Tadpole Equations
During the evaluation of a model, SARAH calculates ’on the fly’ all minimum conditions of the tree-level potential, the so called tadpole equations. In the case of no CP violation, in which complex scalars are decomposed as
S_i \to \frac{1}{\sqrt{2}}(v_i + \phi_i + i \sigma_i) \,,
the expressions
0 = \frac{\partial V}{\partial \phi_i} \equiv T_i
are calculated. These are equivalent to \frac{\partial V}{\partial v_i}. For models with CP violation in the Higgs sector, i.e. where either complex phases appear between the real scalars or where the VEVs have an imaginary part, SARAH calculates the minimum conditions with respect to the CP-even and CP-odd components:
0 = \frac{\partial V}{\partial \phi_i} \equiv T_{\phi_i} \,,\hspace{1cm} 0 = \frac{\partial V}{\partial \sigma_i} \equiv T_{\sigma_i}
The set of all tadpole equations is in this case Ti = {Tϕi, Tσi}.
Getting the tadpole equations from SARAH
All tadpole equations for given eigenstates are returned by
TadpoleEquations[$EIGENSTATES]The order of the tadpole equations in this array corresponds to the order of the definition of VEVs in the model file, see VEVs. There is also the shorter command
TadpoleEquation[X]to obtain the tadpole equations corresponding to a specific VEV or state.
Example
The tadpole equation for vd after EWSB is saved in
TadpoleEquations[EWSB][[/1|1]]and reads
mHd2*vd + (g1^2*vd^3)/8 + (g2^2*vd^3)/8 - (g1^2*vd*vu^2)/8 -
(g2^2*vd*vu^2)/8 + vd*\[Mu]^2 - vu*B[\[Mu]]The same result can be obtained by
TadpoleEquation[vd]or
TadpoleEquation[phid]How to use these equations is explained in some more detail here.
Output
The tadpole equations are exported into LaTeX format as well as in Fortrancode used by SPheno. This ensures that all parameter points evaluated by SPheno are at least sitting at a local minimum of the scalar potential. Moreover, the tadpole equations are included in the model files for Vevacious which is used to find all possible solutions of them with respect to the different VEVs.