Cross sections for inelastic meson-meson scattering via quark-antiquark annihilation

We study inelastic meson-meson scattering that is governed by quark-antiquark annihilation and creation involving a quark and an antiquark annihilating into a gluon, and subsequently the gluon creating another quark-antiquark pair. The resultant hadronic reactions include for $I=1$: $\pi \pi \to \rho \rho$, $K\overline{K}\to K^{*}{\overline{K}}^{*}$, $K{\overline{K}}^{*}\to K^{*}{\overline{K}}^{*}$, $K^{*}\overline{K}\to K^{*}{\overline{K}}^{*}$, as well as $\pi \pi \to K\overline{K}$, $\pi \rho \to K{\overline{K}}^{*}$, $\pi \rho \to K^{*}\overline{K}$, and $K\overline{K}\to \rho \rho$. In each reaction, one or two Feynman diagrams are involved in the Born approximation. We derive formulas for the unpolarized cross section, the transition amplitude, and the transition potential for quark-antiquark annihilation and creation. The unpolarized cross sections for the reactions are calculated at six temperatures, and prominent temperature dependence is found. It is due to differences among mesonic temperature dependence in hadronic matter.

Figure 1

Left diagram with $q_1+{\overline{q}}_2\to q_3+{\overline{q}}_4$ and right diagram with ${\overline{q}}_1+q_2\to q_3+{\overline{q}}_4$ for $A+B\to C+D$. The vertical lines represent spectator quarks (antiquarks) of the mesons $A$, $B$, $C$, $D$.

Figure 2

Quark-antiquark annihilation and creation: $q(p_1)+\overline{q}(-p_2)\to q^{\prime }(p_3)+{\overline{q}}^{\prime }(-p_4)$, where $k$ denotes the gluon four-momentum, $e$ its color index, and $\tau$ its space-time index [cf. Eq. (39)].

Figure 3

$P$-wave $I=1$ elastic phase shifts for $\pi \pi$ scattering. The solid curve is the present theoretical result. Experimental data: $⊲$, Ref. [28]; $⊳$, Ref. [29]; $♢$, Ref. [30]; $△$, Ref. [31]; $▽$, Ref. [33]; $○$, Ref. [34]; $\times$, Ref. [36].

Figure 4

“Prior” diagrams with antiquark interchange. Solid (dot-dashed) lines represent quarks or antiquarks (interaction).

Figure 5

“Post” diagrams with antiquark interchange. Solid (dot-dashed) lines represent quarks or antiquarks (interaction).

Figure 6

$S$-wave $I=0$ elastic phase shifts for $\pi \pi$ scattering. The solid curve is the present theoretical result. Experimental data: $♢$, Ref. [30]; $△$, Ref. [31]; $\square$, Ref. [32]; $▽$, Ref. [33]; $○$, Ref. [34]; +, Ref. [35]; $\times$, Ref. [36].

Figure 7

Cross sections for $\pi \pi \to \rho \rho$ for $I=1$ at various temperatures.

Figure 8

Cross sections for $K\overline{K}\to K^{*}{\overline{K}}^{*}$ for $I=1$ at various temperatures.

Figure 9

Cross sections for $K\overline{K}\to K^{*}{\overline{K}}^{*}$ for $I=0$ at various temperatures.

Figure 10

Cross sections for $K{\overline{K}}^{*}\to K^{*}{\overline{K}}^{*}$ for $I=1$ at various temperatures.

Figure 11

Cross sections for $K{\overline{K}}^{*}\to K^{*}{\overline{K}}^{*}$ for $I=0$ at various temperatures.

Figure 12

Cross sections for $\pi \pi \to K\overline{K}$ for $I=1$ at various temperatures.

Figure 13

Cross sections for $\pi \rho \to K{\overline{K}}^{*}$ for $I=1$ at various temperatures.

Figure 14

Cross sections for $\pi \rho \to K^{*}\overline{K}$ for $I=1$ at various temperatures.

Figure 15

Cross sections for $K\overline{K}\to \rho \rho$ for $I=1$ at various temperatures.

Figure 16

Cross sections for $\pi \pi \to \rho \rho$ for $I=0$ at various temperatures.

Figure 17

“prior” and “post” diagrams with quark interchange. Solid (dot-dashed) lines represent quarks or antiquarks (interaction).