Cross Sections for Inelastic 2-to-2 Meson-Meson Scattering in Hadronic Matter
Ting-Ting Wang Xiao-Ming Xu





Cross Sections for Inelastic 2-to-2 Meson-Meson 

 Scattering in Hadronic Matter
    Ting-Ting Wang and Xiao-Ming Xu
    
================================================================================



Department of Physics, Shanghai University, Baoshan,
Shanghai 200444, China


With quark-antiquark annihilation and creation in the first Born approximation,
we study the reactions: K K̅→ K K̅^∗, 
 K K̅→ K^* K̅,  π K →π K^∗,  π K →ρ K,
 ππ→ K K̅^∗,  ππ→ K^∗K̅,
 ππ→ K^∗K̅^∗,  πρ→ K K̅,
 πρ→ K^∗K̅^∗,  ρρ→ K^∗K̅^∗,
 K K̅^∗→ρρ, and K^* K̅→ρρ. 
Unpolarized cross sections for the reactions are obtained from transition
amplitudes that are composed of mesonic quark-antiquark relative-motion
wave functions and the transition potential for quark-antiquark annihilation
and creation. From a quark-antiquark potential that is equivalent to the 
transition potential, we prove that the total spin of the two final mesons
may not equal the total spin of the two initial mesons. Based on flavor
matrix elements, cross sections for some isospin channels of reactions can be
obtained from the other isospin channels of reactions. Remarkable temperature
dependence of the cross sections is found.


Keywords: Inelastic meson-meson scattering, Quark-antiquark annihilation,
Quark potential model.

PACS: 25.75.-q; 24.85.+p; 12.38.Mh




I. INTRODUCTION



In hadronic matter that is created in ultrarelativistic heavy-ion collisions,
various kinds of meson-meson scattering take place. The meson-meson scattering
can be studied in quark degrees of freedom or meson degrees of freedom.
Elastic ππ scattering for I=2 and elastic π K scattering for
I=3/2 have been studied in the quark interchange mechanism in the first
Born approximation <cit.> and in
nonperturbative schemes together with chiral perturbation theory
<cit.>.
The reactions ππ→ρρ for I=2, KK → K^*K^* for I=1,
π K →ρ K^* for I=3/2, and so on have also been studied in the quark
interchange mechanism <cit.>. 
The reactions ππ→ K
K̅, ρρ→ K K̅, πρ→ K K̅^*, and
πρ→ K^* K̅ can be studied by quark-antiquark
annihilation and creation in the first Born approximation <cit.> or
through one-meson exchange
in effective meson Lagrangians <cit.>. Furthermore, the four
isospin channels, π K →ρ K^∗ for I=1/2, π K^∗→ρ K
for I=1/2, π K^∗→ρ K^∗ for I=1/2, and
ρ K →ρ K^∗ for I=1/2, are studied in the assumption that quark
interchange as well as quark-antiquark annihilation and creation are dominant
mechanisms <cit.>. These studies have revealed interesting features of 
these reactions. For example, the cross
section for the inelastic meson-meson scattering governed by quark interchange
increases very rapidly to a maximum value and then decreases rapidly while the
center-of-mass energy of the two colliding mesons increases from threshold; 
however, the cross section for the inelastic meson-meson scattering governed by
quark-antiquark annihilation and creation may decrease very slowly from the
maximum value.

In the present work we are interested in the inelastic meson-meson scattering
among π, ρ, K, and K^∗ mesons, which is assumed to be
dominated
by quark-antiquark annihilation and creation. The meson-meson reactions studied
in Ref. <cit.> include ππ→ρρ, K K̅→ K^* K̅^∗, K K̅^∗→ K^* K̅^∗, K^∗K̅→ K^* K̅^∗, ππ→ K K̅, πρ→
K K̅^∗, πρ→ K^* K̅, and K K̅→ρρ.
But these reactions do not exhaust all the 2-to-2 meson-meson reactions among
π, ρ, K, and K^∗ mesons. Hence,
in the present work we study these reactions:
K K̅→ K K̅^∗,  K K̅→ K^* K̅,
 π K →π K^∗, π K →ρ K,
 ππ→ K K̅^∗, ππ→ K^∗K̅,
 ππ→ K^∗K̅^∗,  πρ→ K K̅,
 πρ→ K^∗K̅^∗,  ρρ→ K^∗K̅^∗,
 K K̅^∗→ρρ, and K^* K̅→ρρ.
These reactions have not been studied elsewhere, and
complement those reactions studied in Ref. <cit.>.

This paper is organized as follows. In Sec. II we present formulas of
unpolarized cross sections for 2-to-2 meson-meson reactions that are governed
by annihilation of a quark-antiquark pair and creation of 
another quark-antiquark pair. In Sec. III we calculate transition amplitudes
with mesonic quark-antiquark wave functions and
the transition potential for quark-antiquark annihilation and creation.
In Sec. IV we show unpolarized cross sections for 
inelastic meson-meson scattering, and give relevant discussions. In Sec. V we
summarize the present work.




II. CROSS-SECTION FORMULAS



It is shown by the two Feynman diagrams in Fig. 1 that the reaction 
A+B → C+D is caused by
quark-antiquark annihilation and creation in the Born approximation.
In the left diagram of Fig. 1 the quark of meson A and the antiquark of meson
B annihilate into a gluon, this gluon creates a new quark-antiquark pair, 
and the new quark and the antiquark in meson A combine into 
meson C as well as the new antiquark and the quark in meson B combine into
meson D. In the right diagram of Fig. 1 the antiquark of meson A and the
quark of meson B annihilate into a gluon, the gluon creates a new
quark-qnatiquark pair, and the new antiquark and the quark of
meson A form meson C as well as the new quark and the antiquark of meson
B form meson D. These are the processes that we consider in the present 
work.

Let E_A (E_B, E_C, E_D) and J_Az (J_Bz, J_Cz, J_Dz) 
denote the energy and the magnetic projection quantum number of the angular 
momentum J_A (J_B, J_C, J_D) of meson A ( B, C, D), 
respectively. From the four-momenta of mesons A and B, P_A and P_B,
the Mandelstam variable s=(P_A+P_B)^2 is defined. The unpolarized cross 
section for A+B → C+D depends on √(s) and temperature T. Let
θ be the angle between P⃗ and P⃗' which are the 
three-dimensional momenta of mesons A and C in the center-of-mass frame, 
respectively. The unpolarized cross section is <cit.>

    σ^ unpol(√(s),T)     =    1/(2J_A+1)(2J_B+1)1/32π s|P⃗^'(√(s))|/|P⃗(√(s))|
       ×   ∫_0^πdθ∑_J_AzJ_BzJ_CzJ_Dz| M_ aq_1q̅_2+ M_ aq̅_1q_2|^2
    sinθ ,

where  M_ aq_1q̅_2 and
M_ aq̅_1q_2 are the transition amplitudes corresponding 
to the left diagram and the right diagram in Fig. <ref>, respectively. 
The transition amplitudes are given by

    M_ aq_1q̅_2    =    (m_q_3+m_q̅_1)^3/m_q̅_1^3√(2E_A2E_B2E_C2E_D)∫ dr⃗_q_1q̅_1 dr⃗_q_2q̅_4 
    dr⃗_q_3q̅_1,q_2q̅_4
          ψ_q_3q̅_1^+ (r⃗_q_3q̅_1)
    ψ_q_2q̅_4^+ (r⃗_q_2q̅_4)
    V_ aq_1q̅_2ψ_q_1q̅_1 (r⃗_q_1q̅_1)
    ψ_q_2q̅_2 (r⃗_q_2q̅_2)
           
          
    e^ip⃗_q_1q̅_1,q_2q̅_2·r⃗_q_1q̅_1,q_2q̅_2
    -ip⃗_q_3q̅_1,q_2q̅_4·r⃗_q_3q̅_1,q_2q̅_4,


    M_ aq̅_1q_2    =    (m_q_1+m_q̅_4)^3/m_q_1^3√(2E_A2E_B2E_C2E_D)∫ dr⃗_q_1q̅_1 dr⃗_q_3q̅_2 
    dr⃗_q_1q̅_4,q_3q̅_2
          ψ_q_1q̅_4^+ (r⃗_q_1q̅_4)
    ψ_q_3q̅_2^+ (r⃗_q_3q̅_2)
    V_ aq̅_1q_2ψ_q_1q̅_1 (r⃗_q_1q̅_1)
    ψ_q_2q̅_2 (r⃗_q_2q̅_2)
           
          
    e^ip⃗_q_1q̅_1,q_2q̅_2·r⃗_q_1q̅_1,q_2q̅_2
    -ip⃗_q_1q̅_4,q_3q̅_2·r⃗_q_1q̅_4,q_3q̅_2,

where m_a is the mass of constituent a;
V_ aq_1q̅_2 and V_ aq̅_1q_2 are 
the transition potentials for q_1+q̅_2 → q_3+q̅_4
in the left diagram and q̅_1+q_2 → q_3+q̅_4 in the right 
diagram of Fig. <ref>, respectively; 
ψ_ab and r⃗_ab are the wave function and the relative
coordinate of constituents a and b, respectively.
The relative coordinate and the relative momentum of q_1q̅_1 and 
q_2q̅_2 are denoted by r⃗_q_1q̅_1,q_2q̅_2 and
p⃗_q_1q̅_1,q_2q̅_2, respectively;
similar meanings apply to r⃗_q_3q̅_1,q_2q̅_4,
p⃗_q_3q̅_1,q_2q̅_4, 
r⃗_q_1q̅_4,q_3q̅_2, and
p⃗_q_1q̅_4,q_3q̅_2. 




III. TRANSITION AMPLITUDE



In order to calculate the transition amplitudes, we need the mesonic
quark-antiquark wave functions and the transition 
potential for quark-antiquark annihilation and creation. 
The quark-antiquark annihilation and creation is shown in Fig. 2 with 
q (p_1) + q̅ (-p_2) → q^ ' (p_3) + q̅^ ' (-p_4), 
where p_1 and p_3 (p_2 and p_4) are the four-momenta of quarks 
(antiquarks).
The transition potential shown below is given in Ref. <cit.>,

    V_ aqq̅(k⃗)     =    g_ s^2/k^2λ⃗(34)/2·λ⃗(21)/2(
    σ⃗(34) ·k⃗σ⃗(21) ·k⃗/4m_q^' m_q -σ⃗(34) ·σ⃗(21)  .
                           
        -    . σ⃗(21) ·p⃗_2 σ⃗(34)
    ·σ⃗(21) σ⃗(21) ·p⃗_1/4m_q^2
    -σ⃗(34) ·p⃗_3 σ⃗(34)
    ·σ⃗(21) σ⃗(34) ·p⃗_4/4m_q^'^2),

where k⃗ is the three-dimensional momentum of gluon, 
g_s is the gauge coupling constant, m_q (m_q') is the mass of 
the initial (final) quark, λ⃗ are the Gell-Mann matrices, and
σ⃗ are the Pauli matrices. 
λ⃗(21) (σ⃗(21)) mean that they have matrix elements 
between the color (spin) wave functions of the initial quark and the initial
antiquark, and λ⃗(34) (σ⃗(34)) mean that they have
matrix elements between the color (spin) wave functions of the final quark and
the final antiquark.

The wave functions of mesons A , B, C, and D are individually given by

    ψ_A =ϕ_A relϕ_A colorϕ_A flavorχ_S_A S_Az,


    ψ_B =ϕ_B relϕ_B colorϕ_B flavorχ_S_B S_Bz,


    ψ_C =ϕ_C relϕ_C colorϕ_C flavorχ_S_C S_Cz,


    ψ_D =ϕ_D relϕ_D colorϕ_D flavorχ_S_D S_Dz,

where S_i is the spin of meson i, and S_iz is its magnetic projection
quantum number. According to the two diagrams in Fig. 1, 
ψ_A = ψ_q_1q̅_1, ψ_B = ψ_q_2q̅_2, 
ψ_C = ψ_q_3q̅_1 = ψ_q_1q̅_4, and
ψ_D = ψ_q_2q̅_4 = ψ_q_3q̅_2. The wave function of 
meson i is made up of the quark-antiquark relative-motion wave
function ϕ_i rel, the color wave function ϕ_i color, 
the flavor wave function ϕ_i flavor, and the spin wave function 
χ_S_i S_iz.

The transition amplitudes contain color, spin, and flavor matrix elements.
The color matrix elements related to the Gell-Mann matrices are 4/9 for the two
diagrams in Fig. 1. The spin matrix elements related to the Pauli matrices have
been provided in Ref. <cit.>. Let M_ aq_1q̅_2 f
and M_ aq̅_1q_2 f represent the flavor matrix elements
that correspond to the left and right diagrams in Fig. 1, respectively.
The values of the flavor matrix elements are listed in Table <ref>, 
where I is the total isospin of the two initial or final mesons for the 
reactions:

    K K̅→ K K̅^∗,  K K̅→ K^* K̅,
     π K →π K^∗,  π K →ρ K,
     ππ→ K K̅^∗,  ππ→ K^∗K̅,
    
    ππ→ K^∗K̅^∗,  πρ→ K K̅,
     πρ→ K^∗K̅^∗,  ρρ→ K^∗K̅^∗, 
     K K̅^∗→ρρ,  K^* K̅→ρρ.


The quark-antiquark relative-motion wave functions are given by
the Schrödinger equation with a temperature-dependent potential.
The experimental masses of ground-state mesons <cit.> are reproduced 
by the Schrödinger equation, while the up and 
down quark masses are 0.32 GeV and the strange quark mass is 0.5 GeV 
<cit.>. The temperature-dependent potential between constituents a
and b is given by <cit.>
 
    V_ab(r⃗) = V_ si(r⃗) + V_ ss(r⃗),

where r⃗ is the relative coordinate of a and b. The first term 
V_ si(r⃗) is the central spin-independent potential and depends on 
temperature:

    V_si(r⃗) =
    - λ⃗_a/2·λ⃗_b/23/4 D [ 1.3- ( T/T_ c)^4 ]
    tanh (Ar) + λ⃗_a/2·λ⃗_b/26π/25v(λ r)/rexp (-Er),

where D=0. 7 GeV, T_ c=0.175 GeV,
A=1.5[0.75+0.25 (T/T_ c)^10]^6 GeV, E=0. 6 GeV,
λ=√(25/16π^2 α') with α'=1.04 GeV^-2,
λ⃗_a (λ⃗_b) are the Gell-Mann matrices for the color
generators of constituent a (b), and the dimensionless function v(x) is
given by Buchmüller and Tye in Ref. <cit.>.
	
This potential V_ si(r⃗) is relevant to the temperature of 
hadronic matter and the distance r.
It shows some characteristics as follows. At very short distances r < 0.01 
fm, the potential arises from one-gluon exchange plus perturbative one- and 
two-loop corrections. At large distances and blow the 
QCD phase-transition temperature T_ c, the color screening produced by 
high-temperature medium may be strong. Karsch et al. <cit.> have
provided such a numerical quark-antiquark potential at r ≥ 0.3 fm
in a temperature region from lattice QCD calculations. When the distance 
between the quark and the antiquark becomes large, the quark-antiquark 
potential at a given temperature becomes a constant value, which decreases with
increasing temperature.

The second term V_ ss(r⃗) in Eq. (9) is the spin-spin interaction
which originates from one-gluon exchange plus perturbative one- and two-loop 
corrections <cit.>, depends on constituent masses,
and includes relativistic effects <cit.>:

    V_ ss(r⃗)=
    -λ⃗_a/2·λ⃗_b/216π^2/25d^3/π^3/2exp(-d^2r^2) s⃗_a ·s⃗ _b/m_am_b
    + λ⃗_a/2·λ⃗_b/24π/251/rd^2v(λ r)/dr^2s⃗_a ·s⃗_b/m_am_b ,

where s⃗_a (s⃗_b) is the spin of 
constituent a (b), and d is given by

    d^2=d_1^2[1/2+1/2(4m_a m_b/(m_a+m_b)^2)^4]+d_2^2(2m_am_b/m_a+m_b)^2,

where d_1=0.15 GeV and d_2= 0.705.




IV. NUMERICAL CROSS SECTIONS AND DISCUSSIONS 



We consider the following inelastic meson-meson scattering processes that 
mainly take the two Feynman diagrams in Fig. 1:

    K K̅→ K K̅^∗ ,    K K̅→ K^* K̅,
    
    π K →π K^∗,   π K →ρ K,
    
    ππ→ K K̅^∗,   ππ→ K^∗K̅,
      ππ→ K^∗K̅^∗,
    
    πρ→ K K̅,   πρ→ K^∗K̅^∗,
    
    ρρ→ K^∗K̅^∗,    K K̅^∗→ρρ, 
       K^* K̅→ρρ.

The total spin of π and K mesons does not equal the total spin of π
and K^* mesons or of ρ and K mesons, i.e., 
the total spin in either π K →π K^* or
π K →ρ K is not conserved. Quark interchange thus does
not happen in the two reactions. M_ aq_1q̅_2 and 
M_ aq̅_1q_2 are proportional to the flavor matrix 
elements. If the transition amplitudes equal zero, the unpolarized cross
section given in Eq. (1) is zero. As seen in Table 1, the flavor matrix 
elements for the two reactions for I=3/2
are zero. Quark-antiquark annihilation and creation
does not happen in the two reactions for I=3/2 too. Therefore, cross sections
for π K →π K^* for I=3/2 and π K →ρ K for I=3/2 are zero
in the present work, but we still investigate π K →π K^* for I=1/2 
and π K →ρ K for I=1/2 of which M_ aq̅_1q_2f
are not zero. The other reactions must involve
quark-antiquark annihilation and creation, but do not involve quark 
interchange.

It is shown in Table <ref> that only the right diagram in 
Fig. <ref>
contributes to the reactions: ππ→ K K̅^*,
 ππ→ K^∗K̅,  ππ→ K^∗K̅^*,
 πρ→ K K̅,  πρ→ K^* K̅^∗,
 ρρ→ K^* K̅^*,  K K̅^* →ρρ, and
 K^∗K̅→ρρ. Since the flavor matrix elements for the
reactions for I=0 are √(6)/2 times the ones for I=1,
the cross sections for the reactions for I=0 are 1.5 times the cross sections
for I=1. The cross section for KK̅→ K^*K̅
(ππ→ K^* K̅, K^*K̅→ρρ) equals the one for 
KK̅→ KK̅^* (ππ→ KK̅^*,
KK̅^* →ρρ). We thus do not plot the cross sections for
KK̅→ K^*K̅,
ππ→ K^* K̅, and K^*K̅→ρρ.

The gauge coupling constant is 2√(6)π/5 for quark-antiquark
annihilation and creation <cit.>. According to Eq. (1),
we calculate unpolarized cross sections at the six temperatures
T/T_ c =0, 0.65, 0.75, 0.85, 0.9, and 0.95. In 
Figs. <ref>-<ref>
we plot the unpolarized cross sections for the following ten channels:

    I=1  K K̅→ K K̅^∗ ,    I=0  K
    K̅→ K K̅^∗,
    
    
    I=1/2 π K →π K^∗,    I=1/2 π K →ρ K,
    
    
    I=1 ππ→ K K̅^∗,    I=1 ππ→ K^∗K̅^∗,
    
    
    I=1 πρ→ K K̅,    I=1 πρ→ K^∗K̅^∗,
    
    
    I=1 ρρ→ K^∗K̅^∗,    I=1  K K̅^∗→ρρ.

The last channel is endothermic at T/T_ c=0 and exothermic at
T/T_ c=0.65, 0.75, 0.85, 0.9, and 0.95. The other nine channels are
endothermic.
The numerical cross sections for endothermic reactions are parametrized as

    σ^ unpol(√(s),T)
       =   a_1 ( √(s) -√(s_0)/b_1)^e_1exp[ e_1 ( 1-√(s) -√(s_0)/b_1) ]
    
          + a_2 ( √(s) -√(s_0)/b_2)^e_2exp[ e_2 ( 1-√(s) -√(s_0)/b_2) ],

where √(s_0) is the threshold energy, and a_1, b_1, e_1, a_2,
b_2, and e_2 are parameters. The numerical cross sections
for exothermic reactions are parametrized as

    σ^ unpol(√(s),T)
       =   P⃗^' 2/P⃗^2{a_1 ( √(s) -√(s_0)/b_1)^e_1exp[ e_1 ( 1-√(s) -√(s_0)/b_1) ]
    .
    
          + .
    a_2 ( √(s) -√(s_0)/b_2)^e_2exp[ e_2 ( 1-√(s) -√(s_0)/b_2) ]
    }.

The parameter values are listed in Tables 2-4. In the three tables
the quantity d_0 is the separation 
between the peak's location on the √(s)-axis and the threshold energy. 
The smaller d_0 is, the faster the cross section increases from zero to the
peak cross section. The quantity √(s_z) is the square root of the 
Mandelstam variable at which the cross section is 1/100 of the peak cross 
section. The quantity
√(s_z)-√(s_0)-d_0 is the difference between √(s_z) and the
peak's location on the √(s)-axis. The smaller 
√(s_z)-√(s_0)-d_0 is, the faster the cross section decreases from
the peak cross section to zero.

The potential given in Eq. (9) depends on temperature. The Schrödinger
equation with the potential yields temperature-dependent meson masses. For
any endothermic (exothermic) 2-to-2 meson-meson reaction the threshold energy
is the sum of the masses of the two final (initial) mesons. Since the meson 
masses decrease with increasing temperature, the threshold energy decreases
with increasing temperature.

Let m_A (m_B, m_C, m_D) be the mass of meson A (B, C, D). In 
terms of the meson masses we have

|P⃗ |=1/2 
√((s-m_A^2-m_B^2)^2-4m_A^2m_B^2/s),


|P⃗^'|=1/2 
√((s-m_C^2-m_D^2)^2-4m_C^2m_D^2/s).

For endothermic reactions the threshold energy is √(s_0)=m_C+m_D, and 
we obtain

|P⃗^'|/|P⃗ | =
√((√(s)-√(s_0))(√(s)+√(s_0))[s-(m_C-m_D)^2]/[s-(m_A+m_B)^2][s-(m_A-m_B)^2]).

When √(s) is close to √(s_0), |P⃗^'|/|P⃗| is sensitive to √(s) - √(s_0). Since Eq. (1)
contains |P⃗^'|/|P⃗|, the
unpolarized cross section in Eq. (13) may be proportional to
(√(s) - √(s_0))^0.5. Nevertheless, the factor
√((√(s)+√(s_0))[s-(m_C-m_D)^2]/[s-(m_A+m_B)^2][s-(m_A-m_B)^2]) in the expression of 
|P⃗^'|/|P⃗| and the factor
1/s| M_ aq_1q̅_2+ M_ aq̅_1q_2|^2
in Eq. (1) modify the dependence of the unpolarized cross section on 
√(s) - √(s_0). We thus use (√(s) - √(s_0))^e_1 and
(√(s) - √(s_0))^e_2 in Eq. (13)
instead of (√(s) - √(s_0))^0.5. Indeed, it is shown in Tables 
2-4 that the values of e_1 and/or e_2 are near 0.5.

Denote by σ_ num^ unpol(√(s),T) the cross sections 
calculated from
Eq. (1), which are plotted in Figs. 3-12. By comparison we denote by 
σ_ para^ unpol(√(s),T) the cross sections given by Eqs.
(13) and (14). We change a_1, b_1, e_1, a_2, b_2, and e_2 to make
| (σ_ para^ unpol(√(s),T)
-σ_ num^ unpol(√(s),T))/
σ_ num^ unpol(√(s),T) | as small as possible. The 
parameter values which make | (σ_ para^ unpol(√(s),T)
-σ_ num^ unpol(√(s),T))/
σ_ num^ unpol(√(s),T) | smallest are provided in
Tables 2-4.

A feature of the endothermic reactions in Figs. 3-11 is that 
the peak cross section decreases first and then increases as the
temperature goes up. As the temperature increases from zero, 
confinement shown by the potential in Eq. (10) becomes weaker and weaker,
the Schrödinger equation produces increasing meson radii, and mesonic
quark-antiquark states become looser and looser. On one hand the weakening
confinement with increasing temperature makes combining final 
quarks and antiquarks into final mesons more difficult, and thus reduces
cross sections; On the other hand the increasing radii of initial
mesons cause increasing cross sections as the temperature goes up. 
The two factors determine the change in peak cross section with respect to the
temperature. Another feature is that the cross section increases rapidly from
zero to a maximum value when the total energy of the two initial mesons in the
center-of-mass frame increases from the threshold energy, and the cross
section further decreases from the maximum value or exhibits a plateau
on the right of the peak as seen in Figs. 8 and 10.

The unpolarized cross sections for K K̅→ K K̅^∗ for I=1 and
for I=0 are shown in Figs. <ref> and <ref>, respectively. 
Quark-antiquark annihilation and creation takes place in the two 
isospin channels.
If M_ aq_1q̅_2 (M_ aq̅_1q_2) for
a reaction equals zero, the left (right) diagram does not contribute to the
reaction.
It is shown from Table <ref> that the two diagrams in Fig. <ref> 
contribute to K K̅→ K K̅^∗ for I=0, and only the right 
diagram contributes to K K̅→ K K̅^∗ for I=1.
The peak cross section of 
K K̅→ K K̅^∗ for I=0 at a given temperature is more than 4 
times the one for I=1, and √(s_z) for I=0 in Table 2 is roughly 2
times that for 
I=1. When T/T_ c =0, 0.65, and 0.75, d_0 of K K̅→ K 
K̅^∗ for I=0 is equal to d_0 for I=1; when T/T_ c=0.85,
0.90, and 0.95, d_0 of K K̅→ K K̅^∗ for I=0 is less 
than d_0 for I=1. Therefore, the cross section for 
KK̅→ KK̅^* for I=0 is larger than the one for 
KK̅→ KK̅^* for I=1.

The unpolarized cross section for KK̅→ρρ for I=1 has been 
shown in Fig. 15 of Ref.<cit.>, and the one for KK̅^∗→ρρ for I=1 in Fig. 12 in the present work. At zero temperature 
the peak cross
section of KK̅→ρρ for I=1 is smaller than the one of
KK̅^*→ρρ for I=1; the cross section for 
KK̅→ρρ for I=1 decreases from the peak cross section
slower than for
KK̅^*→ρρ for I=1 since √(s)_z of the former is larger
than that of the latter. When the two reactions are exothermic, the cross 
section for KK̅→ρρ for I=1 
decreases slower than for KK̅^*→ρρ for I=1 with
increasing center-of-mass energy of the two initial mesons
from the threshold energy plus 10^-4 GeV. 

In the present work and in Ref. <cit.> we have studied the reactions:
KK̅→ KK̅^*, K^*K̅, and K^*K̅^*; 
ππ→ KK̅,
KK̅^*, K^*K̅, and K^*K̅^*; πρ→ KK̅,
KK̅^*, K^*K̅, and K^*K̅^*. As an example we compare the
production of KK̅^* with the production of K^*K̅^* in the
K+K̅ reaction. The unpolarized cross sections for KK̅→ K^∗K̅^∗ for I=1 and for I=0 have been shown in Figs. 8 and 9 of 
Ref. <cit.>, respectively. Quark-antiquark annihilation and 
creation takes place in the two isospin channels. 
The peak cross section of K K̅→ K K̅^∗ for 
I=1 ( I=0 ) at a given temperature is larger than the one of 
K K̅→ K^* K̅^∗ for I=1 ( I=0 ).
The Mandelstam variable √(s) corresponding to the peak cross section of
KK̅→ KK̅^* is smaller than the one corresponding to the peak 
cross section of KK̅→ K^*K̅^* at a given temperature. Since the
two reactions have the same initial mesons, the initial mesons in 
KK̅→ KK̅^* have a smaller value of |P⃗| 
corresponding to the peak cross section than
in K K̅→ K^∗K̅^∗. The cross section given in Eq. (1)
is proportional to the inverse of s|P⃗|.
Therefore, the peak cross section of 
K K̅→ K K̅^∗ is larger than the one
of K K̅→ K^∗K̅^∗. 

The total spin of the two final mesons may not equal the total spin of the two
initial mesons in the reactions in the present work. This can be accounted for
from a quark-antiquark potential which is equivalent to the transtion potential
in Eq. (4). The quark-antiquark potential is obtained from 
the transition potential under the Fierz transformations in Ref. <cit.>,

    V_ aqq̅ F(k⃗)   =   -g^2_s/k^2[1/3λ^0_f(31)λ^0_f(42)
    +1/2λ⃗_f(31)·λ⃗^T_f(42)]
     
       ×   [4/9λ^0(31)λ^0(42)
    -1/12λ⃗(31)·λ⃗^T(42)]
    [-3/2-1/2σ⃗(31)·σ⃗(42) .
    
       +   σ⃗(42)·p⃗_4σ⃗(42)·p⃗_2/8m_qm_q'
    +σ⃗(31)·p⃗_3σ⃗(31)·p⃗_1/8m_qm_q'
    +3σ⃗(31)·p⃗_1σ⃗(42)·p⃗_2/8m^2_q
       -   σ⃗(31)·p⃗_1σ⃗(42)·p⃗_4/8m_qm_q'
    -σ⃗(31)·p⃗_3σ⃗(42)·p⃗_2/8m_qm_q'
    +3σ⃗(31)·p⃗_3σ⃗(42)·p⃗_4/8m^2_q'
       +   σ⃗(31)σ⃗(31)·p⃗_1σ⃗(42)σ⃗(42)·p⃗_2/8m^2_q
    +σ⃗(31)σ⃗(31)·p⃗_1σ⃗(42)·p⃗_4σ⃗(42)/8m_qm_q'
       +   σ⃗(31)·p⃗_3σ⃗(31)·σ⃗(42)σ⃗(42)·p⃗_2/8m_qm_q' 
    +σ⃗(31)·p⃗_3σ⃗(31)
    σ⃗(42)·p⃗_4σ⃗(42)/8m^2_q'
       -   . σ⃗(42)·p⃗_4σ⃗(31)·σ⃗(42)σ⃗(42)·p⃗_2/8m_qm_q'
    -σ⃗(31)·p⃗_3σ⃗(31)·σ⃗(42)σ⃗(31)·p⃗_1/8m_qm_q'],

where λ^0_f is a 3×3 unit matrix in flavor space, 
λ⃗_f
are the Gell-Mann matrices that operate in flavor space, and λ^0 is 
a 3×3 unit matrix in color space. The superscript T in Eqs. (15),
(16), and (19)-(21) means transposition <cit.>.

For the left diagram in Fig. <ref>, the quark-antiquark potential 
corresponding to q_1+q̅_2 → q_3 + q̅_4 is between q_1 and 
q̅_2,

    V_ aq_1q̅_2 F(k⃗)
       =   -g^2_s/k^2[1/3λ^0_f(31)λ^0_f(42)
    +1/2λ⃗_f(31)·λ⃗^T_f(42)]
     
       ×   [4/9λ^0(31)λ^0(42)-1/12λ⃗(31)·λ⃗^T(42)]
    [-3/2-1/2σ⃗(31)·σ⃗(42) .
    
       +   σ⃗(42)·p⃗_q̅_4σ⃗(42)·p⃗_q̅_2/8m_q_1m_q_3
    +σ⃗(31)·p⃗_q_3σ⃗(31)·p⃗_q_1/8m_q_1m_q_3
    +3σ⃗(31)·p⃗_q_1σ⃗(42)·p⃗_q̅_2/8m^2_q_1
       -   σ⃗(31)·p⃗_q_1σ⃗(42)·p⃗_q̅_4/8m_q_1m_q_3
    -σ⃗(31)·p⃗_q_3σ⃗(42)·p⃗_q̅_2/8m_q_1m_q_3
    +3σ⃗(31)·p⃗_q_3σ⃗(42)·p⃗_q̅_4/8m^2_q_3
       +   σ⃗(31)σ⃗(31)·p⃗_q_1σ⃗(42)σ⃗(42)·p⃗_q̅_2/8m^2_q_1
    +σ⃗(31)σ⃗(31)·p⃗_q_1σ⃗(42)·p⃗_q̅_4σ⃗(42)/8m_q_1m_q_3
       +   σ⃗(31)·p⃗_q_3σ⃗(31)·σ⃗(42)σ⃗(42)·p⃗_q̅_2/8m_q_1m_q_3 
    +σ⃗(31)·p⃗_q_3σ⃗(31)σ⃗(42)
    ·p⃗_q̅_4σ⃗(42)/8m^2_q_3
       -   . σ⃗(42)·p⃗_q̅_4σ⃗(31)·σ⃗(42)σ⃗(42)·p⃗_q̅_2/8m_q_1m_q_3
    -σ⃗(31)·p⃗_q_3σ⃗(31)·σ⃗(42)σ⃗(31)·p⃗_q_1/8m_q_1m_q_3] .

While we carry out the Fierz transformations, we need that quark 3 (antiquark
4) and quark 1 (antiquark 2) have the same flavor. Then, quark and antiquark
masses possess
m_q_3=m_q_1 and m_q̅_4=m_q̅_2. For convenience
we use p⃗_q_1^ '=p⃗_q_3 and 
p⃗_q̅_2^ '=p⃗_q̅_4.
The Hamiltonian corresponding to the left diagram in Fig. <ref> is

    H_1    =    V_ aq_1q̅_2 F(k⃗)+V_q_1q̅_1
    +V_q_2q̅_2
       +   √(m^2_q_1+p⃗^ 2_q_1)
    +√(m^2_q̅_1+p⃗^ 2_q̅_1)
    +√(m^2_q_2+p⃗^ 2_q_2)
    +√(m^2_q̅_2+p⃗^ 2_q̅_2) ,

where V_q_1q̅_1 and V_q_2q̅_2 are the Fourier transform
of V_ab(r⃗) in Eq. (9). The total spin of the two mesons is

    S⃗=S⃗_q_1+S⃗_q̅_1+S⃗_q_2+S⃗_q̅_2 .


The commutator of the z component S_z of the total spin and the
Hamiltonian is

    [S_z,H_1]   =   -g^2_s/k^21/8m^2_q_1(1/3λ^0_q_1fλ^0_q̅_2f
    +1/2λ⃗_q_1f·λ⃗^T_q̅_2f)
    (4/9λ^0_q_1λ^0_q̅_2
    -1/12λ⃗_q_1·λ⃗^T_q̅_2)
    
       ×   {σ_q̅_2y[p'_q̅_2zp_q̅_2y-p'_q̅_2yp_q̅_2z -p_q_1yp_q̅_2z+p_q_1zp_q̅_2y+p_q_1yp'_q̅_2z
    -p_q_1zp'_q̅_2y-p'_q_1yp_q̅_2z
       +   p'_q_1zp_q̅_2y+p'_q_1yp'_q̅_2z
    -p'_q_1zp'_q̅_2y+p'_q_1zp_q_1y-p'_q_1yp_q_1z]
    
       +   σ_q̅_2x[p'_q̅_2zp_q̅_2x-p'_q̅_2xp_q̅_2z -p_q_1xp_q̅_2z+p_q_1zp_q̅_2x+p_q_1xp'_q̅_2z
    -p_q_1zp'_q̅_2x-p'_q_1xp_q̅_2z
       +   p'_q_1zp_q̅_2x+p'_q_1xp'_q̅_2z
    -p'_q_1zp'_q̅_2x+p'_q_1zp_q_1x-p'_q_1xp_q_1z]
    
       +   σ_q_1y[p'_q_1zp_q_1y-p'_q_1yp_q_1z
    -p_q_1zp_q̅_2y+p_q_1yp_q̅_2z-p_q_1zp'_q̅_2y +p_q_1yp'_q̅_2z+p'_q_1zp_q̅_2y
       -   p'_q_1yp_q̅_2z+p'_q_1zp'_q̅_2y
    -p'_q_1yp'_q̅_2z+p'_q̅_2zp_q̅_2y-p'_q̅_2yp_q̅_2z]
    
       +   σ_q_1x[p'_q_1zp_q_1x-p'_q_1xp_q_1z
    -p_q_1zp_q̅_2x+p_q_1xp_q̅_2z-p_q_1zp'_q̅_2x +p_q_1xp'_q̅_2z+p'_q_1zp_q̅_2x
       -   p'_q_1xp_q̅_2z+p'_q_1zp'_q̅_2x
    -p'_q_1xp'_q̅_2z +p'_q̅_2zp_q̅_2x-p'_q̅_2xp_q̅_2z]
    
       +   iσ_q_1xσ_q̅_2z[-3p_q_1yp_q̅_2z +p_q_1yp'_q̅_2z+p'_q_1yp_q̅_2z-3p'_q_1yp'_q̅_2z -p_q_1zp_q̅_2y+p'_q_1zp_q̅_2y
       +   p_q_1zp'_q̅_2y-p'_q_1zp'_q̅_2y
    +p'_q̅_2yp_q̅_2z
    +p'_q̅_2zp_q̅_2y+p'_q_1zp_q_1y 
    +p'_q_1yp_q_1z]
    
       +   iσ_q_1yσ_q̅_2z[3p_q_1xp_q̅_2z -p_q_1xp'_q̅_2z-p'_q_1xp_q̅_2z+3p'_q_1xp'_q̅_2z +p_q_1zp_q̅_2x-p_q_1zp'_q̅_2x
       -   p'_q_1zp_q̅_2x+p'_q_1zp'_q̅_2x
    -p'_q̅_2zp_q̅_2x-p'_q̅_2xp_q̅_2z-p'_q_1zp_q_1x -p'_q_1xp_q_1z]
    
       +   i(σ_q_1xσ_q̅_2y
    +σ_q_1yσ_q̅_2x) [4p_q_1xp_q̅_2x-4p_q_1yp_q̅_2y -2p_q_1xp'_q̅_2x+2p_q_1yp'_q̅_2y
       -   2p'_q_1xp_q̅_2x +2p'_q_1yp_q̅_2y
    +4p'_q_1xp'_q̅_2x-4p'_q_1yp'_q̅_2y
    -2p'_q̅_2xp_q̅_2x +2p'_q̅_2yp_q̅_2y
       -   2p'_q_1xp_q_1x+2p'_q_1yp_q_1y]
    
       +   i(σ_q_1yσ_q̅_2y
    -σ_q_1xσ_q̅_2x)
    [4p_q_1yp_q̅_2x+4p_q_1xp_q̅_2y -2p_q_1yp'_q̅_2x-2p_q_1xp'_q̅_2y
       -   2p'_q_1yp_q̅_2x -2p'_q_1xp_q̅_2y
    +4p'_q_1yp'_q̅_2x+4p'_q_1xp'_q̅_2y
    -2p'_q̅_2yp_q̅_2x-2p'_q̅_2xp_q̅_2y
       -   2p'_q_1yp_q_1x-2p'_q_1xp_q_1y]
    
       +   iσ_q_1zσ_q̅_2y[3p_q_1zp_q̅_2x-p_q_1zp'_q̅_2x -p'_q_1zp_q̅_2x+3p'_q_1zp'_q̅_2x+p_q_1xp_q̅_2z -p_q_1xp'_q̅_2z
       -   p'_q_1xp_q̅_2z+p'_q_1xp'_q̅_2z
    -p'_q̅_2zp_q̅_2x-p'_q̅_2xp_q̅_2z
    -p'_q_1zp_q_1x-p'_q_1xp_q_1z]
    
       +   iσ_q_1zσ_q̅_2x[-3p_q_1zp_q̅_2y+p_q_1zp'_q̅_2y +p'_q_1zp_q̅_2y-3p'_q_1zp'_q̅_2y-p_q_1yp_q̅_2z +p_q_1yp'_q̅_2z
       +   p'_q_1yp_q̅_2z-p'_q_1yp'_q̅_2z
    +p'_q̅_2zp_q̅_2y+p'_q̅_2yp_q̅_2z
    +p'_q_1zp_q_1y+p'_q_1yp_q_1z]} ,

where p_ix, p_iy, and p_iz are the three components of the 
momentum of incoming quark i (i=q_1, q̅_2); p'_ix, p'_iy, and
p'_iz are the three components of the momentum of outgoing quark i 
(i=q_1, q̅_2); σ_q_1x, σ_q_1y, and σ_q_1z 
are the three components of σ⃗_q_1≡σ⃗(31); 
σ_q̅_2x, σ_q̅_2y, and σ_q̅_2z 
are the three components of σ⃗_q̅_2≡σ⃗(42);
λ^0_q_1f≡λ^0_f(31), 
λ^0_q̅_2f≡λ^0_f(42),
λ⃗_q_1f≡λ⃗_f(31), 
λ⃗_q̅_2f≡λ⃗_f(42),
λ^0_q_1≡λ^0(31), 
λ^0_q̅_2≡λ^0(42),
λ⃗_q_1≡λ⃗(31), and
λ⃗_q̅_2≡λ⃗(42).
Applying the momentum conservation p⃗_q̅_2^ '=p⃗_q_1
+p⃗_q̅_2-p⃗_q_1^ ', we get

    [S_z,H_1]   =   -g^2_s/k^21/8m^2_q_1(1/3λ^0_q_1fλ^0_q̅_2f
    +1/2λ⃗_q_1f·λ⃗^T_q̅_2f)
    (4/9λ^0_q_1λ^0_q̅_2
    -1/12λ⃗_q_1·λ⃗^T_q̅_2)
    
       ×   {σ_q̅_2y[p'_q_1yp_q_1z+p'_q_1yp_q̅_2z
    -p'_q_1zp_q_1y-p'_q_1zp_q̅_2y+p_q̅_2yp_q_1z -p_q̅_2zp_q_1y]
    
       +   σ_q̅_2x[p'_q_1xp_q_1z+p'_q_1xp_q̅_2z -p'_q_1zp_q_1x-p'_q_1zp_q̅_2x+p_q̅_2xp_q_1z -p_q̅_2zp_q_1x]
    
       +   σ_q_1y[p'_q_1zp_q_1y-p'_q_1yp_q_1z
    -p_q_1zp_q̅_2y+p'_q_1zp_q̅_2y+p_q_1yp_q̅_2z -p'_q_1yp_q̅_2z]
    
       +   σ_q_1x[p'_q_1zp_q_1x-p'_q_1xp_q_1z
    -p_q_1zp_q̅_2x+p'_q_1zp_q̅_2x+p_q_1xp_q̅_2z -p'_q_1xp_q̅_2z]
    
       +   iσ_q_1xσ_q̅_2z[-p_q_1yp_q̅_2z-3p'_q_1yp_q_1z -3p'_q_1yp_q̅_2z+4p'_q_1yp'_q_1z-p'_q_1zp_q̅_2y
       +   p_q_1zp_q̅_2y+2p_q̅_2zp_q̅_2y
    +2p_q_1yp_q̅_2z-p'_q_1zp_q_1y]
    +iσ_q_1yσ_q̅_2z[p_q_1xp_q̅_2z
       +   3p'_q_1xp_q_1z+3p'_q_1xp_q̅_2z-4p'_q_1xp'_q_1z +p'_q_1zp_q̅_2x-p_q_1zp_q̅_2x-2p_q̅_2zp_q̅_2x
       -   2p_q_1zp_q_1x+p'_q̅_2zp_q_1x]
    +i(σ_q_1xσ_q̅_2y
    +σ_q_1yσ_q̅_2x)[4p'_q_1xp_q_1x
    +4p'_q_1xp_q̅_2x
       -   4p'_q_1xp'_q_1x-4p'_q_1yp_q_1y
    -4p'_q_1yp_q̅_2y+4p'_q_1yp'_q_1y
    +2p_q_1yp_q_1y-2p_q_1xp_q_1x
       +   2p_q̅_2yp_q̅_2y-2p_q̅_2xp_q̅_2x]+i(σ_q_1yσ_q̅_2y
    -σ_q_1xσ_q̅_2x)
    [4p'_q_1xp_q_1y+4p'_q_1xp_q̅_2y
       -   8p'_q_1xp'_q_1y+4p'_q_1yp_q_1x+4p'_q_1yp_q̅_2x
    -4p_q_1xp_q_1y -4p_q̅_2yp_q̅_2x]
    +iσ_q_1zσ_q̅_2y[p_q_1zp_q̅_2x
       +   3p'_q_1zp_q_1x +3p'_q_1zp_q̅_2x-4p'_q_1zp'_q_1x+p'_q_1xp_q̅_2z
    +p'_q_1xp_q_1z-p_q_1xp_q̅_2z
       -   2p_q̅_2xp_q̅_2z-2p_q_1xp_q_1z]
    +iσ_q_1zσ_q̅_2x[-p_q_1zp_q̅_2z-3p'_q_1zp_q_1y-3p'_q_1zp_q̅_2y
       +   4p'_q_1zp'_q_1y-p'_q_1yp_q̅_2z
    -p'_q_1yp_q_1z+2p_q_1zp_q_1y+2p_q̅_2yp_q̅_2z+p_q_1yp_q̅_2z]} .

Replacing the subscripts, x in Eq. (20) with y (z), y with z 
(x), and z with x (y), we get [S_x,H_1] ([S_y,H_1]). Obviously,
[S_x,H_1], [S_y,H_1], and [S_z,H_1] may not be zero. Therefore, S_x, 
S_y, and S_z may not be conserved, and the total spin may not be conserved,
i.e., the total spin of the two final mesons
may not equal the total spin of the two initial mesons.

For the right diagram in Fig. <ref>, the quark-antiquark
potential corresponding to 
q̅_1+q_2 → q_3 + q̅_4 is between q_2 and q̅_1,

    V_ aq_2q̅_1 F(k⃗)   =   
    -g^2_s/k^2[1/3λ^0_f(32)λ^0_f(41)
    +1/2λ⃗_f(32)·λ⃗^T_f(41)]
     
       ×   [4/9λ^0(32)λ^0(41)-1/12λ⃗(32)·λ⃗^T(41)]
    [-3/2-1/2σ⃗(32)·σ⃗(41) .
    
       +   σ⃗(41)·p⃗_q̅_4σ⃗(41)·p⃗_q̅_1/8m_q_2m_q_3
    +σ⃗(32)·p⃗_q_3σ⃗(32)·p⃗_q_2/8m_q_2m_q_3
    +3σ⃗(32)·p⃗_q_2σ⃗(41)·p⃗_q̅_1/8m^2_q_2
       -   σ⃗(32)·p⃗_q_2σ⃗(41)·p⃗_q̅_4/8m_q_2m_q_3
    -σ⃗(32)·p⃗_q_3σ⃗(41)·p⃗_q̅_1/8m_q_2m_q_3
    +3σ⃗(32)·p⃗_q_3σ⃗(41)·p⃗_q̅_4/8m^2_q_3
       +   σ⃗(32)σ⃗(32)·p⃗_q_2σ⃗(41)
    σ⃗(41)·p⃗_q̅_1/8m^2_q_2
    +σ⃗(32)σ⃗(32)·p⃗_q_2σ⃗(41)
    ·p⃗_q̅_4σ⃗(41)/8m_q_2m_q_3
       +   σ⃗(32)·p⃗_q_3σ⃗(32)·σ⃗(41)σ⃗(41)·p⃗_q̅_1/8m_q_2m_q_3 
    +σ⃗(32)·p⃗_q_3σ⃗(32)σ⃗(41)
    ·p⃗_q̅_4σ⃗(41)/8m^2_q_3
       -   . σ⃗(41)·p⃗_q̅_4σ⃗(32)·σ⃗(41)σ⃗(41)·p⃗_q̅_1/8m_q_2m_q_3
    -σ⃗(32)·p⃗_q_3σ⃗(32)·σ⃗(41)σ⃗(32)·p⃗_q_2/8m_q_2m_q_3] .

Required by the Fierz transformations, quark 3 (antiquark 4) and quark 2
(antiquark 1) have the same flavor, and the quark and antiquark masses
satisfy m_q_3=m_q_2 and m_q̅_4=m_q̅_1.
The Hamiltonian corresponding to the right diagram in Fig. <ref> is

    H_2    =    V_ aq_2 q̅_1 F(k⃗)+V_q_1q̅_1
    +V_q_2q̅_2
       +   √(m^2_q_1+p⃗^ 2_q_1)
    +√(m^2_q̅_1+p⃗^ 2_q̅_1)
    +√(m^2_q_2+p⃗^ 2_q_2)
    +√(m^2_q̅_2+p⃗^ 2_q̅_2) .

We may obtain the commutator [S_z,H_2] from [S_z,H_1] in Eq. (20) by the
replacement, q_1 ↔ q_2 and 
q̅_1 ↔q̅_2. We may also 
get [S_x,H_2] and [S_y,H_2] from [S_x,H_1] and [S_y,H_1] by the
replacement. [S_x,H_2], [S_y,H_2], and [S_z,H_2] may not be zero. This
indicates that S_x, S_y, and S_z may not be conserved, and the total spin
may not be conserved, i.e., the total spin of the two final mesons
may not equal the total spin of the two initial mesons.

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).

The development of the relative-motion wave function of mesons C and D
(aside from a normalization constant) is

    e^ip⃗_q_3q̅_1,q_2q̅_4·r⃗_q_3q̅_1,q_2q̅_4    =     4π∑_L_ f=0^∞∑_M_ f=-L_ f^L_ f 
    i^L_ f j_L_ f (|p⃗_q_3q̅_1,q_2q̅_4|
    r_q_3q̅_1,q_2q̅_4) 
    
       ×    Y_L_ fM_ f^∗
    (p̂_q_3q̅_1,q_2q̅_4) Y_L_ fM_ f
    (r̂_q_3q̅_1,q_2q̅_4),

in M_ aq_1q̅_2, and 

    e^ip⃗_q_1q̅_4,q_3q̅_2·r⃗_q_1q̅_4,q_3q̅_2    =     4π∑_L_ f=0^∞∑_M_ f=-L_ f^L_ f 
    i^L_ f j_L_ f (|p⃗_q_1q̅_4,q_3q̅_2|
    r_q_1q̅_4,q_3q̅_2) 
    
       ×    Y_L_ fM_ f^∗
    (p̂_q_1q̅_4,q_3q̅_2) Y_L_ fM_ f
    (r̂_q_1q̅_4,q_3q̅_2),

in M_ aq̅_1q_2. Y_LM(r̂) are the spherical 
harmonics with the orbital-angular-momentum quantum number L and the magnetic
quantum number M, and r̂ denotes the polar angles of r⃗.
Let S (S^') and S_z (S_z^') be the total spin of mesons A
(C) and B (D) and its z component, respectively. In the transition
amplitudes we have

    Y_L_ iM_ iχ_S_AS_Azχ_S_BS_Bz    =    ∑_S=| S_A-S_B |^S_A+S_B∑_S_z=-S^S (S_AS_AzS_BS_Bz| SS_z)
    
       ×   ∑_J=| L_ i-S |^L_ i+S∑_J_z=-J^J
    (L_ iM_ iSS_z | JJ_z) φ_JJ_z^ i,


    Y_L_ fM_ fχ_S_CS_Czχ_S_DS_Dz    =    ∑_S^' =| S_C-S_D |^S_C+S_D∑_S_z^' =-S^'^S^'
    (S_CS_CzS_DS_Dz| S^' S_z^')
    
       ×   ∑_J^'=| L_ f
    -S^'|^L_ f+S^'∑_J_z^'=-J^'^J^'
    (L_ fM_ fS^' S_z^'| J^' J_z^') φ_J^' J_z^'^ f,

where the Clebsch-Gordan coefficients are used.
The spherical harmonics and the spin wave functions are coupled to
the wave functions φ_JJ_z^ i and 
φ_J^' J_z^'^ f, where
J (J^') is the total-angular-momentum quantum number of the 
two initial (final) mesons, and J_z (J_z^') is its z component.
Even though the total spin of the two final mesons may not equal the total
spin of the two initial mesons, in the reactions listed in Table 1 the total 
angular momentum of the two final mesons equals the total
angular momentum of the two initial mesons. In addition, parity
is conserved. The parity conservation connects the orbital-angular-momentum
quantum numbers in the relative-motion wave functions of the initial and final
mesons. The orbital-angular-momentum quantum numbers are selected to
satisfy the parity conservation, symmetrization of wave functions of identical
bosons, and J=J^'. For example, L_ i=L_ f=J excluding 
L_ i=L_ f=0 is required in KK̅→ KK̅^*, 
KK̅→ K^*K̅, π K →π K^*,
π K →ρ K, and πρ→ KK̅; L_ i=L_ f=J with 
L_ i>0, odd L_ i for I=1, and even L_ i for I=0 in 
ππ→ KK̅^* and ππ→ K^* K̅. In practical 
calculations the summations over L_ i in Eq. (23) and over L_ f
in Eqs. (24) and (25) are from 0 to 3.



V. SUMMARY 



From the transition potential and the mesonic quark-antiquark relative-motion
wave functions, we have calculated the unpolarized cross sections for the 
2-to-2 meson-meson reactions that arise from quark-antiquark annihilation and 
creation in the first Born approximation. The reactions include
K K̅→ K K̅^∗, KK̅→ K^∗K̅,
π K →π K^∗, π K →ρ K, ππ→ K K̅^∗,
ππ→ K^* K̅, ππ→ K^∗K̅^∗,
πρ→ K K̅, πρ→ K^∗K̅^∗,
ρρ→ K^∗K̅^∗, K K̅^∗→ρρ, and
K^* K̅→ρρ.
The Hamiltonian of the two mesons contains the quark-antiquark potential which
is equivalent to the transition potential. We have derived the commutation 
relations of the total spin of the two mesons and the Hamiltonian. Due to the 
quark-antiquark potential the commutators may not be zero, 
and the total spin may 
not be conserved in the reactions. With increasing center-of-mass energy of
the two initial mesons from the threshold energy, the
cross sections for the endothermic reactions increase very rapidly to the peak
cross sections first, and then decrease or display plateaus for 
ππ→ K^∗K̅^∗ and πρ→ K^∗K̅^∗
at some temperatures. The cross sections exhibit remarkable temperature
dependence. To use the cross sections conveniently,
we have parametrized the numerical cross sections for the ten isospin channels 
of reactions. Based on the flavor matrix elements, the cross sections for the
other isospin channels of reactions can be obtained from the cross sections
for the ten channels.



ACKNOWLEDGEMENTS



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


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