Determining Strong Phase of $a_1$ Meson Decay Amplitude using $W \to \nu \tau(\to \nu a_1(\to\pi^\mp \pi^\mp \pi^\pm))$ Process.

To measure the helicity of a spin-1 meson from the triple vector product of the three-momenta of its decay products, one needs information about the strong phase of the decay amplitude. In this paper, taking $a_1(1260)$ meson as an example, we present a method to extract information about the strong phase from the triple vector product of the pion momenta in $W \to \nu \tau(\to \nu a_1(\to\pi^\mp \pi^\mp \pi^\pm))$ process, where the $a_1$ helicity is known a priori from electroweak theory. This process is advantageous in that highly-boosted $a_1^-$ mesons from $\tau_L^-$ decays have nearly maximal helicity asymmetry and thus most reflect the strong phase. We revisit the theoretical calculation of the $a_1$ meson helicity in $W \to \nu \tau(\to \nu a_1)$ process. We formulate the differential decay rate of polarized $a_1$ mesons in a manner convenient for the study of the $a_1$ meson helicity asymmetry. Finally, we present the method for extracting information about the strong phase, and assess its feasibility at the LHC.