On the Large $R$-charge Expansion in ${\mathcal N} = 2$ Superconformal Field Theories
Simeon Hellerman, Shunsuke Maeda
TL;DR
The paper develops a universal large-${\cal J}$ expansion for two-point functions of Coulomb-branch chiral ring generators in 4D ${\cal N}=2$ SCFTs with a one-complex-dimensional moduli space. Using the moduli-space effective field theory, it shows ${\cal Y}_{n} = {\cal J}! \left(\frac{|{\mathbf N}_{\cal O}|}{2\pi}\right)^{2{\cal J}} {\cal J}^{\alpha} \tilde{\cal Y}_{n}$ with ${\cal J}=n\Delta_{\cal O}$ and $\tilde{\cal Y}_{n}$ approaching a constant; the exponent $\alpha$ is fixed by anomaly matching via the Wess–Zumino term. The leading nontrivial dependence on ${\cal J}$ arises from the Wess–Zumino coupling to the Euler density, yielding $\alpha$ proportional to the difference in the $a$-anomaly between the UV CFT and the moduli-space EFT, and is verified against exact localization results for ${\cal N}=4$ SYM and ${\cal N}=2$ SCQCD. The work builds a bridge between conformal data and low-energy moduli dynamics, offering precise predictions for rank-one theories and providing sum rules to test large-charge behavior. The results illuminate how macroscopic large-charge sectors are governed by universal EFT structures and anomaly physics, with localization serving as a nonperturbative check.
Abstract
In this note we study two point functions of Coulomb branch chiral ring elements with large $R$-charge, in quantum field theories with ${\mathcal N} = 2$ superconformal symmetry in four spacetime dimensions. Focusing on the case of one-dimensional Coulomb branch, we use the effective-field-theoretic methods of arXiv:1706.05743, to estimate the two-point function $${\mathcal Y}_n \equiv |x-y|^{2nΔ_{\mathcal O}}\left<({\mathcal O}(x))^n(\bar{\mathcal O}(y))^n\right>$$ in the limit where the operator insertion On has large total $R$-charge ${\mathcal J} = nΔ_{\mathcal O}$. We show that ${\mathcal Y}_n$ has a nontrivial but universal asymptotic expansion at large ${\mathcal J}$, of the form $${\mathcal Y}_n = {\mathcal J}! \left(\frac{\left| {\mathbf N}_{\mathcal O}\right|}{2π}\right)^{2{\mathcal J}}{\mathcal J}^α{\tilde{\mathcal Y}}_n$$ where ${\mathcal Y}_n$ approaches a constant as $n\to\infty$, and ${\mathbf N}_{\mathcal O}$ is an $n$-independent constant describing on the normalization of the operator relative to the effective Abelian gauge coupling. The exponent $α$ is a positive number proportional to the difference between the $a$-anomaly coefficient of the underlying CFT and that of the effective theory of the Coulomb branch. For Lagrangian SCFT, we check our predictions against exact results from supersymmetric localization of Baggio et. al. and Gerchkovitz et. al., and find precise agreement for the logarithm ${\mathcal B}_n = \log{\mathcal Y}_n$, up to and including order $\log{\mathcal J}$. We also give predictions for the growth of two-point functions in all rank-one SCFT in the classification of Argyres et. al. In this way, we show the large-$R$-charge expansion serves as a bridge from the world of unbroken superconformal symmetry, OPE data, and bootstraps, to the world of the low-energy dynamics of the moduli space of vacua.