Production of $\psi (4040)$, $\psi (4160)$, and $\psi (4415)$ mesons in hadronic matter

We present the first study of production of $\psi (4040)$, $\psi (4160)$, and $\psi (4415)$ mesons in hadronic matter. Quark interchange between two colliding charmed mesons leads to the production of the three mesons. We calculate unpolarized cross sections for the reactions, $D\bar{D} \to \rho R$, $D\bar{D}^* \to \pi R$, $D\bar{D}^* \to \rho R$, $D^*\bar{D} \to \pi R$, $D^*\bar{D} \to \rho R$, $D^*\bar{D}^* \to \pi R$, and $D^*\bar{D}^* \to \rho R$, where $R$ stands for $\psi (4040)$, $\psi (4160)$, and $\psi (4415)$. In the temperature region covering hadronic matter the peak cross sections of producing $\psi (4040)$ are similar to or larger than the ones of producing $\psi (4415)$, and the latter are generally larger than those of producing $\psi (4160)$. With the cross sections we establish new master rate equations for $\psi (4040)$, $\psi (4160)$, and $\psi (4415)$. The equations are solved for central Pb-Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV at the Large Hadron Collider. Solutions of the equations show that the $\psi (4040)$ number density at kinetic freeze-out of hadronic matter is larger than the $\psi (4415)$ number density which is larger than the $\psi (4160)$ number density.

In the present work, we study the production of ψ(4040), ψ(4160), and ψ(4415) via quark interchange between two colliding charmed mesons in hadronic matter.Since the temperature of hadronic matter varies during its expansion, we need to study temperature dependence of cross sections for the production of the three charmonia.Furthermore, we establish new master rate equations to account for charmonium number densities that result from the reactions of charmed mesons.Number densities are obtained from the equations for central Pb-Pb collisions at the center-of-mass energy per nucleon-nucleon pair √ s N N = 5.02 TeV at the Large Hadron Collider (LHC).
This paper is organized as follows.In Sect.II we provide cross-section formulas for 2-to-2 scattering of charmed mesons.In Sect.III we give master rate equations for ψ(4040), ψ(4160), and ψ(4415) mesons.In Sect.IV we show numerical cross sections and number densities of the three mesons produced in hadronic matter.Relevant discussions are given.
In Sect.V we summarize the present work.

COLLISIONS
Production of ψ(4040), ψ(4160), and ψ(4415) mesons in hadronic matter concerns the reaction A(cq 2 )+B(q 1 c) → C(q 1 q2 )+D(cc) where c is a charm quark, q 1 a light quark, and q2 a light antiquark.This reaction is caused by interchange of c and q 1 between mesons Two forms are involved in the Born-order meson-meson scattering in the quark interchange mechanism [16].Scattering in the prior form as seen in Fig. 1 means that gluon exchange takes place prior to quark interchange.Scattering in the post form shown in Fig. 2 means quark interchange is followed by gluon exchange.In the two figures the interaction between constituents a and b is indicated by the dot-dashed line, and its potential is V ab .Scattering in the prior form needs interactions between the two constituents of meson A and the ones of meson B, which are described by the potentials V cc , V q2 q 1 , V cq 1 , and V q2 c.The transition amplitude for scattering in the prior form is where where m a is the mass of constituent a; P and P ′ are the three-dimensional momenta of mesons A and C in the center-of-mass frame of the two initial mesons, respectively.The first, second, third, and fourth terms enclosed by the brackets correspond to the upper left, upper right, lower left, and lower right diagrams in Fig. 1, respectively.Scattering in the post form needs interactions between the two constituents of meson C and the ones of meson D, which are described by the potentials V q 1 c, V q2 c , V cq 1 , and V q2 c.The transition amplitude for scattering in the post form is With the wave functions of the initial mesons and the final mesons, we have The first, second, third, and fourth terms enclosed by the brackets correspond to the upper left, upper right, lower left, and lower right diagrams in Fig. 2, respectively.In Eqs. ( 4) and ( 6), φ Arel , φ Brel , φ Crel , and φ Drel are functions of p cq 2 , p q 1 c, p q 1 q2 , and p cc , respectively.These relative momenta equal the expressions enclosed by the parentheses that follow φ Arel , φ Brel , φ Crel , and φ Drel .In Eqs. ( 3)-( 6), V q2 c are momentum-space potentials that depend on momenta attached to the dot-dashed lines in Figs. 1 and 2. The momenta equal the expressions enclosed by the parentheses As seen in Eqs. ( 4) and ( 6 Denote by T the temperature and by T c the critical temperature between quark-gluon plasmas and hadronic matter.Let s a be the spin of constituent a.The mesonic quarkantiquark relative-motion wave functions, φ Arel , φ Brel , φ Crel , and φ Drel , are the Fourier transform of the solutions of the Schrödinger equation with the potential between constituents a and b in coordinate space [17], where r ab is the relative coordinate of constituents a and b; ξ 1 = 0.525 GeV, ξ 2 = 1.5[0.75+0.25(T /T c ) 10 ] 6 GeV, ξ 3 = 0.6 GeV, T c = 0.175 GeV, and λ = 25/16π 2 α ′ with α ′ = 1.04 GeV −2 ; v is a function of r ab , and is given by Buchmüller and Tye in Ref. [18]; the quantity d is related to constituent masses by where d 1 = 0.15 GeV and d 2 = 0.705.The short-distance part of the potential is obtained from perturbative quantum chromodynamics [18], and the temperature dependence from lattice gauge calculations [19].The lattice calculations gave a temperature-dependent quark potential at intermediate and long distances.The potential at long distances has a distance-independent value that decreases with increasing temperature.The first term in Eq. ( 7) is the confining potential that corresponds to the lattice results.The expres- in the second term arises from one-gluon exchange plus perturbative one-and two-loop corrections in vacuum [18], and the factor exp(−ξ 3 r ab ) is a medium modification factor.The third term is the smeared spin-spin interaction that comes from one-gluon exchange between constituents a and b [11].The fourth term is the spin-spin interaction that originates from perturbative one-and two-loop corrections to one-gluon exchange [20].The fifth term is the tensor interaction that arises from one-gluon exchange plus perturbative one-and two-loop corrections [20].
In Fig. 3 we plot the central spin-independent potential which is the sum of the first and second terms on the right-hand side of Eq. ( 7), the spin-spin interaction which is the sum of the third and fourth terms, the tensor interaction, and the potential V cc between the charm quark and the charm antiquark inside the ψ(4160) meson at zero temperature by the dotted, dashed, dot-dashed, and solid curves, respectively.The tensor interaction dominates V cc at very short distances, but is short-range.At short distances, V cc increases with increasing distance, while the spin-spin interaction is positive and decreases rapidly.
At long distances, V cc approaches 0.91 GeV, while the spin-spin interaction almost becomes zero.
The Schrödinger equation with the potential V ab ( r ab ) at zero temperature gives meson masses that are close to the experimental masses of π, ρ, K, K * , J/ψ, χ c , ψ ′ , ψ(3770), ψ(4040), ψ(4160), ψ(4415), D, D * , D s , and D * s mesons [21].The experimental data of S-and P -wave elastic phase shifts for ππ scattering in vacuum [22,23] are reproduced in the Born approximation [24,25].In the calculations of the meson masses and the phase shifts, the masses of the up quark, the down quark, the strange quark, and the charm quark are kept as 0.32 GeV, 0.32 GeV, 0.5 GeV, and 1.51 GeV, respectively.In Eqs. ( 4) and (6) , and V q2 c , respectively.From m A , m B , m C , and m D , which are the masses of mesons A, B, C, and D, P and P ′ are given by where the Mandelstam variable s = (P A + P B ) 2 is defined from the four-momenta of mesons A and B, P A and P B .Let J Az (J Bz , J Cz , J Dz ) denote the magnetic projection quantum number of the total angular momentum J A (J B , J C , J D ) of meson A (B, C, D).When the transition amplitude for scattering in the prior form equals the one for scattering in the post form, i.e., no post-prior discrepancy exists, the unpolarized cross section may be calculated from M prior fi , or from M post fi , where θ is the angle between P and P ′ .When the transition amplitude for scattering in the prior form does not equal the one for scattering in the post form, i.e., the post-prior discrepancy occurs, we treat M prior fi and M post fi on a completely equal footing, and the unpolarized cross section for In fact, this equation gives σ unpol = σ prior unpol = σ post unpol when M prior fi = M post fi .Hence, Eq. ( 13) is used to calculate the unpolarized cross section.The cross section depends on temperature and the total energy of the two initial mesons in the center-of-mass frame.In hadronic matter number densities for ψ(4040), ψ(4160), and ψ(4415) mesons change with respect to time and space according to the following rate equations,

III. MASTER RATE EQUATIONS
where µ is the space-time index, and u µ = (u 0 , u) is the four-velocity of a fluid element in hadronic matter.n ψ(4040) , n ψ(4160) , and n ψ(4415) are the number densities of ψ(4040), ψ(4160), and ψ(4415), if R stands for ψ(4040), ψ(4160), and ψ(4415), respectively.In the present work, we take into account only these reactions, where light mesons in these final states are limited to pions and ρ mesons.The source terms are given by The thermal-averaged cross section with the relative velocity of two initial mesons is defined as where f i (k i ) and f j (k j ) are the momentum distribution functions of mesons i and j with the four-momenta k i and k j in the rest frame of hadronic matter, respectively; is the isospin-averaged unpolarized cross section for ij → i ′ j ′ , and is obtained from the unpolarized cross section given in Eq. ( 13) according to formulas given in Appendix A; v ij is the relative velocity of meson i with mass m i and meson j with mass m j , The momentum distribution functions in Eq. ( 16) with the subscripts suppressed are expressed as where the term ∞ l=1 c l (k • u) l indicates deviation from equilibrium.
Denote by k ′ the meson momentum in the local reference frame established on the fluid element, and the meson energy is k ′0 = k • u.The number density of the charmed meson is where g is the degeneracy factor, and m is the mass of the charmed meson.The thermalaveraged cross section is with The present work relates to hadronic matter produced in central nucleus-nucleus collisions.Denote by (x, y, z) the Cartesian coordinates of the fluid element (the origin of the local reference frame) in hadronic matter.The four-velocity of the fluid element takes the form u µ = γ( t τ , v r cos φ, v r sin φ, z τ ) where t is the time, τ = √ t 2 − z 2 the proper time, v r the transverse velocity, and γ = 1/ 1 − v 2 r the Lorentz factor.The z-axis in the rest frame of hadronic matter is set along the moving direction of a nucleus, and goes through the nuclear center.In central collisions hadronic matter has only radial flow and longitudinal flow.In terms of the proper time and the cylindrical polar coordinates (r, φ, z), the left-hand side in Eq. ( 14) becomes Hadronic matter possesses cylindrical symmetry.Equation ( 24) is then reduced to Combining Eqs. ( 14), (15), and (25), we get The temperature and the transverse velocity involved in Eq. ( 26) are given by the relativistic hydrodynamic equation, where T µν is the energy-momentum tensor, where ǫ is the energy density, P the pressure, g µν the metric, η the shear viscosity, and A parametrization of the shear viscosity is given in Ref. [26].

IV. NUMERICAL RESULTS AND DISCUSSIONS
Hadronic matter created in ultrarelativistic heavy-ion collisions changes during expansion.Numerical cross sections obtained from Eq. ( 13) show remarkable temperature dependence.ψ(4040), ψ(4160), and ψ(4415) mesons produced in hadronic matter are appropriate to measurements.

A. Cross sections
The potential given in Eq. ( 7) at large distances decreases with increasing temperature, i.e., confinement becomes weaker and weaker.We solve the Schrödinger equation with the potential to obtain meson masses and mesonic quark-antiquark relative-motion wave functions in coordinate space.The wave functions depend on temperature, and give meson radii which increase with increasing temperature.From the Fourier transform we get the potential and the wave functions in momentum space, which are used in Eqs. ( 4) and (6).
We calculate transition amplitudes in the prior form and in the post form, which lead to For convenience they are parametrized as for endothermic reactions and for exothermic reactions.√ s 0 is the threshold energy, which equals the sum of the masses of the two initial (final) mesons for the exothermic (endothermic) reaction.The temperature-dependent charmonium masses obtained from the Schrödinger equation are parametrized as In hadronic matter where the temperature is constrained by 0.6T c ≤ T < T c , the masses of ψ(4040), ψ(4160), and ψ(4415) are smaller than the sum of two open-charm mesons.Therefore, the production of the three charmonia from fusion of two open-charm mesons is not allowed [17].
The transition amplitudes for scattering in the prior form and in the post form include the Fourier transform of the coordinate-space potential given in Eq. ( 7): where Q is the momentum attached to the dot-dashed lines in Figs. 1 and 2, K = 3/16π 2 α ′ , ρ(q 2 ) is the physical running coupling constant [18], and s az (s bz ) is the magnetic projection quantum number of s a ( s b ).The expressions of M prior fi and M post fi with this momentum-space potential involve seven-dimensional integrals.Because of the two sines and the cosine in the fourth and fifth terms, in a short computational time to carry out integration at a given temperature and a given value of √ s is impossible.In order to reduce the computational time, the two integrals in the fourth and fifth terms are parametrized as where the unit of Q is fm −1 .The momentum-space potential with the two parametrizations causes the post-prior discrepancy.To know the contribution from scattering in the prior form or in the post form to the unpolarized cross section σ unpol , in Fig. 19 we compare the cross sections (σ prior unpol and σ post unpol ) for scattering in the prior form and in the post form with the unpolarized cross section (σ unpol ) for the exothermic reaction GeV) has a maximum (minimum, maximum), and σ prior unpol and σ post unpol deviate from σ unpol by 5.6% (87%, 59.7%).Contributions from scattering in the prior form and in the post form change with increasing √ s.
The potential used by Barnes and Swanson in Ref. [16] includes the color Coulomb interaction, the linear confining interaction, and the one-gluon-exchange spin-spin hyperfine interaction.Describing quark-antiquark relative motion in all mesons by a Gaussian wave function, they obtained that the cross section for elastic ππ (KK) scattering for total isospin I = 2 (I = 1) in the post form equals the one in the prior form.This means that the scattering amplitude in the post form equals the scattering amplitude in the prior form even though the Gaussian wave function is not the exact quark-antiquark relative-motion wave functions that are solutions of the Schrödinger equation with the potential.Matrix elements of the spin-spin hyperfine interaction in scattering in the prior form are given by Eqs. ( 71)-(74) in Ref. [16] when the four constituents of scattering mesons have the same mass.However, the four equations are not enough to help us in understanding that the scattering amplitude in the post form equals the one in the prior form.Therefore, in Ref. [27] we present analytic expressions of all matrix elements of the spin-spin interaction in scattering in the post form and in the prior form allowing that the four constituents have different masses and that different Gaussian wave functions are used to describe quark-antiquark relative motion of different mesons.Every matrix element is the product of a flavor matrix element, a color matrix element, a spin matrix element, and a spatial matrix element.The two upper diagrams in Fig. 1  When the sum of the two matrix elements corresponding to the capture diagrams in the post form does not equal the sum in the prior form, the post-prior discrepancy occurs.
The same flavor matrix element is applied to the eight diagrams in Figs. 1 and 2. The color matrix elements associated with the four capture diagrams are all -4/9.For elastic scattering of two pseudoscalar mesons, the spin matrix elements corresponding to the four capture diagrams are the same.Hence, the matrix element of the spin-spin interaction in a (the other) capture diagram in the post form equals the one in a (the other) capture diagram in the prior form, if the Gaussian wave functions of the initial and final mesons are identical and the quarks or the antiquarks of scattering mesons have the same flavor [27].
Numerical calculations show that the matrix element of the central spin-independent potential in the post form equals the one in the prior form [27]. Therefore, the scattering amplitude in the post form equals the scattering amplitude in the prior form, and no post-prior discrepancy happens in elastic scattering of two pseudoscalar mesons [16,27], when the quark-antiquark relative motion of the initial and final mesons are described by the same Gaussian wave function and if the quarks or the antiquarks of scattering mesons have the same flavor.No post-prior discrepancy is also true for elastic scattering between two vector mesons or between a pseudoscalar meson and a vector meson [27].
In Ref.If two or more mesons in elastic meson-meson scattering are described by different Gaussian wave functions, the scattering amplitudes in the post form and in the prior form are not the same, and the post-prior discrepancy takes place [27].For inelastic scattering of two mesons, the spin matrix elements associated with the four capture diagrams can not guarantee that the sum of the two matrix elements corresponding to the capture diagrams in the post form equals the sum in the prior form, the scattering amplitudes in the two forms are thus not identical, and the post-prior discrepancy occurs [16].If the exact mesonic quark-antiquark wave functions are used, no post-prior discrepancy exists.However, the exact wave functions are not amenable to analytic calculations.
To get analytic expressions of spatial matrix elements, the sum of several Gaussian wave functions was suggested in Ref. [16] to approach the exact quark-antiquark relative-motion wave functions.The more Gaussian wave functions that are used, the smaller post-prior discrepancy that is observed.

B. Number densities
Since charmed mesons are well measured in Pb-Pb collisions at the LHC, we get their distribution functions f (k) from experimental data.The Cooper-Frye formula [28] is where E, k µ , and N are the energy, the four-momentum, and the number of the charmed meson, respectively; σ f is the freeze-out surface with the normal vector dσ µ .With the space-time rapidity η = 1 2 ln t+z t−z , we get where y, k T , and m T are the rapidity, the transverse momentum, and the transverse mass of the charmed meson, respectively; ϕ is the angle between the transverse momentum and the x-axis; R c and R fz are the radii of hadronic-matter surface at hadronization and at kinetic freeze-out, respectively.In Ref. Quark-gluon matter initially produced in Pb-Pb collisions at LHC energies is not a thermal state.Undergoing elastic parton-parton-parton scattering and parton-parton scattering, quark-gluon matter acquires a temperature in a short time and becomes a quark-gluon plasma [30][31][32].Hydrodynamic models are applied to the quark-gluon plasma [33,34].For central Pb-Pb collisions at √ s N N =5.02TeV we get 0.82 GeV as the initial temperature of the quark-gluon plasma at a time of the order of 0.65 fm/c and 10.05 fm/c as the proper time at which hadronization of the quark-gluon plasma occurs.
We start solving the hydrodynamic equation (Eq.( 27)) for hadronic matter at the time 10.05 fm/c with T c = 0.175 GeV.Since ψ(4040), ψ(4160), and ψ(4415) mesons are dissolved in hadronic matter when the temperature is larger than 0.97T c , 0.95T c and 0.87T c , respectively, their number densities are zero above the three dissociation temperatures.
From the dissociation temperatures, Eq. ( 26) is solved until kinetic freeze-out to get number densities that are functions of the proper time and the radius.Variation of the number densities with respect to the proper time at r = 0 fm is drawn in Fig. 20, and radius dependence at kinetic freeze-out is plotted in Fig. 21.Hadronic matter produced from the quark-gluon plasma expands, and its temperature decreases from the critical temperature.When the temperature arrives at 0.97T c , 0.95T c , and 0.87T c , production of ψ(4040), ψ(4160), and ψ(4415) get started from reactions of charmed mesons, respectively.Therefore, when the proper time increases from 10.83 fm/c, 11.38 fm/c, and 13.95 fm/c, respectively, the number densities of ψ(4040), ψ(4160), and ψ(4415) increase.However, the three mesons produced at r = 0 fm spread out, and this reduces the number densities.
When the reduced amount exceeds the increased amount, the number densities decrease as seen in Fig. 20.In Fig. 21 the number densities decrease slowly when r increases from zero.From the last subsection, we have already known that at T = 0.65T c , 0.  4160)).Therefore, the number density of ψ(4040) is larger than the one of ψ(4415) which is larger than that of ψ(4160).At kinetic freeze-out, hadronic matter has a volume of the order of 6 × 10 4 fm 3 .As a consequence of the number densities, the numbers of ψ(4040), ψ(4160), and ψ(4415) at kinetic freeze-out are 0.25, 0.1, and 0.18 , respectively.

V. SUMMARY
We have studied the production of ψ(4040), ψ(4160), and ψ(4415) mesons in collisions of charmed mesons in the quark interchange mechanism and space-time distribution of the three charmonia in hadronic matter created in ultrarelativistic heavy-ion collisions with the master rate equations.Formulas of the transition amplitudes are given explicitly.The temperature dependence of the quark potential, the meson masses, and the mesonic quark- R denotes ψ(4040), ψ(4160), or ψ(4415).In a collision of two charmed mesons, the peak cross sections of producing ψ(4040) at T = 0.65T c , 0.75T c , 0.85T c , 0.9T c , and 0.95T c are similar to or larger than the ones of producing ψ(4415), and the latter are generally larger than those of producing ψ(4160).The numerical cross sections are parametrized so that they can be conveniently used in the master rate equations.For hadronic matter created in central nucleus-nucleus collisions, the master rate equations are given in terms of the proper time and the cylindrical polar coordinates.Below the dissociation temperatures of ψ(4040), ψ(4160), and ψ(4415) the twenty-one reactions produce the three charmonia, and the master rate equations give number densities that first increase with increasing time.Among the number densities of ψ(4040), ψ(4160), and ψ(4415), at kinetic freezeout the one of ψ(4040) is largest and that of ψ(4160) is smallest.One can find the three charmonia in the large volume of hadronic matter created in central Pb-Pb collisions at √ s N N = 5.02 TeV.
where the symbol P qi is the operator that implements quark interchange in flavor space.
Because of isospin conservation, I ′ = I and I ′ z = I z , Since flavor matrix elements are independent of I z , Finally, the average over the isospin states of the two initial mesons and the sum over the isospin states of the two final mesons lead to the isospin-averaged unpolarized cross where σ unpol (I, √ s, T ) is given in Eq. ( 13) for the I channel.The spin-independent potential, the spin-spin interaction, and the tensor interaction are shown by the dotted, dashed, and dot-dashed curves, respectively.The solid curve indicates the sum of the spin-independent potential, the spin-spin interaction, and the tensor interaction.
and B. Denote the spin of meson A (B, C, D) by S A (S B , S C , S D ) and its magnetic projection quantum number by S Az (S Bz , S Cz , S Dz ).The wave function of mesons A and B isψ AB = φ Arel φ Brel φ Acolor φ Bcolor χ S A S Az χ S B S Bz ϕ ABflavor ,(1)and the wave function of mesons C and D isψ CD = φ Crel φ Drel φ Ccolor φ Dcolor χ S C S Cz χ S D S Dz ϕ CDflavor ,(2)where φ Arel (φ Brel , φ Crel , φ Drel ), φ Acolor (φ Bcolor , φ Ccolor , φ Dcolor ), and χ S A S Az (χ S B S Bz , χ S C S Cz , χ S D S Dz ) are the mesonic quark-antiquark relative-motion wave function, the color wave function, and the spin wave function of meson A (B, C, D), respectively; the flavor wave function ϕ ABflavor of mesons A and B possesses the same isospin as the flavor wave function ϕ CDflavor of mesons C and D.
denotes the energy of meson A (B, C, D), and p ab is the relative momentum of constituents a and b.The three-dimensional momentum of constituent a in the initial (final) mesons is labeled as p a ( p ′ a ).Substituting Eqs.(1) and (2) in Eq. (3), we obtain ), each of M prior fi and M post fi contains four integrals.The first and second terms of M prior fi look different than the first and second terms of M post fi , but the third and fourth terms of M prior fi are identical with the third and fourth terms of M post fi .If the sum of the first and second terms of M prior fi does not equal the one of M post fi , M prior fi does not equal M post fi , and the post-prior ambiguity appears.


for the vector isospin doublets.

D
D * → πψ(4040) at T /T c = 0.85 and the endothermic reaction D D * → ρψ(4040).At few energies, for example, √ s = 3.765 GeV in the right panel, the dotted curves and the dashed curves cross, i.e., σ prior unpol = σ post unpol = σ unpol , which indicates that scattering in the prior form and in the post form make the same contribution to σ unpol .When √ s increases from 3.6 GeV, the dotted curves approach the dashed curves, which means that the post-prior discrepancy becomes smaller and smaller.The threshold energy of D D * → πψ(4040) at T /T c = 0.85 is 3.45709 GeV.At √ s = 3.4571 GeV the cross section for scattering in the prior form is by 81.7% smaller than σ unpol , while the cross section for scattering in the post form is by 81.7% larger than σ unpol .At √ s = 3.47209 GeV the solid curve exhibits a maximum, and σ prior unpol (σ post unpol ) is by 68.7% larger (smaller) than σ unpol .In the right panel, σ unpol for D D * → ρψ(4040) at √ s = 3.51319 GeV (3.52569 GeV, 3.55569 (Fig. 2) are capture diagrams in which the interacting quark-antiquark pair scatter into (come from) the same final (initial) meson.The four lower diagrams in Figs. 1 and 2 are transfer diagrams in which the two interacting quarks or antiquarks scatter into different final mesons.The transfer diagrams in the post and prior forms are identical, and the capture diagrams in the two forms look different.A matrix element is associated with a diagram.
[16] H 0 (A), H 0 (B), H 0 (C), and H 0 (D) of mesons A, B, C, and D in the scattering A + B → C + D individually consist of the kinetic energies of the quark and the antiquark and the potential between the quark and the antiquark.The Schrödinger equation with H 0 provides masses and exact quark-antiquark relative-motion wave functions of the four mesons.The interaction H I (A, B) between the two constituents of meson A and the two constituents of meson B turns the color-singlet states A and B into color-octet states.During propagation of quarks and antiquarks, quark interchange causes a quark and an antiquark to form the color-singlet state C as well as the other quark and the other antiquark to form the color-singlet state D. This decomposition of the Hamiltonian of the four constituents, H = H 0 (A) + H 0 (B) + H I (A, B), reflects the prior form of scattering.Certainly, quark interchange between mesons A and B can take place before the interaction H I (C, D) between the two constituents of meson C and the two constituents of meson D, and breaks up mesons A and B to yield two color-octet states.The interaction H I (C, D) makes the two color-octet states colorless so that the bound states C and D are formed.Scattering in the post form thus gives rise to this decomposition H = H 0 (C) + H 0 (D) + H I (C, D).Therefore, the post-prior discrepancy (originally the H I (A, B) matrix element does not equal the H I (C, D) matrix element) is related to the decomposition of the Hamiltonian.
[29] the D-meson production yields are measured at midrapidity (| y |< 0.5) as functions of transverse momentum.y min and y max are thus set as -0.5 and 0.5, respectively.Fits to the experimental data of dN/dp T of prompt D + , D 0 , and D * + mesons at p T < 8 GeV/c in central Pb-Pb collisions at √ s N N = 5.02 TeV with T = 0.1686 GeV give l = 11, c l = 6 × 10 −12 , for D + meson; l = 9, c l = 3 × 10 −9 , for D 0 meson; l = 14, c l = 5 × 10 −16 , for D * + meson; c l equal zero for other l values.This means that only one term in the sum ∞ l=1 c l (k • u) l is needed for each D meson.
75T c , 0.85T c , 0.9T c , and 0.95T c the peak cross sections of D D → ρψ(4040) (D D * → πψ(4040), D D * → ρψ(4040), D * D * → πψ(4040), D * D * → ρψ(4040)) are similar to or larger than the ones of D D → ρψ(4415) (D D * → πψ(4415), D D * → ρψ(4415), D * D * → πψ(4415), D * D * → ρψ(4415)), and the latter are generally larger than those of D D → ρψ(4160) (D D * → πψ(4160), D D * → ρψ(4160), D * D * → πψ(4160), D * D * → ρψ( antiquark relative-motion wave functions lead to remarkable temperature dependence of unpolarized cross sections.We have obtained the unpolarized cross sections for D D → ρR, D D * → πR, D D * → ρR, D * D → πR, D * D → ρR, D * D * → πR, and D * D * → ρR, where APPENDIX A: DERIVE THE ISOSPIN-AVERAGED CROSS SECTION Denote the isospin of meson A (B, C, D) by I A (I B , I C , I D ) and its z component by I Az (I Bz , I Cz , I Dz ).Let φ I A I Az Aflavor , φ I B I Bz Bflavor , φ I C I Cz Cflavor , and φ I D I Dz Dflavor be the flavor wave functions of mesons A, B, C, and D, respectively.φ I A I Az Aflavor and φ I B I Bz Bflavor are coupled to the flavor wave function ϕ IIz ABflavor with the total isospin I and its z component I z .φ I C I Cz Cflavor and φ I D I Dz Dflavor are coupled to the flavor wave function ϕ I ′ I ′ zCDflavor with the total isospin I ′ and its z component I ′ z :φ I A I Az Aflavor φ I B I Bz Bflavor = IIz (I A I Az I B I Bz | II z )ϕ IIz ABflavor ,(39)φ I C I Cz Cflavor φ I D I Dz Dflavor = I ′ I ′ z (I C I Cz I D I Dz | I ′ I ′ z )ϕ I ′ I ′ z CDflavor ,(40)where (I A I Az I B I Bz | II z ) and (I C I Cz I D I Dz | I ′ I ′ z ) are the Clebsch-Gordan coefficients.The isospin part of the transition amplitudes is

Figure 3 :
Figure 3: Potentials as functions of the distance between the quark and the antiquark.

Figure 19 :
Figure 19: Cross sections for D D * → πψ(4040) in the left panel and for D D * → ρψ(4040) in the right panel at T /T c = 0.85.σ unpol , σ prior unpol , and σ post unpol are shown by the solid, dotted, and dashed curves, respectively.

Figure 20 :
Figure 20: Number densities as functions of τ at r = 0 fm.

Figure 21 :
Figure 21: Number densities as functions of r at kinetic freeze-out.

Table 3 :
The same as Table1except for D D * reactions.

Table 4 :
The same as Table2except for D * D * reactions.

Table 5 :
The same as Table1except for D * D * reactions.