Branch-cut in the shear-stress response function of massless λφ4 with Boltzmann statistics

Abstract

Using an analytical result for the eigensystem of the linearized collision term for a classical system of massless scalar particles with quartic self-interactions, we show that the shear-stress linear response function possesses a branch-cut singularity that covers the whole positive imaginary semiaxis. This is demonstrated in two ways: (1) by truncating the exact, infinite system of linear equations for the rank-two tensor modes, which reveals the cut touching the origin; and (2) by employing the Trotterization techniques to invert the linear response problem. The former shows that the first pole tends toward the origin and the average separation between consecutive poles tends toward zero as power laws in the dimension of the basis. The latter allows one to obtain the response function in closed form in terms of Tricomi hypergeometrical functions, which possess a branch-cut on the above-mentioned semiaxis. This suggests that the presence of a cut along the imaginary frequency axis of the shear stress correlator, inferred from previous numerical analyses of weakly coupled scalar $\lambda {\phi }^4$ theories, does not arise due to quantum statistics but instead emerges from the fundamental properties of this system’s interactions.