Regularization Of Breit Quark Potential Model And Bound States Of Mesons

Non-relativistic quark potential model is a simpler and more useful phenomeno-logical model among the various theories for describing the quark motion in hadrons.In the quark potential model, the quark motion in hadrons is assumed to be gov-erned by non-relativistic dynamics and can be very well described by the Schro¨dingerequation. Many successive progresses are made in studying the bound states, finestructure, and scattering of hadrons by using the quark potential model, although thenon-relativistic assumption may not be very valid for light quarks. The theoreticalresults are in fairly good agreement with the experimental data. These successes pro-mote the more extensive study on the quark potential model.The complete one-gluon exchange potential up to the second order in velocity isthe Fermi-Breit potential (or Breit potential), which is also known as the scattering-channel potential or t-channel potential. In the previous studies of the bound statesand scattering of hadrons using the quark potential model, the scattering-channel po-tential is always improved by canceling or changing some terms in the Breit poten-tial for simplicity and meanwhile getting high accuracy. Especially, the momentum-dependent orbit-orbit coupling term in the Breit potential is omitted in these improve-ments, which leads to the localization of the original non-local quark potential. Theinappropriate omissions and changes to the scattering-channel potential may breakthe completeness and hermiticity of the potential. Moreover, the non-locality of thepotential is an important character of the strong interaction. Therefore, studying thequark potential model with the complete Breit potential which includes the non-localterms and investigating the contribution of each term in the Breit potential for thequark potential model are great of significances for people to understand the quarkpotential model deeply and develop it. These study and investigation will be alsoas the bases for the further study of the theories and experimental phenomena of thehadron scatterings.In the previous calculations of meson bound states, people resolved theSchro¨dinger equation with only the l = l' matrix elements. However, the matrixelements for l'≠l are not zero when calculating with the complete Breit potential. It is because the contribution from the tensor force. In this case there are the couplingsbetween the states with different angle momenta. In this thesis a technique of solvingthe equation group of the related coupled matrix equations is developed on the basisof the previous work for independent matrix equations. It allows one to calculate themeson mass spectra with the complete Breit potential.Because there are the singular terms with r^-3 in the Breit potential, it is foundthat in the numerical calculations the results of the meson mass spectra are stronglydependent on the width parametersβin the bases of wave function as well as the num-ber N of the bases. They will change much whenβor N with small changes. So thesolutions are instable. In this thesis several regularizations are performed to the singu-lar terms in the Breit potential in order to get the stable spectra of the meson masses.First, the singular terms are modified by the Yukawa potential e^-μr/r in coordinatespace. Then, the singular terms are tried to be regularized in momentum space bymultiplying by the form factorμ²/(q² +μ²) one or two times simultaneously, or withdifferent times for different the terms. The calculated results indicate that the singu-larity in the Breit potential can be removed with the regularizations. One can obtainthe mass spectra of the mesons with stable and high accuracy and correspondingly theuseful quark potential models after the regularizations.The study of the splittings between the masses of the mesons with different spinand orbit quantum numbers is important for checking the quark potential model. Inthe previous calculations with quark potential models, the splitting betweenη_c andJ/ψis however too small. In this thesis the mass splitting ofη_c– J/ψand the split-ting between the new mesonη_b andÎ¥(1s) are investigated with the complete Breitquark potential model. In addition, the splittings among X_c0,χ_c1 andχ_c2 are alsoinvestigated using the Breit potential which includes the spin-orbit interaction. Theinvestigated results indicate that the screen massμin the form factor of regulariza-tion is related to the reduced mass of meson. The splittings ofη_c–J/ψ,η_b–Υ, andχ_c0–χ_c1–χ_c2 can reach to the experimental results with high accuracy when the screenmass is expanded to the third-order polynomial of meson reduced mass.