1. Introduction
It is shown in [1–3] that ψ(3770), ψ(4040), ψ(4160), and ψ(4415) mesons are the 13D1, 33S1, 23D1, and 43S1 states of a charm quark and a charm antiquark. The four $c\bar{c}$ mesons have been widely studied in e+e− annihilation that produce hadrons [4–7], ππJ/ψ [8–10], ηJ/ψ [9–12], K+K−J/ψ [10], γχcJ (J = 1, 2) [10, 13], two charmed mesons [14, 15], D0D*−π+ [16], two charmed strange mesons [17, 18], ππhc [19], ωχc2 [20], μ+μ−[21], and ${\rm{\Lambda }}\bar{{\rm{\Lambda }}}$ [22]. Electron-positron annihilation produces a virtual photon which splits into a charm quark and a charm antiquark, and this quark-antiquark pair becomes a $c\bar{c}$ meson nonperturbatively. Production of the ψ(3770) meson in e+e− annihilation was studied in the nonrelativistic quantum chromodynamics (NRQCD) factorization formalism that includes color-singlet and color-octet contributions [23]. In [24, 25] the conversion of the photon to the ψ(4040) or ψ(4160) meson is indicated by a constant factor.
Au-Au collisions at the Relativistic Heavy Ion Collider (RHIC) and Pb-Pb collisions at the Large Hadron Collider (LHC) produce quark-gluon plasmas. At the critical temperature Tc, the quark-gluon plasma becomes hadronic matter. Since ψ(4040), ψ(4160), and ψ(4415) mesons are dissolved in hadronic matter when the temperature is larger than 0.97Tc, 0.95Tc, and 0.87Tc, respectively [26], they can only be produced in hadronic matter. Therefore, the production of ψ(4040), ψ(4160), and ψ(4415) can be taken as probes of hadronic matter that results from the quark-gluon plasma created in ultrarelativistic heavy-ion collisions. In hadronic matter, they are produced in the following reactions: $D\bar{D}\to \rho R$, $D{\bar{D}}^{* }\to \pi R$, $D{\bar{D}}^{* }\to \rho R$, ${D}^{* }{\bar{D}}^{* }\to \pi R$, ${D}^{* }{\bar{D}}^{* }\to \rho R$ and so on, where R stands for ψ(4040), ψ(4160), or ψ(4415). Charmed mesons have been well measured in Pb-Pb collisions at the LHC. It is shown in [27] that numbers of ψ(4040), ψ(4160), and ψ(4415) produced in a central Pb-Pb collision at the center-of-mass energy per nucleon-nucleon pair $\sqrt{{s}_{{NN}}}=5.02$ TeV are 0.25, 0.1, and 0.18, respectively. Therefore, it is interesting to measure ψ(4040), ψ(4160), and ψ(4415) mesons produced in Pb-Pb collisions at the LHC.
Production of D-wave charmonia in nucleon-nucleon collisions was studied in NRQCD in [28]. Production cross sections depend on parton distribution functions, short-distance processes, and nonperturbative matrix elements of four-fermion operators. In proton-nucleus reactions a charmonium produced in a proton-nucleon collision further interacts with other nucleons. The nucleon-charmonium collisions may break the charmonium, and thus reduce the charmonium number. Therefore, in the present work we study the dissociation of ψ(3770), ψ(4040), ψ(4160), and ψ(4415) mesons in collisions with nucleons. Since many experiments on pA reactions have been carried out at the RHIC and the LHC, it is interesting to study the dissociation processes.
ψ(3770), ψ(4040), ψ(4160), and ψ(4415) mesons are of special interest because they are easily produced at electron-positron colliders. The mechanism of producing them in proton-nucleus reactions is different from the mechanism of producing them in electron-positron collisions. The mesons are influenced by cold nuclear matter due to the dissociation processes and nuclear modification of parton distribution functions. Therefore, it will be interesting to compare the production of the mesons in proton-nucleus reactions with the production in electron-positron collisions in both experiment and theory in future.
This paper is organized as follows. In section 2 we derive formulas of transition amplitudes which are used to calculate unpolarized cross sections for the dissociation of $c\bar{c}$ mesons in collisions with nucleons. In section 3 we present numerical cross sections along with relevant discussions. In section 4 we summarize the present work.
2. Formalism
We consider the reaction A + B → C + D where A and C represent baryons and B and D are mesons. Denoted by Ei and ${\vec{P}}_{{\rm{i}}}$ (Ef and ${\vec{P}}_{{\rm{f}}}$), the total energy and the total momentum of the initial (final) baryon and the initial (final) meson, respectively. If EA (EB, EC, ED) stands for the energy of hadron A (B, C, D), Ei = EA + EB and Ef = EC + ED. Let HI be the interaction potential between two constituents of hadrons in the reaction $A({q}_{1}{q}_{2}{q}_{3})+B(c\bar{c})\to C({q}_{1}{q}_{2}c)+D({q}_{3}\bar{c})$, where q1, q2, and q3 represent light quarks. Since the quark flavors inside baryon A differ from the charm flavor inside meson B, the quark interchange (for example, q3 and c) between baryon A and meson B gives rise to the reaction. The S-matrix element for A + B → C + D is
$\begin{eqnarray}{S}_{\mathrm{fi}}={\delta }_{\mathrm{fi}}-2\pi {\rm{i}}\delta ({E}_{{\rm{f}}}-{E}_{{\rm{i}}})\langle C,D| {H}_{{\rm{I}}}| A,B\rangle .\end{eqnarray}$
Let ${\vec{P}}_{{q}_{1}{q}_{2}{q}_{3}}$ (${\vec{P}}_{{q}_{1}{q}_{2}c}^{{\prime} }$) and ${\vec{R}}_{{q}_{1}{q}_{2}{q}_{3}}$ (${\vec{R}}_{{q}_{1}{q}_{2}c}$) be the total momentum and the center-of-mass coordinate of q1, q2, and q3 (q1, q2, and c) in baryon A (C), respectively. Let ${\vec{P}}_{c\bar{c}}$ (${\vec{P}}_{{q}_{3}\bar{c}}^{{\prime} }$), ${\vec{R}}_{c\bar{c}}$ (${\vec{R}}_{{q}_{3}\bar{c}}$), and ${\vec{r}}_{c\bar{c}}$ (${\vec{r}}_{{q}_{3}\bar{c}}$) be the total momentum, the center-of-mass coordinate, and the relative coordinate of c and $\bar{c}$ (q3 and $\bar{c}$) of meson B (D), respectively. In the case that quarks q1 and q2 have the same mass, we define
$\begin{eqnarray}\vec{\rho }=\displaystyle \frac{1}{\sqrt{2}}({\vec{r}}_{{q}_{1}}-{\vec{r}}_{{q}_{2}}),\end{eqnarray}$
and $\begin{eqnarray}{\vec{\lambda }}_{{\rm{i}}}=\displaystyle \frac{1}{\sqrt{6}}({\vec{r}}_{{q}_{1}}+{\vec{r}}_{{q}_{2}}-2{\vec{r}}_{{q}_{3}}),\end{eqnarray}$
for baryon A, and $\begin{eqnarray}{\vec{\lambda }}_{{\rm{f}}}=\displaystyle \frac{1}{\sqrt{6}}({\vec{r}}_{{q}_{1}}+{\vec{r}}_{{q}_{2}}-2{\vec{r}}_{c}),\end{eqnarray}$
for baryon C, where ${\vec{r}}_{{q}_{1}}$, ${\vec{r}}_{{q}_{2}}$, ${\vec{r}}_{{q}_{3}}$, and ${\vec{r}}_{c}$ are the position vectors of quarks q1, q2, q3, and c, respectively. The wave function ∣A, B⟩ of baryon A and meson B is $\begin{eqnarray}\begin{array}{rcl}{\psi }_{{AB}} & = & \displaystyle \frac{{{\rm{e}}}^{{\rm{i}}{\vec{P}}_{{q}_{1}{q}_{2}{q}_{3}}\cdot {\vec{R}}_{{q}_{1}{q}_{2}{q}_{3}}}}{\sqrt{V}}{\psi }_{{q}_{1}{q}_{2}{q}_{3}}(\vec{\rho },{\vec{\lambda }}_{{\rm{i}}})\\ & & \times \displaystyle \frac{{{\rm{e}}}^{{\rm{i}}{\vec{P}}_{c\bar{c}}\cdot {\vec{R}}_{c\bar{c}}}}{\sqrt{V}}{\psi }_{c\bar{c}}({\vec{r}}_{c\bar{c}}),\end{array}\end{eqnarray}$
and the wave function ∣C, D⟩ of baryon C and meson D is $\begin{eqnarray}\begin{array}{rcl}{\psi }_{{CD}} & = & \displaystyle \frac{{{\rm{e}}}^{{\rm{i}}{\vec{P}}_{{q}_{1}{q}_{2}c}^{{\prime} }\cdot {\vec{R}}_{{q}_{1}{q}_{2}c}}}{\sqrt{V}}{\psi }_{{q}_{1}{q}_{2}c}(\vec{\rho },{\vec{\lambda }}_{{\rm{f}}})\\ & & \times \displaystyle \frac{{{\rm{e}}}^{{\rm{i}}{\vec{P}}_{{q}_{3}\bar{c}}^{{\prime} }\cdot {\vec{R}}_{{q}_{3}\bar{c}}}}{\sqrt{V}}{\psi }_{{q}_{3}\bar{c}}({\vec{r}}_{{q}_{3}\bar{c}}),\end{array}\end{eqnarray}$
in which V is the volume where every hadron wave function is normalized. ${\psi }_{{q}_{1}{q}_{2}{q}_{3}}(\vec{\rho },{\vec{\lambda }}_{{\rm{i}}})$ (${\psi }_{{q}_{1}{q}_{2}c}(\vec{\rho },{\vec{\lambda }}_{{\rm{f}}})$) is the product of the color wave function, the flavor wave function, the spin wave function, and the space wave function of the three quarks. ${\psi }_{c\bar{c}}({\vec{r}}_{c\bar{c}})$ (${\psi }_{{q}_{3}\bar{c}}({\vec{r}}_{{q}_{3}\bar{c}})$) is the product of the color wave function, the flavor wave function, the spin wave function, and the quark-antiquark relative-motion wave function.With the wave functions we have
$\begin{eqnarray}\begin{array}{l}\langle C,D| {H}_{{\rm{I}}}| A,B\rangle \\ \quad =\displaystyle \int {{\rm{d}}}^{3}{R}_{{q}_{1}{q}_{2}{q}_{3}}{{\rm{d}}}^{3}\rho {{\rm{d}}}^{3}{\lambda }_{{\rm{i}}}{{\rm{d}}}^{3}{R}_{c\bar{c}}{{\rm{d}}}^{3}{r}_{c\bar{c}}{\psi }_{{CD}}^{+}{H}_{{\rm{I}}}{\psi }_{{AB}}\\ \quad =\displaystyle \int {{\rm{d}}}^{3}\rho {{\rm{d}}}^{3}{\lambda }_{{\rm{i}}}{{\rm{d}}}^{3}{r}_{c\bar{c}}{{\rm{d}}}^{3}{r}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}{{\rm{d}}}^{3}{R}_{\mathrm{total}}\\ \qquad \times \displaystyle \frac{{\psi }_{{q}_{1}{q}_{2}c}^{+}(\vec{\rho },{\vec{\lambda }}_{{\rm{f}}})}{\sqrt{V}}\displaystyle \frac{{\psi }_{{q}_{3}\bar{c}}^{+}({\vec{r}}_{{q}_{3}\bar{c}})}{\sqrt{V}}\\ \qquad \times \,\exp (-{\rm{i}}{\vec{P}}_{{\rm{f}}}\cdot {\vec{R}}_{\mathrm{total}}-{\rm{i}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\cdot {\vec{r}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}})\\ {H}_{{\rm{I}}}\displaystyle \frac{{\psi }_{{q}_{1}{q}_{2}{q}_{3}}(\vec{\rho },{\vec{\lambda }}_{{\rm{i}}})}{\sqrt{V}}\displaystyle \frac{{\psi }_{c\bar{c}}({\vec{r}}_{c\bar{c}})}{\sqrt{V}}\\ \times \,\exp ({\rm{i}}{\vec{P}}_{{\rm{i}}}\cdot {\vec{R}}_{\mathrm{total}}+{\rm{i}}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}\cdot {\vec{r}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}})\\ \quad ={\left(2\pi \right)}^{3}{\delta }^{3}({\vec{P}}_{{\rm{f}}}-{\vec{P}}_{{\rm{i}}})\displaystyle \int {{\rm{d}}}^{3}\rho {{\rm{d}}}^{3}{\lambda }_{{\rm{i}}}{{\rm{d}}}^{3}{r}_{c\bar{c}}{{\rm{d}}}^{3}{r}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}\\ \qquad \times \displaystyle \frac{{\psi }_{{q}_{1}{q}_{2}c}^{+}(\vec{\rho },{\vec{\lambda }}_{{\rm{f}}})}{\sqrt{V}}\displaystyle \frac{{\psi }_{{q}_{3}\bar{c}}^{+}({\vec{r}}_{{q}_{3}\bar{c}})}{\sqrt{V}}{H}_{{\rm{I}}}\\ \qquad \times \displaystyle \frac{{\psi }_{{q}_{1}{q}_{2}{q}_{3}}(\vec{\rho },{\vec{\lambda }}_{{\rm{i}}})}{\sqrt{V}}\displaystyle \frac{{\psi }_{c\bar{c}}({\vec{r}}_{c\bar{c}})}{\sqrt{V}}\\ \,\times \,\exp (-{\rm{i}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\cdot {\vec{r}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}+{\rm{i}}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}\cdot {\vec{r}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}})\\ \quad ={\left(2\pi \right)}^{3}{\delta }^{3}({\vec{P}}_{{\rm{f}}}-{\vec{P}}_{{\rm{i}}})\displaystyle \frac{{{ \mathcal M }}_{\mathrm{fi}}}{{V}^{2}\sqrt{2{E}_{A}2{E}_{B}2{E}_{C}2{E}_{D}}},\end{array}\end{eqnarray}$
where ${\vec{r}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}$ (${\vec{r}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}$) and ${\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}$ (${\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }$) are the relative coordinate and the relative momentum of q1q2q3 and $c\bar{c}$ (q1q2c and ${q}_{3}\bar{c}$), respectively; ${\vec{R}}_{\mathrm{total}}$ is the center-of-mass coordinate of the two initial hadrons, i.e., of the two final hadrons; ${\psi }_{{CD}}^{+}$ is the Hermitean conjugate of ψCD; ${{ \mathcal M }}_{\mathrm{fi}}$ is the transition amplitude given by $\begin{eqnarray}\begin{array}{l}{{ \mathcal M }}_{\mathrm{fi}}=\sqrt{2{E}_{A}2{E}_{B}2{E}_{C}2{E}_{D}}\\ \quad \times \displaystyle \int {{\rm{d}}}^{3}\rho {{\rm{d}}}^{3}{\lambda }_{{\rm{i}}}{{\rm{d}}}^{3}{r}_{c\bar{c}}{{\rm{d}}}^{3}{r}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}{\psi }_{{q}_{1}{q}_{2}c}^{\dagger }(\vec{\rho },{\vec{\lambda }}_{{\rm{f}}})\\ \quad \times {\psi }_{{q}_{3}\bar{c}}^{+}({\vec{r}}_{{q}_{3}\bar{c}}){H}_{{\rm{I}}}{\psi }_{{q}_{1}{q}_{2}{q}_{3}}(\vec{\rho },{\vec{\lambda }}_{{\rm{i}}}){\psi }_{c\bar{c}}({\vec{r}}_{c\bar{c}})\\ \quad \times \,\exp (-{\rm{i}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\cdot {\vec{r}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}\\ \quad +\,{\rm{i}}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}\cdot {\vec{r}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}).\end{array}\end{eqnarray}$
The wave function of baryon A and meson B is9 ) ${\phi }_{{{BJ}}_{B}{J}_{{Bz}}}={R}_{{L}_{B}}({r}_{c\bar{c}})$ ${\sum }_{{M}_{B}=-{L}_{B}}^{{L}_{B}}{\sum }_{{S}_{{Bz}}=-{S}_{B}}^{{S}_{B}}({L}_{B}{M}_{B}{S}_{B}{S}_{{Bz}}| {J}_{B}{J}_{{Bz}}){Y}_{{L}_{B}{M}_{B}}{\chi }_{{S}_{B}{S}_{{Bz}}}$ where ${R}_{{L}_{B}}({r}_{c\bar{c}})$ is the radial wave function of the relative-motion of c and $\bar{c}$, (LBMBSBSBz∣JBJBz) are the Clebsch–Gordan coefficients, ${Y}_{{L}_{B}{M}_{B}}$ are the spherical harmonics, and ${\chi }_{{S}_{B}{S}_{{Bz}}}$ are the spin wave functions.
$\begin{eqnarray}\begin{array}{l}{\psi }_{{AB}}={\phi }_{A\mathrm{color}}{\phi }_{B\mathrm{color}}{\phi }_{A\mathrm{flavor}}{\phi }_{B\mathrm{flavor}}\\ \quad {\phi }_{A\mathrm{space}}{\chi }_{{S}_{A}{S}_{{Az}}}{\phi }_{{{BJ}}_{B}{J}_{{Bz}}},\end{array}\end{eqnarray}$
and the wave function of baryon C and meson D is $\begin{eqnarray}\begin{array}{l}{\psi }_{{CD}}={\phi }_{C\mathrm{color}}{\phi }_{D\mathrm{color}}{\phi }_{C\mathrm{flavor}}{\phi }_{D\mathrm{flavor}}\\ \quad {\phi }_{C\mathrm{space}}{\phi }_{D\mathrm{rel}}{\chi }_{{S}_{C}{S}_{{Cz}}}{\chi }_{{S}_{D}{S}_{{Dz}}},\end{array}\end{eqnarray}$
where SA (SC, SD) is the spin of hadron A (C, D) with its magnetic projection quantum number SAz (SCz, SDz); φAcolor (φCcolor, φDcolor), φAflavor (φCflavor, φDflavor), and ${\chi }_{{S}_{A}{S}_{{Az}}}$ (${\chi }_{{S}_{C}{S}_{{Cz}}}$, ${\chi }_{{S}_{D}{S}_{{Dz}}}$) are the color wave function, the flavor wave function, and the spin wave function of hadron A (C, D), respectively; φAspace (φCspace) is the space wave function of baryon A (C); φDrel is the quark-antiquark relative-motion wave function of meson D; φBcolor, φBflavor, and ${\phi }_{{{BJ}}_{B}{J}_{{Bz}}}$ are the color wave function, the flavor wave function, and the space-spin wave function of meson B with the total angular momentum JB and its z component JBz, respectively. Denoted by LB and SB, the orbital angular momentum and the spin of meson B, respectively, and by MB and SBz the magnetic projection quantum numbers of LB and SB. In equation (The interaction that governs scattering in the prior form shown in figure 1 is8 )–(29 ) we obtain the transition amplitude for scattering in the prior form,
$\begin{eqnarray}{H}_{{\rm{I}}}={V}_{{q}_{1}\bar{c}}+{V}_{{q}_{2}\bar{c}}+{V}_{{q}_{3}\bar{c}}+{V}_{{q}_{1}c}+{V}_{{q}_{2}c}+{V}_{{q}_{3}c},\end{eqnarray}$
and the interaction that governs scattering in the post form shown in figure 2 is $\begin{eqnarray}{H}_{{\rm{I}}}={V}_{{q}_{1}\bar{c}}+{V}_{{q}_{2}\bar{c}}+{V}_{c\bar{c}}+{V}_{{q}_{1}{q}_{3}}+{V}_{{q}_{2}{q}_{3}}+{V}_{{q}_{3}c},\end{eqnarray}$
where Vab is the potential between constituents a and b. Let ${\vec{r}}_{\bar{c}}$ be the position vector of antiquark $\bar{c}$. We take the Fourier transform of the potentials and wave functions: $\begin{eqnarray}{V}_{{q}_{1}\bar{c}}({\vec{r}}_{{q}_{1}}-{\vec{r}}_{\bar{c}})=\int \displaystyle \frac{{{\rm{d}}}^{3}Q}{{\left(2\pi \right)}^{3}}{V}_{{q}_{1}\bar{c}}(\vec{Q}){{\rm{e}}}^{{\rm{i}}\vec{Q}\cdot ({\vec{r}}_{{q}_{1}}-{\vec{r}}_{\bar{c}})},\end{eqnarray}$
$\begin{eqnarray}{V}_{{q}_{2}\bar{c}}({\vec{r}}_{{q}_{2}}-{\vec{r}}_{\bar{c}})=\int \displaystyle \frac{{{\rm{d}}}^{3}Q}{{\left(2\pi \right)}^{3}}{V}_{{q}_{2}\bar{c}}(\vec{Q}){{\rm{e}}}^{{\rm{i}}\vec{Q}\cdot ({\vec{r}}_{{q}_{2}}-{\vec{r}}_{\bar{c}})},\end{eqnarray}$
$\begin{eqnarray}{V}_{{q}_{3}\bar{c}}({\vec{r}}_{{q}_{3}}-{\vec{r}}_{\bar{c}})=\int \displaystyle \frac{{{\rm{d}}}^{3}Q}{{\left(2\pi \right)}^{3}}{V}_{{q}_{3}\bar{c}}(\vec{Q}){{\rm{e}}}^{{\rm{i}}\vec{Q}\cdot ({\vec{r}}_{{q}_{3}}-{\vec{r}}_{\bar{c}})},\end{eqnarray}$
$\begin{eqnarray}{V}_{{q}_{1}c}({\vec{r}}_{{q}_{1}}-{\vec{r}}_{c})=\int \displaystyle \frac{{{\rm{d}}}^{3}Q}{{\left(2\pi \right)}^{3}}{V}_{{q}_{1}c}(\vec{Q}){{\rm{e}}}^{{\rm{i}}\vec{Q}\cdot ({\vec{r}}_{{q}_{1}}-{\vec{r}}_{c})},\end{eqnarray}$
$\begin{eqnarray}{V}_{{q}_{2}c}({\vec{r}}_{{q}_{2}}-{\vec{r}}_{c})=\int \displaystyle \frac{{{\rm{d}}}^{3}Q}{{\left(2\pi \right)}^{3}}{V}_{{q}_{2}c}(\vec{Q}){{\rm{e}}}^{{\rm{i}}\vec{Q}\cdot ({\vec{r}}_{{q}_{2}}-{\vec{r}}_{c})},\end{eqnarray}$
$\begin{eqnarray}{V}_{{q}_{3}c}({\vec{r}}_{{q}_{3}}-{\vec{r}}_{c})=\int \displaystyle \frac{{{\rm{d}}}^{3}Q}{{\left(2\pi \right)}^{3}}{V}_{{q}_{3}c}(\vec{Q}){{\rm{e}}}^{{\rm{i}}\vec{Q}\cdot ({\vec{r}}_{{q}_{3}}-{\vec{r}}_{c})},\end{eqnarray}$
$\begin{eqnarray}{V}_{c\bar{c}}({\vec{r}}_{c}-{\vec{r}}_{\bar{c}})=\int \displaystyle \frac{{{\rm{d}}}^{3}Q}{{\left(2\pi \right)}^{3}}{V}_{c\bar{c}}(\vec{Q}){{\rm{e}}}^{{\rm{i}}\vec{Q}\cdot ({\vec{r}}_{c}-{\vec{r}}_{\bar{c}})},\end{eqnarray}$
$\begin{eqnarray}{V}_{{q}_{1}{q}_{3}}({\vec{r}}_{{q}_{1}}-{\vec{r}}_{{q}_{3}})=\int \displaystyle \frac{{{\rm{d}}}^{3}Q}{{\left(2\pi \right)}^{3}}{V}_{{q}_{1}{q}_{3}}(\vec{Q}){{\rm{e}}}^{{\rm{i}}\vec{Q}\cdot ({\vec{r}}_{{q}_{1}}-{\vec{r}}_{{q}_{3}})},\end{eqnarray}$
$\begin{eqnarray}{V}_{{q}_{2}{q}_{3}}({\vec{r}}_{{q}_{2}}-{\vec{r}}_{{q}_{3}})=\int \displaystyle \frac{{{\rm{d}}}^{3}Q}{{\left(2\pi \right)}^{3}}{V}_{{q}_{2}{q}_{3}}(\vec{Q}){{\rm{e}}}^{{\rm{i}}\vec{Q}\cdot ({\vec{r}}_{{q}_{2}}-{\vec{r}}_{{q}_{3}})},\end{eqnarray}$
$\begin{eqnarray}{\phi }_{A\mathrm{space}}(\vec{\rho },\vec{{\lambda }_{{\rm{i}}}})=\int \displaystyle \frac{{{\rm{d}}}^{3}{p}_{\rho }}{{\left(2\pi \right)}^{3}}\displaystyle \frac{{{\rm{d}}}^{3}{p}_{\lambda }}{{\left(2\pi \right)}^{3}}{\phi }_{A\mathrm{space}}({\vec{p}}_{\rho },{\vec{p}}_{\lambda }){{\rm{e}}}^{{\rm{i}}{\vec{p}}_{\rho }\cdot \vec{\rho }+{\rm{i}}{\vec{p}}_{\lambda }\cdot {\vec{\lambda }}_{{\rm{i}}}},\end{eqnarray}$
$\begin{eqnarray}{\phi }_{{{BJ}}_{B}{J}_{{Bz}}}({\vec{r}}_{c\bar{c}})=\int \displaystyle \frac{{{\rm{d}}}^{3}{p}_{c\bar{c}}}{{\left(2\pi \right)}^{3}}{\phi }_{{{BJ}}_{B}{J}_{{Bz}}}({\vec{p}}_{c\bar{c}}){{\rm{e}}}^{{\rm{i}}{\vec{p}}_{c\bar{c}}\cdot {\vec{r}}_{c\bar{c}}},\end{eqnarray}$
$\begin{eqnarray}{\phi }_{C\mathrm{space}}(\vec{\rho },\vec{{\lambda }_{{\rm{f}}}})=\int \displaystyle \frac{{{\rm{d}}}^{3}{p}_{\rho }^{{\prime} }}{{\left(2\pi \right)}^{3}}\displaystyle \frac{{{\rm{d}}}^{3}{p}_{\lambda }^{{\prime} }}{{\left(2\pi \right)}^{3}}{\phi }_{C\mathrm{space}}({\vec{p}}_{\rho }^{{\prime} },{\vec{p}}_{\lambda }^{{\prime} }){{\rm{e}}}^{{\rm{i}}{\vec{p}}_{\rho }^{{\prime} }\cdot \vec{\rho }+{\rm{i}}{\vec{p}}_{\lambda }^{{\prime} }\cdot {\vec{\lambda }}_{{\rm{f}}}},\end{eqnarray}$
$\begin{eqnarray}{\phi }_{D\mathrm{rel}}({\vec{r}}_{{q}_{3}\bar{c}})=\int \displaystyle \frac{{{\rm{d}}}^{3}{p}_{{q}_{3}\bar{c}}^{{\prime} }}{{\left(2\pi \right)}^{3}}{\phi }_{D\mathrm{rel}}({\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }){{\rm{e}}}^{{\rm{i}}{\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }\cdot {\vec{r}}_{{q}_{3}\bar{c}}},\end{eqnarray}$
where $\vec{Q}$ is the momentum attached to the dot-dashed lines in figures 1 and 2, ${\vec{p}}_{c\bar{c}}$ is the relative momentum of c and $\bar{c}$ in meson B, and ${\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }$ is the relative momentum of q3 and $\bar{c}$ in meson D. In momentum space the normalizations are $\begin{eqnarray}\int \displaystyle \frac{{{\rm{d}}}^{3}{p}_{\rho }}{{\left(2\pi \right)}^{3}}\int \displaystyle \frac{{{\rm{d}}}^{3}{p}_{\lambda }}{{\left(2\pi \right)}^{3}}{\phi }_{A\mathrm{space}}^{+}({\vec{p}}_{\rho },{\vec{p}}_{\lambda }){\phi }_{A\mathrm{space}}({\vec{p}}_{\rho },{\vec{p}}_{\lambda })=1,\end{eqnarray}$
$\begin{eqnarray}\int \displaystyle \frac{{{\rm{d}}}^{3}{p}_{c\bar{c}}}{{\left(2\pi \right)}^{3}}{\phi }_{{{BJ}}_{B}{J}_{{Bz}}}^{+}({\vec{p}}_{c\bar{c}}){\phi }_{{{BJ}}_{B}{J}_{{Bz}}}({\vec{p}}_{c\bar{c}})=1,\end{eqnarray}$
$\begin{eqnarray}\int \displaystyle \frac{{{\rm{d}}}^{3}{p}_{\rho }^{{\prime} }}{{\left(2\pi \right)}^{3}}\int \displaystyle \frac{{{\rm{d}}}^{3}{p}_{\lambda }^{{\prime} }}{{\left(2\pi \right)}^{3}}{\phi }_{C\mathrm{space}}^{+}({\vec{p}}_{\rho }^{{\prime} },{\vec{p}}_{\lambda }^{{\prime} }){\phi }_{C\mathrm{space}}({\vec{p}}_{\rho }^{{\prime} },{\vec{p}}_{\lambda }^{{\prime} })=1,\end{eqnarray}$
$\begin{eqnarray}\int \displaystyle \frac{{{\rm{d}}}^{3}{p}_{{q}_{3}\bar{c}}^{{\prime} }}{{\left(2\pi \right)}^{3}}{\phi }_{D\mathrm{rel}}^{+}({\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }){\phi }_{D\mathrm{rel}}({\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} })=1.\end{eqnarray}$
When quarks q1 and q2 have equal masses, their masses are indicated by m. Let ${m}_{{q}_{3}}$, mc, and ${m}_{\bar{c}}$ stand for the q3, c, and $\bar{c}$ masses, respectively. From equations ( $\begin{aligned} \mathcal{M}_{\mathrm{fi}}^{\text {prior }}= & \sqrt{2 E_A 2 E_B 2 E_C 2 E_D} \phi_{C \text { color }}^{+} \phi_{D \text { color }}^{+} \phi_{C \text { flavor }}^{+} \phi_{D \text { flavor }}^{+} \chi_{S_C}^{+} S_{C Z} \chi_{S_D S_{D Z}}^{+} \\ & \times \int \frac{\mathrm{d}^3 p_\rho^{\prime}}{(2 \pi)^3} \frac{\mathrm{d}^3 p_\lambda^{\prime}}{(2 \pi)^3} \frac{\mathrm{d}^3 p_{q_3 \bar{c}}^{\prime}}{(2 \pi)^3} \phi_{C \text { space }}^{+}\left(\vec{p}_\rho^{\prime}, \vec{p}_\lambda^{\prime}\right) \phi_{D \text { rel }}^{+}\left(\vec{p}_{q_3 \bar{c}}^{\prime}\right) \\ & \times\left\{V_{q_1 \bar{c}}\left(\frac{2}{\sqrt{6}} \vec{p}_\lambda^{\prime}+\vec{p}_{q_3 \bar{c}}^{\prime}-\vec{p}_{q_1 q_2 q_3, c \bar{c}}+o_{\mathrm{r}} \vec{p}_{q_1 q_2 c, q_3 \bar{c}}^{\prime}\right)\right. \\ & \phi_{\text {Aspace }}\left(\vec{p}_\rho^{\prime}-\frac{1}{\sqrt{3}} \vec{p}_\lambda^{\prime}-\frac{\sqrt{2}}{2} \vec{p}_{q_3 \bar{c}}^{\prime}+\frac{\sqrt{2}}{2} \vec{p}_{q_1 q_2 q_3, c \bar{c}}\right. \\ & -\frac{\sqrt{2} o_{\mathrm{r}}}{2} \vec{p}_{q_1 q_2 c, q_3 \bar{c}}^{\prime}, \\ & \left.-\displaystyle \frac{\sqrt{6}}{2}{\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }+\displaystyle \frac{\sqrt{6}{m}_{{q}_{3}}}{2(2m+{m}_{{q}_{3}})}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+\displaystyle \frac{\sqrt{6}{m}_{{q}_{3}}}{2({m}_{{q}_{3}}+{m}_{\bar{c}})}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right) \\ & \left.-\displaystyle \frac{\sqrt{6}}{2}{\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }+\displaystyle \frac{\sqrt{6}{m}_{{q}_{3}}}{2(2m+{m}_{{q}_{3}})}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+\displaystyle \frac{\sqrt{6}{m}_{{q}_{3}}}{2({m}_{{q}_{3}}+{m}_{\bar{c}})}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right) \\ & {\phi }_{{{BJ}}_{B}{J}_{{Bz}}}\left(-\displaystyle \frac{2}{\sqrt{6}}{\vec{p}}_{\lambda }^{{\prime} }+\displaystyle \frac{{m}_{c}}{{m}_{c}+{m}_{\bar{c}}}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+\displaystyle \frac{{m}_{c}}{2m+{m}_{c}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right) \\ & +{V}_{{q}_{2}\bar{c}}\left(\displaystyle \frac{2}{\sqrt{6}}{\vec{p}}_{\lambda }^{{\prime} }+{\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }-{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+{o}_{{\rm{r}}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right) \\ & {\phi }_{A\mathrm{space}}\left({\vec{p}}_{\rho }^{{\prime} }+\displaystyle \frac{1}{\sqrt{3}}{\vec{p}}_{\lambda }^{{\prime} }+\displaystyle \frac{\sqrt{2}}{2}{\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }\right. \\ & \,-\displaystyle \frac{\sqrt{2}}{2}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}} \\ & \,+\displaystyle \frac{\sqrt{2}{o}_{{\rm{r}}}}{2}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }, \\ & \left.-\displaystyle \frac{\sqrt{6}}{2}{\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }+\displaystyle \frac{\sqrt{6}{m}_{{q}_{3}}}{2(2m+{m}_{{q}_{3}})}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+\displaystyle \frac{\sqrt{6}{m}_{{q}_{3}}}{2({m}_{{q}_{3}}+{m}_{\bar{c}})}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right) \\ & {\phi }_{{{BJ}}_{B}{J}_{{Bz}}}\left(-\displaystyle \frac{2}{\sqrt{6}}{\vec{p}}_{\lambda }^{{\prime} }+\displaystyle \frac{{m}_{c}}{{m}_{c}+{m}_{\bar{c}}}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+\displaystyle \frac{{m}_{c}}{2m+{m}_{c}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right) \\ & +{V}_{{q}_{3}\bar{c}}\left(\displaystyle \frac{2}{\sqrt{6}}{\vec{p}}_{\lambda }^{{\prime} }+{\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }-{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+{o}_{{\rm{r}}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right) \\ & {\phi }_{A\mathrm{space}}\left({\vec{p}}_{\rho }^{{\prime} },{\vec{p}}_{\lambda }^{{\prime} }-\displaystyle \frac{\sqrt{6}m}{2m+{m}_{{q}_{3}}}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}\right. \\ & \,+\left.\displaystyle \frac{\sqrt{6}m}{2m+{m}_{c}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right) \\ & {\phi }_{{{BJ}}_{B}{J}_{{Bz}}}\left(-\displaystyle \frac{2}{\sqrt{6}}{\vec{p}}_{\lambda }^{{\prime} }+\displaystyle \frac{{m}_{c}}{{m}_{c}+{m}_{\bar{c}}}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+\displaystyle \frac{{m}_{c}}{2m+{m}_{c}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right) \\ & +{V}_{{q}_{1}c}\left(\displaystyle \frac{2}{\sqrt{6}}{\vec{p}}_{\lambda }^{{\prime} }+{\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }-{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+{o}_{{\rm{r}}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right) \\ & \phi_{\text {Aspace }}\left(\vec{p}_{\rho}^{\prime}+\frac{1}{\sqrt{3}} \vec{p}_{\lambda}^{\prime}+\frac{\sqrt{2}}{2} \vec{p}_{q_{3} \bar{c}}^{\prime}\right. \\ & +\frac{\sqrt{2} o_{\mathrm{r}}}{2} \vec{p}_{q_{1} q_{2} c, q_{3} \bar{c}}^{\prime}, \\ & \left.-\frac{\sqrt{6}}{2} \vec{p}_{q_{3} \bar{c}}^{\prime}+\frac{\sqrt{6} m_{q_{3}}}{2\left(2 m+m_{q_{3}}\right)} \vec{p}_{q_{1} q_{2} q_{3}, c \bar{c}}+\frac{\sqrt{6} m_{q_{3}}}{2\left(m_{q_{3}}+m_{\bar{c}}\right)} \vec{p}_{q_{1} q_{2} c, q_{3} \bar{c}}^{\prime}\right) \end{aligned}$$\begin{aligned} \\ & {\phi }_{{{BJ}}_{B}{J}_{{Bz}}}\left({\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }-\displaystyle \frac{{m}_{\bar{c}}}{{m}_{c}+{m}_{\bar{c}}}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+\displaystyle \frac{{m}_{\bar{c}}}{{m}_{{q}_{3}}+{m}_{\bar{c}}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right)\\ & +{V}_{{q}_{2}c}\left(\displaystyle \frac{2}{\sqrt{6}}{\vec{p}}_{\lambda }^{{\prime} }+{\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }-{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+{o}_{{\rm{r}}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right)\\ & {\phi }_{A\mathrm{space}}\left({\vec{p}}_{\rho }^{{\prime} }+\displaystyle \frac{1}{\sqrt{3}}{\vec{p}}_{\lambda }^{{\prime} }+\displaystyle \frac{\sqrt{2}}{2}{\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }-\displaystyle \frac{\sqrt{2}}{2}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}\right.\\ & \,+\displaystyle \frac{\sqrt{2}{o}_{{\rm{r}}}}{2}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} },\\ & \left.-\displaystyle \frac{\sqrt{6}}{2}{\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }+\displaystyle \frac{\sqrt{6}{m}_{{q}_{3}}}{2(2m+{m}_{{q}_{3}})}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+\displaystyle \frac{\sqrt{6}{m}_{{q}_{3}}}{2({m}_{{q}_{3}}+{m}_{\bar{c}})}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right)\\ & {\phi }_{{{BJ}}_{B}{J}_{{Bz}}}\left({\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }-\displaystyle \frac{{m}_{\bar{c}}}{{m}_{c}+{m}_{\bar{c}}}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+\displaystyle \frac{{m}_{\bar{c}}}{{m}_{{q}_{3}}+{m}_{\bar{c}}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right)\\ & +{V}_{{q}_{3}c}\left(\displaystyle \frac{2}{\sqrt{6}}{\vec{p}}_{\lambda }^{{\prime} }+{\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }-{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+{o}_{{\rm{r}}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right)\\ & {\phi }_{A\mathrm{space}}\left({\vec{p}}_{\rho }^{{\prime} },{\vec{p}}_{\lambda }^{{\prime} }-\displaystyle \frac{\sqrt{6}m}{2m+{m}_{{q}_{3}}}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+\displaystyle \frac{\sqrt{6}m}{2m+{m}_{c}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right)\\ & {\phi }_{{{BJ}}_{B}{J}_{{Bz}}}\left.\left({\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }-\displaystyle \frac{{m}_{\bar{c}}}{{m}_{c}+{m}_{\bar{c}}}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+\displaystyle \frac{{m}_{\bar{c}}}{{m}_{{q}_{3}}+{m}_{\bar{c}}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right)\right\}\\ & {\phi }_{A\mathrm{color}}{\phi }_{B\mathrm{color}}{\phi }_{A\mathrm{flavor}}{\phi }_{B\mathrm{flavor}}{\chi }_{{S}_{A}{S}_{{Az}}},\end{aligned}$
with ${o}_{{\rm{r}}}=(2{{mm}}_{\bar{c}}-{m}_{{q}_{3}}{m}_{c})/[(2m+{m}_{c})({m}_{{q}_{3}}+{m}_{\bar{c}})]$, and the transition amplitude for scattering in the post form, $\begin{aligned} {{ \mathcal M }}_{\mathrm{fi}}^{\mathrm{post}} = & \sqrt{2{E}_{A}2{E}_{B}2{E}_{C}2{E}_{D}}{\phi }_{C\mathrm{color}}^{+}{\phi }_{D\mathrm{color}}^{+}{\phi }_{C\mathrm{flavor}}^{+}{\phi }_{D\mathrm{flavor}}^{+}{\chi }_{{S}_{C}{S}_{{Cz}}}^{+}{\chi }_{{S}_{D}{S}_{{Dz}}}^{+} \\ & \times \,\displaystyle \int \displaystyle \frac{{{\rm{d}}}^{3}{p}_{\rho }^{{\prime} }}{{\left(2\pi \right)}^{3}}\displaystyle \frac{{{\rm{d}}}^{3}{p}_{\lambda }^{{\prime} }}{{\left(2\pi \right)}^{3}}\displaystyle \frac{{{\rm{d}}}^{3}{p}_{{q}_{3}\bar{c}}^{{\prime} }}{{\left(2\pi \right)}^{3}}{\phi }_{C\mathrm{space}}^{+}({\vec{p}}_{\rho }^{{\prime} },{\vec{p}}_{\lambda }^{{\prime} }){\phi }_{D\mathrm{rel}}^{+}({\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }) \\ & \left\{{V}_{{q}_{1}\bar{c}}\left(\displaystyle \frac{2}{\sqrt{6}}{\vec{p}}_{\lambda }^{{\prime} }+{\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }-{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+{o}_{{\rm{r}}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right)\right. \\ & {\phi }_{A\mathrm{space}}\left({\vec{p}}_{\rho }^{{\prime} }-\displaystyle \frac{1}{\sqrt{3}}{\vec{p}}_{\lambda }^{{\prime} }-\displaystyle \frac{\sqrt{2}}{2}{\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }+\displaystyle \frac{\sqrt{2}}{2}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}\right. \\ & \,-\displaystyle \frac{\sqrt{2}{o}_{{\rm{r}}}}{2}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }, \\ & \left.-\displaystyle \frac{\sqrt{6}}{2}{\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }+\displaystyle \frac{\sqrt{6}{m}_{{q}_{3}}}{2(2m+{m}_{{q}_{3}})}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+\displaystyle \frac{\sqrt{6}{m}_{{q}_{3}}}{2({m}_{{q}_{3}}+{m}_{\bar{c}})}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right) \\ & {\phi }_{{{BJ}}_{B}{J}_{{Bz}}}\left(-\displaystyle \frac{2}{\sqrt{6}}{\vec{p}}_{\lambda }^{{\prime} }+\displaystyle \frac{{m}_{c}}{{m}_{c}+{m}_{\bar{c}}}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+\displaystyle \frac{{m}_{c}}{2m+{m}_{c}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right) \\ & +{V}_{{q}_{2}\bar{c}}\left(\displaystyle \frac{2}{\sqrt{6}}{\vec{p}}_{\lambda }^{{\prime} }+{\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }-{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+{o}_{{\rm{r}}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right) \\ & {\phi }_{A\mathrm{space}}\left({\vec{p}}_{\rho }^{{\prime} }+\displaystyle \frac{1}{\sqrt{3}}{\vec{p}}_{\lambda }^{{\prime} }+\displaystyle \frac{\sqrt{2}}{2}{\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }-\displaystyle \frac{\sqrt{2}}{2}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+\displaystyle \frac{\sqrt{2}{o}_{{\rm{r}}}}{2}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} },\right. \\ & \left.-\displaystyle \frac{\sqrt{6}}{2}{\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }+\displaystyle \frac{\sqrt{6}{m}_{{q}_{3}}}{2(2m+{m}_{{q}_{3}})}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+\displaystyle \frac{\sqrt{6}{m}_{{q}_{3}}}{2({m}_{{q}_{3}}+{m}_{\bar{c}})}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right) \end{aligned}$$\begin{aligned} \\ & {\phi }_{{{BJ}}_{B}{J}_{{Bz}}}\left(-\displaystyle \frac{2}{\sqrt{6}}{\vec{p}}_{\lambda }^{{\prime} }+\displaystyle \frac{{m}_{c}}{{m}_{c}+{m}_{\bar{c}}}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+\displaystyle \frac{{m}_{c}}{2m+{m}_{c}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right) \\ & +{V}_{{q}_{3}c}\left(\displaystyle \frac{2}{\sqrt{6}}{\vec{p}}_{\lambda }^{{\prime} }+{\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }-{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+{o}_{{\rm{r}}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right) \\ & {\phi }_{A\mathrm{space}}\left({\vec{p}}_{\rho }^{{\prime} },{\vec{p}}_{\lambda }^{{\prime} }-\displaystyle \frac{\sqrt{6}m}{2m+{m}_{{q}_{3}}}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+\displaystyle \frac{\sqrt{6}m}{2m+{m}_{c}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right) \\ & {\phi }_{{{BJ}}_{B}{J}_{{Bz}}}\left.\left({\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }-\displaystyle \frac{{m}_{\bar{c}}}{{m}_{c}+{m}_{\bar{c}}}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+\displaystyle \frac{{m}_{\bar{c}}}{{m}_{{q}_{3}}+{m}_{\bar{c}}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right)\right\} \\ & {\phi }_{A\mathrm{color}}{\phi }_{B\mathrm{color}}{\phi }_{A\mathrm{flavor}}{\phi }_{B\mathrm{flavor}}{\chi }_{{S}_{A}{S}_{{Az}}} \\ & +\sqrt{2{E}_{A}2{E}_{B}2{E}_{C}2{E}_{D}}{\phi }_{C\mathrm{color}}^{+}{\phi }_{D\mathrm{color}}^{+}{\phi }_{C\mathrm{flavor}}^{+}{\phi }_{D\mathrm{flavor}}^{+}{\chi }_{{S}_{C}{S}_{{Cz}}}^{+}{\chi }_{{S}_{D}{S}_{{Dz}}}^{+} \\ & \displaystyle \times \,\int \displaystyle \frac{{{\rm{d}}}^{3}{p}_{\rho }}{{\left(2\pi \right)}^{3}}\displaystyle \frac{{{\rm{d}}}^{3}{p}_{\lambda }}{{\left(2\pi \right)}^{3}}\displaystyle \frac{{{\rm{d}}}^{3}{p}_{c\bar{c}}}{{\left(2\pi \right)}^{3}} \\ & \times \,\left\{{\phi }_{C\mathrm{space}}^{+}\left({\vec{p}}_{\rho },{\vec{p}}_{\lambda }+\displaystyle \frac{\sqrt{6}m}{2m+{m}_{{q}_{3}}}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}-\displaystyle \frac{\sqrt{6}m}{2m+{m}_{c}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right)\right. \\ & {\phi }_{D\mathrm{rel}}^{+}\left(-\displaystyle \frac{2}{\sqrt{6}}{\vec{p}}_{\lambda }+\displaystyle \frac{{m}_{{q}_{3}}}{2m+{m}_{{q}_{3}}}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+\displaystyle \frac{{m}_{{q}_{3}}}{{m}_{{q}_{3}}+{m}_{\bar{c}}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right) \\ & {V}_{c\bar{c}}\left(-\displaystyle \frac{2}{\sqrt{6}}{\vec{p}}_{\lambda }-{\vec{p}}_{c\bar{c}}-{o}_{{\rm{t}}}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right) \\ & +{\phi }_{C\mathrm{space}}^{+}\left({\vec{p}}_{\rho }-\displaystyle \frac{1}{\sqrt{3}}{\vec{p}}_{\lambda }-\displaystyle \frac{\sqrt{2}}{2}{\vec{p}}_{c\bar{c}}-\displaystyle \frac{\sqrt{2}{o}_{{\rm{t}}}}{2}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+\displaystyle \frac{\sqrt{2}}{2}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} },\right. \\ & \left.-\displaystyle \frac{\sqrt{6}}{2}{\vec{p}}_{c\bar{c}}+\displaystyle \frac{\sqrt{6}{m}_{c}}{2({m}_{c}+{m}_{\bar{c}})}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+\displaystyle \frac{\sqrt{6}{m}_{c}}{2(2m+{m}_{c})}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right) \\ & {\phi }_{D\mathrm{rel}}^{+}\left({\vec{p}}_{c\bar{c}}+\displaystyle \frac{{m}_{\bar{c}}}{{m}_{c}+{m}_{\bar{c}}}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}-\displaystyle \frac{{m}_{\bar{c}}}{{m}_{{q}_{3}}+{m}_{\bar{c}}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right) \\ & {V}_{{q}_{1}{q}_{3}}\left(-\displaystyle \frac{2}{\sqrt{6}}{\vec{p}}_{\lambda }-{\vec{p}}_{c\bar{c}}-{o}_{{\rm{t}}}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right) \\ & +{\phi }_{C\mathrm{space}}^{+}\left({\vec{p}}_{\rho }+\displaystyle \frac{1}{\sqrt{3}}{\vec{p}}_{\lambda }+\displaystyle \frac{\sqrt{2}}{2}{\vec{p}}_{c\bar{c}}+\displaystyle \frac{\sqrt{2}{o}_{{\rm{t}}}}{2}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}-\displaystyle \frac{\sqrt{2}}{2}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} },\right. \\ & \left.-\displaystyle \frac{\sqrt{6}}{2}{\vec{p}}_{c\bar{c}}+\displaystyle \frac{\sqrt{6}{m}_{c}}{2({m}_{c}+{m}_{\bar{c}})}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+\displaystyle \frac{\sqrt{6}{m}_{c}}{2(2m+{m}_{c})}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right) \\ & {\phi }_{D\mathrm{rel}}^{+}\left({\vec{p}}_{c\bar{c}}+\displaystyle \frac{{m}_{\bar{c}}}{{m}_{c}+{m}_{\bar{c}}}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}-\displaystyle \frac{{m}_{\bar{c}}}{{m}_{{q}_{3}}+{m}_{\bar{c}}}{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right) \\ & {V}_{{q}_{2}{q}_{3}}\left.\left(-\displaystyle \frac{2}{\sqrt{6}}{\vec{p}}_{\lambda }-{\vec{p}}_{c\bar{c}}-{o}_{{\rm{t}}}{\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}+{\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }\right)\right\} \\ & {\phi }_{A\mathrm{space}}({\vec{p}}_{\rho },{\vec{p}}_{\lambda }){\phi }_{{{BJ}}_{B}{J}_{{Bz}}}({\vec{p}}_{c\bar{c}}){\phi }_{A\mathrm{color}}{\phi }_{B\mathrm{color}}{\phi }_{A\mathrm{flavor}}{\phi }_{B\mathrm{flavor}}{\chi }_{{S}_{A}{S}_{{Az}}}.\end{aligned}$
with ${o}_{{\rm{t}}}=(2{{mm}}_{\bar{c}}-{m}_{{q}_{3}}{m}_{c})/[({m}_{c}+{m}_{\bar{c}})(2m+{m}_{{q}_{3}})]$. The variables ${\vec{p}}_{\rho }$ and ${\vec{p}}_{\lambda }$ in ${\phi }_{A\mathrm{space}}({\vec{p}}_{\rho },$ ${\vec{p}}_{\lambda })$, ${\vec{p}}_{c\bar{c}}$ in ${\phi }_{{{BJ}}_{B}{J}_{{Bz}}}({\vec{p}}_{c\bar{c}})$, ${\vec{p}}_{\rho }^{{\prime} }$ and ${\vec{p}}_{\lambda }^{{\prime} }$ in ${\phi }_{C\mathrm{space}}({\vec{p}}_{\rho }^{{\prime} },{\vec{p}}_{\lambda }^{{\prime} })$, and ${\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} }$ in ${\phi }_{D\mathrm{rel}}({\vec{p}}_{{q}_{3}\bar{c}}^{{\prime} })$ equal the expressions enclosed by the parentheses that follow φAspace, ${\phi }_{{{BJ}}_{B}{J}_{{Bz}}}$, φCspace, and φDrel.
Figure 1. Scattering in the prior form. Solid lines with triangles right (left) represent quarks (antiquarks). Dot-dashed lines indicate interactions. |
Figure 2. Scattering in the post form. Solid lines with triangles right (left) represent quarks (antiquarks). Dot-dashed lines indicate interactions. |
With the transition amplitudes the unpolarized cross section for A + B → C + D is
$\begin{eqnarray}\begin{array}{l}{\sigma }^{\mathrm{unpol}}(\sqrt{s})=\displaystyle \frac{1}{(2{J}_{A}+1)(2{J}_{B}+1)}\displaystyle \frac{1}{64\pi s}\displaystyle \frac{| {\vec{P}}^{{\prime} }(\sqrt{s})| }{| \vec{P}(\sqrt{s})| }\\ \quad \times {\displaystyle \int }_{0}^{\pi }{\rm{d}}\theta \displaystyle \sum _{{J}_{{Az}}{J}_{{Bz}}{J}_{{Cz}}{J}_{{Dz}}}(| {{ \mathcal M }}_{\mathrm{fi}}^{\mathrm{prior}}{| }^{2}+| {{ \mathcal M }}_{\mathrm{fi}}^{\mathrm{post}}{| }^{2})\sin \theta ,\end{array}\end{eqnarray}$
where s is the Mandelstam variable obtained from the four-momenta PA and PB of hadrons A and B by $s={\left({P}_{A}+{P}_{B}\right)}^{2};$ JA (JB, JC, JD) and JAz (JBz, JCz, JDz) of hadron A (B, C, D) are the total angular momentum and its z component, respectively; θ is the angle between $\vec{P}$ and ${\vec{P}}^{{\prime} }$ which are the three-dimensional momentum components of baryons A and C in the center-of-momentum frame of the initial baryon and the initial meson, respectively. We calculate the cross section in the center-of-momentum frame.3. Numerical cross sections and discussions
We use the notation $D=\left(\begin{array}{c}{D}^{+}\\ {D}^{0}\end{array}\right)$, $\bar{D}=\left(\begin{array}{c}{\bar{D}}^{0}\\ {D}^{-}\end{array}\right)$, ${D}^{* }=\left(\begin{array}{c}{D}^{* +}\\ {D}^{* 0}\end{array}\right)$, and ${\bar{D}}^{* }=\left(\begin{array}{c}{\bar{D}}^{* 0}\\ {D}^{* -}\end{array}\right)$. We consider the following reactions:32 ). As seen in equations (30 ) and (31 ) ${{ \mathcal M }}_{\mathrm{fi}}^{\mathrm{prior}}$ and ${{ \mathcal M }}_{\mathrm{fi}}^{\mathrm{post}}$ used in equation (32 ) involve ${\phi }_{{{BJ}}_{B}{J}_{{Bz}}}$ and φDrel. The two wave functions are obtained from solutions of the Schrödinger equation with the potential between constituents a and b in coordinate space,
$\begin{eqnarray*}{pR}\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{0},\qquad {pR}\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{* 0},\end{eqnarray*}$
$\begin{eqnarray*}{pR}\to {{\rm{\Sigma }}}_{c}^{++}{D}^{-},\qquad {pR}\to {{\rm{\Sigma }}}_{c}^{++}{D}^{* -},\end{eqnarray*}$
$\begin{eqnarray*}{pR}\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{0},\qquad {pR}\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{* 0},\end{eqnarray*}$
$\begin{eqnarray*}{pR}\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{-},\qquad {pR}\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{* -},\end{eqnarray*}$
$\begin{eqnarray*}{pR}\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{0},\qquad {pR}\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{* 0},\end{eqnarray*}$
where R stands for ψ(3770), ψ(4040), ψ(4160), or ψ(4415). By replacing the up quark with the down quark and vice versa in these ten reactions, they give ten reactions of a neutron and a $c\bar{c}$ meson. Since the cross section for ${nR}\to {{\rm{\Lambda }}}_{c}^{+}{D}^{-}$ (${nR}\,\to {{\rm{\Lambda }}}_{c}^{+}{D}^{* -}$, ${nR}\to {{\rm{\Sigma }}}_{c}^{0}{\bar{D}}^{0}$, ${nR}\to {{\rm{\Sigma }}}_{c}^{0}{\bar{D}}^{* 0}$, ${nR}\to {{\rm{\Sigma }}}_{c}^{+}{D}^{-}$, ${nR}\,\to {{\rm{\Sigma }}}_{c}^{+}{D}^{* -}$, ${nR}\to {{\rm{\Sigma }}}_{c}^{* 0}{\bar{D}}^{0}$, ${nR}\to {{\rm{\Sigma }}}_{c}^{* 0}{\bar{D}}^{* 0}$, ${nR}\,\to {{\rm{\Sigma }}}_{c}^{* +}{D}^{-}$, ${nR}\to {{\rm{\Sigma }}}_{c}^{* +}{D}^{* -}$) equals the one for ${pR}\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{0}$ (${pR}\,\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{* 0}$, ${pR}\to {{\rm{\Sigma }}}_{c}^{++}{D}^{-}$, ${pR}\to {{\rm{\Sigma }}}_{c}^{++}{D}^{* -}$, ${pR}\,\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{0}$, ${pR}\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{* 0}$, ${pR}\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{-}$, ${pR}\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{* -}$, ${pR}\,\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{0}$, ${pR}\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{* 0}$), it is enough to only discuss reactions of the proton and the $c\bar{c}$ meson in this section. We calculate unpolarized cross sections for these reactions with equation ( $\begin{eqnarray}\begin{array}{l}{V}_{{ab}}({\vec{r}}_{{ab}})=-\displaystyle \frac{{\vec{\lambda }}_{a}}{2}\cdot \displaystyle \frac{{\vec{\lambda }}_{b}}{2}\displaystyle \frac{3}{4}{{kr}}_{{ab}}\\ \quad +\displaystyle \frac{{\vec{\lambda }}_{a}}{2}\cdot \displaystyle \frac{{\vec{\lambda }}_{b}}{2}\displaystyle \frac{6\pi }{25}\displaystyle \frac{v(\lambda {r}_{{ab}})}{{r}_{{ab}}}\\ \quad -\displaystyle \frac{{\vec{\lambda }}_{a}}{2}\cdot \displaystyle \frac{{\vec{\lambda }}_{b}}{2}\displaystyle \frac{16{\pi }^{2}}{25}\displaystyle \frac{{d}^{3}}{{\pi }^{3/2}}\exp (-{d}^{2}{r}_{{ab}}^{2})\displaystyle \frac{{\vec{s}}_{a}\cdot {\vec{s}}_{b}}{{m}_{a}{m}_{b}}\\ \quad +\displaystyle \frac{{\vec{\lambda }}_{a}}{2}\cdot \displaystyle \frac{{\vec{\lambda }}_{b}}{2}\displaystyle \frac{4\pi }{25}\displaystyle \frac{1}{{r}_{{ab}}}\displaystyle \frac{{d}^{2}v(\lambda {r}_{{ab}})}{{{dr}}_{{ab}}^{2}}\displaystyle \frac{{\vec{s}}_{a}\cdot {\vec{s}}_{b}}{{m}_{a}{m}_{b}}\\ \quad -\displaystyle \frac{{\vec{\lambda }}_{a}}{2}\cdot \displaystyle \frac{{\vec{\lambda }}_{b}}{2}\displaystyle \frac{6\pi }{25{m}_{a}{m}_{b}}\left[\Space{0ex}{3.5ex}{0ex}v(\lambda {r}_{{ab}})\right.\\ \quad \left.-{r}_{{ab}}\displaystyle \frac{{dv}(\lambda {r}_{{ab}})}{{{dr}}_{{ab}}}+\displaystyle \frac{{r}_{{ab}}^{2}}{3}\displaystyle \frac{{d}^{2}v(\lambda {r}_{{ab}})}{{{dr}}_{{ab}}^{2}}\right]\\ \quad \times \left(\displaystyle \frac{3{\vec{s}}_{a}\cdot {\vec{r}}_{{ab}}{\vec{s}}_{b}\cdot {\vec{r}}_{{ab}}}{{r}_{{ab}}^{5}}-\displaystyle \frac{{\vec{s}}_{a}\cdot {\vec{s}}_{b}}{{r}_{{ab}}^{3}}\right),\end{array}\end{eqnarray}$
where ${\vec{r}}_{{ab}}$ is the relative coordinate of constituents a and b; k = 0.153 GeV2 and λ = 0.39 GeV; ma, ${\vec{s}}_{a}$, and ${\vec{\lambda }}_{a}$ are individually the mass, the spin, and the Gell–Mann matrices for the color generators of constituent a; the function v is given by Buchmüller and Tye in [29]; the quantity d is $\begin{eqnarray}\begin{array}{rcl}{d}^{2} & = & {d}_{\alpha }^{2}\left[\displaystyle \frac{1}{2}+\displaystyle \frac{1}{2}{\left(\displaystyle \frac{4{m}_{a}{m}_{b}}{{\left({m}_{a}+{m}_{b}\right)}^{2}}\right)}^{4}\right]\\ & & +{d}_{\beta }^{2}{\left(\displaystyle \frac{2{m}_{a}{m}_{b}}{{m}_{a}+{m}_{b}}\right)}^{2},\end{array}\end{eqnarray}$
where dα = 0.34 GeV and dβ = 0.45. The potential originates from quantum chromodynamics (QCD) [29]. The first two terms are the Buchmüller–Tye potential, and the other terms come from one-gluon exchange plus perturbative one- and two-loop corrections [30].The function v(x) manifests one-gluon exchange plus perturbative one- and two-loop corrections between constituents a and b. It increases from 0 to 1 when x increases from 0 to the positive infinity. Consequently, the second term is not a color Coulomb potential.
One-gluon exchange between two constituents gives rise to the Fermi contact term $-\displaystyle \frac{{\vec{\lambda }}_{a}}{2}\cdot \tfrac{{\vec{\lambda }}_{b}}{2}\tfrac{16{\pi }^{2}}{25}{\delta }^{3}({\vec{r}}_{{ab}})\tfrac{{\vec{s}}_{a}\cdot {\vec{s}}_{b}}{{m}_{a}{m}_{b}}$. The ${\delta }^{3}({\vec{r}}_{{ab}})$ function fixes the positions of the two constituents to ${\vec{r}}_{{ab}}=0$. However, the constituent positions fluctuate in the presence of one- and two-loop corrections. To allow the position fluctuation, ${\delta }^{3}({\vec{r}}_{{ab}})$ is replaced with $\tfrac{{d}^{3}}{{\pi }^{3/2}}\exp (\mbox{-}{d}^{2}{r}_{{ab}}^{2})$ so as to arrive at the third term on the right-hand side of equation (33 ), which is the smearing of the Fermi contact term [1]. The Gaussian has a width of $2\sqrt{\mathrm{ln}2}/d$, and d−1 indicates the fluctuation size. The larger is d, the smaller is the fluctuation size. d depends on constituent masses. When ma = mb, ${d}^{2}={d}_{\alpha }^{2}+{d}_{\beta }^{2}{m}_{b}^{2}$. When ma ≫ mb, ${d}^{2}\approx {d}_{\alpha }^{2}/2\,+4{d}_{\beta }^{2}{m}_{b}^{2}$. In the two cases the dα term gives a constant value to d, and the dβ term is proportional to mb2. The two terms provide different mass dependence. Since ${d}^{2}\gt {d}_{\alpha }^{2}/2$, the parameter dα reflects the fact that in a confined system the smearing must be limited.
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 Vab, we obtain meson masses that are close to the experimental masses of π, ρ, K, K*, D, D*, Ds, ${D}_{s}^{* }$, J/ψ, χc, $\psi ^{\prime} $, ψ(3770), ψ(4040), ψ(4160), and ψ(4415) mesons listed in [31]. The experimental data of S-wave I = 2 elastic phase shifts for ππ scattering [32] are reproduced in the Born approximation.
${{ \mathcal M }}_{\mathrm{fi}}^{\mathrm{prior}}$ and ${{ \mathcal M }}_{\mathrm{fi}}^{\mathrm{post}}$ involve the space wave functions φAspace and φCspace. The space wave functions of ground-state baryons are usually assumed to be harmonic-oscillator wave functions [33, 34]:5 ) is6 ) is given from ${\psi }_{{q}_{1}{q}_{2}{q}_{3}}$ by replacing q3 (${\vec{\lambda }}_{{\rm{i}}}$, A) with c (${\vec{\lambda }}_{{\rm{f}}}$, C). Masses of baryons in the baryon octet and the baryon decuplet are given by37 ) is used to calculate masses of ground-state charmed baryons. Let mp, ${m}_{{{\rm{\Lambda }}}_{c}^{+}}$, ${m}_{{{\rm{\Sigma }}}_{c}^{++}}$, ${m}_{{{\rm{\Sigma }}}_{c}^{+}}$, ${m}_{{{\rm{\Sigma }}}_{c}^{* ++}}$, and ${m}_{{{\rm{\Sigma }}}_{c}^{* +}}$ represent the experimental masses of p, ${{\rm{\Lambda }}}_{c}^{+}$, ${{\rm{\Sigma }}}_{c}^{++}$, ${{\rm{\Sigma }}}_{c}^{+}$, ${{\rm{\Sigma }}}_{c}^{* ++}$, and ${{\rm{\Sigma }}}_{c}^{* +}$ baryons, respectively. Fits to the experimental masses of the six baryons give
$\begin{eqnarray}\begin{array}{l}{\phi }_{A\mathrm{space}}(\vec{\rho },{\vec{\lambda }}_{{\rm{i}}})={\left(\displaystyle \frac{{\alpha }_{\rho }{\alpha }_{{\lambda }_{{\rm{i}}}}}{\pi }\right)}^{1.5}\exp \left(-\displaystyle \frac{{\alpha }_{\rho }^{2}{\vec{\rho }}^{\,2}+{\alpha }_{{\lambda }_{{\rm{i}}}}^{2}{{\vec{\lambda }}_{{\rm{i}}}}^{2}}{2}\right),\end{array}\end{eqnarray}$
and ${\phi }_{C\mathrm{space}}(\vec{\rho },{\vec{\lambda }}_{{\rm{f}}})$ is obtained from ${\phi }_{A\mathrm{space}}(\vec{\rho },{\vec{\lambda }}_{{\rm{i}}})$ by replacing λi with λf. The wave function ${\psi }_{{q}_{1}{q}_{2}{q}_{3}}(\vec{\rho },{\vec{\lambda }}_{{\rm{i}}})$ in equation ( $\begin{eqnarray}{\psi }_{{q}_{1}{q}_{2}{q}_{3}}(\vec{\rho },{\vec{\lambda }}_{{\rm{i}}})={\phi }_{A\mathrm{color}}{\phi }_{A\mathrm{flavor}}{\phi }_{A\mathrm{space}}(\vec{\rho },{\vec{\lambda }}_{{\rm{i}}}){\chi }_{{S}_{A}{S}_{{Az}}},\end{eqnarray}$
and ${\psi }_{{q}_{1}{q}_{2}c}$ in equation ( $\begin{eqnarray}\begin{array}{l}{m}_{{\rm{B}}}=2m+{m}_{{q}_{3}}+\displaystyle \int {d}^{3}\rho {d}^{3}{\lambda }_{{\rm{i}}}{\psi }_{{q}_{1}{q}_{2}{q}_{3}}^{+}(\vec{\rho },{\vec{\lambda }}_{{\rm{i}}})\left[\displaystyle \frac{{\vec{{\rm{\nabla }}}}_{\vec{\rho }}^{2}}{2m}+\displaystyle \frac{{\vec{{\rm{\nabla }}}}_{{\vec{\lambda }}_{{\rm{i}}}}^{2}}{2{m}_{{\lambda }_{{\rm{i}}}}}\right.\\ \quad \left.+{V}_{{q}_{1}{q}_{2}}({\vec{r}}_{{q}_{1}{q}_{2}})+{V}_{{q}_{2}{q}_{3}}({\vec{r}}_{{q}_{2}{q}_{3}})+{V}_{{q}_{3}{q}_{1}}({\vec{r}}_{{q}_{3}{q}_{1}})\Space{0ex}{3.9ex}{0ex}\right]{\psi }_{{q}_{1}{q}_{2}{q}_{3}}(\vec{\rho },{\vec{\lambda }}_{{\rm{i}}}),\end{array}\end{eqnarray}$
with ${m}_{{\lambda }_{{\rm{i}}}}=\tfrac{3{{mm}}_{{q}_{3}}}{2m+{m}_{{q}_{3}}}$. Replacing q3 (${\vec{\lambda }}_{{\rm{i}}}$) with c (${\vec{\lambda }}_{{\rm{f}}}$), equation ( $\begin{eqnarray*}\begin{array}{l}{\alpha }_{\rho }=0.3\,\mathrm{GeV},\qquad {\alpha }_{{\lambda }_{{\rm{i}}}}=0.3\,\mathrm{GeV},\\ \qquad {m}_{p}=0.938272\,\mathrm{GeV};\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{\alpha }_{\rho }=0.222594\,\mathrm{GeV},\qquad {\alpha }_{{\lambda }_{{\rm{f}}}}=0.43\,\mathrm{GeV},\\ \qquad {m}_{{{\rm{\Lambda }}}_{c}^{+}}=2.28646\,\mathrm{GeV};\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{\alpha }_{\rho }=0.196273\,\mathrm{GeV},\qquad {\alpha }_{{\lambda }_{{\rm{f}}}}=0.43\,\mathrm{GeV},\\ \qquad {m}_{{{\rm{\Sigma }}}_{c}^{++}}=2.45397\,\mathrm{GeV};\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{\alpha }_{\rho }=0.19642\,\mathrm{GeV},\qquad {\alpha }_{{\lambda }_{{\rm{f}}}}=0.43\,\mathrm{GeV},\\ \qquad {m}_{{{\rm{\Sigma }}}_{c}^{+}}=2.4529\,\mathrm{GeV};\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{\alpha }_{\rho }=0.19092\,\mathrm{GeV},\qquad {\alpha }_{{\lambda }_{{\rm{f}}}}=0.43\,\mathrm{GeV},\\ \qquad {m}_{{{\rm{\Sigma }}}_{c}^{* ++}}=2.51841\,\mathrm{GeV};\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{\alpha }_{\rho }=0.19105\,\mathrm{GeV},\qquad {\alpha }_{{\lambda }_{{\rm{f}}}}=0.43\,\mathrm{GeV},\\ \qquad {m}_{{{\rm{\Sigma }}}_{c}^{* +}}=2.5175\,\mathrm{GeV}.\end{array}\end{eqnarray*}$
Using the mesonic quark-antiquark relative-motion wave functions and the space wave functions of the baryons, we obtain unpolarized cross sections for dissociation of ψ(3770), ψ(4040), ψ(4160), and ψ(4415) mesons in collisions with protons. The cross sections are plotted in figures 3–12, and are parametrized as32 ) equals zero, but $| {\vec{P}}^{{\prime} }| $ does not. The cross section is thus infinite at the threshold energy. The cross sections in figures 3–12 are plotted as functions of $\sqrt{s}$ which equals or is larger than the threshold energy plus 5 × 10−4 GeV.
$\begin{eqnarray}\begin{array}{l}{\sigma }^{\mathrm{unpol}}(\sqrt{s})=\displaystyle \frac{{\vec{P}}^{{\prime} 2}}{{\vec{P}}^{2}}\left\{{a}_{1}{\left(\displaystyle \frac{\sqrt{s}-\sqrt{{s}_{0}}}{{b}_{1}}\right)}^{{c}_{1}}\right.\\ \quad \times \,\exp \left[{c}_{1}\left(1-\displaystyle \frac{\sqrt{s}-\sqrt{{s}_{0}}}{{b}_{1}}\right)\right]\\ \quad \left.+{a}_{2}{\left(\displaystyle \frac{\sqrt{s}-\sqrt{{s}_{0}}}{{b}_{2}}\right)}^{{c}_{2}}\exp \left[{c}_{2}\left(1-\displaystyle \frac{\sqrt{s}-\sqrt{{s}_{0}}}{{b}_{2}}\right)\right]\right\},\end{array}\end{eqnarray}$
where $\sqrt{{s}_{0}}$ is the threshold energy, and a1, b1, c1, a2, b2, and c2 are parameters. The parameter values are listed in tables 1–2. The threshold energy of inelastic p + ψ(3770) (p + ψ(4040), p + ψ(4160), p + ψ(4415)) scattering is the sum of the proton and ψ(3770) (ψ(4040), ψ(4160), ψ(4415)) masses. At the threshold energy $| \vec{P}| $ in equation (
Figure 3. Solid, dashed, dotted, and dot-dashed curves stand for cross sections for $p\psi (3770)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{0}$, $p\psi (4040)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{0}$, $p\psi (4160)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{0}$, and $p\psi (4415)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{0}$, respectively. |
Figure 4. Solid, dashed, dotted, and dot-dashed curves stand for cross sections for $p\psi (3770)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{* 0}$, $p\psi (4040)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{* 0}$, $p\psi (4160)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{* 0}$, and $p\psi (4415)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{* 0}$, respectively. |
Figure 5. Solid, dashed, dotted, and dot-dashed curves stand for cross sections for $p\psi (3770)\to {{\rm{\Sigma }}}_{c}^{++}{D}^{-}$, $p\psi (4040)\to {{\rm{\Sigma }}}_{c}^{++}{D}^{-}$, $p\psi (4160)\to {{\rm{\Sigma }}}_{c}^{++}{D}^{-}$, and $p\psi (4415)\to {{\rm{\Sigma }}}_{c}^{++}{D}^{-}$, respectively. |
Figure 6. Solid, dashed, dotted, and dot-dashed curves stand for cross sections for $p\psi (3770)\to {{\rm{\Sigma }}}_{c}^{++}{D}^{* -}$, $p\psi (4040)\to {{\rm{\Sigma }}}_{c}^{++}{D}^{* -}$, $p\psi (4160)\to {{\rm{\Sigma }}}_{c}^{++}{D}^{* -}$, and $p\psi (4415)\to {{\rm{\Sigma }}}_{c}^{++}{D}^{* -}$, respectively. |
Figure 7. Solid, dashed, dotted, and dot-dashed curves stand for cross sections for $p\psi (3770)\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{0}$, $p\psi (4040)\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{0}$, $p\psi (4160)\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{0}$, and $p\psi (4415)\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{0}$, respectively. |
Figure 8. Solid, dashed, dotted, and dot-dashed curves stand for cross sections for $p\psi (3770)\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{* 0}$, $p\psi (4040)\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{* 0}$, $p\psi (4160)\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{* 0}$, and $p\psi (4415)\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{* 0}$, respectively. |
Figure 9. Solid, dashed, dotted, and dot-dashed curves stand for cross sections for $p\psi (3770)\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{-}$, $p\psi (4040)\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{-}$, $p\psi (4160)\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{-}$, and $p\psi (4415)\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{-}$, respectively. |
Figure 10. Solid, dashed, dotted, and dot-dashed curves stand for cross sections for $p\psi (3770)\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{* -}$, $p\psi (4040)\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{* -}$, $p\psi (4160)\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{* -}$, and $p\psi (4415)\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{* -}$, respectively. |
Figure 11. Solid, dashed, dotted, and dot-dashed curves stand for cross sections for $p\psi (3770)\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{0}$, $p\psi (4040)\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{0}$, $p\psi (4160)\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{0}$, and $p\psi (4415)\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{0}$, respectively. |
Figure 12. Solid, dashed, dotted, and dot-dashed curves stand for cross sections for $p\psi (3770)\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{* 0}$, $p\psi (4040)\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{* 0}$, $p\psi (4160)\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{* 0}$, and $p\psi (4415)\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{* 0}$, respectively. |
Table 1. Values of the parameters. a1 and a2 are in units of millibarns; b1 and b2 are in units of GeV; c1 and c2 are dimensionless. |
| Reaction | a1 | b1 | c1 | a2 | b2 | c2 |
|---|---|---|---|---|---|---|
| $p\psi (3770)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{0}$ | 0.13 | 0.01 | 0.51 | 0.23 | 0.09 | 1.08 |
| $p\psi (4040)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{0}$ | 0.065 | 0.068 | 0.41 | 0.017 | 0.18 | 62.4 |
| $p\psi (4160)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{0}$ | 0.026 | 0.032 | 0.55 | 0.018 | 0.18 | 4.67 |
| $p\psi (4415)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{0}$ | 0.0164 | 0.08 | 0.42 | 0.005 | 0.07 | 11.7 |
| $p\psi (3770)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{* 0}$ | 6.4 | 0.03 | 0.54 | 3.18 | 0.18 | 4.15 |
| $p\psi (4040)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{* 0}$ | 0.14 | 0.31 | 1.15 | 1.12 | 0.06 | 0.47 |
| $p\psi (4160)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{* 0}$ | 0.58 | 0.02 | 0.54 | 0.66 | 0.12 | 2.35 |
| $p\psi (4415)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{* 0}$ | 0.034 | 0.02 | 0.08 | 0.3 | 0.08 | 0.65 |
| $p\psi (3770)\to {{\rm{\Sigma }}}_{c}^{++}{D}^{-}$ | 0.017 | 0.07 | 0.93 | 0.021 | 0.05 | 0.35 |
| $p\psi (4040)\to {{\rm{\Sigma }}}_{c}^{++}{D}^{-}$ | 0.0067 | 0.071 | 0.51 | 0.004 | 0.19 | 38.6 |
| $p\psi (4160)\to {{\rm{\Sigma }}}_{c}^{++}{D}^{-}$ | 0.0012 | 0.01 | 0.35 | 0.0042 | 0.08 | 0.66 |
| $p\psi (4415)\to {{\rm{\Sigma }}}_{c}^{++}{D}^{-}$ | 0.0013 | 0.024 | 0.58 | 0.0022 | 0.168 | 4.29 |
| $p\psi (3770)\to {{\rm{\Sigma }}}_{c}^{++}{D}^{* -}$ | 0.012 | 0.04 | 0.01 | 5.4 | 0.058 | 0.48 |
| $p\psi (4040)\to {{\rm{\Sigma }}}_{c}^{++}{D}^{* -}$ | 0.34 | 0.03 | 0.64 | 0.53 | 0.19 | 2.33 |
| $p\psi (4160)\to {{\rm{\Sigma }}}_{c}^{++}{D}^{* -}$ | 0.13 | 0.017 | 5.92 | 0.85 | 0.058 | 0.5 |
| $p\psi (4415)\to {{\rm{\Sigma }}}_{c}^{++}{D}^{* -}$ | 0.007 | 0.0025 | 0.01 | 0.27 | 0.079 | 0.6 |
| $p\psi (3770)\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{0}$ | 0.005 | 0.1 | 0.38 | 0.014 | 0.04 | 0.47 |
| $p\psi (4040)\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{0}$ | 0.0009 | 0.01 | 0.29 | 0.0033 | 0.12 | 1.6 |
| $p\psi (4160)\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{0}$ | 0.001 12 | 0.16 | 2.11 | 0.002 | 0.03 | 0.49 |
| $p\psi (4415)\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{0}$ | 0.0008 | 0.07 | 0.437 | 0.0006 | 0.162 | 9.3 |
Table 2. The same as table 1, but for twenty other reactions. |
| Reaction | a1 | b1 | c1 | a2 | b2 | c2 |
|---|---|---|---|---|---|---|
| $p\psi (3770)\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{* 0}$ | 1.4 | 0.01 | 0.52 | 2.6 | 0.1 | 1.21 |
| $p\psi (4040)\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{* 0}$ | 0.041 | 0.0035 | 0.53 | 0.39 | 0.117 | 1.16 |
| $p\psi (4160)\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{* 0}$ | 0.14 | 0.008 | 0.53 | 0.42 | 0.072 | 0.82 |
| $p\psi (4415)\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{* 0}$ | 0.025 | 0.008 | 0.46 | 0.131 | 0.1 | 0.66 |
| $p\psi (3770)\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{-}$ | 0.06 | 0.03 | 0.85 | 0.29 | 0.06 | 0.43 |
| $p\psi (4040)\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{-}$ | 0.026 | 0.027 | 0.55 | 0.044 | 0.17 | 2.86 |
| $p\psi (4160)\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{-}$ | 0.0446 | 0.054 | 0.45 | 0.003 | 0.234 | 82.8 |
| $p\psi (4415)\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{-}$ | 0.0037 | 0.24 | 0.21 | 0.013 | 0.07 | 0.63 |
| $p\psi (3770)\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{* -}$ | 0.4 | 0.09 | 3.24 | 1 | 0.06 | 0.4 |
| $p\psi (4040)\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{* -}$ | 0.0058 | 0.0038 | 0.38 | 0.0967 | 0.158 | 1.34 |
| $p\psi (4160)\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{* -}$ | 0.05 | 0.02 | 0.61 | 0.17 | 0.08 | 0.52 |
| $p\psi (4415)\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{* -}$ | 0.034 | 0.059 | 2.62 | 0.038 | 0.29 | 0.32 |
| $p\psi (3770)\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{0}$ | 0.005 | 0.013 | 9.57 | 0.178 | 0.059 | 0.49 |
| $p\psi (4040)\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{0}$ | 0.012 | 0.212 | 8.38 | 0.021 | 0.066 | 0.45 |
| $p\psi (4160)\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{0}$ | 0.0073 | 0.0161 | 1.5 | 0.019 | 0.061 | 0.4 |
| $p\psi (4415)\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{0}$ | 0.0002 | 0.001 | 0.5 | 0.0082 | 0.08 | 0.57 |
| $p\psi (3770)\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{* 0}$ | 0.148 | 0.186 | 4.97 | 0.583 | 0.0477 | 0.514 |
| $p\psi (4040)\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{* 0}$ | 0.0057 | 0.008 | 0.55 | 0.065 | 0.164 | 2.08 |
| $p\psi (4160)\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{* 0}$ | 0.037 | 0.013 | 0.33 | 0.1 | 0.09 | 1.25 |
| $p\psi (4415)\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{* 0}$ | 0.003 | 0.014 | 0.18 | 0.026 | 0.1 | 0.71 |
The reactions considered in the present work are all exothermic. When $\sqrt{s}$ increases from threshold, the cross sections decrease rapidly, and then change slowly. In the slowly-changing region the cross sections may be tens of millibarns. For example, the cross sections for $p\psi (3770)\to {{\rm{\Sigma }}}_{c}^{++}{D}^{* -}$, $p\psi (4040)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{* 0}$, $p\psi (4160)\,\to {{\rm{\Sigma }}}_{c}^{++}{D}^{* -}$, and $p\psi (4415)\,\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{* 0}$ can reach 20 mb, 30 mb, 21 mb, and 15 mb, respectively. According to the quantum numbers of ψ(3770), ψ(4040), ψ(4160), and ψ(4415) mesons, the numbers of their radial nodes are 0, 2, 1, and 3, respectively. If there is a node in the radial wave function ${R}_{{L}_{B}}({r}_{c\bar{c}})$, cancellation between the wave functions on both sides of the node occurs in the integration involved in the transition amplitudes, thus cross sections are reduced. The ψ(4040) mass is near the ψ(4160) mass. Since the ψ(4040) meson has one node more than the ψ(4160) meson, the integration related to ψ(4040) should have more cancellation than that related to ψ(4160). However, the wave function of ψ(4160) contains the spherical harmonics ${Y}_{2{M}_{B}}$ (MB = − 2, − 1, 0, 1, 2), and the wave function of ψ(4040) contains the constant spherical harmonics Y00. Then, the integration related to ψ(4160) may have more cancellation than that related to ψ(4040). Therefore, at the threshold energy plus 5 × 10−4 GeV, the cross sections for p + ψ(4040) reactions are larger in figures 3 and 7 or smaller in figures 4–6 and figures 8–12 than the ones for p + ψ(4160) reactions.
The mesonic quark-antiquark relative-motion wave functions are decreasing functions of the quark-antiquark relative momentum. ${\phi }_{A\mathrm{space}}({\vec{p}}_{\rho },{\vec{p}}_{\lambda })$ (${\phi }_{C\mathrm{space}}({\vec{p}}_{\rho }^{{\prime} },{\vec{p}}_{\lambda }^{{\prime} })$) in the transition amplitudes is an exponentially decreasing function of ${\vec{p}}_{\rho }$ and ${\vec{p}}_{\lambda }$ (${\vec{p}}_{\rho }^{{\prime} }$ and ${\vec{p}}_{\lambda }^{{\prime} }$). The quark-antiquark relative momenta, ${\vec{p}}_{\rho }$, ${\vec{p}}_{\lambda }$, ${\vec{p}}_{\rho }^{{\prime} }$, and ${\vec{p}}_{\lambda }^{{\prime} }$ are given by the expressions enclosed by the parentheses that follow ${\phi }_{{{BJ}}_{B}{J}_{{Bz}}}$, φDrel, φAspace, and φCspace in equations (30 ) and (31 ). These expressions may have ${\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}$ and ${\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }$. In the center-of-momentum frame of the proton and the charmonium, ${\vec{p}}_{{q}_{1}{q}_{2}{q}_{3},c\bar{c}}$ and ${\vec{p}}_{{q}_{1}{q}_{2}c,{q}_{3}\bar{c}}^{{\prime} }$ equal $\vec{P}$ and ${\vec{P}}^{{\prime} }$, respectively. Therefore, the quark-antiquark relative momenta, ${\vec{p}}_{\rho }$, ${\vec{p}}_{\lambda }$, ${\vec{p}}_{\rho }^{{\prime} }$, and ${\vec{p}}_{\lambda }^{{\prime} }$ bear linear relation to $\vec{P}$ and ${\vec{P}}^{{\prime} }$. At the threshold energy plus 5 × 10−4 GeV, $| \vec{P}| $ almost equals zero, and $| {\vec{P}}^{{\prime} }| $ of any p + ψ(3770) reaction is smaller than $| {\vec{P}}^{{\prime} }| $ of p + ψ(4040), p + ψ(4160), and p + ψ(4415) reactions with the same final charmed baryon and the same final charmed meson in any of figures 3–12. The transition amplitudes for the p + ψ(3770) reaction are larger than those for the p + ψ(4040), p + ψ(4160), and p + ψ(4415) reactions. Therefore, at the threshold energy plus 5 × 10−4 GeV, the cross section for $p\psi (3770)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{0}$ in figure 3 is larger than those for $p\psi (4040)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{0}$, $p\psi (4160)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{0}$, and $p\psi (4415)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{0};$ similar results are displayed in figures 4–12. From the ten figures, we can also understand that the cross sections for p + ψ(4160) reactions at the threshold energy plus 5 × 10−4 GeV are larger than the ones for p + ψ(4415) reactions.
The measured proton mass has a very small uncertainty, and the uncertainty is neglected here. The measured masses of charmed baryons have uncertainties [31], for example, the ${{\rm{\Lambda }}}_{c}^{+}$ mass has an error of 0.14 MeV. The measured mass of every charmed baryon has a maximum and a minimum, for example, the ${{\rm{\Lambda }}}_{c}^{+}$ mass has the maximum mass 2286.60 MeV and the minimum mass 2286.32 MeV. Fits to the maximum experimental masses of ${{\rm{\Lambda }}}_{c}^{+}$, ${{\rm{\Sigma }}}_{c}^{++}$, ${{\rm{\Sigma }}}_{c}^{+}$, ${{\rm{\Sigma }}}_{c}^{* ++}$, and ${{\rm{\Sigma }}}_{c}^{* +}$ baryons give ${\alpha }_{{\lambda }_{{\rm{f}}}}=0.43\,\mathrm{GeV}$ and
$\begin{eqnarray*}{\alpha }_{\rho }=0.222571\,\mathrm{GeV},\qquad {m}_{{{\rm{\Lambda }}}_{c}^{+}}=2.28660\,\mathrm{GeV};\end{eqnarray*}$
$\begin{eqnarray*}{\alpha }_{\rho }=0.196254\,\mathrm{GeV},\qquad {m}_{{{\rm{\Sigma }}}_{c}^{++}}=2.45411\,\mathrm{GeV};\end{eqnarray*}$
$\begin{eqnarray*}{\alpha }_{\rho }=0.196365\,\mathrm{GeV},\qquad {m}_{{{\rm{\Sigma }}}_{c}^{+}}=2.4533\,\mathrm{GeV};\end{eqnarray*}$
$\begin{eqnarray*}{\alpha }_{\rho }=0.190894\,\mathrm{GeV},\qquad {m}_{{{\rm{\Sigma }}}_{c}^{* ++}}=2.51862\,\mathrm{GeV};\end{eqnarray*}$
$\begin{eqnarray*}{\alpha }_{\rho }=0.19076\,\mathrm{GeV},\qquad {m}_{{{\rm{\Sigma }}}_{c}^{* +}}=2.5198\,\mathrm{GeV}.\end{eqnarray*}$
Using these values of αρ and ${\alpha }_{{\lambda }_{{\rm{f}}}}$, we obtain unpolarized cross sections which are denoted as ${\sigma }_{\mathrm{lm}}^{\mathrm{unpol}}$. The differences between ${\sigma }_{\mathrm{lm}}^{\mathrm{unpol}}$ and σunpol shown in figures 3–12 are plotted as the lower solid, dashed, dotted, and dot-dashed curves in figures 13–22. Fits to the minimum experimental masses of ${{\rm{\Lambda }}}_{c}^{+}$, ${{\rm{\Sigma }}}_{c}^{++}$, ${{\rm{\Sigma }}}_{c}^{+}$, ${{\rm{\Sigma }}}_{c}^{* ++}$, and ${{\rm{\Sigma }}}_{c}^{* +}$ baryons give ${\alpha }_{{\lambda }_{{\rm{f}}}}=0.43\,\mathrm{GeV}$ and $\begin{eqnarray*}{\alpha }_{\rho }=0.222617\,\mathrm{GeV},\qquad {m}_{{{\rm{\Lambda }}}_{c}^{+}}=2.28632\,\mathrm{GeV};\end{eqnarray*}$
$\begin{eqnarray*}{\alpha }_{\rho }=0.196292\,\mathrm{GeV},\qquad {m}_{{{\rm{\Sigma }}}_{c}^{++}}=2.45383\,\mathrm{GeV};\end{eqnarray*}$
$\begin{eqnarray*}{\alpha }_{\rho }=0.196475\,\mathrm{GeV},\qquad {m}_{{{\rm{\Sigma }}}_{c}^{+}}=2.4525\,\mathrm{GeV};\end{eqnarray*}$
$\begin{eqnarray*}{\alpha }_{\rho }=0.190946\,\mathrm{GeV},\qquad {m}_{{{\rm{\Sigma }}}_{c}^{* ++}}=2.51822\,\mathrm{GeV};\end{eqnarray*}$
$\begin{eqnarray*}{\alpha }_{\rho }=0.19136\,\mathrm{GeV},\qquad {m}_{{{\rm{\Sigma }}}_{c}^{* +}}=2.5152\,\mathrm{GeV}.\end{eqnarray*}$
Using these values of αρ and ${\alpha }_{{\lambda }_{{\rm{f}}}}$, we obtain unpolarized cross sections which are denoted as ${\sigma }_{\mathrm{sm}}^{\mathrm{unpol}}$. The differences between ${\sigma }_{\mathrm{sm}}^{\mathrm{unpol}}$ and σunpol are plotted as the upper solid, dashed, dotted, and dot-dashed curves in figures 13–22. In every figure the orange (green, red, blue) band between the lower and upper solid (dashed, dotted, dot-dashed) curves show uncertainties of the unpolarized cross sections, which are labeled as σuncer and are caused by the uncertainties of the αρ value. However, the cross section uncertainties are too small to be shown if the bands are attached to those curves in figures 3–12. The cross section uncertainties are small because of the small uncertainties of the αρ values, which correspond to errors of measurement of the baryon masses.
Figure 13. The error band between the two solid (dashed, dotted, and dot-dashed) curves indicate uncertainties of the unpolarized cross sections for $p\psi (3770)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{0}$ ($p\psi (4040)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{0}$, $p\psi (4160)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{0}$, and $p\psi (4415)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{0}$). |
Figure 14. The error band between the two solid (dashed, dotted, and dot-dashed) curves indicate uncertainties of the unpolarized cross sections for $p\psi (3770)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{* 0}$ ($p\psi (4040)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{* 0}$, $p\psi (4160)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{* 0}$, and $p\psi (4415)\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{* 0}$). |
Figure 15. The error band between the two solid (dashed, dotted, and dot-dashed) curves indicate uncertainties of the unpolarized cross sections for $p\psi (3770)\to {{\rm{\Sigma }}}_{c}^{++}{D}^{-}$ ($p\psi (4040)\to {{\rm{\Sigma }}}_{c}^{++}{D}^{-}$, $p\psi (4160)\to {{\rm{\Sigma }}}_{c}^{++}{D}^{-}$, and $p\psi (4415)\to {{\rm{\Sigma }}}_{c}^{++}{D}^{-}$). |
Figure 16. The error band between the two solid (dashed, dotted, and dot-dashed) curves indicate uncertainties of the unpolarized cross sections for $p\psi (3770)\to {{\rm{\Sigma }}}_{c}^{++}{D}^{* -}$ ($p\psi (4040)\to {{\rm{\Sigma }}}_{c}^{++}{D}^{* -}$, $p\psi (4160)\to {{\rm{\Sigma }}}_{c}^{++}{D}^{* -}$, and $p\psi (4415)\to {{\rm{\Sigma }}}_{c}^{++}{D}^{* -}$). |
Figure 17. The error band between the two solid (dashed, dotted, and dot-dashed) curves indicate uncertainties of the unpolarized cross sections for $p\psi (3770)\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{0}$ ($p\psi (4040)\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{0}$, $p\psi (4160)\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{0}$, and $p\psi (4415)\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{0}$). |
Figure 18. The error band between the two solid (dashed, dotted, and dot-dashed) curves indicate uncertainties of the unpolarized cross sections for $p\psi (3770)\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{* 0}$ ($p\psi (4040)\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{* 0}$, $p\psi (4160)\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{* 0}$, and $p\psi (4415)\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{* 0}$). |
Figure 19. The error band between the two solid (dashed, dotted, and dot-dashed) curves indicate uncertainties of the unpolarized cross sections for $p\psi (3770)\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{-}$ ($p\psi (4040)\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{-}$, $p\psi (4160)\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{-}$, and $p\psi (4415)\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{-}$). |
Figure 20. The error band between the two solid (dashed, dotted, and dot-dashed) curves indicate uncertainties of the unpolarized cross sections for $p\psi (3770)\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{* -}$ ($p\psi (4040)\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{* -}$, $p\psi (4160)\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{* -}$, and $p\psi (4415)\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{* -}$). |
Figure 21. The error band between the two solid (dashed, dotted, and dot-dashed) curves indicate uncertainties of the unpolarized cross sections for $p\psi (3770)\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{0}$ ($p\psi (4040)\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{0}$, $p\psi (4160)\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{0}$, and $p\psi (4415)\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{0}$). |
Figure 22. The error band between the two solid (dashed, dotted, and dot-dashed) curves indicate uncertainties of the unpolarized cross sections for $p\psi (3770)\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{* 0}$ ($p\psi (4040)\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{* 0}$, $p\psi (4160)\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{* 0}$, and $p\psi (4415)\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{* 0}$). |
In [1–3], ψ(3770), ψ(4040), ψ(4160), and ψ(4415) mesons are identified with the 13D1, 33S1, 23D1, and 43S1 states of a charm quark and a charm antiquark. This identification is also true with the potential given in equation (33 ), and we then study inelastic scattering of a nucleon by the four $c\bar{c}$ mesons in the present work. However, we note that the quantum states of ψ(3770) and ψ(4415) mesons are open to debate. The 13D1 $c\bar{c}$ state of the ψ(3770) meson is suggested to be mixed with the 23S1 $c\bar{c}$ state in [35], and may contain a four-quark component with the up- and down-quarks and antiquarks in [36]. The ψ(4415) meson may be a 53S1 $c\bar{c}$ state given in the screened potential model [37], a 33D1 $c\bar{c}$ state obtained with a quark potential derived from a Lagrangian with chiral symmetry breaking in [38], or a $c\bar{c}$ hybrid recognized in lattice calculations of meson masses [39], from the nonrelativistic reduction of the QCD Hamiltonian in the Coulomb gauge [40], and in the flux-tube model [41].
4. Summary
Flavor interchange between a nucleon and a $c\bar{c}$ meson breaks the meson. According to the quark interchange mechanism, we have derived formulas of the transition amplitudes that include wave functions and constituent-constituent potentials. The transition amplitudes are used to calculate unpolarized cross sections for the reactions: ${pR}\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{0}$, ${pR}\to {{\rm{\Lambda }}}_{c}^{+}{\bar{D}}^{* 0}$, ${pR}\to {{\rm{\Sigma }}}_{c}^{++}{D}^{-}$, ${pR}\to {{\rm{\Sigma }}}_{c}^{++}{D}^{* -}$, ${pR}\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{0}$, ${pR}\,\to {{\rm{\Sigma }}}_{c}^{+}{\bar{D}}^{* 0}$, ${pR}\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{-}$, ${pR}\to {{\rm{\Sigma }}}_{c}^{* ++}{D}^{* -}$, ${pR}\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{0}$, and ${pR}\to {{\rm{\Sigma }}}_{c}^{* +}{\bar{D}}^{* 0}$, where R represents ψ(3770), ψ(4040), ψ(4160), or ψ(4415). These reactions are exothermic, and the $\sqrt{s}$ dependence of their cross sections is so that the cross sections decrease rapidly near threshold and change slowly when the center-of-mass energy of the nucleon and the $c\bar{c}$ meson is not close to the threshold. In the slowly-changing region the cross sections may be tens of millibarns. The cross sections also depend on nodes in the radial wave functions of the $c\bar{c}$ mesons. Numerical cross sections are parametrized. Cross sections for reactions of a neutron and a $c\bar{c}$ meson are obtained from those of a proton and the $c\bar{c}$ meson.