MPIH{V11{1995. in Proton-Nucleus and. Jorg Hufner. Instut fur Theoretische Physik der Universitat, Philosophenweg 19, Heidelberg, Germany

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1 MPIH{V{995 The Relative J to Suppression in Proton-Nucleus and Nucleus-Nucleus Collisions Jorg Hufner Instut fur Theoretische Physik der Universitat, Philosophenweg 9, 692 Heidelberg, Germany huefner@zooey.mpi-hd.mpg.de Boris Kopeliovich Max-Planck Institut fur Kernphysik Postfach 398, 6929 Heidelberg, Germany and Joint Institute for Nuclear Research, Laboratory of Nuclear Problems, Dubna, 498 Moscow Region, Russia bzk@dxnhd.mpi-hd.mpg.de HEP-PH Abstract We calculate the nuclear suppression for J and production within a coupled channel approach in the subspace of the J and states. We are able to explain, why (i) the J and show the same suppression from 2 GeV to 8 GeV in proton-nucleus collisions and why (ii) the is absorbed more strongly than the J in nucleus-nucleus collisions at 2 GeV. The numerical result which includes only interactions with nucleons acconts for half of the observed suppression in sulphururanium collisions.

2 The E772 collaboration [] was the rst who claimed that (we use hereafter this abbreviation for J ) and produced in proton-nucleus (p A) collisions at 8 GeV and x F : experience the same nuclear suppression. This result was conrmed by the NA38 collaboration [2, 3, 4] at an energy of 2 GeV and for x F :, though with larger error bars. The two observations contradict the simple-minded expectation, namely that the should be more strongly suppressed, because absorption cross section scales with the mean square radius hr 2 i of the meson [5, 6] and the is much larger than the. To our opinion, no satisfactory explanation of the data has yet been proposed. Recently the situation became even more mysterious by the observation [2, 4] that in Sulfur-Uranium (S-U) collisions at 2 GeVA the is signicantly more strongly absorbed than the. Does this result signal dense hadronic gas or quark-gluon plasma formation? In this letter we treat the propagation of a cc pair through nuclear matter as a coupled system of the and states. In addition to elastic collisions N N and N N with amplitudes f( ; ) and f( ; ) respectively, we consider the conversion amplitudes f( ; ) and f( ; ) for the processes N *) N during propagation through nuclear matter. The inelastic amplitudes turn out to be nearly as big as the elastic ones. We dene as nuclear suppression factor of a meson in pa collisions the ratio S pa (x F ;E) (pa X; x F ;E)dx F ; () A(pN X; x F ;E)dx F where E is the lab. energy. We also introduce the relative to nuclear suppression function by S pa S pa S pa (2) 2

3 Then the results of the E772 and NA38 experiments can be summarized by S pa for the values of x F and E considered. In order to exhibit the physics of our coupled channel approach, we rst calculate eq. (2) perturbatively, i.e. restricting ourselves to only one N ( N)interaction after the creation of the cc pair. This interaction can be an elastic one as well as a conversion event. Then S pa N 2 tot (r + R)hT i A ; (3) N 2 tot ( + R)hT i A where r f( )f( ), f( )f( ) and N tot 2Imf( ; ). Furthermore, R denotes the relative amplitude of to production in the initial pn collision, and ht i A A R d 2 bt 2 (b), where T (b) R dz A (b; z) is the nuclear thickness. If one neglects the conversion rate ( ), eq. (3) reduces to the conventional result, namely that the suppression depends only on tot N, while r tot N N tot suppression of the. determines the The ratios r and in eq. (3) can be fairly reliably calculated since at high energies the scattering amplitudes f(; ), where and stand for and, are proportional to hjrt 2 ji, as long as the meson radius in the transverse direction, hr2 T i, is small. With S and 2S harmonic oscillation functions for and, respectively, one has r 73 and q 23. The ratio R of initial to production cannot be calculated in this way. We turn the argument around and calculate from eq. (3) that value of R cal which leads to the observed relative suppression S pa and compare it to R exp deduced from pn collisions. A quadratic equation for R leads to R cal q 53 q 23, the smaller one being R cal :47. If one uses the experimental intensities of and produced in pp collisions and corrects 3

4 them for the feedings and, one arrives at an amplitude ratio jr exp j :48 :6 [7, 4], which agrees well with the calculated one, indicating that the initially produced state j cc i i (j i+rj i) p +R 2 is such as to lead to the same nal state attenuation for and. Mathematically spoken, j cc i i is an eigenstate of the nal state interaction matrix, ^f r C A f( ; ) : (4) The property ofj cc i ibeing eigenstate of ^f is equivalent to the statement that j cc i i has an extreme value for its transverse size: d dr hcc i jrtj 2 cc i i : (5) The experimental value jr exp j selects the physical state as that with minimal transverse extension. Strictly speaking the result eq. (3) is only valid for energies E. For nite lab. energies, especially for pa and AA collisions at 2 GeV one has to include the eect of the longitudinal momentum transfer q associated with the conversion reaction *) : q M 2 M 2 ; (6) 2Ex q where x (x F + x 2 F+4M 2 s)2 with p s the c.m. energy. One arrives at an expression like eq. (3) where is replaced by F A (q) with F A (q) 2 Z Z dz A (b; z) dz A (b; z ) exp(iqz ) (7) AhTi z 4

5 being a kind of nuclear formfactor. For the E772 experiment q :6 fm for x F :2 and F A isavery good approximation. Thus, the observed result S pa is reproduced. For the NA38 experiment at 2 GeV A especially for the nucleus{nucleus collisions, the eect of the formfactor becomes crucial: In the S U collision the produced (or ) moves with fractional momentum x F (in the c.m. system) with respect to the target nucleus U, but moves with x F relative to the projectile nucleus S (inverse kinematics). Therefore the suppression arises from two sources, S SU S ps ( x F ;E)S pu (x F ;E) : (8) For 2 GeV and x F :2wehaveq:6 fm to be used in the second factor of eq. (8) and q :56 fm in the rst one. This is the inverse kinematics which leads to a much stronger suppression. For the detailed comparison with experiment we have to go beyond the perturbative expression (3) and use numerical methods. We describe the propagation of cc pair through nuclear matter by a dierential equation [8, 9] i d dz jcc (z; ~ b)i b U(z; ~ b)j cc (z; ~ b)i ; (9) where the limitation to the ; subspace invites the use of matrix notations: j cc i and bu q C A i 2 N tot A ( ~ B b; r C A () with initial condition j cc ini p +R2 at the point ( ~ b; z )ofcccreation. At z + R 5

6 the wave function j cc i is projected on the and states. The result is squared and averaged over the coordinates ( ~ b;z ) of the production point. Note that this is only true if the longitudinal momentum transfer at the production point, q M 2 2Ex q,ismuch larger than the reversed mean internucleon distance in a nucleus. Otherwise one should take into account coherence between dierent production points as well (see discussion in terms of production and formation times in [, ]). This would be equivalent to an eective increase of the length of path of the qq pair in nuclear matter. However it does not aect the relative production rate if j cc i is an eigenstate of interaction. The relative suppressions are calculated for p A collisions and with the help of eq. (8) also for A A collisions. The numerical results shown in Figs. and 2 are calculated with realistic nuclear densities and with the values of and r as given by the harmonic oscillator model and for the absolute value of the N total cross section tot N ChrTi 2 5:7mb, where we use the perturbative QCD estimate of [] or systematics of [6] for a value of C. While at 8 GeV the values for S pa are measured directly, the values given at 2 GeV for B B in nuclear collisions were renormalized by us using the value :8 : % for this quantity from pp and pd collisions [4]. Fig. shows the suppression functions S for p W at 8 GeV and for p U and p W at 2 GeV. Although we predict for 2 GeV some reduction of the relative suppression it is still in agreement with the data of the NA38 experiment within rather large error bars. Fig. 2 shows the predicted and observed [2, 4] suppression S SU for the nucleus-nucleus case. The solid curve is the product of the suppression curves for the p U and the S p collisions (each dashed). Experiment and calculation agree in that the should be signicantly more strongly suppressed than the. However, the measured suppression seems to be stronger than our expectation, 6

7 indicating that there may be room for other eects. The NA38 group has published [2, 4] also four points for the relative to suppression in S U collisions as a function of the transverse energy E T. The suppression is larger for larger values of E T, corresponding to more central collisions. On the basis of our model we expect such abehaviour, but we lack precise quantitative information for the association of a value of E T to a denite geometric conguration. In Fig. 3we show the suppression as a function of x F for Au Au collisions calculated for RHIC and LHC energies. The result is predicted to be energy independent at high energies and the two curves coincide. The model of coupled channels presented in this letter "naturally" explains, why at high energies of the charmonia and should be similarly suppressed as observed in proton-nucleus collisions at 8 GeV, and why in nucleus-nucleus collisions one should see signicant dierences in and suppressions as was reported at 2 GeVA. While the model is free of adjustable parameters, the precision of the two-channel approximation is questionable. In order to evaluate the corrections to S from inclusion of higher charmonium excitations we switch from the hadronic basis to the quark one in coordinate representation. In this case the nuclear suppression of production is calculated as [] S pa h cc j b V (b; z; r T )j cc ini 2 jh j cc inij 2 () and the same for. Here the evolution operator b V at high energy (when the uctuations in r T are frozen by Lorentz time dilation) reads, b V (b; z; r T ) exp[ Cr 2 T2 R z dz A (b; z )]. The 7

8 averaging h:::i A in eq. () denotes the integration over the coordinates of the production point weighted with nuclear density. Eq. () (valid only for q ) generalizes the two-channel approach in that j cc ini may have other components in addition to the and states (see an alternative interpretation of enhancement of the production rate on nuclei in [, 2]). We have evaluated the suppression eq. () for and and the relative suppression S pa for various trial functions for h~rj cc ini like exp( r 2 ), r 2 T exp( r 2 ), r 2 TK (r T )(K is a modied Bessel function), where the constants,, have been chosen to give the same amplitude ratio R for the content ofj cc ini relative to the one. The resulting relative nuclear suppression exceeds the prediction of the two-channel model S pa only by 5 % for heavy nuclei, a deviation which is still compatible with the data. Another correction of the same order of magnitude is expected arising from the -component of the initial cc state. A more complete analysis of these eects as well as recalculation of the nuclear suppression using path-integral methods [] will be presented in a forthcoming paper. Note that the initially produced cc wave packet, which isacombination of and, attenuates in the nucleus less than each of two charmonia. This fact is important for the nuclear suppression of, taken separately. This should be checked with available experimental data. In the case of nucleus-nucleus collisions, we have assumed that charmonia attenuate only due to interaction with the projectile nucleons, having x F. However, particle production (and possibly a dense hadronic gas or a quark-qluon plasma) should cause an additional suppression of S A A down from our prediction and may explain the deviation of our calculations from the results of the NA38 experiment depicted in Fig. 2. Recently, Satz 8

9 [3] has proposed to use the value of S as a probe for dense matter formation. Acknowledgement: We are grateful to P. Giubellino and H. Satz, who stimulated our interest to the problem under discussion. We thank Dr. C. Gerschel for several discussions on the experimental situation. B.K. thanks E. Predazzi for useful discussion and MPI fur Kernphysik, Heidelberg, for nancial support. The work was partially supported by a grant from the BMFT, grant number 6HD742(). References [] D.M. Adle et al., Phys. Rev. Lett. 66 (99) 33 [2] The NA38 Collaboration, B. Ronceux et al., Nucl. Phys. A566 (994) 37c [3] The NA38 Collaboration, C. Lourenco et al., Nucl. Phys. A566 (994) 77c [4] The NA38 Collaboration, C. Baglin et al., Phys. Lett. B345 (995) 67 and M.C. Abreu et al., presented at QUARK MATTER'95 [5] J.F. Gunion and D. Soper, Phys. Rev. D5 (977) 267 [6] J. Hufner and B. Povh, Phys. Lett. B245 (99) 653 [7] The E75 Collaboration, L. Antoniazzi et al. Phys. Rev. Lett. 7 (993) 383 [8] B.Z. Kopeliovich and L.I. Lapidus, Sov. Phys. JETP Lett. 32 (98) 592 [9] B.K. Jennings and B.Z. Kopeliovich, Phys. Rev. Lett. 7 (993)

10 [] S.J. Brodsky and A. Mueller, Phys. Lett. B26 (988) 685 [] B.Z. Kopeliovich and B.G. Zakharov, Phys. Rev. D44 (99) 3466 [2] O. Benhar, B.Z. Kopeliovich, Ch. Mariotti, N.N. Nikolaev and B.G. Zakharov, Phys. Rev. Lett. 69 (992) 56 [3] H. Satz, In "Aachen 992, Proceedings, QCD: 2 Years Later", v. 2, p. 748 Figure capture Fig. The relative nuclear suppression in p W collisions. The full circles and the solid curve are the data of the E772 experiment [] at 8 GeV and our calculation, respectively. The open circle and the dashed curve are the result of the NA38 experiment [4] at 2 GeV and our prediction, respectively. Fig. 2 The relative nuclear suppression in S U collisions at 2GeV. The dashed curves are the relative suppression in p U and S p (inverse kinematics) collisions at 2 GeV. The solid curve, which is the two dashed lines describes our prediction for S U collisions. The data-point is the result of NA38 experiment [4]. Fig. 3 Prediction for the relative suppression in Au Au collisions for the expected energies of the RHIC and LHC accelerators.