The Large-$N$ Limit with Vanishing Leading Order Condensate for Zero Pion Mass

Abstract

It is conventionally assumed that the negative mass squared term in the linear sigma model version of the pion Lagrangian is M 2 ∼ Λ 2 QCD in powers of N c [1]. We consider the case where M 2 ∼ Λ 2 QCD /N c so that to leading order in N c this symmetry breaking term vanishes. We present some arguments why this might be plausible. One might think that such a radical assumption would contradict lattice Monte Carlo data on QCD as function of N c . We show that the linear sigma model gives a fair description of the data of DeGrand and Liu both for N c = 3, and for variable N c . The values of quark masses considered by DeGrand and Liu, and by Bali et. al. turn out to be too large to resolve the case we consider from that of the conventional large N c limit [2,3]. We argue that for quark masses m q Λ QCD /N 3/2 c , both the baryon mass and nucleon size scale as √ N c . For m q Λ QCD /N 3/2 c N c limit, where nuclear matter is bound with binding energies of order the nucleon mass and forms a Skyrme crystal.