Partial-wave unitarity and long-range interactions

Technical Summary: Partial-wave unitarity and long-range interactions

Problem Statement
The paper addresses a foundational obstruction in applying partial-wave unitarity constraints to effective field theories (EFTs) involving massless particles (e.g., photons and gravitons). In theories with long-range $1/r$ interactions, tree-level $t$-channel exchanges generate singularities of the form $1/t$, while loop corrections introduce $\log(-s/t)$ singularities. These forward scattering singularities render standard fixed-order expressions for partial-wave amplitudes ($a_\ell$) ill-defined.

Specifically, the conventional definition of partial-wave amplitudes via the Legendre transform of the scattering amplitude $M(s,t)$ fails because the integral over the scattering angle $\theta$ diverges at $\theta \to 0$ (forward direction). Introducing an infrared (IR) regulator, such as a photon mass $\lambda$, leads to amplitudes scaling as $\alpha \log(\lambda/p_{CM})$, resulting in IR-divergent cross-sections ($\sigma_\ell \propto |a_\ell|^2$). This ambiguity prevents the systematic derivation of unitarity bounds for extensions of the Standard Model (SMEFT) involving charged particles, as the renormalizable Standard Model contribution itself contains these singular $t$-channel photon exchanges.

Methodology
The authors employ a modified perturbation theory based on the work of Dollard, which resolves the non-convergence of standard Møller operators in the presence of long-range potentials. The methodology proceeds through the following steps:

1. Dollard Operator Construction: The standard S-matrix definition is replaced by one utilizing modified Møller operators, $\Omega_D^{(\pm)}$, which include a phase factor scaling as $\log \tau$ (where $\tau$ is time). This phase accounts for the asymptotic Coulomb dynamics that prevent the standard free-evolution limit from converging.
2. Matching and Resummation: Using the method of regions, the authors analyze the scattering wavefunction in the soft/eikonal limit ($|t| \ll p_{CM}^2$). They demonstrate that IR divergences in the naive amplitude, computed in dimensional regularization, resum into a universal phase factor. This phase depends on a time scale $T$ and the "Coulomb parameter" $\eta = z_1 z_2 \alpha / \beta$.
3. Forward Scattering EFT: In the forward scattering region, the authors construct an effective field theory where the dominant contributions are captured by eikonal Feynman rules. They show that the leading-power amplitude in this region is exactly described by the eikonal approximation to all orders in perturbation theory.
4. Subtraction Scheme: A practical subtraction scheme is proposed to isolate the singular forward region. The total S-matrix element is decomposed into a resummed eikonal part ($S_\ell^E$) and a remainder ($a'_\ell$) derived from the difference between the full amplitude and the eikonal approximation ($M - M_E$).

Key Results

• Universal Forward Description: The authors derive a universal description of the forward scattering region that is independent of the renormalization scale. The resulting partial-wave amplitudes are well-defined, single-scale objects free of spurious dependence on IR regulators.
• Reinterpretation of Partial-Wave Amplitudes: A critical finding is that for long-range theories, the Legendre transform of the leading eikonal amplitude does not yield the standard partial-wave coefficient $a_\ell$ (where $S_\ell = 1 + i a_\ell$). Instead, it yields a unitary phase $S_\ell^E = \Gamma(1+\ell+i\eta)/\Gamma(1+\ell-i\eta)$.
• Modified Unitarity Condition: The physical S-matrix is reconstructed as $S_\ell = S_\ell^E + i a'_\ell$, where $a'_\ell$ is the partial-wave projection of the finite difference $[M - M_E]$. The unitarity condition becomes $|S_\ell^E + i a'_\ell| \leq 1$, or equivalently $|a'_\ell|^2 \leq 2\text{Im}(a'^*_\ell S_\ell^E)$.
• Cancellation of Divergences: The subtraction scheme ensures that the quantity $[M - M_E]$ is free of non-integrable forward singularities. Logarithmic IR divergences are shifted to $\log(-t/|t|_{\text{max}})$ via the Dollard construction, rendering the integrals for $a'_\ell$ finite order-by-order in perturbation theory.
• Concrete Application: The authors explicitly demonstrate this procedure for high-energy electron scattering in a Coulomb field, showing that the subtraction of the eikonal amplitude from the tree-level and one-loop results yields finite integrals for the partial-wave coefficients.

Significance and Claims
The paper claims to resolve the conceptual and technical obstructions preventing the systematic application of partial-wave unitarity bounds to theories with massless particles. By establishing a constructive connection between dimensionally regulated amplitudes and Dollard's modified perturbation theory, the authors provide a practical method for computing unitary partial-wave amplitudes order-by-order.

The work sharpens recent conjectures regarding $t$-channel singularities, demonstrating that they do not preclude the definition of a finite and unitary S-matrix for relativistic field theories with massless particles. The authors position this framework as a necessary tool for:

1. SMEFT Constraints: Enabling the derivation of robust unitarity bounds for operators involving electromagnetically charged particles, where previous interpretations were ambiguous due to the Standard Model photon contribution.
2. S-Matrix Bootstrap: Providing a consistent formalism for the S-matrix bootstrap program when applied to theories with long-range interactions.

The paper emphasizes that the Coulomb singularity is distinct from real photon emission issues and that the proposed subtraction scheme allows for the isolation of the long-range dynamics without requiring a specific partitioning of the Hilbert space into "soft" and "hard" sectors for the purpose of defining the S-matrix in vacuum.