Peak locations and relative phase of different decay modes of the $a_1$ axial vector resonance in diffractive production

We show that a single $I=1$ spin-parity $J^{PC}=1^{++}$ $a_1$ resonance can manifest itself as two separated mass peaks, one decaying into an S-wave $\rho \pi$ system and the second decaying into a P-wave $f_0(980)\pi$ system, with a rapid increase of the phase difference between their amplitudes arising mainly from the structure of the diffractive production process. This study clarifies questions related to the mass, width, and decay rates of the $a_1$ resonance raised by the recent high statistics data of the COMPASS collaboration on $a_1$ production in $\pi N\to \pi \pi \pi N$ at high energies.