Mass Sum Rule of the Effective Action on Vacua with Broken Rigid N=1 Supersymmetry

TL;DR

This paper extends the mass sum rule to N=1 supersymmetric effective actions with broken supersymmetry, providing a universal relation applicable to various vacua and including phenomenological implications for gaugino masses.

Contribution

It introduces a generalized mass sum rule for N=1 supersymmetric effective actions, applicable to vacua with broken supersymmetry and including matter fields, with conditions for supertrace vanishing.

Findings

Supertrace vanishes in special Kähler and anomaly-free cases.

Derived an upper bound for gaugino masses based on condensate effects.

Extended analysis to theories with fundamental matter fields.

Abstract

We present an extension of the mass sum rule that applies to renormalizable rigid supersymmetric field theories to the case of the N=1 supersymmetric effective action (the gauged non-linear sigma model) consisting of adjoint scalar superfields and vector superfields possessing a K"ahler potential, a set of gauge coupling functions (second prepotential derivatives) and a superpotential, which respectively set their energy scales. The mass sum rule derived is valid for any vacua, including the (metastable) one of broken supersymmetry with the condensates of D-term and/or F-term. We manage to extend these analyses to the cases where superfields in (anti-)fundamental representation are present. The supertrace is shown to vanish in those cases where underlying geometry is special K"ahler and theory under concern is anomaly free. Simple phenomenological application is given, providing an upper bound for gaugino masses. We discuss that the effects of the D and/or F condensates can be represented as a set of soft breaking terms with their strengths predicted by the scales.