Revisiting the $φ^6$ Theory in Three Dimensions at Large $N$

TL;DR

The paper reexamines the three-dimensional $O(N)$-symmetric $φ^6$ theory in the large-$N$ limit using dimensional regularisation and MSbar, clarifying the fate of the tricritical line, BMB phenomenon, and Pisarski’s UV fixed point. By constructing the effective potential at leading and next-to-leading order through both 2PI composite-operator methods and standard large-$N$ resummations, the authors show that the apparent tricritical line is an artefact of the strict $N\to\infty$ limit and that the BMB mechanism does not persist at NLO. They compute the NLO effective potential explicitly, including nontrivial momentum-dependent ring diagrams, and demonstrate that quantum corrections stabilize the potential and that scale invariance does not spontaneously break away from the UV fixed point. The results indicate the Pisarski fixed point survives large-$N$ corrections, while the tricritical line and BMB endpoint cease to constrain the RG flow in a fully consistent $1/N$ expansion, aligning with (and potentially extendable to) FRG analyses.

Abstract

We investigate the $O(N)$--symmetric $φ^6$ theory in three spacetime dimensions using dimensional regularisation and minimal subtraction. The predictions of other methods are scrutinised in a large-$N$ expansion. We show how the tricritical line of fixed point emerges in a strict $N\to\infty$ limit but argue that it is not a physical manifestation. For the first time in this explicit manner, we compute the effective potential at next-to-leading order in the $1/N$-expansion and discuss its stability. The Bardeen-Moshe-Bander phenomenon is also analysed at next-to-leading order, and we demonstrate that it disappears without breaking the scale invariance spontaneously. Our findings indicate that the UV fixed point found by Pisarski persists at large $N$.

Figures (5)

• Figure 1: Schematic overview of the phase diagram for the theory \ref{['eq:lag']}. On a slice of a fixed value of $\eta$, the unbroken $O(N)$--symmetric phase (light grey surface) and broken $O(N-1)$ symmetry phase (light blue) are shown at tree level. A second-order phase transitions occurs at $\lambda > 0$, $m^2=0$ (solid blue line) and a first-order one at $\lambda < 0$, $m^2 = \tfrac{5}{8} \lambda^2/\eta$ (dashed blue line). Taking quantum corrections into account, a tricritical line of fixed points for $m^2 = \lambda = 0$ and $\eta > 0$ is reported David:1984weDavid:1985zzLitim:2017cnl in a strict $N \to \infty$ limit (red line). However, the line terminates at the BMB endpoint (red) Bardeen:1983rv. Beyond that line, there is a potential UV fixed point (yellow) found in a $1/N$-expansion Townsend:1976syAppelquist:1981sfPisarski:1982vzHager:2002uq.
• Figure 2: $V_\text{2PI}(M)$ for a fixed value $\varphi > 0$ and $\bar{\eta} < 1$ (top panel), $\bar{\eta} = 1$ (middle panel) and $\bar{\eta} > 1$ (bottom panel). Only values $M \geq 0$ (solid blue line) are valid arguments of $V_\text{2PI}(M)$, while $V_\text{2PI}(M < 0)$ (dashed blue line) is only drawn for convenience. The extrema $M_{\pm}$\ref{['eq:dynamical-mass']} are indicated by blue dots. The minimum $M_+$ always exists, while the maximum $M_-$ is only found for $\bar{\eta} > 1$. At $\varphi = 0$, both points are shrunk together at $M_{\pm} = 0$, and $V_\text{2PI}$ becomes flat for $\bar{\eta} = 1$. The plot is adapted from Matsubara:1987iz.
• Figure 3: Sextic term of the classical (solid grey) and next-to-leading order effective potential \ref{['eq:Veff-NLO']} as a function of the rescaled sextic coupling as a function of $\bar{\eta}$, evaluated at the renormalisation scale $\mu = 4\pi \varphi^2/N$, for various choices of $N$. The limit $N \to \infty$ (blue) singles out the LO effective potential \ref{['eq:Veff-LO']}, which has a smaller but stable sextic coupling compared to the classical case (solid grey). With decreasing $N$, the potential remains stable (green) until it turns unstable at integer values $N < 8$ (yellow). However, the large-$N$ expansion might not be reliable in this case. The BMB endpoint $\bar{\eta}_\text{BMB} = 1$ (grey dashed) and Pisarski's UV fixed point \ref{['eq:Pisarski-bar']}$\bar{\eta}^* \approx 12/\pi^2$ (red) are also shown.
• Figure 4: Leading-order 2PI potential \ref{['eq:BMB-stability']} at $\varphi=0$ and as a function of $M$. For $\bar{\eta} < 1$, the potential is bounded from below and has a minimum at $M=0$, while it is unstable for $\bar{\eta} > 1$. At $\bar{\eta} = 1$, $V_\text{2PI}(0,M)$ is flat and the BMB phenomenon occurs.
• Figure 5: Schematic dependence of the effective potential at next-to leading order in large-$N$\ref{['eq:V-c12']} on the sign of $c_1$. For $c_1 > 0$, there is a global minimum at $\mathscr{M} \neq 0$, and $V_\mathscr{M}$ is bounded from below. With $c_1 < 0$, there is a local minimum $\mathscr{M} = 0$, the global maximum lies at $\mathscr{M} > 0$, and $V_\mathscr{M}$ is not bounded from below for $\mathscr{M}/\mu \to \infty$.