Additional_terms_in_Lagrangian

Additional terms in Lagrangian

Purpose

Sometimes, it might be necessary to define interactions in the Lagrangian, which can not be derived from the superpotential or the kinetic interaction. Furthermore, it might be necessary to change the properties of some vertices by hand. For example, integrating out particles might most likely spoil SUSY relations. For instance, the quartic Higgs coupling in the SM is a free parameter, while it is given in SUSY via the D-Terms.

Implementation in SARAH

Both, defining new interactions and changing existing ones, is done in SARAH with one statement. For each set of eigenstates, new or changed terms can be defined separately by declaring them in the model file via

DEFINITION[$EIGENSTATES][Additonal] = {
{Lag, {Options}},
  ... };

Lagrangian

Lag defines the new or changed parts of the Lagrangian. It must have dimension 4. For writing the terms, the same, short conventions as for defining the Potential in non-supersymmetric models can be used:

Fields are separated by dots
Weyl fermions are used
All indices are automatically added and contracted

The new couplings are handled in a similar way as the couplings of the Superpotential: tensor indices are added automatically and they are assumed to be complex. Further assumptions about the coupling can be made in parameters.m. The manually defined terms will be handled like every other term of the Lagrangian, i.e. they are affected by rotations and replacements if the eigenstates are changed.

The two possible options are:

AddHC -> True/False: Defines if the hermitian conjugated of this term is not added to the Lagrangian
Overwrite -> True/False: Defines if existing couplings involving the same fields are overwritten.

Examples

Define new terms A mixed soft-breaking term of the form m_{l H_d}^2 (\tilde{l}^* H_d + \tilde{l} H_d^*)

is added to the Lagrangian of the gauge eigenstates by
```
DEFINITION[GaugeES][Additional] =
{{mlHd2 conj[Sl].SHd, {Overwrite->False, AddHC->True}}};
```
Adding additional terms to existing couplings With DEFINITION[EWSB][Additional] = {{1/24 Kappa hh.hh.hh.hh, {Overwrite->False, AddHC->False}}};

the Higgs self couplings receive an additional contribution:

\Gamma_{h^4} ~ (a_i g_i^2 + b_i Y_i^2) \hspace{0.5cm} \rightarrow \hspace{0.5cm} \Gamma_{h^4} ~ (a_i g_i^2 + b_i Y_i^2 + \kappa)
Overwriting existing terms To overwrite the former expressions for the Higgs self interactions, DEFINITION[EWSB][Additional] = {{LagNew, {Overwrite->True, AddHC->False}}}; LagNew = 1/24 Kappa hh.hh.hh.hh + 1/24 Lambda hh.hh.Ah.Ah;

is used. This has the following effect:

\Gamma_{h^4} ~ (a_i g_i^2 + b_i Y_i^2) \hspace{0.5cm} \rightarrow \hspace{0.5cm} \Gamma_{h^4} ~ \kappa
Interactions involving derivatives For interactions involving derivatives, Der is used DEFINITION[EWSB][Additional] = {{Kappa Der[SHd,lor3].conj[SHd].VB,{Overwrite->False, AddHC->True}}};

Note, this option has to be used carefully since it was never tested thoroughly.

GitLab