Cross sections for 2-to-1 meson-meson scattering

We study the processes $K\bar{K} \to \phi$, $\pi D \to D^\ast$, $\pi \bar{D} \to \bar{D}^\ast$, and the production of $\psi (4160)$ and $\psi (4415)$ mesons in collisions of charmed mesons or charmed strange mesons. The 2-to-1 meson-meson scattering involves a process where a quark and an antiquark from the two initial mesons annihilate into a gluon and subsequently the gluon is absorbed by the spectator quark or antiquark. Transition amplitudes for the scattering process derive from the transition potential in conjunction with mesonic quark-antiquark wave functions and the relative-motion wave function of the two initial mesons. We derive these transition amplitudes in the partial wave expansion of the relative-motion wave function of the two initial mesons so that parity and total-angular-momentum conservation are maintained. We calculate flavor and spin matrix elements in accordance with the transition potential and unpolarized cross sections for the reactions using the transition amplitudes. Cross sections for the production of $\psi (4160)$ and $\psi (4415)$ generally increase as the colliding mesons go through the cases of $D\bar{D}$, $D^*\bar{D}$, and $D^*\bar{D}^*$ or the cases of $D_s^+D_s^-$, $D_s^{*+}D_s^-$, and $D_s^{*+}D_s^{*-}$. We suggest the production of $\psi (4160)$ and $\psi (4415)$ as a probe of hadronic matter that results from the quark-gluon plasma created in ultrarelativistic heavy-ion collisions.

The reaction KK → φ was studied in Ref. [22] in a mesonic model. The fifteen reactions that lead to ψ(4160) or ψ(4415) as a final state have not been studied theoretically. Now we study KK → φ, πD → D * , πD →D * , and the fifteen reactions using quark degrees of freedom. The production of J/ψ is a subject intensively studied in relativistic heavy-ion collisions. The ψ(4160) and ψ(4415) mesons may decay into the J/ψ meson. Through this decay the fifteen reactions add a contribution to the J/ψ production in relativistic heavy-ion collisions. This is another reason why we study the fifteen reactions here.
This paper is organized as follows. In Sect. II we consider four Feynman diagrams and the S-matrix element for 2-to-1 meson-meson scattering, derive transition amplitudes and provide cross-section formulas. In Sect. III we present transition potentials corresponding to the Feynman diagrams and calculate flavor matrix elements and spin matrix elements. In Sect. IV we calculate cross sections, present numerical results and give relevant discussions. In Sect. V we summarize the present work.

II. FORMALISM
Lowest-order Feynman diagrams are shown in Fig. 1 for the reaction A(q 1q1 ) + B(q 2q2 ) → H(q 2q1 or q 1q2 ). A quark in an initial meson and an antiquark in the other initial meson annihilate into a gluon, and the gluon is then absorbed by a spectator quark or antiquark. The four processes q 1 +q 2 +q 1 →q 1 , q 1 +q 2 + q 2 → q 2 , q 2 +q 1 + q 1 → q 1 , and q 2 +q 1 +q 2 →q 2 in Fig. 1 give rise to the four transition potentials V rq 1q2q1 , V rq 1q2 q 2 , V rq 2q1 q 1 , and V rq 2q1q2 , respectively. Denote by E i and P i (E f and P f ) the total energy and the total momentum of the two initial (final) mesons, respectively; let E A (E B , E H ) be the energy of meson A (B, H), and V the volume where every meson wave function is where in the four processes mesons A and B go from the state vector | A, B > to the state vector | H > of meson H, and M rq 1q2q1 , M rq 1q2 q 2 , M rq 2q1 q 1 , and M rq 2q1q2 are the transition amplitudes given by where r ab is the relative coordinate of constituents a and b; r q 1q1 ,q 2q2 the relative coordinate of q 1q1 and q 2q2 ; p q 1q1 ,q 2q2 the relative momentum of q 1q1 and q 2q2 ; ψ + H the Hermitean conjugate of ψ H . The wave function of mesons A and B is and the wave function of meson H is where S A (S B , S H ) is the spin of meson A (B, H) with its magnetic projection quantum number S Az (S Bz , S Hz ); φ Arel (φ Brel , φ Hrel ), φ Acolor (φ Bcolor , φ Hcolor ), and χ S A S Az (χ S B S Bz , χ S H S Hz ) are the quark-antiquark relative-motion wave function, the color wave function, and the spin wave function of meson A (B, H), respectively; φ Hflavor and ϕ ABflavor are the flavor wave functions of meson H and of mesons A and B, respectively.
The development in spherical harmonics of the relative-motion wave function of mesons A and B (aside from a normalization constant) is given by where Y L i M i are the spherical harmonics with the orbital-angular-momentum quantum number L i and the magnetic projection quantum number M i , j L i are the spherical Bessel functions, andp q 1q1 ,q 2q2 (r q 1q1 ,q 2q2 ) denote the polar angles of p q 1q1 ,q 2q2 ( r q 1q1 ,q 2q2 ). Let χ SSz stand for the spin wave function of mesons A and B, which has the total spin S and its z component S z . The Clebsch-Gordan coefficients (S A S Az S B S Bz |SS z ) couple χ SSz to where S min =| S A − S B | and S max = S A + S B . Y L i M i and χ SSz are coupled to the wave function φ in JJz which has the total angular momentum J of mesons A and B and its z component J z , where J min =| L i − S |, J max = L i + S, and (L i M i SS z |JJ z ) are the Clebsch-Gordan coefficients. It follows from Eqs. (8)-(10) that the transition amplitude given in Eq. (2) becomes where φ J H J Hz = φ Hrel χ S H S Hz . Conservation of total angular momentum implies that J equals the total angular momentum J H of meson H and J z equals the z component J Hz of J H . This leads to Using the relation where (L iMi SS z |J H J Hz ) are the Clebsch-Gordan coefficients, we get Furthermore, we need the identity which is obtained with the help of ∞ 0 j l (pr)j l (p ′ r)r 2 dr = π 2p 2 δ(p−p ′ ) [23,24]. Substituting Eq. (15) in Eq. (14), we get Let r c and m c be the position vector and the mass of constituent c, respectively.
Then φ Arel , φ Brel , and φ Hrel are functions of the relative coordinate of the quark and the antiquark. We take the Fourier transform of V rq 1q2q1 and the mesonic quark-antiquark relative-motion wave functions: for the two upper diagrams in Fig. 1, and for the two lower diagrams. In Eq. (17) k is the gluon momentum, and in Eqs. (18)- (21) p ab is the relative momentum of constituents a and b. The spherical polar coordinates of p irm are expressed as (| p irm |, θ irm , φ irm ). The mesonic quark-antiquark relative-motion wave functions in momentum space are normalized as Integration over | p irm |, r q 1q1 , and r q 2q2 yields in which | p irm |=| p q 1q1 ,q 2q2 |. So far, we have obtained a new expression of the transition amplitude from Eq. (2).
Making use of the Fourier transform of V rq 1q2 q 2 , V rq 2q1 q 1 , and V rq 2q1q2 , from Eqs. (3)- (5) we obtain With these transition amplitudes the unpolarized cross section for A + B → H is where P A , m A , and J A (P B , m B , and J B ) are the four-momentum, the mass, and the total angular momentum of meson A (B), respectively. We calculate the cross section in the center-of-mass frame of the two initial mesons, i.e., with meson H at rest.
We use the notation K = Based on the formulas in Sect. II, we study the following reactions: From the Gell-Mann matrices and the Pauli matrices in the transition potentials, the expressions of the transition amplitudes in Eqs. (22) and (26) (4160) . The flavor matrix elements for KK → φ, πD → D * , and πD →D * are zero for the two lower diagrams, and the ones for the production of ψ(4160) and ψ(4415) are zero for the two upper diagrams.

IV. NUMERICAL CROSS SECTIONS AND DISCUSSIONS
The mesonic quark-antiquark relative-motion wave functions φ Arel , φ Brel , and φ Hrel in Eqs. (6) and (7) are solutions of the Schrödinger equation with the potential between constituents a and b in coordinate space [25], where D = 0.7 GeV, E = 0.6 GeV, T c = 0.175 GeV, A = 1.5[0.75 + 0.25(T /T c ) 10 ] 6 GeV, and λ = 25/16π 2 α ′ with α ′ = 1.04 GeV −2 ; T is the temperature; s a is the spin of constituent a; the quantity d is given in Ref. [25]; the function v is given by Buchmüller and Tye in Ref. [20]. The potential is obtained from perturbative QCD [20] and lattice QCD [26]. The masses of the up quark, the down quark, the strange quark, and the charm quark are 0.32 GeV, 0.32 GeV, 0.5 GeV, and 1.51 GeV, respectively. Solving the Schrödinger equation with the potential at zero temperature, we obtain meson masses that are close to the experimental masses of π, ρ, K, K * , J/ψ, χ c , ψ ′ , ψ(4160), ψ(4415), D, D * , D s , and D * s mesons [27]. Moreover, the experimental data of S-wave and Pwave elastic phase shifts for ππ scattering in vacuum [28,29] are reproduced in the Born approximation [5,25].
From the transition potentials, the color matrix elements, the flavor matrix elements, the spin matrix elements, and the mesonic quark-antiquark relative-motion wave functions, we calculate the transition amplitudes. As seen in Eq. (8), the development in spherical harmonics contains the summation over the orbital-angular-momentum quantum number L i that labels the relative motion between mesons A and B. However, not all orbital-angular-momentum quantum numbers are allowed. The orbital-angularmomentum quantum numbers are selected to satisfy parity conservation and J = J H , i.e., the total angular momentum of the two initial mesons equals the total angular momentum of meson H. The choice of L i thus depends on the total spin S of the two initial mesons. From the transition amplitudes we get unpolarized cross sections at zero temperature. The selected orbital-angular-momentum quantum numbers and the cross sections are shown in Table 7 In Table 7 the cross section for KK → φ equals 5.96 mb. The magnitude 5.96 mb is slightly larger than the peak cross section of KK → K * K * for total isospin I = 0 at zero temperature, and is roughly 5 times the peak cross section of KK → K * K * for I = 1 in Ref. [5]. The case KK → K * K * may be caused by a process where a quark in an initial meson and an antiquark in another initial meson annihilate into a gluon and subsequently the gluon creates another quark-antiquark pair. The magnitude is much larger than the peak cross sections of KK → πKK for I=1 and I f πK = 3/2 and for I=1 and I f πK = 1/2 at zero temperature in Ref. [30], where I f πK is the total isospin of the final π andK mesons. The case KK → πKK is governed by a process where a gluon is emitted by a constituent quark or antiquark in the initial mesons and subsequently the gluon creates a quark-antiquark pair. The magnitude is also much larger than the peak cross section of KK → K * K * for I = 1 at zero temperature in Ref. [31]. The case KK → K * K * for I = 1 can be caused by quark interchange between the two colliding mesons. The cross section for πD → D * is particularly large. This means that the reaction easily happens.
The large cross section is caused by the very small difference between the D * mass and the sum of the π and D masses.
The transition potentials involve quark masses. The charm-quark mass is larger than the strange-quark mass, and the transition potentials with the charm quark are smaller than the ones with the strange quark. The cross section for DD → ψ(4160) is thus smaller than the one for KK → φ. Because the D * radius is larger than the D radius, the cross section for D * D * → ψ(4160) is larger than the one for D * D → ψ(4160), and the cross section for D * D → ψ(4160) is larger than the one for DD → ψ(4160). Since D + s (the antiparticle of D − s ) consists of a charm quark and a strange antiquark, the cross section for D + s D − s → ψ(4160) is smaller than the one for DD → ψ(4160). Since the D * ± s radii are larger than the D ± s radii, the cross section for D * + s D − s → ψ(4160) is larger than the one for D + s D − s → ψ(4160). As seen in Table 7, we have the increasing order of the cross sections for D + s D − s → ψ(4415), D * + s D − s → ψ(4415), and D * + s D * − s → ψ(4415). The radial part of the quark-antiquark relative-motion wave function of ψ(4415) has three nodes. The radial wave function on the left of a node has a sign different from the one on the right of the node. The nodes lead to cancellation between the positive radial wave function and the negative radial wave function in the integration involved in the transition amplitudes. Consequently, the cross section for D * D * → ψ(4415) is much smaller than that for D * D → ψ(4415).
The cross sections for A + B → ψ(4160) and for A + B → ψ(4415) are compared in Table 7. Since the ψ(4160) mass is smaller than the ψ(4415) mass, the production of ψ(4160) is easier than the production of ψ(4415). The cross section for the former [for example, DD → ψ(4160)] is larger than the cross section for the latter [for example, In the present work the ψ(4160) and ψ(4415) mesons come from fusion of D,D, D * , D * , D s , and D * s mesons. The reason why we are interested in the reactions is that ψ(4160) and ψ(4415) may decay into J/ψ which is an important probe of the quark-gluon plasma produced in ultrarelativistic heavy-ion collisions. We do not investigate the χ c0 (2P ) and χ c2 (2P ) mesons because they cannot decay into the J/ψ meson. The χ c1 (2P ) meson may decay into the J/ψ meson, but DD → χ c1 (2P ) allowed by energy conservation does not simultaneously satisfy the parity conservation and the conservation of the total angular momentum. Therefore, we do not consider DD → χ c1 (2P ). The production of χ c1 (2P ) from fusion of other charmed mesons is forbidden by energy conservation. πD ± s → D * ± s is not allowed because of violation of isospin conservation, and KD → D * + s andKD → D * − s because of violation of energy conservation.
As seen in Eq. (49), the potential between two constituents depends on temperature.
The meson mass obtained from the Schrödinger equation with the potential thus depends on temperature. The temperature dependence of meson masses is shown in Figs. 2-4. In vacuum the φ mass is larger than two times the kaon mass, and so KK → φ takes place. Since the φ mass in Fig. 2 decreases faster than the kaon mass with increasing temperature, the φ mass turns smaller than two times the kaon mass. The reaction thus does not occur in the temperature region 0.6T c ≤ T < T c . In Fig. 3 the D * mass decreases faster than the pion and D masses, and the D * mass is smaller than the sum of the pion mass and the D mass. πD → D * and πD →D * also do not occur for 0.6T c ≤ T < T c . It is shown in Refs. [20,21] that ψ(4160) and ψ(4415) can be individually interpreted as the 3 3 S 1 and 4 3 S 1 quark-antiquark states, but they are dissolved in hadronic matter when the temperature is larger than 0.97T c and 0.87T c , respectively [32]. Their masses are thus plotted only for 0.6T c ≤ T < 0.97T c and for 0.6T c ≤ T < 0.87T c in Fig. 4, and are smaller than the sum of the masses of the two initial mesons that yield them. Therefore, in hadronic matter where the temperature is constrained by 0.6T c ≤ T < T c , we cannot see the production of ψ(4160) and ψ(4415) from the fusion of two charmed mesons and of two charmed strange mesons. T c is the critical temperature at which the phase transition between the quark-gluon plasma and hadronic matter takes place. Since ψ(4160) and ψ(4415) are dissolved in hadronic matter when the temperature is larger than 0.97T c and 0.87T c , respectively, they cannot be produced in the phase transition, but they can be produced in the following reactions in hadronic matter: (4160) Therefore, ψ(4160) and ψ(4415) may provide us with information on hadronic matter, and are a probe of hadronic matter that results from the quark-gluon plasma.

V. SUMMARY
With the process where one quark annihilates with one antiquark to create a gluon and subsequently the gluon is absorbed by a spectator quark or antiquark, we have studied 2-to-1 meson-meson scattering. In the partial wave expansion of the relative-motion