A Null-Test Parameterization for Non-Cubic Scaling in Vacuum Four-Wave Mixing

This article is accepted(Under Review)by the Journal of Optics and Photonics Research (eISSN 3029-1348), Nov.2025. Vacuum four-wave mixing provides a sensitive probe of quantum electrodynamics (QED) in extreme-field regimes. In standard QED, the signal photon yield follows a strictly cubic intensity scaling. In this work, we investigate a constrained phenomenological extension of this prediction, formulated as an effective test hypothesis for possible deviations from cubic scaling at ultra-high intensity. The electromagnetic field dynamics are described at the level of standard QED, supplemented by a non-dynamical modulation parameter ψ used only to parametrize structured phase-dependent corrections in the observable response. This parameter does not introduce additional propagating degrees of freedom, does not modify the fundamental equations of motion, and is not presented as a microscopic theory of the vacuum. Within this phenomenological formulation, the effective third-order vacuum response acquires a weak intensity-dependent correction at the level of observable scaling behavior. As a result, the signal yield is parameterized by a modified power law Ns ∝ I3+δ in the ultraintense regime, where δ denotes a small phenomenological deviation and vanishes continuously in the recovery limit δ → 0, restoring the standard QED prediction. The framework makes no claim of theoretical necessity or microscopic derivation and is intended purely as an operational null-test parameterization. We formulate experimentally accessible criteria for distinguishing such deviations from known QED nonlinearities and realistic experimental systematics using log–log intensity analyses, and we outline parameter ranges in which forthcoming high-power laser facilities could constrain or exclude the hypothesis. The results provide a minimal and falsifiable phenomenological extension for testing non-cubic scaling in vacuum nonlinear optics.