We employ the two independent Casimir operators of the Poincaré group, the squared fourmomentum, p 2 , and the squared Pauli-Lubanski vector, W 2 , in the construction of a covariant mass-m, and spin-3 2 projector in the four-vector-spinor, ψµ. This projector provides the basis for the construction of an interacting Lagrangian that describes a causally propagating spin-3 2 particle coupled to the electromagnetic field by a gyromagnetic ratio of g 3 2 = 2.
We briefly review the five possible real polynomial solutions of hypergeometric differential equations. Three of them are the well known classical orthogonal polynomials, but the other two are different with respect to their orthogonality properties. We then focus on the family of polynomials which exhibits a finite orthogonality. This family, to be referred to as the Romanovski polynomials, is required in exact solutions of several physics problems ranging from quantum mechanics and quark physics to random matrix theory. It appears timely to draw attention to it by the present study. Our survey also includes several new observations on the orthogonality properties of the Romanovski polynomials and new developments from their Rodrigues formula.
The analytic solutions of the one-dimensional Schrödinger equation for the trigonometric RosenMorse potential reported in the literature rely upon the Jacobi polynomials with complex indices and complex arguments. We first draw attention to the fact that the complex Jacobi polynomials have non-trivial orthogonality properties which make them uncomfortable for physics applications.Instead we here solve above equation in terms of real orthogonal polynomials. The new solutions are used in the construction of the quantum-mechanical superpotential.
Abstract. We make the case that the Coulomb-plus linear quark confinement potential predicted by lattice QCD is an approximation to the exactly solvable trigonometric Rosen-Morse potential that has the property to interpolate between the Coulomb-and the infinite wells. We test the predictive power of this potential in the description of the nucleon (considered as a quark-diquark system) and provide analytic expressions for its mass spectrum and the proton electric form factor. We compare the results obtained in this fashion to data and find quite good agreement. We obtain an effective gluon propagator in closed form as the Fourier transform of the potential under investigation.PACS. 12.39.Jh Non-relativistic quark models -13.40.Gp Electromagnetic form factors The quark potential from lattice QCDThe strong interactions of quarks, the fundamental constituents of hadrons, are governed by the Quantum Chromodynamics (QCD), the non-Abelian gauge theory with the gluons as gauge bosons. As a consequence of the non-Abelian character of QCD, the quark interactions run from one-to many-gluon exchanges over gluon selfinteractions, the latter being responsible for the so-called quark confinement, where highly energetic quarks remain trapped but behave as (asymptotically) free particles at high energies and momenta. The QCD equations are nonlinear and complicated due to the gluon self-interaction processes and their solution requires employment of highly sophisticated techniques such as discretization of space time, so-called lattice QCD. Lattice QCD calculations have established themselves as a reliable tool for the non-perturbative analysis of QCD. The outcome (in the quenched approximation) is a linear confinement potential with energy increase, be it quark-anti-quark (QQ),or two-body potential between quarks (so-called ∆ type),a e-mail: mariana@ifisica.uaslp.mx in obvious notations. The Coulomb-like piece is associated with short-range one-gluon exchange in the perturbative regime, whereas the long-range linear part relates to nonperturbative effects and is attributed to flux-tube QQ, or QQ links. Its strength is then associated with the respective string tension [1]. Detailed analysis of the values of the constants of V QQ and V 3Q has been performed in ref.[2]. The three-quark (3Q) Coulomb-plus linear potential, or, versions of it, has found repeatedly application to baryon spectroscopy [3,4]. In view of this, its generalization to an exactly solvable potential is of interest. We begin with first drawing attention to the proximity of the two-body Coulomb-plus linear potential (be it for QQ, QQ, or Q(QQ) systems), to (− cot |r|). Indeed, this is immediately seen from the corresponding Taylor expansion,This expression shows that the absolute values of the strengths of the linear to Coulomb potentials are in ratio 1 : 3, a value that fits quite reasonably into the range of the σ ∆ : A ∆ ratio of approximately 1/4-1/2 reported by the lattice QCD analysis [2]. In fact, (− cot |r|) is part of the more general and exactly solv...
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