





title: Tadpole Equations



permalink: /Tadpole_Equations/










[Category:Calculations](/Category:Calculations "wikilink") During the evaluation of a model, SARAH calculates ’on the fly’ all minimum conditions of the treelevel 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 CPeven and CPodd 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 *T*<sub>*i*</sub> = {*T*<sub>*ϕ*<sub>*i*</sub></sub>, *T*<sub>*σ*<sub>*i*</sub></sub>}.






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](/VEVs "wikilink").



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 *v*<sub>*d*</sub> after EWSB is saved in






TadpoleEquations[EWSB][[/11]]






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](/Usage_of_tadpoles_equations "wikilink").






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.






See also










 [Usage of tadpoles equations](/Usage_of_tadpoles_equations "wikilink") 