Walking, Weak first-order transitions, and Complex CFTs

Abstract

We discuss walking behavior in gauge theories and weak first-order phase transitions in statistical physics. Despite appearing in very different systems (QCD below the conformal window, the Potts model, deconfined criticality) these two phenomena both imply approximate scale invariance in a range of energies and have the same RG interpretation: a flow passing between pairs of fixed point at complex coupling. We discuss what distinguishes a real theory from a complex theory and call these fixed points complex CFTs. By using conformal perturbation theory we show how observables of the walking theory are computable by perturbing the complex CFTs. This paper discusses the general mechanism while a companion paper [1] will treat a specific and computable example: the two-dimensional Q-state Potts model with Q > 4. Concerning walking in 4d gauge theories, we also comment on the (un)likelihood of the light pseudo-dilaton, and on non-minimal scenarios of the conformal window termination.

Figures (10)

• Figure 1: Structure of RG flow for real coupling for $y<0$ (left) and $y>0$ (right).
• Figure 2: Structure of RG flow in the complex coupling plane, in the approximation of dropping the higher order terms in (\ref{['eq:RGwalk']}). Notice that including those terms will generically change the nature of RG flow trajectories around ${\cal C}$ and $\overline{\cal C}$, since the RG eigenvalue will then acquire a small real part $O(y^2)$, making the flow spirally in- or unwinding. See part2 for an example.
• Figure 3: An example of a random subset $X$ of bonds on a $4 \times 4$ square lattice. Here $b(X)=11$ and $c(X)=6$. Notice that isolated points count as clusters.
• Figure 4: Left: schematic view of the space of existing fixed points of 4d gauge theories as a function of $x=N_f/N_c$. The trivial fixed point (free) exists for any $x$. The line of BZ fixed points branches off from the free theory line at $x=x_{\rm AF}$. At some smaller $x=x_c$ it annihilates with the line of QCD$^*$ fixed points. This latter line should exist for $x$ close to $x_c$ but it's not a priori clear where it starts. Right: schematic view of the space of fixed points for the 2d Potts model. No theory merges with the free theory, at least in the range $Q>0$ we are interested in.
• Figure 5: Conformal window for the ${\cal N}=1$ SUSY case. In this case the BZ-like fixed point disappears by merging with another free theory. Compare with Fig. \ref{['fig:QCDvsPotts']} in the non-SUSY case.