Thermal Field Theory
This note discusses the details of the thermal quantum field theory capabilities implemented in Matchete. We consider the Matsubara or imaginary-time formalism in which time is compactified in a circle of radius 
(where Τ usually represents the finite temperature of the theory) and bosonic/fermionic fields acquire periodic/anti-periodic boundary conditions. This way, fields are split in a tower of Fourier—usually referred to as Matsubara—modes. Each mode 
becomes a field living in the Euclidean 3-dimensional space and acquires a thermal mass given by its Matsubara frequency 
.
(where Τ usually represents the finite temperature of the theory) and bosonic/fermionic fields acquire periodic/anti-periodic boundary conditions. This way, fields are split in a tower of Fourier—usually referred to as Matsubara—modes. Each mode becomes a field living in the Euclidean 3-dimensional space and acquires a thermal mass given by its Matsubara frequency .
In the limit of high temperature all Matsubara modes except for the bosonic zero-modes become heavy, so that one can build a static EFT for the latter. The process of obtaining the 3-dimensional (3D) EFT is known as the Dimensional Reduction program.
Instead of the usual Lorentz index, after Dimensional Reduction our Lagrangian theory will be spatial and so composed of Euclidean spatial index. The user can use the Spatial index the same way as with Lorentz, introducing it in Vector fields and CD.
It is a new property of the fields. If set to true, it means that this field corresponds to the Matsubara zero mode of a 4D- dimensional field. This way derivatives over it are completely spatial
On top of that we can determine ever zero mode present in the already defined fields using DefineZeroModes. Particularly this will define the temporal component of every gauge group and an associated spatial gauge group.
It is a new property of the couplings. We assume that every coupling will scale with the finite temperature depending on the region of the space where you want to study the phase transition. We assume every coupling scales linearly to the temperature by default
One can set the thermal power counting of a particular coupling when defining the coupling, similar to other properties
To truncate an expression to certain thermal power counting, the user can take the function SeriesThermalEFT that works similar to the more conventional SeriesEFT.
In the spatial EFT one can define an even lower-energy EFT that will work at the usually know SuperSoft scale. In this scale the temporal components of the gauge bosons (whose mass is known as Debye mass) are considered heavy and are integrated out.
This process of integration can be performed with the usual tool of Match, although the user must set by hand the temporal gauge boson heavy by hand. For practical purposes the user can use the function SuperSoftMatch, which already takes the spatial Lagrangian and performs the necessary matching.
Matchete is a tool that is hardcoded to work for 4 spacetime dimensions. However since this program of Dimensional Reduction opens the door to work in other spacetime dimensions, we have developed the tool SetDimensions, that allow the user to perform computations in any dimension greater than 1 for the Lorentz index.