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Near-Null Limits and Contact Distributions for Maxwell Composite Operators

Primarily human-written textHuman understanding: all partsmath-ph — Mathematical Physics

Contributed by Zhenyu Sun ↗

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

We investigate the simultaneous near-null, longitudinal-continuum, and ultraviolet limits of four-point correlation functions of quadratic composite operators in Maxwell theory. Taking the external momenta to be collinear and spacelike, we weight each loop integral by a smooth test function of the routed longitudinal momentum. All three connected Wick contraction cycles are generated from a common Gaussian source functional and regulated by a cutoff on the total proper time. For the resulting integrals restricted to a fixed ultraviolet proper-time interval, we establish the uniform bound ∣Iδ,Mϵ[ϕ]−I0,∞ϵ[ϕ]∣≤C∥ϕ∥Cb8(δϵ−3+M−2δ−3ϵ−1)\left|\mathcal I_{\delta,M}^{\epsilon}[\phi]-\mathcal I_{0,\infty}^{\epsilon}[\phi]\right|\le C\|\phi\|_{C_b^8}\left(\delta\epsilon^{-3}+M^{-2}\delta^{-3}\epsilon^{-1}\right), where δ\delta denotes the dimensionless near-null tilt parameter and MM specifies the longitudinal lattice resolution. This estimate determines the joint scaling regimes under which a universal counterterm subtraction yields an identical renormalized finite part. We further show that the longitudinal logarithmic divergence of the alternating chiral cycle is governed by the measure 81(0,1)dξ+3δ0+3δ18\mathbf 1_{(0,1)}\mathrm d\xi+3\delta_0+3\delta_1, whereas each of the other two cycles contributes 81(0,2)dξ8\mathbf 1_{(0,2)}\mathrm d\xi. These distributional coefficients emerge from the complete covariant numerators, including the instantaneous longitudinal terms. Finally, for the alternating cycle and any smooth compactly supported test function coinciding with ξ2\xi^2 in a neighborhood of [0,1][0,1], the critical trajectory δ=cϵ3\delta=c\epsilon^3 with M2δ3ϵ→∞M^2\delta^3\epsilon\to\infty leaves a finite discrepancy of 30c30c, demonstrating the sharpness of the tilt scaling. The proof combines exact Gaussian moment reductions, whole-line quadrature error bounds, and an explicit evaluation of boundary jets.

Provenance statement

The manuscript is primarily human-written. Large language models were used for proofreading, language editing, and assistance with checking calculations and proofs. Symbolic identities and numerical quadrature were checked with SymPy 1.14.0 and mpmath 1.3.0. The contributor confirms that all parts have been checked and understood by a human.

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Version history

  1. v1Submitted by Zhenyu SunInitial depositCurrentOct 29, 2026