Question concerning scalar mixing

[BjHerrmann]

Dear all,

we are currently implementing a non-supersymmetric model including, in addition to the Standard Model, a couple of scalars and new fermions. One important feature of this model is that neutrino masses are generated radiatively at the one-loop level. In a simple case, it turns out that only two neutrinos will have masses, while the first one remains massless (roughly, this is due to the number of degrees of freedom available to generate the masses).

We have implemented the model in SARAH and generated a SPHENO version from it. While testing, we had a look into certain couplings on the Mathematica level, after running the model in SARAH. In particular, we looked at the couplings which allow to generate the neutrino masses, which are the couplings between Standard Model fermions (leptons, neutrinos), a new fermion, and a new scalar.

While the coupling structure looks allright, we are confused about the following point. We first ask to display the coupling in the form:

In[4]:= Vertex[{Fe, bar[FXc], P0}]

Out[4]= {{Fe[{gt1}], bar[FXc], P0[{gt3}]}, {I conj[ZSc[gt3, 1]] sum[j1, 1, 3, cg1[j1] conj[Ve[gt1, j1]]], PL}, {-((I conj[ZSc[gt3, 2]] sum[j1, 1, 3, conj[cg3[j1]] Ue[gt1, j1]])/Sqrt[2]), PR}}

In this example, "Fe" is a Standard Model lepton, "FXc" a new charged fermion, and "P0" a neutral scalar. "cg1", "cg2" are coupling parameters, while "Ue", "Ve", "ZSc" are mixing matrices associated to the leptons and new scalars.

We then ask to display the conjugated coupling, which gives:

In[6]:= Vertex[{bar[Fe], FXc, P0}]

Out[6]= {{bar[Fe[{gt1}]], FXc,
P0[{gt3}]}, {-((
I conj[ZSc[gt3, 2]] sum[j1, 1, 3, cg3[j1] conj[Ue[gt1, j1]]])/Sqrt[
2]), PL}, {I conj[ZSc[gt3, 1]] sum[j1, 1, 3,
conj[cg1[j1]] Ve[gt1, j1]], PR}}

It all looks normal concerning complex conjugation, except the fact that the scalar mixing matrix "ZSc" is not conjugated w.r.t. to the first expression. However, we would expect this matrix also to be conjugated.

Is there something wrong with our model, or are we missing something?

If needed, we can provide more information, of course.
Best regards,
Björn

[FStaub]

Hi Björn,

You don't use conj[PO] in your conjugated vertex, ie I don't expect any change in the corresponding mixing matrix. The question is: is PO and the rotation matrix real in general?

Cheers
Florian

[BjHerrmann]

Hi Florian,
thanks for your reply! (it is good to see that you still read and post here!)
In the above example, indeed I did not conjugate the scalar, but in fact this changes nothing. Asking for the vertex saying P0 or conj[P0] is exactly the same:

In[11]:= Vertex[{bar[Fe], FXc, conj[P0]}]

Out[11]= {{bar[Fe[{gt1}]], FXc,
P0[{gt3}]}, {-((
I conj[ZSc[gt3, 2]] sum[j1, 1, 3, cg3[j1] conj[Ue[gt1, j1]]])/Sqrt[
2]), PL}, {I conj[ZSc[gt3, 1]] sum[j1, 1, 3,
conj[cg1[j1]] Ve[gt1, j1]], PR}}

This makes sense somehow, since P0 is indeed a real scalar.
Moreover, it is indeed true that in the particular model that we have inplemented, the scalar mixing matrix always is real. But how does SARAH know about this? And if it knows about the matrix being real, why does it write "conj[ZSc]" in the first place? All this is somehow confusing to us...

Cheers,
Björn

[FStaub]

Hi Björn,

1) since P0 is real, you get the same for conj[P0] and P0 as it should be
2) the rotation is defined in a way that you have P0_i = Z_ij^* S_j for some gauge eingestate S
3) SARAH doesn't know by its own that Z is real, i.e. it assumed that it is complex.

In principle, there are two possible ways how to evaluate that depending on the moment when you replace the mass by gauge eingestates:

conj[P0] = conj[Z_ij^* S_j] = conj[Z_ij^*] S_j = Z_ij S_j
conj[P0] = P0 = Z_ij^* S_j

SARAH/Mathematica always go the second way. Here, you can also see that Z must be real because the results must be identical.

All in all, the simplest way to remove this confusion is to define ZSc as real in parameters.m

Cheers,
Florian

[BjHerrmann]

Hi Florian,
thanks a lot -- this makes it clear to me!
Cheers,
Björn