Usage of tadpoles equations

The handling of tadpole equations is sometimes a bit tricky. Especially if one wants to calculate the mass spectrum with the help of SPheno for a model, he has to choose a set of parameters which is fixed by the minium conditions of the vacuum. In the MSSM, the standard choice is mu and B_mu, but what can be used for more complicated models? So, it's good to play a bit with the tadpole equations to see what can be done. For this purpose we run the NMSSM with SARAH:

<<SARAH.m;
Start["NMSSM"];

And the first step is to prepare the tadpole equations. Here, we assume for simplicity that we just have real parameters.

equations = (TadpoleEquations[EWSB]==0) /. conj[x_] -> x /. T[x_] :> ToExpression["T" <> ToString[x]];

Now, we can easily solve those expressions with resepct to {mHd^2, mHu^2, ms^2}:

solution=Solve[equations, {mHd2, mHu2, ms2}];

A nice cross check is always to plug this solution into mass matrices involving Goldstones and check the eigenvalues. Here, we use the charged Higgs matrix

Simplify[Eigenvalues[MassMatrix[Hpm] /. solution[[/1|1]] /. conj[x_] -> x]]

and the first eigenvalues shows the correct dependence on the gauge fixing parameter

(g2^2*(vd^2 + vu^2)*RXi[Wm])/4

while the second eigenvalue is independent of RXi. The same can be observed when using MassMatrix[Ah].

If one wants to check for other combinations, there is one caveat: if x and F[x] appear in the equations, Mathematica can't solve the equations with respect to x. Hence, we have to rename T[\[Lambda]] and T[\[Kappa]] to get an “atomic” element. Note, this is usually done automatically by SARAH, when it tries to find the solution during MakeSPheno[] for the given set of parameters. Here, we have to do it by hand:

equations = equations /. T[x_] :> ToExpression["T" <> ToString[x]]

and now we can choose for instance,

sol = Solve[equations, {vS, T\[Lambda], ms2}]

and see that also a solution exists, i.e. this would be another possible choice. Even combinations without any soft-mass term exist, which might be favored by those who want to have some strict unification of all scalars at the GUT scale. Just pick your favorite solution, edit SPheno.m correspondingly and run MakeSPheno[] and you'll have after a couple of minute a spectrum generator fitting to your demands.

See also