Stanislaw D. Glazek

Precise derivation of the logarithmically scale-dependent Hamiltonian eigenstate picture for hadrons in the space of virtual quark and gluon states of the canonical front form of QCD requires addressing first the problem of divergences stronger than logarithmic and especially the small- p + divergences in the dynamics of gluons. We propose to facilitate the regularization and cancellation of these divergences using a gluon mass parameter and an auxiliary color-octet scalar field corresponding to the longitudinally polarized gluons. The auxiliary field decouples from the hadronic constituent dynamics when the mass parameter tends to zero, as required in the gauge theory. The same method applies in the cancellation of the quadratic ultraviolet transverse divergences in the self-interactions. After explaining how the method works in the old-fashioned perturbative computations of the scattering amplitudes to all orders, we describe its application to computing the renormalized, scale-dependent Hamiltonian operator acting in the entire Fock space and to deriving its bound-state eigenvalue problems expressed in terms of the correspondingly scale-dependent creation and annihilation operators for effective quarks and gluons. Our detailed discussion focuses on the weak-coupling expansion of second order for the QCD Hamiltonian of effective heavy quarks and the corresponding eigenvalue problems for quarkonia. The results include the concept of confinement in an effective theory with the gluon mass parameter sent to zero and a heuristic scenario concerning extension of the Hamiltonian approach to the dynamics of light quarks.

Theory of the quantum quartic oscillator is developed with close attention to the energy cutoff one needs to impose on the system in order to approximate the smallest eigenvalues and corresponding eigenstates of its Hamiltonian by diagonalizing matrices of limited size. The matrices are obtained by evaluating matrix elements of the Hamiltonian between the associated harmonic-oscillator eigenstates and by correcting the computed matrices to compensate for their limited dimension, using the Wilsonian renormalization-group procedure. The cutoff dependence of the corrected matrices is found to be described by a spiral motion of a three-dimensional vector. This behavior is shown to result from a combination of a limit-cycle and a floating fixed-point behaviors, a distinct feature of the foundational quantum system that warrants further study. A brief discussion of the research directions concerning renormalization of polynomial interactions of degree higher than four, spontaneous symmetry breaking and coupling of more than one oscillator through the near neighbor couplings known in condensed matter and quantum field theory, is included.

We present an exact computation of effective Hamiltonians for an elementary model obtained from the Yukawa theory by going to the limit of bare fermions being infinitely heavy and bare bosons being at rest with respect to the fermions that emit or absorb them. The coupling constant can be arbitrarily large. The Hamiltonians are computed by solving the differential equation of the renormalization group procedure for effective particles (RGPEP). Physical fermions, defined in the model as eigenstates of the effective Hamiltonians, are obtained in the form of an effective fermion dressed with a coherent state of effective bosons. The model computation illustrates the method that can be used in perturbative computations of effective Hamiltonians for realistic theories. It shows the mechanism by which the perturbative expansion and Tamm-Dancoff approximation increase in accuracy along the RGPEP evolution.

Small-x divergences of Abelian gauge theory in the front form of Hamiltonian dynamics are regulated using a mass parameter for gauge bosons, introduced through a mechanism analogous to the spontaneous breaking of global gauge symmetry. A corresponding family of ultraviolet and infrared finite scale-dependent renormalized Hamiltonians, is calculable order-by-order using the renormalization group procedure for effective particles. The second-order terms described here suggest the magnitude of mass corrections that may be involved in resolving the small-x parton and front-form vacuum and zero-mode problems, assuming that the gauge boson mass that counts does not exceed the current upper bound on the photon mass.

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 counterterms, 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.

Small-x divergences of Abelian gauge theory in the front form of Hamiltonian dynamics are regulated using a mass parameter for gauge bosons, introduced through a mechanism analogous to the spontaneous breaking of global gauge symmetry. A corresponding family of ultraviolet and infrared finite scale-dependent renormalized Hamiltonians, is calculable order-by-order using the renormalization group procedure for effective particles. The second-order terms described here suggest the magnitude of mass corrections that may be involved in resolving the small-x parton and front-form vacuum and zero-mode problems, assuming that the gauge boson mass that counts does not exceed the current upper bound on the photon mass.

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.

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