Superpotential

Syntax

The definition of the superpotential is done via

SuperPotential = Term1 + Term2 + ...

where each term has the form

Term = [Coefficient]*Parameter*[Contraction] Superfield1.[Superfield2.Superfield3]

where the parts in brackets just have to be defined optionally. In general, one can define

Coefficient: a numerical coefficient. If absent, 1 is used
Paramter: a name for the coupling
Contraction: one can fix a contraction of charged indices. Usually, this is not necessary because SARAH derives the contraction by its own
Particles: each interaction in the superpotential can involve up to three superfields (see exception here: Higher dimensionsal terms in superpotential)

Contraction of indices

The indices of the involved particles are automatically contracted by SARAH as explained here. Sometimes, there are more possibilities to contract all indices. Therefore, it is also possible to fix the contraction of each term. The contraction used by SARAH can be seen by

ShowSuperpotentialContractions;

Properties of couplings and soft-breaking terms

If the particles involved in the different interactions have more than one generation, the couplings are in general complex tensors carrying up to three generation indices. Assumptions like diagonality or no CP violation can added by using the parameter file. The corresponding soft-breaking term to each superpotential coupling is automatically added to the Lagrangian. The soft-breaking couplings carry the same indices as the superpotential coupling. They are named as

Quartic terms: Q[“Name of Coupling”]
Trilinear terms: T[“Name of Coupling”]
Bilinear terms: B[“Name of Coupling”]
Linear terms: L[“Name of Coupling”]

Simplifying assumptions for the soft-breaking terms can be made independently of the assumptions for the superpotential parameters in parameters.m

Examples

The superpotential of the MSSM is defined via SuperPotential = Yu u.q.Hu - Yd d.q.Hd - Ye e.l.Hd + [Mu] Hu.Hd;
Soft-breaking terms:
The following soft-breaking terms are automatically added by SARAH
```
T[Yu] Sq.SHu.Su + T[Yd] Sd.Sq.Hd - T[Ye] Se.Sl.SHd + B[\[Mu]] SHu.SHd
```
Index interpretation:
the term involving the up Yukawa coupling
```
 Yu q.Hu.u
```
is interpreted by SARAH as Y^u_{n_1,n_2} \hat{q}^i_{\alpha,n_1} \epsilon^{ij} \hat{H}_u^j\hat{\overline{u}}_{\beta,n_2} \delta_{\alpha,\beta} while
```
T[Yu] Sq.SHu.Su
```
means T(Y_u)_{n_1,n_2} \tilde{q}_{\alpha,n_1}^i \epsilon^{ij} H_u^i \tilde{u}_{\beta,n2}\delta_{\alpha \beta} .
Explicit contraction:
in particular for the SU(2) it might be necessary to adjust the index contraction: there is some ambiguity because of the relation among the fundamental and anti-fundamental representation in this group. For instance, in the seesaw 2 model and might want to define the coupling between the triplet t and the leptons l as follows:
```
SuperPotential =  Yt/Sqrt[2] epsTensor[lef1,lef2] epsTensor[lef2b,lef3] l.t.l
```
The default contraction of SARAH would have been instead
```
epsTensor[lef1,lef2] Delta[lef2b,lef3]
```
(see also automatic index contraction )