['oi r(b;si...sA) 1^0),

and we can write the same relations as before:

aTOT= / d'b I (r) p + j

d'b[2 Re (r)- | (r) |2]

because, due to the same arguments as before, 1— | 1 — (r) 1^ gives the probability (at the impact

parameter b) of losing the incident particle from the elastic channel. It is convenient however to

16
split the second term into two physically different contributions:

/ d^bl2 Re (r)- I (D p]= I

d%\:{T+T)- I (D |2] / =crDT

+ f

d'b[2Re <r)-<r+r>] \ =crpROD

Interpretation

The first contribution (o-dt) comes from processes during which the target gets dissociated

without producing any new particles:

<7DT= f

d'b[J^ (0 I r+ I n){n I r I 0)- I (0 I r I 0) |2]

= /

d^'biZl <nl r|0)p- I (0 1 r|o)ia —the second contribution (ctprod) takes care of production processes coming from the nucleons of

the target nucleus:

0" PRO

= j

d'b[2 Re (r)-(r+r)]

where the "reflection coefficient" t; (b, si . . . sa) (compare formulae (3.5)) is related to r(b, si . . . sa) as follows:

1 — 77 (b, si . . . S4) = r (b, Si . . . Sa) . Hence 1— | ?? (b, si . . . sa) P gives (compare the formulae (3.5) of the standard partial wave

analysis) the production cross section at the impact parameter b~ (l-\-]/i)/k with all nucleons

frozen at the positions si, . . . s^. So, in our model there are three different contributions.

17
But as long as we construct the profiles of the target nucleus from profiles of elastic scattering, the

processes like the one shown in figure 7 (with excited states of the projectile present at intermediate

steps) are excluded.

They are the source of the so-called inelastic screening (or inelastic shadowing) phenomenon [47].

In order to include them we have to ascribe some kind of structure to the incident particle. Earlier

in these notes we gave some examples of such cases. To analyze this problem in more detail, one has to link it with diffractive production processes and we shall postpone such a discussion until our analysis of such processes. Here, let us make only

the following points:

(i) Diffractive production processes are presumably weak (at least at energies of a few GeV)

compared to elastic scattering processes (the cross section is '^3^10 of elastic cross section)

.

(In fact this is one of the very important questions to be answered by the very high energy

experiments of the future: how much cross section goes into diffractive production

processes.)

(ii) Nondiffractive processes are presumably not contributing to the inelastic shadow

because the whole configuration of the target would eventually have to go back to the

initial one—a very complex process in which the whole of the nucleus must take part

(hence it occurs with small probability)

.

(iii) Hence "inelastic shadowing" stands a good chance to contribute little (a few percent) to the elastic cross section.

If this is so, then the three contributions to o- tot discussed above do approximately exhaust the

list of processes contributing to elastic scattering.

From our discussion of the components of crTOT(o'ei, cdt, cprod) it follows that the measure- ments of o-TOT may be a good way of finding out whether the inelastic shadowing (or inelastic screening) corrections are important at very high energies: If one computed cr tot from the Glauber

model (including all possible effects which the model allows for) and then found a definite dis- crepancy with experimentally measured utot—it would very strongly suggest the existence of

inelastic shadowing phenomena described above. In fact such an analysis has recently been done for

ir-d scattering and seems to indicate the existence of such a discrepancy for energies above ^40 GeV

[48].

The remaining important corrections to be discussed (although they are, in principle, included

in the algorithm presented above) are: (i) the Coulomb corrections which play an important role in

elastic scattering from nuclei of charged hadrons, and (ii) the corrections for the cm. motion which

are important for light nuclei but unimportant for heavy ones.

18
Let us consider first the Coulomb corrections for heavy nuclei. Ona can, in principle, use the

individual amplitudes which have Coulomb interactions built into them (this very tedious cal- culation has been done, e.g., in refs. [16, 17], but we shall consider the; effects produced by the

average Coulomb potential produced by the whole nucleus [15] which produces almost identical

results [16, 17].

We shall assume that, in the high energy limit, the total phase shift is the sum of the Coulomb

phase shift (xc, the phase shift one would get if the strong interactions were switched off j and the

strong interaction phase shift (xs, the phase shift we would get if the Coulomb interactions were

switched off; for Xs we have the expression xs= X)j Xu)- This assumption is, of course, obvious in potential scattering.

xib) = -- dzlV. (b, z) + Vc (b, ^) ] = x» (b) +xc (6) ^ •'-=0

3H = zfc fdb bJo {Ab){l- exp [^ (xc (b) +Xa (&) ) ]

}

•'o

Xc is purely real but xs is not Xsib)=im+Ub),

Im3Tl = A; / dfe feJo(A6) { 1 -e-f^"' cos (xc (b) +^(6)

'RemL = k f

d6 6J"o(A6)e-«Wsin (xc(fe)+^(6)). (3.6)

Hence we do not add amplitudes, we add phase shifts. The cross section is

= 1 Im gn |2 + I Re 3n 11

The amplitude (3.6) has some simple properties which show that the Coulomb interaction may

help us in learning about the real part of the strong interaction phase shift, l{b) (which is, as a rule, not well known: we know pretty well the absorption, which is given by ^(6) because this is the

dominating process, but not 1(b)).

If the Coulomb interaction is absent (xc = 0) the elastic cross section is invariant against the

change of sign of I. If, however, Xc^O some drastic changes may be introduced by changing the sign

of I which is equivalent to changing the charge of the incident beam of particles. For isospin zero targets

(^He, ^^0), if one finds no difference between the elastic cross section for 7r+ and ir~ it implies that

there is no real part in the ir^-nucleus elastic scattering strong interaction phase shift. When Xc = 0, 1 — e~^^*> cos ^(6) and e^^'*' sin|(6) go to zero for b>R {R is the radius of the

target). They have, in general, quite different shapes, however—hence Im 311 and Re 911 oscillate differently. They are out of phase and since \ {b) is small in general, | Im 3TI | > ] Re 3TI |. If, however, Xct^O, the situation may change dramatically: x<:(^>) rnay 'stabilize' the arguments

of cos (. . .) and sin (. . .):\{h) decreases; however, the Coulomb phase shift x/ (?>) ^ (^e-A') In {kb)

increases with b. This last expression is the Coulomb phase shift produced by a point charge. If

Xc-\-\ varies around n-ic, the situation is more or less the same as in the case x<; = 0 (scattering of neutral particles). If, however, Xc-\-\ stabilizes around (ra+3^)Tr, the roles of real and imaginary

parts may be interchanged: Re 3n may become large and Im 3TI small. The Conclusion: The Coulomb interactions for large nuclei are, in general, important for all angles

and momentum transfers. In order to compute the amplitude one has to bear in mind that at large b, Xc {b) behaves like a Coulomb phase shift produced by a point charge and hence diverges logarithmically. But we do

know the analytic expression for the Coulomb scattering amplitude of point-like charges

19
gn/'P' = ik j d'b exp (fA • b) [1 -e'^^^^

n exp [—2m In sin {^26) +2/(To]

2k sin2 (i^e)

^ exp [ - in In ( AV4&2) ]e2«>o AV2/b

where n = Ze'^/v, (70= arg r(l+OT).

and hence we get the convergent expression for the complete ampUtude by adding and subtracting a Coulomb point charge amplitude

:

311

/CO y- 00 db bJo (A6) [1 - exp (^x/ (b) )^+ik / db bJo (Ab) exp [?x/ (&) +^xs (b) ]

u

•'0

= 3n.(^)+^A; r dbbJo{Ab)\:ex]){ix/ib))- exp (zxc (6) ) (1- r (6) ) ].

•'0

This last integral has no divergences anymore (although x/(&) and Xc{b) both diverge logarithmically at large b) . In general Xc (b) has to be computed numerically

V J

I r — r I where r = (b, 2) . Note that

limxc(6) = - — / - / dh'pAir')

Hence, for large b, Xc{b)^x/ (b) and the integral for 3TI converges. Let us construct Xs (b) in the case of A large (a large target nucleus) . We assume (for the sake

of simplicity) the independent particle model wave function of the nucleus:

1- exp (zx.(6))= f d%. . .d'sAllpi^j) (1- n (l-7(b-s,)|,

exp [%(&)]= ^1— yd2sp(s)7(b-s)^ > exp —A j dh p{s)y{h — s) .

large (In order to perform a careful limiting procedure one should keep e'x^^''' under the integral

sign of the expression for 3H [12]. Generalizing slightly (allowing for different neutron and proton

profiles and densities) we have (A'' number of neutrons, Z—number of protons)

:

%(6) - - iVTn (0) p„ (fe) - Ztp (0) p, (b)

.

When 7„(6) and yp{b) are very sharp compared with p„(s) and Pp(s) we have

^X. (b) - -Njn (0) pn (b) - Ztp (0)pp (6)

.

ik r As f (d) = —- / d-b exp- (2:5-b)7(6), when jQ)) is very sharp compared to 1/5 (hence we limit our27r J

selves to forward scattering processes) we can approximate 7 (6) =7 (0)5^^' (6) , hence

/(0)=^7(0), and

20
^-Ny2<Jn{l-ian)pn{h)-Zy2ar,{\-iar)pj,{h). f3.7)

where a„ and Up (the ratio of the real to the imaginary part of the forward forward scatt(!ring ampHtude) are defined by /„,p(0) = (i +a„,p)/c(r„,p/47r where Un.p are the total cross section for scattering on either neutron or proton. (Incidentally, one can define an optical potential — (1/f) dz 'Fopt(b,2) = Xs{b) which is equivalent to our multiple scattering description).

From this expression (3.7) one can see that the interplay of Xc{b) and ^(6) is, in this optical

limit, determined by the size and sign of a„ and a^. Some calculations were done [15] with p(r) =

Po(l-exp \_{r-R)/c])-\ For ^ospb, 72 = 6.5 fm and c = 0.523 fm. The densities p„(6), pj,{h) were

obtained by integrating p(r) over z. The parameters of xs(^) were taken from proton-nucleon scat- tering cross sections. For a„ = ap=— 0.33, o-„ = o-j, = 38.9 mb (these parameters are resonable for ~20 GeV protons) , one gets the following table

xc{h)vm h (rad) (fm)

8.43 0.0050

8.45 1.57 8.41 3.28

8.42 4.19

8.32 5.24

8.15 6.15

7.99 7.33

8.15 9.16

Note that ^^7r = 7.85, ^-^n- = 9.42. This means that near to the nuclear boundary sin {xc{b) -\-\{h))

is large. It is amusing that numerically Im SKI with proper xc is approximately the same as Re 911 without Xc! This is true for ^^^Pb. In general one gets all kinds of intermediate situations. In any case, the influence of the Coulomb interaction is very important "everywhere" as the figure below (see

[15]) for a 2°^Pb target and incident neutral-, positive-, and negative- particles which interact

strongly as 20 GeV nucleons.

5 10 15

21
For heavy nuclei there are virtually no experiments with good enough resolution to have only

pure elastic scattering (in which the target nucleus stays in the ground state) . In order to have a genuine elastic scattering one would have to have an energy resolution l^E a fraction of an MeV,

which for £'^20 GeV is still inaccessible. Most experiments (e.g., CERN series-compare [SI])

have poor energy resolution of the incident and outgoing beam ('^50 MeV), hence they sum over

all nuclear excitations (without producing mesons, however). The cross section for such "inclusive" processes is

j

d'bexp iiA'h){<ifn | r(b;si . . . s^) | ^o)

f d% d'b' exp pA- (b-b')]<^o i r+r I ^o).

(2r)

This cross section includes, of course, the elastic cross section. The cross section which, upon integration, gives o-DT is

do, dfi dfl

x[(>i'o i r+(6)r(6) I ^o)-(^o I r+(6) | ^o){^o I r(fe) | >^o)].

It is an interesting fact that while doei/dO, is very strongly influenced by Coulomb interactions

(as we have seen) , da-oT/dO, is influenced very little. In order to make this fact more plausible, let us

consider a collection of neutrons and protons which do not screen each other. Then, we would have

ddDi

do. N\fn{^) Y + z\f,{^) \\

no screening

One may suspect the following:

If we introduce screening there will be, on the average, a certain fraction of nucleons inaccessible

to the incident hadron. Hence the above formula can be applied to a certain "effective" number of nucleons. Indeed one can show (compare ref. [15]—the calculation was done with the Coulomb

interactions present) that to a good approximation (note that since this formula does not exhibit a forward dip, it is not valid for small A)

da-DT

do, a f

|A(A) l^ + ||/p(A) 1^

where the "effective number of nucleons" is

a^A j

d'^b p(b)e-'^''^'>\

where a is an average total hadron-nucleon cross section. If indeed aoT/dil has such a form, the only

place where Coulomb interactions enter are in the individual proton amplitudes, l/p(A) p. But

there we know, e.g., from the proton-proton elastic cross section, that Coulomb interactions are important for very, very small momentum transfers only. In any case, they enter incoherently into

da-DT/dQ.. These two factors make Coulomb corrections insignificant in da-oT/dO.. How important are the details of the target nucleus wave function? Not very important. The

most important are general characteristics: density distributions (hence possible deformations) but not internal correlations. From the published analyses of hadron-nucleus scattering (see e.g., [S2],

[15], [3]) one may conclude that

:

22
(i) the shapes of target nuclei are the most important factors determining the cross sections

(ii) the internal correlations of nucleons in the nucleus are unimportant for da^JcKl or da,.\ d9,. They are of some importance for danr/dQ, (especially at small momentum transfers [15],

[3]). The confrontation with experiment is impressive. (Compare, e.g., the review article by

R. J. Glauber in ref. [S2]).

When we want to discuss light nuclei we have to consider carefully the motion of the center of mass. Take, for example, a deuteron: here taking into account the cm. motion is trivially accomplished by using the wave functions of the relative motion, (j) (r)

For example, the elastic scattering amplitude is

ik

311 (A) = ^ Id% exp (iA-h) j dh I 0(r) \^ [Tp(b-Ks)

+T.(b +i^s)-Tp(b-3^s)7„(b +3^s)].

In the case of more complicated targets the situation is much more involved and often leads to some serious computational problems. Let us introduce the transverse component of the cm. vector

1

and the relative coordinates

Sy — Sjr* T J Sy — I r which are not independent any more

i:s/=o.

3 The operator

ik r 2az(A; si . . . sa) = — / d% exp (?A-b)r(b; si . . . s^)

2ir J

has the following property (see below for the proof)

3n ( A ; si . . . Sa) = exp (i A • r) 911' ( A ; s/ . . . sa') (3.8)

Then we can compute the correction factor to 2fll= (311 (A; si . . . sa) ) assuming the wave function

to be in the form of a product of the cm. wave function and the internal wave function.

311= <(R(r) I exp (zA-r) | (Jl(r) )(*o(ri' . . . ta') \ 311'

| $o(ri' . . . ta') ) (3.9)

This is the corrected amplitude.

23
I

Hence if we can factor out the cm. wave function from the product ^'o= II; (t)j{rj) we can stick to calculating 3H with but we have to multiply it by a correction factor:

<(R(r) I exp (?:At) | (R(r) )-K

This can be done explicitly in the case of oscillator potential wave functions (this is partly the reason why they are so popular!) . There

(R(r) = (A/7r3R6)iM exp {-Ar'-/2W)

where R is the size parameter in the Gaussian factor in harmonic oscillator wave functions:

exp (-rV2R2). Then

((R(r) I exp (iA-r) \ (R(r))-i= exp (A2RV4A).

When one cannot do this factorization the computations become quite involved {s/ are not in- dependent!).

The proof of (3.8)

:

In

m{A;si. . .sa)= ^

f

d'-bexp (iA-h) |l- H [l-7;(b-sy)]

replace

Sj = r +Sy,

then we shall have b — r instead of b. After changing the variable b = r+b' we get the

factor exp(zAT) in front, and the formula (3.9) follows. One can write the general formula which takes into account the interdependence of internal

coordinates by introducing a Dirac 8 function into the amplitudes

:

(A) = ^

y" d% exp {iA-h) I

<Fn . . . dV^^o*(ri, . . . r^) |l- H [1 -7;(b-s,)]

X^o(ri, . . .rA)5(3' T.r)j .

This 5'^' function eliminates redundant excitations of the system of A nucleons. When one cannot

factorize the cm. coordinate and one has to use the above formula the numerical calculations become

much more involved (from trivial—they become difficult [18])

.

An illustrative example: the ground state wave function is a Gaussian [19]. The ground state

densities and the elementary amplitudes are taken in the form

I ^0 P = ilpir,), p{r) =po exp (-r^/R^)

{i+a)k(j

fik) = exp i-y^aS^).

4ir

Then the elastic scattering amplitude (with the cm. motion correction included) reads

(R2 A2\ ^ / (t(1 — ?a)

exp

_27r(R2+2a)_

(R2+2o)A2

4j

24
Many general features of the multiple scattering are included in this formula:

(i) If we neglect a, the amplitude becomes purely imaginary (absorptive). A geometrical

picture of single-, double-, etc. scattering contributions is as follows:

log I m I

(ii) With this picture it is easy to establish the existence of diffractive minima, which are

filled by the real part of 3H. (In order to have Re 911 we have to have a^Q)

(iii) The importance of the cm. motion correction can be seen from the factor exp (R^AV4A)

For small A (say A =2, 3 or 4) it can be a correction of as much as 2 orders of magnitude

for A2^0.3 GeVl

A few concluding remarks about the deuteron target. A lot of attention was concentrated on the deuteron because it is a very important testing

ground for multiple scattering theories (or models)

(i) In experiments (compare deuteron data contained in [Si]), one can clearly see the single

and double scattering.

A

Remark: In fact, this clear distinction between single^and double scattering was used to extract the

p-nucleon total cross section in y-p production experiments on deuterons (see section 4.2)

25
(ii) One can also see (again, compare [SI]) how important the deformation of the target is (existence of the D-state in the deuteron ground state) . Let us discuss this effect in more

detail. The ground state deuteron wave function is

:

4>m{r) = (4T)-i'-2r-i[u(r) +S-'''Suiv(r):\xi.m (3.10)

where u (r) and w (r) are the radial S and D functions and

»Sl2=[3(diT) (d2-r)-di-d2].

di, 62 are the Pauli spin operators, and r is the neutron-proton relative coordinate, xi.m is the spin

function for spin 1 with the magnetic quantum number m. The elastic cross-section is then

dael 1

with

,0 „ Z Km I 311 (A, s) \m')\'

3n(A, s)= ^ f d^exp (?A-b){l - exp pXp(b-Hs)+%(b+Hs)]K

which operator, in this approximation, does not depend on spins. So, if not for the Su term in (3.10)

,

we would have (to | 3H ] m' ) = 0 for m^m'. In fact the {m \ 9TI | m' ) contributions are indeed the most

important but they always lead to a sharp diffractive minimum:

But (m 1 Sia 1 to')?^0 in general (also {m \ m')?^Q) . This matrix element enters

(to I 911 (A, s) I to') and results in spin-flip transitions (classically: rotation of the deuteron spin)

which have completely different "profiles" than (to | 3n ] m), thus resulting in oscillations which are out of phase with oscillations of (to. | 911 1 m) and fill the diffractive minimum:

26
Incidentally, only the spin of the deuteron as a whole is essential. The qualitative effect is independent

of the spin of the incident particle (the 371 (A, s) operator does not act on spin quantum numbers.)

All the other spin effects are presumably not important.

(iii) Calculations such as the one above, as well as more sophisticated calculations, have always

produced cross sections in excellent agreement with experimcmt. (We are not considering

here backward scattering, where the above model does not apply (see also [48]) ) . There

is only one exception: the experiment performed at CERN by Bradamante et al. [20]. In this experiment the discrepancy with theory occurs at a fairly large momentum

transfer (A^~2 GeV^). What is the cause? Perhaps some relativistic effects? There is no

good answer, so far. Without going into any explanation of this discrepancy, let us emphasize the following point:

It is important to realize that when we use the same internal wave function in the initial and

final states, we exclude, by doing this, any possible relativistic deformations of the recoiling target

(we are still discussing only elastic processes). For large momentum transfers {A-/Mt'^'^V) this is probably not a good approximation. Take the deuteron example. In the standard Glauber model,

it is enough to have p(s) = /"^ d20o*(s, z)ct>o{s, z) to compute the cross section. Suppose there is some deformation in the final state:

<^o* (s, 0)-^<^o'*(A, s, z)

(one can assume that the deformation is defined by the momentum transfer A) . Then we should

replace

/+« dz0o'*(A, s, 2)0o(s, z) —

and the amplitude is

ik r 3TI(A) = — j

d=^&d2se^^-b/(A,s){l- exp [fxp(b-Hs) +%(b+Hs)]}.

The interesting fact is that in exactly the same form one can write the Delbriick amplitude

3^Delbruck(^)_ j

(zA-b)/^^, s){l- CXp [%- (b- ^s) +ZXc+ (b +J^s) ] j

where Xc"^ are the Coulomb phase shifts of the electron-positron pair and (A, s) is constructed

from

the "relativistic wave functions" in an analogous way to that shown above in the case of the

deuteron. Here the possibility of a well-defined procedure of introducing relativistic deformations

occurs—modeled on QED! These and other related problems have been discussed in a series of papers by Cheng and Wu [21, 22] (see also [30], [S4])

27
4. DifFractive Dissociation and DifFractive Excitation

Diffradive processes—a brief characterization.

(i) they do not vanish in the limit > <x>

(ii) the target plays a passive role (except in double diffraction, but in any case: no quantum

numbers are exchanged)

.

Examples

:

in QED; elastic electron (positron) scattering from a Coulomb field, Delbriick scattering,

Compton scattering, etc. in hadron physics ; all kinds of elastic hadron-hadron scattering

da_

dA^

=/(A2)

(experiments seem to indicate that the

differential cross sections depend weakly

on energy)

inelastic processes:

-IT—>Sir

TT—>57r p^n+ir

K—>Kinr, Kirinr

etc.

on nucleons

or nuclei.

The nucleon and nuclear targets supplement each other because the nuclear medium amplifies the

scattering of the produced objects.

The model of diffractive processes described below is based on: M. L. Good and W. D. Walker

(1960) [23]. The article which discusses some very early papers on the subject is: E. L. Feinberg

and I. Pomerancuk (1956) [24]. For more recent discussions of many experimental and theoretical

aspects of diffractive processes in hadron physics see refs. [S4] and the article by A. Biafas in [25].

We shall describe diffractive production processes in very close analogy to diffractive dissociation

phenomena which are well known in the case of systems where degeneracy exists. Let us start with an example taken from optics. Consider the absorption of polarized light by

an anisotropic absorber. The incident wave is polarized in the direction n (perpendicular to the

z direction)

.

n = {n^, ny)

'^n = nSfx+ny<^y

where is the wave polarized in the x direction and is the wave polarized in the y direction.

Nicol absorbs completely

this component

28
Suppose the target is a Nicol prism oriented in such a way that it stops all light polarized in the y direction. Hence, the only component which goes through is n^^^. But it can be decomposed into

n and nXe^ components. Hence due to the process of absorption, a new object is created: the wave

which is polarized in the direction nXe^.

Let us compute the elastic and inelastic scattering amplitudes. Since the transmitted wave is

if = the wave which goes into scattering and production is

But

hence

Ki = n^-, Xinei= — nx^j,. Geometrical picture

THE INCIDENT WAVE THE TRANSMITTED WAVE

ABSORBING TARGET

Z = 0

The exact solution is

[(p, z>0.

The "undisturbed" wave is '^n everywhere, hence the scattered wave is

(0, 2<0

i^n-^, Z>0.

Actually it is more important for our purposes to introduce partial absorption (in general different

for the two components (x, y)).

The incident wave : = Ux^x +riy'^y The transmitted wave : (p = tj^nx^x +Vy'^y^y

The scattered wave : '^n — 'P={l — Vx) nx'^x+ {^ — Vy) Vy'^y

and we get

Ainel = nxny('7!/ —

This formula shows that we always produce inelastic scattering, except in two cases:

(i) when the incident wave is polarized either along the x or y axis (hence either iix = 0 or Hy = 0)

(ii) when the absorption coefficients are equal (the absorber is isotropic)

29
We shall extend this description to diffractive production processes of hadronic systems. We

consider the incident hadron to be a superposition of some states which get eaten up at different rates during the passage through the target ; the new combination emerging from the collision then

contains, in general, a new particle (or a collection of new particles)

.

First we introduce the physical states of the system, | X,) (which are analogous to the states \l/n and i/'nxez of the photon). We want to compute \ T \ X,). We expand | X,) into a set of states

I Xi) whose scattering and absorption in the target we assume known:

h)= J2 du I Xy)

j

Xi) = ^ Cij I Xj)

I X,), I Xi) form

orthonormal sets (4.1)

•'' J of states.

I '^i}= H c,7 dji I \i)= I Xi), hence

I 3

Cij dji = bii.

The states | Xi) are assumed to be eigenstates of T in the following sense:

So,

This is the "heat" produced

in the absorber. All these

states are orthogonal to all states which appear in diffractive scattering.

T\\)={h \ Tj^djuil-ri.) \ K}

n

n I = (Xi I dj„{l — r]j +r]j — r]n) Cni \ li}

n I = (Xi I (l-rjj) X J2 djnCnl I ll)- X djnCnl(Vn-Vj) (^i I h) In n I = {l—r]j) XI djnCnl^il—

^

djnC„l{7]n—r]j)dil

I n n I

(h I T I {1-Vj)dij- X djnCniiVn-Vj)- (4-2)

We have obtained the result completely analogous to the one obtained for the optical diffractive

production: for i^^j the production amplitude is proportional to the difference in absorptions of the

i, j components. This is a very general property. All specific models of diffractive production processes I know of exhibit this property. Otherwise the formula is so general that it has virtually no

predictive power.^

The difficulty in applying it to any realistic process is the determination of absorption parameters -,7i because the states | Xi) are not observed in scattering experiments. (The process of Kl°^Ks°

regeneration given below is an example where we know iji's however!) The situation changes when

' One must, however, keep in mind that in the case when the coefficients dij are zero or of the same order of magnitude one does predict some characteristics of production processes from the knowledge of elastic scattering.

Compare the end of this section.

30
we accept that diffractive production processes are weak compared to elastic scattering. This may

mean that the transformation from | Xi) to | X,) differs little from unity:

Cij = 8ij+eij) tij small, hence terms

(4.3)

dij = bij—tij) can be neglected.

Remark : The minus sign guarantees the property

3 j = 5,7 +6,7 — e,7 = 5ii. Then our basic formula takes the form :

{\i I T I \j)= {l — rij)dij— XI iVn — Vj) i8jn—ejn) {8ni +eni)

n = {l—Vj)8ij— {r], —ijj)8ij +eji{r]i-rij)-eji (j]j-rij)^

zero zero = (1 — ^j)5o +«J<(l — -^j) — €>i(l — J?-)- In this approximation

(A. I T\h) = l-r,i.

Hence the absorption parameters rji are determined by elastic scattering of real particles. The inelastic amplitude

7^|X,)=(l-r?,)e,,-(l-r;;)6,y (4.4)

is proportional to the difference between the absorption of the produced particle and the absorption

of the incident particle.

One still faces the problem of specifying the absorption parameters rn and the coefficients e,y. The coefficient tj, of the incident particle is, as a rule, easy because this is a well-known particle which

can form a beam and its scattering (elastic) properties are known reasonably well. The trouble is with the outgoing objects; e.g., when Stt are produced in the ir—>37r reaction: are then its rj's given by

the absorption of 3x in the target? In fad, one usually determines them experimentally (see the end of

this section). So far as the e,; are concerned, they are small—hence some perturbation theory can

be used to compute them. We shall give some examples further in the text. How does one implement this program? There are strongly interacting particles which realize precisely the above outlined scheme and we even know di/s and ri/s: the neutral K mesons. Because

of the relation

charge baryon no. strangeness isospin

Q = y2NB + + T,

the partners of K+, K" particles are A'~, K° antiparticles. Hence there are two different neutral K

mesons which can be produced in the collision of strongly interacting particles : K° and K° (they are

different because they have opposite strangeness, unlike pions where tt" are identical to which

have the same masses, and thus can be considered as a two component degenerate system.

When left in empty space, however, both K° and K°, decay weakly with two different lifetimes as if they were made up of two different particles, which is indeed the case. These two particles are the

following superpositions of | Jv") and | J^") states

I Ko)= [(1+5) I K^)- (1-5) I A'o)]- ^ (I A'o)- 1 K")) = 1 AY)

| Ax°)= [(1+5) |A0)+(l-5) |A°)]-^(|Ao)+1A:°))= |Aa«),

31
where 5 is a small complex number, | d \ '~10-^«1, which gives a measure of CP nonconservation.

I Ki") and | AV) are eigenstates of CP. Indeed

P\K°)=-\K°}, P\K°}=- IK").

C I K'>)= I AO), C I K')= I K")

CP\K'>)=- \K°}, CP\K°)=- \K'>}

but

Similarly

CP - (I A'o)-

I AO)) ^ (lAV- |A'o))= -| (I AO)- I AO)).

CP I A20) = - I A2O).

The particles A^o and Ajr,o have the following lifetimes

T.~ 10-^0 ri~10-«s.

We observe these two different decays by looking at decaying A mesons in the beam.

TARGET WHERE K ARE CREATED

~50% of K° /DECAY

FASTER

-50% of K°

DECAY SLOWER

DISTANCE FROM THE TARGET

So, after some distance only the Kl° beam is in existence. When we let the beam hit another target we can regenerate As" mesons because A° and Ao interact differently with matter, they are absorbed

differently, (e.g., K°(S= -1) +p~^A°{S= -I) +Tr+ while A0(S = +1) cannot produce AO). In

figure 18 the so-called transmission regeneration is sketched.

32
One can, however, also observe diffractive production of A'/' from A/," on individual nuclei. Cross sections for such diffractive coherent production processes wert; rc^centl.y mciasured for copper

and lead nuclei [26]. In order to measure these cross sections, one has to get off the forward direction

where the transmission regeneration (which comes from a coherent process whose coherence extends

over the whole block of matter) dominates. The amplitude (neglecting 5) is

2»KKi«->K/ = K<A'' I T I A°)-H(^° I T I AO).

Hence ^kl'^k/ is given by the elastic scattering amplitudes of A" and A** from the nucleus

(AO

I T I Ao) = 31Zk»(A2) = ^

/" d^bexp (zA-b) {1~ exp [%»(&)]}*

ik C

(7?o| T| Ao) = 9nF(A^)= - / fF6exp (fA-b){l- exp fc« (&)]},

where, as was already shown for copper and lead target nuclei, it is enough to take the large-A

approximation

(6) = - iV ^/^o„ (0) (fe) - Z ^/^o^ (0) (6)

and an analogous expression for A", where

Pn,p{^)=Poj dz 1+ exp ^^"'^^ The final formula is

1 ikt r

-^Ks'-= 2 J ^^^6XP (^'^'b) {exp lixK'ib)'}- exp [tXic»(b)]lThe elementary amplitudes can be gotten from A^i^-nucleon scattering amplitudes assuming

isospin symmetry

kon (0) =fK -p (0) , /^o„ (0) =fK-, (0) ,

fK% (0) =fK *n (0) , f-K^p (0) =/k-„ (0) .

The standard way of calculating these amplitudes is

(i) the imaginary parts are obtained from the optical theorem, e.g.

Im/K±p= —- k.

(ii) the real parts from dispersion relations (for more details compare [26]) . The amplitudes

for neutrons are then obtained using some further acrobatics, as referred to in [26]. In any case, our knowledge of these amplitudes is rather poor.

The real and imaginary parts of the elastic A" and A"—nuclear amplitudes are sketched below.

The uncertainties of our knowledge of these amplitudes are also shown [26].

33
4

Imf (in fermis]

•3

•2 for ~4 Gey/c

Ref (in fermis + 1

From this picture it is clear that neutrons are much more effective in regeneration than protons

(the difference in absorption of K° and A'" is much bigger in the case of neutrons)

.

One can get excellent fits to the differential cross sections daKL^Ks/dA^ by making the neutron

and proton distributions different. One gets the following nuclear parameters from the best fits [26].

Rp

Cn{Rn=-Rp)

Pb

6.60 fm

0.50 fm

(7.29±0.13) fm

(0.68 ±0.04) fm

Cu

4.23 fm

0.57 fm

these are well known from

elastic scat- tering

(4.86db0.10) fm

(0.74±0.03) fm

Discussion of this example:

1. Assuming that we can trust the input data (structure of Ks" and Kl°, fR'n, fROn, /k'p, fWp

amplitudes) we obtained a very important piece of information about the target nuclei: the neutron

distribution. This is so because neutrons are more effective in regeneration than protons. In this case we have not, however, obtained any new information about the elementary processes and the

structure of Ks" and Kl°.

2. The main feature of the K regeneration process seems to be very general, however: The process of diffractive production consists in rearrangement of the "components" (understood in a very broad

sense) of the incident particle (system) : but components undergo only elastic scattering. This description is common to many models of diffractive dissociation (and excitation)

.

3. In describing the regeneration process Kl^^Ks" we drew heavily on the known structure of Kl° and K^": they are superpositions of K° and K°, whose elastic scattering from nucleons is reason- ably well known.

34
4.1. Generalizing to Other Diffractive Production (and Excitation) Processes

First of all, the components of the incident and the produced states are, in g(;neral, not degenerate: their invariant masses differ. This fact may introduce some important corrections at low

energies. But in the limit of very high energies and small momentum transfer, all such effects dis- appear. Let us take, e.g., two such states and give them the same momentum p. Then their energies

differ: E-E*= ^'p'+M^- \/'p-'+M*'

p large 2p

(The only important thing in these approximations is to have very large longitudinal momenta in the initial and final states.) As long as the time of the passage through (or the interaction with)

the target is

we can consider the states to be degenerate because their relative phase factor during the collision is very small and we have exp \_—i{E—E*)T~\^l to very good accuracy. In fact the same argument shows that the incident state and the produced state can also be

considered degenerate in the limit p—> oo (the fact that p will also change slightly during the collision does not change this conclusion) . Note that the degeneracy appears in the laboratory system, where the incident and produced systems move fast. One can also show that in this limit (p-^ <» ) the longitudinal momentum transfer can also be

neglected. So, in the limit p~^0 all the states taking part in diffractive production processes can be

considered degenerate and the longitudinal momentum transfer neglected. As was said, however,

for low p's some important corrections may appear.

Summing up:

Our procedure for evaluating diffractive production in the limit of very high energy consists

of two steps:

(i) Find a "plausible" model and identify the components of the incident and outgoing

systems.

(ii) Compute the transition matrix element by making the components scatter elastically

from the target.

If one can implement such a program, one can treat diffractive production processes on nucleons

(elementary targets) and nuclei (composite targets) on the same footing: the only difference is that

the components scatter elastically from a nucleon in the first and from a nucleus in the second case. We shall start discussing the approach outlined at the beginning of section 4 with restrictions

(4.3) because this scheme contains a large class of known models of diffractive production, including

diffractive processes in QED [21], [22], [27], [28].

Without going into any details let me sketch an example of such an approach in the case of the

process of proton dissociation p^n-\-Tr+. As in QED (compare e.g., lectures by W. Czyz in ref. [S4]) we express the states of the physical proton and the physical neutron-pion pair through the "bare"

states of the proton (p) and the neutron-pion pair (ifn)

Initial state: p=

P

TT

n = di'P

I p) +d2P

I TTW)

small

admixture

35
Final state: 7i-{-r= +

" =di^" \ 7rn)+d2''" 1 p). n

small

admixture

In the sense given earlier in these notes, for the purpose of describing elastic scattering we have

approximately (compare eq (4.4)

)

p~p, TT+p-TT+p.

The production amplitude is

:

37Z/i= <p I T\p)diP d2'"'+ (xn I T \ irn) do* d^"". Suppose the target is a nucleon, then {p\ | p) is taken to be the proton-nucleon elastic scat- tering amplitude, and {irn \ T | 7rn) is constructed (e.g., a la Glauber) from (t | T {it) and {n\ T \n)

pion-nucleon and neutron-nucleon elastic scattering amplitudes [29].

Suppose the target is a nucleus: everything goes the same way except that elastic scattering

amplitudes are taken to be with the nucleus (not a nucleon as before) . Again they can be computed

a la Glauber [29], [S4].

The standard noncovariant perturbation theory is an "obvious" tool to construct such states. Let H' be the interaction Hamiltonian which couples the states ] p) and | wtt). Let this coupling be

weak so that it is sufficient to take only the lowest order corrections

:

^fiVn, P^)) = Z

K, 0, 0)+ E f

"^^

'

' y 1 0, K K}+

0, Pn, p.)- 2^ _ _ I P,, 0, 0)+ • pp ^Pp ^Pn ^P-K (4.6)

where Z is the renormalization constant,

I /cp, 0, 0)= I p) is a state of a bare proton

I 0, k^r, kn)= \ ni:) is a state of a bare neutron-pion pair.

Note that the sign of the second term in \'^f) is "minus" because the order of energies in the denominator was changed. These two states should be orthogonal. Indeed

{^f 1 ^.) = 2^ UVnPAH'\kp) _ (kp I H' I p„p.)*l

because H' is hermitian, hence (kp | H' | p„Pt)*= (p^Ptt | H' \ kp). Note also that if H' is small,

Z ~ 1 to first order. So, we have

:

d,P = d,.«=l and d,''=-(d,''')*= ^^"^^i^^'y. (4.7)

Graphically, dz'^ corresponds to the vertex

:

36
If we accept our "diffractive elastic scattering" operator T to be constructed a la Glauber

(although, in principle, one can take something else for it, we prefer to use the Glauljcr model

prescription because it works so well for elastic diffractive scattering) we have

!r = <p(pjL — ki) +i„(p„j. — k„x) +<^(p^x — k^x) — <„(p„x — k„x)i,r(p;r-L — k;rx).

These four pieces of T produce the following contributions:

(i) <*,l<.i*.>=-E#=^^4^'.(px-kx), E —Fj — Fj p ^Pn '-'Vn graphically

^ , {VnK \H'\k)

(iii) {^f I U I •^i)= Z i.(p.^-k.x)

graphically
(iv) (^/|U,r|^i)= ^U{p^J.-K±)tnipn±

graphically

{KK \H'\h)

k

One can summarize the situation as follows: the coefficients ^2'' (or ^2"") give the amplitude for the neutron-pion fluctuation of the incoming proton. The total production amplitude is the difference

of the two main contributions. This represents both single and double scattering of the n — ir system

in which the diffractive elastic scattering occurs either before or after the fluctuation takes place.

Note that in these considerations we can have any target we want! The target is specified through the scattering operators t-,r and tn. Hence one can use the same technique to describe the processes on

simple and composite (e.g., nuclei) targets [S4].

Relation of the above model to some well established techniques of describing diffractive dissociation. First of all, our description is quite similar to that applied by Cheng and Wu [21], Bjorken, Kogut,

and Soper [27], and Jaroszewicz [30], for high energy bremsstrahlung and pair production processes

in QED (see also [S4]). It is also very closely related to some one particle exchange models of diffractive production processes first suggested by Drell and Hilda [31] and continued by Deck

[32], and M. Ross and Y. Y. Yam [33]. Our description also contains, as a special case, the so-called vector meson dominance models which are employed to describe interactions with hadronic targets

at high energy, a process which will be discussed later.

Example 1 : The processes of QED

:

(a) Bremsstrahlung : the Feynman diagrams

This represents both single and double

scattering of the n - ir system.

AND

38
go over to

where the dot • means that complete (to all orders) elastic scattering amplitudes are to be inserted

and the vertices are given by dij coefficients analogous to the ones in eqs (4.5), (4.6), and (4.7).

(b) Pair production: the Feynman diagrams

go over to

and in our formulation a double scattering process should be added:

which makes our description different from the above two Feynman diagrams (in fact, more complete) but in total agreement with the Bethe and Maximon formulae [34] for the high energy limit

of the pair production cross section in a strong Coulomb field [35].

39
Example 2: One -particle exchange process:

Drell and Hiida [31] started with the diagram

(a)

(a)

In such a model, the exchanged virtual pion scatters elastically (diffractively) off the target proton.

Later (see e.g., Ross and Yam [33]), two more diagrams were added:

(b)

These diagrams (except for the "diffractive" vertex, they are just Feynman diagrams) are intimately

related to our description. In order to see it, one should do some kinematics.

Let us work out some kinematical expressions associated with the vertex di" of the processes

(i)-(iv):

{0 < p <\)

where the four-vectors are denoted (pj., Pz, po) and we employed conservation of the three momen- tum at the vertex. (Note that since we are using noncovariant perturbation theory the energy is not conserved at the vertex.)

Let us evaluate all expressions in the limit w-^- ^ (longitudinal component of the incident

momentum very large). There are two independent variables which may be chosen as pj., /3. (Here

/? is an arbitrary parameter, 0 <;8< 1)

.

40
^ 20^/3(1-/3)

One can compute similarly the invariant mass of the n — ir system:

Mn.*'= ( \//3-w2 +p x2 +m„2 + V( 1 - /3

) ^c^^ +/J x2 +m;-= ) ^ - (px-px)^- (/3,o+(l-/3)a;)2

and the four-momentum transfers: -<p,= -m;'=+,3(M„*'-TOp2)

Hence, the following relations are valid

0^ ^ (1-/3)0' ^ fi^ _ ^/ 1

Since the Feynman diagrams (a)-(c) give the following contributions

V

9n(a)^A^P(SpO

tpn V

311(b) --^Pf (Spp)

. . ^ y

3H(e)-'A"P(s,p) -— ^,

where A's are elastic scattering amplitudes and V's are the vertex functions, we can see, using the

relation

A'^uatot and above proven equalities, that (assuming the vertices identical, which is the case for forward

amplitudes) 2ni(b)= — 3Tl(c), and that they cancel to a large extent (they cancel exactly in the forward

direction if app = apn)

We can also see the correspondence between our diagrams and the "one pion exchange"

diagrams

(i)?±(b), (ii)^(c), (iii)?^(a).

However, our diagram (iv) has no analogue in the Drell-Hiida-Ross-Yam-Deck model. The other

difference is the lack of four-momentum conservation in (i)-(iv) (only three-momentum conservation). This last difference may sometimes be relevant (see e.g. A. Bialas, W. Czyz, and A. Kotanski [29]).

Example 3 : Photoprocesses and vector meson dominance

It is now a well established fact that, at high energy, photons exhibit shadowing. The total photo cross sections for complex nuclei vary approximately as A°-^ in the few GeV energy range.

41
It is very suggestive then to accept that photons have in them strongly interacting components.

Again, we can use a perturbation expansion, where the interaction Hamiltonian gives the photonhadron interaction. We can write the physical photon state as follows:

^iK)) = z(i K 0)+ E I 0, n)+ ^ n I^y —lin

where 'n' labels all possible hadronic states which can be coupled to a photon. We can also write, as

before, the expansion of a hadronic state n

l^(n)) = z{|0,n)- E^^|^^1^.0)+.. j.

y Note that here the state | 0, w) is "almost physical." More precisely: as far as strong interactions go,

it is physical. The situation is very similar to that in the Kl-^Ks regeneration problem: we are expanding our physical states into states which are "physical in their strong interactions."

Since we know strongly interacting vector mesons with the same quantum numbers as the

photon, the simplest possible assumption one can make is to accept that the | 0, n) states are dominated by a vector meson (or mesons) . In fact, there is a well-known model of photo-hadronic inter- actions (Vector Meson Dominance model) which assumes just that. Some important consequences can be inferred from the above expansions even without specifying the nature of the hadronic components. Let us call m„ the invariant mass of the hadronic

component | n). Then

I Ey-En I = I CO- Vco2+m„2 I ^ — > 0.

2aj

ky (u),0,0.(j ) "^^^^^-'^^'^ = {w, 0, 0, + )

The time during which the hadronic vacuum fluctuation lives is A<=^ | (Ey —En) and this also gives the distance it travels {l = cAt). Hence, when l^2cc/mJ':^R (i2 = nuclear radius) we shall have

shadowing fully developed. We can see, therefore, that full shadowing occurs at high enough energies.

At low photon energies \ Ey — En \ is large and the corresponding fluctuation cannot interact with

the whole nucleus.

How can one test the VMD hypothesis? Let us denote the vector mesons by the letter V. Then

(note that Z = 1 in our approximation)

I ^,{ky)}= I ky, 0)+ E ^^"i^^'^^^ 1 0, kv), Y

Uly hlV

iLy — tLv Then the photoproduction of V (on a hadronic target) has the forward amplitude

<kv\H' I ky) an (7-^7) = (0, kv\tv\ 0,'kv)

Ey Ey

42
Similarly, the elastic scattering of a high energy photon (Compton scattering) from a hadronic

target has the forward amplitude*

91Z(7->7) = Z (0, kv \tv\0, kv)

Graphically

(A;^

I H'\kv){kv\H' I ky)

{Ey-Ey) (Ey-Ev)

(There are three known vector mesons: p, w, <^ with masses

Wp^765 MeV, m„^784 MeV, to^^1020 MeV)

One usually writes (a is the fine structure constant)

{kv\H'\ky)

Ey —Ev yV

Then

91^(7-^7)= E91^(7->F)

^

V But

ira yv

* Note that restricting the high energy photon-nucleus interaction to only vector meson interactions (VMD model)

is a very drastic step. There are other possible interactions of the same order of magnitude whose role one

should discuss: for instance all kinds of such Compton-like processes which do not belong to the VMD model, where

the photon is absorbed by the target (or part of it) and re-emitted. E.g., when the total Compton amplitude is a sum

of individual (photon-nucleon) Compton amplitudes

A

3TlcoMPTON(g) = 2Z exp (?qTj)3n:c0MPT0N<'H9)

where q is the three momentum transfer and r, are position vectors of the nucleons in the target nucleus, it gives the

following contribution to the total photon-nucleus cross section (which is proportional to A) :

47r \ Z N — Im 3n:coMPTON(0) = A — arhv) + ^ (^riyn)

where CT^yp) and <JT{yn) are the total photon-proton and photon-neutron cross sections, respectively. When the VMD

model does reproduce the correct total photon-nwcZeon cross section the contribution given above is just a single

scattering contribution of the multiple scattering of vector mesons and is properly taken care of by the VMD model

description of photon-nucleus interactions. If, however, the VMD model fails to reproduce the total photon-ni/c/eon

cross section, the balance between the single and multiple scattering contributions given by the VMD model is dis- turbed and an additional contribution to the total cross section appears which is not screened (~A) (compare eq

(4.10)). In fact, it is very likely that something like that may indeed take place. For instance in ref. [36] the authors working with the parton model find a not screened (^A) contribution amounting to -^20 percent of ariyPb).

43
and hence

47r or (7, hadron) = — Im 911(7—^7)

TT -r—V 47ra - E Im9n(7^7)

/C-y 7V" where (. . .) |o denotes the forward value.

(Tr (7, hadron) = 7- Z —^ ^= 91^ (7->F) jo

|91l(7^F) |o|2 =(l+rjv2) (Im9E(7^F) |o)2= ~ail

where rjy is the ratio of the real to the imaginary part of 3E (7—>F) . So, finally, the relation which can

be tested is as follows :

ar(7, hadron) =- 2:-^ + . (4.8)

ky V yv \dil 0/

By measuring independently the forward vector meson production cross sections and the total photo

cross section one can check the internal consistency of the VMD. There are some other tests but the

above equation was used in the analysis published recently by D. 0. Caldwell et al. [37]. Another

possible test is, e.g., the equation

(JT (7, hadron) = — £ —^ Im 3TI (F-*F)

.

Ky Y 7fAssuming 9TI (F—>F) purely imaginary (in the high energy limit) this becomes

(JT (7, hadron) = Z) — (jvn. (4.9) y yv

First, let us discuss the formula (4.9) which is cruder and contains some nondirectly measurable

parameters. The table below gives an idea of accuracy with which it is possible to test it (see K.

Gottfried report in [S5]) . At 7 GeV incident energy the left hand side is ot (7, nucleon) = 118±4 jub.

V (TFw(mb) —- o-Fjv(Mb) energy (GeV)

p 27.5±2 0.62 ±0 05 81±14 6 25.8±8 4.8 ±±0 5 10± 4 7

13 ±3 2.8 ±0 2 8db 3 23

sum =99 ±21 Mb

The sum 99 ±21 ;ub checks reasonably well with the value given above, 118 ±4 /ib. All these parameters are obtained from a host of yarious experiments: yv^/'^ir from e+e~ storage

rings where the following process is observed:

44
V

(Tyif and Tjyff (not shown above) from the A dependence of V photoproduction, rjvN was also extracted

from Compton scattering and leptonic decays. From the table above one can also see that the p meson contribution is more important than the contribution of the other two vector mesons. Let us go back to formula (4.8). It had been checked both for nucleon and nuclear targets

(nuclear targets: Pb, Cu, C) [37].

Nucleon targets: The authors assumed the w and 0 contributions to or (7, nucleon) to be 20±2 fih (remember: this is just a small contribution) . Then, by measuring err (7, nucleon) one can determine

{da/dQ) (7p—>p"p) |o for which one gets much too high a value. One may get agreement if one reduces

7pV47r from 0.62 to 0.37! But yp-/4:w is well known from colliding beam experiments. So, it is unlikely that one should reduce it by a factor of almost 2! (Unless there is a strong dependence on the

invariant mass of a virtual photon)

One can also get the correct answer when one accepts that there is a contribution from

a Compton-like process (not given by the VMD model!) which does not show any screening and is proportional to A (see footnote 4)

Nuclear target: One could test (4.8) directly against experimental data for nuclear targets if the

energy were high enough. Remember, however, that in order to have VMD active in its full strength one has to have l~2co/m,v'^'^R, where R is the nuclear radius, which condition is not well satisfied at

existing photon energies: e.g., 1^ = 2.1 fm at 6 GeV. Hence one has to use a more sophisticated description which in fact allows for the hadronic fluctuation to fold back into a photon inside of the nucleus:

We shall come back to this point later. For the moment let us simply state the results of such a

"sophisticated" description (the low energy version, in which A||?^0, is given below in eq (4.10))

which was presented in the paper by Caldwell et al. [37]. By investigating the A dependence of the

ariyA) cross section they found less screening than demanded by VMD but more than demanded

by purely electromagnetic interactions. Hence the discrepancy is related to a partial breakdown of VMD, rather than to smaller 7p (which would not change the screening) . Indeed it seems that a Compton-like (non-VMD) contribution ~A which would give ~20 percent of or could make the

theory and experiment agree. In fact one does not need a purely electromagnetic interaction to obtain a contribution ^A. It could come from a heavy vector boson whose

NUCLEUS

2cu „ My2

is still short because its rest mass is large.

45
Conclusion: The VMD model is only approximately correct. There is no commonly accepted

explanation of the discrepancies described above. Perhaps the Compton-like contributions to 311(7-^7) (as suggested by Brodsky, Close and Gunion [36]) should be added to VMD to explain

the recent photoabsorption data.

4.2. Photoproduction of Vector Bosons

The breakdown of VMD which one sees from the results of Caldwell et al. [37] does not elimi- nate the possibility that the previously worked out relations between 9TI(7-^p) and 311 (p^p) are good approximations. It is in fact commonly accepted that they form a sound basis for analysis of production of vector mesons on various nuclear targets and, since it is comparatively well documented experimentally, it is instructive to outline it here. The production amplitude of a vector boson is (in the limit co

)

"V/ TTQ! y^{^^V)=- (0, ky I I 0, k/ = k^) (4.9)

7f

where (| | ) is just the elastic scattering amplitude of the vector meson V from the target nucleus

(which we may take over from our previous discussion of hadronic elastic scattering from nuclei)

.

So, in the high energy limit

{0,kv\tv\0,kv')= / d^fcexp [z(kF-k/).b]{l- exp [?:xF(fe)]l

d'bexp (iA-b)[l- (1-<F I r I V))^2-

3H(7^F) is given graphically below

In the high energy limit (w-^00) one can also describe this process as follows: the photon

penetrates the nucleus up to a certain point where it converts into a V meson which scatters elasti- cally from the other nucleons and then leaves the nucleus. Graphically

46
In computing the amplitude we have to sum over all nucleons of th(! target nucleus because the

conversion of 7 into V can occur on any one of them.

So, the production amplitude is in this case:

3n(7->F)= — / d^feexp (?A-b)(F| r I 7) E I r 1 v)y-\

Summing the geometric series we then obtain

ikv C (V \ V \ '^\ 91I(t^F) = —j

exp (^A • b)

| ^ | [1 - (1 - (F | T | F))

But

(F I r 1 7) _

(F| r I F) ~

because, on one nucleon

^^nucleon (7-^F) = aflnucleon (F—>F)

and

ikv

2nTnucleon= ^

f d^b GXp ( j6 • b) Tnucleon (?>)

for both the 7-^-F and F—>F processes. So, we get again the formula (4.9)

In all the formulae above we have used the profiles of nucleons smeared over the interior of the

nucleus with the single nucleon density functions p (r)

r(6)= IdVp(r)7(b-s),

where r = (s, z) , as always.

The conclusion of the above discussion: The two seemingly different pictures give the same

results in the high energy limit. As we saw in the example of our discussion of ffriy, A), present experiments are not quite in

this limit (at several GeV 1<R). Hence if we want to analyze the existing experimental data we

have to keep the longitudinal momentum transfer different from zero

Then the graphical description of figure 38 is not valid any more. One has to allow the amplitude

sninucieon (t^F) to osclUate wlth the function

This factor partly destroys the coherence of the process of producing F's over the whole nuclear

volume. We construct the production amplitude as the (previously introduced) picture of multiple

scattering tells us to. For the independent particle ground state wave function we have

31I(7->F) = — /" d'b exp (iA-h) f

dhi . . . dhA H p(^i)

X n [l-TF(b-S;)e(3y-0,)]7F^(b-s,) exp {i^\\Zi)

Note that this expression satisfies the correct "weak interaction" limit: when the F interaction is

47
negligible the amplitude reduces to

3Tl(7->F) = E f

dh,pirj)yyy{h-sj) exp (zA • b+zA, |Z)

For large A and the profiles 7 much narrower than the density p, we have

31^^/77(0) f

d^hexp (zA-b) / p (b, 2)e'^ii^

' •'-co

X exp j^-K(r^^A(l-r/^jv) y

d2'p(b, 2') ,

where /v7(0) is the forward production amplitude on one nucleon corresponding to the profile jyy.

The above formulae can also be obtained from the expression^

il^V) = I

dh, . . . dhA I ^(ri, . . . r^) 1^ X exp (fq-r,)

where q = (A, An) , and are the amplitudes for the production of a F meson on thejth nucleon

with the screening of the other nucleons taken into account. For a given configuration of the nucleons we have^

5 When we want to shift the scatterer by the distance r, the amplitude acquires the phase e'"-'. In other words, we

have to transform the amphtudes as follows:

3n(g)^e''<3-'3Tr(g),

where q is the three-momentum transfer vector. As long as there is only one scattering center this phase factor is irrelevant, but when there are more scattering centers we obtain

gn(g) = eii-mj{q),

i which is the formula used here.

^ The argument in yy, which gives the elastic scattering vector meson-nucleon profiles, should be the transverse

distance between a target nucleon and the incident particle. The geometry below shows that the argument of yy in the expression for 311(7-^ F) should be: b'+Si— sy.

PROJECTION ON THE SCATTERING PLANE

' \ X INCIDENT PARTICLE

b'

. t

1 SYMMETRY AXIS h 1

PROJECTION ON A PLANE PERPENDICULAR TO THE

INCIDENT PARTICLE DIRECTION

48

X = b'+ S:

i 'J
2]2ni,>-^(g) exp 0-q-r,)= ~j d'-b exp (fA-b) E IT [l-TvCb-s^OeCz-zODTvyCH-s.Oe'^ll" ^^'-j d'bexp (zA-b) , 2 exp (zA-s,)e'^ii"'^ f d^b

i 27r J Xexp (iA-b) n[l-7v(b'+Si-S;)0(0-2.)]TKv(n

which, upon averaging over the ground state

becomes the formula given above. Note that the same reasoning gives, in the limit of large A,

3TC(7->F)=A jd'bdzexp (/A.b+iA||z)p(fe, z)fyy{0) exp [-^av^^v (1-2^)^/(6, 2)]

with

,+00 TAb,z)=A J

dz'p(b,z').

This is again the same formula as before. Note that for nucleons at different positions z, the attenuation of the outgoing vector meson beam is different. So, in the last two expressions for "SR. (t^F) our Sdlj'^"'^ (q) depend on the position of

all the other nucleons. These formulae are now adapted to take care of nonnegligible A||. They are being used in all standard analyses of photoproduction of vector bosons on nuclei [S3].

Let us also quote, for the sake of completeness (without giving derivation) , the amplitude for

elastic scattering of photons from a nucleus derived from the multiple scattering model, with An nonnegligible, in the limit of large A and VMD assumed:

311(7-^7)= ^ /

^'^exp (zA-b) |a IdVp(r)7„(b-s)- Z(^)

XfyviO)fvy{0) j

dzidz2Ap{b,Z2)e-'^\\'^e(z2-Zi)

X exp (l-V2<TvN{l-ir,vN)lA I

'dz'pih, z')^ Ap{h, 0i)e'^ii^i| . (4.10)

In this formula the single scattering is separated out and one should set = (7ra/7v-)7rF if we

apply VMD for the single scattering too. These two contributions in 9^(7—>7) can be sketched as follows

49
SINGLE SCATTERING

IN THE CASE OF V M D =

Note that the single scattering contribution always goes as A (just as any multiple scattering process)

but it becomes progressively less important as A||—*0 with increasing energy. Recall that the experiments discussed above [37] had shown that when arijA) is computed from (4.10), the single

scattering contribution is not the one given by the VMD model (compare also footnote 1). Now let us go back to photoproduction of vector mesons. The formulae discussed above were

applied to analyze a multitude of experimental data. The first suggestion that one can obtain some

important information on the properties of F-nucleon interactions came from S. D. Drell and J. S. Trefil [38]]. Then a flood of papers followed. The references can be found in the review articles we

referred to at the very beginning of these notes [SI, S3, S5].

A few general comments can be made by inspecting the formulae for photoproduction of vector mesons :

(i) They depend very strongly on A (in the optical limit the dependence on A is exponential)

.

(ii) There may be a very important interplay between the "phase factors"

e'^w' and exp [iayj^riyj^j^T {b, z)^.

So, the differential cross section for photoproduction of vector mesons should, in general, be sensitive to both avff and rjyff. This is very important because these quantities cannot be obtained directly

from any other experiments because there are no vector meson beams available due to their short

lifetime. Here such indirect "measurement" is possible because the vector mesons interact with

nucleons before they decay. The differential photoproduction cross sections look very much like

elastic hadron-nucleus cross sections. They exhibit a steep slope at small and then a flat part at

large with, possibly, some diffractive minima.

50
Just to give some idea of how the results look, let us describe briefly the results of p" photoproduction on deuterium (R. L. Anderson et al. [39]) and a DESY-MIT experiment (H. Alvcnslehf;:!

et al. [40]) of p" photoproduction on light, medium and heavy imclei.

First, the deuterium target. We can use our formulae derived above after specifying thf-m for A =2. We get

ariDeut. (7-^p°)

ik = '—f d^exp (zA-b) /dVp(r){[l-Tppf-)(b-Hs)0(2)]7„('')(b+Ks) exp (zAmH^)

+ [l-Tp'"Hb +Hs)e(-2)]7p.<'"(b-Hs) exp (-?-A||3^2)} — f d^&exp {iA'h) f p(r)7p^<"' (b +3/^s) exp (iAnJ^z)

27r J

+ ^

/" d'bexp (iA-h) f #r p(r)7pyf^'(b-Hs) exp (-fAn^z)

/" d^bexp iiA'h) f

p(r)7pp(''Ub-3^s)e(z)7p/"'(b +3^s)

X exp

ik

2ir J J X exp ( —fAll 1^2)

The cross section experimentally measured looks as follows

single

scattering

contributions

double

scattering

contributions

From the above formulae we can see that at small momentum transfers we are essentially measuring

the amplitudes for photoproduction of p° on neutrons and protons, modulated by the deuteron form

factor F. For example,

51
DEUTERON DEUTERON

and the analogous expression for the proton (fig. 44). Hence this part of the cross section, where

these contributions dominate, can only confirm what we know from p° photoproduction on free nucleons.

At large momentum transfers, however, where the double scattering dominates, the amplitude

is given approximately by (see also fig. 45)

- j (^^5F(6)/,p<->(MA + 5)/„<")(3^A-5)+ j

d^5F(5)/,/"'(i^A-5)/p,<-'(3^A + 8).

So, by measuring the cross section at large momentum transfers we can extract /pp""*^'^"" because we know /p^""'=''=°" quite well. In fact, since F {b) is much steeper than /pp and /p^, it acts as a Dirac

function, and the measured differential cross section is proportional {the proportionality factor being

known) to the elastic p-nucleon cross section. In this experiment [39], the recoiling deuteron was detected. This guaranteed the coherence of the process and excluded any excitations of the target nucleus.

There are a few points which should be stressed again at this stage:

(i) This is the most direct measurement of the elastic p-nucleon cross section in existence.

(ii) If one wants to extract a^N or 7pV47r, one can do it very safely because the analysis, at

large enough g^, depends insensitively on r?p. First one extracts (da/dt) \pn^pn and then

by taking the ratio

52
4ir

da

dt pN->pN

yN^pN

one determines the coupling y^. (Here a is the fine structure constant.)

(iii) The results of this analysis check beautifully with the results obtained from p" phfjtf)-

production on larger nuclei (whose cross sections, incidentally, do depend sensitively on

7]p as we have already indicated before)

(iv) The extraction of (da/dt) \pN-.pN does not depend on the VMD hypothesis. However, it assumes the multiple scattering model of Glauber completely and literally. For more

details we refer to ref. [39].

Let us now go briefly to photoproduction of p" on various nuclei and take as an example the

DESY-MIT experiment mentioned already [40]. Here, the target was not detected and the measured cross section contained contributions from nonelastic processes. The reaction measured was 7+A-^p°+A

The following targets were used: H2, Be, C, Al, Ti, Cu, Ag, Cd, In, Ta, W, Au, Pb, U. The parameters

obtained from fitting the differential cross sections to their very extensive numerical data are [40],

a,N = 27.7zLl.7 mb, 7pV47r = 0.59±0.08,

in good agreement with the more recent numbers already quoted. In analysis they assumed the ratio

of the real to the imaginary part to be rjf, = 0.2 (in agreement with the dispersion relations calcula- tions of the total photon cross sections.)

From the above description and the inspection of the data shown in ref [40] one sees that there

are several points which bring about uncertainties in such an analysis. These uncertainties arise because one has to make some corrections in order to obtain the coherent p-production cross sections

(i) nuclear excitations should be removed

(ii) processes which lead to tt+tt" production (other than p-production) should be subtracted,

(iii) one has to decide which invariant masses of tt+tt" are p°'s and which belong to some kind of background (compare the data) . The problems of interference between w+ir" and p productions are also relevant (see T. H. Bauer in ref. [25])

(i) and (ii) can be reliably estimated. Take (i) . The process can be computed in the same manner as the poor energy resolution cross-section. The table below shows some illustrative cal- culations by Yennie (see K. Gottfried report in [S5])

(c ine 1 /fcoherent) 9=0

No correlations

in the target With correlations

in the target

c 0.060 0.013 Mg .034 .009 Cu .014 .003 U .006 .001

53
So, as long as we measure small angle p" photoproduction we can neglect nucleus excitations

(this won't produce more than a few percent error) . The contribution (ii) was also estimated not to exceed -^10 percent. The point (iii) makes the results of our analysis model dependent but again the

extensive work of many experimental groups analysing their experiments seems to show that this model dependence produces '~10 percent uncertainty. This is how far one can trust such numbers as quoted above for ct^n and Tp"/4ir.

Photoproduction of 4> cmd u vector mesons The data on these two mesons are much poorer than on p. There are in existence, however,

several experiments in which they were photoproduced, both on protons and nuclei. The parameters

(o-, 7V4ir, n) were already given earlier in these notes. We shall not go into any details of these

experiments. Let us stress only two points:

(i) 0 is narrow, hence it is much safer to treat it as a well-defined particle.

(ii) w has, however, a mass very near to w?p and these two mesons may "mix."

One can, in fact, suspect that in the "elastic" scattering after a vector meson is produced, some

kind of superposition of the vector mesons propagates through the nuclear matter (C. Rogers and

Colin Wilkin [41]) • We shall not go into these problems now. A summary of the picture of high energy photon interactions ivith hadronic targets {which include nuclei)

.

1. We considered the high energy limit of elastic scattering of photons by nucleons and nuclei. We discussed the evidence for the existence, in the physical high energy photon, of some strongly

interacting components. The condition for the applicability of the high energy limit was that the

characteristic length, I, defined by the incident energy and the lowest available hadronic invariant mass,

be much larger than the target radius, R. At presently available energies this condition is satisfied for nucleon targets but it is not satisfied for nuclear targets.

2. The consequence of this fact is that one cannot use the high energy limit description in interpreting the present experiments of photon interactions with nuclear targets (nucleon targets are OK). We derived the formulae corrected for a non-negligible longitudinal momentum transfer for the case of vector meson production (the formula for elastic photon scattering was also given with- out derivation)

.

3. The Vector Meson Dominance model was briefly discussed and some recent experiments

which seem to show its incompleteness described.

4. Photoproduction of vector mesons from deuterium, and from light, medium, and heavy

nuclei was also analyzed.

5. Diffractive Production of Hadrons in Hadron-Nucleus Collisions

The Standard Analysis

We have already discussed some general features of diffractive production processes by hadrons

(in the limit of very high energy). Let us now look into a few details of such processes with special

emphasis on coherent'production processes on nuclei. As in the case of vector meson photoproduction,

in the existing experiments the incident energy of hadrons was too low to neglect the longitudinal

momentum transfer and the high energy limit description is, at these energies, not applicable.

However, with our experience in photoproduction of vector mesons on nuclear targets we can easily

remedy this situation. So, the only change we should introduce into the formulae given for vector

54
meson photoproduction is an attenuation of the incident beam of hadrons (the incident photons were

not attenuated due to the weakness of electromagnetic interactions) . Let us make this extension

explicitly and obtain the so-called "one step" model for diffractive production of hadrons in hadronnucleus collisions. In this "one step" process, the production takes place on a nucleon inside the

nucleus.

911

Let us take the large A limit formulae for

(a) coherent diffractive photoproduction of p mesons:

(y^p) =fpy{0) A j d^b dz exY) (iA-h + iAuz) p{h, z) exp ^-}4ayfj{l-iriy^)A dz'p{h,z')^

(b) coherent diffractive production of hadrons by hadrons:

3n(l-^2) =/2i(0)A I

d'bdzexp {iA-h+iAuz)p(b,z) exp(^-}4ai{l-iai) j

dz[p{h,z')^

X exp (^-}4a2(l-ia2) dz"p{b,z")^ . (5.1)

In the forward direction production

M*'-M^

All ~ ,

hence the oscillating factor under the integral sign, e^^n^—>1 as the energy increases. The only other

energy dependence in our expression is possibly in /21. When a diffractive production mechanism is effective, this amplitude is experimentally observed to produce a nucleon cross section which is approximately energy independent. Hence one should expect an increase of the dau/dA? cross section

(obtained from the 9H(1^2) amplitude given above) as the incident energy increases and e'^n'-^^L (Note the difference with vector meson photoproduction: there, due to an intricate interference with

y^i'^YNVvN'^ St dz'p, it is hard to predict what the limit An^O will produce.) The existing experimental data on Stt production by pions seem to suppert this conclusion (H. LeSniak and L. LeSniak

[42]).

55
EMULSION DATA

22

18

r(mb)

14

10

6 -

J I l_l I L

20 40 80100 200 400

p(GeV/c)

In computing the curve shown above, they assumed 0-1 = 0-2 = 25 mb and a realistic density dis- tributions for the nuclear targets. In addition, the cross section was weighted with an invariant mass

distribution W (m) obtained from coherent production of Sir on a nucleon.

There are many uncertainties in this calculation but the result seems to indicate that one gets

to the limit An = 0 rather slowly and with presently available experimental data one has to take An

into account. The point of taking the total 3ir-nucleon cross section to be 25 mb, equal to the one tt cross section, needs some explaining and that will be done below. Here, let us say only a few words

about the high energy limit (An =0) and connect it with our earlier discussion. When An = 0 we can

perform the integration over z (0-' = o (1 — ia) )

A j^^ dzp{b,z) exp(^-y2aM jdz'p{b,z')^exp(^-y2a2'A 1^

dz"p{b,z")^

2 —0-1 j_

d

dz —

<si —ai J _^ dz exp(^-3^(7M /

d2'p(fc,2'))exp(|-3^<r2'A f\z"p{h,z")^

= ^^^\^exp(^-y2<J,'A dz'pib,z')^ - exp(^-y2a2'A j^'^ dz"p{b,z")^ .

Hence, in the high energy (An =0) limit the expression for 2(11(1^2) is

31l(l->2)= fd'bexp (iA-b) exp(-y2<xM f dz'p{b,z'))

(T2 —(71 J L \ •'-=0 /

- exp(^-y2a2'A j^'" dz"pib,z")^ . (5.2)

Note that this is precisely the same expression that one would get from our model of the high energy

diffractive process which assumes a weak transition between the initial (one pion) and the final (3 IT states) . Let us repeat the arguments again. •

Initial state =

Final state =

+

+

small admixture

1

small admixture

= \ l)+d\2)

= l2)-d*|i)

56
(the orthogonality condition :

(- (i

I d+{2 I) (I +d I 2)) = -d+d = Q).

Then the transition amplitude is

2flZ(l->2) =[-(! I r| i)+{2 I T\2)']d.

The results of these experiments are very puzzling and still very poorly understood. Lot us

discuss them briefly. The formula for 9tl (1-^2) with An 5^0 was used to interpret the results, cri was

taken as the well-known pion-nucleon total cross section, was a free parameter used to fit the

coherent production cross section. The energy resolutions were too poor to have pure coherent

processes but the incoherent processes were subtracted reasonably reliably because (as we stated

many times) they are rather unimportant at small momentum transfers (where coherent processes

are important) . Realistic nuclear densities were used:

p{r)= , fdhp{r)=l,

1+ exp

c=1.12A^'^ fm, a = 0.545 fm. To compute the integrated coherent production cross section for a given

bin of the invariant masses of the produced systems we use the formula

, dV(coh) dV(coh

cTeoh (^, M„ M,) = / dM* / dq^ ' i ,

where is a cut-off at the first maximum of the distribution, o-coh {-A., Mi, M2) depends very critically on A and this fact enables one to extract ai (in complete analogy with photoproduction of vector- mesons) . The amazing result was that ui (both for Stt production and for bir production came out to be very small) . The tables below give some of the numbers obtained from the CERN experiment

[S4j. The same equation is valid for nucleon targets. Hence we can eliminate d and get

cyT7fi_^21 Ji^i-^^; = — r(2 I T'-^^ I 2)— (i I T'^' 1 1)1

(2 I T(^) I 2)-<i I r(^) I i) ' ^ ' ^ '

'

which gives the production amplitude on the nucleus in terms of the production amplitude on a

nucleon and the elastic scattering amplitudes of the objects 1 and 2 from the target nucleus

where

(2 I I 2)= — f

d'bexp (tA-b){l- exp l-^T(b):\}

(i I T^^^ I i)= — f d%exp (iA-b){l- exp [- H'^i'r(6) ]}, 2x J

T{b)=A r°"dzpib,z), a' = ail-ia).

Let us use some standard parametrization of the nucleon amplitudes, e.g.,

<2 1 TC.)

I 2)= ii±^^ (i I r(-) I i)= li±^^ e-".«\

47r 47r

57
We get, neglecting the (f dependence of the elementary amplitudes (which vary much slower with

gHhan (2 | T^^* | 2)or (1 | T^^^ | I)):

971(1^2)= \ d%exp (^•A.b){exp[-M<TiT(6)-K(7,T(6)]!,

which is precisely the formula (5.2)

.

All these formulae are very useful (because of their simplicity) in discussing some effects which

do not depend dramatically on An =0.

Let us go back, however, to the case A||=0 and discuss some recent experiments in which

k—^Stt and w-^5w processes were measured on various nuclear targets. (A very rich literature on

this subject, both from experimental arjd theoretical points of view, can be found in the Proceedings

of the XII Cracow School of Theoretical Physics, June 1972 [S4].)

Production of the Stt System

0-2 (mb)

Mass bin (GeV) 9 GeV/c 15 GeV/c

0.9-1.1 29±2 28 ±2

1.1-1.3 22±2 21 ±2

1.3-1.5 5+^

^-2 21±3

1.5-1.7 23±4

1.7-1.9 13±8

Production of the 5;r System

0-2 (mb)

Mass bin (GeV) 15 GeV/c

1.5-1.7 10±7

1.7-1.9 13±10

These results are very puzzling because the Stt and Stt systems do not seem to form well defined

particles. For example, at 9 GeV/c the time required by a system of 1.2 GeV mass to cross one-half

of the thickness of the Pb nucleus (6.5 fm) is t -2.9X10-'-* s or r~230 MeV (relativistic dilation

included) . The observed distribution is wider than 230 MeV (perhaps even as much as 500 MeV)

.

Besides, 0-2 should be smaller at 15 GeV than at 9 GeV (due to time dilation) , which is not the case. In fact, it is hard to accept that these systems are resonances—they look more like a group of Btt or 5ir\ But then their cross section should be 3(x^m and 5a-^M, respectively!

5.1. Discussion of the Anomalously Small Absorption of Stt and Stt Systems in Nuclear

Matter

First, let us stress that there is no satisfactory explanation of this phenomenon available. One

can, nevertheless, make a few points which may advance a bit our understanding of the process. Let us start with an analysis of the process of diffractive production in the high energy limit in the language of multiple scattering. (We choose the high energy limit because it is simple ©nd we

believe that the finite longitudinal momentum has little to do with absorption properties of the

outgoing systems.)

58
First, the "one step" mechanism of production:

Let us introduce the profiles of individual nucleons smeared over the volume of the target nucleus:

(1 1 r 1 1)^ ^/n(0) dzp{b, z) = ^/u(0)r(6)

(21 r I 2)^^^(0)7(6),

ik

where we took only the forward amplitudes /ii(0) and/22(0). We proceed exactly the same way as we did in obtaining the high energy limit of the '7 — p amplitude

(c— 1) factors

£fll(l-.2)= ^ \d'bexp (zA-b) 2: (1-(2| r|2)) ... (1- (2 | T | 2)) (2 | | 1)

x(i-(i|r|i)) . .. (i-(i|r|i)).

(A — c) factors

After summing the geometric series we get

/• (i-(i|r|i)^-(i-(2|r|2))^

an (l->2) =— f

d'h exp (iA-b) (2 | r | 1)

2Tr J (i|r|i)-(2|r|2)

ik- I d^bexp (/A-b)(2 | r | 1)[(1 | T | l)-(2 | T | 2)]-^

A large ^ X[exp (-A(l I r I 1))- exp {-A{2 | T \ 2))]

= f

d'bexp (zA.b){exp [- J^cr/r(6)]- exp [-M<T2T(6)]1,

which is the same formula (5.2) which was obtained before. As we have said before, the "weakness" of the production process makes the "one step" description plausible. Let us point out, however, that Kl—^K, regeneration on one nucleon is weak; nevertheless, this process cannot be described correctly as a one-step process. Indeed, from the "one

step" formula (5.2) we get

^{Kl-K,) = J^^-i- [ rf25 exp (/A-b) (exp [- M<^ic/T(6) ]- exp [-K^ic/r(6)]!, (5.3)

which is wrongl We know the correct answer because we know the composition of \Kl) = I /'^{\ K°)+\K'^))

and 1 K,) = \/V2{\ K°}— \ K°}) which is nonperturbative, hence fundamentally different from the y—V

59
structure of the photon. In fact, accepting the fact that K° and K" scatter only elastically from the

nucleus we get the formula which is correct:

31I(Ki-^A%) f d% exp itA-h) }exp exp [-HcrA'»'r(5)]}. (5.4)

8incefKLKL=fKsKs = }4ifK''K''+fK''K'>), we can see from the numerical values given below that the

profiles in (5.3) and (5.4) are different. To complete the point we are making, let us compute fKLKL=fKsK, and compare it with

fKL^K, = /^i fK''K''—fK''K'') We use the data of Foeth et al. [26]:

At 4 GeV/c incident momentum we have:

for neutrons:

for protons

:

and we get

Im /^"clfo ^2.8 fm, Re /^"c]fo ^ - 1 f

m

/|:^o ^4.1 fm, Re /^".^o - 0 f

m

Im /^^^o ===2.9 fm, Re /^'Sjfo - 0.5 fm

Im fil^, - 3.4 fm, Re f^^, - 0 fm.

0-"" cr^"' ^^-0.055, -^^-0.012.

So, the production process on one nucleon is indeed weak! But as we now see clearly (comparing

(5.3) and (5.4)), that is not enough to apply the "one step" formula; one has to know the internal

structure of the objects which undergo diffractive scattering.

Remark :

Let us note that we may write the amplitude (5.4) so that it has the Ki-^Ks regeneration

amplitude on a single nucleon as a factor similarly to that appearing in the "one step" description,

(5.3).

For a nucleon target we have

:

/iCx-K. = /4 ( fK'>K''—fli''K'') For a nucleus target we have

:

DTKXl^A',)- f d^bexp (.•A.b)lexp [-3^(r^«r(6)]- exp [-K<T«or(6)]|.

Thus, using the optical theorem we can write

9TC(Al->A^) = f d'b exp (fA-b) {exp [-H<r:K°r(fe)]- exp C-M<^if»r(6)]}. (5.5)

But, although this formula does seem to have the production taking place on one nucleon, in fact nothing like that takes place: there is only a "smooth scattering" of the two components (A" and A")

.

No one nucleon along the path plays any role distinguished from the others. (From (5.5) one can

see again that the "one step" formula (5.3) is incapable of describing Kl-^Ks regeneration on

nuclei).

Let us go back to the general case of incoming particle 1 and outgoing particle 2. From the

above discussion we can see that (within the framework of the Good and Walker description of the

diffractive production processes) we have to have weak coupling between | 1 ) and | 2) states in order

to have a "one step" description, since, when the coupling is weak, we can replace

60
and

(2 I r(A')

I 2)~ (2 I 7""^'I 2). Perhaps at this stage one should point out that one could easily explain the small abs(;rptioii cross section by abandoning the "one step" description. The price to pay for it would be the loss of the interpretation of the states | 1) and | 2) as "almost" a one physical pion state and "almost" a

three physical pion state, respectively.

For instance, the following scheme (which imitates Kl—^K, regeneration) wrjuld (,'xplain the

observed effects. (Note that we are working in the high energy limit, hence our arguments are based on the assumption that the absorption properties of the objects produced are the same in this limit as at the experimentally available energies.) Let us accept that

o-i~(r,r = o"i and crj; — o-2«cr^ (and positive)

and

u)=^(rD+i2))=ii)

I3:r)=^(li)-I2))=l2). (5.6)

(Note that the values of the o-'s and the coefficients of the transformation (5.6) are independent

quantities.) Then the production amplitude (as in the Kl—^Ks regeneration) is

201(1^2)= f

cPbexp (^A.b){exp [-3^criT(fe)]- exp [-Ko-jT'Cfe)]! (5.7)

o-J a — o"! J

where

/2i = K[(l|7^'^Mi)-(2|7'('^M2)].

The formula (5.7) contains the correct attenuation of the outgoing object (compare the discussion

of experimental results given previously). Then the elastic scattering amplitude of | l)(s | x))

is totally determined

311(1^1) =i^[ (I I I i)+(2 I r(^) I 2)]

= ^

/" d^fe exp (^•A.b)H!l- exp [-3^<rlT(6)] +l- exp [-3^(To:r(6)]|

which is, to a very good approximation,

f- f d% exp (fA-b) {1- exp l-VzT.Tibn] (5.8)

27r J

hence the correct ir-nucleus elastic scattering amplitude.

Then, one has to worry about the fact that 371(2^2) =911(1-^1), as implied by (5.6). One can

argue that 371(2^2) is not measurable since | 2) decays into Stt and hence is not available as an

initial beam, thus making this concern irrevelant. This example shows that one can easily explain the "anomalously" low absorption of diffrac- tively produced objects if one assumes that the initial (one pion) and the final (something decaying

into 3ir) states are just the different configurations of the same components. (In the weak coupling

case" it is physically more accurate to describe the initial and final states as composed, approximately, of one and three particles, respectively.)

Note the following amusing point: Assuming | tt) to be in the form of a superposition of two

scattering eigenstates we determine uniquely the exponentials (the total cross sections) . Once we

have that, the production cross section is completely determined and comes out right!

61
In fact one can give a more general version of the above remark [43]: The transformation (5.6)

may be replaced by a general transformation which preserves normalization and orthogonality

h'>=E«d/),

Y^au*au = 5ii (5.9)

s and the assumption made that all nonzero an are of the same order of magnitude. It introduces

"democracy" among all physical states and treats the ground state of the incident particle on the same footing as the excited states. This makes good sense if one believes (following Good and Walker

[23]) that in the high energy limit the ground and excited states of the incident system (e.g., a

pion) are considered to be approximately degenerate.

The formula which gives all diffractive elastic and production amplitudes is

{f\T^^'\i)= '^Zcifi*aa f d% exp (iA'h){l- exp [-i^cr,A7^(6)]}, (5.10)

where the "total cross sections" ai are free parameters. In order to obtain (5.8), e.g., we have to make all ai approximately equal to o-^ (they cannot be exactly equal to a-^ because we would have no

production!). When we do that, we have approximately (due to (5.9)) the elastic amplitude of the

form (5.8) , and the production amplitudes become approximately (again due to (5.9) ; for an explicit

example see (5.7) )

,

</| 10- [

d'bexp (fA-b){exp [-3^(7,^(6)]- exp [-3^(7/7(5)]}, f^i.

Since ai — af — a-^ we obtain the result obtained in experiment [S4].

Let us make two comments to close this "strong coupling" description of diffractive production

which, in contrast to the "one step" description, is not a perturbative approach:

(i) In the "strong coupling" model there is an internal relation between elastic and production

processes, whereas in the "one step" picture there is none. To put it differently, a definite relation between atteriuations in the entrance and exit channels exists in this model.

(ii) The "strong coupling" description implies the following general prediction: In all coherent

diffractive production processes where there are strongly interacting particles in the entrance and

exit channels, and diffractive production processes are much weaker than elastic scattering

amplitudes, one should see comparable attenuations in the entrance and all-exit channels.

Before indicating other possibilities of interpretation of the low absorption effect, let us consider an example of a two component system (such as was described before, e.g., by eq (5.6) ) penetrating a piece of nuclear matter. The following remark about penetration through a sequence of thin- slabs of nuclear matter

is in order here.

A SLABS

INCIDENT BEAM

jZ--AL

1

L

62
Assume that p (the density) does not depend on the transverse coordinate b (each slab extends to

infinity in the b plane) . The profiles of the elastic scattering amplitud(!S for one slab are

dimensionless

-

I - 27r 2ir iaik

(1 1 r 1 1)= —jupL= — — pL=}4aipL = y2(7ii IK %K 47r

<2| r|2) = K<xip-.

Let us consider the attenuation of the incident wave, e*^: After A slabs we have

(i) e^^-l3^(i-(i I r I i))^+i^(i-(i I r | i))-^} = Ke'«^^[exp {-y2<Tipz) + exp (-Ho-Jp^)].

The exponentials give the attenuation. This is the attenuation in the case of strong coupling between

I 1 ) and I 2). Then, let us take the case where | 1 ) scatters only elastically (hence we forbid the inter- mediate states I 2)) . Then the attenuation of the incident wave is

(ii)

I r I i))^=e^*^[i-K(<i I r | i)+('2 | r | 2))>

When (Ti = (72, the attenuation in case (i) is the same as in case (ii) , but in the case o-j'j^trj the situation

changes. Take, e.g., cr~i = %(T, (ri=}/i(T. Then the attenuation factor in (i) is

Instead, in case (ii) it is

(ii) e-(i/4)-r.

Hence, for large enough T, the attenuation in (ii) is stronger than the attenuation in (i) . A very

important conclusion follows from this observation: by introducing strong coupling between the initial

state of the incident particle and some other states one may reduce considerably the absorption of the initial

state in nuclear matter. One can put it differently: the effect of allowing intermediate states to occur during the multiple

scattering process may be an increased penetrability (or decreased absorption) . Hence the so-called

inelastic shadowing effect increases the penetrability of a specific component.

We have considered only the 2X2 case but one can consider much more involved systems (which

contain more than two components) . Such a scheme was developed by L. Van Hove ([44], see also

There is, therefore, still another possibility of explaining low absorption of the 3ir (Stt) systems

produced coherently on nuclei: to consider it a "one step" process but to assume that the produced

object, Stt (Stt) , is a superposition of several strongly coupled channels. This coupling may reduce the

absorption of a Stt system as we have seen on the example quoted above. Graphically such process would look as follows:

= eii.exp l-}.i{al +a-,)pz^.

(i) 3^(e-(3/8)crr_^g-(i/8)^r)_

[41]).

63
elastic scotting

of the initial particle

scattering of ttie final object

with many different intermediate states

This is, at the moment, the most commonly considered model of coherent diffractive production. In

order to have a satisfactory solution, one would have to know precisely the superposition of states which form the produced Stt (Stt) system and their interactions with the nucleons of the target. In

other words, one has to specify the structure of the produced object. No one has, so far, proposed a model detailed enough and convincing enough. We shall give a few more details after making the

following remark.

Remark

:

One may also look at the problem of the penetration of a many component system through

nuclear matter as a problem of diagonalization of the "profile-matrix." Let us again assume (for

simplicity sake) that just two states are operative. Then the single scattering profile matrix is

Let us assume the following form for it

0, (T-\y=T'% x = r±r'.

and diagonalize it

:

r-x, r'

r, r-x

Take T = ri];+ T22, T' = Tn— and we get two eigenvalues

i2Tu

[2ri.j

Hence the following transformation of the physical states diagonalizes the interaction

|i)=^(|i>+l2))

2)= ^2 (I i)- I 2)).

64
Hence the matrix

r 2

J.

diagonalizes

Note that | 1) and | 2) are the "eigenstates" of the scattering; they scatter only elastically.

Note also that even if the transition between 1-^2 (or 2—>1) is very weak we have a superposition of these physical states propagating through the nuclear matter, a superposition in which

both these states are equally important.

Now let us go back and close our discussion of the coherent diffractive production by pions of Stt and 5x systems on nuclei by formulating more completely the "one step" description which allows

for reduction of absorption for the produced composite systems (L. Van Hove [44], C. Rogers and

C. Wilkin [41], A. Bial'as and K. Zalewski [45]). It is a direct generalization of the multiple scat- tering we have given above for the standard "one step" amplitude.

We want to keep the "one step" production model, hence we assume the incident particle ground

state 1 1 ) to be "well separated" from a set of excited states | m) from which the system | 2) emerges. On the mass scale

m

(mass )

<ni I r''^'|l> weak transition

the ground state which does not "mix" with m-states. Its elastic scattering profile is

Then, the transition amplitude is

311(1-^2)= ^ \d'-b exp (iA-h) <S 2] (1- (2 I r I m)) (1- (to I r I m')) . . .

all possible

"histories"

...<m,|rw|i)(i-(i|r|i))...(i-(i|rli)).

A "history" is a sequence of intermediate states

rric}, ... I m"), \ m'), \ m).

65
One can perform the sum S by diagonalizing (m' | T \ m). In an abstract form:

r I a) = X„ 1 a)

I «)= Z C{m, a) I m). (5.11) m The states | m) are physical states. The states | a) are their linear combinations. (E.g., in the process

of Kl—*Ks regeneration they are degenerate and identical with | K°} and | K°)). In general they do

not have a well defined energy. The states | d) are decoupled, that is to say only diagonal elements

(d I r I d) are different from zero. Thus

A S = Z Z (2|d)(i-x„)-i<d| rw i i)(i-<i ! r| 1))^-

all " <^=i "histories"

(i-(iir|i))^-(i->.)^

^ ^ f (i|ru)-x.

In the limit of large A{(l —X) ^^e~^) we get

ik (1-.2) =— f

d'b exp (iA-h) Z (2 I I r I 1)

exp (-A(l I r I l))-expi-AXa) X

(1 1 r I i)-x„

The attenuation of the outgoing beam is small when the Xa's are small. So, in this approach the

problem is to construct a physically plausible matrix (to' | F | w ) which produces the correct eigenvalues Xa- Hence one has to go deeply into the structure of the produced object. There is no com- monly accepted model of such a structure but there are many examples ([41], [44], [45], [S4])

which show that one may obtain the desired low absorption in many ways.

6. Shadowing Effects in Inelastic Electron-Nucleus Scattering

Experiments performed to see shadowing effects in inelastic electron-nucleus scattering failed to show it—see H. Kendall in ref. [S5]. Inelastic electron scattering cross sections were measured at:

^ = 6°, the incident electron energies were 4.5, 7, 10, 13.5, 16, 19.5 GeV and the energy losses, p, were

0.1 Gev<f <17 GeV. The targets were Be, Cu, and Au nuclei. They plotted

^nucleus (exp)

Z(Tp-{-N(rn

versus energy loss for two bins of the four-momentum squared

0.25<g2<0.75(GeV/c)2

and

0.75<52<1.50(GeV/c)l

All data points were consistent with S=l for all momentum transfers and energy losses (although the

errors were quite large)

66
0.5 0 9 18 ENERGY LOSS IN GeV

One may say that

1. There is a definite disagreement with vector-meson dominance.

2. This is not very surprising because in the case of real photons, total cross sections do not

exactly follow VMD. How this break-down depends on the "off-shellnoss" of the photon is not known

at all. (Remember -that this inelastic electron scattering experiment sums over all possible inter- actions of the virtual photons consistent with the kinematics of the experiment, hence a total cross section for virtual photons is being measured.)

3. Theoretical analysis of this effect is in a very preliminary stage.

7. Shadowing Effects in Neutrino Reactions on Nuclei

In the following experiment (see K. Borer et al. [46] and also J. S. Bell [S3]) no shadowing was observed

The incident beam was: the CERN neutrino beam which has a very broad spectrum, so the

incident energy was poorly defined but, just to give some idea, the average neutrino energy was about

1.5-2.0 GeV and the width of the spectrum was about 1.5 GeV.

In a spark chamber set-up one could see muons produced, one could also see in which material

the reaction took place, and one could make a rough measurement of the muon momentum.

The following results were obtained {d is the angle of the outgoing n)

Ratio of event rates per nucleon in various Expected ratio if pairs of nuclei no shadowing

e<29°

Pb/C 0.97±0.04 1 Fe/C 0.89d=0.08 1 Pb/Al 0.93±0.07 1

e<5°

Pb/C 0.92±0.15 1

Hence no shadowing was observed.

67
Comments

1. In principle, one may expect some shadowing because

The exchanged particle moy be

a strongly interocting

porticle.

\

\

(Nucleus]

2. Again, for reasons somewhat similar to the case of electron scattering a lack of shadowing is not surprising (for more details of the theoretical analysis, see the lecture by J. S. Bell in ref. [S3])

.

3. The theoretical analysis is at a very primitive stage and when the data improves a lot will remain to be done.

The author wishes to thank I. M. Eisenberg for making his visit to the University of Virginia

possible. He also thanks the students who took this course for their responsive attitude which was

of great help. The help and encouragement from L. C. Maximon are also gratefully acknowledged.

If not for him these lecture notes would have never been completed.

[SI] High Energy Physics and Nuclear Structure, Ed. S. Devons (Plenum Press, 1970).

[S2] Glauber, R. J., High Energy Collision Theory, in Lectures in Theoretical Physics, Ed. W. E. Brittin and L. G. Dunham (Interscience Pub., 1959).

[S3] Hadronic Interactions of Electrons and Photons, Ed., J. Gumming and H. Osborn (Academic Press, 1971).

[S4] Diffractive Production Processes at High Energies, Ed., W. Czyz, Acta<Physica Polonica B3, #6 (1972).

[S5] Proceedings 1971 International Symposium on Electron and Photon Interactions at High Energies, Gornell

University, Ed., N. B. Mistry, Gornell, 1971.

[S6] Landau, L. D., and Lifshitz, E. M., Quantum Mechanics (McGraw-Hill, New York, 1964).

[S7] Bjorken, J. D., and Drell, S. D., Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964).

[1] Friedes, J. L., Palevsky, H., Sutter, R. J., Bennet, G. W., Igo, G. J., Simpson, W. D., and Corley, D. M., Nucl. Phys. A104, 294 (1967).

[2] Scipione, D., Mehlhop, W., Garland, R., Piccioni, O., Kirk, P., Bowles, P., Sebek, J., Murty, S., Kobrak, H., Marraffino, J., and Allen, P., Phys. Letters 42B, 489 (1972).

[3] Bleszyhski, M., Bleszynska, E., Malecki, A., and Picci, P., Phys. Letters B43, 355 (1973).

[4] Kerman, A. K., and Kisslinger, L. S., Phys. Rev. 180, 1483 (1969).

[5] Bethe, H. A., Phys. Rev. Letters 30, 105 (1973).

[6] Eisenberg, J. M., and Weber, H. J., Exclusion Principle Effects in Pion-Nucleus Scattering in the Region of 3,3 Resonances, Univ. of Virginia preprint (to be published) (1973).

[7] Adler, S. L., Phys. Rev. 135, B963 (1964).

[8] Cheng, H., and Wu, T. T., Phys. Rev. D3, 2394 (1971). The same results were obtained independently by

J. Jurkiewicz and communicated to the author (unpublished).

[9] Cheng, H., and Wu, T. T., Phys. Rev. D5, 445 (1972).

[10] Ta-chung, Meng, Phys. Rev. D6, 1169 (1972).

[11] Lee, T. D., and Yang, C. N., Phys. Rev. 128, 885 (1962).

[12] Czyz, W., and Maximon, L. C, Annals of Physics ("USA) 52, 59 (1969).

[13] Chou, T. T., and Yang, C. N., Phys. Rev. Letters 20, 1213 (1968); Phys. Rev. 170, 1591 (1968); Phys. Rev.

175, 1853 (1968).

[14] Cheng, H., Walker, J. K., and Wu, T. T., Phys. Letters 44B, 97 (1973).

[15] Czyz, W., Lesniak, L., and Wolek, H., Nucl. Phys. B19, 125 (1970); Nucl. Phys. B25, 638 (1971).

[16] Lesniak, H., and Lesniak, L., Nucl. Phys. B38, 221 (1972).

8. List of Standard References

9. References

68
[17] Lesniak, H., Doctoral Dissertation, Institute of Nuclear Physics in Krakow (unpublished) (1972).

[18] Bassel, R. H., and Wilkin, C, Phys. Rev. 174, 1179 (1968).

[19] Czyz, W., and Lesniak, L., Physics Letters 24B, 227 (1967).

[20] Bradamante, F., et al., Nucl. Phys. B33, 16.5 (1971).

[21] Cheng, H., and Wu, T. T., Phys. Rev. Dl, 4.56, 1069, 1083 (1970).

[22] Cheng, H., and Wu, T. T., Phys. Rev. D6, 2637 (1972).

[23] Good, M. L., and Walker, W. D., Phys. Rev. 120, 18.57 (I960).

[24] Feinberg, E. L., and Pomerancuk, I., Suppl. Nuovo Cimiento 3, 652 (1956).

[251 Coherent Processes at High Energies, Acta Physica Polonica B3, January 1972 issue.

[26] Foeth, H., et al., Phys. Letters 31B, 544 (1970).

[27] Bjorken, J. D., Kogut, J. B., and Soper, D. E., Phys. Rev. D3, 1382 (1971).

[28] High Energy Scattering of Composite Objects, Acta Physica Polonica B2, #1 (1971).

[29] Bialas, A., Czyz, W., and Kotanski, A., Annals of Physics (USA)_73, 439 (1972); Nucl. Phys. B46. 109 (1972).

[30] Jaroszewicz, T., Positronium Photoproduction in the Coulomb Field: Bound States in the Infinite Moment

Frame and Multi-Photon Exchange, Institute of Nuclear Physics in Krakow, preprint (to be published)

(1973).

[31] Drell, S. D., and Hilda, L., Phys. Rev. Letters 7, 199 (1961).

[32] Deck, R. T., Phys. Rev. Letters 13, 169 (1964).

[33] Ross, M., and Yam, Y. Y., Phys. Rev. Letters 19, 546 (1967).

[34] Bethe, H. A., and Maximon, L. C, Phys. Rev. 93, 768 (1954).

[35] Jaroszewicz, T., and Wosiek, J., (to be published).

[36] Brodsky, S. J., Close, F. E., and Gunion, J. F., Phys. Rev. D6, 177 (1972).

[37] Caldwell, D. 0., et al., Phys. Rev. D7, 1368 (1973).

[38] Drell, S. D., and Trefil, J. S., Phys. Rev. Letters 16, 552 (1966).

[39] Anderson, R. L., et al., Phys. Rev. D4, 3245 (1971).

[40] Alvensleben, H., et al., Nucl. Phys. B18, 333 (1970).

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^ U.S. GOVERNMENT PRINTING OFFICE: 1975 0—553-206

69

NBS-lUA (REV, 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS MN 139

2. Cjov *t Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Interactions of High Energy Particles With Nuclei

5. Publication Date September 1975

6. Performing Organi?.ation Code

7. AUTHOR(S) _ , Wieslaw Czyz8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2400104

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP) 13. Type of Report & Period Covered Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number: 74-13725

16. ABSTRACT (A 200-word or /ess factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

Elastic scattering and diffractive production processes induced in nuclear

targets by high energy projectiles are discussed in this article. Special attention

is paid to the interaction of high energy hadrons and photons. Interactions of

high energy electrons and neutrinos are briefly mentioned. The common features of

all these processes are emphasized throughout the article : The multiple scattering

and shadowing processes inside of the target nuclei. An effort is made to develop

a unified way of treating nuclear interactions of particles which are either

hadrons or exhibit some hadronic components in such interactions.

This article is divided into five sections: 1) Introduction, 2) Description

of multiple scattering, 3) Elastic scattering of hadrons from nuclei, 4) Diffractive dissociation and diffractive excitation, 5) Diffractive production of hadrons

in hadron-nucleon collisions.

17. KEY WORDS (six to twelve entries; alphabetical order: capitalize only the first letter of the first key word unless a proper name; separated by semicolons

Diffractive production; diffractive scattering; Glauber model; hadronic components of

photons; high energy scattering; multiple scattering; neutrino-nucleus interactions;

shadowing effects.

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