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
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.
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.
For vector bosons we have similar simple expressions as for scalar. The one-loop masses of real or complex vector bosons are given by
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 .
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.
- Loop functions
- Calculation of the mass spectrum with SPheno
- Using SPheno for two-loop masses
- Helpful conventions to know to work with SARAH
- Calculation of tree level masses
- Calculation of Vertices
- Calculation of tadpole equations
- Calculation of RGEs
- Calculation of One-Loop Self-Energies and Tadpoles
- Calculation of Two-Loop Self-Energies and Tadpoles
- Calculation of loop masses