Account for axial vector mesons in the $η\toπ^+π^-γ$ and $η' \toπ^+π^-γ$ decays

Abstract

The rates and spectra of the anomalous η → π + π − γ and η → π + π − γ decays are calculated. The approach is based on the effective meson Lagrangian obtained in the Nambu-Jona-Lasinio model with vector and axial-vector mesons by integrating out quark fields. The resulting action is affected by mixing between members of pseudoscalar J P C = 0 −+ and axial-vector 1 ++ nonets that violates some low-energy theorems. In this note we point out that a gauge covariant procedure to diagonalize this mixing allows for consistent description of the η → π + π − γ and η → π + π − γ decays.

Gauge-covariant diagonalization of $π$-$a_1$ mixing and the resolution of a low energy theorem

“…This is not a trivial task, in [8] it has been reported that in a number of well-known models [9]- [16] the πa 1 -mixing breaks low-energy theorems (LET) for some anomalous processes, e.g., γ → 3π, K + K − → 3π. In [17], based on the gauge covariant treatment of πa 1 -mixing, only recently addressed [18]- [22], we show precisely how the deviation of the complete VMD occurs in the framework of the NJL Lagrangian, fulfilling the LET…”

“…where π = τ i π i , a µ = τ i a i µ and τ i are the SU (2) Pauli matrices; g ρ √ 12π is the coupling of the ρ meson to two pions and f π 93MeV the pion weak decay constant. In extensions of the model that couple to the electroweak sector this replacement violates gauge invariance [18]- [22] in anomalous processes, leaving however the real part of the action invariant [20,21]. For example the anomalous low energy amplitude describing the a 1 → γπ + π − decay is not transverse [18,19].To restore gauge invariance the gauge covariant derivative D µ π must be used instead of ∂ µ π [18]- [22]…”