Breakdown of Cluster Decomposition in Instanton Calculations of the Gluino Condensate

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.