Relating β∗′ and γq∗′ in the N=1 SQCD conformal window

In this paper we show that β∗′, the β-function slopes in the electric and magnetic theories, are equal at the corresponding infrared fixed points. This follows from the scaling of the correlators of the trace of the energy momentum tensors. The slopes β∗′ determine the scaling dimensions. Our paper can be considered as a commentary to D. Anselmi, M. T. Grisaru, and A. Johansen [Nucl. Phys. B491, 221 (1997)NUPBBO0550-321310.1016/S0550-3213(97)00108-9]; it proposes an improved derivation not based on a rather contrived construction by D. Kutasov [Phys. Lett. B 351, 230 (1995)PYLBAJ0370-269310.1016/0370-2693(95)00392-X], D. Kutasov and A. Schwimmer [Phys. Lett. B 354, 315 (1995)PYLBAJ0370-269310.1016/0370-2693(95)00676-C], and D. Kutasov, A. Schwimmer, and N. Seiberg, [Nucl. Phys. B459, 455 (1996)NUPBBO0550-321310.1016/0550-3213(95)00599-4]. As a byproduct we note that γQ∗′ - the slopes of the matter superfield anomalous dimension - vanish at both edges of the conformal window where one of the dual theories is strongly coupled. Finally, we determine the two-coupling magnetic fixed point at weak coupling correcting the result of I. I. Kogan, M. A. Shifman, and A. I. Vainshtein [Phys. Rev. D 53, 4526 (1996)PRVDAQ0556-282110.1103/PhysRevD.53.4526; Phys. Rev. D59, 109903(E) (1999)PRVDAQ0556-282110.1103/PhysRevD.59.109903].