0IC/68/99

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICALPHYSICS

GLAUBER SHADOW

AND INELASTIC CONTRIBUTIONS

TO ?rd SCATTERING

G. ALBER1

and

L. BERTOCCH1

1968MIRAMARE - TRIESTE

J Si! Ifc il'l I f in ii.

IC/68/99

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONA L CENTRE FOR THEORETICAL PHYSICS

GLAUBER SHADOW AND INELASTIC CONTRIBUTIONS

TO Jrd SCATTERING * +

G. ALBERI

Istituto dLFisica Teorica delltlniversita, Trieste, Italy,

and

L. BERTOCCHI

International Centre for Theoretical Physics. Trieste, Italy,

and

Istituto di Fisica Teorica dell'Universita, Trieste, Italy.

ABSTRACT

The. contribution of inelastic intermediate states in

the elastic ird scattering is estimated using a Regge pole

model. It is shown that this effect can fill the dip which

otherwise should be present in the differential cross-section.

MIRAMARE - TRIESTE

November 1968

* To be submitted for publication.

T Supported in part by the Istituto Nazionale di Fisica Nucleare.

GLAUBER SHADOW AND INELASTIC CONTRIBUTIONS

TO Td SCATTERING

1. INTRODUCTION

An interesting feature of a number of recent elastic

Td and pd scattering experiments at intermediate energies is the

fact that the qualitative predictions of the Glauber multiple scattering

theory are definitely confirmedt after a quick decrease of the

differential cross-section from the forward direction, ascribed to

the form factor effect in the single scattering term, the cross-

section has a much gentler slope, characteristic of the double

scattering term. However, one of the precise predictions of the

theory, the appearance of a pronounced dip where single and2)double scattering terms strongly interfere, seems to be almost

absent in all the experiments performed up to now (while it is present4

in pHe scattering and in the elastic scattering of protons from other

light nuclei (C ,0 ) ,

We refer in particular to

4)

(a) the pd elastic scattering at 1 GeV

(b) the ?rd elastic scattering ' at 895 MeV/c

(c) the *d elastic scattering at 2. 75 GeV/c.

A number of different explanations can be invoked for this

absence of the dipj among them we quote:

i) The non-asymptoticity of the energy (which, for the measured

values of the momentum transfer t, means not very small angles);

not only does all the Glauber theory rest on the assumptions of large7)

energy and small angles, but it has also been conjectured that

in this situation the "principal part" term in the double.scattering

contribution, which is usually neglected in the Glauber theory,

would have the effect of changing the relative phase between the single

and doable scattering contributions*

-1-

ii) A strong contribution of spin-flip amplitudes, which do not

interfere with the leading part of single scattering and so could fill

the dip.

iii) An effect of the energy dependence of the elementary (jrN or JNTN)

scattering amplitudes, due to strong resonances in the direct

channel which, via the Fermi motion, would smooth any wild

variation in the shape of the differential cross-section.

iv) A very rapid t-dependence of the phase of the scattering

amplitude, defined by f(s,t) = |f(s,t)| e1 < p ( s ' t J .

However, neither of these explanations could work if we want

to explain by only one effect all three experiments (a)-(c). In fact,

while i) can be invoked for both (a) and (b), it is probably wrong

for (c), since 3. 65 GeV/c is already quite a high momentum; ii)

would be valid for (a), but is untenable both for (b) (where the

scattering amplitude is-known from phase shift analysis) and

for (c) (where good representations of the *N scattering amplitudes12)can be obtained from finite energy sum rules ); and the same

argument applies to iv).

On the other hand, iii) is certainly very important for (b),

but should have a negligible effect both for (a) (there are no resonances

in the nucleon-nucleon channel) and for (c), where the resonances

are no longer important for *N scattering near the forward direction.

Of course, a mixture of different effects can always be invoked;

however, an explanation which involves many different effects would

be very unpleasant and complicated.

There are, moreover, another set of experimental data which

are also relevant to Glauber theory, namely the experimental results

for the cross-section defect, defined as the quantity

6

where x is the incident projecti le.

As before, we consider only pd and rd scat tering, where

the experimental information is more copious, and we a r e faced

with the following situations

For ?rd scat ter ing in the few-GeV region outside the

resonance region, the cross-sec t ion defect agrees ra ther

well with the theory,computed using Gartenhaus form factor;13)

moreover , it has been recently shown by Faldt and Ericson

that most of the discrepancy existing in the resonance region

can be resolved, if the F e r m i motion is included, at least in

the single scat ter ing te rm. At energies higher than 8 GeV

or so, there is some evidence of l a rger values of the cross -7)

section defect, but e r r o r s a r e quite large.

For pd scat ter ing the situation is not very clear , but

again there is some evidence that the cross-sec t ion defect

increases with the energy, even where

For pd scattering, 6

We shall therefore use the following.form for the elastic

x-d scattering amplitude:

V

"•SSv^ct,). a.D

Here F _(s,t) is the elastic xN scattering amplitude, normalized soe da

thatdt

el 2 (Y)F

el i nis the amplitude for the inelastic

transition x + N —» y + N, again normalized so that indt

F (y)in

2 .

p is a two-dimensional vector orthogonal to the incident momentum,

t. and t the momentum transfers between the initial, or final, and

the intermediate particle in Fig. 1. The total inelastic differential

cross-section for quasi-two-body production is given by

(v)da. _ da:1" _-, / \ 2

in = V1 m r (y)

dt F i n(1.2)

Formula (1.1) does not mean, however, that we can write for the

cross-section defect the expression

- + r Ar at * [ e + l (1-3)27* J

(T = -t)

In fact, while the part of the double scattering term containing the

elastic cross-section can be considered as a reasonable

approximation, since the elastic amplitudes are, at high energy

and small t, almost completely imaginary, the part containing the

inelastic cross-section can be wrong by a large factor, since in

general the real part of the inelastic amplitude is important. The

correct formula for the cross-section defect is therefore

- 5 -

•» m *» * m.

We see therefore that at small t the elastic amplitudes are

(almost completely) imaginary since their dominant contribution

is given by the exchange of the Pomeranchuk, which has a (0) = 1,P

and positive signature.

We notice, on the contrary, that trajectories with a(0) = —

would produce amplitudes whose real and imaginary parts have,

at t = 0, the same absolute value. Since all the integral

in the expression of the cross-section defect is dominated by the

small t region, due to the strong peaking of the form factor, the

contributions of these amplitudes to. the cross-section defect are

strongly depressed.

A strong depression is also obtained whenever the production

amplitudes vanish at t = 0 due to the presence of kinematical factors

arising from parity and angular momentum conservation, as for

spin-flip amplitudes.

The important point is now that in TT initiated reactions, except

for the case of A production(which, as we said, contributes very

little to the inelastic cross-section) all the important resonances are

produced either via the exchanges of Regge poles with a(0) — — ,

or the cross-section vanishes at t = 0.

In the first class can be included the x? production, which

goes through the A exchange, the w production, through the p

exchange; the typical reaction of the second class is p production,

which goes through the peripheral ir exchange, and so has a[Q) — 0,

but whose amplitude, barring T conspiracy, vanishes in the forward2

direction, due to the presence of the A coefficient which comes from

the 7 matri x. Since we neglect the excitation of the target nucleon,5

we shall not consider reactions such that

/ 2

which, going through P exchange, have imaginary amplitudes. As a

- 7 -

general conclusion, we should expect small contributions from the

inelastic states to the cross-section defect in jrd scattering.

On the contrary, in Nd scattering one must consider the

amplitudes of the reactions

NN -» N*. N

which can be mediated by P exchange, giving rise therefore to

imaginary amplitudes at t = 0 and — .roughly constant in s. We

can therefore expect a sizeable contribution to the cross-section

defect in Nd scattering. The ratio of the cross-section defect,

including elastic and inelastic contribution, over the same 6 a

but including only elastic effect, can be estimated as given by

e l m

where a . is the elastic (integrated) crdss-section, and Let. ' isel inthe sum of all the integrated cross-section for production of those

+N , resonances whose channels are open at the energy which is1/1considered.

However, the argument which asserts the lack of effect in

6 a for ird scattering is no longer valid when we consider the real

part of the rd double scattering amplitude and when we look at t j- 0.

It is therefore possible that the inclusion of inelastic intermediate

states would alter the angular distribution of jrd scattering.

Our aim here is not to perform an exact calculation, but to

construct a model in which the inelastic amplitudes are described

by reasonable assumption and which, in the absence of inelastic

contributions, has a narrow, deep interference minimum; we shall

therefore show that the inclusion of a reasonable amount of inelastic

contribution will not alter 6 a , will not significantly change the

cross-section in the second maximum region, but will completely

fill up the minimum.

- 8 -

2. THE MODEL

We shall use the following model:

The elastic amplitude is described by a fixed pole at19)a =• 1, plus a Regge pole with positive signature and a

trajectory of the form a ft) = . 5 + t {the P1); for simplicity

we neglect charge exchange and we take both residua of the

same exponential dependence in t. Therefore we have

bt bt -i~a(t)T = i ^ e -JZle e . (2.1)

el 1 ^

In the inelastic amplitude, we neglect the contribution of possible

negative signature trajectories with a(0) — - j the reason is

that in general they produce amplitudes which vanish in the

forward direction (for example, the p exchange in to production).

Therefore we typify the production amplitude by a TU'ime pou-

exchange with positive signature and a (t) = 0. 5 + t,

T ( 3 ) = -sf^e 2 e 2 2 (2.2)in 3

plus a term with a(0) = 0, but whose amplitude vanishes like

t at t = 0. This last term could represent the peripheral

exchange of the it in p production, and therefore we choose it

in the form

Since this term is not very important, we neglect the dependence

of its phase on t; moreover we have added an exponential

decrease in order to have a narrower peak for " p" production

(form factor effect).

- 9 -

3. RESULTS AND CONCLUSIONS

We never include the factors s in our formulae, since we

work at fixed energy. An energy variation would therefore only alter

the ratios of the different c ' s ,i

In Fig. 2 we show the result of calculations including the inelastic

effects (continuous line) and excluding the inelastic effects (dotted-2line)] for simplicity we have chosen b = b = b = 4 (GeV/c) ;

A A

the value of the constants are c = 69. 6 GeV " ; c = 2 c /25 GeV" ;-4 -4

c3 = 40 GeV ; c4 - 25 GeV . They correspond to a total *Ncross-section of 27.6 mb, which would tend asymptotically to 23 mb

(3)(the P contribution). The inelastic cross-section due to T. is(4) *) i n

1.9 mb, and that due to T. ' is 0.9 mb.in

We see that in the absence of inelastic contribution we do have

a narrow, deep dip in the angular distribution; the inclusion of a

reasonable amount of inelastic contribution, of the form discussed

before,does not alter the cross-section defect very much (the variation

of the imaginary part of the double scattering amplitude is of the order

of 10%), does not significantly change the cross-section in the

secondary maximum region, but completely alters the shape just

before and in the region of the minimum, completely smoothing out

the differential cross-section. In Fig. 3 we report the same calculation

using values of c^eaual to 20 and 80 (the result is not sensitive to c ).

One can see .that the general features remain the same, except that

with a large inelastic contribution even the cross-section on the

second maximum can be altered; in the first case (cQ" 20) a minimum

appears. This happens because an amplitude which gives an inelastic

cross-section of 2 mb is not very much smaller than the elastic one

(in our case or = 5. 6 mb) therefore where there are cancellationsel

in the elastic amplitude, the inelastic part can give a substantial

contribution.

*/ These values of the inelastic cross-sections should not be taken to represent the cross-section

of one definite channel, but rather the sum of the cross-sections of the channels which can be described

by the same production mechanism.

-10-

to beWe do not consider our examplelmore than a general indication)

in a more refined treatment one should include carefully all the

inelastic channels with all their kinematical complications. However

we feel that an effect like that which we have discussed can be the20)

origin of the filling of the dip.

We shall also briefly discuss what is the energy dependence of

the effect we are introducing here. It is true that every inelastic

channel shows a cross-section which decreases with the energy;

however, the number of open inelastic channels also increases with

the energy, so that the two effects could compensate each other.

Moreover, at intermediate energies for the production of a definiteminimum

state, one must remember that a]value of j t | exists for the

production of an inelastic state; since this 111 . decreases with the

energy (as l /s in our case), we include in the double scattering

integral a larger fraction of the exponential e , thus partiallyfor

compensatingjthe energy decrease of the cross-section. We should

therefore expect the effect of inelastic channels to be present at

rather high energies a l so . Whether this effect would persist at

asymptotic energies can at present be only a matter of speculation

and is connected with the question: do resonant states of higher and

higher mass exist and are they produced with sizeable cross-

sections?

For what concerns the problem of the energy dependence of the

cross-section defect in nucleon-deuteron scattering, if we assume

that at high energy the elastic cross-section tends to a finite fraction

of the total cross-section (as implied by our fixed pole in (2. 2)), we

can have the following two alternatives:

*a) At high energies the number of N , resonances which are

produced increases more and more, and the sum of their production

cross-section tends to a l imi t ) or,,* —> KCT, = K{cr - or ), with/ j N , u in tot el

K < 1 ; then the cross-section defect will increase continuously,tending to a finite limit; this finite limit is, however, reached only

at infinite energy.

-11-

b) There are a finite number of N . resonances; then, if the

threshold for producing the N* . with the largest mass is s , at

energies rather larger than s ( so as to neglect kinematical effects)

6 or will already approach its asymptotic value.

ACKNOWLEDGMENTS

The authors are grateful to Professors Abdus Salam and

P. Budini and the International Atomic Energy Agency for

hospitality at the International Centre for Theoretical Physics,

Trieste.

-12-

REFERENCES

1) R. J. Glauber in Lectures in Theoretical Physics, vol. I, ed.

W. E. Brittin.et al. (Inters cience Publishers, Inc., New

York, 1959} p. 315;

V. Franco and R. J. Glauber, Phys. Rev. 142_, 944 (1966);

R. J. Glauber, Proceedings of the II. International Conference

on High-Energy Physics and Nuclear Structure, Rehovoth,

Israel, February 1967.

2) V. Franco and E. Coleman, Phys. Rev. Letters 17, 827 (1966);

L. Bertocchi, F. Bradamante, G°. Fidecaro, M. Fidecaro,

M. Giorgi, F. Sauli and P. Schiavon, "Proposal to Measure

the Phase of Pion-Nucleon Scattering at Various Energies and

at Non-Zero Momentum Transfer", Internal Note INFN/AE -

67/5 (1967);

T.T. Chou, Phys. Rev. 168., 1594 (19fi8).

3) H. Palevsky, J. L. Friedes, R. J. Sutter, G. W. Bennett,

G.J . Igo, W.D. Simpson, G. C. Phillips, D. M. Corley,

N.S. Wall, R .L. Stearns and B. Gottschalk, Phys. Rev.

Letters 18_, 1200 (1967).

4) G. Bennett, J. Friedes, H. Palevsky, R. Sutter, G.J. Igo,

W. Simpson, G, Phillips, R. Stearns and D. Corley, Phys.

Rev. Letters 19, 387 (1967).

5) F. Bradamante, *S. Conetti, G. Fidecaro, M. Fidecaro,

M. Giorgi, A. Penzo, L. Piemontese, F. Sauli and P. Schiavon,

"ird coherent scattering at 895 MeV/c" CERN preprint (1968).

-13-

6) R.C. Chase, E. Coleman, T.G. Rhoades, M. Fellinger,

E. Gutman, R.C. Lamb and L. S. Schroeder, XIV'

International Conference on High-Energy Physics, Vienna (1968).

7) J. Pumplin, Phys. Rev. 173, 1651 (1968).

8) One has to remember that the singular contribution of the produot

of the principal part is already taken into account in the9)usual formulation of Glauber theory ; what we mean here is

the effect of the remaining non-singular part.

9) L. Bertocchi and A. Capella, Nuovo Cimento 51A, 33 (1967).

10) V. Franco (Los Alamos preprint, 1968) has been able to show

that with reasonable assumptions on nucleon-nucleon amplitudes

it is possible to fill the dip in pd scattering.

11) A. Donnachie, R.G. Kirsopp and C. Lovelace, CERN

preprint TH 838 addendum (1967)j

P. Bar eyre, C. Bricman and G. Villet, Phys. Rev. 165,

1730 (1968);

C.H. Johnson, UCRL preprint 17683 (1967).

12) C. Fer ro Fontan, R. Odorico and L. Masperi, ICTP, Trieste,

preprint IC/68/90'(to appear in Nuovo Cimento);

V. Barger a n d R . J . N . Phillips, Phys. Letters JJ6B, 730 (1968).

13) G. Faldt and T. Ericson, CERN preprint TH 938 (1968).

14) M. J. Longo, paper presented at the XIV. International Conference

on High-Energy Physics, Vienna (19 68).

15) The inclusion of double charge exchange terms in Glauber

formula (C. Wilkin, Phys. Rev. Letters 17, 561 (1966) is

already an example of inelastic contributions; we speak here

of genuine elastic terms, where the intermediate state in the

double scattering has a mass different from that of tie incident particle.

-14-

16) K .S . Kolbig and B. Margol is , Nucl. .Phys . JB6, 85 (1968).

17) It has been recent ly estimated that the coupling of the deuteron

to other channels is only of the o rde r of a few per centj

see A .K . Kerman and L. S. Kiss l inger , MIT prepr int 1968.

18) F o r s implici ty we neglect charge exchange, so that xp and xn

ampli tudes a r e equal.

19) It is general ly found that a'p is r a t h e r sma l l (in the second

paper of Ref. 12 it is shown that a'p

FIGURE CAPTIONS

Fig. 1 The double scattering diagram with elastic (a) and

inelastic (b) intermediate state.

Fig. 2 The elastic ird differential cross-section —-. The

dotted line is the cross-section when only elastic

intermediate states are included in the double scattering

term, the continuous line is the cross-section when

the inelastic states are also taken into account. The-4

value of the constant c is cQ = 40 GeV

Fig. 3 The same as Fig. 2, but here the dot-dashed (- • - ) and the

dashed { ) lines contain the inelastic contributions_4 ' -4

with c_ - 80 GeV and c = 20 GeV , respectively.a o

-16-

\x

/ x

17-

' • * • • * * * ' * • • * • * •

- 1.

- ,.1

- .01

1\\

V*t

t

t

ft

+

. I 1

in mb/(GeV/c)2

•\ *** •* •

* *

* •

% *

\ t, *

''Jl 1

-1 in (GeV/c)2

• *

.1 .4 , ) . fi . 7

- 1 2 -

X

\

-1. \

X\

in mb/(GeV/c)2 \

- . 1

- . 0 1

i i

' ' \ \

i

*

•. \• . »

*. *

: *

• \

*. *

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* s . >

c = 20

/

+

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* ' * * * • • ' " *

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-t in (GeV/c)2

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.1 . 2 . 3 . 7

Kitt. 3