|
|
---
|
|
|
title: Usage of tadpoles equations
|
|
|
permalink: /Usage_of_tadpoles_equations/
|
|
|
---
|
|
|
|
|
|
[Category:Calculations](/Category:Calculations "wikilink") 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
|
|
|
-------- |
|
|
\ No newline at end of file |