External Mass Continuation and Commutation Relations inπ−πScattering

Abstract

Single-variable, subtracted dispersion relations in the external mass and energy are used to calculate the difference between the $\pi -\pi$ amplitude evaluated at physical threshold and at the unphysical zero point which occurs in the application of current-algebra techniques. All differences are relatively small. For one of the amplitudes this is in disagreement with the result obtained using expansion techniques. Further, the pseudoscalar density commutator, which lies outside the ${\mathrm{SU}}_3\times {\mathrm{SU}}_3$ algebra, is evaluated from a model using only conventional fields. The resulting current-algebra relations differ from those given by the quark model. However, the scattering lengths obtained, $a_0=+0.18\mathrm{}$ and $a_2=-0.04\mathrm{}$, are consistent with those obtained by Weinberg using expansion techniques together with the quark model. This appears to be nontrivial, since our mass extrapolation and quark-model commutation relations for the pseudoscalar density give a different result, $a_0=+0.05\mathrm{}$, $a_2=-0.09\mathrm{}$. The $\pi -\pi$ sum rule involving an unsubtracted dispersion relation is evaluated. The $\rho$ meson gives the dominant contribution, in contrast to the subtracted dispersion integrals where it gives only minor contributions. Crossing symmetry implies large background contributions from the $\rho$ in the crossed channel and when these are taken into account, the sum rule is approximately satisfied with our scattering lengths.