Integrability of the evolution equations for heavy-light baryon distribution amplitudes

We consider evolution equations describing the scale dependence of the wave function of a baryon containing an infinitely heavy quark and a pair of light quarks at small transverse separations, which is the QCD analogue of the helium atom.The evolution equations depend on the relative helicity of the light quarks. For the aligned helicities, we find that the equation is completely integrable, that is it has a nontrivial integral of motion, and obtain exact analytic expressions for the eigenfunctions and the anomalous dimensions. The evolution equation for anti-aligned helicities contains an extra term that breaks integrability and creates a "bound state" with the anomalous dimension separated from the rest of the spectrum by a finite gap. The corresponding eigenfunction is found using numerical methods. It describes the momentum fraction distribution of the light quarks in, e.g., $\Lambda_b$-baryon at large scales.