[Paper Review] Bifundamental Multiscalar Fixed Points in $d=3-ε$

Samarth Kapoor, Shiroman Prakash|arXiv (Cornell University)|Dec 2, 2021

Black Holes and Theoretical Physics4 citations

TL;DR

This paper studies bifundamental multiscalar fixed points in $d=3-\epsilon$ dimensions with $O(N)\times O(M)/\mathbb{Z}_2$ symmetry, generalizing the tricritical sextic vector model. Using large-$N$ and large-$M/N$ expansions, it identifies a unique infrared fixed point in $d=3-\epsilon$ to $\mathcal{O}(\epsilon^2)$, and finds a UV fixed point merging with it at $\epsilon \sim O(M/N)$, suggesting a potential nonsupersymmetric CFT dual to Einstein gravity.

ABSTRACT

We study fixed-points of scalar fields that transform in the bifundamental representation of $O(N) imes O(M)$ in $3-ε$ dimensions, generalizing the classic tricritical sextic vector model. In the limit where $N$ is large but $M$ is finite, we determine the complete beta function to order $1/N$ for arbitrary $M$. We find a rich collection of large-$N$ fixed-points in $d=3$, as well as fixed-points in $d=3-ε$, that can be studied to all orders in the parameter $\hatε=Nε$. With the goal of defining a large-$N$ nonsupersymmetric conformal field theory dominated by a web of planar diagrams, we also study fixed-points in the ``bifundamental'' large-$N$ limit, in which $M$ and $N$ are both large, but the ratio $M/N$ is held fixed. We find a unique infrared fixed-point in $d=3-ε$, which we determine to order $ε^2$. When $M/N \ll 1$, we also find an ultraviolet fixed-point in $d=3$ and $d=3-ε$ that merges with the infrared fixed-point at $ε\sim O(M/N)$. We expect at least one of two candidate fixed-points in integer dimensions -- the infrared fixed-point in $d=2$ and the ultraviolet fixed-point in $d=3$ -- to survive for finite values of $M/N$.

Motivation & Objective

• To investigate the existence and structure of conformal fixed points in $O(N)\times O(M)/\mathbb{Z}_2$-symmetric scalar theories in $d=3-\epsilon$ dimensions.
• To explore the phase diagram of such theories in the large-$N$ limit with finite $M$, and in the bifundamental large-$N$ limit where $M/N$ is held fixed.
• To test the conjecture that nonsupersymmetric CFTs with planar diagram dominance and holographic duals to Einstein gravity may exist, by analyzing fixed points in a controlled $\epsilon$-expansion.
• To identify unconventional fixed points with complex or imaginary eigenvalues in the stability matrix, which could signal limit cycles or exotic RG flows.

Proposed method

• Derives the complete beta function to $\mathcal{O}(1/N)$ in the large-$N$ limit with finite $M$, using perturbative $\epsilon$-expansion and large-$N$ techniques.
• Applies a gradient flow formalism by expressing the beta functions as the gradient of a potential, using a metric $T$ derived from the inverse of the two-point function.
• Performs a systematic analysis of fixed points in both the $N \to \infty$ and $M/N$ finite limits, including $\mathcal{O}(\epsilon^2)$ corrections to the fixed-point structure.
• Uses the $\hat{\epsilon} = N\epsilon$ parameter to rescale the theory and study fixed points to all orders in $\hat{\epsilon}$, enabling a controlled expansion.
• Analyzes the stability matrix $\partial\beta_i / \partial g_j$ to detect unconventional fixed points with complex or purely imaginary eigenvalues, indicating potential limit cycles.
• Extends the analysis to non-integer $M$ to probe non-unitary regimes and the possibility of exotic RG flows, using category-theoretic formulations of $O(M)$ symmetry.

Experimental results

Research questions

• RQ1 Does a unique infrared fixed point exist in $d=3-\epsilon$ for the $O(N)\times O(M)/\mathbb{Z}_2$ bifundamental scalar theory in the large-$N$ limit with finite $M/N$?
• RQ2 Can an ultraviolet fixed point emerge in $d=3$ when $M/N \ll 1$, and does it merge with the infrared fixed point at $\epsilon \sim O(M/N)$?
• RQ3 Are there fixed points with complex or purely imaginary eigenvalues in the stability matrix, indicating limit cycles or unconventional RG flows?
• RQ4 Can the theory support a nonsupersymmetric CFT with a planar diagram web and a holographic dual to Einstein gravity, as suggested by the $\epsilon$-expansion and fixed-point structure?

Key findings

• A unique infrared fixed point exists in $d=3-\epsilon$ for finite $M/N$, determined to $\mathcal{O}(\epsilon^2)$, with couplings that depend on $M/N$ and $\epsilon$.
• When $M/N \ll 1$, an ultraviolet fixed point emerges in $d=3$ and $d=3-\epsilon$, which merges with the infrared fixed point at $\epsilon \sim O(M/N)$.
• For non-integer $M$ in the range $0 < M < 2$, the theory exhibits fixed points with complex eigenvalues in the stability matrix, indicating spiral-like RG trajectories, but no purely imaginary eigenvalues are found, so no limit cycles occur.
• In the range $-3 < M < 0$, no fixed points with purely imaginary eigenvalues are found, and for $M < -3$, the inverse metric becomes negative definite, ruling out unconventional fixed points.
• The fixed points in integer dimensions ($d=3$ and $d=2$) are expected to survive for finite $M/N$, with at least one of the infrared ($d=2$) or ultraviolet ($d=3$) fixed points being physical.
• The theory supports a rich structure of large-$N$ fixed points in $d=3$, including non-unitary fixed points with exotic stability matrix properties, though no limit cycles are realized.