- Email: [email protected]
Asymptotic behaviour of essentially inelastic collisions
Nuclear Physics 68 (1964) 591--608; (~) North-Holland Publishing Co., A m s t e r d a m Not to be reproduced by photoprint or microfilm without written permission from the publisher
ASYMPTOTIC
BEHAVIOUR
OF
ESSENTIAIJ,Y
INELASTIC
COLLISIONS
K. A. T E R - M A R T I R O S Y A N Theoretical and Experimental Physics Institute,
Moscow, USSR
Received 15 February 1964
Abstract: The asymptotic value of particle production amplitudes at s ~ oo is investigated for "essentially inelastic" collisions when by the production in the c.m. system each moving nearly parallel to the The momenta of all particles in each
it is especially large. These collisions are characterized of two groups of particles at high primary energies, momenta p~ and Pb = --P~ of the colliding particles. group differ considerably by their order of magnitude: Pl >>Ps >> • • • >>Pnl, Pn >>Pn-1 >> • • • >>Phi+l, where the subscripts 1, 2 . . . nx refer to the particles of the first group and nx+ 1, n x + 2 . . , n--1, n to those of the second. The mathematically corresponding limit of a particle production amplitude is determined by the requirement that the squares sik of the energies of any pair of particles in their c.m. system should increase (or remain large as compared with particle masses squared) as the primary energy increases, s ~ oo. It is also necessary that the transverse components/¢~ of all momenta Pi of the particles produced and momentum transfers squared tt = (Pa--Px-- • • • --pi) ~, i = 1, 2 . . . . . n should be small. It is shown that in this limit the asymptotic value of the amplitude takes the form of a product of Regge factors I I n - ~ ( s i , i+ 1)Y°(t0Fi (where jo(t i) is a vacuum pole) and can in a sense be compared with the contribution of Feynman-type graphs with n-- I "reggeons". The quantities ~ = 7~(K'~, K'~+I) depending on the projections ;¢'~ and K'~+I of the transverse momenta of reggeons before and after the emission of a particle correspond to the vertex parts designating in these graphs the emission of the particle by a reggeon. In a preliminary report on this question 2,s) the dependence of the quantities ~ on the angle between ;¢'~ and ;¢'~+~was overlooked. Thus the coefficient of the product of the Regge factors in the asymptotic value of the amplitude depends on the transverse components of the momenta of all the particles produced. As a result the asymptotic value of a particle production amplitude proves to be dependent on the same number of variables as in the general case (i.e. is sensitive to any change of the momenta). The asymptotic behaviour of mixed "essentially inelastic" and "quasi-elastic" collisions is also considered, when not particles but n groups of particles are produced and the energy of each particle in the c.m.s, system of a group is low. Collisions of this type are mainly responsible for multiple production at high energies. In conclusion brief consideration is given to the "complicated" types of asymptotic behaviour of "essentially inelastic" processes when the extreme right singularity in t h e j plane is a branching point or a point of accumulation of singularities etc. It is shown that in this case instead of the factor (s~, t+l)JO(tl) o n e ought to expect in the asymptotic values of essentially inelastic amplitudes expressions like ~v ;tv(t;~, ~ + l ) ( S t , t+1) iV(it)' where ~e, t+1 = In(s~, t+l/m 2) and ;tv(t,~, ~+1) is a function depending on the character of the vth singularity and perfectly analogous to those which determine the corresponding asymptotic values of elastic scattering amplitudes.
1. Introduction I n a p r e v i o u s p a p e r 1) t h e a s y m p t o t i c v a l u e o f t h e a m p l i t u d e f o r t h e r e a c t i o n a+b
~ c+d+e
(1)
w a s o b t a i n e d i n t h e c a s e w h e n t h e e n e r g y Sed o f t h e p a i r o f p a r t i c l e s c a n d d r e m a i n s s m a l l a t Sab -* o0. 591
592
1~.. A. TER-MARTIROSYAN
Considered below is the case when all the three energies Sod, Sde and see are high and the transverse components of the momenta of the particles c, d and e are small at sab ~ o o . We 2.3). Several other authors 4, 6) have obtained for such essentially inelastic collision amplitudes asymptotic values of the form of contributions of simple Feynman graphs (see figs. 4, 6 and 7). However, the dependence o f the coefficient in front of the product of the powers s w) of the energies on some transverse momentum components was left out of consideration. As shown below, the asymptotic value of an amplitude in its correct form depends on the same number of variables as the amplitude itself and is sensitive to any change of the particle momenta. Brief consideration is given to the asymptotic values in the case when the extreme right singularity in t h e j plane is not an isolated pole, but a singularity like a branching point, a point of condensation of poles or accumulation of branching points. S=Sab
t=Sae
a
e~sl
"d
Fig. 1. Amplitude of the reaction e + d ~ a-t-b-l-c.
To use all the notations introduced earlier i) it is convenient to consider not reaction (1) but its analogue (fig. 1) e+d ~ a+b+c,
(2)
which differs from reaction (1) only in the permutation of the particles a and e, and b and d. For the transition from the asymptotic value of the amplitude for reaction (1) to that for reaction (2) (or vice versa), it is sufficient to transpose the subscripts of these particles. b Pa~ .........
.
1 _ ~~ a
. Pd
Fig. 2. Configuration of the momenta of the particles involved in the e-I-d ~ a-I-b-/-c reaction in the Sae channel, Pe and Pa are the momenta of primary particles, Pa, Pb, Pc are the momenta of particles produced, and ~:a, ~b and ~:c are the transverse projections on a plane perpendicular to Pe and --Pc-
2. Limiting Values of Invariants and cosines of s Channel Angles
After this permutation in fig. (2) and eqs. (37) and (38) of rcf. 1), we can determine the configuration of the momenta (see fig. 2 of this paper) and the limits of the in-
ESSENTIALLY INELASTIC COLLISIONS
593
variants in the physical region of the essentially inelastic process (2) at Sac latter is in the form s++ +
kc /
2
-
kb ~mb
+++)
S.b ~ 4 k a k b .~ m 2 Sac ,~ Sde ~
= m +
°de
o0.
The
++,
m2
Sdc
~
~
o0,
(3)
00~
t -- Sa+ = -Xa2 = const, sl
=
Sod
=
kd
Sbd ~-
--X~2 = const,
(4)
--Sb° -+ --
kb
In fig. 2 and in eqs. (3) and (4) it is assumed that k b << ko. For the following this condition is sufficient. However, similar to the presentation ofeq. (37) of the previous paper 1), we assume in eq. (3) also that the ratio y' = k o / k b increases when Sd. ~ O0 as (Sa+/m2) ~ where 0 < g < ½ and m is a quantity of the order of particle masses. When k b << k o we obtainpc ~ Pd ~ ka "~ ko = ½x/Sdc. The condition g < ½ means that we do not consider the case when k b decreases with the growth of primary energy t (it is assumed that when Sdc ~ oo all momentap~ in fig. 2 are ultrarelativistic). Besides the configuration indicated in fig. 2 there is another perfect analogue o f it for which all energies Sbc, Sab and sac in the limit Sd+ ~ O0 are high and the transverse components x~, x b and go (and momentum transfers) are small. It only differs from fig. 2 in that the momentum Pb o f the particle b is nearly parallel to p~ and not to Pc. In both cases the energies satisfy the relation Sab Sbc = Sde(mb2 q- t¢~).
(5)
For the investigation of the asymptotic value (3) of the amplitude for'reaction (2) it is convenient to use its expansion in partial waves in the t channel (see eqs. (3), (4) and (7) of ref. 1)). What are the values o f the cosines z, z; and cos ~b contained by these expansions in the region (3) unphysical for the t channel? They can be expressed in terms of the invariants Sab ~--- (qa--qb) 2,
Sb¢ = (qb+q¢) 2,
See = (qc--qo) 2,
which are determined in the t channel according to fig. 2 of ref. 1). As a result, we * The reg ion o f s m a l l k a ~ m i n t r o d u c e s at sac -+ oo a v a n i s h i n g c o n t r i b u t i o n ¢) to t he t o t a l cross section o f r e a c t i o n (2). A l l the results o b t a i n e d b e l o w are a c t ua l l y v a l i d for this re gi on as well.
594
K . A . TER-MARTIROSYAN
obtain 5ab ~
m az -- m 2 = 2qaqbZ_20~aO~b
bo--m2
2 = --mo
2 a f (~b Oj~+ q2) +
m~ q-Sab--Sl--Sde
_
2q__~
V2
2qaq, z,,
, ,
x/St (t'o:q~z~'Cq:oJ'c)Z
(6)
2m+
-- ~/Sl ((O:OJ"q-qaq~z')+ Eq:q'~/(1--Z2)(1--Z ;2) COS ~, q~ -- qb,
as q~ = --qb
(q~ is written everywhere instead of [qil for brevity). In the right-hand side o f eq. (6) instead o f s , c - m,2 - m o2 the identical quantity m~+s,b--Sl--Sa, is written (the three invariants Sod = Sl, Soc and sa, in fig. 1 are connected by the identity st +s~, +sd, = m~2 + m~ + m~ + S,b). The energies and toomenta o f particles in the t channel are designated by qi = (oh, qs); they depend on t = sa, and Sl -- S~d. The energy and m o m e n t u m o f the particle c in the c.m. system o f the t channel o~c, q, = [qJ, zo are expressed by their values 0~, q" and z" in the c.m.system of the particles c and d with the aid o f the Lorentz transformation* 1
t
t
t
q,z, = ~/st (m, qcz,+q,m,),
q~x/1--z~2
t
=
q~x/1--zo
t2 •
Here ~ : = (t+Sx--m2)/2x/t and q: = ~/o~2~ - s x are the overall energy and m o m e n t u m o f the particles c and d in the c.m.system of the t channel. The quantities o~ and q" obviously depend only on s 1 and the masses. Neglecting in eq. (6) the quantities which do not increase in the limit of eqs. (3), (4) we find that in this limit z~,
Sab ~ O 0 , 2qaqb
Z'c ~
Sbc
* oo,
~b-~b
o
2qb q"
(qb = q~, as in c.m.system o f the t channel qb = --q~), where
o= o:O+, + V z,,,: *
in the direction q~ = qe+qd with the velocity
p = q~/~/~+sl = ~/VL--(sdo~?).
S.b1 /
(7)
K. A. TER-MARTIROSYAN
595
According to eq. (3) all terms in square brackets can be neglected except sd.. Therefore taking into account eqs. (4) and (5) and substituting the values o9, and qa, we obtain cos 40 ~ V/st +
V--T2q2 St
Sd___L Sab
Sbo
2
_ K b 2( m b 2 ..~/£a2 .1_ X°2 ) .jl_(Ka2 .+. Kc2 ) "{-
2+
(s)
2x a i%(m~ + 1c~)" This quantity may, in general, be larger than unity in absolute value since the region (3), (4) is unphysical for the t channel in which the angle 4 is determined. If, however, the values of s I and t are small as compared with m 2, i.e. x 2a < m 2 9 x~2 << m 2, and x~ (tq + ~c) 2 << m 2 (this region is the most essential in computing 7) the total cross section o f reaction (9)), we obtain from eq. (8) 2
2
2
cos
,
or
rC-4o - ~oao, where ~O.ois the physical angle between the s channel transversal components o f the momenta r , and xc (see fig. 2). Thus in the region of small x a and re cos (rC-4o) ~ cos t#,c = < 1.
3. Asymptotic Value of the AmpBtude of Fig. 1 when z ~ ~
and z~--~
The asymptotic value o f the amplitude of fig. 1 in the limit (7) can be determined from eqs. (27)-(29) o f ref. 1) (valid in the region z ~ oo) by putting there 4 = 4o and going over to the limit z" --. oo. To this end, expansion (29) of ref. t) should be written in the form of the Sommerfeld-Watson integral over l' (9)
i)2, = ,v )(,+ ) a7_- , ~( -2)'
1 fc
F ( I ' - 2 ' + 1) ×
!
Zo)___
t
21' + 1 sin
hi'
dl'.
(10)
Here C is the contour in the I' plane such as in fig. 9 of ref. 1) and v~?(st, t) is an analytical continuation into the l' plane of the quantities vv, x,(sl, t) defined for even
596
K.A.
TER-MARTIROSYAN
and odd integral l'. Does there exist such an analytical continuation, and if it exists what are its properties? According to eqs. (26) and (27) of ref. 1) the quantities vra, differ from the amplitudes =
(11)
Ax,(s,, t, z, z;)e,,x,(z'o)dz'+ -1
only by the /'-independent factor #(t)Io(t)(Sab/mz)l°CO(m/2qb)J°(Ocx,(jo(t))i x'. Therefore to answer the question it is sufficient to construct an analytical continuation of the amplitudes avx, into the l' plane. This can be done exactly in the same way as used earlier for obtaining the analytical functions of the variable j determined by eqs. (16) and (18) of ref. 1). The discussion is based on the assumption that the quantity Ax, (sl, t 1 z, z~)/(z'o 2 - 1)ix' can be represented in the z~ plane by the dispersion relation with respect to z" with complex integration contours ?', and ?'o2 similar to those indicated in figs. 5 and 6 of ref. 1). Just as in the case 1) of the amplitude avx,(sl, t, z)/(z 2 - 1 ) +x' we do not consider in this dispersion relation the part AAx,(s 1 , t, z, z')/(z'c 2 - 1)+x' of the integrals over complex contours as inessential t for the asymptotic value A x, at z~ ~ oo. In other words we write the Ax, part of the function Ax, essential for the asymptotic value when z'c --+ oo, in the form
A'~(sl , t, z, z') (z~2-1)+ x'
1 [~ A~÷)(sl, t, z, z'~')dz'~' = .
.
.
.
1 (-zo~°A~2,)(sl, t, z, z')dz'o'
(z,J-1)+X'(z'o'-Z'o)
~ _ _ +
o and z°z are the points where the integration contours in the precise relation where Zct for A x, = A'x,+ AAx, go from the real axis into the complex plane, and All,) and A(~,2) are the absorption parts of the amplitude Ax,. Substituting the right-hand side of this relation into eq. (11) instead of Ax,/(z'¢ 2 - 1 ) ½x' we obtain (just as the functions in eqs. (16) and (18) were obtained in ref. 1)) two analytical functions a~,~) of the variable l' a(+) I'X" ~ bvx,___CI'X' b,w(st, t, z)
1 f zo~ zc o
t --i~X'
t
t
cvx'(sl, t, z) = l f z ~° [--A(~,2)(sl, t, z, Zo)]e Qrz,(zc)dzo, , -i~x' , , t It is more difficult to justify this neglect here than in the case x) a;,x,(sx, t, z) since the perimeters o f the complex contours in the dispersion relation for Ax,/(zc "2- l)tX' with respect to z'o may prove not to be finite, but increasing with z--+ co. Actually this neglect seems to be justified since in all the cases investigated e) the absorption parts in such anomalous complex contours always decrease rapidly when moving along them, towards the singularities.
597
ESSENTIALLY INELASTIC COLLISIONS
which coincide with amplitudes (11) for integral even and odd l' (plus or minus sign) respectively. Assuming that A~, when z~ ~ oo does not increase more rapidly titan z "¢ we conclude directly 1) from these formulae that (i) the functions ,,vx"t5)are analytical functions of l' in the right-hand half-plane Re l' > v', (ii) they decrease exponentially when l' ~ oo along any ray in this half-plane, and (iii) they vanish at all integer physical values of l' = n smaller than 1' (and than v'). ,,t5) only by an l' independent Evidently, the functions ~rx"ts)which differ from ,-,'x, factor (see eqs. (26) and (27) of ref. 1)) will also possess all these properties. The asymptotic value at z~ ~ ~ of the integral (10) is determined by the extreme right singularity in the l' plane of the functions •L,I,(5) 2 , . From the unitarity condition in the l' channel it follows (similarly to eq. (24) of ref. a)) that
t) =
t)
1-2ip(sl)2
,
)(sl-)
'
where Sl • = Sx _ iz, p(sl) = 2k'(s2)/~/sl and 2},~)(Sl) is an analytical continuation into the l' plane of the elastic scattering amplitude for the particles c and d with momentum k'(sl) in their c.m. system. Hence"~ lt5) , ) . , has a pole at the point l' = floS)(sl) at which
1 - 2ip(s1)2[,5 )(sl) = O. Near the pole vvx, can be written as
t) =
)(Sl' t) r -j oS )(s, ) '
(12)
where u(+)(Sl) is the same function as in eqs. (25) of ref. 1), and w]~)(Sl, t) is a new unknown function (it is real in the region Sl and t below the elastic scattering threshold; the residue in the right-hand side of eq. (12) can be factorized in the conventional manner). Assuming that the extreme right singularity of V,(+)to l ' M k O l , t) in the l' plane is a vacuum polejo = jot+)(Sl), we obtain from eqs. (10) and (12) the following asymptotic value of v~, when z" ~ oo (i.e. when She ~ o0):
vx,(s,, t, sbc) = i L \ ~ q b r t l
i~'c~,(jo(Sl))w~+)(Sl, t
O(sl)Io(sl) ~-~!
'
where g(sl) is the vertex part introduced previously 1). Substituting this value vx, into eq. (28) of ref. 1) and putting in them q~ = tko, we obtain the following asymptotic value of the amplitude of fig. 1 in the asymptotic region determined by eqs. 0 ) , (4) and (7): A (3 *-- 2) ~, Y(sl)iy(sl, t, cos Cko)a(t)Io(sl)Io(t),
(13)
598
K.A.
TER-MARTIROSYAN
where ? is the value o f the function oo
~(s,, t, cos ~b) = Z ' (2 cos 2'(n-~b))?a,(s,, t),
(14)
~'=0
7~,(sl t) = ( m2sl ~rJ°(s')( m2t ~½jo(O ' \ k ( s ~ , t, m 2 ) ] \ r ( s 2 , t, m2)] cx,(jo(s~))cx,(jo(t))w(a,+)(sl, t), (15) K ( m 2 , s l , t) = 4tq 2 = (m2) 2 - 2 m 2 ( s l + t) + (sl - t) z
for the unphysical value (8) o f the angle q~ = ~bo. The series (15) converges only when Icos 4[ < 1. Therefore it is assumed that the dependence o f T on cos ~b is obtained f r o m eq. (14) in the region Icos~l < 1, and then analytical continuation is obtained into the region Icos~l > 1. The m e t h o d o f such an analytical continuation and explicit calculation o f 7 when q~ = ~bo is not indicated here (V cannot be calculated a n y w a y since eq. (15) contains an u n k n o w n function wta,+)). It is essential that cos~b o depends, according to eq. (8), on all transverse c o m p o n e n t s r , , Xb and to, o f the m o m e n t a o f the particles produced. Therefore the value 7 when ~b = ~bo also depends on these c o m p o n e n t s V = V(x., Xb, /¢¢) o r ~ ( x c , r a ) , and the asymptotic value (14) can be represented as / o \ jo(x¢) / Sab~Jo(x.) A(3 ~ 2) = g(x~)iy(xo, r,)g(x,)I(x~) I ~--~| I(x,) , \m-I
(16)
where it is taken into account that sl ~ - g 2 and t ,~ - g 2 . This asymptotic value depends, as might well be expected, on five variables referring to any change o f the m o m e n t a in fig. 2. It can be regarded in a sense as the contribution o f the graph o f
d
xe
Fig. 3. Graph with reggeons determining the asymptotic value of the amplitude for the reaction e-t-d ~ a-t-bq-c when Sea ~ oo, S~b ~ oo and sbo ~ oo. fig. 3 with two reggeons. T h e internal vertex in this graph corresponds to the contribution i7 dependent on the transverse particle m o m e n t u m c o m p o n e n t s o f all the three lines reaching it t * In other words, the vertex 7 in fig. 3 depends not only on the squares of the momenta of the par= ticles involved, sl 1¢c: and t = --/eL2, as in the case of the usual Feynman graphs, but also on the quantity sab sbc/sae (as 4o depends on it). In the physical region of the essentially inelastic proeess (2), this quantity is mbZq-Kb~, according to eq. (5). =
- -
ESSENTIALLYINELASTICCOLLISIONS
599
Transposing in eq. (16) the subscripts of the particles a and e, and b and d, we obtain an analogous asymptotic value of reaction (1) which obviously corresponds to the graph of fig. 4.
tl b=\ g e,
S
i t~.
g/a
|d
,e
Fig. 4. Graph with reggeons determining the asymptotic value of the amplitude for the reaction
a+b ~ eWd+e when sab ~ o0, sea ~ co and
sac "," oo.
4. Asymptotic ValUe of More Complicated Essentially Inelastic Processes The asymptotic value of more complicated essentially inelastic processes in which four, five etc. particles are produced can be obtained by the same method (by using the expansion in rotation group coefficients d~,~x),,(z) instead 3) of the polynomials P~, i,(z) in the t channel).
Pi
__
a)
Pb
"~_'..~___
_v
Pa
p
Fig. 5. Configurations o f the m o m e n t a o f the particles involved in the a + b ---> 1 + 2 + 3 + 4 reaction in the t channel; Ps and Pb are the m o m e n t a o f primaries, p t , p s , p s a n d p , are the m o m e n t a o f particles produced, and ~q, t¢~, 1¢~ and ;Ca are the transverse projections o n a plane perpendicular to Pa and Pb.
Fig. 5 represents two of three possible configurations of the momenta of the particles produced in the essentially inelastic process a+b~
1+2+3+4
600
K . A . TER-MARTIROSYAN
(fig. 6). In these configurations all energies Sag, s12 and s23 when S.b ---, oO are high and the three-momentum transfers Sa I =
__/£2,
Sal2 =
__ (/£2 + / ~ 2 ) 2 =
- - (/(~¢ + / ¢ 4 ) 2,
$4.b =
--/~?
are small. There is a third similar configuration which differs f r o m that of fig. 5b) only in that the m o m e n t u m px has one direction (almost parallel with p . ) and the three m o m e n t a P2, P3 and P4 are almost parallel to Pb in the opposite direction (while P4 >> Pa >> P2)" In all the three cases when Sab ~ o0, the relation $3,
+
analogous to eq. (5), holds in the physical region. The asymptotic value of the amplitude in these configurations is determined by
2-----1 i.¥
Fig. 6. Graph with reggeons determining the asymptotic value of the amplitude for the a + b ~ 1 + 2 + 3 + 4 reaction when ssb ~ oo, s~2 ~ oo, s2a -+ oo and s84 ~ oo.
the contribution of the graph of fig. 6, depends on eight variables and has a form analogous to eq. (16): A(4 *- 2) ~ g(xl)iT(xl, ~'2)i7(r'2, x'3)g(x,~)
/s
r
p
\ Jo(~)
/
Jo(~'2)
[sa4"~Jo(~'3)
t
where r~ = t¢2 + r t , and r 3 = I¢3 + x2 - x 4. Each of the vertices 7 ( r l , I¢2) and 7(t¢2, r a ) depends on the length of the above vectors and the angle between them. Similarly in the essentially inelastic process a + b ---, 1 + 2 + 3 + 4 + 5 , there are four-momentum eontigurations t in which at Sab --, oo all the energies s12, s23, sa4 and s4s are high, and the four-momentum transfers Sal, sa12, SbS and SbS4 are small. In all cases when S~b ---' oo, the relation 2 2 2 2 2 2 $12 $23 $34.$4S = s(m2 + / ~ 2 ) ( m 3 +/¢3)(D24 • + / ¢ 4 )
t They are similar to fig. 5 and correspond in pairs by the division of the particles produced into two groups, of the type 2 + 3 or 1 + 4 flying in opposite directions.
ESSENTIALLY INELASTIC COLLISIONS
601
is fulfilled, and the asymptotic value of the amplitude A (5 ~- 2) is given by the contribution of the graph of fig. 7. Obviously these results allow of a generalization for the production of any number of particles. Actually they were obtained earlier 5, 3) but in these preliminary reports the dependence of the quantities y on the asymptotic values of types (16) and (17) on the angle between the vectors ~ and ~, was overlooked. Hence the asymptotic values of the amplitudes proved to be dependent on a similar number of variables as the amplitudes themselves. 1 y a
S/~b Fig. 7. G r a p h s with reggeons d e t e r m i n i n g the a s y m p t o t i c value o f the a m p l i t u d e for t h e a + b ~ lq-2+3--}-4q-5 reaction w h e n sab --~ oo, si,~ ~ m , s2s ~ m , sa~ --* oo, s ~ - - * oo.
Let us consider in conclusion a mixed process, essentially inelastic with respect to some groups of particles and quasi-elastic with respect to others. One of the graphs of such a process is given as an example in fig. 8. The corresponding reaction is characterized by the production o f three groups of fast particles, the first of which
Fig. 8. G r a p h with reggeons determining t h e a s y m p t o t i c value o f t h e a m p l i t u d e for the a + b ~ 1 + 2 q - 3 - 1 - 4 + 5 reaction with t h e p r o d u c t i o n o f o n e particle 1 a n d two s t r e a m s consisting o f particles 2, 3 a n d 4, 5 w h e n sa~ --~ oo, sx, 8a ' ~ oo a n d s~a, 4~ ~ oo.
contains one particle and the two others contain two particles each with low energies s53 and s45 in their c.m.system. When sa, ~ ~ , the energies sx, 53 = (P, +P2 +P3) 2 and s23,45 = ( P z + P 3 + P 4 + p s ) 5 are high. Treating two particles 2 and 3 in fig. 8 as one particle b, particles 4 and 5 as one particle c and particle 1 as a (in accordance with fig. 3), we obtain for the contribution of the graph of fig. 8 a value perfectly analogous to eq. (16) in which x a = x l , Xb = 1¢2+X3, Xc = X4+1¢5, Scb = S~5, 3, %° = S23,,S and instead o f the factor gOc¢)iy(x:xa)g(xa), we have
G(s.;
)ir05,;
=
(18)
Similarly we can write the asymptotic value of the amplitude o f the more general form of fig. 9 corresponding to the case when many showers of several particles in
602
K. A. TER-MARTIROSYAN
each of them are produced, the overall momenta of the shower particles being almost parallel and differing from each other in order of magnitude and the energy of the particles in the c.m.system of each shower being small. Our derivation of the asymptotic value of forms (16) and (17) is based on definite assumptions (formulated above and in ref. 1)) about the analytical properties of the part of the amplitude essential for the asymptotic value. The correctness of these assumptions have not been proved and therefore the derivation cannot be regarded as a rigorous demonstration.
~
a
G iF
'b Fig. 9. G r a p h with reggeons determining t h e a m p l i t u d e for a process with the p r o d u c t i o n o f n g r o u p s o f particles with low relative energies o f t h e particles inside a group.
Nevertheless, the results (16) and (17) seem to be correct since they can be confirmed independently by a direct summation of the contribution of the ladder graphs s). Having performed such a summation, Halliday and Polkinghorne 9) obtained results which at first seem to contradict our own results. Actually, however, they were not interested in the kinematics and hence have not considered the only t limiting case (3), (4) of physical interest. It is precisely in this case that the summation of the ladder graphs yields s) the asymptotic value (1'6). 5. Branching Point etc. as Extreme Right Singularity
So far we have assumed that the extreme right singularity in the j, 1',... planes is the same, an isolated vacuum pole (in particular the asymptotic values (16) and (17) are correct only under this assumption). How will the asymptotic values of the amplitides change if this singularity is a branching point, a point of condensation of singularities, etc.? The question is not purely academic since experimental data xo) and theoretical analysis 11) show that an isolated (e.g. vacuum) pole cannot be at t < 0 the extreme right singularity in t h e j plane. t A n exception is t h e case o f quasi-elastic collisions, for which t h e results o f Halliday a n d Polkingh o m e coincide with o u r previous results. T h e case (i; i) w h i c h they regard as physical c a n n o t occur in reality; in o u r n o t a t i o n s it c o r r e s p o n d s to sac = ks, she = kxs a n d sab = k : s (see fig. 3) a n d s ~ oo. But in this case, in t h e physical region, t h e quantities st = Sod a n d t = S~e increase as s --+ oo a n d are n o t fixed.
ESSENTIALLY INELASTIC COLLISIONS
603
Let us consider at first as an example the simplest case (fig. 10), when the extreme right singularity in t h e j plane is a point of condensation of p o l e s j , ( t ) a t j = 1. We assume that all poles in fig. I0 are of the vacuum type, i.e.jv(0 ) = 1 for all v, and as jr! t<0
i = j " + ij'
1
J0 Jl J2-.. J.j
J'
Fig. I0. Location o f an infinite number o f j plane poles condensing towards the point j = 1.
s ja
Fig. 11. Location of an infinite number of branching points in the j plane condensing towards the point j = 1.
tc2 = - t increases the poles shift all the slower to the left in fig. 11 the larger their number v. In other words, for small r2 = _ t we have
j,(t) = 1 - j ' ~ c 2,
(19)
where all j" = [dj,(t)[dt]t=o are positive and decrease with increasing v. The asymptotic value o f the elastic scattering amplitude A (2 ~ 2) has in this case the f o r m of a sum o f contributions of pole graphs of fig. 11: A(2 *-- 2 ) = ~ ' where
/ s \Jv(o
c,(t)~)
,
c,(t) = #~(t)I,(t). In the region x 2 << m 2 we obtain m2 A(2 ~- 2) = X S
c°e-J"~2¢,
v
where c,o = (1/m2)c,(O) and 2~ = In (s/m2). Taking into account when s ~ oo the contribution o f only the poles at the extreme right in fig. 11 (i.e. poles in very large number v) and passing f r o m the summation over v (or over j~) to an integration over the variable x = 1 - j r ( t ) ~ j~x 2, we obtain A(2 ~ 2) ~ c(~, t) ( - - 3 ,
\m'l
(20)
604
K. A. TER-MARTIROSYAN
where
G(~, t) = -~
'fo °PI(x' t)e-X2¢dx
=
27~
'fo'' e
P2
t
dy.
Here pl(x, t) denotes the quantity (g/Aj,)e~(t) = (g/AjOc(j,, t), and AL = L + 1 - J , is the distance between adjacent poles in fig. 11. The same form o f the asymptotic value A (2 ~ 2) is obtained in the case when the extreme right singularity in the j plane is a branching point at j = 1. In this ease p(x, t) is a discontinuity of the partial amplitude ~pj(t) = ~p(x, t); x = 1 - y across the cut indicated in fig. 12 (by the line from the point j = 1 to j = - oo). If the
j = j' + i j "
1 Jo
J'
h J2 J3-.~-J~
Fig. 12. Graph with reggeons determining the asymptotic value of the elastic scattering amplitude A(2 +- 2). branching point j = j~(t) is a moving one (the cut being drawn f r o m A ( t ) to - o o ) the contribution from it is given by a formula analogous to eq. (20)
c,(¢, t) [,m2 /
,
(22)
where cv(~, t) has the same meaning as eq. (21): 1
c,(~, t) = ~ f o p,(x, t)e-2¢Xdx,
(23)
while x = Yo(t)-Y. Actually in t h e j plane, except the vacuum pole (jo(t) in fig. 12) there is a set o f branching points al) situated between the pole and condensed at the point j = 1. Therefore the asymptotic value of the amplitude will be given by a sum of the form
[ s ~ jv
A(2 ~ 2) = Ev c,(~, t) [m2]
(24)
the value v referring to the contribution of the vacuum pole and Co = Co(t) being independent of ~ = In (x/s/m).
ESSENTIALLY INELASTIC COLLISIONS
60~
The asymptotic values of the functions Co(l, t) when ~ --+ oo are given by the character of the branching pointsjv(t ), i.e. the course of pv(x, t) when x = j r ( t ) - j --+0. If, for example
(25)
p.(x, t) ... x ~'-1,
where v = 1, 2 . . . . (this value o f Pv is indicated by a preliminary study 11) of the character of the branchings in t h e j plane), then we have when ~ -+ m c~(¢, t) ,.~
c•(t)
(2~)~,--' where c" are certain constants. In this case the "complicated" asymptotic value of the elastic scattering amplitude is of a form similar to the case of accumulation of poles (fig. 10): A(2 ~ 2 ) = £
c'(t)
[s'~
(2¢)2(,_1) kin2/
J~(') ,
(26)
or in the region o f small x z = - t oo
m2 A(2 ~ 2) ~ £ s
Cry0
(27)
e-j'°~22~
~=o (2~) ~"-1~
"
When eq. (26) was written it was assumed that the position of the vth branching point in fig. 12 is given by eq. (19), where j" decreases with the growth of v (probably 11) as j~/(v+l)). The asymptotic behaviour of the sum (26)when ~ ~ oo depends essentially on the coefficients c"°. When ~ --+ oo and x 2 # 0, large values of v, the larger as the quantity x 2 becomes larger, are essential in the sum (26). In this case, similarly to eq. (21), the sum over v can be written as an integral over x = 1 - j , . However, a further study of the complicated asymptotic value (26) of the elastic scattering amplitude A (2 ~ 2) is beyond the scope of this paper. What are the asymptotic values of particle production amplitudes of types (16) and (17) in the case of "complicated" singularities o f the form of fig. 12 in the j plane? The asymptotic value o f fig. 1 in the limit (3) corresponding to the contribution of branching points j = j~(t) in the j plane and l ' = j¢(sz)in the l' plane is, evidently, given by a value analogous to eq. (22)
( Sbc)Jv"xc)( Sab)jr(tea)
A,¢(3 +-- 2) = c,,,(¢b~, ¢,b, X¢, X~) ~ i where t =--t¢a,
2
s1 =
~
2
--tCc,
O3 c.., ~ - ~ f f p . , ( ¢ , x ,
0
xo,x.)e-~"¢'~+~¢b°'dxdx '.
(28)
606
K.A..TER-MARTIROSYAN
Here pv. ~. is the double discontinuity of the partial amplitudes ?~, ~. of the variables x = j , ( t ) - j , x' = j ¢ ( s l ) - l ' (in the region x > 0; x' > 0). If the character of branching points is given by eq. (25) the limiting form of p,,, when x ~ 0 and x' ---, 0 is Pvv" '~ x 2 V - l x t 2 v ' - l ,
while Cvv' ~'~
(29)
Jc2v,~2v' :~ ~ab bbc
where cv,¢(ror.) is a function the form of which is not given by the unitarity conditions used above in the Sx and t channels. I f the singularities in the j and l' planes are condensations of branching points of fig. 12 the asymptotic value of the amplitude A (3 ~- 2) is given by the sum of the contributions (28) and (29) from all singularities h ( 3 4-- 2) =
~ cv'"(/Ce/Ca) [ sbc'~jv'0ce)[ Sab~jv(t:a), v, v'=0 ~ab bbe
(30)
or for small transverse momenta in the approximation (19) 1 A(3 +-- 2) S
i
o e - u " ~°2¢'°+J"~"¢"1 cv'"
$:2v,~2v'
v, v'=0 bab~bc
)
(31)
where c~,°¢ ,~, (m2+r2)/m4c~,¢(O, 0), and just as in eq. (27), jr" = j~/(v+ 1), where j~ = [djo(t)/dt]t=o (the value v = 0 or v' = 0 refers in sum (31) to the contribution from the vacuum polejo(t) in fig. 12). Asymptotic values (26) and (31) can in a sense be regarded as sums of the contributions from graphs of the form of fig. 11 or fig. 3 with reggeons of different types (v or v'), where the quantity
mV corresponds to the reggion line (of the "type" v) on the diagram 2¢ = ln(s/mZ), and the function ).v(2~) depends on the character of branching at the p o i n t j = j,(x). Eqs. (24) and (29) were written under the assumption that 2,(~) = 1/2~2". It is clear that the asymptotic values of the amplitudes of more complicated processes corresponding to the graphs of figs. 6--9 can be written quite similarly using the universal function 2~(O. 6. Conclusion
The calculation of the cross sections for different inelastic processes with the aid of the asymptotic values of the amplitudes of forms (16) and (17) has shown 7,12)
ESSENTIALLY INELASTIC "COLLISIONS
607
that in all cases all the powers o f the transverse components o f the m o m e n t a of the particles produced (increasing with primary energy) are cancelled out and the cross sections prove to be dependent only on the transverse components of the m o m e n t a o f the particles and on the logarithms of their longitudinal components. A rather sensible picture o f inelastic processes arises on the whole. Further calculations, however, have shown ta) that the unitarity condition in the s channel, and in particular, the condition o f the equality of the sum of the cross sections for all inelastic processes to the imaginary part of the amplitude .4 (2 .-- 2) for zero angle scattering, cannot be satisfied ta) if it is assumed that the vertex parts (15) ?(sl, t, cos 40) -= ?(wa, ~:c) corresponding to the emission of a particle by a reggeon do not vanish at ra = r c --- 0. Are there any reasons to assume that the vertex ~(~a, ~c) m a y vanish at Ka --. 0 and i¢c ~ 07 At present it is difficult to answer this question. In the region j - - , j o ( t ) and 1' ~ Jo (Sl), essential for the high-energy behaviour, the partial amplitude Zj; r a, (sl, t) was written according to (12) and formula (25) of the previous paper 1) in the form (~7[)2~(j; I,A,(Sl, t) =
u(st)iwz'(st' t)u(t) ( j - j o ( t))( l'-jo(sl)) "
For the asymptotic behaviour of the amplitude, the value (13) was obtained. Its vertex part ?(x,, ~ ) proves, according to (15), to be proportional to the value
x/st, two, (st, t) at t ~ 0 and st "-" 0; ~(7¢a, 7¢c) = y(St, t, qbo) "" x/st, t w~,(St, t). All other factors in (15) at Sl --. 0 and t --. 0 when jo(t) --. 1 have finite non-zero values. The point st = 0 or t 1 = 0 is not a singular one for the amplitude A (2 -~ 3); therefore, it cannot, in particular, be a branching point of the asymptotic behaviour of the amplitude. Therefore, in the region of small s t and t the function w~(s 1 , t) must be proportional either to 2) 1/x/st ' t or to the positive odd power x/s-~. In other words, at r~ --. 0 and r~ --. 0 ~
0%
=
where k = 0 in the first case, k = 1, 2 , . . . in the second case and ?o is a certain coefficient. The assumption that k = 0 seems to be the most natural, since in the opposite case (k = 1, 2 . . . . ) the contribution f r o m the total cross section of essentially inelastic processes proves to be logarithmically small at ~ = l n ( x / s / m ) ~ oo compared t with the contribution f r o m quasi-elastic processes where the same number o f particles is produced but in the f o r m of two showers. t At k = 1 the essentially inelastic part of the ~ than the two-shower quasi-vlastic part ~2).
total cross
section proves to he smaller by the factor
608
K.A. TER-MARTIROSYAN
At k - 0 the essentially inelastic contribution proves on the contrary to be large; the total cross section an of the production of n particles or n showers proves to be proportional ~2. t3) to the value ~, ~ ( , - 2) (in ¢)"- ~
It is assumed that in all cases only vacuum Regge poles give finite contributions. Thus, in this case the total cross sections of the production, e.g. of three, four, five. . . . particles decrease with the growth o f the energy slower than the total cross section of the elastic scattering. However, at k = 0 the total cross section ~ , a , of the production of any number o f particles proves, as is pointed out above, to be increasing with increasing ~ due to the production of a number of particles of the order n - ~ which is still very small compared with the number N = x/s/m of the particles whose production is allowed by the conservation laws. This growth of the total cross section at ~ --} oo results perhaps from the fact that the asymptotic behaviour of types (16) (17) etc. here obtained is incorrect in the region where n, the number o f particles (or of showers, as in the fig. 9) is large, when n ~ ~. The above investigation practically referred to the case n = 3 or to the case when n is any fixed number not increasing at ~ ~ oo.
Note added in proof." At very small sl and t the cosines z and z'c become small (for any large fixed Sab and Sb~ in eq. (6)) and the Regge type formulae like eq. (13) are not valid at all. Therefore, the general assumptionmade above on the function w~(s,, t) proves to be incorrect, i.e., w~(sl, t) may be constant and not ,,/s,, t or 1/x/s,, t as s, ~ 0 and t ~ 0. Taking this into account and computing ~, (just as in refs. 7, ,2,13)) with 7(ra, re) in the form y ~ Xa XoyO, where ~'o = const., we obtain 0% -~ cn/~n; this value of tT, does not give the difficulty mentioned above, i.e. does not lead to the growth of the total cross sections a = ~n an at ~ -} o0. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)
A. M. Popova and K. A. Ter-Martirosyan, Nuclear Physics 60 (1964) 107 K. A. Ter-Martirosyan, Proc. Int. Conf. on High Energy Physics, CERN (1962) p. 556 K. A. Ter-Martirosyan, JETP 44 (1963) 341 D. Amati, S. Fubini, A. Stanghellini and M. Tonin, Nuovo Cim. 22 (1961) 569; D. Amati, S. Fubini and A. Stanghellini, Phys. Rev. Lett. 1 (1962) 29 S. C. Frautshi, in the press V. N. Gribov and I. T. Dyatlov, JETP 15 (1962) 140 I. G. Ivanter, A. M. Popova and K. A. Ter-Martirosyan, JETP 46 (1964) 568 Y. A. Simonov and K. A. Ter-Martirosyan, Nuclear Physics 66 (1965) 641 I. G. Halliday and J. C. Polkinghorne, preprint K. J. Foley et al., Phys. Rev. Lett. 10 (1963) 376 S. Mandelstam, Nuovo Cim. 30 (1963) 1113 I. A. Verdiyev, A. M. Popova and K. A. Ter.Martirosyan, JETP 46 (1964) 1295 I. A. Verdiyev, et aL, JETP 46 (1964) 1700
ASYMPTOTIC
BEHAVIOUR
OF
ESSENTIAIJ,Y
INELASTIC
COLLISIONS
K. A. T E R - M A R T I R O S Y A N Theoretical and Experimental Physics Institute,
Moscow, USSR
Received 15 February 1964
Abstract: The asymptotic value of particle production amplitudes at s ~ oo is investigated for "essentially inelastic" collisions when by the production in the c.m. system each moving nearly parallel to the The momenta of all particles in each
it is especially large. These collisions are characterized of two groups of particles at high primary energies, momenta p~ and Pb = --P~ of the colliding particles. group differ considerably by their order of magnitude: Pl >>Ps >> • • • >>Pnl, Pn >>Pn-1 >> • • • >>Phi+l, where the subscripts 1, 2 . . . nx refer to the particles of the first group and nx+ 1, n x + 2 . . , n--1, n to those of the second. The mathematically corresponding limit of a particle production amplitude is determined by the requirement that the squares sik of the energies of any pair of particles in their c.m. system should increase (or remain large as compared with particle masses squared) as the primary energy increases, s ~ oo. It is also necessary that the transverse components/¢~ of all momenta Pi of the particles produced and momentum transfers squared tt = (Pa--Px-- • • • --pi) ~, i = 1, 2 . . . . . n should be small. It is shown that in this limit the asymptotic value of the amplitude takes the form of a product of Regge factors I I n - ~ ( s i , i+ 1)Y°(t0Fi (where jo(t i) is a vacuum pole) and can in a sense be compared with the contribution of Feynman-type graphs with n-- I "reggeons". The quantities ~ = 7~(K'~, K'~+I) depending on the projections ;¢'~ and K'~+I of the transverse momenta of reggeons before and after the emission of a particle correspond to the vertex parts designating in these graphs the emission of the particle by a reggeon. In a preliminary report on this question 2,s) the dependence of the quantities ~ on the angle between ;¢'~ and ;¢'~+~was overlooked. Thus the coefficient of the product of the Regge factors in the asymptotic value of the amplitude depends on the transverse components of the momenta of all the particles produced. As a result the asymptotic value of a particle production amplitude proves to be dependent on the same number of variables as in the general case (i.e. is sensitive to any change of the momenta). The asymptotic behaviour of mixed "essentially inelastic" and "quasi-elastic" collisions is also considered, when not particles but n groups of particles are produced and the energy of each particle in the c.m.s, system of a group is low. Collisions of this type are mainly responsible for multiple production at high energies. In conclusion brief consideration is given to the "complicated" types of asymptotic behaviour of "essentially inelastic" processes when the extreme right singularity in t h e j plane is a branching point or a point of accumulation of singularities etc. It is shown that in this case instead of the factor (s~, t+l)JO(tl) o n e ought to expect in the asymptotic values of essentially inelastic amplitudes expressions like ~v ;tv(t;~, ~ + l ) ( S t , t+1) iV(it)' where ~e, t+1 = In(s~, t+l/m 2) and ;tv(t,~, ~+1) is a function depending on the character of the vth singularity and perfectly analogous to those which determine the corresponding asymptotic values of elastic scattering amplitudes.
1. Introduction I n a p r e v i o u s p a p e r 1) t h e a s y m p t o t i c v a l u e o f t h e a m p l i t u d e f o r t h e r e a c t i o n a+b
~ c+d+e
(1)
w a s o b t a i n e d i n t h e c a s e w h e n t h e e n e r g y Sed o f t h e p a i r o f p a r t i c l e s c a n d d r e m a i n s s m a l l a t Sab -* o0. 591
592
1~.. A. TER-MARTIROSYAN
Considered below is the case when all the three energies Sod, Sde and see are high and the transverse components of the momenta of the particles c, d and e are small at sab ~ o o . We 2.3). Several other authors 4, 6) have obtained for such essentially inelastic collision amplitudes asymptotic values of the form of contributions of simple Feynman graphs (see figs. 4, 6 and 7). However, the dependence o f the coefficient in front of the product of the powers s w) of the energies on some transverse momentum components was left out of consideration. As shown below, the asymptotic value of an amplitude in its correct form depends on the same number of variables as the amplitude itself and is sensitive to any change of the particle momenta. Brief consideration is given to the asymptotic values in the case when the extreme right singularity in t h e j plane is not an isolated pole, but a singularity like a branching point, a point of condensation of poles or accumulation of branching points. S=Sab
t=Sae
a
e~sl
"d
Fig. 1. Amplitude of the reaction e + d ~ a-t-b-l-c.
To use all the notations introduced earlier i) it is convenient to consider not reaction (1) but its analogue (fig. 1) e+d ~ a+b+c,
(2)
which differs from reaction (1) only in the permutation of the particles a and e, and b and d. For the transition from the asymptotic value of the amplitude for reaction (1) to that for reaction (2) (or vice versa), it is sufficient to transpose the subscripts of these particles. b Pa~ .........
.
1 _ ~~ a
. Pd
Fig. 2. Configuration of the momenta of the particles involved in the e-I-d ~ a-I-b-/-c reaction in the Sae channel, Pe and Pa are the momenta of primary particles, Pa, Pb, Pc are the momenta of particles produced, and ~:a, ~b and ~:c are the transverse projections on a plane perpendicular to Pe and --Pc-
2. Limiting Values of Invariants and cosines of s Channel Angles
After this permutation in fig. (2) and eqs. (37) and (38) of rcf. 1), we can determine the configuration of the momenta (see fig. 2 of this paper) and the limits of the in-
ESSENTIALLY INELASTIC COLLISIONS
593
variants in the physical region of the essentially inelastic process (2) at Sac latter is in the form s++ +
kc /
2
-
kb ~mb
+++)
S.b ~ 4 k a k b .~ m 2 Sac ,~ Sde ~
= m +
°de
o0.
The
++,
m2
Sdc
~
~
o0,
(3)
00~
t -- Sa+ = -Xa2 = const, sl
=
Sod
=
kd
Sbd ~-
--X~2 = const,
(4)
--Sb° -+ --
kb
In fig. 2 and in eqs. (3) and (4) it is assumed that k b << ko. For the following this condition is sufficient. However, similar to the presentation ofeq. (37) of the previous paper 1), we assume in eq. (3) also that the ratio y' = k o / k b increases when Sd. ~ O0 as (Sa+/m2) ~ where 0 < g < ½ and m is a quantity of the order of particle masses. When k b << k o we obtainpc ~ Pd ~ ka "~ ko = ½x/Sdc. The condition g < ½ means that we do not consider the case when k b decreases with the growth of primary energy t (it is assumed that when Sdc ~ oo all momentap~ in fig. 2 are ultrarelativistic). Besides the configuration indicated in fig. 2 there is another perfect analogue o f it for which all energies Sbc, Sab and sac in the limit Sd+ ~ O0 are high and the transverse components x~, x b and go (and momentum transfers) are small. It only differs from fig. 2 in that the momentum Pb o f the particle b is nearly parallel to p~ and not to Pc. In both cases the energies satisfy the relation Sab Sbc = Sde(mb2 q- t¢~).
(5)
For the investigation of the asymptotic value (3) of the amplitude for'reaction (2) it is convenient to use its expansion in partial waves in the t channel (see eqs. (3), (4) and (7) of ref. 1)). What are the values o f the cosines z, z; and cos ~b contained by these expansions in the region (3) unphysical for the t channel? They can be expressed in terms of the invariants Sab ~--- (qa--qb) 2,
Sb¢ = (qb+q¢) 2,
See = (qc--qo) 2,
which are determined in the t channel according to fig. 2 of ref. 1). As a result, we * The reg ion o f s m a l l k a ~ m i n t r o d u c e s at sac -+ oo a v a n i s h i n g c o n t r i b u t i o n ¢) to t he t o t a l cross section o f r e a c t i o n (2). A l l the results o b t a i n e d b e l o w are a c t ua l l y v a l i d for this re gi on as well.
594
K . A . TER-MARTIROSYAN
obtain 5ab ~
m az -- m 2 = 2qaqbZ_20~aO~b
bo--m2
2 = --mo
2 a f (~b Oj~+ q2) +
m~ q-Sab--Sl--Sde
_
2q__~
V2
2qaq, z,,
, ,
x/St (t'o:q~z~'Cq:oJ'c)Z
(6)
2m+
-- ~/Sl ((O:OJ"q-qaq~z')+ Eq:q'~/(1--Z2)(1--Z ;2) COS ~, q~ -- qb,
as q~ = --qb
(q~ is written everywhere instead of [qil for brevity). In the right-hand side o f eq. (6) instead o f s , c - m,2 - m o2 the identical quantity m~+s,b--Sl--Sa, is written (the three invariants Sod = Sl, Soc and sa, in fig. 1 are connected by the identity st +s~, +sd, = m~2 + m~ + m~ + S,b). The energies and toomenta o f particles in the t channel are designated by qi = (oh, qs); they depend on t = sa, and Sl -- S~d. The energy and m o m e n t u m o f the particle c in the c.m. system o f the t channel o~c, q, = [qJ, zo are expressed by their values 0~, q" and z" in the c.m.system of the particles c and d with the aid o f the Lorentz transformation* 1
t
t
t
q,z, = ~/st (m, qcz,+q,m,),
q~x/1--z~2
t
=
q~x/1--zo
t2 •
Here ~ : = (t+Sx--m2)/2x/t and q: = ~/o~2~ - s x are the overall energy and m o m e n t u m o f the particles c and d in the c.m.system of the t channel. The quantities o~ and q" obviously depend only on s 1 and the masses. Neglecting in eq. (6) the quantities which do not increase in the limit of eqs. (3), (4) we find that in this limit z~,
Sab ~ O 0 , 2qaqb
Z'c ~
Sbc
* oo,
~b-~b
o
2qb q"
(qb = q~, as in c.m.system o f the t channel qb = --q~), where
o= o:O+, + V z,,,: *
in the direction q~ = qe+qd with the velocity
p = q~/~/~+sl = ~/VL--(sdo~?).
S.b1 /
(7)
K. A. TER-MARTIROSYAN
595
According to eq. (3) all terms in square brackets can be neglected except sd.. Therefore taking into account eqs. (4) and (5) and substituting the values o9, and qa, we obtain cos 40 ~ V/st +
V--T2q2 St
Sd___L Sab
Sbo
2
_ K b 2( m b 2 ..~/£a2 .1_ X°2 ) .jl_(Ka2 .+. Kc2 ) "{-
2+
(s)
2x a i%(m~ + 1c~)" This quantity may, in general, be larger than unity in absolute value since the region (3), (4) is unphysical for the t channel in which the angle 4 is determined. If, however, the values of s I and t are small as compared with m 2, i.e. x 2a < m 2 9 x~2 << m 2, and x~ (tq + ~c) 2 << m 2 (this region is the most essential in computing 7) the total cross section o f reaction (9)), we obtain from eq. (8) 2
2
2
cos
,
or
rC-4o - ~oao, where ~O.ois the physical angle between the s channel transversal components o f the momenta r , and xc (see fig. 2). Thus in the region of small x a and re cos (rC-4o) ~ cos t#,c = < 1.
3. Asymptotic Value of the AmpBtude of Fig. 1 when z ~ ~
and z~--~
The asymptotic value o f the amplitude of fig. 1 in the limit (7) can be determined from eqs. (27)-(29) o f ref. 1) (valid in the region z ~ oo) by putting there 4 = 4o and going over to the limit z" --. oo. To this end, expansion (29) of ref. t) should be written in the form of the Sommerfeld-Watson integral over l' (9)
i)2, = ,v )(,+ ) a7_- , ~( -2)'
1 fc
F ( I ' - 2 ' + 1) ×
!
Zo)___
t
21' + 1 sin
hi'
dl'.
(10)
Here C is the contour in the I' plane such as in fig. 9 of ref. 1) and v~?(st, t) is an analytical continuation into the l' plane of the quantities vv, x,(sl, t) defined for even
596
K.A.
TER-MARTIROSYAN
and odd integral l'. Does there exist such an analytical continuation, and if it exists what are its properties? According to eqs. (26) and (27) of ref. 1) the quantities vra, differ from the amplitudes =
(11)
Ax,(s,, t, z, z;)e,,x,(z'o)dz'+ -1
only by the /'-independent factor #(t)Io(t)(Sab/mz)l°CO(m/2qb)J°(Ocx,(jo(t))i x'. Therefore to answer the question it is sufficient to construct an analytical continuation of the amplitudes avx, into the l' plane. This can be done exactly in the same way as used earlier for obtaining the analytical functions of the variable j determined by eqs. (16) and (18) of ref. 1). The discussion is based on the assumption that the quantity Ax, (sl, t 1 z, z~)/(z'o 2 - 1)ix' can be represented in the z~ plane by the dispersion relation with respect to z" with complex integration contours ?', and ?'o2 similar to those indicated in figs. 5 and 6 of ref. 1). Just as in the case 1) of the amplitude avx,(sl, t, z)/(z 2 - 1 ) +x' we do not consider in this dispersion relation the part AAx,(s 1 , t, z, z')/(z'c 2 - 1)+x' of the integrals over complex contours as inessential t for the asymptotic value A x, at z~ ~ oo. In other words we write the Ax, part of the function Ax, essential for the asymptotic value when z'c --+ oo, in the form
A'~(sl , t, z, z') (z~2-1)+ x'
1 [~ A~÷)(sl, t, z, z'~')dz'~' = .
.
.
.
1 (-zo~°A~2,)(sl, t, z, z')dz'o'
(z,J-1)+X'(z'o'-Z'o)
~ _ _ +
o and z°z are the points where the integration contours in the precise relation where Zct for A x, = A'x,+ AAx, go from the real axis into the complex plane, and All,) and A(~,2) are the absorption parts of the amplitude Ax,. Substituting the right-hand side of this relation into eq. (11) instead of Ax,/(z'¢ 2 - 1 ) ½x' we obtain (just as the functions in eqs. (16) and (18) were obtained in ref. 1)) two analytical functions a~,~) of the variable l' a(+) I'X" ~ bvx,___CI'X' b,w(st, t, z)
1 f zo~ zc o
t --i~X'
t
t
cvx'(sl, t, z) = l f z ~° [--A(~,2)(sl, t, z, Zo)]e Qrz,(zc)dzo, , -i~x' , , t It is more difficult to justify this neglect here than in the case x) a;,x,(sx, t, z) since the perimeters o f the complex contours in the dispersion relation for Ax,/(zc "2- l)tX' with respect to z'o may prove not to be finite, but increasing with z--+ co. Actually this neglect seems to be justified since in all the cases investigated e) the absorption parts in such anomalous complex contours always decrease rapidly when moving along them, towards the singularities.
597
ESSENTIALLY INELASTIC COLLISIONS
which coincide with amplitudes (11) for integral even and odd l' (plus or minus sign) respectively. Assuming that A~, when z~ ~ oo does not increase more rapidly titan z "¢ we conclude directly 1) from these formulae that (i) the functions ,,vx"t5)are analytical functions of l' in the right-hand half-plane Re l' > v', (ii) they decrease exponentially when l' ~ oo along any ray in this half-plane, and (iii) they vanish at all integer physical values of l' = n smaller than 1' (and than v'). ,,t5) only by an l' independent Evidently, the functions ~rx"ts)which differ from ,-,'x, factor (see eqs. (26) and (27) of ref. 1)) will also possess all these properties. The asymptotic value at z~ ~ ~ of the integral (10) is determined by the extreme right singularity in the l' plane of the functions •L,I,(5) 2 , . From the unitarity condition in the l' channel it follows (similarly to eq. (24) of ref. a)) that
t) =
t)
1-2ip(sl)2
,
)(sl-)
'
where Sl • = Sx _ iz, p(sl) = 2k'(s2)/~/sl and 2},~)(Sl) is an analytical continuation into the l' plane of the elastic scattering amplitude for the particles c and d with momentum k'(sl) in their c.m. system. Hence"~ lt5) , ) . , has a pole at the point l' = floS)(sl) at which
1 - 2ip(s1)2[,5 )(sl) = O. Near the pole vvx, can be written as
t) =
)(Sl' t) r -j oS )(s, ) '
(12)
where u(+)(Sl) is the same function as in eqs. (25) of ref. 1), and w]~)(Sl, t) is a new unknown function (it is real in the region Sl and t below the elastic scattering threshold; the residue in the right-hand side of eq. (12) can be factorized in the conventional manner). Assuming that the extreme right singularity of V,(+)to l ' M k O l , t) in the l' plane is a vacuum polejo = jot+)(Sl), we obtain from eqs. (10) and (12) the following asymptotic value of v~, when z" ~ oo (i.e. when She ~ o0):
vx,(s,, t, sbc) = i L \ ~ q b r t l
i~'c~,(jo(Sl))w~+)(Sl, t
O(sl)Io(sl) ~-~!
'
where g(sl) is the vertex part introduced previously 1). Substituting this value vx, into eq. (28) of ref. 1) and putting in them q~ = tko, we obtain the following asymptotic value of the amplitude of fig. 1 in the asymptotic region determined by eqs. 0 ) , (4) and (7): A (3 *-- 2) ~, Y(sl)iy(sl, t, cos Cko)a(t)Io(sl)Io(t),
(13)
598
K.A.
TER-MARTIROSYAN
where ? is the value o f the function oo
~(s,, t, cos ~b) = Z ' (2 cos 2'(n-~b))?a,(s,, t),
(14)
~'=0
7~,(sl t) = ( m2sl ~rJ°(s')( m2t ~½jo(O ' \ k ( s ~ , t, m 2 ) ] \ r ( s 2 , t, m2)] cx,(jo(s~))cx,(jo(t))w(a,+)(sl, t), (15) K ( m 2 , s l , t) = 4tq 2 = (m2) 2 - 2 m 2 ( s l + t) + (sl - t) z
for the unphysical value (8) o f the angle q~ = ~bo. The series (15) converges only when Icos 4[ < 1. Therefore it is assumed that the dependence o f T on cos ~b is obtained f r o m eq. (14) in the region Icos~l < 1, and then analytical continuation is obtained into the region Icos~l > 1. The m e t h o d o f such an analytical continuation and explicit calculation o f 7 when q~ = ~bo is not indicated here (V cannot be calculated a n y w a y since eq. (15) contains an u n k n o w n function wta,+)). It is essential that cos~b o depends, according to eq. (8), on all transverse c o m p o n e n t s r , , Xb and to, o f the m o m e n t a o f the particles produced. Therefore the value 7 when ~b = ~bo also depends on these c o m p o n e n t s V = V(x., Xb, /¢¢) o r ~ ( x c , r a ) , and the asymptotic value (14) can be represented as / o \ jo(x¢) / Sab~Jo(x.) A(3 ~ 2) = g(x~)iy(xo, r,)g(x,)I(x~) I ~--~| I(x,) , \m-I
(16)
where it is taken into account that sl ~ - g 2 and t ,~ - g 2 . This asymptotic value depends, as might well be expected, on five variables referring to any change o f the m o m e n t a in fig. 2. It can be regarded in a sense as the contribution o f the graph o f
d
xe
Fig. 3. Graph with reggeons determining the asymptotic value of the amplitude for the reaction e-t-d ~ a-t-bq-c when Sea ~ oo, S~b ~ oo and sbo ~ oo. fig. 3 with two reggeons. T h e internal vertex in this graph corresponds to the contribution i7 dependent on the transverse particle m o m e n t u m c o m p o n e n t s o f all the three lines reaching it t * In other words, the vertex 7 in fig. 3 depends not only on the squares of the momenta of the par= ticles involved, sl 1¢c: and t = --/eL2, as in the case of the usual Feynman graphs, but also on the quantity sab sbc/sae (as 4o depends on it). In the physical region of the essentially inelastic proeess (2), this quantity is mbZq-Kb~, according to eq. (5). =
- -
ESSENTIALLYINELASTICCOLLISIONS
599
Transposing in eq. (16) the subscripts of the particles a and e, and b and d, we obtain an analogous asymptotic value of reaction (1) which obviously corresponds to the graph of fig. 4.
tl b=\ g e,
S
i t~.
g/a
|d
,e
Fig. 4. Graph with reggeons determining the asymptotic value of the amplitude for the reaction
a+b ~ eWd+e when sab ~ o0, sea ~ co and
sac "," oo.
4. Asymptotic ValUe of More Complicated Essentially Inelastic Processes The asymptotic value of more complicated essentially inelastic processes in which four, five etc. particles are produced can be obtained by the same method (by using the expansion in rotation group coefficients d~,~x),,(z) instead 3) of the polynomials P~, i,(z) in the t channel).
Pi
__
a)
Pb
"~_'..~___
_v
Pa
p
Fig. 5. Configurations o f the m o m e n t a o f the particles involved in the a + b ---> 1 + 2 + 3 + 4 reaction in the t channel; Ps and Pb are the m o m e n t a o f primaries, p t , p s , p s a n d p , are the m o m e n t a o f particles produced, and ~q, t¢~, 1¢~ and ;Ca are the transverse projections o n a plane perpendicular to Pa and Pb.
Fig. 5 represents two of three possible configurations of the momenta of the particles produced in the essentially inelastic process a+b~
1+2+3+4
600
K . A . TER-MARTIROSYAN
(fig. 6). In these configurations all energies Sag, s12 and s23 when S.b ---, oO are high and the three-momentum transfers Sa I =
__/£2,
Sal2 =
__ (/£2 + / ~ 2 ) 2 =
- - (/(~¢ + / ¢ 4 ) 2,
$4.b =
--/~?
are small. There is a third similar configuration which differs f r o m that of fig. 5b) only in that the m o m e n t u m px has one direction (almost parallel with p . ) and the three m o m e n t a P2, P3 and P4 are almost parallel to Pb in the opposite direction (while P4 >> Pa >> P2)" In all the three cases when Sab ~ o0, the relation $3,
+
analogous to eq. (5), holds in the physical region. The asymptotic value of the amplitude in these configurations is determined by
2-----1 i.¥
Fig. 6. Graph with reggeons determining the asymptotic value of the amplitude for the a + b ~ 1 + 2 + 3 + 4 reaction when ssb ~ oo, s~2 ~ oo, s2a -+ oo and s84 ~ oo.
the contribution of the graph of fig. 6, depends on eight variables and has a form analogous to eq. (16): A(4 *- 2) ~ g(xl)iT(xl, ~'2)i7(r'2, x'3)g(x,~)
/s
r
p
\ Jo(~)
/
Jo(~'2)
[sa4"~Jo(~'3)
t
where r~ = t¢2 + r t , and r 3 = I¢3 + x2 - x 4. Each of the vertices 7 ( r l , I¢2) and 7(t¢2, r a ) depends on the length of the above vectors and the angle between them. Similarly in the essentially inelastic process a + b ---, 1 + 2 + 3 + 4 + 5 , there are four-momentum eontigurations t in which at Sab --, oo all the energies s12, s23, sa4 and s4s are high, and the four-momentum transfers Sal, sa12, SbS and SbS4 are small. In all cases when S~b ---' oo, the relation 2 2 2 2 2 2 $12 $23 $34.$4S = s(m2 + / ~ 2 ) ( m 3 +/¢3)(D24 • + / ¢ 4 )
t They are similar to fig. 5 and correspond in pairs by the division of the particles produced into two groups, of the type 2 + 3 or 1 + 4 flying in opposite directions.
ESSENTIALLY INELASTIC COLLISIONS
601
is fulfilled, and the asymptotic value of the amplitude A (5 ~- 2) is given by the contribution of the graph of fig. 7. Obviously these results allow of a generalization for the production of any number of particles. Actually they were obtained earlier 5, 3) but in these preliminary reports the dependence of the quantities y on the asymptotic values of types (16) and (17) on the angle between the vectors ~ and ~, was overlooked. Hence the asymptotic values of the amplitudes proved to be dependent on a similar number of variables as the amplitudes themselves. 1 y a
S/~b Fig. 7. G r a p h s with reggeons d e t e r m i n i n g the a s y m p t o t i c value o f the a m p l i t u d e for t h e a + b ~ lq-2+3--}-4q-5 reaction w h e n sab --~ oo, si,~ ~ m , s2s ~ m , sa~ --* oo, s ~ - - * oo.
Let us consider in conclusion a mixed process, essentially inelastic with respect to some groups of particles and quasi-elastic with respect to others. One of the graphs of such a process is given as an example in fig. 8. The corresponding reaction is characterized by the production o f three groups of fast particles, the first of which
Fig. 8. G r a p h with reggeons determining t h e a s y m p t o t i c value o f t h e a m p l i t u d e for the a + b ~ 1 + 2 q - 3 - 1 - 4 + 5 reaction with t h e p r o d u c t i o n o f o n e particle 1 a n d two s t r e a m s consisting o f particles 2, 3 a n d 4, 5 w h e n sa~ --~ oo, sx, 8a ' ~ oo a n d s~a, 4~ ~ oo.
contains one particle and the two others contain two particles each with low energies s53 and s45 in their c.m.system. When sa, ~ ~ , the energies sx, 53 = (P, +P2 +P3) 2 and s23,45 = ( P z + P 3 + P 4 + p s ) 5 are high. Treating two particles 2 and 3 in fig. 8 as one particle b, particles 4 and 5 as one particle c and particle 1 as a (in accordance with fig. 3), we obtain for the contribution of the graph of fig. 8 a value perfectly analogous to eq. (16) in which x a = x l , Xb = 1¢2+X3, Xc = X4+1¢5, Scb = S~5, 3, %° = S23,,S and instead o f the factor gOc¢)iy(x:xa)g(xa), we have
G(s.;
)ir05,;
=
(18)
Similarly we can write the asymptotic value of the amplitude o f the more general form of fig. 9 corresponding to the case when many showers of several particles in
602
K. A. TER-MARTIROSYAN
each of them are produced, the overall momenta of the shower particles being almost parallel and differing from each other in order of magnitude and the energy of the particles in the c.m.system of each shower being small. Our derivation of the asymptotic value of forms (16) and (17) is based on definite assumptions (formulated above and in ref. 1)) about the analytical properties of the part of the amplitude essential for the asymptotic value. The correctness of these assumptions have not been proved and therefore the derivation cannot be regarded as a rigorous demonstration.
~
a
G iF
'b Fig. 9. G r a p h with reggeons determining t h e a m p l i t u d e for a process with the p r o d u c t i o n o f n g r o u p s o f particles with low relative energies o f t h e particles inside a group.
Nevertheless, the results (16) and (17) seem to be correct since they can be confirmed independently by a direct summation of the contribution of the ladder graphs s). Having performed such a summation, Halliday and Polkinghorne 9) obtained results which at first seem to contradict our own results. Actually, however, they were not interested in the kinematics and hence have not considered the only t limiting case (3), (4) of physical interest. It is precisely in this case that the summation of the ladder graphs yields s) the asymptotic value (1'6). 5. Branching Point etc. as Extreme Right Singularity
So far we have assumed that the extreme right singularity in the j, 1',... planes is the same, an isolated vacuum pole (in particular the asymptotic values (16) and (17) are correct only under this assumption). How will the asymptotic values of the amplitides change if this singularity is a branching point, a point of condensation of singularities, etc.? The question is not purely academic since experimental data xo) and theoretical analysis 11) show that an isolated (e.g. vacuum) pole cannot be at t < 0 the extreme right singularity in t h e j plane. t A n exception is t h e case o f quasi-elastic collisions, for which t h e results o f Halliday a n d Polkingh o m e coincide with o u r previous results. T h e case (i; i) w h i c h they regard as physical c a n n o t occur in reality; in o u r n o t a t i o n s it c o r r e s p o n d s to sac = ks, she = kxs a n d sab = k : s (see fig. 3) a n d s ~ oo. But in this case, in t h e physical region, t h e quantities st = Sod a n d t = S~e increase as s --+ oo a n d are n o t fixed.
ESSENTIALLY INELASTIC COLLISIONS
603
Let us consider at first as an example the simplest case (fig. 10), when the extreme right singularity in t h e j plane is a point of condensation of p o l e s j , ( t ) a t j = 1. We assume that all poles in fig. I0 are of the vacuum type, i.e.jv(0 ) = 1 for all v, and as jr! t<0
i = j " + ij'
1
J0 Jl J2-.. J.j
J'
Fig. I0. Location o f an infinite number o f j plane poles condensing towards the point j = 1.
s ja
Fig. 11. Location of an infinite number of branching points in the j plane condensing towards the point j = 1.
tc2 = - t increases the poles shift all the slower to the left in fig. 11 the larger their number v. In other words, for small r2 = _ t we have
j,(t) = 1 - j ' ~ c 2,
(19)
where all j" = [dj,(t)[dt]t=o are positive and decrease with increasing v. The asymptotic value o f the elastic scattering amplitude A (2 ~ 2) has in this case the f o r m of a sum o f contributions of pole graphs of fig. 11: A(2 *-- 2 ) = ~ ' where
/ s \Jv(o
c,(t)~)
,
c,(t) = #~(t)I,(t). In the region x 2 << m 2 we obtain m2 A(2 ~- 2) = X S
c°e-J"~2¢,
v
where c,o = (1/m2)c,(O) and 2~ = In (s/m2). Taking into account when s ~ oo the contribution o f only the poles at the extreme right in fig. 11 (i.e. poles in very large number v) and passing f r o m the summation over v (or over j~) to an integration over the variable x = 1 - j r ( t ) ~ j~x 2, we obtain A(2 ~ 2) ~ c(~, t) ( - - 3 ,
\m'l
(20)
604
K. A. TER-MARTIROSYAN
where
G(~, t) = -~
'fo °PI(x' t)e-X2¢dx
=
27~
'fo'' e
P2
t
dy.
Here pl(x, t) denotes the quantity (g/Aj,)e~(t) = (g/AjOc(j,, t), and AL = L + 1 - J , is the distance between adjacent poles in fig. 11. The same form o f the asymptotic value A (2 ~ 2) is obtained in the case when the extreme right singularity in the j plane is a branching point at j = 1. In this ease p(x, t) is a discontinuity of the partial amplitude ~pj(t) = ~p(x, t); x = 1 - y across the cut indicated in fig. 12 (by the line from the point j = 1 to j = - oo). If the
j = j' + i j "
1 Jo
J'
h J2 J3-.~-J~
Fig. 12. Graph with reggeons determining the asymptotic value of the elastic scattering amplitude A(2 +- 2). branching point j = j~(t) is a moving one (the cut being drawn f r o m A ( t ) to - o o ) the contribution from it is given by a formula analogous to eq. (20)
c,(¢, t) [,m2 /
,
(22)
where cv(~, t) has the same meaning as eq. (21): 1
c,(~, t) = ~ f o p,(x, t)e-2¢Xdx,
(23)
while x = Yo(t)-Y. Actually in t h e j plane, except the vacuum pole (jo(t) in fig. 12) there is a set o f branching points al) situated between the pole and condensed at the point j = 1. Therefore the asymptotic value of the amplitude will be given by a sum of the form
[ s ~ jv
A(2 ~ 2) = Ev c,(~, t) [m2]
(24)
the value v referring to the contribution of the vacuum pole and Co = Co(t) being independent of ~ = In (x/s/m).
ESSENTIALLY INELASTIC COLLISIONS
60~
The asymptotic values of the functions Co(l, t) when ~ --+ oo are given by the character of the branching pointsjv(t ), i.e. the course of pv(x, t) when x = j r ( t ) - j --+0. If, for example
(25)
p.(x, t) ... x ~'-1,
where v = 1, 2 . . . . (this value o f Pv is indicated by a preliminary study 11) of the character of the branchings in t h e j plane), then we have when ~ -+ m c~(¢, t) ,.~
c•(t)
(2~)~,--' where c" are certain constants. In this case the "complicated" asymptotic value of the elastic scattering amplitude is of a form similar to the case of accumulation of poles (fig. 10): A(2 ~ 2 ) = £
c'(t)
[s'~
(2¢)2(,_1) kin2/
J~(') ,
(26)
or in the region o f small x z = - t oo
m2 A(2 ~ 2) ~ £ s
Cry0
(27)
e-j'°~22~
~=o (2~) ~"-1~
"
When eq. (26) was written it was assumed that the position of the vth branching point in fig. 12 is given by eq. (19), where j" decreases with the growth of v (probably 11) as j~/(v+l)). The asymptotic behaviour of the sum (26)when ~ ~ oo depends essentially on the coefficients c"°. When ~ --+ oo and x 2 # 0, large values of v, the larger as the quantity x 2 becomes larger, are essential in the sum (26). In this case, similarly to eq. (21), the sum over v can be written as an integral over x = 1 - j , . However, a further study of the complicated asymptotic value (26) of the elastic scattering amplitude A (2 ~ 2) is beyond the scope of this paper. What are the asymptotic values of particle production amplitudes of types (16) and (17) in the case of "complicated" singularities o f the form of fig. 12 in the j plane? The asymptotic value o f fig. 1 in the limit (3) corresponding to the contribution of branching points j = j~(t) in the j plane and l ' = j¢(sz)in the l' plane is, evidently, given by a value analogous to eq. (22)
( Sbc)Jv"xc)( Sab)jr(tea)
A,¢(3 +-- 2) = c,,,(¢b~, ¢,b, X¢, X~) ~ i where t =--t¢a,
2
s1 =
~
2
--tCc,
O3 c.., ~ - ~ f f p . , ( ¢ , x ,
0
xo,x.)e-~"¢'~+~¢b°'dxdx '.
(28)
606
K.A..TER-MARTIROSYAN
Here pv. ~. is the double discontinuity of the partial amplitudes ?~, ~. of the variables x = j , ( t ) - j , x' = j ¢ ( s l ) - l ' (in the region x > 0; x' > 0). If the character of branching points is given by eq. (25) the limiting form of p,,, when x ~ 0 and x' ---, 0 is Pvv" '~ x 2 V - l x t 2 v ' - l ,
while Cvv' ~'~
(29)
Jc2v,~2v' :~ ~ab bbc
where cv,¢(ror.) is a function the form of which is not given by the unitarity conditions used above in the Sx and t channels. I f the singularities in the j and l' planes are condensations of branching points of fig. 12 the asymptotic value of the amplitude A (3 ~- 2) is given by the sum of the contributions (28) and (29) from all singularities h ( 3 4-- 2) =
~ cv'"(/Ce/Ca) [ sbc'~jv'0ce)[ Sab~jv(t:a), v, v'=0 ~ab bbe
(30)
or for small transverse momenta in the approximation (19) 1 A(3 +-- 2) S
i
o e - u " ~°2¢'°+J"~"¢"1 cv'"
$:2v,~2v'
v, v'=0 bab~bc
)
(31)
where c~,°¢ ,~, (m2+r2)/m4c~,¢(O, 0), and just as in eq. (27), jr" = j~/(v+ 1), where j~ = [djo(t)/dt]t=o (the value v = 0 or v' = 0 refers in sum (31) to the contribution from the vacuum polejo(t) in fig. 12). Asymptotic values (26) and (31) can in a sense be regarded as sums of the contributions from graphs of the form of fig. 11 or fig. 3 with reggeons of different types (v or v'), where the quantity
mV corresponds to the reggion line (of the "type" v) on the diagram 2¢ = ln(s/mZ), and the function ).v(2~) depends on the character of branching at the p o i n t j = j,(x). Eqs. (24) and (29) were written under the assumption that 2,(~) = 1/2~2". It is clear that the asymptotic values of the amplitudes of more complicated processes corresponding to the graphs of figs. 6--9 can be written quite similarly using the universal function 2~(O. 6. Conclusion
The calculation of the cross sections for different inelastic processes with the aid of the asymptotic values of the amplitudes of forms (16) and (17) has shown 7,12)
ESSENTIALLY INELASTIC "COLLISIONS
607
that in all cases all the powers o f the transverse components o f the m o m e n t a of the particles produced (increasing with primary energy) are cancelled out and the cross sections prove to be dependent only on the transverse components of the m o m e n t a o f the particles and on the logarithms of their longitudinal components. A rather sensible picture o f inelastic processes arises on the whole. Further calculations, however, have shown ta) that the unitarity condition in the s channel, and in particular, the condition o f the equality of the sum of the cross sections for all inelastic processes to the imaginary part of the amplitude .4 (2 .-- 2) for zero angle scattering, cannot be satisfied ta) if it is assumed that the vertex parts (15) ?(sl, t, cos 40) -= ?(wa, ~:c) corresponding to the emission of a particle by a reggeon do not vanish at ra = r c --- 0. Are there any reasons to assume that the vertex ~(~a, ~c) m a y vanish at Ka --. 0 and i¢c ~ 07 At present it is difficult to answer this question. In the region j - - , j o ( t ) and 1' ~ Jo (Sl), essential for the high-energy behaviour, the partial amplitude Zj; r a, (sl, t) was written according to (12) and formula (25) of the previous paper 1) in the form (~7[)2~(j; I,A,(Sl, t) =
u(st)iwz'(st' t)u(t) ( j - j o ( t))( l'-jo(sl)) "
For the asymptotic behaviour of the amplitude, the value (13) was obtained. Its vertex part ?(x,, ~ ) proves, according to (15), to be proportional to the value
x/st, two, (st, t) at t ~ 0 and st "-" 0; ~(7¢a, 7¢c) = y(St, t, qbo) "" x/st, t w~,(St, t). All other factors in (15) at Sl --. 0 and t --. 0 when jo(t) --. 1 have finite non-zero values. The point st = 0 or t 1 = 0 is not a singular one for the amplitude A (2 -~ 3); therefore, it cannot, in particular, be a branching point of the asymptotic behaviour of the amplitude. Therefore, in the region of small s t and t the function w~(s 1 , t) must be proportional either to 2) 1/x/st ' t or to the positive odd power x/s-~. In other words, at r~ --. 0 and r~ --. 0 ~
0%
=
where k = 0 in the first case, k = 1, 2 , . . . in the second case and ?o is a certain coefficient. The assumption that k = 0 seems to be the most natural, since in the opposite case (k = 1, 2 . . . . ) the contribution f r o m the total cross section of essentially inelastic processes proves to be logarithmically small at ~ = l n ( x / s / m ) ~ oo compared t with the contribution f r o m quasi-elastic processes where the same number o f particles is produced but in the f o r m of two showers. t At k = 1 the essentially inelastic part of the ~ than the two-shower quasi-vlastic part ~2).
total cross
section proves to he smaller by the factor
608
K.A. TER-MARTIROSYAN
At k - 0 the essentially inelastic contribution proves on the contrary to be large; the total cross section an of the production of n particles or n showers proves to be proportional ~2. t3) to the value ~, ~ ( , - 2) (in ¢)"- ~
It is assumed that in all cases only vacuum Regge poles give finite contributions. Thus, in this case the total cross sections of the production, e.g. of three, four, five. . . . particles decrease with the growth o f the energy slower than the total cross section of the elastic scattering. However, at k = 0 the total cross section ~ , a , of the production of any number o f particles proves, as is pointed out above, to be increasing with increasing ~ due to the production of a number of particles of the order n - ~ which is still very small compared with the number N = x/s/m of the particles whose production is allowed by the conservation laws. This growth of the total cross section at ~ --} oo results perhaps from the fact that the asymptotic behaviour of types (16) (17) etc. here obtained is incorrect in the region where n, the number o f particles (or of showers, as in the fig. 9) is large, when n ~ ~. The above investigation practically referred to the case n = 3 or to the case when n is any fixed number not increasing at ~ ~ oo.
Note added in proof." At very small sl and t the cosines z and z'c become small (for any large fixed Sab and Sb~ in eq. (6)) and the Regge type formulae like eq. (13) are not valid at all. Therefore, the general assumptionmade above on the function w~(s,, t) proves to be incorrect, i.e., w~(sl, t) may be constant and not ,,/s,, t or 1/x/s,, t as s, ~ 0 and t ~ 0. Taking this into account and computing ~, (just as in refs. 7, ,2,13)) with 7(ra, re) in the form y ~ Xa XoyO, where ~'o = const., we obtain 0% -~ cn/~n; this value of tT, does not give the difficulty mentioned above, i.e. does not lead to the growth of the total cross sections a = ~n an at ~ -} o0. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)
A. M. Popova and K. A. Ter-Martirosyan, Nuclear Physics 60 (1964) 107 K. A. Ter-Martirosyan, Proc. Int. Conf. on High Energy Physics, CERN (1962) p. 556 K. A. Ter-Martirosyan, JETP 44 (1963) 341 D. Amati, S. Fubini, A. Stanghellini and M. Tonin, Nuovo Cim. 22 (1961) 569; D. Amati, S. Fubini and A. Stanghellini, Phys. Rev. Lett. 1 (1962) 29 S. C. Frautshi, in the press V. N. Gribov and I. T. Dyatlov, JETP 15 (1962) 140 I. G. Ivanter, A. M. Popova and K. A. Ter-Martirosyan, JETP 46 (1964) 568 Y. A. Simonov and K. A. Ter-Martirosyan, Nuclear Physics 66 (1965) 641 I. G. Halliday and J. C. Polkinghorne, preprint K. J. Foley et al., Phys. Rev. Lett. 10 (1963) 376 S. Mandelstam, Nuovo Cim. 30 (1963) 1113 I. A. Verdiyev, A. M. Popova and K. A. Ter.Martirosyan, JETP 46 (1964) 1295 I. A. Verdiyev, et aL, JETP 46 (1964) 1700