Abstract
Carrying to higher precision the large- mathcal{J} expansion of [1], we calculate to all orders in 1/mathcal{J} the power-law corrections to the two-point functions {mathcal{Y}}_nequiv {left|x-yright|}^{2n{varDelta}_{mathcal{O}}}cdot le leftlangle {mathcal{O}}^n(x){overline{mathcal{O}}}^n(y)rightrangle for generators mathcal{O} of Coulomb branch chiral rings in four-dimensional mathcal{N}=2 superconformal field theories. We show these correlators have the universal large-n expansionlogYn≃JA+B+logΓJ+α+1,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\log \\left({\\mathcal{Y}}_n\\right)\\simeq \\mathcal{J}\\mathbf{A}+\\mathbf{B}+\\log \\left(\\Gamma \\left(\\mathcal{J}+\\alpha +1\\right)\\right), $$\\end{document}where mathcal{J}equiv n{Delta}_{mathcal{O}} is the total R-charge of mathcal{O} n, the A and B are theory-dependent coefficients, α is the coefficient of the Wess-Zumino term for the Weyl a-anomaly, and the ≃ denotes equality up to terms exponentially small in mathcal{J} . Our methods combine the structure of the Coulomb-branch eft with the supersymmetric recursion relations. However, our results constrain the power-law corrections to all orders, even for non-Lagrangian theories to which the recursion relations do not apply. For the case of mathcal{N}=2 sqcd, we also comment on the nature of the exponentially small corrections, which can be calculated to high precision in the double-scaling limit recently discussed by Bourget et al. in [2]. We show the exponentially small correction is consistent with the interpretation of the eft breaking down due to the propagation of massive bps particles over distances of order of the infrared scale |x − y|.
Highlights
Log(Yn) J A + B + log(Γ(J + α + 1)), where J ≡ n∆O is the total R-charge of On, the A and B are theory-dependent coefficients, α is the coefficient of the Wess-Zumino term for the Weyl a-anomaly, and the denotes equality up to terms exponentially small in J
The wz terms arising both for the Weyl symmetry and U(1) R-charge are needed to compensate the difference between the anomaly coefficients of the underlying cft and the eft of the Coulomb branch
We have studied N = 2 superconformal field theories with a one-complexdimensional Coulomb branch in a sector of fixed and large R-charge J
Summary
Since two-point functions can in principle be computed by supersymmetric localization, they are necessarily independent of D-terms It follows that all power-law 1/nm corrections to the logarithm of the correlator are necessarily independent of the details of the microscopic theory, depending only on the α-coefficient. In a Lagrangian theory, φhol is the superfield whose effective kinetic term is Lkinetic = −i × (const.) × d4θN =2 τ Φ2hol + (h.c.). For purposes of quantizing the effective theory of the Coulomb branch, it is more convenient to work in terms of the field φunit ≡ Im(τ ), whose kinetic term is Lkinetic = (const.) × d4θN =2 Φ2unit + (h.c.) = |∂φunit|2 + fermion kinetic + gauge kinetic (2.5).
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