A counterexample to the a-'theorem'

TL;DR

This paper challenges the proposed monotonicity of the conformal central charge $a$ along four-dimensional RG flows by constructing RG trajectories between maximal-rank Argyres-Douglas points in $N=2$ SU$(N_c)$ gauge theories with $N_f$ flavors. Using the Shapere-Ting (ST) method, it computes the central charges $a$ and $c$ for these SCFTs from $R$-symmetry anomalies and Coulomb-branch operator dimensions derived from Seiberg-Witten curves, obtaining explicit formulas for various parity cases and a special $N_f=2N_c$ limit. The core result is an explicit flow from the maximal-rank AD point of SU$(N{+}1)$ with $2N$ flavors to the SU$(N)$ theory with the same flavor content at infinite coupling, for which $a_{UV} < a_{IR}$ when $N\ge4$, thereby violating the a-theorem under standard assumptions. The authors argue that this does not contradict the a-theorem in known formulations because accidental U(1) symmetries arise and holographic proofs do not apply to these theories, and discuss implications for the search for a universal monotone quantity in 4D QFTs and for the dynamics of AD points. The work highlights subtle IR dynamics in AD theories and motivates further study of RG flows among 4D SCFTs with large $N_f/N_c$.

Abstract

The conclusion of the original paper was wrong, due to the incorrect assumption that the low-energy limit at the strongly-coupled point consists of a single, coupled SCFT. By taking into account the fact that the low-energy limit consists of multiple decoupled parts, it was later shown in arXiv:1011.4568 that there is no violation of the a-theorem in this system. Furthermore, the a-theorem itself was convincingly demonstrated in arXiv:1107.3987, and the argument presented there has been further refined. The rest of this paper is kept as it was, for some parts of the discussions might still be of interest. Original abstract: We exhibit a renormalization group flow for a four-dimensional gauge theory along which the conformal central charge 'a' increases. The flow connects the maximally superconformal point of an N=2 gauge theory with gauge group SU(N+1) and N_f=2N flavors in the ultraviolet, to a strongly-coupled superconformal point of the SU(N) gauge theory with N_f=2N massless flavors in the infrared. Our example does not contradict the proof of the a-theorem via a-maximization, due to the presence of accidental symmetries in the infrared limit. Nor does it contradict the holographic a-theorem, because these gauge theories do not possess weakly-curved holographic duals.