PROBLEMS OF DIFFRACTION SCATTERING IN RELATIVISTIC THEORY *

V. N. Gribov Leningrad Institute for Nuclear Research

1. Introduction

One of the main characteristic features of strong interactions is approximate constancy of

the total cross sections. It is the great challenge for a theorist to describe this phenomenon in a

simple, general. and selfconsistent way. The possibility of such a description. despite the absence

of a strong interaction theory. is the belief that this phenomenon does not depend on the features of

strong interactions at small distances which we, of course, do not know from the spectrum of

hadrons and even from symmetries of hadron interaction, but only on the fact that the interaction

is strong enough and the energy is high enough as compared with all characteristic masses.

At first sight this phenomenon looks too general to be interesting. In 1958 I. Ya. Pomeranchuk

made the first attempt to look at this description from the point of view of relativistic theory and

found out that the cross sections of particles and antiparticles must be equal if the radius of inter­

actions does not increase with the energy like In s.

Another attempt to look at the diffraction phenomena from the relativistic point of view was

made in 1960 (I. Ya. Pomeranchuk. V. B. Berestetsky (1960), V. N. Gribov ( 1960)J. It was shown

that a simple diffraction picture with the radius of interaction not depending on the energy is in con­

tradiction with t-channel unitarity. The idea is very simple. In relativistic theory all the hadron

interactions are due to the exchange of other hadrons. Therefore. if we suppose that any hadron has

its total cross sections not depending asymptotically on energy. we must look what will happen if we

consider all the processes due to the exchange of some hadron and take into account that this

exchanged hadron can trans fer different energies. The processes correspond to the diagram of Fig. 1.

Fig. 1

The cross section of all these processes integrated on s1 and s2 over the region s1 s2 s is the'U

order of magnitude

dS1ds2 a( s) 'V --s-- cr( s 1) cr( s2) 'V In s; (1 )f

that is. after summing on all transfered energies this particular cross section becomes larger

than the total cross section.

This contradiction has led to the hypothesis that the radius of interactions must increase

with the energy like (In s) 1/2, because in the case of the radius of interactions increasing with

energy, different exchanges (one, two, three etc. of hadrons) can interfere strongly to avoid the

*Although Dr. Gribov accepted our invitation to give a plenary talk on the dynamics of strong interactions, at the last minute he was unable to attend. ELG

-491 ­

contradiction. The particular choice of the dependence of the radius on energy was connected with

Regge I s discovery of analyticity in the angular momentum plane in nonrelativistic theory. After

ten years of experimental study of diffraction scattering we know now that the radius of interactions 3

increases in this way in the region from 10 to 10 GeV for p-p interactions.

It is clear that the relativistic description of diffraction must be different from the non­

relativistic one. This relativistic diffraction theory was formulated [V. N. Gribov (1961L

G. F. Chew and S. C. Frautschi (f96f») in terms of the so-called Pomeranchuk pole. But some

problems have appeared again when it was understood that the pomeron and other Regge poles in

many aspects must be considered as usual hadrons [K. A. Ter-Martirosyan (1963), S. Mandelstam,

Polltinghorne. V. N. Gribov. I. Ya. Pomeranchuk, K. A. Ter-Martirosyan]. There are many ways

to observe these problems [E. Verdiev, O. V. Kancheli. S. G. Matinyan. A. M. Popova and

K. A. Ter-Martirosyan (1963). O. V. Kancheli (1968). J. Finkelstein, K. Kajantie (1968),

V. N. Gribov, A. A. Migdal (1968), H. D. 1. Abarbanel, G. F. Chew, M. L. Goldberger, and

L. M. Saunders (1971)].

One of the simplest ways to see it is to consider all the possible processes which are due

to the exchange of the pomeron. In particular consider the diagram of Fig. Z.

Fig. Z

In the kinematical region s/s »1, Sf fixed, and small perpendicular momentum transfer q #1 l.

these processes do not interfere with the others. The summation on all s1 J even over this small

region.. gives cross sections which are increasing as in in s, if we suppose that the total cross

section of the pomeron at ql. = 0 with any hadron does not decrease with energy. Therefore, it

was supposed [V. N. Gribov and A. A. Migdal (f968)] that this total cross section (which corre­

sponds to diagram. of Fig. 3)

Fig. 3

is really decreasing with energy; that is, the forward-scattering hadron-ql. = 0 pomeron amplitude

does not contain the vacuum. pole of Fig. 4.

pp ~ ql.

=0

Fig. 4

-492­

In other words this means that the three-pomeron interaction vertex must be zero if the perpendicular

momenta of the pomerons are zero.

Now we come to the problem: is it possible that all hadrons have constant total cross section

but one of them, the pomeron, has a decreasing one? It is clear that if it is possible, it puts severe

restrictions on all pomeron interactions.

During the last year it has become clear [V. N. Gribov, O. V. Kancheli, unpublished,

H. D. I. Abarbanel, S. D. Ellis, M. B. Green, A. Zee. NAL-THY-71 preprint. Apri11972,

H. D. 1. Abarbanel, V. N. Gribov. O. V. Kancheli. NAL-THY-76, August 1972, C. E. Jones,

F. E. Low, H. Tye, G. Veneziano, J. E. Young. Phys. Rev. D6, 1033 (1972), R. Brower and

J. Weis, MIT preprint. June t972] that these restrictions are very severe. Namely. it is possible

only if the amplitude of any inelastic process due to pomeron exchange is zero at zero perpendicular

momentum of pomeron. What is the reason?

The first observation: if the total cross section (Fig. 3) decreases as a function of s1 the

amplitude for any particular process (Fig. 5) decreases as a function of 8 1,

Fig. 5

because of the unitarity condition. From unitarity

~d b1m I ~~cd

it is clear that

L (2)

The first factor in (2) is a decreasing one. the second is a constant. It means that the amplitude

for any particular process decreases.

The second observation: if the state "c" has vacuum quantum numbers, then the asym.ptotic

behavior of the amplitude (Fig. 5) is governed by pomeron exchange (Fig. 6) which does not decrease.

Fig. 6

-493 ­

It means that vertex of Fig. 7 (two pomeron - any hadron state) is zero if the perpendicular momen­

tum of one of pomerons is zero.

Fig. 7

The third observation: let us suppose that the state " c " contains two particles. c c2. with1 ,

the total.center of mass energy M2.. As the vertex of Fig. 7 must be zero identically in M2. then

the contribution of any Reggeon exchange in this vertex at large M2. must be zero (Fig. 8).

p,""",~~ ~CIIRI C

p~C2

Fig. 8

This means that any Reggeon, hadron, pomeron vertex must be zero at q ::: O. 1.

Fig. 9

The last step of the proof consists in putting the Reggeon of momentum k1. on the mass shell to be

a particle h . Thus we will reach the amplitude (Fig. to) of two hadrons - pomeron, which mustt be ze ro at q ::: O.

1.

b I ---,r----C,

~~=O Fig. to

At first sight we can put particle hi equal to particle c and prove that the forward scatter­t ing amplitude, that is the total cross section, is zero. But as we shall see later on it is not so

2simple. The vertex of Fig. 10 could be singular as a function of q and mb21 - m C because, att this particular point (mb~ = mc~. ql. = 0) the contribution of differtnt exchanges to the amplitude

Fig. 8 can compensate each other. In any case the amplitude Fig. 10 for inelastic processes must

be zero at ql. ::: O. The simplest form for this vertex g(q1., ... ) at small q.L is

-494­

where 1 is some vector characteristic of states b f and Ct' .L This result is most important from an experimental point of view. It predicts that the

pomeron contribution vanishes to any inelastic amplitude at q I = O. And one is left with the two­

pomeron-exchange contribution (Fig. 11L which decreases with energy. In the region of small

Fig. 1 t

q the cross section for most inelastic processes which can go through the vacuum exchange have 1.

asymptotically (after averaging over polarization of states b 1. and c 1) the following structure

do(q ) 2 2 __1.._ = a q 2 e -2£l1 tql.. ~ + a. i... e -atq1.. ~ (3)

dq1.. 1 1. l. S4

2~ = In s/m , a' is the slope of the pomeron trajectory. It follows from (3) that if the minimum at

q ::: 0 is observed in any cross section, it must become deeper with increasing energy. 1.

From the theoretical point of view we still have questions. Do all these restrictions on

pomeron interactions really exist? Is it possible to satisfy them? And there is another interesting

question: does there exist any other object in Nature which has zero inelastic interaction and non­

zero elastic one at zero perpendicular momentum? The answer is, of course, yes. It is the

photon. But there is a very important difference. Photons have all inelastic interactions equal to

zero, while elastic interactions are nonzero only for charged states. The pomeron must have all

elastic interactions not zero. The second example is the graviton. It has all elastic interactions

nonvanishing, but its interactions increase with energy. The most interesting analogy~ from my

point of view, is neither photons nor gravitons but the process of diffraction scattering on a target

with large radius R.

II. Diffraction Dissociation

Consider for example a fast deuteron dissociating into a proton and a neutron in the process

of scattering on a heavy object of radius R »r , where r is the deuteron radius. Let 0t(P1' R)d d

and 0Z( P ' R) be complex phase shifts correspondingly of proton and neutron scattering on thisZ

heavy object (p is the impact parameter). Then the dissociation amplitude is well knoJNll to be

representable in the form:

f iq Pc 2 f 3 (-) * 1 fd ;n. p::: e 1.. d Pcdr12 4Jnp ( r 12) IT [S1( Pi' R) S2( p2' R) - 1] 4Ji r 1 2) ( 4)

with S(P. R) :: exp[2i6( p, R)]. Suppose S( P. R) ::: S[( p /R), R]. Then on writing down Pi = Pc ± (1/ 2 )P12

and taking into account that P12 « R one can expand Si[(Pi/R). RJ in powers of P12 JR. The first

term of this series will not contribute to fd. np because of the orthogonality of wave functions. and

for qR « 1 we shall obtain:

( 5)

-495 ­

where

and

2Generally speaking~ f

1 and £2 are of the order of R . Thus we have obtained in the square of (5)J

averaged over azimuth, an expression analogous to (3)~ where ~2 acts as the square of the inter­2

action radius. It is obvious that the vanishing of amplitudes at q = 0 in the zeroeth-order in l/R1

takes place for any inelastic scattering~ and from this viewpoint the above discussed properties of

the Pomeranchuk pole (which had in fact been invented for diffraction scattering) seem to be quite

natural. There is another evident feature of scattering on a large object that is closely connected

to the vanishing of inelastic properties. Let us suppose a system of n~ p to have several bound

states, d, d t -- and calculate the total cross sections with which these d~ d t , ••• , interact with the 2

target. All of these cross sections to zero order in 1/R are equal, and given by

(6)

It is obvious that the vanishing of amplitudes and equality of cross sections are closely related

as are orthogonality and closure properties of a system of wave functions. From the relativistic

point of view, though, it is essential. to view this relationship in terms of analytic properties as a

function of 8 --the relative energy squared of neutron and proton.12

To achieve this let us repre sent 8 8 - 1 as1 2

(7)

and

(_) ik12 ' r12 ~np = e + 4Jn'p'

The amplitude (4) may then be represented as a sum of six terms, corresponding to the graphs of

Fig. 12.

~~ P K2 ~KI

P K2

~KI P K2

(0) (b) (e)

n

~"p p r"p p ~" p P

(d) (e) (f)

Fig. 12

-496­

2As stated above, the main term in the amplitude (of the order of R ) becomes zero at

ql = 0 and this is true for all s1Z' The fnp, np amplitude, entering graphs d, e, f of Fig. 12 has

singularities in s1.2 corresponding to continuum and bOWld states, but contributions of all states

other than the deuteron are zero at q I = 0, since they contain inelastic amplitudes. The only con­

tributions left come from the deuteron state, hence at q = 0 graphs d, e, f of Fig. 12 coincide with 1­

those of Fig. 13 containing the deuteron pole with nonzero residue. In order to make the whole l

f amplitude zero at q = 0 (i. e., the main term in powers of l/& ) the graphs of Fig. 13 mustd,np I

cancel those of Fig. 1la,1)" c at q = O. 1.

~n P f P

(0) ( b) (d)

Fig. 13

Such a compensation of graphs of Fig. 12a, b and Fig. 13 is obviously possible, since at 2ql = 0 all of them have poles at St2 :; m . The Fig. 13c graph, corresponding to screening, hasd

nonpole singularities and at first sight cannot be compensated. Asymptotically (R - (0), though,

(S1 - 1)(5 - 1) is independent of P1Z' Hence its Fourier transform contains a O(k - k 'l)' leading2 1z 1

to a pole singularity of Fig. 13c graph. If the total contribution of graphs of Fig. 12a, b, c is

represented by Fig. 14, the vertex may be interpreted as the interaction amplitude of two particles

Fig. 14

with the target (ignoring their mutual interaction) and at q ::: 0 equals C' 6(k12 - ki I

2)· The l.

amplitude

~n fd;np = p~p

then equals zero if C = CJd t , i. e., as a result of compensation of initial and final state interactions.

III. Cross Section Relationships in Relativistic Theory

The situation in relativistic theory is analogous to the one just considered. If we suppose

the coupling constant of the Pomeranchuk. pole fa, bc (see Fig. 15) to be equal to zero at q.L = 0

Fig. 15

-497­

c

for all inelastic processes of the a - b + c type. a legitimate question is: how does this fact fit

with the analytic properties of fa, bc and nonzero total cross sections? Let us put s12 =(~ + k )2

equal to m 2. The amplitude then contains a pole term (Fig. 16a) which is not equal to zero since a

f determines the total cross section crt. At q = 0, though, there are two other pole terms which a, a 2 a l.

diverge at 8 = m (Fig. i6b. c). * Since the complete amplitude of f b must be equal to zero12 a a, c

(0) (b) (e)

Fig. 16

at q for all s1Z' contributions of graphs Fig. 16a, b, c must compensate. It is easy to make sure

theyl.actually do compensate if u t = Cft + Cf t. Nevertheless. it is evident that such a way of vanish­

a b c ing is impossible in relativistic theory. By applying this reasoning to the 1'1' - 31'1' process one

obtains crTI' = 3uTI" and hence un = O. This means that in relativistic theory there must also be pole

contributions of the graph Fig. 1Zc type, the singularity of which is determined by the growth of

the interaction range with the increase of energy. From here it follows that there must be a con­

tribution of C· o(k - 1(1Z) type in the f • bc amplitude (Fig. 17) which would not come from just12 bc

b~b c ~ c

Fig. 17

the sum of interactions with each separate particle. We shall discuss the possible existence of

such contributions later. remembering that the case of classical diffraction we can state the equality.

a a =::::::b ~b c + (8)e ~ c·~

The contribution from all nonpole terms equals zero since all the inelastic amplitudes are zero.

It follows from (8) that in order to have the right-hand side of (8) equal to zero at q = 0 we must

put CI t = C. If now a different particle at capable of transition into the same final-~ate particlesa

*At first sight singularities in graphs Fig. 16a, b, c are independent as is usual for singularities in s, t and u. This independence is caused by the fact that even for q2 = 0 there are three independent components of the vector q which enter s, t and u differently. In our case. though. the Reggeon interaction amplitude has been obtained by calculating the asymrtotics of a certain process in which it was asswned from the beginning that q2 = -q-tZ i. e., ql =qz. This means that from the begin­ning Reggeon amplitudes contain only 3 indepenaent components of q. On putting ql = 0 there is only one independent component in the Reggeon amplitude left. By putting now s1Z = ml we fix even this last component to be equal to zero (it becomes obvious in the system where particle a is at rest). Then. evidently. (Pa - kb)2 = ml. (Pa - kc)Z = mbZ.

h, c on a vacuum pole with q = 0 is considered instead of particle a we get C1 t. =C. Thus total1 . a-­cross sections for all particles transformable to the same final state particles by a vacuum pole

with 9 = 0 are necessarily equal. To come to this conclusion we have assumed C to be independent1

of 8 12" i. e., the same for different masses rna and rna'. We think such assumption is a very

natural one from both formal and physical points of view. Formally any dependence of C and sil

must be connected to certain intermediate states, the amplitudes of transition into which are equal

to zero when q1 = O. Hence particles b and c can get no information on the spectrum corresponding

to states a and at etc. Physically the situation is even clearer. The amplitude of b and c inter­

action with the target having the form C6(k - k ' ) in momentum space corresponds to an amplitudei2 f l independent of relative separation Pi2 in impact parameter space. This means that in calculating C

one can consider the interaction with the target of particles spaced by p f2 much larger than all

characteristic Compton wave lengths, but much smaller than the interaction radius at In s = a'~.

In such circumstances particles b and c can exchange nothing and their scattering by the target may

be treated as that of independent particles. This dictates the amplitude of their scattering to have

the structure of the type represented by graphs of Fig. f8a, b j c

fbc,bC = fbo(k - k I) + fc6(~ -~) + fbfc'V (9)

c C

here V is independent of the properties of the particles b anc c. The amplitude (9) being independent

Kc K~ Ke K~ Kc

~b ~ ~b (a) ( b) (e)

Fig. 18

of P12 is also independent of relative energy si2' Moreover" due to factorization, it follows from

Fig. f8 that C equals

(iO)

here gb' gc are constants of coupling to the Pomeranchuk pole and f3 is a universal quantity, de­

termined by Reggeon interactions. The condition of vanishing of the am.plitude for a - b + C then

means

(:11)

If an a - a + a type reaction is possible on a vacuum pole, it follows from (i1) that

f = l + ga~. (i2)

Then ga is independent of the type of particle and, consequently, the total cross section for any "a"­

particle interaction, determined by ga' must be a universal value, independent of particle a. If

the a - a + a reaction is iInpossible, following O. V. Kanchelits suggestion, one can consider the

reaction a - a + a + a which is always possible. Then. because ga = ga' a similar reasoning will

lead to (Fig. f9)

(i2a)

-499 ­

T Fig. 19

From which it follows that ga is universal, i. e. I in order to make inelastic amplitudes at ql =0

asymptotically zero it is essential that the total cross sections of all hadron interactions must go

to the same limit.

IV. Differences from Diffraction on a Large Object

At first sight it seems that the conclusion drawn above about asymptotic equality of total

cross sections is in accord with the equality of the diffraction cross sections of a large object. It

is important, however J to bring out an essential difference. If in (6) 6{ Z[( P /R) I R] is of the order Z 2 ' c

of unity [8 = exp(2io)) J then O't "" R and diverges as R - CD. which corresponds to low transparency

of the target.

In relativistic theory we assume cross sections not to be growing with energy (interaction

range), which corresponds to increasing transparency. In classical theory it is only possible if

S - 1 • 2i6 '" (l/Rz

)tP(p/R). But in such a case 8 - 1 and 8 - 1 are expandable in powers of 6 oz.1 ,1 2

O'a = 4 1m JdZPc[61 + 6z] (13)

and the cross section of scattering on a composite object thus equals the sum. of cross sections of

scatterings on its components. This is an evident dilemma of classical scattering: either cross

sections diverge 8S R - CD or the screening (Fig. 12c) is unimportant, and as we have seen above

it cannot be correct in a relativistic theory. A possible source of such an essential difference may

be in the fact that a relativistic particle approaching the target may be in different virtual states.

The resulting scattering is then obtained as a coherent average over such states. Equation (6)

perhaps should be written with average values of 8 - 1:1

2 - ­a = -ZRe f d p [8 - f + S - 1 + (S - 1)( S - 1)] (14)c :i 2 1 Z

and assume Si - 1 not to be small. but 8 - 1 and (S1 - :i)(Sz - 1) to be of the same order of magnitude.i

V. The Possibility of Asymptotic Equality of Total Cross Sections

As has been demonstrated above, the inelastic amplitude's vanishing which requires asymp­

totic equality of total cross sections is possible only in case the graph of Fig. 1Sc, corresponding Ito screening. contains a term of the C6(k - k ) type. Let us now consider the feasibility of12 1Z

such a term I s emergence and consequent requirements on the Reggeon interaction vertex V. We

shall start with the calculation of Fig. 1Sc on the assumption that V = constant in momentum space.

The amplitude corresponding to this graph is analogous to the one describing inclusive cross sec­

tions in the three-Reggeon limit and may be represented as:

(:i5)

-500­

Zwhere M =lpq = lpq1. 2

Herep is the target's momentum. At q.L = 0, for example, in the laboratory system (po =m) the

amplitude f is representable as: s

3 f la' q/' lnf3t f ~gV-e 1- (16) s ~1.

where fl = s/Zmq is the fraction of longitudinal momentum transfered by particle fli fl. Thei 1z

amplitude f is largest for small ~t and qt • Hence in all integrals over 131 and q1 it will effec­s _ _ 1. 1 tivelyact as O(~i)O(qil.) '" 6(kiZ - k t2.) substituting values of kt'z equal to k iZ . The problem boils

down to the magnitude of the contribution. Using (16) we find

lf ~idZqlfs '" In In 8/m ,

i. e., the 6 -function type contribution turns out to be logarithmically growing with the energy. This

additional In In s is well known and is one of the reasons for assuming the three Reggeon interac­

tion vertex to be equal to zero at q = 0, V:: yq/I. But if one uses this latter form of V in (16),1. _ _

the resultant f is not large at small qll , ~1 and will not act as O(kil - ). A possible way out s k 1.•l of this contradiction may lie in the following observation: up to now the three Reggeon interaction

vertex has entered only in studies of either branch cut contributions or of inclusive reactions. In

our case we calculate the interaction vertex of two particles with a pomeron, i. e., a quantity

analogous to the coupling constant, which cannot be expressed in terms of renormalized Reggeon

interaction vertices. Hence the quantity being calculated may include this vertex taken in a dif­

ferent kinematic regime which makes its value different from the one entering the branch cut con­

tribution and inclusive reaction calculations. Such a possibility looks more realistic if f is s

considered in impact parameter space rather than momentum space. We are interested in 2 Z

fs(P i , PZ' ~1) for large PtZ =(Pi - pz) and large k ' ~z (k1z and klz are of the samek iz ' k Zz' tz order of magnitude); at PiZ of the order of t there is no reason to treat the interaction as that of

Reggeons. It should be noted that the three-Reggeon interaction amplitude under such conditions

has not been studied before. In inclusive reactions P = o. Though amplitudes with large PtZ can12 be encountered in Reggeon graphs, the In k iz Iltzz is always considered to be large. Let us trace

this difference for the example of Fig. 1Bc, writing it out in p-representation for the simplest case

of V =constant.

(17)

~Z = In ltzz' z

_..f. ­_ t 4a'e,

G(p, g) - 411'at~ e • (tS)

The normalization integral for f ' i. e., the magnitude of the contribution is s

f f s~ t ~ V f a(Pi - pi, ~ - ~ I ) a{ Pz - pi, ~ - ~ I ) G( p', ~ I ) d ZP t d~ I , (19)

The meaning of Eq. (1.9) is easily understood within the framework of the pictures of diffusive slowing

-sot ­

of virtual particles inside a high-energy hadron. S. 6 This picture is an immediate interpretation

of processes corresponding to Feynman ladder graphs (Fig. 20) and boils down to the following:

Fig. 20

a fast particle of energy E ("rapidity" ; = In E) and large impact parameter cannot directly interact

with the target at rest in the center of coordinates. It can, however.. lower its energy by virtual­

particle emission and approach the target. In processes like Fig. 20 such a lowering of energy

takes place in successive steps, each one changing the rapidity by a quantity of the order of unity

and the impact parameter by that of a characteristic Compton wave length 1. 1m. As a result, after

n steps. n 1\0 ;, the particle slows down and finds itself separated from its initial position by a

distance of the order of ~ due to the random character of motion in the impact parameter plane.

The probability of such a process is determined by the diffusion Green function (18) with rapidity

~ acting as time. The situation is similar to the slowing of neutrons in reactor.

Let us now discuss the interaction with the target of two particles of high energy of the same

order of magnitude (; 1 ::: ;2 ::: ;) spaced by a large p12. Each one of them will be slowing down

and diffusing in the impact parameter plane. If at a certain stage of this process they find them­

selves with rapidity e in the vicinity of point p' spaced by about one Compton wave length. theyl

will interact and continue the slowing down as a single particle (Fig. 21). The probability of such

0,0

Fig. 21

a process is given by the expression under the integral sign in Eq. (19), composed of a product of

two Green functions. which determine the probability of an encounter, and a third function, de­

termining subsequent slowing down into the center of coordinates. If the encounter probability is

small, events taking place at different;' are independent and the integral in Eq. (19) determines

the total interaction probability. The probability of an encounter at ~' somewhere between ~ and

;0 is given by

-502­

(20)

Given 0"(; - ;0) « P1~' the probability w{g, SOJ Pi2) is small, since approaching requires a

"time" of the order of p/z. If ()II(; - ;0) rv P1~' w(~, ~O' Pill is of the o:der of unity. For

al(~ - so) » Pi~ the probability grows proportionally to lnl(; - so)/P1~J. This means that addi­

tion of probabilities for such S - So is meaningless since the encounter is sure to have taken place

"at an earlier time. It Interactions "at an earlier time" are formally accounted for through a non­

constant Reggeon interaction vertex.

For ar(S - SO) « P1~' however, the "preceding interactions" are obviously of no importance Z

and the vertex may be kept a constant. At ()I1(g - SO) » P the vertex should cut off the integral,12 2

L e., become small. If the integral in Eq. (19) is cut off at a'e g - gl ) rv P12 by a form factor of

the type f[P1~ 10"(; - Sl)]. a straightforward calculation shows Eq. (19) to be equal to

(C/ 41TaI S) exp(-pc2/4als) where C is independent of Pi2' just what we need for the vanishing of

inelastic amplitude.

Let us now examine from this point of view the vertex entering the branch point contribution

and inclusive reactions. The contribution of the simplest branch point containing a Reggeon inter­. . 2 2

actIon vertex corresponds to the graph of Fig. 22, where P12 "'" 11m and a'( g - g') must be much

larger than unity. From the diffusion slowing point of view, Fig. 22 gives the probability that two

0,0

Fig. 22

slowing objects that are close together nevertheless do not interact during a long "time 11 S - £I.

That probability being small may be described by the smallness of the Reggeon interaction vertex.

As an example of possible behavior of the graphs of Fig. fBc and Fig. 22 as a function of

Pi2' a simple probability calculation may be suggested. Suppose two slowing objects with "rapiditylr

g have been created in points P1 and Pz' Let us calculate the probability w(s, SI, P12' v) to find

these two "for the first time" within a two-dimensional valume v located anywhere with "rapidity"

g'. Let p(g, g', Pi2' v) be the probability that these objects are within volume v with "rapidity"

~ t, their history being ignored:

v) =Jd2p~ Jd2ph G(P1 - P~ - P~2, ; -;' )G(P2 - P~

P12 +-2-'

v 2 2

(P12 - P1Z) 2 -~ d Pf2 e 8a's"81Ta l

; " (21 )= f e 81ral~ II = gnals 11 v

v

-503 ­

where

It is easy to show then that

where 'V characterizes the rate of the diffusion process. Substituting (2f) into the exponent in (22)

one gets:

l S S V- I P(S", P ' v) dS". =-8,1CO e -x dx Zi zo 1rit P f 2/8a'(; _ g I ) x

Thus the factor in (22) making w and p different actually features the cutting-off property in the

above-stated sense. If 'Vv/81fat = f, then for small p2 2!8a'(; - Sf ) , we havet

Z P'P t 2

w""'----­8a'(; - ; I)

This is equivalent to a linear vanishing of the interaction vertex in Reggeon graphs and inclusive

processes.

In ending this section one should stress that all this reasoning is highly qualitative and may

only indicate the possibility of a noncontradictory description, but cannot serve as a proof. In

particular this description, being of a diffusion character, looks irreversible in time and is hardly

applicable in a coordinate system where particles b and c have small. energies.

VI. Asymptotic Behavior of Total Cross Sections

Let us calculate how total cross sections approach their asymptotic values in the first

approximation in t /In s. In this approximation total cross section asymptotics are determined by

contributions from the pole and two-Reggeon branch cuts (graphs of Fig. 23). The two-Reggeon

Fig. 23

cut contribution to the total cross section has the form

(23)

2 k -0 1.

where N a

, b(w, k1.) may be presented as

tfCO ds' 2.Na,b(w, k ) = - --- A (8' k ) (24)1. 11' 2 (s,)2+w a. b • 1.

m

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with A(s'J k 2) denoting the imaginary part of the amplitude of forward scattering of a Reggeon with

transverse ~omentum k on a particle. A(s', k Z) satisfies the usual unitarity condition of Fig. 2.4. 1. 1.

1m )=( = >--< >:< +••. +

Fig. 24

Since all inelastic amplitudes are zero at k1 = 0, the only contribution left from the region of finite

8' in (2.4) comes for small k from elastic scattering. As part of this contribution it is important1

to include the region of large Sf _ since for (,l) - 0 the integral in (24) diverges. In this latter region

A( s·. k1.) may be represented as (Fig. 25)

A = gyk 2 S' . (25)1

x Fig. ZS

Substituting (25) and the first term of (15) into (24) one gets for small CtJ and k.L:

gyk Z Z 1

N , b = g + -;c:;- (26)a

i. e.. Na. ~(",. ki) at '" - O. k; - 0 does not depend on the type of the particle and in the limit

(a) = -la'k 1.

Thus not only the total cross sections must be asymptotically equal, but also the first terms of

their series in powers of 1 lIn s:

(Z8)

This last conclusion may turn out to be meaningless, however, if g - y I lYra I =0 and at has no

terms of the order of t /~. It is interesting that such a vanishing of g - y/2.1Ta l looks highly natural

within the framework of the diffusion slowing picture. If the two-Reggeon cut contribution actually

determines the probability that two jets leaving a small volume do not interact at all afterwards,

this contribution must contain an extra smallness and decrease faster than t /~, which is what

occurs if g - y /Zwa' = O. If this is actually so, the asymptotic behavior will not be universal and

a = gZ[t - ~ a~b 1 (29) t ~3

Finally one should discuss a realistic value of the expansion parameter in the power series

in 1/In s. From the physical part of the above reasoning it is clear that cross sections will (if� 2�

ever) become equal only after the growing radius of the interaction (al In s = R ) becomes much� 2 2�

larger than characteristic sizes roof particles at small energies. If a' =: f 12m and r 0 '" t/4ID-rr2,� 2 2 2 2�

the condition R /r0 » 1 is essentially unattainable. If r 0 '" i 1m the situation is more hopeful.� 2 2�

However, the parameter R /r0 is not really large even at the CERN ISR, but some trend in the

behavior of total cross sections toward equality might be seen.

Yshould like to end this paper with expressions of deep gratitude to O. V. Kancheli,

L. N. Lipatov, E. M. Levin, K. A. Ter-Martirosyan, and A. P. Kaidalov for numerous en­

lightening discussions.

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