Two loop soft function for secondary massive quarks

We present the calculation of the $\mathcal{O}(α_s^2 C_F T_F)$ massive quark corrections to the soft function for the double hemisphere jet mass distribution in $e^+ e^-$ collisions, a necessary ingredient for the calculation of several event shape distributions at N${}^3$LL order. Using the mass as an infrared regulator allows us to derive an expression for the massless momentum space limit, which has not been given so far in the literature. Furthermore, we compute the corresponding corrections in the soft function for thrust, the most prominent projection of the double hemisphere mass distribution. Finally we give expressions for the corresponding renormalon subtractions in the gap scheme.

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