Abstract
Renormalization group procedure for effective particles (RGPEP) is applied in terms of a second-order perturbative computation to an Abelian gauge theory, as an example of application worth studying on the way toward derivation of a dynamical connection between the spectroscopy of bound states and their parton-model picture in the front form of Hamiltonian dynamics. In addition to the ultraviolet transverse divergences that are handled using the RGPEP in previously known ways, the small-x divergences are handled by introducing a mass parameter and a third polarization state for gauge bosons using a mechanism analogous to spontaneous breaking of global gauge symmetry, in a special limit that simplifies the theory to Soper's front form of massive QED. The resulting orders of magnitude of scales involved in the dynamics of effective constituents or partons in the simplified theory are identified for the fermion and boson mass counter terms, effective masses and self-interactions, as well as for the Coulomb-like effective interactions in bound states of fermions. Computations in orders higher than second are mentioned but not described in this article.
Highlights
Particle theory singularities that are associated with wee partons of the parton model of hadrons [1] or with field quanta that carry small kinematic momenta in the front form (FF) of Hamiltonian dynamics [2], require a renormalization group procedure that is capable of simultaneous handling of the ultraviolet and infrared divergences in combination with the bound-state problem, which is a complex issue [3]
The canonical Hamiltonian leads to difficulties with unambiguous handling of small x and large k⊥ singularities because the singular terms involve the ratio k⊥2=x and the ultraviolet divergences are mixed with the small x divergences
Once the mass parameter for gauge bosons is introduced according to the principles of local gauge symmetry and spontaneous violation of the global gauge symmetry, a mass gap is introduced and one achieves unambiguous control on the divergences
Summary
Particle theory singularities that are associated with wee partons of the parton model of hadrons [1] or with field quanta that carry small kinematic momenta in the front form (FF) of Hamiltonian dynamics [2], require a renormalization group procedure that is capable of simultaneous handling of the ultraviolet and infrared divergences in combination with the bound-state problem, which is a complex issue [3]. We start from a different Lagrangian than massive QED and in the manner analogous to spontaneous breaking of global gauge invariance arrive at Soper’s theory as a helpful simplification in a special limit. Instead of aiming at reproducing or predicting observables directly in terms of the degrees of freedom (d.o.f.) that appear in canonical FF Hamiltonian in a diverging way, our goal is to compute the equivalent effective FF Hamiltonian operators that are written in terms of apparently more adequate d.o.f [18,19] Computation of such Hamiltonians is hoped to eventually lead to a sequence of successive approximations for relativistic description of strongly bound states because. Appendixes provide details of our notation and the canonical Hamiltonian of Soper’s theory
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