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arxiv: 2607.00136 · v1 · pith:FP3VA3HBnew · submitted 2026-06-30 · ✦ hep-ph · hep-th

The left-cut for partial waves in terms of physical amplitudes

Alexandre Salas-Bern\'ardez This is my paper

Pith reviewed 2026-07-02 18:45 UTC · model grok-4.3

classification ✦ hep-ph hep-th
keywords left-hand cutpartial wavesdispersion relations2-to-2 scatteringunitarizationright-hand cutInverse Amplitude Methodanalyticity
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0 comments X

The pith

The left-hand cut of partial waves equals an integral over right-hand cut imaginary parts.

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

The paper derives an integral representation for the left-hand cut of partial-wave amplitudes in 2-to-2 scattering processes. It shows that this cut, for any isospin and angular momentum, can be written directly as an integral involving only the imaginary parts along the right-hand cut. The representation isolates the logarithmic branch points explicitly without requiring extra subtractions. A reader would care because the result supplies a model-independent way to handle left-hand cuts inside unitarization procedures such as the Inverse Amplitude Method or N/D equations.

Core claim

We derive a novel representation of the partial wave amplitude over the left-hand cut for 2 to 2 scattering. We express the left-hand cut of arbitrary isospin and angular momentum partial waves as an integral of right-hand cut imaginary parts. This formulation provides an explicit, exact extraction of the logarithmic branch cut structures.

What carries the argument

An integral representation that maps the discontinuity across the right-hand cut to the value of the partial-wave amplitude on the left-hand cut, obtained from crossing symmetry and analyticity of the 2-to-2 amplitude.

If this is right

  • Quantifies left-hand cut uncertainties exactly inside the Inverse Amplitude Method without model inputs.
  • Supplies the same exact left-cut term for N/D unitarization approaches.
  • Isolates logarithmic branch points for any isospin and angular momentum.
  • Removes the need for auxiliary model-dependent parametrizations of the left cut.

Where Pith is reading between the lines

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

  • The formula could be used to propagate experimental uncertainties from right-cut data into left-cut contributions when fitting resonances.
  • It offers a route to compare dispersive partial-wave results directly with lattice determinations of scattering amplitudes.
  • The representation might extend to processes with more than two particles once suitable crossing relations are available.

Load-bearing premise

The partial-wave projected amplitude admits a dispersion relation whose left-cut discontinuity can be written solely through the right-cut imaginary parts with no additional subtractions.

What would settle it

Direct numerical comparison in a solvable model such as the linear sigma model, where the left-cut can be computed independently, would falsify the claim if the integral fails to reproduce the known left-cut values.

read the original abstract

We derive a novel representation of the partial wave amplitude over the left-hand cut for $2 \to 2$ scattering. We express the left-hand cut of arbitrary isospin and angular momentum partial waves as an integral of right-hand cut imaginary parts. This formulation provides an explicit, exact extraction of the logarithmic branch cut structures, offering a valuable tool to systematically quantify left-hand cut uncertainties in unitarization methods such as the Inverse Amplitude Method or $N/D$ approaches.

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

1 major / 0 minor

Summary. The manuscript derives a representation for the left-hand cut of partial-wave amplitudes in 2→2 scattering. It expresses the left-cut discontinuity for arbitrary isospin and angular momentum as an integral over the imaginary parts on the right-hand cut, using standard analyticity and crossing properties, with the goal of providing an exact extraction of logarithmic branch points to quantify left-cut uncertainties in unitarization methods such as the Inverse Amplitude Method or N/D approaches.

Significance. If the central result holds without hidden assumptions, the representation would supply a model-independent tool for handling left-hand cuts in dispersion relations for partial waves. This could improve systematic control of uncertainties in phenomenological applications to hadron scattering and resonance physics.

major comments (1)
  1. [Abstract / central derivation] Abstract / central derivation: The claim of an exact, parameter-free integral expression for the left-cut discontinuity assumes that the partial-wave amplitude obeys an unsubtracted dispersion relation whose only singularities are the standard right- and left-hand cuts. Standard Regge asymptotics for fixed-L partial waves imply that low-L (especially s-wave) or certain isospin channels generally require one or more subtractions; these would contribute polynomial terms that alter the left-cut value. The manuscript does not discuss or incorporate subtraction constants, which is load-bearing for the asserted exactness.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for raising the important issue of subtraction constants. We address the concern point by point below and will incorporate a clarification in the revised version.

read point-by-point responses

  1. Referee: [Abstract / central derivation] Abstract / central derivation: The claim of an exact, parameter-free integral expression for the left-cut discontinuity assumes that the partial-wave amplitude obeys an unsubtracted dispersion relation whose only singularities are the standard right- and left-hand cuts. Standard Regge asymptotics for fixed-L partial waves imply that low-L (especially s-wave) or certain isospin channels generally require one or more subtractions; these would contribute polynomial terms that alter the left-cut value. The manuscript does not discuss or incorporate subtraction constants, which is load-bearing for the asserted exactness.

    Authors: We thank the referee for this observation. The manuscript does not explicitly discuss subtraction constants. However, the central result is an exact integral representation specifically for the discontinuity of the partial-wave amplitude across the left-hand cut, obtained from the standard analytic structure (right- and left-hand cuts), unitarity on the right-hand cut, and crossing symmetry. Subtraction constants enter dispersion relations for the amplitude as polynomial terms. These polynomials are entire functions and therefore possess vanishing discontinuities on every cut. It follows that the left-hand-cut discontinuity itself is unaffected by the presence, number, or values of subtraction constants; the integral expression over right-hand-cut imaginary parts remains exact and parameter-free. The polynomials would be fixed separately (e.g., by low-energy theorems or phenomenology), but they do not modify the left-cut discontinuity or the logarithmic branch points it encodes. We will add a short clarifying paragraph after the statement of the main result and in the introduction to make this independence explicit. revision: yes

Circularity Check

0 steps flagged

Derivation from standard analyticity and crossing; no reduction to inputs by construction

full rationale

The paper presents a derivation of the left-hand cut discontinuity for partial waves as an integral over right-hand cut imaginary parts, based on unsubtracted dispersion relations and crossing for 2→2 amplitudes. No quoted equations or steps reduce the claimed result to a fitted parameter, self-definition, or load-bearing self-citation. The central expression follows from assumed analytic properties without renaming known results or smuggling ansatze. This matches the most common honest finding of self-contained derivations from first principles.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of dispersion relations linking left- and right-hand cuts for 2→2 amplitudes; no free parameters, invented entities, or ad-hoc axioms are mentioned in the abstract.

axioms (1)
  • domain assumption Scattering amplitudes for 2→2 processes possess standard analytic properties including a right-hand cut from unitarity and a left-hand cut from crossing symmetry.
    This background property is required for any dispersion relation that isolates the left-cut contribution as an integral over right-cut imaginary parts.

pith-pipeline@v0.9.1-grok · 5592 in / 1224 out tokens · 30522 ms · 2026-07-02T18:45:35.932950+00:00 · methodology

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