O ?*13* T™* IT1

t % i _ i I icM/ioi

INTERNATIONAL CENTRE FOR

THEORETICAL PHYSICS

ON THE PROPAGATION OF THROUGH NUCLEAR MATTER

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL.

SCIENTIFICAND CULTURALORGANIZATION

J . Leon

1974 MIRAMARE-TRIESTE

i)

• ! rit

it

iiItit

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERHATIONAL CENTRE FOR THEORETICAL PHYSICS

ON THE PROPAGATION OF ^ THROUGH NUCLEAR MATTER •

J. Leon ••

ABSTRACT

The relativistic propagation of the A^, considered aB a irp

composite, is analysed in two extreme situations: tee TTp system being

interacting or asymptotic. Mechanisms lowering the absorption are predicted.

The off-shell effects and deviations from Glauber theory are- taken into

account and discussed.

KRAMARE - TRIESTE

September 197U

• To be submitted for publication.

•• On leave of absence from Laboratorio de Particulas Elementalee, Instituto

Hocasolano, OSIC, Madrid,and GIFT, Spain.

I,

Coherent nuclear production has been videly used as the method for

measuring the interaction cross-sections of the produced systems through their

absorption in nuclear matter.

The problem was first stated by Goldhaber et al. when there was no

precise knowledge about the nature of the produced states they were looking for

(i.e. A. "bump"). As is well known, the small experimental cross-section

25mb 1)along with the actual evidence for the non-resonant nature of

"1"

the A, system , constitute a contradiction from which it is difficult toIt)come out in the framework of the usual Glauber theory

Van Hove proposed a novel and physical-minded explanation for that

paradox: what nuclear matter probes after dissociation of the incident pion is

a non-asymptotic hadronic system with interactions among constituents. There

are, thus, different states accessible to the system (labelled by their masses).

Collision with the nueleons produce a quick oscillation of the system among these

states in such a way that certain coherent superpositions of these will propagate

with amain absorption,That can all be described by means of an optical potential

V{m,m ) non-diagonal in the mass. Moreover, for a fast system,the time

interval t between successive collisions and the interval t n necessary to

establish energy conservation are related by (t0A J^E. This implies

and thenoe the system does not have enough time to settle to a definite mass

value betveen successive collisions. This fact is described by a singularity-free

optical potential, i.e.

V{a<n') ?* c(a) S(m-m') .

The two possible optical potential structures give very different

results for the absorption in nuclear matter. In fact, for an optical potential

with a 6-type singularity, the volume term in the amplitude disappears, giving

increased absorption. For a singularity-free potential, however, the volume

term holds; this originates smaller absorption, Moreover, components with

different mass will be differently absorbed,and this predicts a narrowing of the

mass spectrum.

The model works in the right direction and, what seems of extreme

importance, its origin rests in the consideration that the dissociate system is

neither elemental nor asymptotic. In fact the recent analyses about the

-2-

nature of those states {A ,A ,.,.) by means of pion dissociation off protons,

are in fair agreement with such a type of non-resonant structure. What cannot

be clarified with the above analyses is the origin of that structure {an 5 wave

TTp system for the A ^ if it is merely kinematical, as stated In the Drell-Hiida

Deck model , or if there Is some dynamical component. This should be the case

by imposing a non-linear realization of the axial vector meson in chiral

SU(3) x SU(3), which,on the other hand, seems bent by the above-mentioned non-

resonant character. At this stage the usefulness of the coherent production in

nuclear matter appears evident because the nueleons check the system shortly

after creation, giving information on how it looks like before asymptotia is

reached.

The knowledge about the origin and range of validity of the Glauber

description has been greatly improved by the use of the Feynman diagram technique.

Gribov and Bertocchi have shown that the Glauber formula and its generaliz-

ations are obtained by summing up all the Feynman diagrams for the scattering

with n (nuclear) nueleons (where n = 1 A) if the following approximations

are used:11)a) one-channel coupling and impulse approximation (in Gross's scheme

for the nuclear part,

b) relativistic eikonal approximation for the propagation of the

projectile and/or the produced states, and

c) terms of second and higher order in the momentum transfers are

neglected.

We have at hand, in this way, a relativistic description of the scattering which

can account for the Internal degrees of freedom of the dissociated system.

In this paper we shall restrict ourselves to a irp system created by

diffractive dissociation of an Incident pion inside nuclear matter. We shall use

Feynman diagrams as indicated above and two extreme situations will be analysed.

In Sec. II the propagation of a purely kinematical Trp section will be considered,

and a Glauber formula obtained for its cross-section with a value o. ,„ = 1*0 mb,_ (nTp;H

which is very close to the experimental value for the JT = 0 irp Bystem. In

Sec, III we focus our attention on the more interesting situation in which there

are strong interactions between TT and p during their flight through the nucleus.

In this case the experimental result a. . = 25 mb for the .T = 1* system can

easily be accommodated. As will be discussed in Sec. IV, our treatment 1B

closely parallel to that in Van Hove's second paper . The main differences are

our use of Feynman diagrams and the fact that Van Eove uses a general interacting

composite while we give a more restricted structure to the system.

-3-

II. PROPAGATION OF AH UNCOREELATED TTp SYSTEM

The problem of the inclusion of inelastic intermediate states in the

Glauber theory has been solved with the use of Feynman diagrams * or the12)

equivalent coupled channel optical model . Let us consider a particle "l"

which couples coherently with the Btates of the set {a} . The amplitudes for

coherent elastic scattering and production are given in terms of the nuclear

density along with the amplitudes and momentum transfers in the reactions

H, B + U.

taken in the forward direction. However, as Gribov pointed out , problems

appear when the states a,6 are not single-particle states. In this case,

F(a3 -*• 8N) are many-particle amplitudes to be determined in terms of the usual

two-body amplitudes. There is not yet an explicit treatment of this problem;

in the calculations only single-particle states are considered in the set {a}.

It seems convenient at this stage to give a detailed description of the propagation

after production of a two-body composite; by the way; focusing our attention on

the uncorrelated Trp Bystem. We shall obtain in this caBe the amplitude for

Ctrp) + H -*• (Trp) + H appearing in the multiple scattering series and to be

compared vith the experimental value coming from the up absorption in nuclear

matter.

In what follows we shall consider, as usual , that the pion

dissociation takes place on a single nucleon and, assuming the dissociation

amplitude is small, we shall neglect effects coming from coherent regeneration of

the pion, etc. These approximations are in order to concentrate ourselves on

Up propagation, neglecting small effects coming from well-known phenomena.

Let us consider the case vhere n nucleons take part in the interaction

and the production of the irp system takes place on the i-th nucleon, after

(i-1) elastic collisions of the incident pion. To account for the scattering

of the TTp system with the n-i remaining nucleons, we need to sum up the

diagrams corresponding to the following cases (Fig.l);

a) the ir(p) scatters elastically (n-i) times while the p{v) leaves

the nucleons without interactions;

b) the ir scatters elastically J times while the p does it (n-i-j)

times before leaving the nucleus (0 < J <n-i) ;

c) the IT scatters elastic ally J times while p does it SL times

with 0 < i $ n-i and Jl > n-i-J . In this case J + I - (n-i)

-k-

nucleons will be doubly scattered, first by it(p) and then

by the p(tr) .

We are using here a boosted version of the Eiaenberg result for the scattering

on a two-body composite (the Tp system actually), so that only single-scattering

and the on-shell nucleon part of the double scattering have been accounted for

in cases a,b and c, respectively.

In type c diagrams a new problem appears because of the double scattering

of a nucleon. Taking into account the definition of the generalized form

factor, it is straightforward to associate to a process vith n dispersions, one

of these being double (the J for instance), the amplitude

where the nucleon J has been considered on-mass-shell between both collisions,

the ^ are the momentum transfers by the nueleons and the modified form factor

is given by

where $ is the Fourier transform of the nucleus wave function. (2.2)

The momenta, initial (P ), final (pj) and intermediate P^11* , of the double-

scattered nucleon are gives by

(£.3)

-5-

To first order in the momentum transfers, and "working in the Breit system as

in Ref. 10. P „ » P1 - P J? « m and (2.2) coincides with the generalized formJQ JO J°

PJO *

factor used only for single-scattered nueleoos with q +2q1: + cj'i i

If there are3 i

several double—scattered nucleons, we sum over all possible orderings including

a factor (n f)~ where n is the number of double scatterings. In the caseJ J

where there are two of these, for instance, we sum the direct and crossed diagramsof Fig. S, obtaining, for the equivalent to (2.2),

The internal momenta for the direct term G(n) will he

as in (2.3)• However, for the crossed term 0(n)

it

and again to firBt order in momentum transfers (.(»).„•(»}0 Grs rs

cancels out. This result can be extended to the general ease.

and the factor (2!)"

On the other

hand, the two possibilities: first scattering with the ir(p) and second with

the P(IT), introduce a factor 2 in every double scattering term.

We could calculate separately the different diagrams corresponding to

sets a,h and c, and then sum up to obtain the n-th order term in the multiple

scattering series. However, it is straightforward to build an expression for

the n-th order term which accounts for those possibilities. In fact, using the

notation explained below, we can write

ml , ' « irlrf ,1 ., f

A-

{2.4)

We are using the Breit frame kinematicB in which the external momenta are

The k's have been chosen along the z axis and the i's are transverse

momenta. The longitudinal momentum transfer is fixed by energy conservation

_•» ^ '

where

(2.T)

-7-

We have several relations involving internal and external momenta along with

internal momenta transfers

fc.-*, i?: • ( 2. a a )

The momentum of the incident pion after r collisions is

and for the produced system after collision with nuoleon s, it is

(2.9a)

!*!,...,* ", 1.2.9b)

vherethe indices 1 and 2 stand for pion and rho, respectively. We also

define for the longitudinal components of the momentum transfers

(2.10)

So that, to first order in momentum transfer, the propagators are given by:

(2.11)

which are the expressions used in {.Z.k}.

Taking into account Eqs. (2.10), ire have vritten

(2.12)

and thence, using (2.5) and (2.8) and denoting hy s and z the transverse and

longitudinal components of x , we ohtaln

(2.13)

Carrying the result (2.13) to (2.14) and performing there the Integrations

over longitudinal momenta, we obtain

L-A

n-t

J

We can perform the summation over i in (S.lU); in fact, using

.n-L.

ln-D! (2.15)

-9-

we obtain, in schematic notation, for the n-th order term and the summed

multiple scattering series the expressions:

*F r £ ,n-±

0.16)

The exponentiated optical model expression, used in the data analysis, can be

obtained introducing the optical model density p(x) = Ap(x) and taking the

limit A •+ <• in F as given in (2.16).

The amplitude (2.lit) gives a precise prediction far the cross-section

0/ \JJ . If we use a gaussian density,

p(?) - p(0) e" r 2 / R 2, E 2 = R2A"/3, E 2 = 12 mb ,

and parametrize the high-energy amplitudes,

(2.17)

ti) =10 (2.18)

ve obtain for the profile functions the value

The term in (2.16) giving the wave distortion due to absorption can be -written,

in terms of the profiles (2.19), in the form:

^ i(2.20a)

-10-

2 2But using (2.19) ve obtain in the wide nucleus approximation B .'.«R

neglecting T!' s) :

and the predicted cross-section is

(2.20b)

_ JLfi. = 1,0 mb , (£.21)

with a change of the order of 1 mb if we consider r| = ±0.2 .

The result (2.21) strongly disagrees with the experimental value

a = 25 mb . Moreover, in order to keep the experimental value in (2.21), 6exp „

should be $ = 1.98 mb, which is one half of the experimental value. This

suggests that pion and rho should be BO near one another that they partially

overlap. But then strong interactions between them should be important. In

the next section we shall study the propagation of the irp system under such

conditions.

III. PEOPAGATIOS OF AH IlflERACTIHG TTp SYSTEM

In Sec. I I we considered an asymptotic irp system without in t e rac t ions

between cons t i tuen t s . In other words, pion and rho were so far separated tha t

they were outside each o t h e r ' s in te rac t ion range. The r e su l t obtained for

o, . „ , inconsis tent with c r . 6 ^ , ind ica tes t ha t the A system iB not BO(TTp)N* A,H ' 1

simple during the flight through nuclear matter. Let us consider for the moment

that pion and rho are less separated than the range of mutual interaction. It is

clear that the description in Sec. II is not valid for this case because both

constituents will be interchanging quanta not only in their propagation between

successive collisions, but also during the collision time. Intermediate state

interactions will be present and the system will be typically non-asymptotic in

the interaction with nucleons. This last fact will reflect in the impossibility

of applying to the system the one-channel and impulse approximation in the usual

way 11'. virtuality of constituents along with the presence of mutual interactions-11-

will play an important role in the scattering with aucleons.

In the following we shall be concerned with the propagation of this

non-asymptotic system. The nuclear part will be described as before and ve

shall suppose again that the system moves fast enough for us to apply theXk)

relativistic eikonal approximation to rho ana pion propagators. We shall

assume, for simplicity, that the internucleon distance is longer than the iT-p

interaction range. Let us take, as in Sec. II, the n-th order contribution

to the multiple scattering aeries; the production taking place on nucleoo i.

We again neglect initial pion regeneration along with rho decay inside the nucleus.

In Feynman diagram language, propagation looks as illustrated in Fig. 3, where

the F's are nueleon-hare system amplitudes. There is no interaction (wavy

lines) crossing more than one F due to the interaction range being shorter

than the nucleon-micleon distance as supposed above.

Let us call H the number of interactions that the pion (which

vill be denoted as constituent "l") undergoes betveen successive collisions

vith nucleons J and j + 1 . The product of propagators along line 1 betveen

these collisions vill be (see Fig. It)

(3.1)

W-.Where v is the pion momentum when it leaves the J-th nucleon. TheJ

are the momenta emitted from the pion line in the interactions, and the symbol

...) denotes summation over the permutations of k ,k ,... By defining

A » m - ("O and using the eikonal approximation in the other propagators,J TT J

that i s

we can write (3-l) in an eikonalized integral form

(3.2)

-12-

In (3.2) we denoted by ^ k e the ej momenta of the set G^,,,.,^ }

absorbed by the rho (constituent "2") between collisions with nucleons J-l and J;

the f momenta absorbed betveen collisions j andby t ,,.,, k

Jj + 1 ; and by k",...,k" those g absorbed after tbiu laBt collision.

The product of propagators aloag line 2 betveen the same collisions

J and i + 1, can also be vritten in the form

W£Wi . -.1

n i

where k , k and k correspond to momenta emitted from line 1 before collision

J, between this and the following collision and after the latterjrespectively.

By grouping the terms "surrounding" each collision as shown in Fig.U,

it is possible to write the overall product of propagators in the factorized

form

C3.lt)

In (3.1) to (3.3) the sum over all the possible diagrams obtained by

all possible attachments (permutations) of the k's to the lines 1 and 2 has been

considered, so that we must include a factor (N,!)~ to avoid double counting.J

For H fixed we must also sum over all the topolagically different FeynmanJ

diagrams leading to the sets (e,,f, ,g ) . This introduces a factorJ J j—1

-13-

(3.5)

where M = e, + f + g ia the number of interactions surrounding the j-th

collision.

1 2

Let TI and u te the momenta of pion and rho JuBt before reaching

the nueleon J. The amplitude for the dispersion off the J-th nueleon including

interaction "between constituents will be

(3.6)

•where, apart from the f a c t o r s and suns i n d i c a t e d i n ( 3 . 5 ) , T i s g i v e n byJ

H'

(3.7)

6(ks) ia the propagator of the interaction carrying momentum k and F(u ,q }

i s the bare lrp eystem nueleon amplitude.

Expression (3.7) can be reduced to a form similar to that of the

model a la Glauber. To do i t , we introduce the following definitions:

1131V X E ( X V (1 12) J 11 '3with

-Hi-

UBing (3.6) and (3.9), we can write C3.7) in the form

(3.10)

The amplitude F(u ,a ) depends explicitly on the momenta transfer to pion andJ d

rho In the eollisiocij that ia, on

This allows us to define a generalized profile function in'the form

(3.12)

writing

We now escploit the explicit dependence of the A on the q., by

t f a*1and rearranging

<^"« •(3.13a)

Similarly, we haTe

The expresEion (3.H) can thus be written, using the variables

-15-

in the form:

A: =4-1

V bi

i \

where we used final state kinematics for simplicity, and

end C3.15)

Using (3.12), (3.1U) and (3.15) in (3.10), we can reduce the integrations over

1 21j Bni q,j to

1 2so that we can remove the integrals over X, and 1., obtaining

S I 3 1 T ,(3. IT)

-16-

vhere, for instance,

and I and I are the corresponding factors in (3.1k).

The last part of (3.17) describes the interaction betweenconstituents. It can be written in the exponential form usual in the eikonaldescription . First , by using (3.9) we have

4 l

so that

r

If we now define the Fourier transform of the propagator

-17-

(3.18)

and use (3-18), we obtain

1 b»

V »*

Introducing now the sums and factors indicated in (3.5) in the form

(3.19)

1 ¥ i53

' I?',

But the sum of a l l momenta emitted from line 1 equals that of momenta absorted

in line 2, that is

1 2 •*So that v. + v i s fixed Tjy the final momentum and the a,

1 J J

In the same way, is fixed by p"(p") , q and J|V2'kJ B

(3.22)

(3.23)

amplitude for production on nucleon i (once the term corresponding to the inter-

action of constituents has teen included in (3.11*)),can then he written in the

form:

ve mist sum over all possible values of M , that is,

Using (3.19) and (3.20} In (3,17), we have

H

(3.20)

f^Lfe) 2x*rt* -2?}t>)\ •(3.21)

The amplitude for the production process taking place on nucleon irequires some special considerations. In fact, for j > 1 we had therestrictions

-O .

f ̂ e"

where

(3.2!*)

(3.25)

The terms exp(iq n) must be placed in expressions like (3.T). Their effect is

a shift in the arguments of the generalized profile functions to

(3.26)

In (3.2!t) we only considered the explicit dependence of the amplitude in the

momentum q_̂ transfer to the i-th nueleon. It determines completely the

reaction of nuclear matter to the production process. We could use separated1 2

dependences on v and v , thence ottaining a more general distribution on1 2Ti and ri .

-18-

-19-

ii.

At this point, by collecting results C3.2M, (3.21) ,(3.6) and aftar using

for the first i-1 elastic collisions of the pion the methods of Sec. II, we

can vrite for the n—th order term of the multiple scattering series, the

following:

L-i

(3.27)

•where

at C3.26)

In C3.27) ve use different classes of profile functions depending only

on the impact parameter or, also, an the longitudinal components and times.

However, for fast propagation, as in our case, the matter distributions are

Lorentz contracted in the direction of motion Oz.

Then for the production profile we may replace

(3.29,

This corresponds to "punctual" production of the two-"body composite in ri as

considered in (3.21*). To reduce the TTp profile, we first define the times of

IT and p in our frame

so that,

(3.30)

(3.31*)

-20-

and the velocities 6 ,B will be taken parallel to the Oz ails.

Let us first consider no interaction between constituents, that is,

"i> = 0 . Then the system is free and we can define tbe centre of mass

>

2,4-= (3.32)

which will propagate forward with constant velocity. In this case, we can

describe the system in a surface x = const, and also translate the contraction

to the centre of mass in such a way that

Using (3.33) we re-o'btain the result of See. II:C3.33J

where y (b) should tie the profile for an asymptotic irp system.TfP

When 2># 0 the situation is very different. The off-shell effects

due to the presence of interactions render meaningless the surface i = const,

and the centre of mass is no longer (3.32). We must renounce to the surface

x m const, and vrite

(3.35)

-21-

Instead of (3.3*0, ve now obtain

\fa I--,-.

(3.36)

where u indicates the energy transfer in the collision to pion and rho; T^ J

is the relative time, and the eikonals depend on the z.'» and t/s in a way

fixed by (3.28), (3.29) and (3.32).r.

IV. COHCLOSIOHS

It seems worthwhile to spend a short time to analyse the meaning and

consequences of the modifications to the absorption obtained in Sec. III. First

of all, the eikonals and x.

absorptioa. In fact, one expects

in (3.36) could largely modify the

Be e"* < 1 ,

due to two-body unitarity (as used here). This fact by itself can explain the

unexpected experimental low absorption. It is interesting to remark that these

eikonals are a realization of those interactions lacking non-relativistic

analogue indicated by Gottfried as possible lowering absorption mechanisms.

They are missing in a SehrBdinger-type analysis of the interaction. They appear

here due to the very special features of the system (i.e. Trp distance short in

front of the nucleon size).

Let us assume that the nucleon system is so dilute that the up

interaction range is much shorter than nucieon-nucleon distances. Then the

integral over the relative time T gives S((i> ) (in the measure that weJ J

substitute the limits of integration Is the eikonals by the asymptotic values).We have recovered energy conservation for a system of nucleons diluted enough!

The energy transfer u has a two-fold, off-shell connection.J

Firstly, by looking to the Trp phase space, one realizes that ui • 0 correspondsJ

to a read TTp system scattered in the forward direction, Jis thus related to the distances from the pion and rho energy shells.

Secondly, for u ^ 0 the Intermediate nucleon has an energy given by (2,3):J

ninS10 * 10 + U1 ' ^ n ^ e ^outlle scattering term we now have a sum over the

-22-

inelastic intermediate nueleon states in.a form similar to that of the correction

to the Glauber formula . For a "frozen" micleon, the distance between the

nucleon pole and the pion-nucleon threshold in the propagator is AE "> (it u) - m,

so that for ID m u the contribution from this effect will be measurable.

In conclusion ve show that for an interacting composite, low

absorption should be expected. Of course interchange of scalar quanta is the

simplest model for the interaction and important phenomena could be missing. InI1*)particular, we neglected radiative corrections in the hope that they factorise,

along with non-commuting interactions. But the information extracted from

coherent production experiments is so restricted that inclusion of such effects

would be redundant by now. There remains the achievement of the improved

agreement with the experimental data and the better understanding of the A

structure.

ACKNOWLEDCMEHTS

I should like to thank Professor Abdus Salam, the International

Atomic Energy Agency and UKESCO for hospitality at the International Centre for

Theoretical Physics, Trieste. I am also indebted to Professor L. Bertocchi

for introducing me to the problem and for many useful discussions.

-23-

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13) V.H. Gribov, Bov. Phys.-JETP 2g, U83 (1969).

lit) M, Islam, in Lectures in Theoretical Physics, vol . l!*, eds. B&rut and B r i t t i n ,

(Colorado A.U.P., Boulder 1972), p,101;

B.M. Barbashov, S.P. Kuleshov, V.A. Matveev and A.H. Sissakian, JIHE, Dubna

prepr int E2-it9S3 (1970);

C.A. Orsa les i , ICTP, Tr ies te preprint 10/73/1^8.

FIGURE CAPTIOES

Fig. 1 Diagrams for the production and sca t t e r ing of the asymptotic irp

system.

Fig. 2 Direct and crossed diagrams for two double-nueleon sca t t e r ings .

Fig. 3 Production and propagation of an in te rac t ing Up system.

Fig. It Col l is ion with nucleon J dressed with the surrounding in t e rac t ions .

- 2 5 -

it- * ~

l1^ Hflj.

(a) . Fip.2

31 • -

JT •

(b)

(c)

Pig.3

. i- 2 6 - - 2 7 -

•# 4»i >!'-