Spontaneous symmetry breaking

Matchete provides functionality to automate the transition from a symmetric-phase Lagrangian to its broken-phase counterpart. This is the primary route to obtaining a Lagrangian that contains massive gauge bosons, for example as a starting point for computing Feynman rules or UFO models (and, once implemented, for integrating out heavy vectors).

Workflow overview

Spontaneous symmetry breaking in Matchete is a semi-automated three-step process:

1. Define the broken-phase fields and couplings. The physical fields of the broken phase (e.g., the Higgs boson, massive gauge bosons, mass-eigenstate fermions, and Goldstone bosons) and any new parameters (vacuum expectation values, mixing angles, physical masses) are introduced with the standard DefineField and DefineCoupling calls.

2. Specify the symmetry-breaking pattern. SetSymmetryBreakingPattern declares which gauge groups are broken and what the residual symmetry is. The electroweak symmetry breaking pattern $SU(2)_L \times U(1)_Y \to U(1)_{em}$ in the SM(EFT) is represented by:

SetSymmetryBreakingPattern[{SU2L, U1Y}, {U1em}];

RepresentationDecomposition then specifies how each representation of the original group decomposes into representations of the stability group, and FieldDecomposition maps each symmetric-phase field onto its broken-phase components. Coupling substitutions (e.g., expressing the $W$-boson coupling in terms of the EM coupling and weak mixing angle) are set with SetSSBReplacements.

3. Convert the Lagrangian. ToBrokenPhase applies all decompositions and coupling substitutions to produce the broken-phase Lagrangian from the symmetric-phase one.

Notes

  • Matchete does not automatically determine the vacuum of the scalar potential; the user must parametrize scalar fields with a consistent vacuum expectation value.
  • The DefineCoSinePair function introduces a pair of couplings representing the cosine and sine of a mixing angle, allowing Matchete’s simplification routines to apply trigonometric identities automatically.
  • A detailed tutorial covering the full SM electroweak symmetry breaking is available in the documentation.