[Paper Review] The Light-Cone Effective Potential

T. Heinzl|ArXiv.org|Dec 17, 2002

Quantum and Classical Electrodynamics1 references5 citations

TL;DR

This paper resolves a paradox in light-cone quantum field theory by demonstrating that zero modes—specifically the $k^+ = 0$ mode—are essential for correctly computing the one-loop effective potential in $ar{ ho}^4$ theory. Using Yan's formula for the $k^-$-integral, it shows that the standard covariant result for the effective potential is recovered only when zero modes are included, and discarding them via a cutoff leads to unphysical results, such as a vanishing one-loop contribution after renormalization.

ABSTRACT

It is shown how to calculate simple vacuum diagrams in light-cone quantum field theory. As an application, I consider the one-loop effective potential of phi^4 theory. The standard result is recovered both with and without the inclusion of zero modes having longitudinal momentum k^+ = 0.

Motivation & Objective

• To resolve the discrepancy between light-cone quantization and covariant perturbation theory in computing the one-loop effective potential.
• To clarify the role of zero modes—specifically $k^+ = 0$—in light-cone field theory, particularly in vacuum amplitudes.
• To demonstrate that excluding zero modes via a $k^+$-cutoff leads to unphysical results, such as a vanishing one-loop contribution after renormalization.
• To show that the correct effective potential can be recovered in light-cone formalism only when zero mode contributions are properly included.
• To provide a consistent regularization and renormalization scheme for the effective potential in light-cone quantization that preserves symmetry breaking effects.

Proposed method

• Derives the light-cone effective potential using the standard one-loop formula for the effective action, expressed as a sum over 1PI diagrams with vanishing external momenta.
• Applies Yan's formula for the $k^-$-integral: $\int dk^- (k^+k^- - m^2 + i\epsilon)^{-2} = \frac{2\pi i}{m^2} \delta(k^+)$, which isolates the contribution from the zero mode at $k^+ = 0$.
• Performs the calculation in light-cone coordinates, showing that the standard covariant result for the effective potential is recovered only when this delta-function contribution is included.
• Compares results with and without zero modes: excluding them via a $|k^+| > \delta$ cutoff leads to $M_\delta(0) = 0$, contradicting the covariant result.
• Uses a derivative trick: differentiating the effective potential with respect to $\phi_c$ maps the log-determinant to a tadpole diagram with mass $U''$, which is then evaluated using known regularization techniques.
• Introduces a modified regularization scheme where the mass $m^2$ is replaced by $U'' = m^2 + \frac{\lambda}{2}\phi_c^2$ to recover the correct logarithmic dependence, showing that zero mode resummation effectively modifies the cutoff.

Experimental results

Research questions

• RQ1 Why does light-cone perturbation theory with a $k^+$-cutoff yield a vanishing scattering amplitude for $p^2 = 0$, while covariant theory gives a finite result?
• RQ2 What is the role of the zero mode ($k^+ = 0$) in the light-cone calculation of vacuum amplitudes?
• RQ3 Can the standard one-loop effective potential in $\phi^4$ theory be consistently reproduced in light-cone quantization?
• RQ4 What happens to the effective potential if zero mode contributions are strictly excluded via regularization?
• RQ5 How can a consistent regularization and renormalization scheme be constructed in light-cone field theory that preserves spontaneous symmetry breaking?

Key findings

• The standard one-loop effective potential in $\phi^4$ theory is correctly reproduced in light-cone quantization only when the zero mode contribution—proportional to $\delta(k^+)$—is included.
• Excluding zero modes via a $|k^+| > \delta$ cutoff leads to an unphysical result: the one-loop contribution vanishes after renormalization, yielding $V_{\text{nZM,R}} = U$, the tree-level potential.
• The correct logarithmic dependence in the effective potential arises from the resummation of zero mode contributions, which effectively modifies the regularization cutoff by replacing $m^2 \to U'' = m^2 + \frac{\lambda}{2}\phi_c^2$.
• The derivative trick—differentiating the effective potential to reduce the log-determinant to a tadpole diagram—allows consistent evaluation of the effective potential in light-cone coordinates.
• The result confirms that the light-cone vacuum is not trivial in the presence of zero modes, and that their inclusion is essential for capturing physical effects like spontaneous symmetry breaking.
• The paper provides a resolution to the apparent paradox in light-cone field theory by showing that the $k^+$-cutoff breaks the continuity of the $k^-$-integral, and that the delta function at $k^+ = 0$ is the source of the missing amplitude.