Role of resonances in ${\rho }^0\to {\pi }^{+}{\pi }^{-}\gamma$

We study the effect of the $\sigma (600)$ and $a_1(1260)$ resonances in the ${\rho }^0\to {\pi }^{+}{\pi }^{-}\gamma$ decay within the meson dominance model. Major effects are driven by the mass and width parameters of the $\sigma (600)$, and the usually neglected contribution of the $a_1(1260)$, although small by itself, may become sizable through its interference with pion bremsstrahlung, and the proper relative sign can favor the central value of the experimental branching ratio. We present a procedure, using the gauge invariant structure of the resonant amplitudes, to kinematically enhance the resonant effects in the angular and energy distribution of the photon. We also elaborate on the coupling constants involved.

Figure 1

Feynman diagrams that contribute to the decay ${\rho }^0\to {\pi }^{+}{\pi }^{-}\gamma$.

Figure 2

Di-pion invariant mass due to the pion bremsstrahlung for the ${\rho }^0\to {\pi }^{+}{\pi }^{-}\gamma$ decay.

Figure 3

Photon energy spectrum due to the Low emission for $\theta =85°$ and 5° (solid and dashed lines, respectively).

Figure 4

Photon energy spectrum due to the interference between electric and $\sigma$ emission for $\theta =85°$ and 5° (solid and dashed lines, respectively). Here $m_{\sigma }={\Gamma }_{\sigma }=400\mathrm{MeV}$.

Figure 5

Photon energy spectrum due to the interference between electric and $a_1$ emission for $\theta =85°$ and 5° (solid and dashed lines, respectively).