Journal of High Energy Physics (Dec 2017)
On the Large R-charge Expansion in N=2 $$ \mathcal{N}=2 $$ Superconformal Field Theories
Abstract
Abstract In this note we study two point functions of Coulomb branch chiral ring elements with large R-charge, in quantum field theories with N=2 $$ \mathcal{N}=2 $$ superconformal symmetry in four spacetime dimensions. Focusing on the case of one-dimensional Coulomb branch, we use the effective-field-theoretic methods of [1], to estimate the two-point correlation function Yn≡x−y2nΔOOxnO¯yn $$ {\mathcal{Y}}_n\equiv {\left|x-y\right|}^{2n{\Delta}_{\mathcal{O}}}\left\langle {\left(\mathcal{O}(x)\right)}^n{\left(\overline{\mathcal{O}}(y)\right)}^n\right\rangle $$ in the limit where the operator insertion On $$ {\mathcal{O}}^n $$ has large total R-charge J=nΔO $$ \mathcal{J}=n{\Delta}_{\mathcal{O}} $$. We show that Yn $$ {\mathcal{Y}}_n $$ has a nontrivial but universal asymptotic expansion at large J $$ \mathcal{J} $$, of the form Y n = J ! N O 2 π 2 J J α Y ˜ n , $$ {\mathcal{Y}}_n=\mathcal{J}!{\left(\frac{\left|{\mathbf{N}}_{\mathcal{O}}\right|}{2\uppi}\right)}^{2\mathcal{J}}{\mathcal{J}}^{\alpha }{\tilde{\mathcal{Y}}}_n, $$ where Y˜n $$ {\tilde{\mathcal{Y}}}_n $$ approaches a consstant as n → ∞, and NO $$ {\mathbf{N}}_{\mathcal{O}} $$ 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 for the logarithm ℬn=logYn $$ {\mathrm{\mathcal{B}}}_n= \log \left({\mathcal{Y}}_n\right) $$, up to and including order log J $$ \mathcal{J} $$ against exact results from supersymmetric localization [2-5]. In the case of N=4 $$ \mathcal{N}=4 $$ we find precise agreement and in the case N=2 $$ \mathcal{N}=2 $$ we find reasonably good numerical agreement at J≃60 $$ \mathcal{J}\simeq 60 $$ using the no-instanton approximation to the S 4 partition function. We also give predictions for the growth of two-point functions in all rank-one SCFT in the classification of [6-9]. 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 ssof the moduli space of vacua.
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