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To measure the helicity of a spin-1 meson from the triple vector product ofthe three-momenta of its decay products, one needs information about the strongphase of the decay amplitude. In this paper, taking $a_1(1260)$ meson as anexample, we present a method to extract information about the strong phase fromthe triple vector product of the pion momenta in $W \to \nu \tau(\to \nua_1(\to\pi^\mp \pi^\mp \pi^\pm))$ process, where the $a_1$ helicity is known apriori from electroweak theory. This process is advantageous in thathighly-boosted $a_1^-$ mesons from $\tau_L^-$ decays have nearly maximalhelicity asymmetry and thus most reflect the strong phase. We revisit thetheoretical calculation of the $a_1$ meson helicity in $W \to \nu \tau(\to \nua_1)$ process. We formulate the differential decay rate of polarized $a_1$mesons in a manner convenient for the study of the $a_1$ meson helicityasymmetry. Finally, we present the method for extracting information about thestrong phase, and assess its feasibility at the LHC.

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