Breakdown of Cluster Decomposition in Instanton Calculations of the Gluino Condensate

TL;DR

This paper resolves a long-standing discrepancy in ${\cal N}=1$ SU(N) gauge theory by showing that strong-coupling instanton (SCI) calculations violate cluster decomposition in multi-instanton sectors, casting doubt on the SCI value for the gluino condensate $\langle {\rm tr}\lambda^2/16\pi^2\rangle=c\Lambda^3$. Using multi-instanton correlators, it demonstrates that a Kovner–Shifman vacuum cannot restore clustering for topological number $k>1$, and provides a large-$N$ analytic evaluation of $kN$-point functions that yields a linear-in-$k$ scaling incompatible with both SCI and KS reconciliations. A dedicated SU(2) calculation of the 4-point function at two instantons finds $c\approx0.50$, reinforcing the argument that instanton-based clustering fails in the strongly coupled regime. The authors argue that additional nonperturbative configurations (instanton partons/monopole constituents) are needed to restore clustering and align with the weak-coupling result, a perspective coherent with Seiberg–Witten analyses and pointing toward a more complete nonperturbative framework for SUSY gauge dynamics.

Abstract

A longstanding puzzle concerns the calculation of the gluino condensate <{trλ^2\over 16π^2}> = cΛ^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 <{trλ^2\over 16π^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 <{trλ^2\over 16π^2}> = 0) cannot straightforwardly resolve this mismatch.