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arxiv: 2512.02097 · v2 · submitted 2025-12-01 · ✦ hep-ph · hep-th

Unitarizing non-relativistic scattering

Marcos M. Flores , Kalliopi Petraki This is my paper

Pith reviewed 2026-05-17 02:45 UTC · model grok-4.3

classification ✦ hep-ph hep-th
keywords non-relativistic scatteringunitarityinelastic channelsoptical theoremseparable potentialsanti-Hermitian contributionsbound statesrenormalization
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The pith

Inelastic channels generate anti-Hermitian contributions that unitarize non-relativistic scattering through separable potentials.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to satisfy unitarity constraints on both elastic and inelastic amplitudes by resumming self-energy contributions from all open channels. Inelastic processes supply anti-Hermitian pieces that are extracted directly from the optical theorem, which produces non-local separable potentials and a compact unitarization procedure valid throughout the non-relativistic regime. The same construction incorporates bound states and supplies renormalization rules for amplitudes that fail to converge in the complex plane. Readers would care because the resulting cross sections remain finite and respect probability conservation, supplying a practical tool for low-energy scattering calculations.

Core claim

Unitarity imposes coupled constraints on elastic and inelastic amplitudes. Satisfying them requires resummation of the self-energy contributions from both elastic and inelastic channels. Inelastic channels generate anti-Hermitian contributions that can be consistently deduced from the unitarity relation underlying the optical theorem, leading to non-local separable potentials and a compact, unique and complete unitarization scheme in the non-relativistic regime. Two alternative derivations of the anti-Hermitian kernel are given, from the continuity equation combined with LSZ reduction and by integrating out inelastic channels. The framework is extended to non-analytic and non-convergent inel

What carries the argument

Non-local separable potentials generated by an anti-Hermitian kernel extracted from the optical theorem, which resums self-energy corrections from inelastic channels while preserving unitarity.

If this is right

  • Elastic and inelastic amplitudes remain coupled through the shared separable potentials without further assumptions.
  • Bound states are included inside the same unitarization procedure.
  • Non-convergent inelastic amplitudes are rendered finite by Hermitian counterterms induced by the anti-Hermitian potentials.
  • The resulting cross sections stay consistent with unitarity at all energies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may serve as a default unitarization prescription in low-energy effective theories where multiple channels open.
  • Direct comparison with exactly solvable models could test whether the scheme is indeed unique.
  • Similar kernel constructions might be explored for other non-relativistic systems such as few-body nuclear reactions.
  • Dark-matter calculations that employ this framework could produce tighter unitarity-derived limits on interaction strengths.

Load-bearing premise

The resummation of self-energy contributions from elastic and inelastic channels via non-local separable potentials derived from the optical theorem yields a unique and complete scheme without additional inconsistencies in the non-relativistic regime.

What would settle it

Numerical solution of the Lippmann-Schwinger equation for a concrete non-relativistic potential that includes inelastic channels, followed by a direct check that the unitarized total cross section exactly equals the imaginary part of the forward amplitude as required by the optical theorem.

Figures

Figures reproduced from arXiv: 2512.02097 by Kalliopi Petraki, Marcos M. Flores.

Figure 1
Figure 1. Figure 1: Example sketch of a contour (red line) required to compute the integral in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗
read the original abstract

Unitarity imposes coupled constraints on elastic and inelastic amplitudes. Satisfying them requires resummation of the self-energy contributions from both elastic and inelastic channels. Inelastic channels generate anti-Hermitian contributions that can be consistently deduced from the unitarity relation underlying the optical theorem, leading to non-local separable potentials and a compact, unique and complete unitarization scheme in the non-relativistic regime. We present two alternative derivations of the anti-Hermitian kernel, from the continuity equation combined with LSZ reduction, and by integrating out inelastic channels. We further extend the unitarization framework to treat non-analytic and non-convergent behavior of inelastic amplitudes in the complex momentum plane and to incorporate bound states. For non-convergent amplitudes, we demonstrate two renormalization procedures in which anti-Hermitian separable potentials necessarily induce Hermitian separable counterterms, yielding finite cross-sections consistent with unitarity. These results provide a general tool for non-relativistic scattering, with clear applications to dark-matter phenomenology.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper claims that unitarity constraints on elastic and inelastic amplitudes in non-relativistic scattering can be satisfied by resumming self-energy contributions, with inelastic channels generating anti-Hermitian terms deduced from the optical theorem (and alternatively from the continuity equation plus LSZ reduction). This leads to non-local separable potentials that furnish a compact, unique and complete unitarization scheme, which is further extended to non-analytic/non-convergent amplitudes (via two renormalization procedures inducing Hermitian counterterms) and bound states, with applications to dark-matter phenomenology.

Significance. If the derivations are free of hidden scheme dependence and the uniqueness of the separable ansatz is established, the framework could supply a systematic, parameter-light tool for enforcing unitarity in non-relativistic effective theories. The dual derivations and explicit treatment of non-convergent cases are potentially useful strengths for phenomenology.

major comments (2)
  1. [Abstract and derivations section] Abstract and the section presenting the two alternative derivations of the anti-Hermitian kernel: the central claim that the resulting scheme is 'unique and complete' is load-bearing, yet the manuscript does not demonstrate that different choices of projection onto the separable non-local form or different regularizations when integrating out inelastic channels produce identical on-shell amplitudes after renormalization; this leaves open the possibility of residual scheme dependence.
  2. [Renormalization section] Section on renormalization for non-convergent amplitudes: the statement that anti-Hermitian separable potentials 'necessarily induce' Hermitian separable counterterms yielding finite, unitary cross-sections requires explicit verification that the counterterm basis is uniquely determined by the unitarity relation and does not introduce new free parameters or alter the on-shell uniqueness.
minor comments (1)
  1. [Notation and definitions] Clarify the precise definition of the non-local separable kernel (e.g., its momentum dependence and cutoff procedure) in the main text to facilitate reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the two major comments point by point below. Where appropriate, we have revised the manuscript to incorporate additional clarifications and explicit verifications.

read point-by-point responses

  1. Referee: [Abstract and derivations section] Abstract and the section presenting the two alternative derivations of the anti-Hermitian kernel: the central claim that the resulting scheme is 'unique and complete' is load-bearing, yet the manuscript does not demonstrate that different choices of projection onto the separable non-local form or different regularizations when integrating out inelastic channels produce identical on-shell amplitudes after renormalization; this leaves open the possibility of residual scheme dependence.

    Authors: We appreciate this observation. The two derivations in the manuscript (from the continuity equation plus LSZ reduction, and from integrating out inelastic channels) produce identical anti-Hermitian kernels, which supports the uniqueness of the resulting separable form. The separable projection is fixed by the requirement that the resummed amplitude exactly satisfies the optical theorem on-shell; off-shell differences arising from alternative projections are absorbed into the renormalization counterterms without affecting physical observables. For regularization dependence, the renormalization procedures ensure cutoff-independent finite unitary cross sections. We have added a clarifying paragraph in the derivations section and a new appendix with an explicit comparison of two regularization schemes demonstrating equivalence of the on-shell results. revision: yes

  2. Referee: [Renormalization section] Section on renormalization for non-convergent amplitudes: the statement that anti-Hermitian separable potentials 'necessarily induce' Hermitian separable counterterms yielding finite, unitary cross-sections requires explicit verification that the counterterm basis is uniquely determined by the unitarity relation and does not introduce new free parameters or alter the on-shell uniqueness.

    Authors: We thank the referee for this suggestion. The revised manuscript expands the renormalization section with an explicit derivation showing that the Hermitian counterterms are uniquely fixed by the unitarity relation (optical theorem) together with the finiteness condition. No additional free parameters are introduced; the counterterms are completely determined by these requirements. We demonstrate that the resulting on-shell amplitude remains unique and satisfies unitarity. This explicit verification has been added to confirm that the induced counterterms preserve the scheme's properties. revision: yes

Circularity Check

0 steps flagged

Derivation from optical theorem and continuity equation shows no reduction to inputs by construction

full rationale

The paper grounds its unitarization in the standard optical theorem for anti-Hermitian contributions from inelastic channels, with two alternative derivations (continuity equation + LSZ reduction; integrating out channels). No quoted equations or steps in the provided text demonstrate that the separable non-local potentials, uniqueness claim, or renormalization procedures reduce by definition to fitted parameters or prior self-citations. The scheme is presented as extending standard non-relativistic scattering tools, remaining self-contained against external benchmarks like unitarity relations. Minor self-citation risk noted but not load-bearing per available description.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard QFT principles such as unitarity and the optical theorem, plus domain assumptions about the non-relativistic limit and LSZ applicability; no free parameters or new entities with independent evidence are identified in the abstract.

axioms (2)
  • standard math Unitarity of the S-matrix and validity of the optical theorem
    Invoked as the underlying relation for deducing anti-Hermitian contributions from inelastic channels.
  • domain assumption Applicability of LSZ reduction formula and continuity equation in the non-relativistic regime
    Used in one of the two derivations of the anti-Hermitian kernel.
invented entities (1)
  • non-local separable anti-Hermitian potentials no independent evidence
    purpose: To encode inelastic channel contributions into the unitarized amplitudes
    Introduced as the output of the resummation procedure; no external falsifiable prediction is stated in the abstract.

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Forward citations

Cited by 2 Pith papers

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Reference graph

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