SPheno mass calculation
After the iterative calculation of the parameters described here, the pole masses are calculated as follows:
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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.
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The running parameters are used to solve the minimisation conditions of the vacuum (the tadpole equations Ti) 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.
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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 mih, (T) are the eigenvalues of the tree-level mass matrix Mh, (T) defined by $M^{h,(T)} = \frac{\partial^2 V^{(T)}}{\partial \phi_i \partial \phi_j}$
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Similarly, the tree-level masses of all other particles present in the model are calculated.
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Using the tree-level masses the one-loop corrections δ**MZ to the Z boson are calculated
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The electroweak VEV v is expressed by the measured pole mass of the Z, MZpol**e, the one-loop corrections and a function of the involved gauge couplings gi. $v = \sqrt{\frac{M_Z^{2,\text{pole}} + \delta M^2_Z}{f(\{g_i\})}} \label{eq:electroweakv}$
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.
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The tree-level masses are calculated again with the new values for the VEVs.
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The one- (δ**ti(1)) and two-loop (δ**ti(2)) corrections to the tadpole equations are calculated. These are used to solve the loop-corrected minimisation conditions Ti + δ**ti(1) + δ**ti(2) ≡ 0.
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The one-loop self-energies for all particles including the external momentum p are calculated. For the Higgs, we call them in the following Πh, (1L)(p2).
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For the Higgs states, the two-loop self-energies (with zero external momentum) Πh, (2L)(0) are calculated as explained here. The possible flags to steer these calculations are explained here
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The physical Higgs masses are then calculated by taking the real part of the poles of the corresponding propagator matrices Det[pi21−Mh, (2L)(p2)] = 0,
where
M2, (2L)(p2)=M̃h, (T) − Πh, (1L)(p2)−Πh, (2L)(0).
Here, M̃h, (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 = mi2 in an iterative way.