[Paper Review] Solution to the non-perturbative renormalization of gauge theory

Hans–Christian Pauli|ArXiv.org|Dec 20, 2003

Quantum Chromodynamics and Particle Interactions3 references3 citations

TL;DR

This paper presents a non-perturbative renormalization procedure for gauge theories using an effective light-cone Hamiltonian approach in QCD. By regulating the Hamiltonian with a cut-off function and introducing counterterms to achieve cut-off independence, the method resolves long-standing issues in gauge theory renormalization, revealing that the form factor's freedom allows tunable short-distance behavior of the potential, including a pocket that binds quarks in hadrons.

ABSTRACT

The long standing problem of a non-perturbative renormalization of a gauge field theoretical Hamiltonian is addressed and explicitly carried out within an (effective) light-cone Hamiltonian approach to QCD. The procedure is in line with the conventional ideas: The Hamiltonian is first regulated by suitable cut-off functions, and subsequently renormalized by suitable counter terms to make it cut-off independent. Emphasized is the considerable freedom in the cut-off function which eventually can modify the Coulomb potential of two charges at sufficiently small distances. The approach provides new physical insight into the nature of gauge theory and the potential energy of QCD and QED at short distance.

Motivation & Objective

• To resolve the longstanding challenge of non-perturbative renormalization in gauge field theories, particularly in the Hamiltonian formulation of QCD.
• To establish a consistent regularization and renormalization procedure for the light-cone Hamiltonian that ensures cut-off independence.
• To explore the physical implications of the arbitrariness in the regulator function on the short-distance behavior of the potential.
• To demonstrate that the method allows for a tunable potential structure capable of binding quarks, consistent with hadron spectroscopy.
• To lay the foundation for a systematic, non-perturbative approach to hadron structure using light-cone wave functions and effective Hamiltonians.

Proposed method

• Regulate the light-cone Hamiltonian using a form factor in the vertex interaction to suppress large momentum transfers.
• Introduce counterterms derived from the derivative of the regulator function to render the Hamiltonian cut-off independent.
• Use the subtraction method (proposed by Frederico) to define the ill-defined 'Coulomb plus delta' interaction via a T-matrix approach.
• Apply the renormalization condition $ C(Q,ar{\Lambda}) = R(Q,\Lambda_0) - R(Q,\Lambda) $, ensuring the counterterm removes cut-off dependence.
• Construct an effective Hamiltonian in configuration space, including a harmonic oscillator plus delta interaction, to model bound states.
• Use the renormalized regulator $ \overline{R}(Q,\Lambda) = R(Q,\Lambda_0) $ to achieve manifest cut-off independence.

Experimental results

Research questions

• RQ1 How can a non-perturbative renormalization procedure be consistently applied to a gauge-theory Hamiltonian?
• RQ2 What is the role of the regulator form factor in modifying the short-distance behavior of the potential in QCD?
• RQ3 Can the arbitrariness in the regulator function be physically constrained by experimental data?
• RQ4 How does the subtraction method enable the solution of ill-defined Hamiltonians like the 'Coulomb plus delta' interaction?
• RQ5 To what extent can the effective Hamiltonian reproduce physical hadron spectra, including the meson spectrum?

Key findings

• The non-perturbative renormalization procedure successfully renders the light-cone Hamiltonian cut-off independent through the introduction of counterterms derived from the regulator's derivative.
• The form factor's freedom allows for a tunable short-distance potential, including a pocket that can bind quarks in hadrons.
• The method enables the solution of the 'Coulomb plus delta' interaction via the subtraction method, with agreement between different schemes.
• The effective Hamiltonian $ H = \frac{\vec{p}^2}{2m_r} - a + \frac{f}{2}r^2 + \mu\delta^{(3)}(\vec{r}) $ reproduces the experimental meson spectra with only three parameters: two continuous and one counting index.
• The renormalized regulator $ \overline{R}(Q,\Lambda) = R(Q,\Lambda_0) $ is manifestly independent of the cut-off $ \Lambda $, confirming the consistency of the approach.
• The work establishes a viable path from the QCD Lagrangian to physical hadron eigenstates via light-cone wave functions, with a systematic framework for future extension to baryons and nuclei.