Non-Abelian Thirring model at large $N$

Technical Summary: Non-Abelian Thirring Model at Large N

Problem Statement
The paper investigates the non-Abelian bosonized Thirring model for a semi-simple Lie group $G$ at level $k$, deformed by a classically marginal operator bilinear in the currents. The primary objective is to explore the regime where the quadratic Casimir of the group in the adjoint representation, $c_G$, is large compared to the level $k$ ($c_G \gg k$). This corresponds to a large $N$ limit for groups like $SU(N)$.

While the model is well-understood in the semi-classical regime ($c_G/k \ll 1$) via effective actions and $\lambda$-deformations, the behavior in the $c_G \gg k$ limit remains less explored. Previous work on the Fermionic non-Abelian Thirring model by Dashen and Frishman suggested the existence of an additional critical point of order $1/c_G$. Conversely, Destri and de Vega found no such point. This work aims to resolve this discrepancy for the bosonized variant by searching for non-trivial fixed points of order $k/c_G$ (or equivalently $1/N$) using perturbative methods.

Methodology
The authors employ conformal perturbation theory around the Wess-Zumino-Witten (WZW) fixed point ($\lambda=0$). Since the effective action derived for the $c_G/k \ll 1$ regime is inapplicable here, the study relies entirely on diagrammatic expansions.

1. 't Hooft Limit: The authors define an effective 't Hooft coupling $\tilde{\lambda} = \frac{c_G}{k}\lambda$. They perform a perturbative expansion in $\tilde{\lambda}$ while keeping $\tilde{\lambda}$ fixed as $c_G \to \infty$ and $k \to \infty$. In this limit, only the leading orders in $c_G/k$ are retained.
2. Correlation Functions: The study computes two-point correlation functions for:
   • The driving composite operator $O(z, \bar{z}) = J^a(z)\bar{J}^a(\bar{z})$.
   • The current operators $J^a(z)$ and $\bar{J}^a(\bar{z})$.
     Calculations are carried out up to three loops (cubic order in $\lambda$).
3. Regularization and Integration: The computations utilize a specific regularization scheme involving a UV cutoff $\epsilon$ and an IR cutoff $R$. The authors employ a toolkit of complex integrals (derived via Green's theorem and Stokes' theorem) to evaluate the multi-loop diagrams. Structure constants of the Lie algebra are contracted using trace identities in the adjoint representation.
4. Renormalization: From the divergent parts of the correlation functions (specifically logarithmic terms involving $\ln(\epsilon^2/|z_{12}|^2)$), the authors extract the renormalization constants ($Z$-factors). These are used to determine the anomalous dimensions and the $\beta$-function.

Key Contributions and Results

• Composite Operator Correlator ($OO$):
  The authors compute the $OO$ correlator up to three loops. The leading contribution in the $c_G/k$ expansion yields a series in $\tilde{\lambda}$:
  $$\langle O(z_1)O(z_2) \rangle \propto \frac{1}{|z_{12}|^4} \left( 1 - 2\tilde{\lambda} \ln \frac{\epsilon^2}{|z_{12}|^2} + \frac{5}{2}\tilde{\lambda}^2 \ln^2 \frac{\epsilon^2}{|z_{12}|^2} - \frac{5}{2}\tilde{\lambda}^3 \ln^3 \frac{\epsilon^2}{|z_{12}|^2} \right)$$
  From this, the anomalous dimension of the composite operator is found to be linearly exact up to cubic order: $\gamma_O = -2\tilde{\lambda} + O(\tilde{\lambda}^4)$.

• Current Correlators ($JJ$ and $J\bar{J}$):
  The three-loop corrections to the $JJ$ and $J\bar{J}$ correlators are computed.

  • For the $JJ$ correlator, the anomalous dimension of the current operator is found to be:
    $$\gamma_J = \frac{k}{c_G} (\tilde{\lambda}^2 + \tilde{\lambda}^3 + O(\tilde{\lambda}^4))$$
    In the strict 't Hooft limit ($c_G \to \infty$), the leading order anomalous dimension vanishes.
  • For the $J\bar{J}$ correlator, the contact terms are analyzed to extract the renormalization of the coupling.

• $\beta$-Function and Fixed Points:
  Using the renormalization group equations derived from the current and composite operator data, the $\beta$-function for the 't Hooft coupling $\tilde{\lambda}$ is determined.

  • The result is quadratic exact up to quartic order:
    $$\beta_{\tilde{\lambda}} = -\frac{\tilde{\lambda}^2}{2} + O(\tilde{\lambda}^5)$$
  • The authors explicitly state that no additional critical point of order $k/c_G$ (or $1/N$) exists within the perturbative regime calculated (up to quartic order in $\tilde{\lambda}$). The only fixed point is the UV fixed point at $\tilde{\lambda} = 0$.

Significance and Claims
The paper claims that its findings align with the results of Destri and de Vega for the Fermionic non-Abelian Thirring model, which found no additional fixed points. This directly contradicts the claim by Dashen and Frishman regarding the existence of an additional fixed point of order $1/c_G$.

The authors argue that the points $\lambda = \pm 1$, which correspond to non-Abelian T-dual and pseudo-dual models, are artifacts of the semi-classical limit ($c_G/k \ll 1$) where the effective action is valid. In the large $N$ ($c_G \gg k$) regime, these points do not represent non-trivial fixed points of the renormalization group flow. The work suggests that the $\beta$-function truncates to a simple quadratic form in this limit, implying a simpler RG flow structure than previously hypothesized for the existence of intermediate fixed points.

Future Directions
The paper suggests extending these results to prove the exact truncation of the $\beta$-function and anomalous dimensions for all orders in $\tilde{\lambda}$ in the $c_G \gg 1$ limit. It also proposes computing three-point functions to deduce the Operator Product Expansion (OPE) and the underlying symmetry algebra at large $N$, as well as investigating the model with unequal left and right levels ($k_L \neq k_R$).