SPheno mass calculation

After the iterative calculation of the parameters described here, the pole masses are calculated as follows:

1. The starting point for all loop calculations is the set of running parameters at the renormalization scale Q. This scale can be either be a fixed value or a variable which depends on other parameters of the model. For instance, in SUSY models it is common to choose Q to be the geometric mean of the stop masses.

2. The running parameters are used to solve the minimisation conditions of the vacuum (the tadpole equations T_i) at tree-level $T_i = \frac{\partial V^{(T)}}{\partial v_i} \equiv 0.$

   These equations are solved for a set of parameters, one per equation. This set is determined by the user; typically these are mass-squared parameters, which can be solved for linearly, but SARAHalso allows non-linear tadpole equations.

3. The running parameters as well as the solutions of the tadpole equations are used to calculate the tree-level mass spectrum. The tree-level Higgs masses m_i^h, (T) are the eigenvalues of the tree-level mass matrix M^h, (T) defined by $M^{h,(T)} = \frac{\partial^2 V^{(T)}}{\partial \phi_i \partial \phi_j}$

4. Similarly, the tree-level masses of all other particles present in the model are calculated.

5. Using the tree-level masses the one-loop corrections δ**M_Z to the Z boson are calculated

6. The electroweak VEV v is expressed by the measured pole mass of the Z, M_Z^pol**e, the one-loop corrections and a function of the involved gauge couplings g_i. $v = \sqrt{\frac{M_Z^{2,\text{pole}} + \delta M^2_Z}{f(\{g_i\})}} \label{eq:electroweakv}$

   In the case of the MSSM $f(\{g_i\}) = f(g_1, g_2) = \frac{1}{4} (g_1^2 + g_2^2)$ holds. Together with the value of the running tan β, the values for the VEVs of the up- and down Higgs can be calculated.

7. The tree-level masses are calculated again with the new values for the VEVs.

8. The one- (δ**t_i⁽¹⁾) and two-loop (δ**t_i⁽²⁾) corrections to the tadpole equations are calculated. These are used to solve the loop-corrected minimisation conditions T_i + δ**t_i⁽¹⁾ + δ**t_i⁽²⁾ ≡ 0.

9. The one-loop self-energies for all particles including the external momentum p are calculated. For the Higgs, we call them in the following Π^h, (1L)(p²).

10. For the Higgs states, the two-loop self-energies (with zero external momentum) Π^h, (2L)(0) are calculated as explained here. The possible flags to steer these calculations are explained here

11. The physical Higgs masses are then calculated by taking the real part of the poles of the corresponding propagator matrices Det[p_i²1−M^h, (2L)(p²)] = 0,

    where

    M^2, (2L)(p²)=M̃^h, (T) − Π^h, (1L)(p²)−Π^h, (2L)(0).

    Here, M̃^h, (T) is the tree-level mass matrix where the parameters solving the loop-corrected tadpole equations are used. Eq. ([eq:propagator]) is solved for each eigenvalue p² = m_i² in an iterative way.

See also