Dispersion Relation Bounds for pi pi Scattering

Abstract

Axiomatic principles such as analyticity, unitarity and crossing symmetry constrain the second derivative of the pi 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 \bar{l}_1 and \bar{l}_2. We compare our predictions with those derived previously in the literature using similar methods. We compute the one-loop pi pi scattering amplitude in the linear sigma model (LSM) using the MS-bar scheme, a result hitherto absent in the literature. The LSM values for \bar{l}_1 and \bar{l}_2 violate the bounds for small values of m_sigma/m_pi. We show how this can occur, while still being consistent with the axiomatic principles.

12 pages, 8 figures. Two references added, a few minor changes. Published version

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• ππscattering
• Elastic $\pi \pi$ scattering to two loops
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• Consistency of the Chiral Pion-Pion Scattering Amplitudes with Axiomatic Constraints
• Pion-pion scattering at low energy
• The Chiral Lagrangian parameters, ${\overline{\ell }}_1$, ${\overline{\ell }}_2$, are determined by the $\rho$--resonance
• Falsifying Models of New Physics Via WW Scattering
• Constraints on Chiral Perturbation Theory Parameters from QCD Inequalities
• Positivity Constraints on Chiral Perturbation Theory Pion-Pion Scattering Amplitudes
• Universal Bounds for SU(3) Low Energy Constants
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