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.

Renormalization of twist-four operators in QCD

“…It is interesting to note that only these two Hamiltonians appear in supersymmetric extensions of QCD, see Refs. [46,47].…”

N=8supergravity on the light cone

“…The integration overθ in this case leads to special kind of differential operators acting on two superfields, see e.g. [20], [22] in the YM case. It is a particular combination of space-time ∂ + and spinorial derivatives.…”

“…Meanwhile, N =8 SG in the light-cone superspace with 16 Grassmann coordinates developed in [18], [19] has an unconstrained chiral scalar superfield, which in a chiral basis depends only on 8 Grassmann coordinates. Since the chiral light-cone superfield is off shell, there is a possibility to identify the supergraph Feynman rules in the light-cone superspace as it was done in the past for N =4 SYM in [20]- [22]. For N =8 SG it may be more difficult technically; moreover, only 3-and 4-coupling vertices are known so far.…”

Qcd-like properties of anomalous dimensions in the $$ \mathcal{N} $$=4 supersymmetric Yang-Mills theory

“…12 The excitation numbers K i of the Bethe roots u i can be computed [4] from the quantum numbers of the superconformal state associated with the twist-3 gluonic operator under consideration. To identify the correct superconformal primary describing this sector, we can use the superconformal properties of the (maximally symmetric) tensor product of three singletons [49].…”

“…Integrability itself, as a basis for the evolution of composite operators, was first discovered in studying planar QCD [11]. Conformal symmetry, unbroken in QCD at the one-loop level, does not seem a necessary condition for integrability, as discussed in [12]- [15], but it plays an important role by imposing selection rules and multiplet structures. Moreover, a (somewhat hidden) consequence of conformal symmetry can explain the structure of the large-spin expansion of the anomalous dimensions of twist operators.…”