6533b851fe1ef96bd12aa035

RESEARCH PRODUCT

Dispersion relation bounds forππscattering

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description

Axiomatic principles such as analyticity, unitarity, and crossing symmetry constrain the second derivative of the $\ensuremath{\pi}\ensuremath{\pi}$ scattering amplitudes in some channels to be positive in a region of the Mandelstam plane. Since this region lies in the domain of validity of chiral perturbation theory, we can use these positivity conditions to bound linear combinations of ${\overline{l}}_{1}$ and ${\overline{l}}_{2}$. We compare our predictions with those derived previously in the literature using similar methods. We compute the one-loop $\ensuremath{\pi}\ensuremath{\pi}$ scattering amplitude in the linear sigma model (LSM) using the $\overline{\mathrm{MS}}$ scheme, a result hitherto absent in the literature. The LSM values for ${\overline{l}}_{1}$ and ${\overline{l}}_{2}$ violate the bounds for small values of ${m}_{\ensuremath{\sigma}}/{m}_{\ensuremath{\pi}}$. We show how this can occur, while still being consistent with the axiomatic principles.