Counting 4 pi 's in Strongly Coupled Supersymmetry

TL;DR

This paper develops a naive dimensional analysis (NDA) framework for strongly coupled supersymmetric theories to estimate operator coefficients in low-energy EFTs, anchored by a single scale $\Lambda$ and the guideline $g\sim 4\pi$. It extends NDA to SUSY by rescaling superfields and incorporating spurion analysis and operator matching, enabling systematic counting of $4\pi$ factors in both superpotential and Kähler sectors. Through explicit examples, including an $Sp(4)$-based composite model with a calculable dynamical superpotential and SUSY-breaking scenarios for gaugino and sfermion masses, the authors demonstrate concrete scaling laws and the appearance of order-$4\pi$ couplings. The framework provides a practical method to construct natural EFTs from strongly coupled dynamics and to assess the relative sizes of weakly coupled sectors and spurions, with clear guidance on when NDA remains reliable (single-scale) and when it may fail (multiple comparable scales).

Abstract

We extend the "naive dimensional analysis" arguments used in QCD for estimating the strengths of operators in chiral Lagrangians to strongly coupled supersymmetric theories. In particular, we show how to count factors of 4 pi ---an inexact science, but nevertheless a useful art when such theories are used to model real particle physics.