Branch-cut in the shear-stress response function of massless $\lambda \varphi^4$ with Boltzmann statistics

TL;DR

This paper analytically demonstrates that the shear-stress response function of a classical massless scalar field with quartic interactions exhibits a branch-cut singularity on the positive imaginary axis, indicating fundamental interaction effects.

Contribution

It provides an analytical derivation of the branch-cut in the shear-stress response function for a classical scalar system, clarifying its origin beyond quantum statistical effects.

Findings

Response function has a branch-cut on the positive imaginary axis.

Poles approach the origin with decreasing separation as basis dimension increases.

Closed-form expression involves Tricomi hypergeometrical functions.

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 semi-axis. This is demonstrated in two ways: (1) by truncating the exact, infinite linear 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 towards the origin and the average separation between consecutive poles tends towards 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 semi-axis. 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 $\lambda \varphi^4$ theories, does not arise due to quantum statistics but instead emerges from the fundamental properties of this system's interactions.