Hadron scattering in an additive eikonal quark model

Abstract Total and differential cross sections for pp, pp, π ± p, K +- p and Ψp at high energies are analysed in an additive eikonal quark model. It is shown that the exceptionally small Ψp slope contradicts the Chou-Yang model at present energies but can be accounted for by a quark-quark potential with different ranges for different pairs of quarks. This leads to smaller rms radii for hadrons than those derived from e.m. formfactors, implying that the constituent quarks have formfactors. Predictions for hadron-neutron and strange and charmed hadron-nucleon scattering are derived.

The purpose of this letter is to extend the additive quark model, which has been quite successful m relating total hadronic cross sections [1][2][3], to describe the differential cross sections as well. Our model, besides being phenomenologically very useful, gives information on the structure of hadrons and on quark-quark interactions. While the systematics of total cross sections O~p > Opp > oTr-p > oTr+p > OK-p > O K+p > O¢9 > O¢ p could be simply attributed to different strengths of quark-quark scattering amplitudes, a more complicated mechanism is necessary to explain the identical inequalities among the forward slopes of differential cross sections, Bw,, > Bn,, > ... etc.
where n A is the number of quarks (antiquarks) of type i in particle A and N is the number of flavours. This "additive eikonal quark model" could, for example, be justified using a potential model in elkonal approximation in which case X AB is given by oo aB(s,b) = .f dz /d3rl d3r2 PA(rl) PB(rl) PB(r2)//AB(s, r --r 1 + r2).
(2) Xi/ __oo Here ph ~2 is the density distribution of quarks in hadron h and IiAB is the effective quark-quark potential, which may also depend on the interacting hadrons, as will be discussed later. If we classify the quark eikonals similarly to Lipkin [2] 1 Permanent address: Phystcs Department, Ben Gurion Universtty, Beer Sheva, Israel. 2 On leave of absence from Institute of Nuclear Research, Warsaw, Poland. ,l The eikonal ×(s, b) ts defined by the usual relation ,2 From now on we shall use the following subscrtpts: h -hadron, m -meson, b -baryon, a -annihilation, d -non strange, c -charm.
Xd~ =Xu ~Xa, Xu u =Xd d =Xu d =Xu ~ ~Xd, and assume that Xa,d,s,c are independent of AB, we get immediately Lipkin's sum rules but at the level of eikonals instead of total cross sections. Unfortunately, the eikonal is rather difficult to calculate directly from experiment [5] ; in particular, do/dt must be known for large Itl and with high accuracy. Therefore, Instead of checking the eikonal relations directly, we prefer to parametrize the eikonals and compare the resulting differential and total cross sections with experiment.
One possIbe way out is to assume that different pairs of quarks have different interaction ranges, in particular 2 2 1 2 2 (R 2) > 2 (R2). However, since (rp 2) ~-{Bpp/g(Ypp) -1 (R 2) = (0.7) fm -~(~Rd) , a large (R d) would lead to a smaller rms radius for proton than the one following from its e.m. formfactor, 6dF(t)/dtlt=o ~-(0.8) 2 fm 2. To account for the difference one has to assume that constituent quarks themselves have their own formfactors. Then the observed e.m. formfactor of hadron F h would be given roughly by the product where Qq are the charges of the constituent quarks and Fq(t) are their e.m. formfactors. This point will be discussed in more detail elsewhere [7].

Rij-r A +r 2 +R 2,
and similarly for the core.
To explain the meaning and the role of different parameters in the above ansatz, we shall start by assuming drastic simplifying relations among them and then relax some of these gradually, showing the improvement of the Table 1 The total cross sections a tot and the slopes B (calculated at t = -0.2 GeV 2) at Plab = 50 GeV/c. The upper part of the table includes the input data, the lower one, predictions. The nomenclature of the charmed particles (except ¢) corresponds to the one in ref. [19]. In fig. 1 we illustrate the improvement of our fit by plotting o t°t and B together with ;(2 for all fitting steps and in fig. 2 the differential cross sections resulting from the final fit. Now, we shall shortly review successive approximations ~:3 .
,3 Note that the first two steps correspond to naive [1 ] and sophisticated [2] quark model, while the other 3 steps are the natural extension of the former, indicated by experiment.
Step 1 (2 parameters): Step 2 (4 parameters). Allow for different ranges of quark interactions according to classification (3): R a >R d > R s. In particular, the inequality R a > R d leads to ~p > pp, 7r-p > zr+p, K-p > K+p for both o t°t and B. Note that although ~ is still common for all quarks, the corresponding g's will split, ga > gd > gs" Step 3 (5 parameters). The observation that o~ °t is always smaller whereas Otm °t is always larger than experiment (see fig. 1) lead us to introduce a phenomenological parameter 3% > 7m -1. This parameter is responsible for the difference between effective quark-quark couplings in meson-baryon and baryon-baryon scattering and could be attributed to the dependence of quark interactions on velocity and different velocity distributions of quarks within mesons and baryons. If this is the case, then 7 should approach unity as s ~ oo. The additional parameter 7b improves the fit enormously (compare ×2 in fig. 1).
Step 4 (7 parameters). Although the above 5 parameter fit leads to very good results for o t°t and B, the theoretical differential cross sections show a prominent dip at too low values of t (~--1GeV 2) for all 6 reactions. By introducing a core term, with Reore., "~ Rii, we can push the dip to larger values of t. For simplicity we add only 2 new core parameters: common couphng xeore and a common ratio a -Rcore ' a/Ra -Rcore,d/R d -Reore,s/R s and assume 7~ °re = 3%, 7~re = 1.
Finally, we list in table 1 predictions for the total cross sections and slopes for several reactions and compare them with experiment when possible. Note the remarkable agreement with the experimental neutron data: less than 2% deviation from Fermilab data [12] and even better agreement with Serpukhov data [13]. Unfortunately, the hyperon-nucleon scattering is measured only up to Plab = 21 GeV/c so that direct comparison is not possible. However, the rough agreement is visible if one remembers that Ap and An decrease faster with energy than Ap and An, due to the strongly varying annihilation channel dd and ~u. The predictions for charmed particles are intended only as rough estimates.
We conclude that the additive eikonal quark model describes successfully the different hadron scattering data. Our good fit shows implicitly that the quark model relations (3) are well satisfied within mesons and baryons respectively and indicates the necessity of breaking of simple additwity by introducing some additional difference be-,4 The distribution Oh for different quarks should be unequal if some quarks are much heavier than the others (e.g. charmed quark).
Nevertheless, we use a universal Ph for simplicity. tween mesons and baryons (Tb parameter). The interpretation given to the various parameters has important implications for the quark dynamics and the structure of hadrons. To gain a deeper understanding of this model we used it also to study the energy dependence of hadronic interactions. The results will be discussed elsewhere [ 18].
It is a pleasure to thank Dr. F. Gutbrod, Prof. H. Joos, Dr. M. Krammer, Dr. M. Kuroda and Dr. T. Walsh for very fruitful discussions.