aa r X i v : . [ h e p - t h ] J a n RUP-21-1
On the trace anomaly of Chaudhuri-Choi-Rabinovici model
Yu NakayamaDepartment of Physics, Rikkyo University, Toshima, Tokyo 171-8501, Japan
Abstract
Recently a non-supersymmetric conformal field theory with an exactly marginaldeformation in the large N limit was constructed by Chaudhuri-Choi-Rabinovici.On a non-supersymmetric conformal manifold, c coefficient of the trace anomaly infour dimensions would generically change. In this model, we, however, find that itdoes not change at the first non-trivial order given by three-loop diagrams. n four-dimensional conformal field theories, the trace anomaly has the form T µµ = c Weyl − a Euler (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 theoryexample where c changes (except for the effective holographic constructions in [2]). Themain obstruction has been that we have no good examples of non-supersymmetric con-formal 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 confor-mal field theory with an exactly marginal deformation in the large N limit [9]. Thistheory may serve as a first non-trivial check if c can really change under exactly marginaldeformations. In this short note, we, however, show that it does not change at the firstnon-trivial order given by three-loop diagrams.The model (called complex bifundamental model in [9] is given by four SU ( N c ) gaugetheories with names 1 , ′ , ′ , each of which has N f Dirac fermions in the fundamentalrepresentation. We have two complex scalars in the bifundamental representations Φ (under gauge group 1 and 1 ′ ) and Φ (under gauge group 2 and 2 ′ ). It has no Yukawainteraction, absence of which is protected by chiral symmetry, but it has a scalar potential V = ˜ h Tr[Φ † Φ Φ † Φ ] + ˜ h Tr[Φ † Φ Φ † Φ ]+ ˜ f Tr[Φ † Φ ]Tr[Φ † Φ ] + ˜ f Tr[Φ † Φ ]Tr[Φ † Φ ] + 2 ˜ ζ Tr[Φ † Φ ]Tr[Φ † Φ ] . (2)We take the Veneziano limit of N c , N f → ∞ with fixed x = N f N c and consider the limit x → to make the theory weakly coupled.In terms of rescaled coupling constants ( i = 1 , λ i = N c g i π , h i = N c ˜ h i π , f i = N c ˜ f i π , ζ = N c ˜ ζ π , (3)the renormalization group beta functions in the Veneziano limit are expressed as (no sumover i unless explicitly shown) β λ i = − − x λ i + −
54 + 26 x λ i See also [10][11] for other recently constructed examples of non-supersymmetric field theories withexactly marginal deformations in different dimensions than four. h i = 8 h i − λ i h i + 32 λ i β f i = 4 f i + 16 f i h i + 12 h i + 4 ζ − λ i f i + 92 λ i β ζ = ζ X i =1 (4 f i + 8 h i − λ i ) . (4)The zero of the beta functions was studied in [9] and they found that there exists aconformal manifold given by λ = λ = λ = 21 − x −
54 + 26 xh = h = 3 − √ λf p ≡ f + f r λζ + f m = 18 √ − λ , (5)where f m ≡ f − f . 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.We now ask if the coefficient c in the trace anomaly can change on this conformal man-ifold. In addition to the coupling constant independent contributions from the one-loopdiagrams (that count a number of fields), the coupling constant dependent contributionsto the trace anomaly that are relevant for us come from the three-loop diagrams shownin Fig 1. The detailed computation for diagram (A) (as well as other two-loop diagrams)can be found in [12][13][14], but we only need the relative coefficient, so we can simplywork on combinatorics.Up to an overall proportionality factor, the result in the Veneziano limit is summarizedas c = − f m − ζ + c λ λ (6)on the conformal manifold, where c λ is some numerical constant, which is unimportantfor our discussions. Since the relative coefficient appearing here coincides what appears 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. A typo in the two-loop gauge contribution [14] that could affect c λ has been corrected in [15].
2n the last line of (5), we conclude that c does not change on the conformal manifoldalthough the value itself is perturbatively corrected. We also note that these two- andthree-loop diagrams do not change the value of a as anticipated [1][16] (rather triviallywithout cancellation unlike c ).The result is surprising in the sense that we generically expect that c would change onnon-supersymmetric conformal manifold. It is an interesting question to see if the higherloop corrections modify our conclusion. It may be possible to relate the all-loop argumentfor the existence of the exactly marginal deformation in [9] with the computation of c byclosing all the external lines in beta functions to make vacuum diagrams.(A) (cid:1) (B) (cid:2) (C) (cid:3) (cid:4) Fig 1: Three-loop Feynman diagrams that could contribute to c . Acknowledgements
This work is in part supported by JSPS KAKENHI Grant Number 17K14301. It ismotivated from the online talk by Z. Komargodski at YITP workshop on Strings andFields 2020, which the author watched on Youtube later.
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