Extracting light-cone wave functions from covariant amplitudes: a detailed study in scalar field theory

TL;DR

The paper tackles the problem of obtaining light-cone wave functions (LCWFs) from covariant off-shell amplitudes in a cubic scalar theory by proposing a practical mapping that uses covariant inputs $\Sigma_R$ and $\Gamma_{3R}$. It tests the conjectured relation at one loop by explicit comparison with light-cone perturbation theory (LCPT) for both vertex and self-energy contributions, and demonstrates that the resulting $1\to 2$ LCWFs reproduce known LCPT results, including the massless limit. The approach yields compact expressions for $1\to 2$ LCWFs derived directly from covariant amplitudes, bypassing the intricacies of light-cone quantization and standard LCPT calculations, while preserving the correct energy denominators and renormalization structure. The authors argue that the framework can be extended to higher orders and gauge theories, offering a practical route to explore the equivalence of covariant and light-cone quantizations and to simplify perturbative wave-function computations. Overall, the work provides both a concrete method and a set of validation points that underscore the potential of covariant-to-LCWF mappings for broader quantum field theory applications.

Abstract

We propose a conjectured formula that systematically maps covariant off-shell amplitudes to light-cone wave functions in scalar field theory. Through an explicit comparison at one-loop accuracy, we establish its equivalence to the light-cone perturbation theory series, thereby validating the conjecture at this order. Applying this formula, we efficiently re-derive wave functions from known covariant amplitudes, bypassing both the conceptual complexities of light-cone quantization and the technical challenges of perturbative calculations in this framework. In addition to simplifying computations, this approach opens new avenues for applications in gauge theories and deeper explorations of the fundamental equivalence between covariant and light-cone quantization.

Figures (8)

• Figure 1: Order $\bar{\lambda}_R^2$ contributions to the self-energy $-i\Sigma_R(k^2)$.
• Figure 2: Order $\bar{\lambda}_R^3$ contributions to the vertex function $i\Gamma_{3R}$.
• Figure 3: A subset of the one-loop diagrams contributing to $\psi_{\phi\to\varphi\varphi}$. We have labeled the propagators by the $d$-dimensional momenta that flow through them, from left to right, and the trivalent vertices by (possibly primed) numbers.
• Figure 4: One-loop vertex correction diagrams in light-cone perturbation theory, that correspond to the covariant diagram of Fig. \ref{['subfig:scalarvertex']}. Light-cone time increases from left to right, and the $+$ components of the displayed momenta are all positive.
• Figure 5: Complex plane of the $k_1^-$ variable for the diagram in Fig. \ref{['subfig:scalarvertex']}. The initial integration contour runs along the entire real axis. We eventually deform this contour into a contour enclosing the lower half-plane, where it may be shrunk to surround a single pole and a cut (dotted lines). The integral \ref{['eq:vertex-1-loop-integrated']} then reduces to the sum of the residue at this pole and the integral of the discontinuity of the integrand across the cut.