Electromagnetic Properties of Baryons

Abstract

Static observables of bound state systems in field theoretic descriptions are usually extracted from form factors in the limit of vanishing squared four-momentum transfer of the probing exchange particle. On the other hand, static properties in nonrelativistic quantum mechanics can be formulated by means of expectation values involving essentially scalar products of wave functions. The main objective of this work is to show that a synthesis of both approaches is indeed possible - at least if certain restrictions are made to the kind of interactions between the constituents of the bound system - leading to new insights into the structure of static properties. The focus lies especially on the charge radii and magnetic moments of baryons described within a covariant constituent quark model having its field theoretic foundations in the Bethe-Salpeter equation. The results can be generalized to other three-fermion systems in an obvious way. <br /> The current matrix element in the Breit frame between the vertex functions is derived, which is the basis of the following calculations. The charge radius and magnetic moment of a bound three-fermion system is then derived by starting from their usual definition from form factors and in case of the charge radius also from the well-known radius of a charge distribution in classical electrodynamics. In both cases the static limit at the photon point is taken analytically and subsequently the integration over the relative energy variables is done. Finally the vertex functions are replaced by Salpeter amplitudes and the expression is symmetrized over the three fermions. <br /> The final results express the charge radius and magnetic moment of the three-fermion system as expectation values with respect to Salpeter amplitudes. The resulting operators in both cases show an interesting structure: The center of mass motion of the system is accounted for by the subtraction of the relativistic center of mass. Besides a relativistic weight factor the expressions resemble their nonrelativistic counterparts with the particle masses replaced by the relativistic single-particle energies. In addition, the magnetic moment operator allows for a decomposition of the total magnetic moment into contributions from the fermions intrinsic spins and their orbital angular momenta - a result which is hardly to obtain in a form factor calculation. The numerical implementation of the analytic results is done within a covariant constituent quark model with quark confinement and a residual instanton interaction accounting for the fine structure of the observed mass spectra. The Salpeter amplitudes which where obtained by solving the Salpeter equation are used to compute the expectation values of the charge radius and magnetic moment operators. The charge radii and magnetic moments of the baryon octet are thus obtained The proton charge radius is in good agreement with experiment whereas the neutron radius is much too large. A study of the dependence of these quantities on the particular choices of the instanton force parameters reveals a strong variation of the neutron radius with the instanton range parameter. The dependence of the proton radius in contrast is quite moderate and the spectra are hardly affected by a variation, which allows to propose an adapted parameter set describing both radii equally well. The octet magnetic moments are nicely reproduced with an average accuracy of roughly ten percent. A study on the dependence of the contributions of quark spins and orbital angular momenta to the total magnetic moment on the constituent quark mass reveals that at a quark mass of 330 MeV (which is the usual value in the model) roughly ninety percent are due to spin. Decreasing the quark mass, while keeping the ground state masses fixed at the empirical values lowers the spin contribution to roughly sixty percent at a quark mass of 25 MeV. The calculated values for the hyperon radii and magnetic moments of selected exited states as well as mostly the baryon decuplet are predictions. There are however ongoing experiments to measure the magnetic moments of the Δ⁺ and the S¹¹(1535) resonance. <br /> The method presented in this work to compute static properties of three-fermion systems gives new insight into the nature of such observables. When applied in the context of a covariant constituent quark model it allows for unprecedented studies, not least because of a much reduced computation time as compared to form factor calculations and an improved numerical stability. The quark model that was used to conduct those studies describes baryon charge radii and magnetic moments quite well. As has already been touched in this work, the method may also be applied to other moments such as magnetic radii and electric polarizabilities as well in the future.