Conformality Lost

TL;DR

This work investigates how conformal behavior is lost in quantum field theories at zero temperature. By unifying RG insights with concrete models—nonrelativistic 1/r^2 systems, relativistic defect QFTs, holographic BF-bound considerations, and large-N_c, N_f QCD—the authors identify fixed-point merger as a central mechanism (mechanism (iii)) leading to Berezinskii–Kosterlitz–Thouless–like scaling of the mass gap and correlation lengths. They demonstrate that two fixed points can annihilate into the complex plane, producing an exponential sensitivity to parameters and a generated infrared scale, and they speculate that the chiral transition in QCD at the lower edge of the conformal window realizes this universality class (with a possible UV fixed point QCD*). Across the examples, operator dimensions satisfy Δ_+ + Δ_- = d+2 or deviate under certain strong-coupling scalar sectors, linking field theory phenomena to holographic intuition via BF bound considerations. The results have broad implications for understanding conformal windows, walking technicolor scenarios, and the nonperturbative structure of QCD-like theories.

Abstract

We consider zero-temperature transitions from conformal to non-conformal phases in quantum theories. We argue that there are three generic mechanisms for the loss of conformality in any number of dimensions: (i) fixed point goes to zero coupling, (ii) fixed point runs off to infinite coupling, or (iii) an IR fixed point annihilates with a UV fixed point and they both disappear into the complex plane. We give both relativistic and non-relativistic examples of the last case in various dimensions and show that the critical behavior of the mass gap behaves similarly to the correlation length in the finite temperature Berezinskii-Kosterlitz-Thouless (BKT) phase transition in two dimensions, xi ~ exp(c/|T-T_c|^{1/2}). We speculate that the chiral phase transition in QCD at large number of fermion flavors belongs to this universality class, and attempt to identify the UV fixed point that annihilates with the Banks-Zaks fixed point at the lower end of the conformal window.

Figures (6)

• Figure 1: (a) A toy $\beta$-function. For $\alpha>\alpha_*$ there are fixed points at $g_\pm$ which are UV- and IR-stable respectively; these fixed points merge at $g_*$ for $\alpha=\alpha_*$, and disappear for $\alpha<\alpha_*$; (b) The RG flow of the coupling $g$ as a function of $t=\ln\mu$ in the non-conformal phase, with $(t_\text{UV}-t_\text{IR})\propto 1/\sqrt{\alpha_*-\alpha}$.
• Figure 2: The function $\beta(\tau)$ in the vicinity of $\tau=0$ for the BKT transition eq. (\ref{['eq:BKT-RG']}); the gray region is outside the realm of validity of the calculation.
• Figure 3: Two diagrams contributing to the $\beta$-function in Eq. (\ref{['eq:beta1']}). Note that the second diagram is a tree diagram.
• Figure 4: Diagrams contributing to the perturbative $\beta$-function for $\tilde{g}$.
• Figure 5: The one-loop graph that contributes to the gap equation