Pomeron decoupling in inclusive multi-Regge models

Nuclear Physics B90 (1975) 307-312 © North-Holland Publishing Company

POMERON DECOUPLING IN INCLUSIVE MULTI-REGGE MODELS* J.R. FREEMAN** Department of Physzcs, Umverstty of Washington, Seattle, Washington 98195 USA Received 30 September 1974

Inclusive charge conservation sum rules and naive duality are used to constram the parameters of a mtdtl-Regge-Mueller model wath faetonzable J-plane smgulantles mcludtng a leading pomeron trajectory (of intercept not necessarily one) and a countable number of secondary trajectories. Consistency requires the vanlstung of the central couphng of particle ~r+ to two pomeron trajectories, that is, for single parUcle ~r+ spectra, the rapidity plateau must dlsappearvThis unpalatable result may be a warumg not to push overslmplLfied models too far, nevertheless, we show m two examples, a possible way of avoiding the decouplmg through modification of the pomeron propagator to vamsh for small rapldtty separations

A natural first stage m the a t t e m p t to unravel the mysteries o f multiparticle hadronic interactions has been to test various s~mple models o f mcluslve reactions against the growing collection o f experimental data. One such model, the ReggeMueller [1] model, with factorlzable J-plane poles has served as a useful guide for reclusive phenomenology [2] But it does have hrmtatlons since lncluswe spectra do show long-range rapidity correlations [3] which are absent m the factonzable ReggeMueller model. To disentangle the long- and short-range correlations, it is popular to adopt the view that there are two dastlnct mechanisms [4] o f particle production a long-range diffractive component and a short-range multlperlpheral mechamsm. I f these components are indeed distmct, then mcluswe conservation sum rules [5] may be applied separately to each. The purpose o f this note is to point out that a number o f assumptions that are c o m m o n in the inclusive multl-Regge literature and that do have some phenomenologlcal support are in fact Incompatible with mcluswe charge conservation The apparent lesson impled by this result is not to overextend slmphfied models b e y o n d their phenomenologlcal limitations. The assumptions we make are hsted below together with appropriate comments: (1) The short-range component o f inclusive rapidity spectra m a y be described b y a Regge-Mueller model wath factorlzable poles consisting of a leading singularity, the pomeron o f Intercept ap (not necessartly equal to one) and a countable number of * Supported by the US Atomic Energy Commission ** Present address Department of Nuclear Physics, Weizmann Institute of Science, Rehovot, Israel.

308

J R Freeman/Porneron decouphng 0

Z' J

-6

0

I!

-6 ,j

d

i t!

b

-d

b

Fig 1 Manyqeg Regge-Mucller graph TraJectories Y are non-pomeron trajectories, I', I" are arb:trary secondary trajectories J o f intercept o~j < ap. Phenomenologlcal analyses are common in whxch the pomeron is gwen as unit intercept and the secondary trajectories are restricted to the leading meson trajectories of intercept one-half. (i:) In mtegratmg reclusive rapidity spectra over a domain away from the kinematic boundaries, the multl-Regge vers:on of the Mueller model may be used, even for small rapldaty separations. This is m keeping with the observed two-particle correlations m the central region at ISR energies [3] which do seem to extublt exponential fall-off with correlation length ~ 2 (m) Harara-Freund duality [6] may be extended to many leg Mueller gra__phsm the sense that the sum of graphs shown m fig. 1 vanishes when the time-like (cd) channel has exotic quantum numbers Nothing need be assumed about the "space-hke" channels at both ends of the chain Assumption (m) corresponds to the notion that if resonance production is :mpossible m the (c-d) channel, then there must be a corresponding cancellataon among cross channel secondary trajectories. In particular, it suggests there should be no short-range rapidity correlations between negatwe plons m non-diffractive production, a frequent assumption m inclusive models [7]. Of course, as we emphasized earlier, the world is more complicated than pure SRC models. Nevertheless two partitle ( ; r - r r - ) correlations are indeed much smaller than 0r-Tr +) correlations [8] so that in the context of two component models, assumption (lil) may not be unreasonable. Now let us see to where (i), (u) and (m) lead us First we write down the Regge:zed spectra. The total a + b cross sectmn at sufficiently high energles is ~a,~Y(aJ -I ),

•-)-, ~~b~ ,j,jvj~

(1)

J.R Freeman/Pomeron decouphng

309

where Y = ya _ y b as the rapidity separataon of a and b, ~ is the partacle t-traJectory I residue at zero momentum transfer, T1 as the trajectory sagnature For convemence we wall normalize reggeized spectra by the pomeron pmce Op o f ( l ) The smgle particle raplchty spectrum of a + b ~ c + .. m the double Regge domam yac - y a _ y C ~>L a n d y eb ~>L is simply t

t do= ~ ~ e_yaChe_yCbx,~ -- -ov dye x~0 x'~0 I I'

F

b c a 7i, r r h z , F s 7 ? ,

with

,~/--

t3} , ~],

(2) where Y,~ sums over trajectories of common intercept (Otp - X) (thus ~ = 0 corresponds to the pomeron and k > 0 to secondary trajectories); h},/is the central coupling of particle t to the 1' and I trajectones, for example, h~,p gives the height of the rapichty plateau of (2). Smularly the two particle spectrum of a + b ~ t +1 + -. for ~yat,yl,ylb} >~L has the form _

1 do = ~ ~ e_yaZXe_ytlx, e_y]bx,, Op dytdyl ~. k' k" I _

[..

_

I'

]

(3)

I"

The reggelzed spectra may be contralned by equating terms wath the samey c and Y dependence m the mcluswe charge conservation sum rule [5]

dO~QOfdya[1 do

Qc 1 o dyC

d

k o dyC--dyO

1 do d o ] , o2 dyC dyO t

(4)

where Qt is the charge of particle i. Taking ~vac, ycb} 1> L the energy-independent pieces of (4) ymld d _tOChc = ~ a d ~ 2 j ~ t chpjTjhjp PP d ~.>0 ~k

(5)

a result found also by Webber [9]. By combining (5) wath smallar constramts from strangeness and baryon number conservation and restricting the number of traJectories, one might try to solve the conservation equations [9]. Instead, we focus on the addmonal constramt of duahty. Although what dualaty entails for incluswe cross sections as controversial [10,11 ], we wall assume the previously stated analogue of the usual Harari-Freund hypothesas. That is (wath reference to fig 1)

~t c d J h~,yrjhaT,, = 0

for cd exotac.

(6)

To exploat this relatmn, consader the mclusave charge conservation sum rule connectlng the three-particle spectrum of a + b -+ t +/" + d + ... and the two-particle spectrum o f a + b ~ i +] + .... _(QI+Q/)I

do

o dy'dyl

=~]f d"

F1 do dyd [_-°dyidyldyd

1

do

do ] Q d

02 dytdyl d-yd J

'

(7)

JR Freeman/Pomerondecoupling

310

with the restrictions (val,ytl,y lb} >~L In the domain 0 <~yta <<.y~lwe take

1 do = ~ ~ ~ ~ e-yalXle-fldX2e-YdlXae-y/bx4 [ . .], Op dytdy/dya xl h2 ~'3 h4

(8)

wath ssmilar forms m 0 <~yV <<.yzd and 0 <~yd~ <~yO Equatang the constant pieces in (7) that are independent o f y ~,yl and Y leads, of course, to notlung new. For the exp [ - ffyq] terms with ~ > 0 corresponding to a family of common intercept traJectories Z, a straightforward calculation yields i

[

i

t

J

d

_(Qt + Q/) z~ h~zTzh/zp = ~d Qd ~'Z [ x>0 ~ 1X j ~ [hrz'rzhzjrjhjp

1 l d +hrzrzhzphlpp] + hapjrlh)zrzlgzp] __([hpphpzrzhlzp , d +~

X~ ~

1

j~',

d

,

d

[hPdT"YhJzrzhlzP+ hPZ'Czhzj'QhlJP]

}

(9)

X~O There are also other constraints, for example, from the yq e-ytl~ piece of (7) one finds

d

Z

Z' hPzrzhzz";z'hIz'P'

(10)

but let us focus on (9) Summing (9) over ~ > 0 and using (5) lmphes ~' . _(Qt + Qt) ~ h~,z.rzh/zp ~->0 z d

~>0 Z x>O

[h pz r zhlzjr lh jp

d t + hpyrjhJzrzhIzP] = 71 (Qt + Q1)htppl~pp

(11)

Now take i = / = Ir+ and use duality assumption (6). The left-hand side of (11) vanishes. Thus, ~+ = 0 hpp

.

(12)

This rather unpalatable type of pomeron decoupling means that the (short-range) ~r+ spectrum vanishes ill the central region except for non-scalmg terms that die off with increasing Y! To disrmss this predicament one could argue that (i) the extreme multi-Regge assumption is unwarranted; (11) duality is far more complicated [10] than we have assumed; (tis) the idea of a distinct short-range mechanism describable by a Regge-

y R Freeman/Pomerondecouphng

311

MueUer model with factonzable poles but without cuts is overslmphfied and should not be pushed too far. However, m the spirit of the mclusive multt-Regge bootstrap program [ 12], one might try to retain our original assumptions except to modify the pomeron singularity. For example, suppose the pomeron pole propagator exp [ap ytl] is replaced instead by [C + K y zl] exp [ap ytl] where to retain factorlzatlon b o t h K and C are taken to be constants Independent o f / a n d ] The reclusive spectra are easily wntten down. Imposing the conservation sum rule (4), we fred that from terms of ordert (KY), the constraint (5) again follows. Use o f ( 7 ) results m separate constraints for the [Kyayb/y] pieces and the K exp [_yl/~]yayb/y pieces. Straightforward manipulation yields eq. (11) modified by a multlpllcative factor of C on the right-hand side. Nawe duality now reqmres either C = 0

or h~p = 0 .

(13)

So by taking the pomeron propagator to be Lvabeytl~P] the decoupling may be avolded.tt If the corresponding calculations are repeated with the pomeron propagator [C+Ky II + D(yll) 2 ] exp [apyi], the result is again (13). In our examples, the "escape route" from the decouphng result (12) is to modify the pomeron so that it vamshes for small rapi&ty separation but grows with increasing rapidity separation There may be a more general depth to tins posslbihty than our crude examples illustrate. Harari and co-workers [13] have pointed out that for K+p and pp total cross sectaons and ~rN elastic scattering, the pomeron does seem to "disappear" below lab momenta of 1 GeV/c. On the other hand, a modest dose of non-factorlzabillty tends to cripple the utility of conservalaon sum rules, so that a possible moral of our exercises is not to push overslmphfied models [ 14] beyond their phenomenologlcal llmttations. I thank Robert Cahn and Lowell Brown for helpful comments, and Naren Bali for interesting dascusslons.

References [1] A.H. Mueller, Phys Rev. D2 (1970) 2963. [2] D.P Roy, Rutherford report RPP/T/41 (1973). [3] G. BeUettmi, Proceedings of the 5th Int Conf. on high-energy colhslons, Stony Brook, N.Y, 1973. [4] K Wilson, Cornell report CLNS 131 (1970), unpubhshed. [5] C E. De Tar, D.Z. Freedman and G Venezlano, Plays Rev. D4 (1971) 906; B R. Webber, Nucl Phys. B43 (1972) 541 t Note there are non-cancelling L-dependent pieces that are constant m energy. t t For such a propogator, the Mueller correlation parameter f2 is not positive but this poses no problem m two component models

312

J R Freeman/Pomeron decouphng

[6] H. Haran, Phys. Rev. Letters 20 (1968) 1395; P.G.O. Freund, Phys Rev Letters 20 (1968) 235. [7] W.R Frazer, R. Peccel, S Plnsky and C.I Tan, Phys Rev. D7 (1973) 2647 [8] D. Cohen, Report at 4th Int. Symposmm on multl-parUde hadrodynamlcs, Pavia, Italy, 1973. [9] B.R Webber, Argonne preprlnt ANL/HEP 7348. [10] S-H.H Tye and G Veneziano, Plays Letters 38B (1972) 30, M. Emhorn, M.B. Green and M A. Vtrasoro, Phys. Letters 37B (1972) 292 [11] J.R. Freemen and C Qulgg, Phys Letters 47B (1973) 39; J R. Freemen, Nucl. Phys. B71 (1974) 301, Chan Hong-Mo, H.I. Mtettmen, D P. Roy and P. Hoyer, Phys. Letters 40B (1972) 406 [12] M.B. Green and B R Webber, Cambridge preprmt HEP 73/4; W A Bardean and R D Peccei, Phys. Letters 45B (1973) 353 [13] H. Haran and Y. Zarml, Phys Rev 187 (1969) 2230, H. Haran and E Rablnowci, Weizmann preprmt WIS 73/41 Ph [14] S. Plnsky and G.H. Thomas, Argonne report ANL/HEP 7345.