Implications of supersymmetry for QCD conformal operators

Abstract

We prove a set of identities for the anomalous dimensions of the quark and gluon conformal operators in the flavour singlet channel in QCD. These relations arise from the graded commutator algebra of the N = 1 superconformal group. We evaluate the rotation matrices for the quantities under study from the conventional dimensional regularization to the supersymmetry preserving regularization scheme. Using them we verify the equalities in two-loop approximation employing the results for the NLO anomalous dimensions of the conformal operators in the minimal subtraction scheme derived earlier.

Superconformal operators inN=4super-Yang-Mills theory

“…This explains why in the minimal supersymmetric extension of QCD -N = 1 super-Yang-Mills -there are two independent supermultiplets of conformal operators: one of aligned helicities [35,37] and another one of opposite helicities [38,35,39,40]. The former supermultiplet inherits integrability of the one-loop dilatation operator in QCD.…”

“…This should be compared with the N = 1 super-Yang-Mills theory [35,37,38,39,40]. In that case, a similar diagram has two disconnected components and, as a consequence, the two-particle conformal operators form two different supermultiplets.…”

“…Unfortunately, a practical implementation of such perturbative regularization procedure does not exist. The most prominent candidatethe dimensional reduction [44] -preserves supersymmetry to rather high orders of perturbation theory, but breaks conformal boosts starting from two loops [45,39,40]. The reason for this is that the coupling constant acquires a nontrivial scaling dimension in the space-time with d = 4 leading to a modification of the beta-function β d (g) = 1 2 (d − 4)g + β(g).…”

“…From the latter one can easily read off the components of the supermultiplet of conformal operators in this theory [35,39,40] which form the chiral superfield [40].…”

Heisenberg spin chains based on sℓ(2|1) symmetry

“…A particular sℓ(2|1) representation of interest are the infinitesimal conformal transformations in one dimension together with their twofold supersymmetric extensions. This symmetry applies to the Bjorken limit of four-dimensional supersymmetric Yang-Mills theory at least up to one loop [19], [20]. This means, the one-loop renormalization of quasipartonic composite operators can be obtained by sℓ(2|1) symmetric pairwise interactions of partons, the states (light-cone momenta, helicity,fermion number) of which form an infinite-dimensional lowest weight module of this algebra.…”