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Unsubtracted Pion-Pion Dispersion Relation

Jerrold Franklin

Open publisher page 5 citations

Abstract

The de Alfaro-Fubini-Furlan-Rossetti assumption on the rapid falloff of amplitudes corresponding to pure $I=2$, $t$-channel exchange is applied to pion-pion forward dispersion relations, and a sum rule is obtained for the pion-pion scattering lengths. The sum rule is fairly well satisfied using the $\ensuremath{\rho}$, $f$, and low-energy $s$-wave contributions; but the inclusion of the recently discovered ${g}_{1}$ resonance requires the existence of at least one new $I=0$ resonance to satisfy the sum rule.

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What this paper is about

The de Alfaro-Fubini-Furlan-Rossetti assumption on the rapid falloff of amplitudes corresponding to pure $I=2$, $t$-channel exchange is applied to pion-pion forward dispersion relations, and a sum rule is obtained for the pion-pion scattering lengths. The sum rule is fairly well satisfied using the $\ensuremath{\rho}$, $f$, and low-energy $s$-wave contributions; but the inclusion of the recently discovered ${g}_{1}$ resonance requires the existence of at least one new $I=0$ resonance to satisfy the sum rule.

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Available abstract

The de Alfaro-Fubini-Furlan-Rossetti assumption on the rapid falloff of amplitudes corresponding to pure $I=2$, $t$-channel exchange is applied to pion-pion forward dispersion relations, and a sum rule is obtained for the pion-pion scattering lengths. The sum rule is fairly well satisfied using the $\ensuremath{\rho}$, $f$, and low-energy $s$-wave contributions; but the inclusion of the recently discovered ${g}_{1}$ resonance requires the existence of at least one new $I=0$ resonance to satisfy the sum rule.

Key concepts: Pion, Physics, Sum rule in quantum mechanics, Dispersion relation, Resonance (particle physics), Scattering, Particle physics, Fubini's theorem

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