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arxiv: 2506.22546 · v2 · pith:3YEYP3AHnew · submitted 2025-06-27 · ✦ hep-th

Primal S-matrix bootstrap with dispersion relations

Claudia de Rham , Andrew J. Tolley , Zhuo-Hui Wang , Shuang-Yong Zhou This is my paper

Pith reviewed 2026-05-21 23:56 UTC · model grok-4.3

classification ✦ hep-th
keywords S-matrix bootstrapdispersion relationspartial wavesunitaritycrossing symmetryglueball couplingsRegge behaviorscattering amplitudes
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0 comments X

The pith

Parameterizing the imaginary parts of partial waves and feeding them into dispersion relations builds the space of consistent scattering amplitudes that obey unitarity and crossing by construction.

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

The paper develops a method to generate scattering amplitudes by directly specifying the imaginary parts of their partial waves. Dispersion relations then reconstruct the real parts while enforcing analyticity and crossing symmetry, and unitarity is imposed in full. This produces amplitudes that automatically respect high-energy bounds such as the Froissart-Martin limit. The same construction is applied to place numerical bounds on leading couplings and to constrain interactions involving spinning bound states such as glueballs. Different assumptions about the growth rate at high energies can be tested through a new family of fractionally subtracted dispersion relations.

Core claim

By parameterizing the imaginary parts of partial waves and utilizing dispersion relations together with crossing symmetry and full unitarity, the framework constructs the consistent space of scattering amplitudes and computes bounds on leading couplings while automatically satisfying Froissart-Martin/Jin-Martin bounds. The method also readily accommodates spinning bound states, which are used to constrain glueball couplings. Incorporating dispersion relations ensures the amplitudes satisfy the Froissart-Martin/Jin-Martin bounds or softer high-energy behaviors by construction, allowing a new class of fractionally subtracted dispersion relations that probe the sensitivity of coupling bounds to

What carries the argument

The parameterization of the imaginary parts of partial waves, which is used as input to dispersion relations that enforce analyticity, crossing symmetry, and high-energy consistency.

If this is right

  • Numerical upper and lower bounds on the leading low-energy couplings can be extracted for any chosen high-energy behavior.
  • The Regge trajectories and residues of the constructed amplitudes can be read off directly from the partial-wave data.
  • Spinning glueball states can be included to derive concrete limits on their couplings to the scattering particles.
  • Fractionally subtracted dispersion relations make the dependence of the bounds on the asymptotic growth rate quantitatively accessible.

Where Pith is reading between the lines

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

  • The same parameterization might be applied to processes with more particles or to amplitudes in curved backgrounds.
  • Connecting this approach to lattice calculations of glueball spectra could provide independent cross-checks on the coupling bounds.
  • The method offers a route to test whether certain effective field theory operators are excluded by consistency alone.

Load-bearing premise

The chosen parameterization of the imaginary parts of partial waves is general enough to capture every physically allowed amplitude without missing constraints from locality or analyticity.

What would settle it

An explicit amplitude that satisfies unitarity, crossing symmetry, analyticity, and the Froissart-Martin bound yet cannot be reproduced by any choice of the partial-wave imaginary-part parameterization would show the method is incomplete.

read the original abstract

We propose a new method for constructing the consistent space of scattering amplitudes by parameterizing the imaginary parts of partial waves and utilizing dispersion relations, crossing symmetry, and full unitarity. Using this framework, we explicitly compute bounds on the leading couplings and examine the Regge behaviors of the constructed amplitudes. The method also readily accommodates spinning bound states, which we use to constrain glueball couplings. By incorporating dispersion relations, our approach inherently satisfies the Froissart-Martin/Jin-Martin bounds or softer high-energy behaviors by construction. This, in turn, allows us to formulate a new class of fractionally subtracted dispersion relations, through which we investigate the sensitivity of coupling bounds to the asymptotic growth rate.

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 / 2 minor

Summary. The manuscript proposes a primal S-matrix bootstrap framework in which the imaginary parts of partial waves are parameterized and then fed into dispersion relations together with crossing symmetry and full unitarity. This construction is used to map out the space of consistent scattering amplitudes, to derive explicit bounds on leading low-energy couplings, to examine Regge behavior, and to constrain glueball couplings by including spinning bound states. The approach is designed to satisfy the Froissart-Martin and Jin-Martin bounds automatically; fractionally subtracted dispersion relations are introduced to test the sensitivity of the coupling bounds to the assumed high-energy growth.

Significance. If the parameterization is shown to be sufficiently complete, the method would constitute a useful technical advance in the S-matrix bootstrap program by enforcing analyticity and high-energy constraints at the level of the primal variables rather than through post-hoc checks. The built-in compliance with Froissart-Martin/Jin-Martin bounds and the explicit treatment of spinning states are clear strengths. The sensitivity analysis via fractional subtractions provides a concrete way to quantify the impact of asymptotic assumptions on low-energy results.

major comments (2)
  1. [§3] §3 (parameterization of Im partial waves): the finite-basis ansatz adopted for the imaginary parts must be shown to be dense in the space of functions allowed by analyticity, locality, and the high-energy bounds that are imposed by construction. If the chosen functional form excludes certain Regge trajectories or non-perturbative contributions permitted by the axioms, the resulting bounds on leading couplings become conditional on the ansatz rather than universal.
  2. [§5] §5 (fractionally subtracted dispersion relations): the claim that the framework constructs the full consistent space relies on the completeness of the parameterization; without either an analytic argument or numerical evidence that arbitrary high-energy behaviors consistent with the axioms can be reproduced, the sensitivity study in this section risks underestimating the allowed range of low-energy couplings.
minor comments (2)
  1. [Notation] The notation for the partial-wave indices and the subtraction constants could be made more uniform between the main text and the appendices to improve readability.
  2. [Results] A short table summarizing the numerical values of the derived bounds together with the corresponding Regge intercept assumptions would help readers compare results across different growth rates.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive major comments. We address each point below, indicating where revisions will be made to clarify the scope and limitations of our results.

read point-by-point responses

  1. Referee: [§3] §3 (parameterization of Im partial waves): the finite-basis ansatz adopted for the imaginary parts must be shown to be dense in the space of functions allowed by analyticity, locality, and the high-energy bounds that are imposed by construction. If the chosen functional form excludes certain Regge trajectories or non-perturbative contributions permitted by the axioms, the resulting bounds on leading couplings become conditional on the ansatz rather than universal.

    Authors: We agree that the finite-basis parameterization is not proven to be dense in the full space of functions permitted by the axioms. Our ansatz is constructed to incorporate the leading Regge behaviors and to respect the high-energy bounds enforced by the dispersion relations, but it remains a specific functional family. In the revised manuscript we will expand the discussion in §3 to explicitly state that the derived coupling bounds apply within this parameterized class, to describe the criteria used in selecting the basis functions, and to note possible extensions that could accommodate additional non-perturbative contributions. This clarification will make the conditional character of the results transparent. revision: partial

  2. Referee: [§5] §5 (fractionally subtracted dispersion relations): the claim that the framework constructs the full consistent space relies on the completeness of the parameterization; without either an analytic argument or numerical evidence that arbitrary high-energy behaviors consistent with the axioms can be reproduced, the sensitivity study in this section risks underestimating the allowed range of low-energy couplings.

    Authors: We acknowledge that the sensitivity analysis in §5 is performed inside the chosen parameterization and therefore cannot claim to exhaust all high-energy behaviors allowed by the axioms. The fractional subtractions are introduced to vary the assumed asymptotic growth while remaining consistent with the Froissart–Martin and Jin–Martin bounds. In the revision we will add a short numerical test that enlarges the basis in the high-energy region and recomputes a subset of the coupling bounds, thereby providing concrete evidence of the stability (or variation) of the low-energy results under changes in the high-energy parameterization. This will qualify the interpretation of the sensitivity study without altering the central claims of the paper. revision: partial

Circularity Check

0 steps flagged

No significant circularity; parameterization and dispersion relations are independent inputs

full rationale

The paper parameterizes imaginary parts of partial waves as an ansatz for the spectral density, then applies dispersion relations (as an independent integral transform) together with crossing symmetry and unitarity constraints to generate the amplitude and extract bounds on couplings. This is a standard constructive bootstrap procedure in which the output bounds are obtained by optimizing over the allowed parameter space subject to the imposed axioms, rather than reducing tautologically to the input parameterization. The automatic satisfaction of Froissart-Martin bounds follows directly from the choice of dispersion relations and is presented as a methodological feature, not a hidden fit. No load-bearing self-citations, self-definitional equations, or renaming of known results are indicated in the abstract or description; the derivation remains self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Insufficient information in the provided abstract to enumerate specific free parameters or axioms; the method relies on standard dispersion relations and unitarity but the precise parameterization choices are not detailed.

pith-pipeline@v0.9.0 · 5645 in / 1096 out tokens · 30157 ms · 2026-05-21T23:56:27.346754+00:00 · methodology

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Forward citations

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Reference graph

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