  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.
