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

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.

💡 Research Summary

In this work the author addresses a long‑standing practical difficulty in light‑cone quantization: the extraction of light‑cone wave functions (LCWFs) from first‑principles calculations. While covariant perturbation theory provides compact, manifestly Lorentz‑invariant amplitudes, the traditional method for obtaining LCWFs relies on light‑cone perturbation theory (LCPT), which requires a non‑covariant Hamiltonian formulation, intricate time‑ordered diagrams, and careful handling of energy denominators. The paper proposes a conjectured mapping formula that directly converts off‑shell covariant amplitudes into LCWFs for a massive cubic scalar theory (φ³).

The author first sets up the scalar theory in d = 6 − 2δ₆ dimensions, introducing renormalized mass m_R and coupling λ_R, and works in the MS scheme. Detailed one‑loop calculations are presented for the self‑energy Σ_R(k²) and the three‑point vertex Γ₃(k²,k₁²,k₂²). The self‑energy is evaluated (Eqs. 10‑15), its imaginary part above the two‑particle threshold is identified via the discontinuity of the function J(x), and a dispersion relation is written down. The vertex correction is expressed through Feynman parameters (Eq. 25) and its UV divergence is removed by the standard counter‑term.

Next, the paper switches to light‑cone coordinates (v⁺, v⁻, v⊥) and defines the light‑cone energy E_R(𝑘⃗) = (k_⊥² + m_R²)/(2k⁺). The light‑cone Hamiltonian H = H₀ + H_int is constructed, and the Fock basis of multi‑particle states is introduced with the normalization (31)–(33). The LCWF for a transition 1 → n is written as the time‑ordered series (35), where each intermediate state contributes an energy denominator Δ_R that is the difference between the virtual light‑cone energy of the incoming particle and the sum of on‑shell energies of the intermediate particles.

The central conjecture is then stated: to obtain the LCWF one simply takes the covariant off‑shell amplitude, rewrites each external momentum in light‑cone variables, replaces every propagator denominator (k² − m_R²)⁻¹ by the corresponding light‑cone energy denominator Δ_R, and inserts a factor of the “virtual light‑cone energy” ˜k⁻ = E_R(𝑘⃗) for each external line. In this way the entire LCPT series is reproduced automatically, without having to perform the cumbersome time‑ordered expansion.

The conjecture is tested at one‑loop order for the simplest non‑trivial process: a 1 → 2 transition. The author evaluates the self‑energy contribution to the LCWF using the mapping formula and shows that the resulting expression matches exactly the LCPT calculation, including both the real part (mass‑renormalization effects) and the imaginary part (threshold cut). The vertex correction is treated similarly; after translating the covariant vertex integral into light‑cone language, the same Δ_R denominators appear, and the final LCWF coincides with the direct LCPT result. This explicit agreement validates the conjecture at the one‑loop level.

Section 4 extends the discussion to two‑loop contributions. By combining covariant amplitudes that contain nested self‑energy insertions or double‑vertex corrections, the author demonstrates that the same mapping reproduces the known two‑loop LCWF structure. The key technical tool is the use of dispersion relations and cut‑diagram techniques, which allow the author to reconstruct the LCPT energy denominators from the analytic properties of the covariant amplitudes.

The paper concludes by emphasizing several advantages of the new method: (i) it bypasses the need for a non‑covariant light‑cone Hamiltonian, preserving Lorentz symmetry throughout; (ii) it dramatically reduces the algebraic workload, especially at higher orders where LCPT diagrams proliferate; (iii) it provides a clear conceptual bridge between covariant S‑matrix elements and partonic light‑cone wave functions, reinforcing the physical equivalence of the two quantization schemes. The author also outlines future directions, notably the extension to gauge theories such as QCD, where color and spin structures would be incorporated into the same mapping framework, and the potential for applying the method to phenomenological calculations of high‑energy scattering off dense nuclear targets.

Overall, the work offers a concrete, technically verified prescription for extracting LCWFs from covariant amplitudes, opening a promising route to more efficient perturbative calculations in both scalar and, potentially, gauge field theories.