Comments on interactions in the SUSY models

TL;DR

This paper explores special supersymmetry transformations in certain models, analyzing their effects on path integral measures, resulting interactions, and Ward identities, with illustrative examples using supersymmetric harmonic oscillators.

Contribution

It introduces a generalized formulation of SUSY transformations with finite parameters, revealing how they generate trivial interactions and modify Ward identities, and demonstrates limitations in producing non-trivial interactions.

Findings

SUSY transformations induce trivial interactions via Jacobians.

Modified Ward identities depend on transformation parameters.

Non-trivial interactions cannot be generated through these transformations.

Abstract

We consider special supersymmetry (SUSY) transformations with $m$ generators $\overleftarrow{s}_{\alpha }$ for a certain class of models and study some physical consequences of Grassmann-odd transformations which form an Abelian supergroup with finite parameters and respective group-like elements being functionals of field variables. The SUSY-invariant path integral measure within conventional quantization implies the appearance, under a change of variables related to such SUSY transformations, of a Jacobian which is explicitly calculated. The Jacobian implies, first of all, the appearance of trivial interactions in the transformed action, and, second, the presence of a modified Ward identity which reduces to the standard Ward identities in the case of constant parameters. We examine the case of ${N}=1$ and $N=2$ supersymmetric harmonic oscillators to illustrate the general concept by a simple free model with $(1,1)$ physical degrees of freedom. It is shown that the interaction terms $U_{tr}$ have a corresponding SUSY-exact form: $U_{tr}=% \big(V_{(1)}\overleftarrow{s};V_{(2)}\overleftarrow{\bar{s}}\overleftarrow{s}% \big)$ naturally generated in this generalized formulation. We argue that the case of non-trivial interactions cannot be obtained in such a way.