Dilatation operator in (super-)Yang–Mills theories on the light-cone

Abstract

The gauge/string correspondence hints that the dilatation operator in gauge theories with the superconformal SU ( 2 , 2 | N ) symmetry should possess universal integrability properties for different N . We provide further support for this conjecture by computing a one-loop dilatation operator in all (super)symmetric Yang–Mills theories on the light-cone ranging from gluodynamics all the way to the maximally supersymmetric N = 4 theory. We demonstrate that the dilatation operator takes a remarkably simple form when realized in the space spanned by single-trace products of superfields separated by light-like distances. The latter operators serve as generating functions for Wilson operators of the maximal Lorentz spin and the scale dependence of the two are in the one-to-one correspondence with each other. In the maximally supersymmetric, N = 4 theory all nonlocal light-cone operators are built from a single CPT self-conjugated superfield while for N = 0 , 1 , 2 one has to deal with two distinct superfields and distinguish three different types of such operators. We find that for the light-cone operators built from only one species of superfields, the one-loop dilatation operator takes the same, universal form in all SYM theories and it can be mapped in the multi-color limit into a Hamiltonian of the SL ( 2 | N ) Heisenberg (super)spin chain of length equal to the number of superfields involved. For “mixed” light-cone operators involving both superfields the dilatation operator for N ⩽ 2 receives an additional contribution from the exchange interaction between superfields on the light-cone which breaks its integrability symmetry and creates a mass gap in the spectrum of anomalous dimensions.