Charmonium production in $\bar p$-induced reactions on nuclei

The production of charmonia in the antiproton-nucleus reactions at $p_{\rm lab}=3-10$ GeV/c is studied within the Glauber model and the generalized eikonal approximation. The main reaction channel is charmonium formation in an antiproton-proton collision. The target mass dependence of the charmonium transparency ratio allows to determine the charmonium-nucleon cross section. The polarization effects in the production of $\chi_{c2}$ states are evaluated.


Introduction
Investigation of charmonium-nucleon interactions is important for several reasons: (i) Interpretation of J/ψ suppression in relativistic heavy-ion collisions and separation of the charmonium dissociation mechanism in a quark-gluon plasma from cold nuclear matter effects. (ii) Constraining the QCDinspired models of charmonia. (iii) Growing understanding of nonperturbative vs perturbative QCD aspects, such as factorization theorem, color dipole cross section, color transparency phenomenon.
Charmonium formationpp → R (R = J/ψ, ψ ′ , χ c , . . .) on a nuclear target proton has been proposed long ago [1,2] to address the genuine charmonium-nucleon cross section, since the charmonium formation length is short in this case. In this talk we present some results of our detailed studies [3,4] of J/ψ, ψ ′ and χ c production inpA interactions at threshold.

Glauber model calculation of J/ψ and ψ ′ production
Charmonium production cross section in antiproton-nucleus interaction is calculated as where vp = p lab /Ep is the antiproton velocity. Γp →R is the antiproton width with respect to the charmonium production: is the charmonium formation cross section; vp p = s(s/4 − m 2 N )/EpE p is thepp relative velocity. We assume the proton energy E p = m N − B, where B ≃ 8 MeV is the nucleus binding energy per nucleon.
The exponential factors in Eq.(1) account for the antiproton and charmonium survival probabilities. The effective charmonium-nucleon cross section is chosen as σ eff is the total interaction cross section of the entirely formed charmonium with a nucleon; κ(z) is the charmonium position dependent function, κ(z) < 1 for z < l R and κ(z) = 1 for z ≥ l R , which takes into account the charmonium formation length l R according to the color diffusion model [2]. The cross sections σ RN have been studied by several groups of authors (c.f. [5][6][7][8]) and are expected to be within 3.5-7 mb for J/ψ , 0-20 mb for ψ ′ , and 7-16 mb for χ c -states. Fig. 1 shows the J/ψ and ψ ′ production cross sections on the lead target as a function of thep-beam momentum. We have chosen three representative values of the J/ψN cross section: σ J/ψN = 0, 3.5 and 6 mb. For the ψ ′ N cross section, we considered only the two limiting cases, i.e. σ ψ ′ N = 0 and 20 mb [7]. The formation length for J/ψ can be evaluated as GeV/c the J/ψ formation length is 0.4 fm which is even smaller than the internucleon spacing, d NN ≃ 2 fm. As a consequence, the J/ψ production cross sections shown in Fig. 1 are almost insensitive to the formation length effects. For ψ ′ , the formation length is larger (l ψ ′ ≃ 2l J/ψ , c.f. [7]), although even in this case formation length practically does not influence the production cross sections. On the other hand, we observe the sensitivity of the charmonium production cross section to the chosen charmonium-nucleon cross section.
at the on-shell peak beam momentum p lab = 4.07 GeV/c as a function of the target mass number. The scaling factor ∝ A −2/3 accounts for the surface absorption of the antiproton. In general, the MESON2014 -the 13 th International Workshop on Meson Production, Properties and Interaction transparency ratio is expected to be a more robust observable sensitive to charmonium-nucleon cross section, since the possible uncertainties in the production width Γp →R cancel out in Eq.(4). Indeed, we see that the transparency ratio is quite sensitive to the J/ψN cross section, although for the quantitative conclusions one should carefully take into account the empirical details of the nuclear density profiles at the surface region. 3 χ cJ (J = 0, 1, 2) production Since the mass splitting between the different χ cJ states is small (∼ 140 MeV), the nondiagonal transitions χ cJ 1 N → χ cJ N should be easily possible. The amplitudes of such transitions can be calculated from the Clebsch-Gordan decomposition of a physical χ cJ (ν) state with helicity ν in the basis of the cc states with fixed orbital (L z ) and spin (S z ) magnetic quantum numbers [4]: With a help of the optical theorem, the forward scattering amplitude M L z (0) can be expressed via the total interaction cross section σ L z of a cc pair with fixed L z with a nucleon: 30. In Ref. [7], the cross sections σ L z have been calculated by using the nonrelativistic quarkonium model and the QCD factorization theorem. Their values, σ ±1 = 15.9 mb and σ 0 = 6.8 mb, differ by a factor of two, which is the ratio of the transverse size squared for the cc states with L z = ±1 and L z = 0. This directly leads to the finite nondiagonal transition amplitudes: . By applying the generalized eikonal approximation (GEA)(c.f. [9] and refs. therein), we have calculated the helicity ratio for χ c2 production inp 208 Pb collisions. Here, B 0 is the helicity amplitudepp → χ c2 (ν = 0), |B 0 | 2 = 0.13 ± 0.08 [10]. Fig. 3 shows the ratio (5) at small transverse momentum as a function ofp-beam momentum. In the absence of any χ cJ N interactions, one has R = 1. However, the interference term of the directpp → χ c2 (0) and the two-steppp → χ c0 (0), χ c0 (0)N → χ c2 (0)N amplitudes leads to EPJ Web of Conferences ∼ 30% deviations of the helicity ratio from one due to the strong coupling of the χ c0 state withpp channel.