Exact Adler Function in Supersymmetric QCD

Abstract

The Adler function D is found exactly in supersymmetric QCD. Our exact formula relates D(Q 2 ) to the anomalous dimension of the matter superfields γ(αs(Q 2 )). En rout we prove another theorem: the absence of the so-called singlet contribution to D. While such singlet contributions are present in individual supergraphs, they cancel in the sum.PACS numbers: 11.15. 11.30.Pb, 12.60.Jv, FORMULATION OF THE PROBLEM AND RESULTSThe celebrated ratioplays a special role in QCD-based phenomenology. For instance, it can be used for a precise determination of the gauge coupling α s from accurate data on e + e − → hadrons in an appropriate energy range. It is also one of the key objects in various theoretical analyses in QCD, both in perturbation theory and beyond. In perturbation theory the ratio R is defined as a normalized cross section σ(e + e − → quarks + gluons) .It is directly reducible to the imaginary part of the photon polarization operator Π (see (4)),Alternatively, one can define R QCD through a certain analytic continuation (see e.g.In QCD the Adler function and the ratio R are calculatedis only a cousin of QCD since there is still no indication to the existence of supersymmetry in our world. Nevertheless, SQCD is known to be a unique theoretical laboratory in many aspects of gauge dynamics. The O(α s ) correction to R in SQCD was calculated in [4].In this paper we will derive an exact relation between D SQCD and the anomalous dimension γ of the matter superfield(s), valid to all orders in α s ,

Derivation of the exact expression for theDfunction inN=1SQCD

“…As was mentioned in [1], the exact relation (1) is closely related to the exact NSVZ β function [2,3,20,4] …”

“…The sum of these singularities is calculated in Section 4.4, where we relate it to the anomalous dimension of the matter superfields. Finally, in Section 5 we discuss some consequences of the exact relation (1) for SQCD in the conformal window.…”

“…In our brief publication [1] we reported a new exact relation between the Adler function D in supersymmetric QCD (SQCD) with the gauge group SU (N ) on the one hand, and the anomalous dimensions of the matter superfields on the other,…”

“…at the points at which β NSVZ = 0. After our publication [1] it was argued [9] that the above conformal relation can be generalized for the renormalization group (RG) flow by considering R charges as functions of the running α s . The string-based observation [9] is complementary to our derivation.…”

“…Equation (1) assumes that all matter fields are in the fundamental representation of SU (N ), although their electric charges can be different. In calculating γ α s (Q 2 ) one should remember that α s (Q 2 ) runs according to the Novikov-Vainshtein-Shifman-Zakharov (NSVZ) β function [2,3,4].…”