Production Effects and Final-state Interactions in $π_1 \to 3π$

We investigate the $\pi_1(1600)$ signal channel of $3\pi$ in the $J^{PC} = 1^{-+} \,[\pi\pi]_{1^{--}}\pi~P$-wave. This channel has recently been analyzed by the COMPASS collaboration using the ``freed-isobar" technique which provides direct experimental access to the lineshape modifications of the $P$-wave $\pi\pi$ sub-channel arising from both final state interactions and production processes such as the Deck mechanism. We motivate a unified formalism combining production effects and final state interactions using the Khuri-Treiman formalism, which is consistent with low-energy unitarity, analyticity, and crossing symmetry, seeded by an initial production amplitude. We demonstrate that the precise lineshape of the $\pi\pi$ mass spectrum can be used to determine the relative strength and complex phase of different production mechanisms contributing to the $3\pi$ final state. We conduct a global fit of the $\pi_1$ freed-isobar data set, yielding a multi-dimensional parameterization of the $\pi_1\to3\pi$ decay amplitude at COMPASS as a function of both decay and production variables. We identify the contributions from short-range production and the Deck mechanism and extract the total $3\pi$ invariant mass dependence which will be important for a future extraction of the $\pi_1$ pole position in the $\rho\pi$ channel.