Asymptotic safety of scalar field theories

We study $3d$ $O(N)$ symmetric scalar field theories using Polchinski’s renormalization group. In the infinite $N$ limit the model is solved exactly including at strong coupling. At short distances the theory is described by a line of asymptotically safe ultraviolet fixed points bounded by asymptotic freedom at weak, and the Bardeen-Moshe-Bander phenomenon at strong sextic coupling. The Wilson-Fisher fixed point arises as an isolated low-energy fixed point. Further results include the conformal window for asymptotic safety, convergence-limiting poles in the complex field plane, and the phase diagram with regions of first and second order phase transitions. We substantiate a duality between Polchinski’s and Wetterich’s versions of the functional renormalization group, also showing that eigenperturbations are identical at any fixed point. At a critical sextic coupling, the duality is worked out in detail to explain the spontaneous breaking of scale symmetry responsible for the generation of a light dilaton. Implications for asymptotic safety in other theories are indicated.

Figure 1

Polchinski-Wetterich duality and propagator sum rule. Shown are different UV-IR decompositions of the massless propagator into a Polchinski part $P_{\mathrm{UV}}$ where $k$ takes the role of a UV cutoff (blue), and a Wetterich part $P_{\mathrm{IR}}$ where $k$ acts as an IR cutoff (red). Only for the decomposition (29), the respective Polchinski and Wetterich flows (21) are exact duals of each other (right panel). Dualities cannot be found for any other decomposition (30) including the sharp momentum cutoff (left panel).

Figure 2

Global fixed points $\rho (u^{\prime })$ of the Polchinski equation in the large-$N$ limit for all fields $-\infty \le \rho \le \infty$ and potentials $-\infty \le u^{\prime }\le 1$, in dependence on the free parameter $c$ and color-coded according to the legend. Axes are rescaled as $\rho \to \rho /(1+|\rho |)$ and $u^{\prime }\to u^{\prime }/(2-u^{\prime })$ for better display. Thick lines indicate the Gaussian fixed point (black, $|c|=\infty$), the Wilson-Fisher fixed point (white, $c=0$), and the Bardeen-Moshe-Bander (BMB) fixed point (grey, $c=3\pi /4$), thin black lines for constant $c$ are added to guide the eye.

Figure 3

Global Wilson-Fisher fixed point from the Polchinski flow in the entire complex field plane, showing the real (left panel) and imaginary (right panel) part of the field variable $\rho =\frac{1}{2}{\varphi }^a{\varphi }^a/k$ as a function of $u^{\prime }$ in the entire complex $u^{\prime }$ plane. Axes are rescaled as $X\to X/(1+|X|)$ for better display. We observe a pole at $u^{\prime }=1$. Along the real $\rho$ axis, we observe that the Wilson-Fisher fixed point (full red line) cannot be extended beyond the pole at $u^{\prime }=1$ without acquiring an imaginary part (dashed red line).

Figure 4

Shown are the Polchinski mass functions $u^{\prime }(\rho )$ (black lines) for a weakly coupled UV fixed point (left panel), and a strongly coupled UV fixed point (right panel) with $c_{\pm }=c_{\mathrm{crit}}\pm {10}^{-2}$. The strong-coupling solution exhibits a turning point at ${\rho }_s$ where $1/{u_{*}}^{\prime \prime }({\rho }_s)=0$, and a secondary solution where the fixed point has become multivalued (dashed line). In either case, it is observed that the weak and strongly interacting fixed points follow from the Wilson-Fisher solution (red line) by a constant shift into either direction (70), with $\mathrm{\Delta }=c(u^{\prime }{)}^{1/2}(1-u^{\prime }{)}^{-5/2}$.

Figure 5

Global fixed point solutions (47) for the Polchinski mass function $u_{*}^{\prime }(\rho )$ (left panel) and the interaction potential $u_{*}(\rho )$ (right panel) for different values of the free parameter $c$. The red (purple) [black] line corresponds to the Wilson-Fisher (Bardeen-Moshe-Bander) [Gaussian] fixed point. The blue region and blue lines correspond to a family of (tricritical) UV fixed points, and dashed curves denote unphysical multivalued solutions at strong coupling which do not extend over all physical fields. The BMB fixed point sticks out, indicating the border between strongly and weakly coupled domains.

Figure 6

Shown is the exact conformal window with asymptotic safety of large-$N$ scalar field theories, characterized by the bare sextic coupling $\tau$. The regime with asymptotic safety (yellow band) is limited by the Gaussian fixed point ($\tau =0$, green line) and the Bardeen-Moshe-Bander fixed point ($\tau ={\tau }_{\mathrm{crit}}$, red line). Regions with incomplete fixed points (light red band) and unstable vacua (gray band) are also indicated.

Figure 7

Polchinski-Wetterich duality. Shown are the mass functions at the BMB fixed point for the Polchinski interaction potential (left panel) and for the dual quantum effective potential (right panel); axes are rescaled as $x\to x/(1+|x|)$. Both mass functions show a nonanalytic singularity at vanishing field. The dotted (blue), dashed (magenta) and full (red) segments in both plots are mapped onto each other under the duality (22). The physical and unphysical branches of the Wetterich flow are connected via analytical continuation (103). We notice that the divergence in the small-field region of the Polchinski flow originates from the large-field behavior in the unphysical branch of the Wetterich flow. Conversely, the small field BMB singularity of the Wetterich mass function—the fingerprint for the spontaneous breaking of scale invariance—does not correspond to a singularity in the Polchinski mass function.

Figure 8

Polchinski-Wetterich duality (continued). Shown are the Polchinski interaction potential $u_{\mathrm{BMB}}(\rho )$ and the Wetterich quantum effective potential $w_{\mathrm{BMB}}(z)$ at the BMB fixed point; axes are rescaled as in Fig. 7. The dotted (blue), dashed (magenta) and full (red) segments of either potential are mapped onto each other under the duality transformation (22). The small-field region of the Wetterich effective potential, which encodes the breaking of scale invariance, relates to the midfield region of the Polchinski interaction potential. The small-field region of the logarithmically unbounded Polchinski potential is related to the unphysical (because globally unstable) branch of the Wetterich potential (103), which is linearly unbounded for large fields.

Figure 9

UV-IR crossover from any of the asymptotically safe fixed points to the unique Wilson-Fisher fixed point. Shown are the running quartic (red) and sextic (blue) couplings along the UV-IR connecting separatrices as functions of $t=\ln (\mu /\mathrm{\Lambda })$ within the conformal window, with $t_{\mathrm{char}}=\ln ({\mu }_{\mathrm{char}}/\mathrm{\Lambda })$ from (118).

Figure 10

The phase diagram of $3d$ large-$N$ scalar field theories in the plane of the quartic ($\lambda$) and sextic ($\tau$) couplings from the Polchinski flow, with arrows indicating the flow towards the infrared. The UV conformal window with asymptotic safety $0\le \tau \le {\tau }_{\mathrm{crit}}$ (blue line) is limited by the Gaussian (G) and the Bardeen-Moshe-Bander (BMB) fixed points (full dots). Along the UV line ($\lambda =0$), UV fixed points with short-distance vacuum instabilities $(\tau <0)$ or incomplete fixed points $(\tau >{\tau }_{\mathrm{crit}})$ are also indicated (dashed blue lines). The Wilson-Fisher (WF) fixed point takes the role of an IR attractive sink for asymptotically safe trajectories with $\delta \kappa =0$ and $\delta \lambda >0$.