Volume 43B, number 5 PHYSICS LETTERS 5 March 1973

W H A T CAN BE L E A R N E D F R O M I N C L U S I V E A N T I P R O T O N S P E C T R A ?

R. JENGO, A. KRZYWlCKI and B. PETERSSON Laboratoire de Physique Thdorique et Particules Eldmentaires, Orsay, France and Laboratoire de Physique Thdorique et Hautes Energies *, Orsay, France

Received 17 January 1973

Without free parameters, we explain the raise with energy of the p yield and we obtain a correct description of rapidity distributions at ISR energies. We show that N and N have the tendency to be created within a small rapidity interval as expected, e.g. from the multiperipheral model.

The aim of this work is twofold. First, we want to understand the apparently spectacular raise with energy of the antiproton yield between 20 GeV, say, and the ISR (Intersecting Storage Rings at CERN) energies. Second, we will argue that the antiproton one-particle inclusive spectra at ISR energies reveal very important general features of the multiparticle production dynamics.

Concerning the first problem, we should remark that it has already been discussed by other authors [e.g. 1 ]. We are not fully satisfied with this discussion, however, since it is not clear for us to what extent the results merely reflect the specific model and the par- ticular parametrization used. We believe that the main characteristics of baryon pair produc tion should be understandable from known, gross features of multiparticle production phenomena using fairly general arguments. It is in this spirit that we shall try to find whether the raise of the antiproton yield should be considered as fast or slow, or perhaps just normal, when compared to the observed rate of varia- tion of the pion inclusive spectra.

Our second problem is deeper. We observe, that there is, in fact, rather little empirical evidence for the importance of particle (Reggeon) exchange effects in the structure of genuine nultiparticle amplitudes (we shall not consider diffraction dissociation which is a somewhat separate subject). There are indications that these effects are indeed present. Let us mention the "cornering effect" in three-body production [2] and the relative successes of the B 5 phenomenology [3], some results of the LPS analysis of Van Hove and

* Laboratoire associ6 au CNRS. Postal address: B$timent 211, Universit6 Paris-Sud, 91-Orsay, France.

coworkers [4] and the analogies between the energy behaviour of partial cross sections for many-body and two-body processes [5]. Depending on one's belief and taste one can add a few items to this list, but it would not get long anyhow. It is true that the most general predictions of the multiperipheral models, like scaling, are (or seem to be) confirmed by the data. But these general predictions have been deduced from other models too.

We further observe, that even at highest ISR ener- gies one produces, on the average, no more than about 0.2 F/event [6]. In this situation it is presum- ably pointless to interpret antiproton spectra using an asymptotic formalism. Our idea is precisely to take advantage o f the scarcity o f pair production to test the validity o f the exchange aspects o f the multi- particle production.

Within the context of the multiperipheral model, using standard arguments and assuming that no more than one Niq pair is produced one gets

d2o/(dqN) (dq~) = (M2) - ' r (1)

where (dq) = d3q/E and M denotes the mass of the pair (to simplify writing we omitted factors weakly depending onM in eq. (1)). The part of the ladder responsible for the strongM-dependence in eq. (1) (baryon exchanges) is essentially the same as the one encountered in the calculation of the total NN anni- hilation cross section. Thus one expects 3' = 1 - a~o(0 ) ~ ~-. Strictly speaking, eq. (1) is valid at large values of M; whenM is not very far from thresh- old a somewhat larger "effective" 3' can be expected.

Needless to say that at asymptotic energies where it becomes probable to produce several NN pairs in a single event, eq. (1) breaks down. Well known argu-

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ments can be used to predict that d2o/(dqN) (dq~) becomes independent of M for large values of the NN mass.

Since M 2 ~ exp ( l y ( N ) - y ( N ) I ) , the question we ask can be reformulated as follows *: is it true or not that a NN pair is necessarily created within a small rapidity interval? Stated in this manner our problem becomes more general than might be inferred from the preceding argument based on the multiperipheral model.

We shall first work with a "statistical" model which does not involve any constraints on M, except those due to kinematics and kinematic reflections (e.g. of the presence of leading particles) which hold also for a pion pair. Then, we shall check how the introduc- tion of a cut-off on M influences the results. The essence of our "statistical" model can be summarized in the following very simple heuristic assumption: the probability of producing a pp pair is proportional to the probability of producing a 7r+ rr - pair with the same total four-momentum. Thus

d2o d2o (dqp)(dq~) o¢ (dk+)(dk_) (2)

k+ +k_=qp+q~

Since the equation k+ + k_ = qp + q ~ does not deter- mine the four-momenta k+_ uniquely we add supple- mentary conditions on transverse momenta

T k T = qT (3) k T = qp ,

and on rapidities

sgn Lv0r +) - y 0 r - ) ] = sgn [ y ( p ) - y ( p - ) ] . (4)

Obviously, eqs. (3) and (4) ensure that the two pairs are kinematically as similar as possible.

Eq. (2) can be derived within the framework of the independent emission model. We do not attach much importance to this derivation since eq. (2) is likely to be of more general validity. However, we quote the argument below since it helps to better understand the meaning of this equation. Using the simplest ver- sion of the independent emission model we write the inclusive cross section for p + p ~ 27r + X as

* y (A) denotes in this paper the longitudinal rapidity of particle A.

d2° - f (k+T)f(k T ) F ( k + + k ,P, Q) (5) (dk+) (dk_) - -

where

F(k+ + k_,P, Q) n+2

n n : ~ • j=l

} d3p 1 d3pj d3pn+2

The factors corresponding to the leading particles are explicitly exhibited; P and Q denote the incident four-momenta. To simplify notation we treat pions in the above formulae as identical; in fact we mean a cross section for a 7r+Tr - pair and a summation over all charge configurations. Baryon pair production, considered as a second order effect, is neglected. An expression closely analogous to eq. (5) can be written for a p~ pair. Eliminating the common phase space integral F one gets eq. (2).

Of course, eq. (2) does not involve any specific dynamical factor restricting the rapidity separation between p and ~. In an exchange model one expects to have (cf. eq. (1)), instead ofeq . (2):

d20 F(qp + q~)2 ] - ' r d20

(dqp)(dq~-) oc L" ~ J ( d g + ) ( d k ) k++k=qp+q~

(7) A few remarks are in order: Our goal is a semi-quantitative understanding of the

data and not a fit to these data, which are not very precise yet. In this spirit, we find it sufficient to adopt the simplest assumption, namely that the proportio- nality factor in eqs. (2) and (7) is just a constant• This means that we neglect the observed difference between pion and baryon transverse distributions (we have nothing to say about these distributions anyhow)• We focus our attention on antiproton rapidity distribu- tions and we have checked that the shapes and energy dependence of these distributions are not sensitive to the introduction of a transverse momentum dependent corrective factor in eqs. (2) and (7)•

At ISR energies the ~ rapidity distributions depend rather weakly on the details of the input pion spectra• What really determines the shape and energy variation of the g spectra is the fact that pion distributions

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Volume 43B, number 5 PHYSICS LETTERS 5 March 1973

0

o.1

" 0 l a J

0.01

B

P,=O

2 3 . 3 ~"

I

:/ I I 1.0 2.0 3.0 4.0

Ymax- Y

Fig. 1. The invariant inclusive cross section for antiprotons at fixed pT = 0.4 GeV/c, as a function ofYma x - Ycm(P-) where Ymax = In x/~/mN • The data are from a compilation by Liilethun [6] : x/~-(GeV) 23.3 30.5 44.7 53.0 Ref. [91 o ~ • • Ref. [10] o • Ref. [11] • • • • The dotted curve represents data at x ~ -= 7 GeV [ 12]. The full lines are the predictions of our "exchange" model, with 3" = ½ at 7 GeV, 23.3 GeV and 53 GeV.

develop at these energies a rapidiy plateau whose length increases logarithmically with energy.

The ~ rapidity distributions (at fixed transverse momentum) are obtained by integrating eq. (2) or (7) with respect to qp (remember that we neglect production of several pairs in a single event). Notice that (k+ + k_) 2 > 4m~ implies that the two pions are separated by about three units in rapidity. Conse- quently, we use the approximation

d2o 1 do do - - (8)

(dk+)(dg_) Oinel (rig+) (Ok_)

which is good at ISR energies and which provides a rough estimate at 20 GeV.

We use the empirical formulae for pion inclusive spectra in the 20 GeV energy range and at ISR energies given respectively by BC)ggild et al. [7] and Damgaard and Hansen [8].

Typical results for the "exchange" model (eq. (7)

>~ o

~ . o.T

~o - 0

0.01

1 m

m

m

m

m

/

/

,i 1.0

I I 2.0 3.0

Ymax - Y

m

I- 4.0

Fig. 2. The same as in fig. 1, but the theoretical curves refer to the "statistical" model (3' = 0).

with Y = ~) together with the experimental points from ref. [6] are shown in fig. 1. The results of an analogous calculation with the "statistical" model (eq. (2)) are shown in fig. 2. The curves are normal- ized to the average of the x = 0 points at x/~ = 53 GeV (the proportionality constant in eqs. (2) and (7) is of the order of (m~r/mN)2). Our comments and conclu- sions are summarized below:

i) Using the pion spectra as input data the "exchange" model describes correctly and without free parameters the shape and energy variation of the antiproton rapidity spectra at ISR energies. It is definitely better than the "statistical" model, which systematically gives too broad rapidity distributions. Furthermore the "statistical" model predicts a signifi- cantly too rapid raise with energy of the antiproton yield in the fragmentation region.

ii) Our results at low energies should be regarded as a rough estimate only, since our assumptions at 20 GeV imply rather far going idealizations. We notice, however, that we obtain a correct magnitude for the ~ yield at these energies. Therefore, the "exchange"model, with 3' = ½ (again without free parameters), explains the raise of the antiproton yield in the incident laboratory energy interval between 20 GeV and 2000 GeV. The "statistical" model also

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gives a correct order of magnitude for the increase of the ~ yield (in fact, it leads to a somewhat too rapid raise). This means that the rate of increase with energy of the probabil i ty to produce a NN pair is essentially the same as the corresponding rate for pion pairs with the same kinematic characteristics (i.e. the same total four-momentum and individual transverse momenta). In other words, we identify two effects: First, there is a threshold effect which is responsible for the growth with energy of (n~) and which is in- corporated in both models. Second, there is a more specific dynamical effect, an energy independent cut-off on M, which is particularly felt at ISR ener- gies since at 20 GeV large M is anyhow forbidden.

- - 1 iii) We present results corresponding to 7 - ~-, suggested by the multiperipheral arguments. However, as we already mentioned, there is some lati tude in this prediction. A somewhat larger value o fT , e.g. 7 = 1 gives a better fit to the data. Because of large errors in the present data, we have not a t tempted any best fit to determine 7, we feel this would be prema- ture. Better data in the rapidity interval Ymax - Y = 1 to 2 would be helpful to pin down the dynamical content of baryon pair production, since this part of the spectrum is most sensitive to the value of 7.

iv) The superiority of the "exchange" model means that p and ~ have a definite tendency to "keep together". This can be interpreted by invoking baryon exchange as in the multiperipheral model. One can also claim that the NN pair production proceeds via

intermediate boson resonances with proper mass spectrum. If duality arguments can be used the two interpretations are equivalent, however.

References

[ 1] J. Ranft, Phys. Lett. 41B (1972) 613. [ 2] H.M. Chan, K. Kajantie and G. Ranft, Nuovo Cim. A49

(1967) 157. [3] G.H. Thomas, in Proc. Coll. on Multiparticle dynamics,

Helsinki 1971 and references therein. [4] W. Kjttel, S. Ratti and L. Van Hove, Nucl. Phys. B30

(1971) 333, and references therein. [51 4" Wroblewski, XVth Intern. Conf. on High energy

physics, Kiev 1970. [6] E. Lillethun, Charged particle production at ISR, XVI

Ig~ern. Conf. on High energy physics, Batavia 1972, Bergen Report Nr. 47.

[7] H.~Bi0ggild, K.H. Hansen and M. Suk, Nucl. Phys. B27 (197q) 1.

[8] G. Damgaard and K.H. Hansen, An estimate at the multi- plicity of charged particles produced at the ISR, Zako- pane Conf. on Multiparticle dynamics, Niels Bohr Inst. rel~ort (197 2).

[9] A. Bertin et al., XVI Intern. Conf. on High energy physics, Batavia 1972.

[ 10] B. Alper et al., XVI Intern. Conf. on High energy physics, Batavia 1972.

[ 11] M. Banner et al., XVI Intern. Conf. on High energy physics, Batavia 1972.

[12] J.V. Allaby et al., 4th Intern. Conf. on High energy collisions, Oxford 1972.

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