





title: FineTuning calculations with SPheno



permalink: /FineTuning_calculations_with_SPheno/







# FineTuning calculations with SPheno






[Category:Outputs](/Category:Outputs "wikilink") [Category:SPheno](/Category:SPheno "wikilink")









Definition of the electroweak FineTuning










A measure for the electroweak finetuning was introduced in Refs.






$\\label{eq:measure}



\\Delta_{FT} \\equiv \\max {\\text{Abs}}\\big\[\\Delta _{\\alpha}\\big\],\\qquad \\Delta _{\\alpha}\\equiv \\frac{\\partial \\ln M_Z^{2}}{\\partial \\ln \\alpha} = \\frac{\\alpha}{M_Z^2}\\frac{\\partial M_Z^2}{\\partial \\alpha} \\;.$



$` \Delta_{FT} \equiv \max {\text{Abs}}\big\[\Delta _{\alpha}\big\],\qquad \Delta _{\alpha}\equiv \frac{\partial \ln M_Z^{2}}{\partial \ln \alpha} = \frac{\alpha}{M_Z^2}\frac{\partial M_Z^2}{\partial \alpha} \;.`$






*α* is a set of independent parameters.*Δ*<sub>*α*</sub><sup>−1</sup> gives an estimate of the accuracy to which the parameter*α* must be tuned to get the correct electroweak breaking scale . Using this definition the finetuning of a given models depends on the choice what parameters are considered as fundamental and at which scale they are defined. The approach by SARAH is that it takes by default the scale at which the SUSY breaking parameters are set. This corresponds in models where SUSY is broken by gravity to the scale of grand unification (GUT scale), while for models with gauge mediated SUSY breaking (GMSB) the messenger scale would be used. For simplicity, I call both*M*<sub>*B**o**u**n**d**a**r**y*</sub>. The choice of the set of parameters*α* is made by user. Usually, one uses in scenarios motivated by supergravity the universal scalar and gaugino masses (</math>m_0</math>,*M*<sub>1/2</sub>) as well as the parameters relating the superpotential terms and the corresponding softbreaking terms (</math>B</math>,*A*) to calculate the finetuning. However, since also these parameters are related in specific models for SUSY breaking, it might be necessary to consider even more fundamental parameters like the gravitino mass*m*<sub>3/2</sub>. In addition, also the finetuning with respect to the superpotential parameters themselves as well as to the strong coupling*α*<sub>*S*</sub> might be included because they can even supersede the finetuning in the softsusy breaking sector.



To calculate the finetuning in practice, an iteration of the RGEs between*M*<sub>*S**U**S**Y*</sub> and</math>M_{Boundary}</math> happens using the full twoloop RGEs. In each iteration one of the fundamental parameters is slightly varied and the running parameters at*M*<sub>*S**U**S**Y*</sub> are calculated. These parameters are used to solve the tadpole equations numerically with respect to all VEVs and to recalculate the*Z*boson mass. To give an even more accurate estimate, also oneloop corrections to the*Z* mass stemming from*Π*<sub>*Z*</sub><sup>*T*</sup> can be included.

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