# Fine-Tuning calculations with SPheno

## Definition of the electroweak Fine-Tuning

A measure for the electroweak fine-tuning was introduced in Refs.

` \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*_{α}^{−1} 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 fine-tuning 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*_{Boundar**y}. 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 (m_0,*M*_{1/2}) as well as the parameters relating the superpotential terms and the corresponding soft-breaking terms (B,*A*) to calculate the fine-tuning. 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*_{3/2}. In addition, also the fine-tuning with respect to the superpotential parameters themselves as well as to the strong coupling*α*_{S} might be included because they can even supersede the fine-tuning in the soft-susy breaking sector.
To calculate the fine-tuning in practice, an iteration of the RGEs between*M*_{SUS**Y} andM_{Boundary} happens using the full two-loop RGEs. In each iteration one of the fundamental parameters is slightly varied and the running parameters at*M*_{SUS**Y} are calculated. These parameters are used to solve the tadpole equations numerically with respect to all VEVs and to re-calculate the*Z*-boson mass. To give an even more accurate estimate, also one-loop corrections to the*Z* mass stemming from*Π*_{Z}^{T} can be included.

## Calculating the fine-tuning with SPheno

In order to get a prediction for the fine-tuning with SPheno, one needs to add the following lines to SPheno.m

```
IncludeFineTuning = True;
FineTuningParameters={
{Parameter1,Coefficient1},{Parameter2,Coefficient2},...
};
```

Thus, the fine-tuning is calculated with respect to all parameters in `FineTuningParameters`

(`ParameterN`

) and individually weighted with the coefficients `CoefficientN`

.

### Example

The fine-tuning in the CMSSM is calculated via

```
IncludeFineTuning = True;
FineTuningParameters={
{m0,1/2},{m12,1},{Azero,1},{\[Mu],1},{B[\[Mu]],1}
};
```

where*m*_{0} appears with a relative factor 1/2 because all boundary conditions are of the form*m*_{0}^{2}. After producing the SPheno code, the calculation of fine-tuning is switched on via the flag

```
Block SPhenoInput # SPheno specific input
...
550 1. # Calculate Fine-Tuning
```

For an arbitrary MSSM point, the result in `SPheno.spc.MSSM`

reads

```
Block FineTuning #
0 1.05189732E+03 # Overall FT
1 1.49559021E+01 # m0
2 8.27404724E+02 # m12
3 2.89078174E+02 # Azero
4 1.05189732E+03 # \[Mu]
5 1.68368940E+01 # B[\[Mu]]
```