Theta-dependence of the spectrum of SU(N) gauge theories

We study the theta dependence of the spectrum of four-dimensional SU(N) gauge theories, where theta is the coefficient of the topological term in the Lagrangian, for N>=3 and in the large-N limit. We compute the O(theta^2) terms of the expansions around theta=0 of the string tension and the lowest glueball mass, respectively sigma(theta) = sigma (1 + s_2 theta^2 + ...) and M(theta) = M (1 + g_2 theta^2 + ...), where sigma and M are the values at theta=0. For this purpose we use numerical simulations of the Wilson lattice formulation of SU(N) gauge theories for N=3,4,6. The O(theta^2) coefficients turn out to be very small for all N>=3. For example, s_2=-0.08(1) and g_2=-0.06(2) for N=3. Their absolute values decrease with increasing N. Our results are suggestive of a scenario in which the theta dependence in the string and glueball spectrum vanishes in the large-N limit, at least for sufficiently small values of |theta|. They support the general large-N scaling arguments that indicate (theta/N) as the relevant Lagrangian parameter in the large-N expansion.