Diagrammatic approach to meson-meson scattering in the nonrelativistic quark potential model

In this paper we use Born-order quark-exchange diagrams in a nonrelativistic potential model to describe low-energy scattering of $q\overline{q}$ mesons. A formalism for evaluating quark Born diagrams is developed, and as a first application we consider meson-meson scattering in channels in which $q\overline{q}$ annihilation is thought to be unimportant. In particular our results are relevant to $I=2\mathrm{} \pi \pi$ and $I=1\mathrm{} \mathrm{KK}$ elastic scattering. Simple rules for the Born diagrams are given, which allow the evaluation of scattering amplitudes in terms of external meson wave functions by inspection. These techniques are applied to systems having identical quarks, and ${\pi }^{+}{\pi }^{+}$, $K^{+}{K}^{+}$, and ${\rho }^{+}{\rho }^{+}$ elastic scattering phase shifts, cross sections, and equivalent potentials are derived as examples. The $S$-wave $I=2\mathrm{} \pi \pi$ phase shift for a Gaussian $q\overline{q}$ wave function with conventional quark model parameters ${\alpha }_s$, $m_q$, and ${\beta }_{\mathrm{SHO}}$ is found to be in good agreement with experiment and with Weinberg's PCAC (partial conservation of axial-vector current) result. At higher energies the predicted differential cross sections have the characteristic diffractive features of an exponential $t$ peak at small angles and vacuum quantum number exchange. The phase of the predicted amplitude however differs from the experimental diffractive amplitude, so these quark Born diagrams cannot be directly identified with the "Pomeron" of diffractive scattering phenomenology.