On the mass of the world-sheet `axion' in SU(N) gauge theories in 3+1 dimensions

There is numerical evidence that the world sheet action of the confining flux tube in D=3+1 SU(N) gauge theories contains a massive excitation with 0- quantum numbers whose mass shows some decrease as one goes from SU(3) to SU(5). It has furthermore been shown that this particle is naturally described as arising from a topological interaction term in the world-sheet action, so that one can describe it as being `axion'-like. Recently it has been pointed out that if the mass of this `axion' vanishes as N -> oo then it becomes possible for the world sheet theory to be integrable in the planar limit. In this paper we perform lattice calculations of this `axion' mass from SU(2) to SU(12), which allows us to make a controlled extrapolation to N=oo and so test this interesting possibility. We find that the `axion' does not in fact become massless as N -> oo. So if the theory is to possess planar integrability then it must be some other world sheet excitation that becomes massless in the planar limit.