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Log Soft Constraints on KMOC Formalism

Siddhartha Paul, Adarsh Vishwakarma

TL;DR

The paper investigates how the KMOC framework maps on-shell scattering amplitudes to classical electromagnetic observables in four dimensions, with a focus on the memory effect and its log-tail. By constructing the soft-radiation kernel from leading and subleading soft-photon theorems, it shows that infrared observables emerge directly from amplitude data and that macroscopic causality imposes nontrivial consistency conditions on the S-matrix. It proves leading soft constraints hold to all orders and verifies subleading log-soft constraints at one loop, clarifying how classical tails arise from quantum amplitudes. The results illuminate the infrared structure of the S-matrix, connecting universal soft theorems to macroscopic causality and suggesting a framework to generalize to gravitational theories. Overall, the work demonstrates that classical memory and tail phenomena encode non-perturbative information about the S-matrix accessible through inclusive, on-shell methods.

Abstract

The KMOC formalism provides a systematic framework for extracting classical observables perturbatively from on-shell scattering amplitudes. In this work, we apply this formalism to compute electromagnetic observables in four dimensions, focusing in particular on the linear memory effect and its tail contributions. Using the leading and subleading soft-photon theorems to construct the soft radiation kernel, we demonstrate how these infrared observables emerge directly from amplitude data. We further show that demanding the expected non-perturbative properties of memory and tail effects imposes a nontrivial set of consistency conditions on the underlying S-matrix. We interpret these constraints as imposing the requirement of macroscopic causality on the S-matrix via analysis of inclusive observables.

Log Soft Constraints on KMOC Formalism

TL;DR

The paper investigates how the KMOC framework maps on-shell scattering amplitudes to classical electromagnetic observables in four dimensions, with a focus on the memory effect and its log-tail. By constructing the soft-radiation kernel from leading and subleading soft-photon theorems, it shows that infrared observables emerge directly from amplitude data and that macroscopic causality imposes nontrivial consistency conditions on the S-matrix. It proves leading soft constraints hold to all orders and verifies subleading log-soft constraints at one loop, clarifying how classical tails arise from quantum amplitudes. The results illuminate the infrared structure of the S-matrix, connecting universal soft theorems to macroscopic causality and suggesting a framework to generalize to gravitational theories. Overall, the work demonstrates that classical memory and tail phenomena encode non-perturbative information about the S-matrix accessible through inclusive, on-shell methods.

Abstract

The KMOC formalism provides a systematic framework for extracting classical observables perturbatively from on-shell scattering amplitudes. In this work, we apply this formalism to compute electromagnetic observables in four dimensions, focusing in particular on the linear memory effect and its tail contributions. Using the leading and subleading soft-photon theorems to construct the soft radiation kernel, we demonstrate how these infrared observables emerge directly from amplitude data. We further show that demanding the expected non-perturbative properties of memory and tail effects imposes a nontrivial set of consistency conditions on the underlying S-matrix. We interpret these constraints as imposing the requirement of macroscopic causality on the S-matrix via analysis of inclusive observables.
Paper Structure (24 sections, 132 equations, 7 figures)

This paper contains 24 sections, 132 equations, 7 figures.

Table of Contents

  1. Introduction
  2. A brief review on KMOC formalism and soft theorems
  3. KMOC
  4. Soft Theorems
  5. Soft limit in KMOC formalism
  6. Leading Soft Constraints on $\mathcal{R}(k)$
  7. $\langle q^{\mu_1}_a\cdots q^{\mu_N}_a \rangle$ and Contact terms
  8. Sub-leading soft radiation kernel
  9. Computation of Angular Impulse
  10. Computation of $\langle q_1^{\alpha_1}\cdots q_1^{\alpha_N}\hat{J}_{1,\text{out}}^{\mu \nu} \rangle$.
  11. Log soft constraints on radiation kernel
  12. First set of Consistency conditions.
  13. Constraints at L loop
  14. One-loop Calculation
  15. Second set of Consistency Conditions
  16. ...and 9 more sections

Figures (7)

  • Figure 1: The sequence of classical and soft limits. CSPT = Classical Soft Photon Theorem, QSPT = Quantum Soft Photon Theorem.
  • Figure 2: The four classes of $L$ loop diagrams that give log-divergent contributions in the soft region of the photon separated from the $L$ loop.
  • Figure 3: $L$ loop diagram with the outer photon being in the soft region.
  • Figure 4: Example of $L$ loop diagram appearing in the cut contribution.
  • Figure 5: The 1-loop cut diagrams corresponding to the two possibilities of the photon leg being soft.
  • ...and 2 more figures