Dimensional reduction at finite temperature

Matchotter is the finite-temperature module of Matchete. It implements the dimensional reduction (DR) program: at high temperatures $T$, all non-zero Matsubara modes acquire a thermal mass proportional to their Matsubara frequency and become heavy. Integrating them out yields a static three-dimensional (3D) EFT that describes the dynamics of the bosonic zero-mode sector at scales $p \ll \pi T$. This 3D EFT is the natural starting point for studying thermal phase transitions and electroweak baryogenesis.

Matchete works in the Matsubara (imaginary-time) formalism, where the Euclidean time direction is compactified on a circle of circumference $1/T$.

Identifying the zero modes

Before running the dimensional reduction, the bosonic zero-mode fields must be made known to Matchete. DefineZeroModes creates them automatically from the existing field definitions:

DefineZeroModes[];

For every scalar field phi already defined via DefineField, this introduces a ZeroMode-tagged counterpart phi0. For every gauge group with gauge boson A and coupling g, it creates a spatial gauge group together with a spatial gauge boson and a new scalar field A0 for the temporal component, with effective coupling $ig/\sqrt{T}$.

As a concrete example, the Abelian Higgs model requires:

DefineGaugeGroup[U1Y, U1, g1, B, Abelian -> True];
DefineField[phi, Scalar, Charges -> {U1Y[1]}];
DefineZeroModes[];
(* Introduces phi0 (ZeroMode-tagged scalar) and a spatial U1Y
   gauge group with gauge boson B3 and temporal scalar B0 *)

The names of the zero-mode fields can be customised with the ZeroModeNames option of DefineZeroModes.

Running the dimensional reduction

The main matching routine for dimensional reduction is DRMatch:

L3D = DRMatch[L4D,
  EFTOrder  -> 6,
  LoopOrder -> 1
];

DRMatch integrates out all massive Matsubara modes and returns the static 3D EFT Lagrangian. The options mirror those of Match: EFTOrder sets the power-counting truncation and LoopOrder controls the loop order. By default, vacuum renormalization contributions are discarded (RemoveConstantTerms -> True).

Super-soft matching

The 3D EFT produced by DRMatch contains temporal gauge-boson components (the Debye fields) that are heavy at the super-soft scale $\sim g^2 T$. SuperSoftMatch integrates them out in a further matching step:

LSS = SuperSoftMatch[L3D];

SuperSoftMatch must be called after DRMatch. It automatically identifies the temporal gauge-boson fields and captures their Debye masses as effective couplings. The result can then be truncated to a given thermal power counting with SeriesThermalEFT:

SeriesThermalEFT[LSS, ThermalPowerCounting -> 4];

Notes

  • After dimensional reduction, the spatial Lorentz index replaces the four-dimensional one; Matchete introduces a Spatial index type for fields and covariant derivatives in the 3D Lagrangian.
  • The 3D EFT produced by DRMatch can be further simplified with GreensSimplify and EOMSimplify before passing it to SuperSoftMatch.
  • A detailed tutorial covering the thermal field theory formalism is available in the documentation.