Abstract:We show that the Brodsky-Lepage evolution equation for the spin 3/2 baryon distribution amplitude is completely integrable and reduces to the three-particle XXX s=−1 Heisenberg spin chain. Trajectories of the anomalous dimensions are identified and calculated using the 1/N expansion. Extending this result, we prove integrability of the evolution equations for twist 3 quark-gluon operators in the large N c limit.

We develop a new theoretical framework for the description of leading twist light-cone baryon distribution amplitudes which is based on integrability of the helicity λ = 3/2 evolution equation to leading logarithmic accuracy. A physical interpretation is that one can identify a new 'hidden' quantum number which distinguishes components in the λ = 3/2 distribution amplitudes with different scale dependence. The solution of the corresponding evolution equation is reduced to a simple three-term recurrence relation. The exact analytic solution is found for the component with the lowest anomalous dimension for all moments N, and the WKB-type expansion is constructed for other levels, which becomes asymptotically exact at large N. Evolution equations for the λ = 1/2 distribution amplitudes (e.g. for the nucleon) are studied as well. We find that the two lowest anomalous dimensions for the λ = 1/2 operators (one for each parity) are separated from the rest of the spectrum by a finite 'mass gap'. These special states can be interpreted as scalar diquarks.

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We construct a representation of the Separated Variables (SoV) for the quantum SL(2, R) Heisenberg closed spin chain following the Sklyanin's approach and obtain the integral representation for the eigenfunctions of the model. We calculate explicitly the Sklyanin measure defining the scalar product in the SoV representation and demonstrate that the language of Feynman diagrams is extremely useful in establishing various properties of the model. The kernel of the unitary transformation to the SoV representation is described by the same "pyramid diagram" as appeared before in the SoV representation for the SL(2, C) spin magnet. We argue that this kernel is given by the product of the Baxter Q−operators projected onto a special reference state.1 One can choose instead Ψ(z n ) to be holomorphic in the lower half-plane. As we will show below, the two cases, Im z > 0 and Im z < 0, correspond to the different values of the total momentum of the system, p > 0 and p < 0, respectively.2 Performing the conformal mapping w = i(z − i)/(z + i), one can bring this expression to a canonical form involving the integration over an interior of the unit disk in the w−plane [18].

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.