A nonperturbative calculation of the electron's magnetic moment

Abstract

In principle, the complete spectrum and bound-state wave functions of a quantum field theory can be determined by finding the eigenvalues and eigensolutions of its light-cone Hamiltonian. One of the challenges in obtaining nonperturbative solutions for gauge theories such as QCD using light-cone Hamiltonian methods is to renormalize the theory while preserving Lorentz symmetries and gauge invariance. For example, the truncation of the light-cone Fock space leads to uncompensated ultraviolet divergences. We present two methods for consistently regularizing lightcone-quantized gauge theories in Feynman and light-cone gauges: (1) the introduction of a spectrum of Pauli-Villars fields which produces a finite theory while preserving Lorentz invariance; (2) the augmentation of the gauge-theory Lagrangian with higher derivatives. In the latter case, which is applicable to light-cone gauge (A + = 0), the A − component of the gauge field is maintained as an independent degree of freedom rather than a constraint. Finite-mass Pauli-Villars regulators can also be used to compensate for neglected higher Fock states. As a test case, we apply these regularization procedures to an approximate nonperturbative computation of the anomalous magnetic moment of the electron in QED as a first attempt to meet Feynman's famous challenge.

Nonperturbative light-front Hamiltonian methods

“…As the number of basis functions is taken to infinity, there can and will be divergences in general. For example, in QED the wave functions of the dressed electron are known to fall off too slowly to be normalizable [24], but in the BLFQ approach the approximate wave functions are normalizable; the normalization becomes infinite only in the limit of infinite N max . This then requires some care in the regularization of the original Hamiltonian and in the process of taking the continuum limit.…”

“…However, the interaction Lagrangian has been arranged in just such a way that the A-dependent terms can be canceled between the constraints for individual fields [24].…”

“…The distinction between which are kept and which not is the number of constituents required by intermediate states. This prevents complete removal of the regulators and requires a strategy of seeking a range of regulation parameters (e.g., the PV masses) within which the physical observables are insensitive to the regularization [24]. This minimizes the net effect of two sources of error: the truncation itself, which creates the sensitivity to the regulators, and the presence of PV fields with finite masses.…”

“…Hiller and Brodsky [17] used DLCQ combined with sector-dependent renormalization, an invariant-mass cutoff, and a Fock space truncation to two photons. More recent calculations [24,82] use PV regularization. The one-photon truncation of the dressed electron state has also been computed by Karmanov et al [145,18].…”

Quantum fields on the light front, formulation in coordinates close to the light front, lattice approximation

“…the total number of "particles" (the Tamm-Dankoff method on the LF) [15,16]. We can then leave coordinates unrestricted.…”