Volume 50B, number 3 PHYSICS LETTERS 10 June 1974

BINARY-COLLISION T H E O R Y F O R M U L T I H A D R O N

P R O D U C T I O N A M P L I T U D E S *

R.C. ARNOLD High Energy Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA

Received 9 Mai 1974

A model for hadron production amplitudes is proposed which includes dynamical correlations, determined from two-body reactions, between all pairs of particles. Qualitative results are obtained accounting for many of the charac- teristic features recently seen in high energy collisions, including: large transverse momentum inclusive cross-sections, broad multiplicity distributions, and strong clustering effects in rapidity.

In experiments on multiparticle production above 100 GeV several qualitative features appear which are not evident in lower energy data, nor can they be ac- counted for in uncorrelated statistical models. These include:

(a) Multiplicity distributions increasingly broad compared to Poisson distributions [1 ];

(b) Inclusive single-particle cross-sections as a func- tion of transverse momenta dropping much less rapidly [2] than a Planck thermal distribution;

(c) Strong two-particle inclusive rapidity correlation effects of a dynamical origin in the central rapidity plateau [3].

Specific physical models have been proposed which explain each of these features, but not necessarily in a consistent fashion, and usually not in a way which has a natural and smooth - let alone a quantitative - ' connection with lower energy, few-body reactions. Multiplicity distribution broadening might result from a two-component [4] (diffractive plus multipheral) model, but we do not find the diffractive energy beha- viour expected in low multiplicities. Large transverse momentum components might be expected in models containing point-like objects (partons) within hadrons [5], but a mystery remains concerning the eternal con- finement of such objects. Strong rapidity clustering might be the result [6] of independent emission of "clusters" of several GeV in mass, but the structure and mechanism for producing these heavy objects re- mains unexplained, particularly since the heavier stable

* Work performed under the auspices of the U.S. Atomic Energy Commission.

hadrons' individual cross-sections seem very suppressed relative to light particles.

All these features could conceivably come out of a future complete theory of strong interactions, e.g. a realistic dual-resonance model including all orders of loops; but since such a theory does not exist, we are still forced to rely on a phenomenological viewpoint, if we hope to relate our new knowledge of multipar- ticle phenomena to (tentatively) established proper- ties of two-body collisions such as Regge exchanges, resonance formation, and threshold behavior.

Binary-collision mode l We propose an s-matrix theory of multiparticle amplitudes based on: (a) know- ledge of two-particle scattering and reaction ampli: tudes, (b) cluster decomposition in rapidity (factori- zation of leading exclusive singularities), and (c) mini- mal complexity consistent with a qualitatively correct phenomenology. This model has been motivated by extensive study of the analogy between fluids in ther- mal equilibrium and density distributions of hadrons in the final state from high energy collisions [7], and by multipheral [8] and Eikonal [9] models for pro- duction amplitudes.

In this letter we will obtain only illustrative, quali- tative predictions. Further publications will contain detailed calculations concerning multiparticle observ- ables, including quantum number and spin effects, questions of diffractive excitation, and asymptotic be- havior of elastic scattering (which is related to critical behavior in the analog fluid).

For an inelastic collision a + b ~ n final state parti- cles we first write the collision amplitude A n in terms of rapidities Yi and transverse momenta k i, and con-

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Volume 50B, number 3 PHYSICS LETTERS ! 0 June 1974

sider the two-dimensional Fourier transform defined by:

An (s ;Yl, bl ;Y2' b2 ; ""Yn, bn) = fd2kl ... d2kn (1)

X exp (i ~ k ] " b]) "A n (s ;Yl, kl ;"" Yn, bn)" 1

The variables b i are essentially impact parameters for the secondary particles, as discussed by Henyey [10l.

Now the fundamental assumption in our model, which we may call the binary-collision amplitude as- sumption (after Lee and Yang [11 ], who used a similar Ansatz fruitfully in the systematic development of ap- proximations for wave functions of imperfect Bose fluids), is:

~n (s ;Yl, bl ; ' " Yn, bn) (2)

= Ao(S ) " ~.1 F(bj)'.i~<i exp[u(y i - yj, b , - b])],

where u ~ 0 faster than ly i -y / I - I whenever lYi-Y]I-* oo, and F(b) ~ 0 for I b I ~ oo. These conditions guaran- tee factorizability of leading J-plane singularities. The Ansatz (2) exhibits a "zero-particle" factor A 0, a pro- duct of one-particle terms F, and finally a double pro- duct connecting all particle pairs.

If u = 0, we would have an independent emission model. Alternatively, if F is independent of b while exp u is zero unless nearest-neighbors in rapidity are connected, we would obtain a multipheral model.

Such assumptions here are replaced by the condition that the 2 ~ 2 inelastic amplitude A 2 is also o f the form {2). This enables us to determine A 0, F (modulo a con- stant), and u, given complete two-body amplitudes at all energies.

Since amplitude analyses based on experimental data are not available at all energies, we must now proceed by applying phenomenological theories of two-body reactions to the prediction of multiparticle phenomena via relation (2).

Basic" amplitude features and their consequences. As s ~ 0% for fixed b, u (Ins, b) ~ 0, and our assumptions imply that A 2 becomes asymptotically a product of s- dependent and t-dependent functions, except possibly in a neighborhood of t = 0. Thus A 2 has a fixed J-plane singularity dominant for s ~ 0% for t ~ 0.

This behavior is consistent with the predicted asymp- totic behavior of all multiple-scattering or absorptive-

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Regge models; the locus of leading singularities (accu- mulation of cuts) is a horizontal line in the J-plane located at J = ~(0), where t~(t) is the trajectory of the leading Regge pole exchange.

It is further known that F2(b) is the same as the elastic scattering amplitude profile in such models, as in the "droplet model" [12]. The value of t~(0) we will assume here corresponds to 19 or 60 exchange; a(0) ~ 1/2.

At finite energies u will in general give strong correc- tions to the asymptotic fixed-singularity behavior. It will be a good approximation at medium energies where shrinkage is prominent, in absorptive-Regge models, to simply take exp u as a Regge pole. Note the represen- tation (2) is opposite to the appearance of an absorp- tive-correction formula; F is the asymptotic behavior, not the "Born approximation" Regge pole; while exp u is (roughly) the Fourier transform of a Regge pole, considered as a finite-energy correction.

We expect then, for Ay >> 1,

exp [u(Ay, b)] ~ exp [(a - b2)/a'Ay] , (3)

where a > 0, since two-body cross sections (at t ~ 0 ) shrink down to their asymptotic limit. This approxi- mation is not to be considered precise as Ay ~ 0% since u must decrease somewhat faster.

In more realistic treatments of two-body inelastic amplitudes, u will contain (in addition to Regge poles) one-pion exchange terms; these will then occur multi- plicatively with F(b).

Note the behavior of u given by (3) is attractive for small impact parameter and long range in Ay (not ex- ponentially damped). Thus we immediately expect multiparticle inclusive observables based on relation (2) will exhibit strong collective multiparticle effects, e.g. critical behavior, as seen in corresponding fluid theories. This is the origin of the large fluctuations seen in multiplicities [13].

At low energies (Ay <~ 1) the function exp u will contain threshold effects; u will then have a "repulsive core". For moderate Ay ~ 2 we expect resonance peaks followed by a smooth decrease with Ay as in the two- body cross-section. These latter properties will be inhe- rited by the small-rapidity region of the inclusive corre- lation functions. Thus we expect clustering at relatively small rapidity intervals, but the clusters will involve many particles on the average. If u has a long range, as suggested by (3), the average cluster size will increase slowly with increasing collision energy.

Volume 50B, number 3 PHYSICS LETTERS 10 June 1974

Note the qualitative features of u are just the same as those of inter-atomic potentials between complex atoms. Thus, in many-particle reactions, hadron confi- gurations typically will resemble complex molecules [14] in the appropriate variables.

In impact parameter variables, clustering will also be present, since u is most positice for Ab =0 for a wide range of Ay. This will lead to final states at very high energies which are grouped into roughly point-like clusters in b space which are very heavy (containing many particles). Thus, spontaneous fluctuations will resemble naive "par ton" models in some senses.

Now consider large transverse momentum outgoing particles in multiparticle final states. If multiplicity is low, scattering to high p± is rare, since F provides a strong damping as seen in elastic scattering at very high energies. However, if multiplicity is high, many factors of u can come into play, and scattering t o large p± can take place by transferring a number of small Pi values to a set of several other final state hadrons. This means high multiplicities are associated with any large p± events.

Since the probability of fin~ling large multiplicity grows steadily with collision energy, the large Pi cross- sections (and inverse slopes) should increase steadily with increasing collision energy. Semi-quantitative es- timates of this effect can be obtained using relation (3) together with a self-consistent-field approximation [ 15 ] for single particle inclusive densities. It is found that the slope at p± >~ 5 GeV can decrease a factor of 2 be- tween 30 GeV and 1500 GeV lab equivalent momenta, which is the right order of magnitude [2] for this effect.

At low energies, the expression (2) can be used to. gether with 2 ~ 2 unitarity to develop N/D equations and a generalized isobar model for production ampli- tudes. This should allow a bootstrap program to pro- ceed further than possible previously, although the am- plitudes we propose are not manifestly crossing-symme- tric. Since such an approach is known to yield internal symmetry predictions [16], we have some reason to believe that the binary-collision Ansatz (2) may provide the necessary missing link for a comprehensive theory of strong interactions based on S-matrix principles. As we have remarked above, the high energy form of such a theory will necessarily strongly resemble the parton picture, as critical (cluster) fluctuations become increa- singly prominent.

References

[1] G. Charlton et al., Phys. Rev. Lett. 29 (1972) 515; F.T. Dao et al., ibid. 29 (1972) 1627; C. Bromberg et al., ibid 31 (1973) 1563.

[2] D.C. Carey et al., Phys. Rev. Lett. 32 (1974) 24; J.W. Cronin et al., Phys. Rev. Lett. 31 (1973) 1426; M. Banner et al., Phys. Lett. 44B (1973) 537; F.W. Biisser et al., Phys. Lett. 46B (1973) 471; B. Alper et al., Phys. Lett. 44B (1973) 521,527.

[3] R. Singer et al., Argonne preprint ANL/HEP 7368 (to be published in PRL); G. Belletini et al., presentation at 5 th Intern. Conf. on High energy collisions, Stony Brook, 1973.

[4] W.R. Frazer, D.R. Snider and C.-I. Tan, Phys. Rev. D8, (1973) 3180; J.S. Ball and F. Zachariasen, Phys. Lett. 41B (1972) 525.

[5] R. Blankenbecler, S.J. Brodsky, and J.F. Gunion, Phys. Lett. 42B (1972) 461; S.M. Berman, J.D. Bjorken, and J.B. Kogut, Phys. Rev. D4 (1971) 3388.

[6] E.L. Berger and G.C. Fox, Phys. Lett. 47B (1963) 162. [7] R.C. Arnold and G.H. Thomas, Phys. Lett. 47B (1973)

371; G.H. Thomas, Phys. Rev. D8 (1973) 3042; R.C. Arnold, S. Fenster, and G.H. Thomas, Phys. Rev. D8 (1973) 3138.

[8] D. Horn and F. Zaehariasen, High energy collisions; (Benjamin, N.Y. 1973). The present developments have involved including final-state interactions; cf. R.C. Arnold and J.S. Steinhoff, Phys. Lett. 45B (1973) 14.

[9] R. Aviv, R.L. Sugar and R. Blankenbecler, Phys. Rev. D5 (1972) 3252; S. Auerbach, R. Aviv, R. Sugar and R. Blankenbecler, Phys. Rev. D6 (1972) 2216; S.-J. Chang and T.-M. Yah, Phys. Rev. D4 (1971) 537.

[10] F.S. Henyey, Phys. Lett. 45B (1973) 363. [11] T.D. Lee and C.N. Yang, Phys. Rev. 105 (1957) 1119;

Phys. Rev. 113 (1959) 1165. [12] N. Byers and C.N. Yang, Phys. Rev. 142 (1966) 976. [13] M. Kac, G.E. Uhlenbeck and P.C. Hemmer, J. Math. Phys.

4 (1963) 216. For relevance to particle physics cf. refs. (7) above, and D. Sivers and G.H. Thomas, Phys. Rev. D (January 1974).

[14] J.D. Bjorken, in Particles and fields, AlP (Proc. 1972 DPF Meeting at Rochester, New York).

[15] cf. J.K. Percus, The pair distribution function in classical statistical mechanics, in The equilibrium theory of classi- cal fluids (Benjamin, 1964) esp. Appendix D. We utilize an appropriate nonuniform density integral equation such as eqns. D10 and D13 of this reference.

[16] E. Abers, F. Zachariasen, and C. Zemach, Phys. Rev. 132 (1963) 1831; R.H. Capps, Phys. Rev. 134 (1964) B460; ibid, 139 (1965) B421.

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