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---
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title: SPheno mass calculation
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permalink: /SPheno_mass_calculation/
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---
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[Category:SPheno](/Category:SPheno "wikilink")
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After the iterative calculation of the parameters [described here](/Renormalisation_procedure_of_SPheno "wikilink"), the pole masses are calculated as follows:
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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.
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2. The running parameters are used to solve the minimisation conditions of the vacuum (the tadpole equations *T*<sub>*i*</sub>) at tree-level
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$T_i = \\frac{\\partial V^{(T)}}{\\partial v_i} \\equiv 0.$
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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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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*<sub>*i*</sub><sup>*h*, (*T*)</sup> are the eigenvalues of the tree-level mass matrix *M*<sup>*h*, (*T*)</sup> defined by
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$M^{h,(T)} = \\frac{\\partial^2 V^{(T)}}{\\partial \\phi_i \\partial \\phi_j}$
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4. Similarly, the tree-level masses of all other particles present in the model are calculated.
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5. Using the tree-level masses the one-loop corrections *δ**M*<sub>*Z*</sub> to the *Z* boson are calculated
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6. The electroweak VEV *v* is expressed by the measured pole mass of the *Z*, *M*<sub>*Z*</sub><sup>*p**o**l**e*</sup>, the one-loop corrections and a function of the involved gauge couplings *g*<sub>*i*</sub>.
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$v = \\sqrt{\\frac{M_Z^{2,\\text{pole}} + \\delta M^2_Z}{f(\\{g_i\\})}}
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\\label{eq:electroweakv}$
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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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7. The tree-level masses are calculated again with the new values for the VEVs.
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8. The one- (*δ**t*<sub>*i*</sub><sup>(1)</sup>) and two-loop (*δ**t*<sub>*i*</sub><sup>(2)</sup>) corrections to the tadpole equations are calculated. These are used to solve the loop-corrected minimisation conditions
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*T*<sub>*i*</sub> + *δ**t*<sub>*i*</sub><sup>(1)</sup> + *δ**t*<sub>*i*</sub><sup>(2)</sup> ≡ 0.
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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*<sup>*h*, (1*L*)</sup>(*p*<sup>2</sup>).
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10. For the Higgs states, the two-loop self-energies (with zero external momentum) *Π*<sup>*h*, (2*L*)</sup>(0) are calculated as explained [here](/Two-Loop_Self-Energies_and_Tadpoles "wikilink"). The possible flags to steer these calculations are explained [here](/Using_SPheno_for_two-loop_masses "wikilink")
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11. The physical Higgs masses are then calculated by taking the real part of the poles of the corresponding propagator matrices
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*D**e**t*\[*p*<sub>*i*</sub><sup>2</sup>**1**−*M*<sup>*h*, (2*L*)</sup>(*p*<sup>2</sup>)\] = 0,
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where
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*M*<sup>2, (2*L*)</sup>(*p*<sup>2</sup>)=*M̃*<sup>*h*, (*T*)</sup> − *Π*<sup>*h*, (1*L*)</sup>(*p*<sup>2</sup>)−*Π*<sup>*h*, (2*L*)</sup>(0).
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Here, *M̃*<sup>*h*, (*T*)</sup> 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 *p*<sup>2</sup> = *m*<sub>*i*</sub><sup>2</sup> in an iterative way.
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See also
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-------- |
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\ No newline at end of file |