On the mass of the world-sheet 'axion' in SU(N) gauge theories in 3Â +Â 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). Moreover it has been shown that the natural coupling of this pseudoscalar has a topological interpretation making it natural to call it the world-sheet 'axion'. Recently it has been pointed out that if the mass of this 'axion' vanishes as N→∞ 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=∞ and so test this interesting possibility. We find that the 'axion' does not in fact become massless as N→∞. 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.