Breakdown of Cluster Decomposition in Instanton Calculations of the Gluino Condensate

TL;DR

This paper critically examines instanton calculations of the gluino condensate in supersymmetric gauge theories, revealing that the strong-coupling approach's key assumption, cluster decomposition, is invalid, leading to questions about its validity.

Contribution

The paper demonstrates that the cluster decomposition principle fails in multi-instanton sectors, challenging the validity of the strong-coupling instanton calculation of the gluino condensate.

Findings

Weak-coupling instanton calculation yields c=1

Strong-coupling instanton calculation yields a different value

Cluster decomposition is invalid in multi-instanton sectors

Abstract

A longstanding puzzle concerns the calculation of the gluino condensate <{tr\lambda^2\over 16\pi^2}> = c\Lambda^3 in N=1 supersymmetric SU(N) gauge theory: so-called weak-coupling instanton (WCI) calculations give c=1, whereas strong-coupling instanton (SCI) calculations give, instead, c=2[(N-1)!(3N-1)]^{-1/N}. By examining correlators of this condensate in arbitrary multi-instanton sectors, we cast serious doubt on the SCI calculation of <{tr\lambda^2\over 16\pi^2}> by showing that an essential step --- namely cluster decomposition --- is invalid. We also show that the addition of a so-called Kovner-Shifman vacuum (in which <{tr\lambda^2\over 16\pi^2}> = 0) cannot straightforwardly resolve this mismatch.