# What is a holomorphic combination of standard model fields?

I am looking at exotic Higgs decay models like SM + Fermion and MSSM. In SM + Fermion the new fermion might have a three body decay and in MSSM for R-parity violating neutralino decays involve holomorphic combinations of SM fermion fields When I search for what holomorphic means in this case they all redirect me to the mathematical definition. Can somebody explain what this means physically?

## 1 Answer

In this context, you are asking what it means that terms are holomorphic in a superpotential. Superpotentials are combinations of superfields which encode the interaction in a Lagrangian. Superfields encode supermultiplets: particles + their supersymmetric partner (plus auxiliary fields for off-shell supersymmetry).

For your question, it's sufficient to look only at chiral superfields. In the MSSM, these are the superfields that contain SM matter. This is in contrast to vector superfields which contain the SM gauge fields. (There are other types of superfields, but those are not used in the MSSM.)

The Yukawa interactions of the Standard Model and other Yukawa-like interactions are encoded by holomorphic terms in the superpotential. You've written down some R-partiy violating superpotential terms.

The holomorphy of these terms refers to the fact that the superpotential is only a function of the superfield, but not the conjugate superfield. You can write down $LLe^c$, but you cannot write down $L^\dagger L$ in the superpotential. This is because the superpotential is only a holomorphic function of the superfields, and cannot contain the conjugates of any of the superfields. (How to determine which field is chiral and which is anti-chiral is somewhat a matter of convention; we usually define the chiral superfield to be the one to contain the left-handed fermion and the anti-chiral superfield to be the conjugate of this.)

You can learn more about this in the standard SUSY references. Two that I like are the SUSY primer and the Cambridge SUSY lectures. You can just search for the phrase "holomorphic" and read the corresponding sections.