$I=0,1$ $\pi\pi$ and $I=1/2$ $K\pi$ Scattering using Quark Born Diagrams

We extend the quark Born diagram formalism for hadron-hadron scattering to processes with valence $q\bar q$ annihilation, specifically $I=0$ and $I=1$ $\pi\pi$ and $I=1/2$ $K\pi$ elastic scattering. This involves the $s$-channel hybrid annihilation process $q^2\bar q^2 \to q\bar q g \to q^2\bar q^2$ and conventional $s$-channel $q\bar q$ resonances, in addition to the $t$-channel gluon exchange treated previously using this formalism. The strength of the $t$-channel gluon amplitude is fixed by previous studies of $I=2$ $\pi\pi$ and $I=3/2$ $K\pi$ scattering. The $s$-channel resonances $\rho(770)$, $K^*(892)$, $f_0(1400)$ and $K_0^*(1430)$ are incorporated as relativized Breit-Wigner amplitudes, with masses and energy-dependent widths fitted to experimental phase shifts. The strength of the $s$-channel gluon ``hybrid" annihilation diagrams is problematical since the perturbative massless-gluon energy denominator must be modified to account for the effect of confinement on the energy of the virtual hybrid state. Our naive expectation is that near threshold the hybrid diagrams are comparable in magnitude but opposite in sign to the contribution predicted using massless perturbative gluons. Fitting the strength of this amplitude to $I=0$ $\pi\pi$ and $I=1/2$ $K\pi$ S-wave data gives a result consistent with this expectation. We find good agreement with experimental phase shifts from threshold to 0.9 GeV in $\pi\pi$ and 1.6 GeV in $K\pi$ using this approach. We conclude that the most important contribution to low energy S-wave scattering in these channels arises from the nonresonant quark Born diagrams, but that the low-energy wings of the broad $s$-channel resonances $f_0(1400)$ and $K^*_0(1430)$ also give important contributions near threshold. The nonresonant contributions are found to be much smaller in $L\;