N = 1 $$ \mathcal{N}=1 $$ Euler anomaly flow from dilaton effective action

Abstract

We consider N = 1 supersymmetric gauge theories in the conformal window. The running of the gauge coupling is absorbed into the metric by applying a suitable matter superfield-and Weyl-transformation. The computation becomes equivalent to one of a free theory in a curved background carrying the information of the renormalisation group flow. We use the techniques of conformal anomaly matching and dilaton effective action, by Komargodski and Schwimmer, to rederive the difference of the Euler anomaly coefficient ∆a ≡ a UV − a IR for the N = 1 theory. The structure of ∆a is therefore in one-to-one correspondence with the Wess-Zumino dilaton action.

Flow of the □R Weyl anomaly

“…Since propagators of massive fields Φ (s) of spin s contain terms scaling like (k 2 ) s−1 , the representation in (37) diverges in the UV for conformally coupled fields of spin 1 and higher. 14 UV-convergent cases include the previously quoted free spin 0 (conformally coupled) and spin 1/2 particles for which we find results (cf. section 3.4) in accordance with direct R-computations [9].…”

“…appendix C for further remarks. 14 This seems linked to the scheme-dependence found for direct evaluation of the spin 1 term via (3) cf. [9].…”

“…Let us summarise the various ways in which ∆b (14) can be expressed as an integral using (2), (10) and (21)…”

“…1 In this paper the coefficients in front of the geometric invariants are denoted by β-functions, in (dis)accordance with [10][11][12] ( [13,14]). The association of the letters a, b and c with the geometric invariants is variable in the literature.…”

“…last line the formula σ = −β b /(2β)[11] was used along with the formula for β b Taking the IR limit (a s → a s IR ) in (A.15) we getL MS b (a IR s ) = −σ IR,MS − gg (u)du . (A.17) Further using L b (a IR s ) = b UV − b IR (A 14). and taking into account σ UV = 0 one arrives at…”