Computation of effective front form Hamiltonians for massive Abelian gauge theory

Abstract

Renormalization group procedure for effective particles (RGPEP) is applied in terms of a secondorder 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.

Elementary example of exact effective-Hamiltonian computation

“…Emergence of the RGPEP vertex form factors in the Yukawa theory is described in [38]. Analogous appearance of the vertex form factors in the Abelian gauge theory is shown in [16]. The RGPEP form factors that emerge in the third-order computation of the effective vertices in a non-Abelian theory is provided in [39].…”

“…The RGPEP employs the rules of the similarity renormalization group procedure for Hamiltonians [12,13] and takes advantage of the double-commutator feature of Wegner's flow equation for Hamiltonian matrices [14]. In application to local QFT, the RGPEP has been recently illustrated in [16], which also includes references to the previous works.…”

“…In QFT, these form factors regulate singularities of the local interactions, e.g. see [16], and can be thought of as corresponding to a finite size of the effective particles. However, in the model solved here the situation is much simpler because of the static limit.…”

Asymptotic freedom using a gluon mass as a regulator

“…( 22) t = t r , where t r is a small value that acts as a cutoff. Frequently, the notation f t+t r ,ab is used instead of f t,ab f t r ,ab , since it allows to clearly see that for any finite value of t the regularization parameter is "muted" in the limit t r → 0 [21].…”

“…Published works in this direction [13][14][15] reproduce the asymptotic-freedom result obtained from renormalization group techniques in Euclidean space [16]. A finite dependence on the regularization functions used to regulate small momentum fractions (small-x) usually A regularization provided by a canonical gluon mass [5] seems to be more adequate for various reasons 1 : first of all, the same regulating function is used to remove both ultraviolet-and small-x divergences; furthermore, we use the same type of function as the ones introduced by the RGPEP procedure; and finally, it allows one to include a large range of +-component momenta near zero [20,21].…”