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On RG Flow of tau_{RR} for Supersymmetric Field Theories in Three-Dimensions

Tatsuma Nishioka, Kazuya Yonekura

TL;DR

The paper tests whether the 3d N=2 C_T analogue, encoded in tau_RR, decreases along RG flows. Using localization on a squashed S^3 and Z-extremization, the authors compute tau_RR for various theories, including Wess-Zumino models and U(N_c) SQCD with flavors, in both large-N_f and numerical regimes. They find tau_RR generally decreases along RG flows in gauge theories, consistent with a monotonic C_T-like behavior, but reveal a counterexample in a Wess-Zumino model where tau_RR increases along the flow, while the free energy F on S^3 still decreases in line with the F-theorem. This combination of results supports the F-theorem in 3d while ruling out tau_RR/C_T monotonicity as a universal measure of degrees of freedom, at least for 3d CFTs. The study highlights the nuanced relationship between R-current correlators, stress-tensor correlators, and RG flow in diverse 3d supersymmetric theories.

Abstract

The coefficient tau_{RR} of the two-point function of the superconformal U(1)_R currents of N=2 SCFTs in three-dimensions is recently shown to be obtained by differentiating the partition function on a squashed three-sphere with respect to the squashing parameter. With this method, we compute the tau_{RR} for N=2 Wess-Zumino models and SQCD numerically for small number of flavors and analytically in the large number limit. We study the behavior of tau_{RR} under an RG flow by adding superpotentials to the theories. While the tau_{RR} decreases for the gauge theories, we find an N=2 Wess-Zumino model whose tau_{RR} increases along the RG flow. Since tau_{RR} is proportional to the coefficient C_T of the two-point correlation function of the stress-energy tensors for N=2 superconformal field theories, this rules out the possibility of C_T being a measure of the degrees of freedom which monotonically decreases along RG flows in three-dimensions.

On RG Flow of tau_{RR} for Supersymmetric Field Theories in Three-Dimensions

TL;DR

The paper tests whether the 3d N=2 C_T analogue, encoded in tau_RR, decreases along RG flows. Using localization on a squashed S^3 and Z-extremization, the authors compute tau_RR for various theories, including Wess-Zumino models and U(N_c) SQCD with flavors, in both large-N_f and numerical regimes. They find tau_RR generally decreases along RG flows in gauge theories, consistent with a monotonic C_T-like behavior, but reveal a counterexample in a Wess-Zumino model where tau_RR increases along the flow, while the free energy F on S^3 still decreases in line with the F-theorem. This combination of results supports the F-theorem in 3d while ruling out tau_RR/C_T monotonicity as a universal measure of degrees of freedom, at least for 3d CFTs. The study highlights the nuanced relationship between R-current correlators, stress-tensor correlators, and RG flow in diverse 3d supersymmetric theories.

Abstract

The coefficient tau_{RR} of the two-point function of the superconformal U(1)_R currents of N=2 SCFTs in three-dimensions is recently shown to be obtained by differentiating the partition function on a squashed three-sphere with respect to the squashing parameter. With this method, we compute the tau_{RR} for N=2 Wess-Zumino models and SQCD numerically for small number of flavors and analytically in the large number limit. We study the behavior of tau_{RR} under an RG flow by adding superpotentials to the theories. While the tau_{RR} decreases for the gauge theories, we find an N=2 Wess-Zumino model whose tau_{RR} increases along the RG flow. Since tau_{RR} is proportional to the coefficient C_T of the two-point correlation function of the stress-energy tensors for N=2 superconformal field theories, this rules out the possibility of C_T being a measure of the degrees of freedom which monotonically decreases along RG flows in three-dimensions.

Paper Structure

This paper contains 18 sections, 63 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. $\tau_{RR}$ in ${\cal N} = 2$ supersymmeric field theories
  3. Localization on a squashed three-sphere
  4. Calculation of $\tau_{RR}$-coefficient
  5. Wess-Zumino model:
  6. Chern-Simons theory:
  7. Large-$N$ gauge theories:
  8. ${\cal N} = 2$ $U(N_c)$ gauge theory with flavors
  9. Non-chiral theory
  10. Large-$N_f$ limit
  11. Numerical computation for small $N_f$
  12. Chiral theory
  13. More general theories
  14. RG flow with increasing $\tau_{RR}$
  15. Hyperbolic gamma function
  16. ...and 3 more sections

Figures (2)

  • Figure 1: Plots of $\tau_{RR}^{{\cal N}=2}$ and $\tau_{RR}^{{\cal N}=3}$ as functions of $N_f$ for the non-chiral $U(N_c)$ SQCD of $N_c=1$ [Left] and $N_c =2$ [Right]. The solid orange and dotted blue curves are drawn using the large-$N_f$ expansion at the ${\cal N}=2$ and ${\cal N}=3$ fixed points, respectively. The dashed black lines are the values at the UV fixed point. The orange and blue dots are computed numerically. They fit the large-$N_f$ approximation very well even for small $N_f$.
  • Figure 2: A plot of $\tau_{RR}$ as a function of $N$ for the Wess-Zumino model \ref{['eq:WZsuperpot']} [Left]. A plot of $F$ for the same model [Right]. The solid orange curves are calculated using the large-$N$ expansion at the fixed point with Eqs. \ref{['eq:WZlargeN']} and \ref{['WZlargeNF']}. The orange dots are computed numerically. The dashed blue lines are the values at the IR free theory with Eqs. \ref{['WZIR']} and \ref{['WZIRF']}.