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Low-Energy Pion-Photon Interaction: The ( 2 π , 2 γ ) Vertex

1961; American Institute of Physics; Volume: 124; Issue: 4 Linguagem: Inglês

10.1103/physrev.124.1248

1536-6065

Bipin R. Desai,

Particle physics theoretical and experimental studies

In the ($2\ensuremath{\pi}, 2\ensuremath{\gamma}$) problem, the Mandelstam representation is written for the two independent gauge-invariant amplitudes. On the basis of unitarity limitations on the asymptotic behavior of these amplitudes, only a $j=1$ subtraction in the $\ensuremath{\gamma}+\ensuremath{\pi}\ensuremath{\rightarrow}\ensuremath{\gamma}+\ensuremath{\pi}$ channel and a $j=0$ subtraction in the $\ensuremath{\gamma}+\ensuremath{\gamma}\ensuremath{\rightarrow}\ensuremath{\pi}+\ensuremath{\pi}$ channel are allowed. No over-all subtraction constants are required and the Thomson limit is automatically maintained. Only the effect of $2\ensuremath{\pi}$ intermediate states is considered. The odd-$j$ $\ensuremath{\pi}\ensuremath{\pi}$ contribution involves the amplitude for the process $\ensuremath{\gamma}+\ensuremath{\pi}\ensuremath{\rightarrow}2\ensuremath{\pi}$ analyzed by Wong and shown to be proportional to a pseudo-elementary constant $\ensuremath{\Lambda}$. Even with a $\ensuremath{\pi}\ensuremath{\pi}$ $P$ resonance, the correction is negligible (\ensuremath{\lesssim}1%) if we use the value of $\ensuremath{\Lambda}$ estimated by Wong on the basis of ${\ensuremath{\pi}}^{0}$ decay and confirmed by Ball in connection with photopion production on nucleons. A moderately important contribution comes from the $S$-wave interaction if we use a recent estimate of $\ensuremath{\pi}\ensuremath{\pi}$ $S$-wave phase shifts obtained from crossing relations. For the pion-pion coupling constant $\ensuremath{\lambda}$ of order -0.20, this effect is \ensuremath{\sim}10% in $\ensuremath{\gamma}+\ensuremath{\pi}\ensuremath{\rightarrow}\ensuremath{\gamma}+\ensuremath{\pi}$ scattering. For $\ensuremath{\gamma}+\ensuremath{\gamma}\ensuremath{\rightarrow}\ensuremath{\pi}+\ensuremath{\pi}$, the correction for the $I=0$ state at threshold is positive and \ensuremath{\sim}100% of the Born approximation. However, as the energy is increased, the correction quickly changes sign.