Loop Masses

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The information about the one- and two-loop corrections to the one- and two-point functions can be used to calculate the loop corrected mass spectrum. The renormalized mass matrices (or masses) are related to the tree-level mass matrices (or masses) and the self-energies as follows.

Loop corrected masses

Real scalars

For a real scalar , the one-loop, and in some cases also two-loop, self-energies are calculated by SPheno. The loop corrected mass matrix squared is related to the tree-level mass matrix squared and the self-energies via

The one-shell condition for the eigenvalue of the loop corrected mass matrix reads

A stable solution of eq. ([eq:propagator]) for each eigenvalue is usually just found via an iterative procedure. In this approach one has to be careful how is defined: this is the tree-level mass matrix where the parameters are taken at the minimum of the effective potential evaluated at the same loop-level at which the self-energies are known. The physical masses are associated with the eigenvalues . In general, for each eigenvalue the rotation matrix is slightly different because of the dependence of the self-energies. The convention by SARAH and SPheno is that the rotation matrix of the lightest eigenvalue is used in all further calculations and the output.

Complex scalars

For a complex scalar the one-loop corrected mass matrix squared is related to the tree-level mass and the one-loop self-energy via

The same on-shell condition, eq. ([eq:propagator]), as for real scalars is used.

Vector bosons

For vector bosons we have similar simple expressions as for scalar. The one-loop masses of real or complex vector bosons are given by

Majorana fermions

The one-loop mass matrix of a Majorana fermion is related to the tree-level mass matrix and the different parts of the self-energies by

Note, is used to assign tree-level values while denotes a transposition. Eq. ([eq:propagator]) can also be used for fermions by taking the eigenvalues of .

Dirac fermions

For a Dirac fermion one obtains the one-loop corrected mass matrix via

Here, the eigenvalues of are used in eq. ([eq:propagator]) to get the pole masses.

See also