θ dependence of the spectrum of SU(N) gauge theories

We study the θ dependence of the spectrum of four-dimensional SU(N) gauge theories, where θ is the coefficient of the topological term in the Lagrangian, for N ⩾ 3 and in the large-Nlimit. We compute the O(θ²) terms of the expansions around θ = 0 of the string tension and the lowest glueball mass, respectively σ(θ) = σ(1+s₂θ²+...) andM(θ) = M(1+g₂θ²+...), where σ andM are the values at θ = 0. For this purpose we use numerical simulations of the Wilson lattice formulation of SU(N) gauge theories forN = 3,4,6. The O(θ²) coefficients turn out to be very small for allN ⩾ 3. For example, s₂ = −0.08(1) and g₂ = −0.06(2) for N = 3. Their absolute values decrease with increasing N. Our results are suggestive of a scenario in which the θ dependence in the string and glueball spectrum vanishes in the large-N limit, at least for sufficiently small values of |θ|. They support the general large-N scaling arguments that indicate ≡θ/N as the relevant Lagrangian parameter in the large-N expansion.