Amplitude analysis of the anomalous decay ${\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}\gamma$

In this paper we perform an amplitude analysis of ${\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}\gamma$ and confront it with the latest BESIII data. Based on the final-state interaction theorem, we represent the amplitude in terms of an Omnés function multiplied by a form factor that corresponds to the contributions from left-hand cuts and right-hand cuts in the inelastic channels. We also take into account the isospin violation effect induced by $\rho -\omega$ mixing. Our results show that the anomaly contribution is mandatory in order to explain the data. Its contribution to the decay width of $\mathrm{\Gamma }({\eta }^{\prime }\to \pi \pi \gamma )$ is larger than that induced by isospin violation. Finally we extract the pole positions of the $\rho$ and $\omega$ as well as their corresponding residues.

Figure 1

Left: Phase of the $\pi \pi$ scattering amplitude in a $P$-wave with isospin $I=1$. Right: a modulus of the Omnès function.

Figure 2

The Feynmann diagrams of the Wess-Zumino-Witten (WZW) term (a), and the isospin violation amplitudes (b)–(d). The symbol “V” represents the vector resonances $\rho$, $\omega$, and $\varphi$. The odd-intrinsic parity and isospin-violating vertices are denoted by the circled crosses and black squares, respectively. Note that in (b) the black square also represents an odd-intrinsic parity vertex.

Figure 3

The top figure corresponds to the fit to the invariant mass spectrum of ${\eta }^{\prime }\to \pi \pi \gamma$. The black solid line is the one of fit 3, the blue dotted line is of fit 1. As explained in the text, fit 2 is almost indistinguishable from fit 3 and thus not shown. The BESIII data is from [27] and Crystal Ball data from [24]. The bottom figure corresponds to the fit to the experiment data of the $\pi \pi \to \pi \pi P$-wave. The Cern-Munich data are from [58], and the olive and light grey bands in the low energy region are from Refs. [66, 67, 68, 69].

Figure 4

Comparison of our amplitudes with the ones from the Roy equation analysis in the domain where the Roy equations work. On the left side are real and imaginary parts of our amplitudes, and on the right side are those from the Roy equation [56].

Figure 5

Phase of ${\pi }^{+}{\pi }^{-}P$-wave from our amplitude analysis. The blue dashed line is from fit 1 and the black solid line from fit 3. The data sets of are from Ref. [58] (red circles), from Ref. [59] (green triangles) and from Ref. [60] (orange stars), respectively.