On the trace anomaly of Chaudhuri-Choi-Rabinovici model

Recently a non-supersymmetric conformal field theory with an exactly marginal deformation in the large $N$ limit was constructed by Chaudhuri-Choi-Rabinovici. On a non-supersymmetric conformal manifold, $c$ coefficient of the trace anomaly in four dimensions would generically change. In this model, we, however, find that it does not change at the first non-trivial order given by three-loop diagrams.

In four-dimensional conformal field theories, the trace anomaly has the form T µ µ = cWeyl 2 − aEuler (1) and it is known that coefficient a cannot change under exactly marginal deformations, but coefficient c may [1][2] [3][4] [5][6] [7]. However, there has been no explicit field theory example where c changes (except for the effective holographic constructions in [2]). The main obstruction has been that we have no good examples of non-supersymmetric conformal field theories with exactly marginal deformations; in superconformal field theories, while it is easier to realize exactly marginal deformations, c does not change [8].
Recently, Chaudhuri-Choi-Rabinovici have constructed a non-supersymmetric conformal field theory with an exactly marginal deformation in the large N limit [9]. 1 This theory may serve as a first non-trivial check if c can really change under exactly marginal deformations. In this short note, we, however, show that it does not change at the first non-trivial order given by three-loop diagrams.
The model (called complex bifundamental model in [9] is given by four SU(N c ) gauge theories with names 1, 1 ′ , 2 and 2 ′ , each of which has N f Dirac fermions in the fundamental representation. We have two complex scalars in the bifundamental representations Φ 1 (under gauge group 1 and 1 ′ ) and Φ 2 (under gauge group 2 and 2 ′ ). It has no Yukawa interaction, absence of which is protected by chiral symmetry, but it has a scalar potential We take the Veneziano limit of N c , N f → ∞ with fixed x = N f Nc and consider the limit x → 21 4 to make the theory weakly coupled. In terms of rescaled coupling constants (i = 1, 2) the renormalization group beta functions in the Veneziano limit are expressed as (no sum over i unless explicitly shown) The zero of the beta functions was studied in [9] and they found that there exists a conformal manifold given by where f m ≡ f 1 −f 2 2 . From the last line of (5), we see that it has the topology of a circle. As long as λ is small, we may neglect higher order corrections. can be found in [12][13] [14], 2 but we only need the relative coefficient, so we can simply work on combinatorics.
Up to an overall proportionality factor, the result in the Veneziano limit is summarized as on the conformal manifold, where c λ is some numerical constant, which is unimportant for our discussions. 3 Since the relative coefficient appearing here coincides what appears 2 The three-loop diagrams of (B)(C)(D) are not evaluated in the literature, but we see that diagram (B) and (C) do not contribute to c. Diagram (D) may contribute in general, but the contributions to c in our theory do not depend on ζ or f m from the symmetry of the diagrams. 3 A typo in the two-loop gauge contribution [14] that could affect c λ has been corrected in [15].
in the last line of (5), we conclude that c does not change on the conformal manifold although the value itself is perturbatively corrected. We also note that these two-and three-loop diagrams do not change the value of a as anticipated [1][16] (rather trivially without cancellation unlike c).
The result is surprising in the sense that we generically expect that c would change on non-supersymmetric conformal manifold. It is an interesting question to see if the higher loop corrections modify our conclusion. It may be possible to relate the all-loop argument for the existence of the exactly marginal deformation in [9] with the computation of c by closing all the external lines in beta functions to make vacuum diagrams.