$a_1(1420)$ peak as the $\pi f_0(980)$ decay mode of the $a_1(1260)$

We study the decay mode of the $a_1(1260)$ into a ${\pi }^{+}$ in $p$ wave and the $f_0(980)$ that decays into ${\pi }^{+}{\pi }^{-}$ in $s$ wave. The mechanism proceeds via a triangular mechanism where the $a_1(1260)$ decays into $K^{*}\overline{K}$, the $K^{*}$ decays to an external ${\pi }^{+}$ and an internal $K$ that fuses with the $\overline{K}$ producing the $f_0(980)$ resonance. The mechanism develops a singularity at a mass of the $a_1(1260)$ around 1420 MeV, producing a peak in the cross section of the $\pi p$ reaction, used to generate the mesonic final state, which provides a natural explanation of all the features observed in the COMPASS experiment, where a peak observed at this energy is tentatively associated to a new resonance called $a_1(1420)$. On the other hand, the triangular singularity studied here gives rise to a remarkable feature, where a peak is seen for a certain decay channel of a resonance at an energy about 200 MeV higher than its nominal mass.

Figure 1

Feynman diagrams for the process $a_1^{+}(1260)\to {\pi }^{+}{\pi }^{+}{\pi }^{-}$, where the $a_1^{+}(1260)$ decays to ${\overline{K}}^{*}K$ [diagrams (a) and (b)] and to $\rho \pi$ [diagrams (c) and (d)].

Figure 2

Momenta assignment for the decay.

Figure 3

$d\mathrm{\Gamma }/d{M}_{\mathrm{inv}}$ for $a_1^{+}(1260)\to {\pi }^{+}{\pi }^{+}{\pi }^{-}$ decay as a function of $M_{\mathrm{inv}}$ in the region of the $f_0(980)$ considering only the ${\overline{K}}^{*}K$ loop diagrams (solid line), only the $\rho \pi$ loop diagrams (dashed line) and all the contributions (thick line). The widths of the $K^{*}$ and $\rho$ are removed in Eqs. (10) and (11).

Figure 4

$d\mathrm{\Gamma }/d{M}_{\mathrm{inv}}$ for $a_1^{+}(1260)\to {\pi }^{+}{\pi }^{+}{\pi }^{-}$ considering the contributions coming from the four diagrams of Fig. 1 for different values of the mass of the $a_1^{+}$. The widths of the $K^{*}$ and $\rho$ in Eqs. (10) and (11) are not taken into account.

Figure 5

Decay widths for the processes $a_1^{+}(1260)\to {\pi }^{+}{f}_0(980)$ (solid line) and $a_1^{+}(1260)\to {\pi }^{+}{\rho }^0$ (dashed line) as a function of the mass of the $a_1^{+}$.

Figure 6

Production of the $a_1(1260)$ by scattering of a high energy pion off a proton.

Figure 7

Cross section for the decay $a_1^{+}(1260)\to {\pi }^{+}{f}_0(980)$ as a function of the center of mass energy.

Figure 8

Cross section for the decay $a_1^{+}(1260)\to {\pi }^{+}{\rho }^0$ as a function of the center of mass energy.