Journal of High Energy Physics (Nov 2021)
Exact expressions for n-point maximal U(1) Y -violating integrated correlators in SU(N) N $$ \mathcal{N} $$ = 4 SYM
Abstract
Abstract The exact expressions for integrated maximal U(1) Y violating (MUV) n-point correlators in SU(N) N $$ \mathcal{N} $$ = 4 supersymmetric Yang-Mills theory are determined. The analysis generalises previous results on the integrated correlator of four superconformal primaries and is based on supersymmetric localisation. The integrated correlators are functions of N and τ = θ/(2π) + 4πi/ g YM 2 $$ {g}_{YM}^2 $$ , and are expressed as two-dimensional lattice sums that are modular forms with holomorphic and anti-holomorphic weights (w, −w) where w = n − 4. The correlators satisfy Laplace-difference equations that relate the SU(N+1), SU(N) and SU(N−1) expressions and generalise the equations previously found in the w = 0 case. The correlators can be expressed as infinite sums of Eisenstein modular forms of weight (w, −w). For any fixed value of N the perturbation expansion of this correlator is found to start at order ( g YM 2 $$ {g}_{YM}^2 $$ N) w . The contributions of Yang-Mills instantons of charge k > 0 are of the form q k f(g YM ), where q = e 2πiτ and f(g YM ) = O( g YM − 2 w $$ {g}_{YM}^{-2w} $$ ) when g YM 2 $$ {g}_{YM}^2 $$ ≪ 1. Anti-instanton contributions have charge k < 0 and are of the form q ¯ k f ̂ g YM $$ {\overline{q}}^{\left|k\right|}\hat{f}\left({g}_{YM}\right) $$ , where f ̂ g YM = O g YM 2 w $$ \hat{f}\left({g}_{YM}\right)=O\left({g}_{YM}^{2w}\right) $$ when g YM 2 $$ {g}_{YM}^2 $$ ≪ 1. Properties of the large-N expansion are in agreement with expectations based on the low energy expansion of flat-space type IIB superstring amplitudes. We also comment on the identification of n-point free-field MUV correlators with the integrands of (n − 4)-loop perturbative contributions to the four-point correlator. In particular, we emphasise the important rôle of SL(2, ℤ)-covariance in the construction.
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