Evasion of HSR in S-wave charmonium decaying to P-wave light hadrons

Abstract

The S-wave charmonium decaying to a P-wave and S-wave light hadron pairs are supposed to be suppressed by the helicity selection rule in the perturbative QCD framework. With an effective Lagrangian method, we show that the intermediate charmed meson loops can provide a possible mechanism for the evasion of the helicity selection rule, and result in sizeable decay branching ratios in some of those channels. The theoretical predictions can be examined by the forthcoming BES-III data in the near future.

Appendices

Appendix A: Long-distance transition amplitudes for η _c →T+V

The explicit transition amplitudes of η _c (p)→D ^∗(q ₁)D ^(∗)(q ₃)[D ^∗(q ₂)]→T(p ₁)V(p ₂) via charged charmed-meson loops in Fig. 3 are given as follows:

$$\begin{aligned} &M_{DD^* [D^*]} \\ &\quad = (i)^3\int \frac {d^4q_2}{(2\pi)^4}\bigl[ g_{\eta_c D^*D}(p+q_3)_\mu \bigr] \bigl[g_{D^*D^*T} \phi_{\alpha\beta}^*\bigr] \\ &\qquad \times \bigl[-4 g_{D^*DV} \varepsilon_{\rho\sigma\xi\tau} p_{2}^\rho \varepsilon_2^{*\sigma} q_2^\xi \bigr] \\ &\qquad \times \frac {i(-g^{\mu\alpha} +q_1^\mu q_1^\alpha/m_1^2)}{q_1^2-m_1^2} \frac {i(-g^{\beta\tau} +q_2^\beta q_2^\tau/m_2^2)}{q_2^2-m_2^2} \\ &\qquad \times \frac {i}{q_3^2-m_3^2} \mathcal{F} \bigl(m_i, q_i^2\bigr) , \\ &M_{D^*D^* [D^*]}\\ &\quad = (i)^3\int \frac {d^4q_2}{(2\pi)^4}\bigl[ g_{\eta_c D^*D^*}\varepsilon_{\mu\nu\alpha\beta} q_1^\mu p^\beta \bigr] \bigl[g_{D^*D^*T} \phi_{\rho\sigma}^*\bigr] \\ &\qquad \times \bigl[ g_{D^*D^*V}(q_2-q_3)_\eta \varepsilon_2^{*\eta} g_{\xi\tau} \\ &\qquad + 4f_{D^*D^*V} \bigl( p_{2\tau} \varepsilon_{2\xi}^* - p_{2\xi} \varepsilon_{2\tau}^*\bigr)\bigr] \\ &\qquad \times \frac {i(-g^{\nu\rho} +q_1^\nu q_1^\rho/m_1^2)}{q_1^2-m_1^2} \frac {i(-g^{\sigma\xi} +q_2^\sigma q_2^\xi/m_2^2)}{q_2^2-m_2^2} \\ &\qquad \times \frac {i(-g^{\alpha\tau} +q_3^\alpha q_3^\tau /m_3^2)}{q_3^2-m_3^2} \mathcal{F} \bigl(m_i, q_i^2\bigr) . \end{aligned}$$

(A.1)

Fig. 3

Schematic picture for η _c →TV via charmed meson loops. Here, T and V denote the light tensor and vector mesons, respectively. Similar diagrams are for neutral and strange charmed mesons. Similar pictures occur in $\eta_{c}^{\prime}\to TV$

Appendix B: Long-distance transition amplitudes for J/ψ→h+V

The explicit transition amplitudes of J/ψ(p)→D ^(∗)(q ₁)D ^(∗)(q ₃)[D ^(∗)(q ₂)]→ h(p ₁)V(p ₂) via charged charmed-meson loops in Fig. 4 are given as follows:

$$\begin{aligned} &M_{DD [D^*]} \\ &\quad = (i)^3\int \frac {d^4q_2}{(2\pi)^4}\bigl[ -g_{\psi D^*D}(q_3-q_1)_\mu \varepsilon^\mu \bigr] \bigl[g_{D^*D h} \varepsilon_1^{*\nu} \bigr] \\ &\qquad \times \bigl[-4 g_{D^*DV} \varepsilon_{\rho\sigma\xi\tau} p_{2}^\rho \varepsilon_2^{*\sigma} q_2^\xi \bigr] \\ &\qquad \times \frac {i}{q_1^2-m_1^2} \frac {i(-g^{\nu\tau} +q_2^\nu q_2^\tau/m_2^2)}{q_2^2-m_2^2} \frac {i}{q_3^2-m_3^2} \mathcal{F} \bigl(m_i, q_i^2\bigr) , \\ &M_{DD^* [D^*]} \\ &\quad = (i)^3\int \frac {d^4q_2}{(2\pi)^4}\bigl[ g_{\psi D^*D}\varepsilon_{\mu\nu\alpha\beta} p^\mu \varepsilon^\nu (q_3-q_1)^\alpha \bigr] \\ &\qquad \times\bigl[g_{D^*Dh} \varepsilon_1^{*\rho}\bigr] \\ &\qquad \times \bigl[ g_{D^*D^*V}(q_2-q_3)_\eta \varepsilon_2^{*\eta} g_{\kappa\lambda} \\ &\qquad + 4f_{D^*D^*V} \bigl( p_{2\kappa} \varepsilon_{2\lambda}^* - p_{2\lambda} \varepsilon_{2\kappa}^*\bigr)\bigr] \\ &\qquad \times \frac {i}{q_1^2-m_1^2} \frac {i(-g^{\rho\lambda} +q_2^\rho q_2^\lambda/m_2^2)}{q_2^2-m_2^2} \\ &\qquad \times\frac {i(-g^{\beta\kappa} +q_3^\beta q_3^\kappa /m_3^2)}{q_3^2-m_3^2} \mathcal{F} \bigl(m_i, q_i^2\bigr) , \\ &M_{D^*D [D]} \\ &\quad = (i)^3\int \frac {d^4q_2}{(2\pi)^4}\bigl[ g_{\psi D^*D}\varepsilon_{\mu\nu\alpha\beta} p^\mu \varepsilon^\nu (q_3-q_1)^\alpha \bigr] \\ &\qquad \times\bigl[g_{D^*Dh} \varepsilon_1^{*\rho}\bigr] \bigl[ g_{DDV}(q_2-q_3)_\lambda \varepsilon_2^{*\lambda} \bigr] \\ &\qquad \times \frac {i(-g^{\beta\rho} +q_1^\beta q_1^\rho/m_1^2)}{q_1^2-m_1^2} \frac {i}{q_2^2-m_2^2} \frac {i}{q_3^2-m_3^2} \mathcal{F} \bigl(m_i, q_i^2\bigr) , \\ &M_{D^*D [D^*]} \\ &\quad = (i)^3\int \frac {d^4q_2}{(2\pi)^4}\bigl[ g_{\psi D^*D}\varepsilon_{\mu\nu\alpha\beta} p^\mu \varepsilon^\nu (q_3-q_1)^\alpha \bigr] \\ &\qquad \times\bigl[g_{D^*D^*h} \varepsilon_{\rho\sigma\xi\tau} \varepsilon_1^{*\sigma} p_1^\xi \bigr] \bigl[ -4g_{D^*DV} \varepsilon_{\lambda\eta \kappa \theta} p_2^\lambda \varepsilon_2^{*\eta} q_2^\kappa \bigr] \\ &\qquad \times \frac {i(-g^{\beta\rho} +q_1^\beta q_1^\rho/m_1^2)}{q_1^2-m_1^2} \frac {i(-g^{\tau\theta} +q_2^\tau q_2^\theta/m_2^2)}{q_2^2-m_2^2} \\ &\qquad \times \frac {i}{q_3^2-m_3^2} \mathcal{F} \bigl(m_i, q_i^2\bigr) , \\ &M_{D^*D^* [D]} \\ &\quad = (i)^3\int \frac {d^4q_2}{(2\pi)^4}\bigl[ g_{\psi D^*D^*}(g_{\mu\nu}g_{\alpha\beta} \\ &\qquad - g_{\mu\beta}g_{\nu\alpha} + g_{\mu\alpha}g_{\nu\beta}) \varepsilon^\mu (q_3-q_1)^\beta \bigr] \bigl[g_{D^*Dh} \varepsilon_{1\rho}^*\bigr] \\ &\qquad \times \bigl[ -4g_{D^*DV}\varepsilon_{\lambda\eta \kappa \theta} p_2^\lambda \varepsilon_2^{*\eta} q_2^\kappa \bigr]\frac {i(-g^{\alpha\rho} +q_1^\alpha q_1^\rho/m_1^2)}{q_1^2-m_1^2} \\ &\qquad \times \frac {i}{q_2^2-m_2^2} \frac {i(-g^{\nu\theta} +q_3^\nu q_3^\theta/m_3^2)}{q_3^2-m_3^2} \mathcal{F}\bigl(m_i, q_i^2 \bigr) , \\ &M_{D^*D^* [D^*]} \\ &\quad = (i)^3\int \frac {d^4q_2}{(2\pi)^4}\bigl[ g_{\psi D^*D^*}(g_{\mu\nu}g_{\alpha\beta}- g_{\mu\beta}g_{\nu\alpha} \\ &\qquad + g_{\mu\alpha}g_{\nu\beta}) \varepsilon^\mu (q_3-q_1)^\beta \bigr] \bigl[g_{D^*D^*h} \varepsilon_{\rho\sigma\xi\tau} \varepsilon_1^{*\sigma} p_1^\xi \bigr] \\ &\qquad \times \bigl[ g_{D^*D^*V}(q_2-q_3)_\eta \varepsilon_2^{*\eta} g_{\kappa\lambda} \\ &\qquad + 4f_{D^*D^*V} \bigl( p_{2\kappa} \varepsilon_{2\lambda}^* - p_{2\lambda} \varepsilon_{2\kappa}^*\bigr)\bigr] \\ &\qquad \times \frac {i(-g^{\alpha\rho} +q_1^\alpha q_1^\rho/m_1^2)}{q_1^2-m_1^2} \frac {i(-g^{\tau\lambda} +q_2^\tau q_2^\lambda/m_2^2)}{q_2^2-m_2^2} \\ &\qquad \times\frac {i(-g^{\nu\kappa} +q_3^\nu q_3^\kappa/m_3^2)}{q_3^2-m_3^2} \mathcal{F} \bigl(m_i, q_i^2\bigr) . \end{aligned}$$

(B.1)

Fig. 4

Schematic picture for J/ψ→ h+V via charmed meson loops. Similar diagrams are for neutral and strange charmed mesons. Similar pictures occur in ψ′→ h+V. Here h denotes the axial-vector meson ¹ P ₁ states

Appendix C: Long-distance transition amplitudes for J/ψ→A+V

The explicit transition amplitudes of J/ψ(p)→D ^(∗)(q ₁)D ^(∗)(q ₃)[D ^(∗)(q ₂)]→ A(p ₁)V(p ₂) via charged charmed-meson loops in Fig. 5 are given as follows:

$$\begin{aligned} &M_{DD [D^*]} \\ &\quad = (i)^3\int \frac {d^4q_2}{(2\pi)^4}\bigl[ -g_{\psi D^*D}(q_3-q_1)_\mu \varepsilon^\mu \bigr] \\ &\qquad \times\bigl[g_{D^*D A} \varepsilon_1^{*\nu} \bigr] \bigl[-4 g_{D^*DV} \varepsilon_{\rho\sigma\xi\tau} p_{2}^\rho \varepsilon_2^{*\sigma} q_2^\xi \bigr] \\ &\qquad \times \frac {i}{q_1^2-m_1^2} \frac {i(-g^{\nu\tau} +q_2^\nu q_2^\tau/m_2^2)}{q_2^2-m_2^2} \frac {i}{q_3^2-m_3^2} \mathcal{F} \bigl(m_i, q_i^2\bigr) , \\ &M_{DD^* [D^*]} \\ &\quad = (i)^3\int \frac {d^4q_2}{(2\pi)^4}\bigl[ g_{\psi D^*D}\varepsilon_{\mu\nu\alpha\beta} p^\mu \varepsilon^\nu (q_3-q_1)^\alpha \bigr] \\ &\qquad \times\bigl[g_{D^*DA} \varepsilon_1^{*\rho}\bigr] \bigl[ g_{D^*D^*V}(q_2-q_3)_\eta \varepsilon_2^{*\eta} g_{\kappa\lambda} \\ &\qquad + 4f_{D^*D^*V} \bigl( p_{2\kappa} \varepsilon_{2\lambda}^* - p_{2\lambda} \varepsilon_{2\kappa}^*\bigr)\bigr] \\ &\qquad \times \frac {i}{q_1^2-m_1^2} \frac {i(-g^{\rho\lambda} +q_2^\rho q_2^\lambda/m_2^2)}{q_2^2-m_2^2} \\ &\qquad \times \frac {i(-g^{\beta\kappa} +q_3^\beta q_3^\kappa /m_3^2)}{q_3^2-m_3^2} \mathcal{F} \bigl(m_i, q_i^2\bigr) , \\ &M_{D^*D [D]}\\ &\quad = (i)^3\int \frac {d^4q_2}{(2\pi)^4}\bigl[ g_{\psi D^*D}\varepsilon_{\mu\nu\alpha\beta} p^\mu \varepsilon^\nu (q_3-q_1)^\alpha \bigr] \\ &\qquad \times\bigl[g_{D^*DA} \varepsilon_1^{*\rho}\bigr] \bigl[ g_{DDV}(q_2-q_3)_\lambda \varepsilon_2^{*\lambda} \bigr] \\ &\qquad \times \frac {i(-g^{\beta\rho} +q_1^\beta q_1^\rho/m_1^2)}{q_1^2-m_1^2} \frac {i}{q_2^2-m_2^2} \\ &\qquad \times \frac {i}{q_3^2-m_3^2} \mathcal{F} \bigl(m_i, q_i^2\bigr) , \\ &M_{D^*D^* [D]} \\ &\quad = (i)^3\int \frac {d^4q_2}{(2\pi)^4}\bigl[ g_{\psi D^*D^*}(g_{\mu\nu}g_{\alpha\beta}- g_{\mu\beta}g_{\nu\alpha} \\ &\qquad + g_{\mu\alpha}g_{\nu\beta}) \varepsilon^\mu (q_3-q_1)^\beta \bigr] \\ &\qquad \times \bigl[g_{D^*DA} \varepsilon_{1\rho}^*\bigr] \bigl[ -4g_{D^*DV}\varepsilon_{\lambda\eta \kappa \theta} p_2^\lambda \varepsilon_2^{*\eta} q_2^\kappa \bigr] \\ &\qquad \times \frac {i(-g^{\alpha\rho} +q_1^\alpha q_1^\rho/m_1^2)}{q_1^2-m_1^2} \\ &\qquad \times\frac {i}{q_2^2-m_2^2} \frac {i(-g^{\nu\theta} +q_3^\nu q_3^\theta/m_3^2)}{q_3^2-m_3^2} \mathcal{F} \bigl(m_i, q_i^2\bigr) . \end{aligned}$$

(C.1)

Fig. 5

Schematic picture for J/ψ→ A+V via charmed meson loops. Similar diagrams are for neutral and strange charmed mesons. Similar pictures occur in ψ′→ A+V. Here A denotes the axial-vector meson ³ P ₁ states

Fig. 6

The branching ratios of J/ψ→f ₁ ϕ (solid line) and $J/\psi\to f_{1}^{\prime}\omega$ (dashed line) via charmed meson loops in terms of f ₁–$f_{1}^{\prime}$ mixing angle α _A with α=0.78