Matchotter: An Automated Tool for Dimensional Reduction at Finite Temperature

TL;DR

Matchotter automates the dimensional reduction process at finite temperature, enabling precise computation of thermal observables by extracting three-dimensional effective theories from four-dimensional quantum field theories.

Contribution

It introduces Matchotter, a new module integrated into Matchete, that automates finite-temperature matching up to one-loop order for generic Lagrangians, including supersoft matching.

Findings

Successfully automates dimensional reduction for various models.

Efficiently extracts low-energy EFT directly from thermal path integral.

Demonstrates capabilities on Standard Model Effective Field Theory (SMEFT).

Abstract

At finite temperature, the decoupling of heavy Matsubara modes allows a four-dimensional quantum field theory to be matched onto a purely spatial, three-dimensional effective field theory (EFT). This dimensional reduction is a crucial prerequisite for the precise computation of thermal observables, most prominently those related to cosmological phase transitions. In this work, we present Matchotter -- a dedicated finite-temperature module natively integrated into the Matchete package -- which automates this matching process up to one-loop order for generic Lagrangians. By adapting modern functional matching techniques to the finite-temperature formalism, Matchotter efficiently extracts the low-energy EFT directly from the thermal path integral. Furthermore, the module fully automates supersoft matching, where the temporal gauge bosons, which acquire a Debye mass during the dimensional reduction process, are integrated out. We outline the underlying architecture of the program and demonstrate its capabilities across a range of models, including the Standard Model Effective Field Theory (SMEFT).