Martin GabelmannLoop Masses
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 \Phi, the one-loop, and in some cases also two-loop, self-energies are calculated by SPheno. The loop corrected mass matrix squared m_\Phi^{2,(L)} is related to the tree-level mass matrix squared m_\Phi^{2,(T)} and the self-energies via
m_\Phi^{2,(L)}(p^2)= m_\Phi^{2,(T)} - \mathcal{R}\left(\Pi_\Phi^{(1L)}(p^2)\right) -\mathcal{R}\left(\Pi_\Phi^{(2L)}(0)\right)
The one-shell condition for the eigenvalue M_{\Phi_i}^2(p^2) of the loop corrected mass matrix m_\Phi^{2,(L)} reads
Det\left[p_i^2 \mathcal{1} - M_{\Phi_i}^2(p^2) \right]=0
A stable solution of this equation for each eigenvalue M_{\Phi_i}^2\left( p^2 = M_{\Phi_i}^2\right) is usually just found via an iterative procedure. In this approach one has to be careful how m_\Phi^{2,(T)} 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 M_{\Phi_i}^2\left( p^2 = M_{\Phi_i}^2\right). In general, for each eigenvalue the rotation matrix is slightly different because of the p^2 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 scalarη the one-loop corrected mass matrix squared is related to the tree-level mass and the one-loop self-energy via
m_\eta^{2,(L)}(p^2)= m_\eta^{2,(T)} - \Pi_\eta^{(1L)}(p^2)
The same on-shell condition as for real scalars is used (see previous section).
Vector bosons
For vector bosons we have similar simple expressions as for scalar. The one-loop masses of real or complex vector bosonsV are given by
m_V^{2,(L)}(p^2)= m_V^{2,(T)} - \Pi_V^{T,(1L)}(p^2)
Majorana fermions
The one-loop mass matrix of a Majorana fermionχ is related to the tree-level mass matrix m_\chi^{(T)} and the different parts of the self-energies by
m_\chi^{(1L)} (p^2) = m_\chi^{(T)} - \frac{1}{2} \bigg[ \Sigma^\chi_S(p^2) + \Sigma^{\chi,T}_S(p^2) + \left(\Sigma^{\chi,T}_L(p^2)+ \Sigma^\chi_R(p^2)\right) m_\chi^{(T)} + m_{\chi}^{(T)} \left(\Sigma^{\chi,T}_R(p^2) + \Sigma^\chi_L(p^2) \right) \bigg]
Note, (T) is used to assign tree-level values while T denotes a transposition. This equation can also be used for fermions by taking the eigenvalues of m_\chi^{1,(1L)} (p^2)= \left(m_\chi^{(1L)} (p^2)\right)^* m_\chi^{(1L)} (p^2).
Dirac fermions
For a Dirac fermion \Psi one obtains the one-loop corrected mass matrix via
m_\Psi^{(1L)}(p^2) = m_\Psi^{(T)} - \Sigma^+_S(p^2) - \Sigma^+_R(p^2) m_\Psi^{(T)} - m_\Psi^{(T)} \Sigma^+_L(p^2) .
Here, the eigenvalues of (m_\Psi)^* m_\Psi are used in the iterative procedure to get the pole masses.