Cross sections for 2-to-1 meson-meson scattering
Wan-Xia Li Xiao-Ming Xu H. J. Weber
hep-ph nucl-th





Cross sections for 2-to-1 meson-meson scattering
    Wan-Xia Li^1, Xiao-Ming Xu^1, and H. J. Weber^2
    
===================================================



^1Department of Physics, Shanghai University, Baoshan,
Shanghai 200444, China
^2Department of Physics, University of Virginia, Charlottesville,
VA 22904, USA


We study the processes KK̅→ϕ, π D → D^∗, 
πD̅→D̅^∗, and the production of ψ (3770), 
ψ (4040), ψ (4160), and
ψ (4415) mesons in collisions of charmed mesons or charmed strange mesons.
The process of 2-to-1 meson-meson scattering involves a quark 
and an antiquark from the two initial mesons annihilating into a gluon and
subsequently the gluon being 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 ψ (4040), ψ (4160),
and ψ (4415) relate to nodes in their radial wave functions.
We suggest the production of ψ (4040),
ψ (4160), and ψ (4415) as probes of hadronic matter that
results from the quark-gluon plasma created in ultrarelativistic heavy-ion 
collisions.


Keywords: Inelastic meson-meson scattering, quark-antiquark annihilation,
relativistic constituent quark potential model.

PACS: 13.75.Lb; 12.39.Jh; 12.39.Pn




I. INTRODUCTION



Elastic meson-meson scattering produces many resonances. 
Starting from meson-meson scattering amplitudes obtained in
chiral perturbation theory <cit.>, elastic scattering has been
studied within nonperturbative schemes, for example, 
the inverse amplitude method <cit.> and
the coupled-channel unitary approaches <cit.>. 
Elastic meson-meson scattering has also been studied with quark 
interchange in the first Born approximation in Ref. <cit.> and with
quark-antiquark annihilation and creation in Ref. <cit.>. 
Elastic scattering reported in the literature includes ππ
<cit.>, π K <cit.>,
KK̅ <cit.>, πη 
<cit.>, K η <cit.>,
ηη <cit.>, πρ <cit.>,
π D <cit.>, K̅ D <cit.>, 
K̅ D^∗ <cit.>, and D D^∗ <cit.>.
We know that resonances observed in the elastic scattering
are usually produced by a process where two mesons scatter into one meson. 
The 2-to-1 meson-meson scattering includes
ππ→ρ, ππ→ f_0(980), π K → K^∗, 
KK̅→ϕ, πη→ a_0(980), πρ→ a_1(1260),
π D → D^∗, and so on. Since some resonances like f_0(980), 
a_0(980), and a_1(1260) are not quark-antiquark states, we do not study
ππ→ f_0(980), πη→ a_0(980), πρ→ a_1(1260),
etc. in the present work. The reactions ππ→ρ and
π K → K^∗ have been studied in Ref. <cit.> via a process
where a quark in an initial meson and an antiquark in another initial meson
annihilate into a gluon and subsequently the gluon is absorbed by the other
antiquark or quark. The resulting cross sections in vacuum
agree with the empirical data.
Since these two reactions also take place in hadronic matter that
is created in ultrarelativistic heavy-ion collisions at the Relativistic Heavy
Ion Collider and at the Large Hadron Collider, the dependence of 
cross sections for the two reactions 
on the temperature of hadronic matter has also been investigated. 
With increasing temperature the cross sections
decrease. In the present work we consider the reactions: 
KK̅→ϕ, π D → D^*, πD̅→D̅^*, 
DD̅→ψ (3770), DD̅→ψ (4040),
D^*D̅→ψ (4040), DD̅^* →ψ (4040),
D^*D̅^* →ψ (4040), D_s^+D_s^- →ψ (4040),
DD̅→ψ (4160),
D^*D̅→ψ (4160), DD̅^* →ψ (4160),
D^*D̅^* →ψ (4160), D_s^+D_s^- →ψ (4160),
D_s^*+D_s^- →ψ (4160), D_s^+D_s^*-→ψ (4160),
DD̅→ψ (4415), D^*D̅→ψ (4415),
DD̅^* →ψ (4415), D^*D̅^* →ψ (4415),
D_s^+D_s^- →ψ (4415), D_s^*+D_s^- →ψ (4415),
D_s^+D_s^*-→ψ (4415), and D_s^*+D_s^*-→ψ (4415).
The ψ (3770), ψ (4040), ψ (4160), and ψ (4415) mesons
consist of a quark and an antiquark <cit.>. 
All these reactions are governed by the strong interaction.
The reaction KK̅→ϕ was studied in Ref. 
<cit.> in a mesonic model. The twenty-one reactions that lead to  
ψ (3770), ψ (4040), 
ψ (4160), or ψ (4415) as a final state have not been studied 
theoretically. We now study KK̅→ϕ, π D → D^*, 
πD̅→D̅^*, and the twenty-one reactions using quark degrees
of freedom. The production of J/ψ is a subject intensively studied in
relativistic heavy-ion collisions. The ψ (3770), ψ (4040), 
ψ (4160), and ψ (4415) mesons
may decay into the J/ψ meson. Through this decay the twenty-one reactions
add a contribution to the J/ψ production in relativistic heavy-ion 
collisions. This is another reason why we study the twenty-one 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.
In an appendix we study decays of the ψ (3770), ψ (4040), 
ψ (4160), and ψ (4415) mesons to charmed mesons or charmed strange
mesons.




II. FORMALISM



Lowest-order Feynman diagrams are shown in 
Fig. 1 for the reaction A(q_1q̅_1)+B(q_2q̅_2) → 
H(q_2q̅_1  or q_1q̅_2).
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+q̅_2+q̅_1 →q̅_1, 
q_1+q̅_2+q_2 → q_2, q_2+q̅_1+q_1 → q_1, and
q_2+q̅_1+q̅_2 →q̅_2 in Fig. 1 give rise to
the four transition potentials 
V_ rq_1q̅_2q̅_1, V_ rq_1q̅_2q_2,
V_ rq_2q̅_1q_1, and V_ rq_2q̅_1q̅_2, 
respectively.
Denote by E_ i and P⃗_ i (E_ f and P⃗_ 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 normalized.
The S-matrix element for A+B → H is

    S_ fi    =    δ_ fi - 2π i δ (E_ f - E_ i)
    (<H | V_ rq_1q̅_2 q̅_1| A,B> + <H | V_ r
    q_1q̅_2 q_2| A,B> 
          
    + <H | V_ rq_2 q̅_1 q_1| A,B> + <H | V_ r
    q_2 q̅_1 q̅_2| A,B>) 
        =    δ_ fi -(2π)^4 i δ (E_ f - E_ i)
    δ^3 (P⃗_ f - P⃗_ i)
     M_ rq_1q̅_2 q̅_1+
     M_ rq_1q̅_2 q_2+ M_ rq_2 q̅_1 q_1
    + M_ rq_2 q̅_1 q̅_2/V^3/2√(2E_A2E_B2E_H),

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_1q̅_2 q̅_1, 
M_ rq_1q̅_2 q_2, M_ rq_2 q̅_1 q_1,
and M_ rq_2 q̅_1 q̅_2 are 
the transition amplitudes given by

    M_ rq_1q̅_2 q̅_1=
    √(2E_A2E_B2E_H)∫ dr⃗_q_1q̅_1 dr⃗_q_2q̅_2ψ_H^+ V_ rq_1q̅_2q̅_1ψ_AB e^ip⃗_q_1q̅_1,q_2q̅_2·r⃗_q_1q̅_1,q_2q̅_2,


    M_ rq_1q̅_2 q_2=
    √(2E_A2E_B2E_H)∫ dr⃗_q_1q̅_1 dr⃗_q_2q̅_2ψ_H^+ V_ rq_1q̅_2 q_2ψ_AB e^ip⃗_q_1q̅_1,q_2q̅_2·r⃗_q_1q̅_1,q_2q̅_2,


    M_ rq_2 q̅_1 q_1=
    √(2E_A2E_B2E_H)∫ dr⃗_q_1q̅_1 dr⃗_q_2q̅_2ψ_H^+ V_ rq_2q̅_1 q_1ψ_AB e^ip⃗_q_1q̅_1,q_2q̅_2·r⃗_q_1q̅_1,q_2q̅_2,


    M_ rq_2 q̅_1 q̅_2=
    √(2E_A2E_B2E_H)∫ dr⃗_q_1q̅_1 dr⃗_q_2q̅_2ψ_H^+ V_ rq_2 q̅_1 q̅_2ψ_AB e^ip⃗_q_1q̅_1,q_2q̅_2·r⃗_q_1q̅_1,q_2q̅_2,

where r⃗_ab is the relative coordinate of constituents a and b;
r⃗_q_1q̅_1,q_2q̅_2 the relative coordinate 
of q_1q̅_1 and q_2q̅_2; 
p⃗_q_1q̅_1,q_2q̅_2 the relative momentum of 
q_1q̅_1 and q_2q̅_2; ψ_H^+ the Hermitean conjugate of 
ψ_H. 

The wave function of mesons A and B is

    ψ_AB =ϕ_A relϕ_B relϕ_A colorϕ_B colorχ_S_A S_Azχ_S_B S_Bzφ_AB flavor,

and the wave function of meson H is

    ψ_H =ϕ_J_HJ_Hzϕ_H colorϕ_H flavor,

where S_A (S_B) is the spin of meson A (B) with its 
magnetic projection quantum number S_Az (S_Bz); ϕ_A rel 
(ϕ_B rel), ϕ_A color (ϕ_B color), 
and χ_S_A S_Az (χ_S_B S_Bz)
are the quark-antiquark relative-motion wave function, the color wave function,
and the spin wave function of meson A (B), respectively;
ϕ_H flavor and φ_AB flavor are the flavor wave 
functions of meson H and of mesons A and B, respectively;
ϕ_H color is the color wave function of meson H; ϕ_J_HJ_Hz
is the space-spin wave function of meson H with the total angular momentum
J_H and its z component J_Hz.

The development in spherical harmonics of the relative-motion wave function
of mesons A and B (aside from a normalization constant) is given by

    e^ip⃗_q_1q̅_1,q_2q̅_2·r⃗_q_1q̅_1,q_2q̅_2    =     4π∑_L_ i=0^∞∑_M_ i=-L_ i^L_ i
    i^L_ i j_L_ i (|p⃗_q_1q̅_1,q_2q̅_2|
    r_q_1q̅_1,q_2q̅_2) 
    
           Y_L_ iM_ i^∗
    (p̂_q_1q̅_1,q_2q̅_2) Y_L_ iM_ i
    (r̂_q_1q̅_1,q_2q̅_2),

where Y_L_ iM_ 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, and p̂_q_1q̅_1,q_2q̅_2 
(r̂_q_1q̅_1,q_2q̅_2)
denote the polar angles of p⃗_q_1q̅_1,q_2q̅_2
(r⃗_q_1q̅_1,q_2q̅_2).
Let χ_SS_z 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_AS_AzS_BS_Bz|SS_z) couple 
χ_SS_z to χ_S_AS_Azχ_S_BS_Bz,

    χ_S_AS_A_zχ_S_BS_B_z=
    ∑^S_ max_S=S_ min∑^S_S_z=-S
    (S_AS_AzS_BS_Bz|SS_z)χ_SS_z,

where S_ min=| S_A-S_B | and S_ max=S_A+S_B.
Y_L_ iM_ i and χ_SS_z are coupled to the wave function
ϕ^ in_JJ_z which has the total angular momentum J of mesons
A and B and its z component J_z,

    Y_L_iM_iχ_SS_z=
    ∑^J_ max_J=J_ min∑^J_J_z=-J
    (L_iM_iSS_z|JJ_z)ϕ^in_JJ_z,

where J_ min=| L_ i-S |, J_ max=L_ i+S, and
(L_iM_iSS_z|JJ_z) are the Clebsch-Gordan coefficients. It
follows from Eqs. (8)-(10) that the transition amplitude given in Eq. (2)
becomes

    M_rq_1q̅_2q̅_1   =   √(2E_A2E_B2E_H)4π∑^∞_L_i=0∑^L_i_M_i=-L_ii^L_i
    Y^*_L_iM_i(p̂_q_1q̅_1,q_2q̅_2)
    ϕ^+_Hcolorϕ^+_Hflavor
          ∫d^3r_q_1q̅_1d^3r_q_2q̅_2ϕ^+_J_HJ_HzV_rq_1q̅_2q̅_1∑^S_ max_S=S_ min∑^S_S_z=-S
    (S_AS_AzS_BS_Bz|SS_z)
          ∑^J_ max_J=J_ min∑^J_J_z=-J
    (L_iM_iSS_z|JJ_z)
    ϕ^in_JJ_zj_L_i(|p⃗_q_1q̅_1,
    q_2q̅_2|r_q_1q̅_1,q_2q̅_2)
          ϕ_Arelϕ_Brelϕ_Acolorϕ_Bcolorφ_ABflavor,

Denote by L_H and S_H the orbital angular momentum and the spin of meson
H, respectively, and by M_H and S_Hz the magnetic projection
quantum number of L_H and S_H. In Eq. (11)
ϕ_J_HJ_Hz=R_L_H(r_q_2q̅_1) ∑^L_H_M_H=-L_H∑^S_H_S_Hz=-S_H (L_HM_HS_HS_Hz| J_HJ_Hz) Y_L_HM_Hχ_S_HS_Hz where R_L_H(r_q_2q̅_1) is the radial wave function
of the relative motion of q_2 and q̅_1, and 
(L_H M_H S_H S_Hz| J_H J_Hz) are the Clebsch-Gordan coefficients.
Conservation of total angular momentum implies that J equals J_H and J_z
equals J_Hz. This leads to

    M_rq_1q̅_2q̅_1   =   √(2E_A2E_B2E_H)4
    π∑^∞_L_i=0∑^L_i_M_i=-L_ii^L_i
    Y^*_L_iM_i(p̂_q_1q̅_1,q_2q̅_2)
    ϕ^+_Hcolorϕ^+_Hflavor
          ∫d^3r_q_1q̅_1d^3r_q_2q̅_2ϕ^+_J_HJ_HzV_rq_1q̅_2q̅_1∑^S_ max_S=S_ min∑^S_S_z=-S
    (S_AS_AzS_BS_Bz|SS_z)
          
    (L_ iM_ iSS_z|J_HJ_H_z)ϕ^in_J_HJ_Hz
    j_L_i(|p⃗_q_1q̅_1,q_2q̅_2
    |r_q_1q̅_1,q_2q̅_2)
          ϕ_Arelϕ_Brelϕ_Acolorϕ_Bcolorφ_ABflavor.

Using the relation

    ϕ^in_J_HJ_Hz=
    ∑^L_ i_M̅_ i=-L_ i∑^S_S̅_z=-S
    (L_iM̅_iSS̅_z|J_HJ_Hz)
      Y_L_iM̅_iχ_SS̅_z,

where (L_iM̅_iSS̅_z|J_HJ_Hz) are the
Clebsch-Gordan coefficients, we get

    M_rq_1q̅_2q̅_1   =   √(2E_A2E_B2E_H)4π∑^S_ max_S=S_ min∑^S_S_z=-S
    (S_AS_AzS_BS_Bz
    |SS_z)∑^∞_L_i=0∑^L_i_M_i
    =-L_ii^L_i
          
    Y^*_L_iM_i(p̂_q_1q̅_1,q_2q̅_2)
    (L_iM_iSS_z|J_HJ_Hz)
    ∑^L_ i_M̅_ i=-L_ i∑^S_S̅_z=-S
    (L_iM̅_iSS̅_z|J_HJ_Hz)
          ϕ^+_Hcolorϕ^+_Hflavor∫d^3r_q_1q̅_1d^3r_q_2q̅_2ϕ^+_J_HJ_Hz
    V_rq_1q̅_2q̅_1j_L_i
    (|p⃗_q_1q̅_1,q_2q̅_2|r_q_1q̅_1,
    q_2q̅_2)
          
    Y_L_iM̅_i(r̂_q_1q̅_1,q_2q̅_2)
    ϕ_Arelϕ_Brelχ_SS̅_zϕ_Acolorϕ_Bcolorφ_ABflavor.

Furthermore, we need the identity

    j_l(pr)Y_lm(r̂)=∫d^3p^'/(2π)^32π^2/p^2δ (p-p^') i^l (-1)^l Y_lm(p̂^') 
    e^ip⃗^ '·r⃗ ,

which is obtained with the help of 
∫_0^∞ j_l(pr)j_l(p^' r)r^2dr=π/2p^2δ (p-p^') <cit.>. Substituting Eq. (15) in Eq. (14),
we get

    M_rq_1q̅_2q̅_1   =   √(2E_A2E_B2E_H)4π∑^S_ max_S=S_ min∑^S_S_z=-S
    (S_AS_AzS_BS_Bz|SS_z)∑^∞_L_i=0∑^L_i_M_i=-L_ii^L_i
          
    Y^*_L_iM_i(p̂_q_1q̅_1,q_2q̅_2)
    (L_iM_iSS_z|J_HJ_Hz)
    ∑^L_ i_M̅_ i=-L_ i∑^S_S̅_z=-S
    (L_iM̅_iSS̅_z|J_HJ_Hz)
          ϕ^+_Hcolorϕ^+_Hflavor∫d^3p_irm/(2π)^32π^2/p⃗^ 2_q_1q̅_1,q_2q̅_2δ(|p⃗_q_1q̅_1,q_2q̅_2|-|p⃗_irm|)
    i^L_i(-1)^L_i
          
    Y_L_iM̅_i(p̂_irm)∫d^3r_q_1q̅_1d^3r_q_2q̅_2ϕ^+_J_HJ_HzV_rq_1q̅_2q̅_1e^ip⃗_irm·r⃗_q_1q̅_1,q_2q̅_2
          ϕ_Arelϕ_Brelχ_SS̅_zϕ_Acolorϕ_Bcolorφ_ABflavor.


Let r⃗_c and m_c be the position vector and the mass of constituent 
c, respectively. Then ϕ_Arel, ϕ_Brel, and 
R_L_H are functions of the relative coordinate of the quark and
the antiquark. We take the Fourier transform of 
V_ rq_1q̅_2q̅_1 and the mesonic quark-antiquark
relative-motion wave functions:

    V_ rq_1q̅_2q̅_1(r⃗_q̅_1-r⃗_q_1) =
    ∫d^3k/(2π)^3 V_ rq_1q̅_2q̅_1 (k⃗)
    e^ik⃗· (r⃗_q̅_1-r⃗_q_1),


    ϕ_A rel(r⃗_q_1q̅_1) =
    ∫d^3p_q_1q̅_1/(2π)^3ϕ_A rel
    (p⃗_q_1q̅_1)
    e^ip⃗_q_1q̅_1·r⃗_q_1q̅_1,


    ϕ_B rel(r⃗_q_2q̅_2) =
    ∫d^3p_q_2q̅_2/(2π)^3ϕ_B rel
    (p⃗_q_2q̅_2)
    e^ip⃗_q_2q̅_2·r⃗_q_2q̅_2,


    ϕ_J_HJ_Hz(r⃗_q_2q̅_1) =
    ∫d^3p_q_2q̅_1/(2π)^3ϕ_J_HJ_Hz
    (p⃗_q_2q̅_1)
    e^ip⃗_q_2q̅_1·r⃗_q_2q̅_1,

for the two upper diagrams in Fig. 1, and

    ϕ_J_HJ_Hz(r⃗_q_1q̅_2) =
    ∫d^3p_q_1q̅_2/(2π)^3ϕ_J_HJ_Hz
    (p⃗_q_1q̅_2)
    e^ip⃗_q_1q̅_2·r⃗_q_1q̅_2,

for the two lower diagrams. In Eq. (17) k⃗ is the gluon momentum, 
and in Eqs. (18)-(21)
p⃗_ab is the relative momentum of constituents a and b. The
spherical polar coordinates of p⃗_ irm are expressed as 
(|p⃗_ irm|, θ_ irm, ϕ_ irm). In momentum
space the normalizations are 

∫d^3p_q_1q̅_1/(2π)^3
ϕ_Arel^+(p⃗_q_1q̅_1)
ϕ_Arel(p⃗_q_1q̅_1)=1, 


∫d^3p_q_2q̅_2/(2π)^3
ϕ_Brel^+(p⃗_q_2q̅_2)
ϕ_Brel(p⃗_q_2q̅_2)=1,


∫d^3p_q_2q̅_1/(2π)^3
ϕ_J_HJ_Hz^+(p⃗_q_2q̅_1)
ϕ_J_HJ_Hz(p⃗_q_2q̅_1)=1,


∫d^3p_q_1q̅_2/(2π)^3
ϕ_J_HJ_Hz^+(p⃗_q_1q̅_2)
ϕ_J_HJ_Hz(p⃗_q_1q̅_2)=1. 

Integration over
|p⃗_ irm|, r⃗_q_1q̅_1, and
r⃗_q_2q̅_2 yields

    M_rq_1q̅_2q̅_1   =   √(2E_A2E_B2E_H)∑^S_ max_S=S_ min∑^S_S_z=-S
    (S_AS_AzS_BS_Bz|SS_z)∑^∞_L_i=0∑^L_i_M_i=-L_i
          
    Y^*_L_iM_i(p̂_q_1q̅_1,q_2q̅_2)
    (L_iM_iSS_z|J_HJ_Hz)
    ∑^L_ i_M̅_ i=-L_ i∑^S_S̅_z=-S
    (L_iM̅_iSS̅_z|J_HJ_Hz)
          ϕ^+_Hcolorϕ^+_Hflavor∫dθ_irm
    dϕ_irmsinθ_irmY_L_iM̅_i
    (p̂_irm)
          ∫d^3p_q_1q̅_1/(2π)^3∫d^3p_q_2q̅_2/(2π)^3ϕ^+_J_HJ_Hz
    (p⃗_q_2q̅_2-m_q_2/m_q_2+m_q̅_2p⃗_irm)V_rq_1q̅_2q̅_1
          
    [p⃗_q_1q̅_1-p⃗_q_2q̅_2+
    (m_q_2/m_q_2+m_q̅_2-m_q̅_1/m_q_1+m_q̅_1)p⃗_irm]
          ϕ_Arel(p⃗_q_1q̅_1)ϕ_Brel
    (p⃗_q_2q̅_2)χ_SS̅_zϕ_Acolorϕ_Bcolorφ_ABflavor,

in which |p⃗_ irm| = |p⃗_q_1q̅_1,q_2q̅_2|. So far, we have obtained a new
expression of the transition amplitude from Eq. (2).

Making use of the Fourier transform of V_ rq_1q̅_2q_2, 
V_ rq_2q̅_1q_1, and V_ rq_2q̅_1q̅_2,

    V_ rq_1q̅_2q_2(r⃗_q_2-r⃗_q̅_2) =
    ∫d^3k/(2π)^3 V_ rq_1q̅_2q_2 (k⃗)
    e^ik⃗· (r⃗_q_2-r⃗_q̅_2),


    V_ rq_2q̅_1q_1(r⃗_q_1-r⃗_q̅_1) =
    ∫d^3k/(2π)^3 V_ rq_2q̅_1q_1 (k⃗)
    e^ik⃗· (r⃗_q_1-r⃗_q̅_1),


    V_ rq_2q̅_1q̅_2(r⃗_q̅_2-r⃗_q_2) =
    ∫d^3k/(2π)^3 V_ rq_2q̅_1q̅_2 (k⃗)
    e^ik⃗· (r⃗_q̅_2-r⃗_q_2),

from Eqs. (3)-(5) we obtain

    M_rq_1q̅_2q_2   =   √(2E_A2E_B2E_H)∑^S_ max_S=S_ min∑^S_S_z=-S
    (S_AS_AzS_BS_Bz|SS_z)∑^∞_L_i=0∑^L_i_M_i=-L_i
          
    Y^*_L_iM_i(p̂_q_1q̅_1,q_2q̅_2)
    (L_iM_iSS_z|J_HJ_Hz)
    ∑^L_ i_M̅_ i=-L_ i∑^S_S̅_z=-S
    (L_iM̅_iSS̅_z|J_HJ_Hz)
          ϕ^+_Hcolorϕ^+_Hflavor∫dθ_irm
    dϕ_irmsinθ_irmY_L_iM̅_i
    (p̂_irm)
          ∫d^3p_q_1q̅_1/(2π)^3∫d^3p_q_2q̅_2/(2π)^3ϕ^+_J_HJ_Hz
    (p⃗_q_1q̅_1-m_q̅_1/m_q_1+m_q̅_1p⃗_irm)V_rq_1q̅_2q_2
          
    [p⃗_q_1q̅_1-p⃗_q_2q̅_2+
    (m_q_2/m_q_2+m_q̅_2-m_q̅_1/m_q_1+m_q̅_1)p⃗_irm]
          ϕ_Arel(p⃗_q_1q̅_1)ϕ_Brel
    (p⃗_q_2q̅_2)χ_SS̅_zϕ_Acolorϕ_Bcolorφ_ABflavor,


    M_rq_2q̅_1q_1   =   √(2E_A2E_B2E_H)∑^S_ max_S=S_ min∑^S_S_z=-S
    (S_AS_AzS_BS_Bz|SS_z)∑^∞_L_i=0∑^L_i_M_i=-L_i
          
    Y^*_L_iM_i(p̂_q_1q̅_1,q_2q̅_2)
    (L_iM_iSS_z|J_HJ_Hz)
    ∑^L_ i_M̅_ i=-L_ i∑^S_S̅_z=-S
    (L_iM̅_iSS̅_z|J_HJ_Hz)
          ϕ^+_Hcolorϕ^+_Hflavor∫dθ_irm
    dϕ_irmsinθ_irmY_L_iM̅_i
    (p̂_irm)
          ∫d^3p_q_1q̅_1/(2π)^3∫d^3p_q_2q̅_2/(2π)^3ϕ^+_J_HJ_Hz
    (p⃗_q_2q̅_2+m_q̅_2/m_q_2+m_q̅_2p⃗_irm)V_rq_2q̅_1q_1
          
    [-p⃗_q_1q̅_1+p⃗_q_2q̅_2+
    (m_q̅_2/m_q_2+m_q̅_2-m_q_1/m_q_1+m_q̅_1)p⃗_irm]
          ϕ_Arel(p⃗_q_1q̅_1)ϕ_Brel
    (p⃗_q_2q̅_2)χ_SS̅_zϕ_Acolorϕ_Bcolorφ_ABflavor,


    M_rq_2q̅_1q̅_2   =   √(2E_A2E_B2E_H)∑^S_ max_S=S_ min∑^S_S_z=-S
    (S_AS_AzS_BS_Bz|SS_z)∑^∞_L_i=0∑^L_i_M_i=-L_i
          
    Y^*_L_iM_i(p̂_q_1q̅_1,q_2q̅_2)
    (L_iM_iSS_z|J_HJ_Hz)
    ∑^L_ i_M̅_ i=-L_ i∑^S_S̅_z=-S
    (L_iM̅_iSS̅_z|J_HJ_Hz)
          ϕ^+_Hcolorϕ^+_Hflavor∫dθ_irm
    dϕ_irmsinθ_irmY_L_iM̅_i
    (p̂_irm)
          ∫d^3p_q_1q̅_1/(2π)^3∫d^3p_q_2q̅_2/(2π)^3ϕ^+_J_HJ_Hz
    (p⃗_q_1q̅_1+m_q_1/m_q_1+m_q̅_1p⃗_irm)V_rq_2q̅_1q̅_2
          
    [-p⃗_q_1q̅_1+p⃗_q_2q̅_2+
    (m_q̅_2/m_q_2+m_q̅_2-m_q_1/m_q_1+m_q̅_1)p⃗_irm]
          ϕ_Arel(p⃗_q_1q̅_1)ϕ_Brel
    (p⃗_q_2q̅_2)χ_SS̅_zϕ_Acolorϕ_Bcolorφ_ABflavor.


With these transition amplitudes the unpolarized cross section for A+B → H
is

    σ^ unpol    =    πδ(E_ f-E_ i)/4√((P_A · P_B)^2 - m_A^2m_B^2)E_H1/(2J_A+1)(2J_B+1)
          ∑_J_AzJ_BzJ_Hz| M_ rq_1q̅_2 q̅_1+ M_ rq_1q̅_2q_2
    + M_ rq_2q̅_1 q_1
    + M_ rq_2q̅_1 q̅_2|^2 ,

where P_A, m_A, J_A, and J_Az (P_B, m_B, J_B, and J_Bz) of 
meson A (B) are the four-momentum, the mass, the total angular momentum, 
and its z component, respectively. We calculate
the cross section in the center-of-mass frame of the two initial mesons, i.e.,
with meson H at rest.




III. FLAVOR AND SPIN MATRIX ELEMENTS



Let p_c be the four-momentum of constituent c. The two upper diagrams
give q_1(p_q_1)+q̅_1(p_q̅_1)+q_2(p_q_2)
+q̅_2(p_q̅_2) →q̅_1(p_q̅_1^')
+q_2(p_q_2^'), and the two lower diagrams
q_1(p_q_1)+q̅_1(p_q̅_1)+q_2(p_q_2)
+q̅_2(p_q̅_2) → q_1(p_q_1^')
+q̅_2(p_q̅_2^'). 
The transition potentials V_ rq_1q̅_2q̅_1,
V_ rq_1q̅_2q_2, V_ rq_2q̅_1q_1, and
V_ rq_2q̅_1q̅_2 are expressed as 

    V_ rq_1q̅_2q̅_1(k⃗)     =     -
    λ⃗(1)/2·λ⃗(21)/2g_ s^2/k^2(σ⃗(21)·k⃗/2m_q_1.
                   
          . -σ⃗(1)·p⃗_q̅_1σ⃗(1)·σ⃗(21)
    +σ⃗(1)·σ⃗(21)
    σ⃗(1)·p⃗_q̅_1^ '/2m_q̅_1) ,


    V_ rq_1q̅_2q_2(k⃗)     =    λ⃗(2)/2·λ⃗(21)/2g_ s^2/k^2(σ⃗(21)·k⃗/2m_q_1.
                   
          . -
    σ⃗(2)·σ⃗(21)σ⃗(2)·p⃗_q_2+
    σ⃗(2)·p⃗_q_2^ 'σ⃗(2)·σ⃗(21)/2m_q_2) ,


    V_ rq_2q̅_1q_1(k⃗)     =    λ⃗(1)/2·λ⃗(12)/2g_ s^2/k^2(σ⃗(12)·k⃗/2m_q_2.
                   
          . -
    σ⃗(1)·σ⃗(12)σ⃗(1)·p⃗_q_1+
    σ⃗(1)·p⃗_q_1^ 'σ⃗(1)·σ⃗(12)/2m_q_1) ,


    V_ rq_2q̅_1q̅_2(k⃗)     =     -
    λ⃗(2)/2·λ⃗(12)/2g_ s^2/k^2(σ⃗(12)·k⃗/2m_q_2.
                   
          . -
    σ⃗(2)·p⃗_q̅_2σ⃗(2)·σ⃗(12)
    +σ⃗(2)·σ⃗(12)
    σ⃗(2)·p⃗_q̅_2^ '/2m_q̅_2) ,

where g_ s is the gauge coupling constant, k the gluon 
four-momentum, λ⃗ the Gell-Mann matrices,
and σ⃗ the Pauli matrices.
In Eqs. (30) and (31), λ⃗(21) 
(σ⃗(21)) mean that they have matrix elements between the color 
(spin) wave functions of initial antiquark q̅_2 and initial quark q_1.
In Eqs. (32) and (33), λ⃗(12) 
(σ⃗(12)) mean that they have matrix elements between the color 
(spin) wave functions of initial antiquark q̅_1 and initial quark q_2.
In Eqs. (30) and (33), λ⃗(1) and λ⃗(2)
(σ⃗(1) and σ⃗(2)) mean that they 
have matrix elements between the color (spin) wave functions of the initial 
antiquark and the final antiquark. 
In Eqs. (31) and (32), λ⃗(2) and λ⃗(1) 
(σ⃗(2) and σ⃗(1)) mean that they 
have matrix elements between the color (spin) wave functions of the final 
quark and the initial quark.

It is shown in Refs. <cit.> 
that ψ (3770), ψ (4040), ψ (4160), and ψ (4415) can be 
individually interpreted as the 1 ^3D_1, 3 ^3S_1,
2 ^3D_1, and 4 ^3S_1 quark-antiquark states. We use the notation
K= ( [ K^+; K^0 ]),
K̅= ( [ K̅^0;  K^- ]),
K^*= ( [ K^*+; K^*0 ]),
K̅^*= ( [ K̅^*0;  K^*- ]),
D= ( [ D^+; D^0 ]),
D̅= ( [ D̅^0;  D^- ]),
D^*= ( [ D^*+; D^*0 ]), and
D̅^*= ( [ D̅^*0;  D^*- ]).
Based on the formulas in Sect. II, we study the following reactions:

KK̅ →ϕ, πD →D^*, πD̅ →D̅^*,


DD̅ →ψ(3770), DD̅ →ψ(4040),


D^*D̅ →ψ(4040), DD̅^* →ψ(4040),
 D^*D̅^* →ψ(4040), D_s^+D_s^- →ψ(4040),


DD̅ →ψ(4160), D^*D̅ →ψ(4160),
 DD̅^* →ψ(4160), D^*D̅^* →ψ(4160),


D_s^+D_s^- →ψ(4160), D_s^*+D_s^- →ψ(4160),
 D_s^+D_s^*- →ψ(4160),


DD̅ →ψ(4415), D^*D̅ →ψ(4415),
 DD̅^* →ψ(4415), D^*D̅^* →ψ(4415),


D_s^+D_s^- →ψ(4415), D_s^*+D_s^- →ψ(4415),
 D_s^+D_s^*- →ψ(4415), D_s^*+D_s^*- →ψ(4415).

From the Gell-Mann matrices and the Pauli matrices in the transition 
potentials, the expressions of the transition amplitudes in Eqs. (22) and
(26)-(28) involve color matrix elements, flavor matrix elements, and spin 
matrix elements for the above reactions. The color matrix elements in 
M_rq_1q̅_2q̅_1, M_rq_1q̅_2q_2,
M_rq_2q̅_1q_1, and M_rq_2q̅_1q̅_2 are
-4/3√(3), 4/3√(3), 4/3√(3), and
-4/3√(3), respectively.
While we calculate the flavor matrix elements, we keep the total isospin
of the two initial mesons the same as the isospin of the final meson.
The flavor wave functions of charmed mesons and charmed strange mesons are
| D^+>=-| cd̅>, | D^0>=| cu̅>, 
|D̅^0>=| uc̅>, | D^->=| dc̅>, 
| D^*+>=-| cd̅>, | D^*0>=| cu̅>, 
|D̅^*0>=| uc̅>, and | D^*->=| dc̅>.
The flavor matrix element M_ fKK̅→ϕ
(M_ fπ D → D^*,
M_ fDD̅→ψ (3770),
M_ fD_s^+D_s^- →ψ (4040))
for KK̅→ϕ (π D → D^*,
DD̅→ψ (3770), D_s^+D_s^- →ψ (4040)) is shown in 
Table 1. The flavor matrix element for πD̅→D̅^* equals
M_ fπ D → D^*.
The flavor matrix elements for 
DD̅→ψ (4040), D^*D̅→ψ (4040),
DD̅^* →ψ (4040), D^*D̅^* →ψ (4040),
DD̅→ψ (4160), D^*D̅→ψ (4160), 
DD̅^* →ψ (4160), D^*D̅^* →ψ (4160),
DD̅→ψ (4415), D^*D̅→ψ (4415), 
DD̅^* →ψ (4415), and D^*D̅^* →ψ (4415) are the same 
as M_ fDD̅→ψ (3770). 
The flavor matrix elements for D_s^+D_s^- →ψ (4160),
D_s^*+D_s^- →ψ (4160), D_s^+D_s^*-→ψ (4160), 
D_s^+D_s^- →ψ (4415), D_s^*+D_s^- →ψ (4415), 
D_s^+D_s^*-→ψ (4415), and D_s^*+D_s^*-→ψ (4415) equal 
M_ fD_s^+D_s^- →ψ (4040). The flavor matrix elements
for KK̅→ϕ, π D → D^*, and πD̅→D̅^* are
zero for the two lower diagrams, and those for the production of 
ψ (3770), ψ (4040),
ψ(4160), and ψ(4415) are zero for the two upper diagrams. Hence,
every reaction receives contributions only from two Feynman diagrams.

Now we give the spin matrix elements.
Let P_rq_1q̅_2q̅_1i with i=0, ···, and 15 stand
for 1, σ_1(21), σ_2(21), σ_3(21), σ_1(1), 
σ_2(1), σ_3(1), σ_1(21) σ_1(1), 
σ_1(21) σ_2(1), σ_1(21) σ_3(1), 
σ_2(21) σ_1(1), σ_2(21) σ_2(1), 
σ_2(21) σ_3(1), σ_3(21) σ_1(1), 
σ_3(21) σ_2(1), and σ_3(21) σ_3(1), respectively. 
Let P_rq_1q̅_2q_2i with i=0, ···, and 15 correspond 
to 1, σ_1(21), σ_2(21), σ_3(21), σ_1(2), 
σ_2(2), σ_3(2), σ_1(21) σ_1(2), 
σ_1(21) σ_2(2), σ_1(21) σ_3(2), 
σ_2(21) σ_1(2), σ_2(21) σ_2(2), 
σ_2(21) σ_3(2), σ_3(21) σ_1(2), 
σ_3(21) σ_2(2), and σ_3(21) σ_3(2), respectively. 
Let P_rq_2q̅_1q_1i with i=0, ···, and 15 
represent 1, σ_1(12), σ_2(12), σ_3(12), σ_1(1), 
σ_2(1), σ_3(1), σ_1(12) σ_1(1), 
σ_1(12) σ_2(1), σ_1(12) σ_3(1), 
σ_2(12) σ_1(1), σ_2(12) σ_2(1), 
σ_2(12) σ_3(1), σ_3(12) σ_1(1), 
σ_3(12) σ_2(1), and σ_3(12) σ_3(1), respectively. 
Let P_rq_2q̅_1q̅_2i with i=0, ···, and 15
denote 1, σ_1(12), σ_2(12), σ_3(12), σ_1(2), 
σ_2(2), σ_3(2), σ_1(12) σ_1(2), 
σ_1(12) σ_2(2), σ_1(12) σ_3(2), 
σ_2(12) σ_1(2), σ_2(12) σ_2(2), 
σ_2(12) σ_3(2), σ_3(12) σ_1(2), 
σ_3(12) σ_2(2), and σ_3(12) σ_3(2), respectively. 
Set n_A as -1, 0, or 1, and n_B as -1, 0, or 1. In order to easily tabulate
values
of the spin matrix elements, we define ϕ_ iss (S_A,S_Az;S_B,S_Bz) 
≡χ_S_AS_Azχ_S_BS_Bz and 
ϕ_ fss(S_H,S_Hz)≡χ_S_HS_Hz.
The spin matrix 
elements ϕ^+_ fss(S_H,S_Hz)P_rq_1q̅_2q̅_1iϕ_ iss (S_A,S_Az;S_B,S_Bz) are shown in Tables 2-6. Other
spin matrix elements, ϕ^+_ fss (S_H, S_Hz) P_rq_1q̅_2q_2iϕ_ iss (S_A, S_Az; S_B, S_Bz),
ϕ^+_ fss (S_H, S_Hz) P_rq_2q̅_1q_1i
ϕ_ iss (S_A, S_Az; S_B, S_Bz), and
ϕ^+_ fss (S_H, S_Hz) P_rq_2q̅_1q̅_2i
ϕ_ iss (S_A, S_Az; S_B, S_Bz), are related to 
ϕ^+_ fss (S_H, S_Hz) P_rq_1q̅_2q̅_1iϕ_ iss (S_A, S_Az; S_B, S_Bz) by the following equations:

    ϕ^+_ fss(S_H=1,S_Hz)P_rq_1q̅_2q_2iϕ_ iss 
    (S_A=1,S_Az=n_A; S_B=0,S_Bz=0)
    
        =    ϕ^+_ fss(S_H=1,S_Hz)P_rq_1q̅_2q̅_1iϕ_ iss 
    (S_A=0,S_Az=0; S_B=1,S_Bz=n_A) ,

with i=2, 5, 8, 10, 12, and 14;

    ϕ^+_ fss(S_H=1,S_Hz)P_rq_1q̅_2q_2iϕ_ iss 
    (S_A=1,S_Az=n_A; S_B=0,S_Bz=0)
    
        =     -ϕ^+_ fss(S_H=1,S_Hz)P_rq_1q̅_2q̅_1iϕ_ iss 
    (S_A=0,S_Az=0; S_B=1,S_Bz=n_A) ,

with i=0, 1, 3, 4, 6, 7, 9, 11, 13, and 15;

    ϕ^+_ fss(S_H=1,S_Hz)P_rq_1q̅_2q_2iϕ_ iss 
    (S_A=0,S_Az=0; S_B=1,S_Bz=n_B)
    
        =    ϕ^+_ fss(S_H=1,S_Hz)P_rq_1q̅_2q̅_1iϕ_ iss 
    (S_A=1,S_Az=n_B; S_B=0,S_Bz=0) ,

with i=2, 5, 8, 10, 12, and 14;

    ϕ^+_ fss(S_H=1,S_Hz)P_rq_1q̅_2q_2iϕ_ iss 
    (S_A=0,S_Az=0; S_B=1,S_Bz=n_B)
    
        =     -ϕ^+_ fss(S_H=1,S_Hz)P_rq_1q̅_2q̅_1iϕ_ iss 
    (S_A=1,S_Az=n_B; S_B=0,S_Bz=0) ,

with i=0, 1, 3, 4, 6, 7, 9, 11, 13, and 15;

    ϕ^+_ fss(S_H=1,S_Hz)P_rq_1q̅_2q_2iϕ_ iss 
    (S_A=1,S_Az=n_A; S_B=1,S_Bz=n_B)
    
        =    ϕ^+_ fss(S_H=1,S_Hz)P_rq_1q̅_2q̅_1iϕ_ iss 
    (S_A=1,S_Az=n_B; S_B=1,S_Bz=n_A) ,

with i=0, 1, 3, 4, 6, 7, 9, 11, 13, and 15;

    ϕ^+_ fss(S_H=1,S_Hz)P_rq_1q̅_2q_2iϕ_ iss 
    (S_A=1,S_Az=n_A; S_B=1,S_Bz=n_B)
    
        =     -ϕ^+_ fss(S_H=1,S_Hz)P_rq_1q̅_2q̅_1iϕ_ iss 
    (S_A=1,S_Az=n_B; S_B=1,S_Bz=n_A) ,

with i=2, 5, 8, 10, 12, and 14;

    ϕ^+_ fss(S_H=1,S_Hz)P_rq_2q̅_1q_1iϕ_ iss 
    (S_A=1,S_Az=n_A; S_B=0,S_Bz=0)
    
        =    ϕ^+_ fss(S_H=1,S_Hz)P_rq_1q̅_2q̅_1iϕ_ iss 
    (S_A=1,S_Az=n_A; S_B=0,S_Bz=0) ,

with i=2, 5, 8, 10, 12, and 14;

    ϕ^+_ fss(S_H=1,S_Hz)P_rq_2q̅_1q_1iϕ_ iss 
    (S_A=1,S_Az=n_A; S_B=0,S_Bz=0)
    
        =     -ϕ^+_ fss(S_H=1,S_Hz)P_rq_1q̅_2q̅_1iϕ_ iss 
    (S_A=1,S_Az=n_A; S_B=0,S_Bz=0) ,

with i=0, 1, 3, 4, 6, 7, 9, 11, 13, and 15;

    ϕ^+_ fss(S_H=1,S_Hz)P_rq_2q̅_1q_1iϕ_ iss 
    (S_A=0,S_Az=0; S_B=1,S_Bz=n_B)
    
        =    ϕ^+_ fss(S_H=1,S_Hz)P_rq_1q̅_2q̅_1iϕ_ iss 
    (S_A=0,S_Az=0; S_B=1,S_Bz=n_B) ,

with i=2, 5, 8, 10, 12, and 14;

    ϕ^+_ fss(S_H=1,S_Hz)P_rq_2q̅_1q_1iϕ_ iss 
    (S_A=0,S_Az=0; S_B=1,S_Bz=n_B)
    
        =     -ϕ^+_ fss(S_H=1,S_Hz)P_rq_1q̅_2q̅_1iϕ_ iss 
    (S_A=0,S_Az=0; S_B=1,S_Bz=n_B) ,

with i=0, 1, 3, 4, 6, 7, 9, 11, 13, and 15;

    ϕ^+_ fss(S_H=1,S_Hz)P_rq_2q̅_1q_1iϕ_ iss 
    (S_A=1,S_Az=n_A; S_B=1,S_Bz=n_B)
    
        =    ϕ^+_ fss(S_H=1,S_Hz)P_rq_1q̅_2q̅_1iϕ_ iss 
    (S_A=1,S_Az=n_A; S_B=1,S_Bz=n_B) ,

with i=0, 1, 3, 4, 6, 7, 9, 11, 13, and 15;

    ϕ^+_ fss(S_H=1,S_Hz)P_rq_2q̅_1q_1iϕ_ iss 
    (S_A=1,S_Az=n_A; S_B=1,S_Bz=n_B)
    
        =     -ϕ^+_ fss(S_H=1,S_Hz)P_rq_1q̅_2q̅_1iϕ_ iss 
    (S_A=1,S_Az=n_A; S_B=1,S_Bz=n_B) ,

with i=2, 5, 8, 10, 12, and 14;

    ϕ^+_ fss(S_H=1,S_Hz)P_rq_2q̅_1q̅_2iϕ_ iss 
    (S_A=1,S_Az=n_A; S_B=0,S_Bz=0)
    
        =    ϕ^+_ fss(S_H=1,S_Hz)P_rq_1q̅_2q̅_1iϕ_ iss 
    (S_A=0,S_Az=0; S_B=1,S_Bz=n_A) ,

with i=0, ···, and 15;

    ϕ^+_ fss(S_H=1,S_Hz)P_rq_2q̅_1q̅_2iϕ_ iss 
    (S_A=0,S_Az=0; S_B=1,S_Bz=n_B)
    
        =    ϕ^+_ fss(S_H=1,S_Hz)P_rq_1q̅_2q̅_1iϕ_ iss 
    (S_A=1,S_Az=n_B; S_B=0,S_Bz=0) ,

with i=0, ···, and 15;

    ϕ^+_ fss(S_H=1,S_Hz)P_rq_2q̅_1q̅_2iϕ_ iss 
    (S_A=1,S_Az=n_A; S_B=1,S_Bz=n_B)
    
        =    ϕ^+_ fss(S_H=1,S_Hz)P_rq_1q̅_2q̅_1iϕ_ iss 
    (S_A=1,S_Az=n_B; S_B=1,S_Bz=n_A) ,

with i=0, ···, and 15.




IV. NUMERICAL CROSS SECTIONS AND DISCUSSIONS 



The mesonic quark-antiquark relative-motion wave functions ϕ_A rel,
ϕ_B rel, and ϕ_J_HJ_Hz in Eqs. (6) and (7)  
are solutions of the Schrödinger equation with the
potential between constituents a and b in coordinate space <cit.>,

    V_ab(r⃗_ab)     =    
    - λ⃗_a/2·λ⃗_b/2ξ_1 [ 1.3- ( T/T_ c)^4 ] tanh 
    (ξ_2 r_ab) + λ⃗_a/2·λ⃗_b/26π/25v(λ r_ab)/r_abexp (-ξ_3 r_ab)
    
           -λ⃗_a/2·λ⃗_b/216π^2/25d^3/π^3/2exp(-d^2r^2_ab) 
    s⃗_a ·s⃗_b/m_am_b
    +λ⃗_a/2·λ⃗_b/24π/251/r_abd^2v(λ r_ab)/dr_ab^2s⃗_a ·s⃗_b/m_am_b
           -λ⃗_a/2·λ⃗_b/26π/25m_am_b[ v(λ r_ab) 
    -r_abdv(λ r_ab)/dr_ab +r_ab^2/3d^2v(λ r_ab)/dr_ab^2]
    
          ( 3s⃗_a ·r⃗_abs⃗_b ·r⃗_ab/r_ab^5 -s⃗_a ·s⃗_b/r_ab^3) ,

where ξ_1=0.525 GeV, ξ_3=0.6 GeV, T_ c=0.175 GeV,
ξ_2=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;
s⃗_a is the spin of constituent a; the quantity d is given in Ref. 
<cit.>; the function v is given by Buchmüller and Tye in Ref. 
<cit.>. The potential is obtained from perturbative QCD <cit.> and
lattice QCD <cit.>. 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 (3770), ψ (4040), ψ (4160), ψ (4415), D, D^*,
D_s, and D^*_s 
mesons <cit.>. Moreover, the experimental data of S-wave and P-wave
elastic phase shifts for ππ scattering in 
vacuum <cit.> are reproduced in the Born approximation 
<cit.>.

Gluon, quark, and antiquark fields in the thermal medium depend on its 
temperature. The interaction between two constituents
is influenced by gluons, quarks, and antiquarks in the thermal
medium, and thus depends on the temperature. The quark-antiquark potential is 
related to the Polyakov loop correlation function defined from the gluon
field, the quark field, and the antiquark field, and has 
been obtained in lattice gauge calculations. When the temperature is low, the
potential at large distances is modified by the medium. When the temperature is
near the critical temperature T_ c, the potential at intermediate 
distances is also modified. The lattice gauge calculations <cit.>
only provide a numerical spin-independent and temperature-dependent
potential at intermediate and large distances. 
When the distance increases from zero, the numerical potential increases, and
obviously becomes a distance-independent value (exhibits a plateau) at
large distances at T>0.55T_ c. The plateau height decreases with 
increasing temperature. This means that confinement becomes weaker and weaker.
The short-distance part of the first two terms in Eq. (49) originates from
one-gluon exchange plus perturbative one- and two-loop corrections, and the
intermediate-distance and large-distance part fits well the numerical
potential. The third and fourth terms indicate the spin-spin interaction with 
relativistic effects, and the fifth term is the tensor interaction, 
which are obtained from an application of the 
Foldy-Wouthuysen canonical transformation to the gluon propagator with 
perturbative one- and two-loop corrections <cit.>. The potential in
Eq. (49) is valid when the temperature is below the critical temperature.

In Eq. (49) ξ_1 is fixed, but the values of ξ_2 and ξ_3 are not 
unique; ξ_2 is allowed to change by at most 5%, and ξ_3 by at most
10% while the numerical potential obtained in the lattice gauge calculations
can be well fitted. However, an increase (decrease) of ξ_2 must be 
accompanied by a decrease (increase) of ξ_3. The uncertainties of
ξ_2 and ξ_3 cause a change less than 2.7% in meson mass,
a change less than 3.6% in cross section, and a change less than 4.3% in
decay width.

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-angular-momentum 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 Tables 7 and 8. The processes 
D^∗D̅^∗→ψ (4040), 
D^∗D̅^∗→ψ (4160), D^∗D̅^∗→ψ (4415),
and D^∗ +_s D^∗ -_s →ψ (4415) allow S=0, S=1,
and S=2. Including contributions from S=0, S=1, and S=2, the cross
sections for the four reactions are 0.42 mb, 0.57 mb, 1.39 mb, and 0.11 mb, 
respectively. When L_ i=1 is selected, the partial wave for 
L_ i=1 in Eq. (8) is normalized as

4π|p⃗_q_1q̅_1,q_2q̅_2 |i j_1 (|p⃗_q_1q̅_1,q_2q̅_2 |r_q_1q̅_1,q_2q̅_2) Y_1M_i
(r̂_q_1q̅_1,q_2q̅_2).

When L_ i=1 and L_ i=3 are selected together, the partial waves
for L_ i=1 and L_ i=3 are normalized as

4π|p⃗_q_1q̅_1,q_2q̅_2 |i
j_1 (|p⃗_q_1q̅_1,q_2q̅_2 |r_q_1q̅_1,q_2q̅_2) ∑_M_i=-1^1 Y_1M_i
(r̂_q_1q̅_1,q_2q̅_2) √(4π/10)
Y_1M_i^∗(p̂_q_1q̅_1,q_2q̅_2)


+4π|p⃗_q_1q̅_1,q_2q̅_2 |i^3
j_3 (|p⃗_q_1q̅_1,q_2q̅_2 |r_q_1q̅_1,q_2q̅_2) ∑_M_i=-3^3 Y_3M_i
(r̂_q_1q̅_1,q_2q̅_2) √(4π/10)
Y_3M_i^∗(p̂_q_1q̅_1,q_2q̅_2).


In Table 7 the cross section for KK̅→ϕ equals 8.05 mb. The
magnitude 8.05 mb is larger than the peak cross section of
KK̅→ K^*K̅^* for total isospin I=0 at zero temperature,
and is roughly 8 times the peak cross section of 
KK̅→ K^*K̅^* for I=1 in Ref. <cit.>. 
The case KK̅→ K^*K̅^* 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 KK̅→π K K̅ for I=1 and 
I_πK̅^ f=3/2 and for I=1 and I_πK̅^ f=1/2
at zero temperature in Refs. <cit.>, 
where I_πK̅^ f is the total isospin of the final π and 
K̅ mesons. The case KK̅→π K K̅ 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.
<cit.>. The case KK → K^∗ K^∗ for I=1 can be caused by quark 
interchange between the two colliding mesons.
The cross section for π D → 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 sections for DD̅→ψ (3770), DD̅→ψ (4040),
DD̅→ψ (4160), and DD̅→ψ (4415)
are thus smaller than the one for KK̅→ϕ.
Because the D^* radius is larger than the D radius, the cross sections for 
D^*D̅→ψ (4040), D^*D̅→ψ (4160), and
D^*D̅→ψ (4415) are larger than those for 
DD̅→ψ (4040), DD̅→ψ (4160), and
DD̅→ψ (4415), respectively. However, the cross sections for
D^*D̅^* →ψ (4040), D^*D̅^* →ψ (4160), and 
D^*D̅^* →ψ (4415) are smaller than those for 
D^*D̅→ψ (4040), D^*D̅→ψ (4160), and
D^*D̅→ψ (4415), respectively. This is caused by the two nodes
of ψ (4040), the node of ψ (4160), and the three nodes of 
ψ (4415) in the radial part of the 
quark-antiquark relative-motion wave function.
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.
Since D^+_s (the antiparticle of D^-_s) 
consists of a charm quark and a strange antiquark, the cross sections for
D^+_s D^-_s →ψ (4040), D^+_s D^-_s →ψ (4160), and 
D^+_s D^-_s →ψ (4415) are smaller than the ones for
DD̅→ψ (4040), DD̅→ψ (4160), and 
DD̅→ψ (4415), respectively.

In the present work the ψ (3770), ψ (4040), ψ (4160), and 
ψ (4415) mesons come from
fusion of D, D̅, D^*, D̅^*, D_s, and D^*_s mesons.
The reason why we are interested in these reactions is that ψ (3770), 
ψ (4040), 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 D D̅→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 D D̅→χ_c1(2P). The production
of χ_c1(2P) from fusion of other charmed mesons is forbidden by 
energy conservation. Also π D^±_s → D^*±_s is not allowed because
of violation of isospin conservation, and K D → D^*+_s and
K̅D̅→ 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. 2-5. In vacuum the ϕ mass is larger than two
times the kaon mass, and so the reaction KK̅→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 no longer occurs 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. The processes 
π D → D^* and πD̅→D̅^* also do not occur for
0.6T_ c≤ T <T_ c. The mesons 
ψ (4040), ψ (4160), and ψ (4415) are dissolved in 
hadronic matter when the temperature is larger than 0.97T_ c,
0.95T_ c, and 0.87T_ c, respectively <cit.>. 
Their masses are thus plotted only for 0.6T_ c≤ T < 0.97T_ c, 
for 0.6T_ c≤ T < 0.95T_ c,
and for 0.6T_ c≤ T < 0.87T_ c 
in Figs. 4 and 5, 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 (4040), ψ (4160), and ψ (4415)
from the fusion of two charmed mesons and of two charmed
strange mesons. Here T_ c is the critical temperature at which the phase
transition between the quark-gluon plasma and hadronic matter takes place.
Since ψ (4040), ψ (4160), and ψ (4415) are dissolved in hadronic
matter when the temperature is larger than 0.97T_ c, 0.95T_ 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:

D D̅ →ρR;
 D D̅^* →πR, ρR, ηR;
 D^* D̅ →πR, ρR, ηR;
 D^* D̅^* →πR, ρR, ηR;


D_s^+ D̅ →K^∗R;
 D_s^- D →K̅^∗R;
 D_s^+ D̅^* →K R, K^∗R;
 D_s^- D^* →K̅ R, K̅^∗R;


D_s^*+ D̅ →K R, K^∗R;
 D_s^*- D →K̅ R, K̅^∗R;
 D_s^*+ D̅^* →K R, K^∗R;
 D_s^*- D^* →K̅ R, K̅^∗R;


D_s^+ D_s^- →ϕR;
 D_s^+ D_s^*- →ηR, ϕR;
 D_s^*+ D_s^- →ηR, ϕR;
 D_s^*+ D_s^*- →ηR, ϕR.

In these reactions R stands for ψ (4040), ψ (4160), or 
ψ (4415). Hadronic matter undergoes expansion, 
and its temperature decreases until kinetic freeze-out occurs. Every kind 
of hadron in hadronic matter satisfies a momentum distribution function. 
The temperature, the expansion, and the momentum distribution 
functions are needed for a full description of hadronic matter. 
The ψ (4040), ψ (4160), and 
ψ (4415) mesons produced from hadronic matter depend on the
temperature, the expansion, and the distribution functions. 
Inversely, from the production we know the temperature, the expansion, and 
the distribution functions.
Therefore, ψ (4040), 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.

No experimental data on KK̅→ϕ are available.
Our cross section for KK̅→ϕ is about 8 mb, which is smaller than
the experimental cross section 80 mb for ππ→ρ <cit.>.
Assuming pointlike hadron vertices, the mesonic model in Ref. 
<cit.> leads to a large value of 260 mb. 

From Tables 7 and 8 we get ratios of the cross sections,

    σ_DD̅→ψ (4040):
    σ_D^*D̅+DD̅^* →ψ (4040):
    σ_D^*D̅^* →ψ (4040):
    σ_D_s^+D_s^- →ψ (4040)
        =     1:4.6:0.12:0.33 ,


    σ_DD̅→ψ (4160):
    σ_D^*D̅+DD̅^* →ψ (4160):
    σ_D^*D̅^* →ψ (4160):
    
          σ_D_s^+D_s^- →ψ (4160):
    σ_D_s^*+D_s^- + D_s^+D_s^*-→ψ (4160)
        =     1:4.2:0.17:0.15:0.14 .

In an analysis of the production
of charmed mesons and charmed strange mesons in the region of ψ (4040) 
and ψ (4160) resonances in e^+e^- collisions in Ref. <cit.>,
the couplings of
ψ (4040) and ψ (4160) to two charmed mesons and charmed strange 
mesons are factorized. The couplings are assumed to be proportional to a 
coefficient (denoted as g_Ri in the reference), the values of which 
are listed in Table 1 of the reference.
If it is assumed that cross sections for the production of ψ (4040) and 
ψ (4160) from DD̅, D^*D̅, DD̅^*, D^*D̅^*, 
D_s^+D_s^-, D_s^*+D_s^-, D_s^+D_s^*-, and D_s^*+D_s^*- 
differ from each other only by the coefficient, then ratios of the cross 
sections are,

    σ_DD̅→ψ (4040):
    σ_D^*D̅+DD̅^* →ψ (4040):
    σ_D^*D̅^* →ψ (4040):
    σ_D_s^+D_s^- →ψ (4040)
        =     1:4:7:1 ,


    σ_DD̅→ψ (4160):
    σ_D^*D̅+DD̅^* →ψ (4160):
    σ_D^*D̅^* →ψ (4160):
    
          σ_D_s^+D_s^- →ψ (4160):
    σ_D_s^*+D_s^- + D_s^+D_s^*-→ψ (4160)
        =     1:1:7.7:1:1 .

This assumption is correct when the D^* (D̅^*, D_s^*+, D_s^*-)
mass equals the D (D̅, D_s^+, D_s^-) mass. Regarding the ratios
given in Eq. (52) we mention a study of production of charmed mesons in 
e^+ e^- collisions in Ref. <cit.>. An electron and a positron annihilate
into a photon. The photon directly produces a pair of charmed quarks, each
of which becomes a charmed meson in combination with a subsequently produced
u or d quark. Neglecting the difference between the D and D^* masses
and the spin-spin interaction between the charmed quark and the light quark, 
the final states DD̅, D^*D̅ + DD̅^*, and D^*D̅^* are
populated in ratios 1:4:7, which agree with those given in Eq. (52). 
Nevertheless, the agreement does not surprise us in view of heavy quark
symmetry <cit.>. Heavy quark symmetry is that, in the
limit of very large quark mass, strong interactions of the heavy quark
become independent of its mass and spin. A consequence of heavy quark
symmetry is that m_D^*=m_D, m_D̅^*=m_D̅, and
m_D_s^*±=m_D_s^±. Therefore, heavy quark symmetry is assumed
in obtaining the ratios in Eq. (52) and in Ref. <cit.>, and
the ratios 1:4:7 are general in the transition between charmed mesons and 
cc̅ in heavy quark symmetry.

Cross sections for the production of ψ(4040) from the fusion of two
charmed mesons are proportional to not only the couplings squared but also the
factor 1/√((P_A · P_B)^2 -m_A^2m_B^2) as seen in Eq. (29). 
When the colliding mesons go through the cases of DD̅, D^*D̅, and
D^*D̅^*, the factor does not change in heavy quark symmetry, but
changes if the D^* mass is not the same as the D mass. The latter causes 
σ_DD̅→ψ (4040): σ_D^*D̅+DD̅^* →ψ (4040): σ_D^*D̅^* →ψ (4040) to deviate from the 
general ratios 1:4:7.

We have obtained cross sections for the production of ψ (3770),
ψ (4040), ψ (4160), and ψ (4415) from the fusion of two charmed
mesons and of two charmed strange mesons. As seen from the Review of Particle
Physics in Ref. <cit.>, time reversal of these reactions dominate decays
of the four cc̅ mesons. We study the decays in terms of the 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. This part is given 
in the appendix.




V. SUMMARY 



Using 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 wave function of the two initial mesons, we have obtained
the new expressions of the transition 
amplitudes. The orbital-angular-momentum quantum number 
corresponding to the total spin of the two initial mesons is selected to
satisfy parity conservation and conservation of the total angular
momentum. The flavor and spin matrix elements have been calculated. The spin 
matrix elements corresponding to the four transition potentials are presented.
The mesonic quark-antiquark relative-motion wave functions are given by the
Schrödinger equation with the temperature-dependent potential. From
the transition amplitudes we have obtained the cross sections for the 
reactions: 
KK̅→ϕ, π D → D^*, πD̅→D̅^*,
DD̅→ψ (3770), DD̅→ψ (4040), 
D^*D̅→ψ (4040), DD̅^* →ψ (4040), 
D^*D̅^* →ψ (4040), D_s^+D_s^- →ψ (4040),
DD̅→ψ (4160), D^*D̅→ψ (4160),
DD̅^* →ψ (4160), D^*D̅^* →ψ (4160),
D_s^+D_s^- →ψ (4160), D_s^*+D_s^- →ψ (4160),
D_s^+D_s^*-→ψ (4160), DD̅→ψ (4415),
D^*D̅→ψ (4415), DD̅^* →ψ (4415),
D^*D̅^* →ψ (4415), D_s^+D_s^- →ψ (4415),
D_s^*+D_s^- →ψ (4415), D_s^+D_s^*-→ψ (4415), and
D_s^*+D_s^*-→ψ (4415).
The cross sections are affected by radii of initial mesons and quark masses 
that enter the transition potentials. The cross section for KK̅→ϕ
is larger than the ones for DD̅→ψ (3770), 
DD̅→ψ (4040), DD̅→ψ (4160), and 
DD̅→ψ (4415), but smaller than the one for π D → D^*. 
The ψ (4040), ψ (4160), and ψ (4415) mesons resulting from 
ultrarelativistic heavy-ion collisions have been shown to be a probe
of hadronic matter that is produced in the phase transition of the quark-gluon 
plasma.




APPENDIX



We calculate widths of the ψ (3770), ψ (4040), ψ (4160), and 
ψ (4415) decays, which produce two charmed mesons or two charmed strange 
mesons. Four Feynman diagrams at tree level are involved in Z → X+Y, 
and are shown in Fig. 6. The two upper diagrams correspond to 
Z(q_2q̅_1) → X(q_1q̅_1)+Y(q_2q̅_2), and the two lower
diagrams Z(q_1q̅_2) → X(q_1q̅_1)+Y(q_2q̅_2). Denote by
ψ_Z and ψ_XY the mesonic quark-antiquark wave functions of meson
Z and of mesons X and Y, respectively.
Let (E_X, p⃗_X), (E_Y, p⃗_Y), and (E_Z, p⃗_Z)
be the four-momenta of mesons X, Y, and Z, respectively.
The transition amplitudes corresponding to the left upper diagram, the right
upper diagram, the left lower diagram, and the right lower diagram are given by

    M_ cq̅_1 = √(2E_Z2E_X2E_Y)∫ dr⃗_q_1q̅_1 dr⃗_q_2q̅_2ψ_XY^+ V_ cq̅_1ψ_Z e^-ip⃗_q_1q̅_1,q_2q̅_2·r⃗_q_1q̅_1,q_2q̅_2,


    M_ cq_2 = √(2E_Z2E_X2E_Y)∫ dr⃗_q_1q̅_1 dr⃗_q_2q̅_2ψ_XY^+ V_ cq_2ψ_Z e^-ip⃗_q_1q̅_1,q_2q̅_2·r⃗_q_1q̅_1,q_2q̅_2,


    M_ cq_1 = √(2E_Z2E_X2E_Y)∫ dr⃗_q_1q̅_1 dr⃗_q_2q̅_2ψ_XY^+ V_ cq_1ψ_Z e^-ip⃗_q_1q̅_1,q_2q̅_2·r⃗_q_1q̅_1,q_2q̅_2,


    M_ cq̅_2 = √(2E_Z2E_X2E_Y)∫ dr⃗_q_1q̅_1 dr⃗_q_2q̅_2ψ_XY^+ V_ cq̅_2ψ_Z e^-ip⃗_q_1q̅_1,q_2q̅_2·r⃗_q_1q̅_1,q_2q̅_2,

where V_ cq̅_1, V_ cq_2, V_ cq_1, and 
V_ cq̅_2 are the transition potentials for the processes where
a gluon is emitted by a constiutent quark or antiquark in the initial mesons
and subsequently the gluon creates a quark-antiquark pair, and are given in
Eqs. (51) and (52) of Ref. <cit.>. The two upper diagrams give
q̅_1(p_q̅_1^') +q_2(p_q_2^') →
q_1(p_q_1)+q̅_1(p_q̅_1)+q_2(p_q_2) +q̅_2(p_q̅_2),
and the two lower diagrams
q_1(p_q_1^') +q̅_2(p_q̅_2^') → 
q_1(p_q_1)+q̅_1(p_q̅_1)+q_2(p_q_2) +q̅_2(p_q̅_2). 
We have
V_ cq̅_1=V_ rq_1q̅_2q̅_1,
V_ cq_2=V_ rq_1q̅_2q_2, 
V_ cq_1=V_ rq_2q̅_1q_1, and
V_ cq̅_2=V_ rq_2q̅_1q̅_2.

The transition amplitudes squared are given by

    | M_ cq̅_1|^2 = |√(2E_X2E_Y2E_Z)∫ dr⃗_q_1q̅_1 dr⃗_q_2q̅_2ψ_Z^+ V_ cq̅_1ψ_XY e^ip⃗_q_1q̅_1,q_2q̅_2·r⃗_q_1q̅_1,q_2q̅_2|^2,


    | M_ cq_2|^2 = |√(2E_X2E_Y2E_Z)∫ dr⃗_q_1q̅_1 dr⃗_q_2q̅_2ψ_Z^+ V_ cq_2ψ_XY e^ip⃗_q_1q̅_1,q_2q̅_2·r⃗_q_1q̅_1,q_2q̅_2|^2,


    | M_ cq_1|^2 = |√(2E_X2E_Y2E_Z)∫ dr⃗_q_1q̅_1 dr⃗_q_2q̅_2ψ_Z^+ V_ cq_1ψ_XY e^ip⃗_q_1q̅_1,q_2q̅_2·r⃗_q_1q̅_1,q_2q̅_2|^2,


    | M_ cq̅_2|^2 = |√(2E_X2E_Y2E_Z)∫ dr⃗_q_1q̅_1 dr⃗_q_2q̅_2ψ_Z^+ V_ cq̅_2ψ_XY e^ip⃗_q_1q̅_1,q_2q̅_2·r⃗_q_1q̅_1,q_2q̅_2|^2.

It is obvious that | M_ cq̅_1|^2, 
| M_ cq_2|^2, | M_ cq_1|^2, and 
| M_ cq̅_2|^2 equal 
| M_ rq_1q̅_2 q̅_1|^2, 
| M_ rq_1q̅_2 q_2|^2, 
| M_ rq_2 q̅_1 q_1|^2, 
and | M_ rq_2 q̅_1 q̅_2|^2, respectively.
Therefore, Eqs. (22) and (26)-(28) can be used to calculate
| M_ cq̅_1|^2, 
| M_ cq_2|^2, | M_ cq_1|^2, and 
| M_ cq̅_2|^2.

The transition amplitudes lead to the decay width for Z → X+Y:

    W     =    1/2J_Z+1∫d^3p_X/(2π)^3d^3p_Y/(2π)^3(2π)^4δ (E_X+E_Y-E_Z)δ^3 (p⃗_X+p⃗_Y-p⃗_Z)/2E_X2E_Y2E_Z
          ∑_J_ZzJ_XzJ_Yz| M_ cq̅_1 +   M_ cq_2 
    +  M_ cq_1 +  M_ cq̅_2|^2 .

where J_i (i=Z, X, Y) is the total angular momentum of meson i with
its magnetic projection quantum number J_iz.
In Table 9 decay widths are shown for the ψ (3770), ψ (4040), 
ψ (4160), and ψ (4415) decays that produce charmed mesons or
charmed strange mesons. The decay width for ψ (3770) → DD̅ is
18 MeV compared to the experimental value 25.3± 0.09 MeV. The total widths
of the ψ (4040), ψ (4160), and ψ (4415) decays listed in the
Review of Particle Physics <cit.> are 80± 10 MeV, 70± 10 MeV, and
62± 20 MeV, respectively, but the partial widths for the ψ (4040), 
ψ (4160), and ψ (4415) decays to charmed mesons or charmed strange
mesons have not been given. However, since the ψ (4040), 
ψ (4160), and ψ (4415) decays mainly produce charmed mesons and
charmed strange mesons, the measured total widths of the ψ (4040), 
ψ (4160), and ψ (4415) decays may be compared to 70.2 MeV as the sum
of the five ψ (4040) partial widths, 63.3 MeV as the sum
of the seven ψ (4160) partial widths, and 61.1 MeV as the sum
of the eight ψ (4415) partial widths. We note that our partial widths
differ from those obtained in the ^3P_0 model in Ref. <cit.>.




ACKNOWLEDGEMENTS



This work was supported by the National Natural Science Foundation of China
under Grant No. 11175111.


00
CPT J. Gasser and H. Leutwyler, Ann. Phys. (N.Y.) 158, 142 (1984); 
J. Gasser and H. Leutwyler, Nucl. Phys. B 250, 465 (1985); 
J. Gasser and U. G. Meissner, Phys. Lett. B 258, 219 (1991); 
J. Gasser and U. G. Meissner, Nucl. Phys. B 357, 90 (1991); 
I. Bijnens, G. Colangelo, G. Ecker, J. Gasser, and M. E. Sainio, Nucl. Phys. 
B 508, 263 (1997);
F. Guerrero and J. A. Oller, Nucl. Phys. B 537, 459 (1999);
A. G. Nicola and J. Peláez, Phys. Rev. D 65, 054009 (2002).
Hannah T. N. Truong, Phys. Rev. Lett. 67, 2260 (1991); 
T. Hannah, Phys. Rev. D 55, 5613 (1997); 
A. Dobado and J. R. Peláez, Phys. Rev. D 56, 3057 (1997);
M. Boglione and M. R. Pennington, Z. Phys. C 75, 113 (1997);
J. A. Oller, E. Oset, and J. R. Peláez, Phys. Rev. D 59, 074001 (1999); 
J. Nieves, M. P. Valderrama, and E. R. Arriola, Phys. Rev. D 65, 036002 (2002);
J. Nebreda and J. R. Peláez, Phys. Rev. D 81, 054035 (2010); 
M. Döring and U.-G. Meißner, JHEP 01, 009 (2012).
OO J. A. Oller and E. Oset, Nucl. Phys. A 620, 438 (1997); 
F.-K. Guo, R.-G. Ping, P.-N. Shen, H.-C. Chiang, and B. S. Zou, Nucl. Phys. 
A 773, 78 (2006); 
I. V. Danilkin, L. I. R. Gil, and M. F. M. Lutz, Phys. Lett. B 703, 504 (2011);
Z.-H. Guo, L. Liu, U.-G. Meißner, J. A. Oller, and A. Rusetsky, Phys. Rev. 
D 95, 054004 (2017);
I. V. Danilkin and M. Vanderhaeghen, Phys. Lett. B 789, 366 (2019).
BS T. Barnes and E. S. Swanson, Phys. Rev. D 46, 131 (1992);
E. S. Swanson, Ann. Phys. (N.Y.) 220, 73 (1992); 
T. Barnes, E. S. Swanson, and J. Weinstein, Phys. Rev. D 46, 4868 (1992); 
T. Barnes, N. Black, and E. S. Swanson, Phys. Rev. C 63, 025204 (2001).
SXW Z.-Y. Shen, X.-M. Xu, and H. J. Weber, Phys. Rev. D 94, 034030
(2016).
OOP J. A. Oller, E. Oset, and J. R. Peláez, Phys. Rev. D 59, 
074001 (1999).
DEW J. J. Dudek, R. G. Edwards, and D. J. Wilson, Phys. Rev. D 93, 
094506 (2016).
SBGKLN Y. S. Surovtsev, P. Bydžovský, 
T. Gutsche, R. Kamiński, V. E. Lyubovitskij, and M. Nagy, Phys. 
Rev. D 97,
014009 (2018).
AMRR A. T. Aoude, P. C. Magalhães, A. C. dos Reis, and
M. R. Robilotta, Phys. Rev. D 98, 056021 (2018).
BFS D. Black, A. H. Fariborz, and J. Schechter, Phys. Rev. D 61, 
074030 (2000).
GLMOR Z.-H. Guo, L. Liu, U.-G. Meißner, J. A. Oller, and
A. Rusetsky, Phys. Rev. D 95, 054004 (2017).
BHHKM V. Baru, J. Haidenbauer, C. Hanhart, A. Kudryavtsev, and 
U.-G. Meißner, Eur. Phys. J. A 23, 523 (2005).
BN B. Borasoy and R. Nißler, Eur. Phys. J. A 26, 383 (2005).
LLMP C. B. Lang, L. Leskovec, D. Mohler, and S. Prelovsek, JHEP 04, 
162 (2014).
RO L. Roca and E. Oset, Phys. Rev. D 85, 054507 (2012).
FN J. M. Flynn and J. Nieves, Phys. Rev. D 75, 074024 (2007).
ICADH Y. Ikeda, B. Charron, S. Aoki, T. Doi, T. Hatsuda, T. Inoue,
N. Ishii, K. Murano, H. Nemura, and K. Sasaki, Phys. Lett. B 729, 85 (2014).
TDKJO A. M. Torres, L. R. Dai, C. Koren, D. Jido, and E. Oset,
Phys. Rev. D 85, 014027 (2012).
YXW K. Yang, X.-M. Xu, and H. J. Weber, Phys. Rev. D 96, 114025 
(2017).
BT W. Buchmüller and S.-H. H. Tye, Phys. Rev. D 24, 132 (1981).
GI S. Godfrey and N. Isgur, Phys. Rev. D 32, 189 (1985).
BGS T. Barnes, S. Godfrey, and E. S. Swanson, Phys. Rev. D 72, 054026
(2005).
VFV J. Vijande, F. Fernández, and A. Valcarce, J. 
Phys. G 31, 481 (2005); P. G. Ortega, J. Segovia, D. R. Entem, F. 
Fernández, Phys. Lett. B 778, 1 (2018).
LK G. Q. Li and C. M. Ko, Nucl. Phys. A 582, 731 (1995);
W. S. Chung, G. Q. Li, and C. M. Ko, Nucl. Phys. A 625, 347 (1997).
AW G. B. Arfken and H. J. Weber, Mathematical Methods for 
Physicists (Elsevier, Amsterdam, 2006).
joachain C. J. Joachain, Quantum Collision Theory
(North-Holland Publishing Company, Amsterdam, 1983).
JSX S.-T. Ji, Z.-Y. Shen, and X.-M. Xu, J. Phys. G 42, 095110 (2015).
KLP F. Karsch, E. Laermann, and A. Peikert, Nucl. Phys. B 605, 579
(2001).
PDG M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98,
030001 (2018) and 2019 update.
pipiqi E. Colton et al., Phys. Rev. D 3, 2028 (1971);
N. B. Durusoy et al., Phys. Lett. B 45, 517 (1973);
M. J. Losty et al., Nucl. Phys. B 69, 185 (1974);
W. Hoogland et al., Nucl. Phys. B 126, 109 (1977).
pipianni S. D. Protopopescu et al., Phys. Rev. D 7, 1279 
(1973); 
B. Hyams et al., Nucl. Phys. B 64, 134 (1973);
P. Estabrooks and A. D. Martin, Nucl. Phys. B 79, 301 (1974);
V. Srinivasan et al., Phys. Rev. D 12, 681 (1975);
L. Rosselet et al., Phys. Rev. D 15, 574 (1977);
C. D. Froggatt and J. L. Petersen, Nucl. Phys. B 129, 89 (1977);
A. A. Bel'kov et al., JETP Lett. 29, 597 (1979);
E. A. Alekseeva et al., Sov. Phys. JETP 55, 591 (1982);
R. García-Martín,
R. Kamiński, J. R. Peláez, J. R. de Elvira, and
F. J. Ynduráin, Phys. Rev. D 83, 074004 (2011).
Xu2002 X.-M. Xu, Nucl. Phys. A 697, 825 (2002).
LXW W.-X. Li, X.-M. Xu, and H. J. Weber, Phys. Rev. D 101, 014025 
(2020).
XW X.-M. Xu and H. J. Weber, Mod. Phys. Lett. A 35, 2030016 (2020).
SX1 Z.-Y. Shen and X.-M. Xu, J. Korean Phys. Soc. 66, 754 (2015).
JXW S.-T. Ji, X.-M. Xu, and H. J. Weber, Nucl. Phys. A 966, 224 
(2017).
FMMR V. Flaminio, W. G. Moorhead, D. R. O. Morrison, N. Rivoire,
CERN, Geneva Report No. CERN-HERA-84-01, 1984.
BIO M. Bayar, N. Ikeno, and E. Oset, Eur. Phys. J. C 80, 222 (2020).
RGG A. De Rújula, H. Georgi, and S. L. Glashow, Phys. 
Rev. Lett. 37, 398 (1976).
IW N. Isgur and M. B. Wise, Phys. Lett. B 232, 113 (1989); 237, 527
(1990).
Georgi H. Georgi, Phys. Lett. B 240, 447 (1990).
Neubert M. Neubert, Phys. Rep. 245, 259 (1994).



































