Dissociation Cross Sections of Large-Momentum Charmonia with Light Mesons in Hadronic Matter

Momenta of charmonia created in Pb-Pb collisions at the Large Hadron Collider are so large that three or more mesons may be produced when the charmonia collide with light mesons in hadronic matter. We study the meson-charmonium collision in a mechanism where the collision produces two quarks and two antiquarks, the charm quark then fragmenting into charmed mesons, and the other three constituents as well as quarks and antiquarks created from vacuum give rise to two or more mesons. The absolute square of the transition amplitude for the production of two quarks and two antiquarks is derived from the $S$-matrix element, and cross-section formulas are derived from the absolute square of the transition amplitude and charm-quark fragmentation functions. With a temperature-dependent quark potential, we calculate unpolarized cross sections for inclusive $D^+$, $D^0$, $D^+_s$, or $D^{*+}$ production in scattering of charmonia by $\pi$, $\rho$, $K$, or $K^\ast$ mesons. At low center-of-mass energies of the charmonium and the light meson, the cross sections are very small. At high energies the cross sections have obvious temperature dependence, and are comparable to peak cross sections of two-to-two meson-charmonium reactions.


Introduction
Plenty of efforts with quantum chromodynamics (QCD) and effective field theories have been devoted to exploring strong interactions of charmonia with light hadrons. Four main approaches involved in the study are the short-distance approach, the effective meson approach, the quark-interchange approach, and QCD sum rules. In the short-distance approach, methods in perturbative QCD, for example, the operator product expansion, were applied to charmonia of small sizes [1][2][3]. In the effective meson approach, effective meson Lagrangians with different symmetries and Feynman diagrams with various vertex functions have been used to get millibarn-scale cross sections for meson-charmonium reactions [4][5][6][7][8][9][10][11][12][13][14]. Recently, πJ/ψ, ρη c , and DD * (D * D ) reactions have been put together to form coupled channels. The corresponding scattering amplitude derived in SU(4) chiral perturbation theory is unitarized [12][13][14], and cross sections obtained for πJ/ψ dissociation are different from those given in Refs. [4][5][6][7][8][9][10][11]. In the quark-interchange approach, charmonium dissociation in collisions with light mesons is caused by quark interchange [15] between the charmonium and the meson. The dissociation has been studied in the Born approximation with quark potentials which reproduce spectroscopic data and provide mesonic quark-antiquark wave functions [16][17][18][19][20][21]. In the QCD sum rules, πJ/ψ dissociation cross sections were obtained from the general vacuum-pion correlation function with the currents of J/ψ and of charmed mesons in the soft-pion limit [22].
Adopting temperature dependence in meson masses and two-meson Green functions in the coupled-channel unitary approach, temperature-dependent cross sections for J/ψ scattering by light mesons have been obtained in Ref. [25].
For Pb-Pb collisions at the center-of-mass energy per nucleon-nucleon pair √ s N N = 5.02 TeV at the CERN Large Hadron Collider, the prompt-J/ψ transverse momentum measured by the CMS Collaboration [26] and the ATLAS Collaboration [27] goes up to 50 GeV/c. J/ψ mesons with such large transverse momenta may be broken up due to collisions with light mesons in hadronic matter, and three or more mesons can be produced.
Since the reactions studied in Refs. [4-14, 16-22, 25] were limited to two-to-two mesoncharmonium reactions, a mechanism for large-momentum charmonium dissociation is proposed in Ref. [28] to study the production of three or more mesons in pion-charmonium collisions. In this mechanism a collision between a light meson and a charmonium produces quarks and antiquarks first, the charm quark then fragments into charmed hadrons, and finally three or more mesons are produced. Besides pions, ρ mesons, kaons, vector kaons, and so on in hadronic matter also induce charmonium dissociation. This motivates studying the production of three or more mesons in the dissociation of large-momentum charmonia in collisions with light mesons in the present work.
This paper is organized as follows. In the next section we derive cross-section formulas for dissociation of large-momentum charmonia in collisions with light mesons. Numerical results and relevant discussions are presented in Section 3. A summary is in the last section.

Formalism
We obtain the cross section for A(q 1q2 ) + B(cc) → q 1 +q 2 + c +c → H c + X from the cross section for A + B → q 1 +q 2 + c +c and the fragmentation function for c → H c , where H c represents a hadron that contains the charm quark. The fragmentation of the charm quark into hadrons is related to quark-antiquark pairs created from the color field around the charm quark, which is that scenario of Feynman and Field [29]. The charm quark may combine the antiquark of a quark-antiquark pair to form meson H c . Two quark-antiquark pairs may also form another H c . All possible ways to form meson H c are accounted for by the c → H c fragmentation function. Hence, only one H c symbol is in A(q 1q2 )+B(cc) → q 1 +q 2 +c+c → H c +X. Unused quarks and antiquarks combine q 1 ,q 2 , andc to form two or more mesons. The symbol X indicates the two or more mesons that do not include meson H c . We first derive cross-section formulas for A+B → q 1 +q 2 +c+c.
Let E i be the total energy of mesons A and B, and E f be the one of constituents q 1 ,q 2 , c, andc. The S-matrix element for A + B → q 1 +q 2 + c +c is where V ab is the potential between constituents a and b. Let r ab be the relative coordinate of a and b, and denote the momentum and the position vector of meson A (B) by P A ( P B ) and R A ( R B ), respectively. The wave function |A, B of A and B is where every meson wave function is normalized in the volume V , and ψ A (ψ B ) is a wave function of color, flavor, spin, and relative motion of the quark and the antiquark in meson and r q 1 ( rq 2 , r c , rc) denote the momentum and the position vector of q 1 (q 2 , c,c), respectively. The wave function |q 1 ,q 2 , c,c of q 1 ,q 2 , c, andc is where ϕ q 1q2 cccolor , ϕ q 1q2 ccflavor , and ϕ q 1q2 ccspin are the color wave function, the flavor wave function, and the spin wave function of q 1 ,q 2 , c, andc, respectively. Using the wave functions, we get where R total and P f are the center-of-mass coordinate and the total momentum of q 1 ,q 2 , c, andc, respectively; r q 1q2 ,cc and p ′ q 1q2 ,cc the relative coordinate and the relative momentum of the two colored pairs q 1q2 and cc, respectively; p ′ q 1q2 the relative momentum of q 1 and q 2 , and p ′ cc the relative momentum of c andc; p AB ( r AB , P i ) the relative momentum (the relative coordinate, the total momentum) of mesons A and B; c the energies of A, B, q 1 ,q 2 , c, andc, respectively. M ab is the transition amplitude corresponding to V ab : Let φ Acolor (φ Bcolor ), φ Aflavor (φ Bflavor ), φ Arel (φ Brel ), and χ A (χ B ) stand for the color wave function, the flavor wave function, the quark-antiquark relative-motion wave function, and the spin wave function of meson A (B), respectively; denote the total angular momentum, the orbital angular momentum, and the spin of meson A (B) by J A (J B ), L A (L B ), and S A (S B ), respectively. ψ A and ψ B in Eq. (2) are given by where J Az (J Bz ) is the magnetic projection quantum number of J A (J B ), and the symbol indicates the space-spin wave function of meson A (B). The product of ψ A and ψ B is where J is the total angular momentum of mesons A and B, and J z its magnetic projection quantum number; (J A J Az J B J Bz |JJ z ) are the Clebsch-Gordan coefficients. Let L (S) and L z (S z ) denote the total orbital angular momentum (total spin) of mesons A and B and its magnetic projection quantum number, respectively. ψ JJz in comes from the coupling of the space-spin states of meson A and of meson B: where the two braces in each of the second and third expressions indicate the Wigner 9j symbol.
Denote by φ iflavor , φ i , and χ i the flavor wave function, the space wave function, and the spin wave function of constituent quark or antiquark labeled as i (i = q 1 ,q 2 , c,c), and ϕ q 1q2 ccspin = χ q 1 χq 2 χ c χc.
After the spin states of q 1 (c) and ofq 2 (c) are coupled to the spin state with the spin S ′ q 1 +q 2 (S ′ c+c ) and its z component S ′ q 1 +q 2 z (S ′ c+cz ), the spin states with S ′ q 1 +q 2 and with S ′ c+c are coupled to the spin state of q 1 ,q 2 , c, andc, which has the spin S ′ and its z component S ′ z . In this way we have where s i is the spin of constituent i, and s iz its z component; the spin wave function ψ According to Eq. (1), the transition amplitude for A + B → q 1 +q 2 + c +c is where M q 1c , Mq 2 c , M q 1 c , and Mq 2c correspond to V q 1c , Vq 2 c , V q 1 c , and Vq 2c , respectively.
Summing over the states of A, B, q 1 ,q 2 , c, andc gives where ab in ab runs through q 1c ,q 2 c, q 1 c, andq 2c .
The potential V ab consists of the central spin-independent potential V si and the spinspin interaction V ss : Below the critical temperature T c = 0.175 GeV, the spin-independent potential is given by where ξ 1 = 0.525 GeV, ξ 2 = 1.5[0.75 + 0.25(T /T c ) 10 ] 6 GeV, ξ 3 = 0.6 GeV, and λ = 3b 0 /16π 2 α ′ in which α ′ = 1.04 GeV −2 and b 0 = 11 − 2 3 N f with the quark flavor number N f = 4. λ a are the Gell-Mann matrices for the color generators of constituent a. The dimensionless function v(x) is given by Buchmüller and Tye [30].
The spin-spin interaction with relativistic effects [31] is [19,24] where m a is the mass of constituent a, and d is given by where d 1 = 0.15 GeV and d 2 = 0.705.
One-gluon exchange between constituents a and b gives rise to the Fermi contact in the nonrelativistic limit. The δ 3 ( r ab ) function fixes the positions of the two constituents to r ab = 0. However, the constituent positions fluctuate because each constituent is coupled to a gluon field which has vacuum polarization. This is similar to the well-known fact that the fluctuation of an electron position arises from vacuum polarization of its coupled electromagnetic field [32,33]. To take into account this relativistic effect, δ 3 ( r ab ) is replaced with d 3 π 3/2 exp(−d 2 r 2 ab ) so as to arrive at the first term on the right-hand side of Eq. (19). This is the smearing of the one-gluon-exchange spin-spin interaction [31]. The second term on the right-hand side of Eq. (18) comes from one-gluon exchange plus perturbative one-and two-loop corrections. The second term on the right-hand side of Eq. (19) originates from perturbative one-and two-loop corrections to one-gluon exchange [24]. The loop corrections are another relativistic effect embedded in the spin-independent potential and the spin-spin interaction. Therefore, the potential V ab given in Eq. (17) is a relativized potential.
The potential at short distances is dominated by the the second term of the spinindependent potential and the two terms of the spin-spin interaction. When the centerof-mass energy of mesons A and B is large, short distances are reached by constituents, and the three terms with relativistic effects make a contribution to the scattering of mesons A and B.
The total-spin operator of A and B, i.e., of q 1 ,q 2 , c, andc is It is easily proved that the commutator of s and the Hamiltonian that includes V ab equals zero. This leads to S ′ = S and S ′ z = S z . We thus get We take the Fourier transform of the mesonic quark-antiquark relative-motion wave functions and the potentials: The quark-antiquark relative-motion wave functions in momentum space, φ Arel ( p q 1q2 ) and φ Brel ( p cc ), satisfy We finally arrive at where Q is the gluon momentum.
Let dσ free represent the differential cross section corresponding to the factor ψ SSz+ where m A (m B ) is the mass of meson A (B); L z , S ′ q 1 +q 2 , and S ′ c+c . The unpolarized differential cross section for A + B → q 1 +q 2 + c +c is thus Now we give the cross section for A + B → q 1 +q 2 + c +c → H c + X, which includes the fragmentation process c → H c . Denote by z the fraction of energy passed on from quark c to hadron H c . The fragmentation function D Hc c (z, µ 2 ) at the factorization scale µ indicates that D Hc c (z, µ 2 )dz is the number of hadron H c produced at z and within dz. Consequently, the unpolarized differential cross section for The present work involves the three cases: Values of the Wigner 9j symbol in these cases reduce the unpolarized differential cross section to The unpolarized cross section for A + B → q 1 +q 2 + c +c → H c + X is The cross section depends on temperature and the center-of-mass energy √ s of mesons A and B.

Numerical results and discussions
For large-momentum charmonia we consider the following charmonium dissociation reactions: We solve the Schrödinger equation with the potential given in Eq. (17) to obtain φ Arel ( r q 1q2 ), φ Brel ( r cc ), and temperature-dependent meson masses where the up-quark mass, the strange-quark mass, and the charm-quark mass are 0.32, 0.5, and 1.51 GeV, respectively. The momentum-space wave functions [φ Arel ( p q 1q2 ) and φ Brel ( p cc )] appearing in Eqs. (22) and (23) are used in Eq. (29) to calculate dσ free .
According to Eqs. (17)- (19) and (29), we calculate ψ SSz+ final (χ A χ B ) S Sz for the central spin-independent potential and ψ SSz+ final s a · s b (χ A χ B ) S Sz for the spin-spin interaction. These spin matrix elements are listed in Table 1. They are independent of S z , and dσ free is thus independent of S z .
where √ s 0 is the threshold energy and equals the sum of the H c mass, theD mass, the q 1 mass, and theq 2 mass. The values of the parameters, a 1 , b 1 , c 1 , a 2 , b 2 , and c 2 , are listed in Tables 2-10. In these tables, d 0 is the separation between the peak's location on the √ s-axis and the threshold energy, and √ s z is the square root of the Mandelstam variable at which the cross section is 1/100 of the peak cross section that is obtained from the parametrization. We note that the parametrization of a reaction at a given temperature is valid in the √ s region where the cross-section curve for the reaction is displayed below.
Cross sections for the pion-charmonium reactions were obtained with an early version of FORTRAN code in Ref. [28]. After an error is removed, a new version is used to calculate pion-charmonium dissociation cross sections, which are smaller than those shown in Ref. [28]. We do not plot the cross sections, but list values of a 1 , b 1 , c 1 , a 2 and √ s z in Tables 11-13.
A low-energy reaction between a light meson and a charmonium produces two charmed mesons. While √ s increases from threshold, the cross section for every endothermic reaction rises from zero, arrives at a maximum value, and decreases, but the cross section for every exothermic reaction decreases rapidly from infinity and then may increase, reaching a maximum and decreasing. Peak cross sections of J/ψ dissociation in collisions with π, ρ, K, and K * mesons are collected from some references, and are listed in Table 14. In the last row of the table, we show cross sections at √ s = 11 GeV and T = 0 GeV for and K * J/ψ → D + X +D 0 X +D + s X +D * + X +D * 0 X +D * + s X, which are obtained in the present work. Since c → D * 0 and c → D * + s fragmentation functions are unknown, cross sections for πJ/ψ → D * 0 X, πJ/ψ → D * + s X, ρJ/ψ → D * 0 X, ρJ/ψ → D * + s X, KJ/ψ → D * 0 X, KJ/ψ → D * + s X, K * J/ψ → D * 0 X, and K * J/ψ → D * + s X are not calculated. In order to estimate the eight cross sections at √ s = 11 GeV and T = 0 GeV, it is assumed that the ratio of the cross section for inclusive D * 0 (D * + s ) production to the one for inclusive D * + production equals the ratio of the one for inclusive D 0 (D + s ) production to the one for inclusive D + production, that is, and so on. It is shown from the table that the cross sections at high √ s in the present work are comparable to those peak cross sections at low √ s.

Summary
We have obtained temperature-dependent cross sections for dissociation of largemomentum charmonia in collisions with π, ρ, K, and K * mesons in a mechanism where the collision between a light meson and a charmonium produces two quarks and two antiquarks first, then the charm quark fragments into charmed mesons, and the other three constituents combine with quarks and antiquarks created from vacuum to form two or more mesons. According to the mechanism, we have derived the transition amplitude for A + B → q 1 +q 2 + c +c from the wave functions of mesons A and B and of q 1 , Each group element forms a class, and the S 2 × S 2 group has four classes. The class operator is defined as the sum of all group elements in the class. The group has the following four class operators: In the class space, the class operators give where D kj (C i ) form a matrix that defines a representation of C i as Let Q and λ i individually stand for the eigenvector and the eigenvalue of C i in the class space, and Q is expressed as where q j are determined by Solving the above equations, we get the projection operator which differs from Q by a constant: Let P operate on the following six color wave functions: and then add normalized P ϕ 1 , P ϕ 2 , P ϕ 3 , P ϕ 4 , P ϕ 5 , and P ϕ 6 to give the color singlet of q 1 ,q 2 , c, andc, It satisfies Denote by φ q 1 cm (φq 2 cm , φ ccm , φc cm ) the color wave function of q 1 (q 2 , c,c). When m runs from 1 to 3, φ q 1 cm and φ ccm are r, g, and b, and φq 2 cm and φc cm arer,ḡ, andb, respectively.
With this notation the color singlet is given by the short expression, (φ q 1 cm φ ccn φq 2 cm φc cn + φ q 1 cm φ ccn φq 2 cn φc cm ).               Reaction  Reaction  Reaction    Reaction  Reaction    Reaction  Reaction    Reaction  Reaction