Quark-quark potentials from the inversion of baryon-spectra and its application to the Roper resonance

We show that it is in principle possible to determine the quark-quark potential, in the nonrelativistic potential model, from the baryon spectrum. The method we propose is based on the fact that the lowest order of the hyperspherical-harmonic expansion method, the hypercentral approximation, is an excellent approximation for confining quark-quark potentials. However, our method applies to all cases where this assumption is valid. Using standard inverse spectrum techniques adapted to our problem we invert the baryon spectrum to obtain the hypercentral potential in the hyperradius of the three-quark system. By means of a new exact relation based on the Abel integral equation, we can invert the hypercentral potential to determine the quark-quark potential. A first application of this new method to the inversion of the s-state baryon spectrum demonstrates in a model-independent way the inability of nonrelativistic two-quark potentials to reproduce the Roper resonance without violating the QCD-motivated concavity requirement. (authors)