Quark-hadron duality for orbital excitations

We investigate the question of duality between physical and bare quark cross sections for the annihilation of a current into states with high orbital angular momentum. We show the simple nature of the duality which must take place at sufficiently high energies in any theory with an interaction which switches off at small distances. However, the duality is not necessarily present for the lowest levels (in a channel with given spin); moreover, it is absent for high orbital excitations. In quantum chromodynamics the (averaged) cross section for the production of resonances with spin n starts to deviate sharply from the quark cross section at sapprox.μ2n2, where μ is a characteristic hadron mass, while for the lowest levels the masses squared are approx.μ2n (linear Regge trajectories). There exists a broad energy interval from sapprox.μ2n to sapprox.μ2n2 in which the physical spectral density is different from zero but not dual to the quark density. A similar phenomenon exists also in the nonrelativistic harmonic oscillator problem, where the renormalization of the spectral density becomes of order one at Eapprox.ωl/sup 3/2/, but the energy of the lowest levels is approx.ωl (l is the angular momentum)