Multipomeron production of showers at high energy

Volume 44B, number 2

PHYSICS LETTERS

16 April 1973

MULTIPOMERON PRODUCTION OF SHOWERS AT HIGH ENERGY * K.A. TER-MARTIROSYAN

lnstttute for Theoretical and Expertmental Physics, Moscow, USSR Received 27 February 1973 Cross section of the multipomeron producnon of heavy showers ~s calculated and found at very high energy in the form O),B = [1-(So/S)'r] ot~. It is shown that the mulUpenpheral part of the total cross sectmn decreases at s --, 0-: a~l~ = ot~ - O'AB = (So/S)3' .ot~ where the constant "r can be obtained from ISR data on proton spectra. It is found to be numerically very small "r ~ 10 -2 • It follows that at accessible energies the value of tr)~B ~ (~lns/So)'Ot~l ~ must be small, logarithmically increasing with energy. This predlctmn can be checked experimentally on new accelerators. t

Below we calculate the total cross section OAB of two and more showers production at s ~ oo. Its value is compared with the remaining part of the total cross section o ~ = o ~ - a~B corresponding to the multiperipheral particle production mechanism. Both types of processes have different physical nature: at multiperpheral production the density AN/Ay of particles fitting a single interval of their rapidities y = In (p ii/m) is approximate constant (at each inelasuc event, see fig. la). At multipomeron showers production events the particles are devlded into groups with relatwely close "rapidlties" divided on axisy by intervals A i not less than some value ~o = lnL: A i/> ~o (fig. lb). Here L is a timed large number such that at s/s o >~L the Regge approximation is valid. Inside each of the groups AN/Ay is also approximately constant since it is also a multiperpheral shower (we shall also call them "irreducible"). It can be shown that ifs 1 ---P21 and s 2 = p2 are the squares of masses of any showers neighbourmg o n y axis and s12 = (/91 + P2) 2 is their total effective mass square then the condition of applicabihty of muluregge approximation [ 1] is always fulfilled:

SlS 2
(1)

where s o is the value of order o f m 2. The shower consisting of several such groups will be called reducible (or "mulupomeron"). t The total cross section OAB of tWO showers production (see fig. 2b), one of which (the upper in fig. 2b) is irreducible and the other is of any type, includes the Revised version of report submitted to Batavia Conference, USA, 1972.

cases of multipomeron production of any number of irreducible showers (correspondingly, the lower vertex in fig. 2 is represented in the multipomeron form). Let us define it in the form: 0 = ff dsldS 2 f d t l ( d o ' A B / d t l d s l d s 2 ) , (2)

SlS2 < SSo/L

_o.

where* ds I ds 2 dO'd3 = 4rr°(0A)(tl' Sl)rrs--1 °pB(tl' s2) rrs--2X [r/+(ap)[ 2 ~SSo---~ 2(~P(tl)-1)

(3)

\sis2 / is differential cross section of the showers production process shown in fig. 2. Here a p ( t l ) ~ 1 + a ' p ( 0 ) t I , r/+ ( a p ) ~ i is the signature factor and opB(t 1 , s) = f(lVpB(t 1 ;s 2, r2)12/2s2)dr2 is the total cross section of interaction of the pomeron, with the mass x/~l, with the particle B at energy ~ 2 . At large s 2 important m (2) this cross section has the pomeron high energy limiting form apB ~ 7rgBr(t 1),

(4)

where gB Is the particle B to pomeron coupling constant (at t = 0) and * as, by definition, do~B = f(IT'ABI2/2s)drt2 with dr12 = (drldst/2n)(drzds2/2n)(dtl/8ns) the phase space weight of final state and T~,B= 8nso V~°2VpBrl~(~D)(ss,/sls 2)uP(t) the matrix element corresponding to fig. 2b. V ~ , VpB are the contributions of upper and lower vertices m fig. 2b. 179

Volume 44B, number 2

PHYSICS LETTERS

16 April 1973

=

.

/--!,

b

z---q, f--q

e, t

I a,"

a~

a

~:'9, A

r--~..~.a~ ~,,

I

¢

,..- -, Fig. 1. Different types of d~stnbuuon In y = In (pll/m) m each melasnc mulUpartlcle e~'ent: a) mult~penpheral shower b) four showers produced m mult~pomeron process c) bumps (dotted hnes) resulting from parallel productmns of pa~rs of showers at production process of fig. lb.

Fig. 2. a) cross sectmn o ~ of multlperlpheral events b) cross

r(t 1) ~ A o + A 1a ~ ( 0 ) " ( - t 1)

sectmn ~AB of multlpomeron production of heavy showers. The right hand parts of fig. 2 gwe the corresponding imaginary parts of the elastic scattering amphtude.

is the vertex of the pomeron decay into two pomerons [2, 3] determining the cross sections of lncluswe processes. It is well known that here A o = 0, since otherwise for the total inclusive cross section follows an increasing at ~ = In sis o -~ oo value [3] proportional to Aoln ~. Through o(pO)A(t1 , s 1) we denote that part of the total cross section of the pomeron-particle A interaction which is due only to multiperiphal states (i.e. of the type shown m fig. 2a). It differs from the total cross section (of the type of (4)) at large due to multlpomeron showers production effect and has Regge high energy hmltlng form o(0) ~, , Sl) ,~ rrgAr(tl)(So/S)l -ao(O) = pA~.~l

~rgAr(t 1) exp (--7~),

(6)

where the vertlcesg A and r ( t l ) must be (as will be shown) the same as in (4) and 3' = 1 - S o ( 0 ) is some positive non-zero number. Note that it follows from factorization relations that cross sections o(A0{~of the A + B interaction and O(p0t~(t2, t 1 , s) of the pomeronpomeron interaction corresponding only to the multiperipheral intermediate state (fig. 2) have the same high energy hmltlng form as (6):

o(OA{~(S) ~ 8n'gAgB exp(--3'~), 180

o~O~-~ -~r(t 2 )r(q ) exp(-3'~).

(7)

The total cross section of these same interactions (through all the states) do not contain, as (4), the factor (So/S)~ = exp (-3'~): otot A B ~ 87rgAgB,

Opp ~ ~rrr(tl)r(t2).

(8)

To prove that 3' ¢ 0 let us calculate the cross section o)~B making use of (4), (6) and taking into account that in (2) at ~ > > 1 only small t 1 are important. We ob tam: , _tot ~-A I 21 OAB = OAB

f~,., F

exp (-3'~1) d~ld~2

0

×

fexp(2a~,(0)t 1(~ + a - ~ l - ~ 2 ) } ( - a P ( 0 ) t)2 dtl _oo

= ~'AB~t°t.1,r(~) '

(9)

with l~t(~) = A2

8~

~1+~2~--~O

~lfJ~ °

1 (.A 1/4N/r~-Px~2

exp(-3'~l)d~ld~2

(~+a-~l-~2) 2

~-~( ~o~-~ ] (1-exp(~)~,

V0)

Volume 44B, number 2

PHYSICS LETTERS

16 April 1973

t.£ 3~=_oo I

il+

O8 06. OCz.

t t I

2. $ ~ £

t0

i

st [deV9

i

+ i

ii

I

V z

Fig. 3. The pomeron-particle (nucleon) cross sectmn opN(t, s) obtained from experimental data [6, 7] by Kaidalov a ) - t l = 0.01 (GeV/c) 2 , b) - t t = 0.15 (GeV/c) 2 •

where r(tl) was inserted in form (5) with additional factor exp (a'pat 1) introduced for generality and where terms of the order of (to + a)3' were disregarded as we will find 7 to be very small (see below). (The exact value of

(A l l4X/a'P_~2 exp(-Tt) I'r(~) X

= t- ~o Ya -)

apB(tl, ~) -- o(O)B(tl, t) = p

0

4+rf dtl f f -**

t

°(POp)(tl, tl, t l ) OVB(tl ' t2)

~1+~2~-~o

× exp (2t~,t~ (~-~1 -~2)} d~l d~2 •

7

7~o

/~o + a \2 [1 - exp (--h,(t'-2~o)} ]dt~+-+~a__-7~,J

~'=2~o is given by (10) where the expression [1-e -° + oo (e - ° - Vo/(V + %)}+ v2~Uo(O)] exp(-Tto) is substituted for square brackets. Here o = (t - 3~o)% Oo = (to + a)7, ~ > 3t and ~uo(o ) =f~ e °'-u do'/(o'+Oo); • Oo(O) ~ l/v for v > 1 or ~Uo(O) ~ in 0/% for vo < o < 1.) At 7 = 0 the cross section OAB would increase proportlonally to t (it is, in fact, a well known effect found many years ago by Berestetsky and Pomeranchuk [4]. This rising of O~B is not permitted since V ,,tot , + A B _- OAB o(A0~must be constant at ~ + oo. To find 7 let us note that (at large s) the difference opB(t 1 , s) -- o(~(tl, s) is determined by the total cross section a~B(tl, s) of multipomeron production of two and more showers which differs from (2), (3) only by replacing particle A by the pomeron (corresponding a m p l i t u d e VpB are given in fig. 2): ¢

This is a well known equation by Chew, Goldberger and Low [5]. It can easily be shown that at given (in (6), (7)) values of @0B) and o ~ it has a soluUon for opB just of the form (5) with the same residue 7rgBr(tl) that is in @0~ if the condition l.r(~ ) = 1 is fulfilled at + oo. From (10) we thus obtain: (A__I/4~/~PP.~2

7 =~-~o+a

-).

(11)

The same value of'/follows directly from the condition a ~ = o ~ + OAB at ~ -~ oo with values (7)--(10) of all cross sections. Presently known data on proton inclusive spectra obtained on ISR give [6] for cross section opN a value of order (0.1 +- 0.8) mb at It 11 ~ 0.16 (GeV/c) 2. It corresponds to a value A 1 ~ (1.1 + 0 . 8 ) ~ p i.e. to very small 7 ~ 10 -2 (values of opN obtained by Kaidalov from ISR [6] and other [7] data are shown in fig. 3a, b). With value (1 1) of 7 we get from (10): I

t

OAB = { 1 -- exp(--7~)} " °AB-t°t .

(12)

At 7~ < 1, i.e. in all region of available energies O~B is 181

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PHYSICS LETTERS

small and OAB _tot practically coincides with the multiperipheral cross section a ~ ~ O(A0~.Here a~B = Zn=l a)~B (n), where O~B(n ) ~ (1/n!)(7~) n exp(--3,~)o~ is the (n + 1) showers production cross section (it corresponds to distribution of fig. lb with n gaps with A l t> Go). The t cross section of two showers* creation OAB(1) ~-7~" otot AB can be measured more easily and must have at ISR (at ~ ~ 8) the value approximately twice as large as at Serpukhov (where ~ ~ 4). In the region of fantastically large energy, at ~ 1/3' ~ 102 , 1.e. at E ~ 1050 GeV, the situation changes: the multlpenpheral part of the cross section vanishes and the value of o t°t appears to be due to showers t production only cr~°~AB~ ° A B . R e m a r k : So far we have not considered processes of parallel production of pairs of showers giving bumps with double values of densities A N / A y m rapldltles distribution (see fig. lc) just at the place of each gap in fig. lb. As was shown by Abramovsky, Kancheli and Gribov [8], the probability of appearance of such a bump is exactly twice as large as the probability of the gap at the same place (in figs. lb, c). Account of these processes changes slightly our formulae. The summary cross section °AB of obtaining at least one gap or bump is given by (9), (10) with an additional factor 3 in front ofeq. (10); it is equal as (12) to °AB = (1--ex~a(--37~)}o~. Correspondingly, the cross section ~O~ of appearance of distribution of the type of fig. la with no both gaps and bumps wdi be ~A0~ = e x p ( - - 3 7 ~ ) o ~ . Both cross sections O~B(gaps ) of appearance only of gaps (as in * The exact value of it (with terms of all order m vo) at ~>3~o is given by: O~B(1) = o ~ exp (-~,~)f~e-V'o'd(oo/[V+Vo-V '] )2 = oe - v + Uo(U+Uo-1)[e-V-Vo/(O+Oo) ] + Oo2~vo(O), where v, Vo, e~Vo(O) are the same as before.

182

16 April 1973

fig. lb) and O~B(bumps ) of appearance only of bumps turn now to be decreasing at ~ ~ oo. O~B(gaps) = { 1 - - e x p ( - 7 ~ ) } e x p ( - 2 7 ~ ) o ~ , O~B(bumps) = (1--exp(--23,~)}exp(-7~)o~. f

This again gives exactly (12) for the cross section OAB = °AB -- °AB(bumps) of appearance of at least one gap P and any number of bumps: OAB = ( 1 - - e x p ( - - 7 ~ ) } o ~ . So, the account of bumps m fig. ld does not change m essence the results (12). The author is indebted to A.B. Kaldalov and V.N. Gribov for remarks Important for this paper.

References [1] K.A. Ter-Martlrosyan. Nucl. Phys. 68 (1964) 591; I.O. Verdiev et al., Zh. Eksp. i Teor. Fiz. 46 (1964) 1770. [2] C.E. De Tar, D.Z. Freedman and G. Veneziano, Phys. Rev. D4 (1971) 906. [3] V.N. Grlbov and A.A. Migdal. Yad. Fiz. 8 (1968) 1002. [4] V.B. Berestetsky and I.Ya. Pomeranchuk. Zh. Eksp. i Teor Flz. 39 (1960) 1078. [5] G.F. Chew, M.L. Goldberger and F.E. Low, Phys. Rev. Lett. 22 (1969) 208; H. Abarbanel et al., Phys Rev. Lett. 26 (1971) 537. [6] D R.O. Morrison, Review Talk at IX Conf., Oxford (1972), F~g. 16 2; M.C Albrow and O P. Barber et al., Reports no 940,941 at Batavia Conf., USA, 1972. [7] E.W. Anderson et al., Phys. Rev. Lett. 16 (1966) 855. [8] V.A. Abramovsky, V.N. Gnbov and O.V. Kancheli. Rep. at Batavia Conf, USA, 1972.