The $g^6$ pressure of hot Yang-Mills theory: Canonical form of the integrand

TL;DR

This paper advances the calculation of the $g^6$ order pressure in hot Yang-Mills theory by transforming complex sum-integrals into a canonical form, significantly reducing the computational complexity and bringing us closer to a complete theoretical understanding.

Contribution

It introduces a systematic method to convert a vast number of four-loop sum-integrals into a simplified canonical form, streamlining high-order perturbative calculations in thermal field theory.

Findings

Reduced from over 100,000 to 21 sum-integrals at order $g^6$

Mapped 11 sum-integrals onto known lower-loop structures

Identified only 10 genuine four-loop sum-integrals remaining to evaluate

Abstract

We present major progress towards the determination of the last missing piece for the pressure of a Yang-Mills plasma at high temperatures at order $g^6$ in the strong coupling constant. This order is of key importance due to its role in resolving the long-standing infrared problem of finite-temperature field theory within a dimensionally reduced effective field theory setup. By systematically applying linear transformations of integration variables, or momentum shifts, we resolve equivalences between different representations of Feynman sum-integrals. on the integrand level, transforming those into a canonical form. At the order $g^6$, this results in reducing a sum of O(100000) distinct sum-integrals which are produced from all four-loop vacuum diagrams down to merely 21. Furthermore, we succeed to map 11 of those onto known lower-loop structures. This leaves only 10 genuine 4-loop sum-integrals to be evaluated, thereby bringing the finalization of three decades of theoretical efforts within reach.