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Charmonium dissociation in collision with phi meson in hadronic matter

The phi-charmonium dissociation reactions in hadronic matter are studied. Unpolarised cross sections for 12 reactions are calculated in the Born approximation, in the quark-interchange mechanism and with a temperature-dependent quark potential. The potential leads to remarkable temperature dependence of the cross sections. With the cross sections and the phi distribution function we calculate the dissociation rates of the charmonia in the interactions with the phi meson in hadronic matter. The dependence of the rates on temperature and charmonium momentum is meaningful to the influence of phi mesons on charmonium suppression.


I. INTRODUCTION
Hadronic matter affects production of particles in relativistic heavy-ion collisions. Change of J/ψ in hadronic matter is appreciable. The change of J/ψ number is caused by inelastic meson-charmonium scattering [1][2][3][4][5] and spontaneous dissociation that may happen when temperature is higher than the J/ψ dissociation temperature [6]. Studies of charmonium dissociation in collisions with hadrons are important in the field of relativistic heavy-ion collisions. Since the φ-meson production is enhanced in Au-Au collisions at Relativistic Heavy Ion Collider energies, it is interesting to investigate φ-charmonium dissociation in hadronic matter. This is the subject of this paper.
We calculate φ-charmonium dissociation cross sections in the Born approximation and in the quark-interchange mechanism [7]. The quark interchange mechanism between the incident meson and the charmonium breaks the charmonium and produces charmed mesons and/or charmed strange mesons. Dissociation of charmonia in collisions with π, ρ and K in vacuum has been studied in Ref. [3] and dissociation in hadronic matter in Refs. [4,5]. Meson-charmonium dissociation cross sections in hadronic matter differ from those in vacuum. The in-vacuum cross sections obtained in the quark-interchange approach are different from the cross sections obtained in the meson-exchange approach [1] and in the short-distance approach [2]. The φ-charmonium dissociation cross sections have not been calculated in the quark-interchange approach, in the meson-exchange approach and in the short-distance approach.
With the φ-charmonium dissociation cross sections we calculate the dissociation rate of the charmonium in the interaction with the φ meson in hadronic matter. The larger the dissociation rate is, the stronger charmonium suppression φ mesons cause. From the dissociation rate we can understand contribution of the φ meson to the charmonium suppression. Therefore, the second task of the paper is to calculate the dissociation rate.
The remainder of this paper is organized as follows. In Section II formulas for the dissociation cross section and the dissociation rate are presented. In Section III we show numerical results of unpolarised φ-charmonium dissociation cross sections and charmonium dissociation rates. Relevant discussions are given. A summary is in Section IV.

II. CROSS-SECTION AND DISSOCIATION-RATE FORMULAS
The scattering A(ss) + B(cc) → C(sc) + D(cs) takes place due to quark interchange and interactions among constituents (quarks and antiquarks). The scattering has the two forms, the prior form and the post form. The scattering in the prior form means that gluon exchange occurs before quark interchange. The scattering in the post form contains gluon exchange after quark interchange. The scattering in the two forms is depicted in Figs. 1 and 2 in Ref. [5]. Cross-section formulas for the scattering are given in Refs. [4,8]. Here we briefly introduce the formulas. Application of the cross section to get the charmonium dissociation rate follows.
Denote the spin, orbital angular momentum and four-momentum of meson i (i = ss, cc, sc, cs) by S i , L i and P i = (E i , P i ), respectively. The Mandelstam variables are s = (E ss + E cc ) 2 − ( P ss + P cc ) 2 and t = (E ss − E sc ) 2 − ( P ss − P sc ) 2 . Let P and P ′ be the momenta of mesons A and C in the center-of-mass frame of A and B, respectively. The unpolarised cross section for the scattering in the prior form is where M prior fi is the transition amplitude for the scattering in the prior form, S is the total spin of mesons A and B, L ccz is the magnetic projection quantum number of L cc and θ is the angle between P and P ′ . The unpolarised cross section for the scattering in the post form is where M post fi is the transition amplitude for the scattering in the post form. The unpolarised cross section for A(ss) + B(cc) → C(sc) + D(cs) is Let ψ ss (ψ cc , ψ sc , ψ cs ) stand for the product of colour, spin, flavour and relative-motion wave functions of ss (cc, sc, cs) and V ab the interaction between constituents a and b. The transition amplitudes are given by The potential V ab in coordinate space is where r is the relative coordinate of a and b, V si is the central spin-independent potential and V ss the spin-spin interaction. V si is given by where is the product of the Gell-Mann matrices for the colour generators of a and b. The dimensionless function v(x) is given by Buchmüller and Tye [9]. The shortdistance part of V si originates from one-gluon exchange plus perturbative one-and two-loop corrections. The intermediate-distance and large-distance part of V si fits well the numerical potential obtained in the lattice gauge calculations [10]. At large distances V si is independent of r and exhibits a plateau. The plateau height decreases with increasing temperature. This means that confinement becomes weaker and weaker.
The spin-spin interaction with relativistic effects [7,11] is [4,12] where s a ( s b ) and m a (m b ) are the spin and mass of constituent a(b), respectively. The flavour dependence of the interaction is relevant to quark masses as shown in 1 mam b and d, where σ 0 = 0.15 GeV and σ 1 = 0.705. The second term in Eq. (8) arises from the perturbative one-and two-loop corrections to the gluon propagator.
By solving the Schrödinger equation with the central spin-independent potential plus the spin-spin interaction, meson masses and quark-antiquark relative-motion wave functions are obtained. In solving the Schrödinger equation the masses of the charm quark, the up quark and the strange quark are 1.51 GeV, 0.32 GeV and 0.5 GeV, respectively. The experimental masses of φ, J/ψ, ψ ′ , χ c , D s and D * s mesons [13] and the experimental data of S-wave I = 2 elastic phase shifts for ππ scattering in vacuum [14] are reproduced with V ab ( r) at T = 0 and the quark-antiquark relative-motion wave functions.
While V ab ( r) and the quark-antiquark relative-motion wave functions obtained from the This is exactly what is concluded in Ref. [15] from a general argument. With the unpolarised cross sections for charmonium dissociation we calculate the dissociation rate of charmonium in the interaction with φ meson in hadronic matter, where n is the φ number density, v rel is the relative velocity of the φ meson and the charmonium, θ is the angle between the φ momentum k and the charmonium momentum and f ( k) is the Bose-Einstein distribution that φ mesons obey. The thermal average of v rel σ unpol is indicated by v rel σ unpol [12].  Charmonium dissociation in collision with φ meson has four channels. The unpolarised dissociation cross section in the dissociation-rate formula is the sum of the unpolarised cross sections for the four channels. In collision of φ with J/ψ (ψ ′ , χ c ) σ unpol is the unpolarised Numerical results of dissociation rates as functions of charmonium momentum are plotted in Figs. 10-12. Since the φ distribution function, the unpolarised cross section and the relative velocity in Eq. (10) vary with temperature, the dissociation rate depends on temperature. As temperature increases from T /T c = 0.65 to 0.95, the φ mass decreases and the φ distribution function increases. This is a factor that increases the charmonium dissociation rate. At zero charmonium momentum the slope of each curve is zero. Because φ mesons obey the Bose-Einstein distribution, there is a domain of k in Eq. (10), which keeps √ s near the threshold energy so that the unpolarised cross section contributes significantly.
While the charmonium momentum increases, the domain shrinks and the dissociation rate thus decreases. Since the dissociation rates of J/ψ, ψ ′ and χ c with φ are less than 0.00066, 0.002 and 0.0008, respectively, the charmonium suppression caused by the φ-charmonium dissociation is weak.

IV. SUMMARY
Using the temperature-dependent quark potential that is derived from perturbative QCD at short distances and the lattice gauge results at intermediate-and large-distances, we obtain the unpolarised cross sections in the quark-interchange mechanism and in the Born approximation. The reactions that we concern are φJ/ψ → The meson masses, confinement and mesonic quark-antiquark relative-motion wave functions cause the large variation of the cross sections with respect to temperature. Medium effects on the dissociation reactions are prominent. When the confinement gives similar contributions to two reaction channels, the spin-spin interaction makes the difference of the unpolarised cross sections for the two channels. Using the unpolarised cross sections for the 12 reactions, we obtain the dissociation rates of charmonia in the interactions with the φ meson. The rates generally increase with increasing temperature or with increasing charmonium momentum. The rates are quite small.