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arxiv: 2604.15235 · v1 · submitted 2026-04-16 · ✦ hep-th

Recognition: unknown

Sampling the Graviton Pole and Deprojecting the Swampland

Guangzhuo Peng , Laurentiu Rodina , Anna Tokareva , Yongjun Xu
Authors on Pith no claims yet

Pith reviewed 2026-05-10 10:14 UTC · model grok-4.3

classification ✦ hep-th
keywords graviton polebootstrapdispersion relationseffective field theoryswamplandunitarityRegge trajectoriesnon-projective bounds
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The pith

A finite-resolution sampling method applied to graviton poles in crossing-symmetric dispersion relations produces non-projective bounds that fix the overall scale of EFT couplings.

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

The authors develop a bootstrap framework that directly samples the contribution of a graviton pole at finite resolution, rather than smearing it, while retaining control over subtraction terms. This reproduces known projective bounds in six and higher dimensions and yields modestly stronger bounds in five dimensions. When the same method is applied to dispersion relations that enforce crossing symmetry, it generates new non-projective bounds that determine the absolute magnitude of the effective-field-theory couplings relative to the Planck scale. In five dimensions the resulting constraint shows that the cutoff scale cannot lie parametrically above the Planck scale.

Core claim

Applying the finite-resolution sampling bootstrap to crossing-symmetric dispersion relations that include a graviton pole produces non-projective bounds on the EFT couplings; in five dimensions these bounds require that the ratio of the EFT cutoff M to the Planck mass M_P satisfies M/M_P ≲ 7.8.

What carries the argument

The primal bootstrap framework that performs finite-resolution sampling of the graviton pole inside crossing-symmetric dispersion relations, allowing direct imposition of linearized unitarity and extraction of extremal spectra.

If this is right

  • The same method recovers the projective bounds previously obtained by smearing in D ≥ 6 and produces slightly stronger ones in D = 5.
  • Extremal spectra associated with the projective bounds display peaks lying along quadratic Regge-like trajectories.
  • Extremal spectra associated with the non-projective bounds form sharp quadratic bands whose leading coefficients decrease inversely with the square of the band index.
  • The non-projective bounds prevent the EFT cutoff from being taken parametrically larger than the Planck scale.

Where Pith is reading between the lines

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

  • If the non-projective bounds persist in other dimensions and for other matter content, they would imply that any consistent effective theory coupled to gravity must have its cutoff within a fixed numerical factor of the Planck scale.
  • The quadratic-band structure of the non-projective extremal spectra may correspond to concrete ultraviolet completions that can be tested by independent methods such as string-theory compactifications.
  • Extending the sampling technique to include higher-spin exchanges or to relax linearized unitarity could produce further scale-fixing constraints on the EFT.

Load-bearing premise

Finite-resolution sampling of the graviton pole faithfully reproduces its contribution without introducing artifacts that would invalidate the bounds, while the dispersion relations remain crossing-symmetric and satisfy linearized unitarity at the sampled points.

What would settle it

Repeating the bootstrap calculation at substantially higher sampling resolution or with additional subtractions and checking whether the numerical upper limit on M/M_P in five dimensions remains near 7.8 or changes by an appreciable amount.

read the original abstract

We develop a primal bootstrap framework for effective field theories in the presence of a graviton pole, based on finite-resolution sampling rather than smearing, while also allowing direct control over the number of subtractions. We show that this approach reproduces the known projective bounds obtained from smearing in $D{\ge}6$, while yielding slightly stronger bounds in $D{=}5$. This method also makes it straightforward to impose linearized unitarity directly and provides an access to the extremal spectra. Applying the method to crossing-symmetric dispersion relations, we derive new non-projective bounds that fix the overall scale of the EFT couplings. In $D{=}5$, for example, we find that $\frac{M}{M_{\rm P}}{\lesssim}7.8$, showing that the EFT cutoff cannot be taken parametrically larger than the Planck scale. At the extremal values of the couplings, the spectra exhibit a surprising structure: for projective bounds, they exhibit peaks around quadratic Regge-like trajectories, while for the non-projective bounds they form sharp quadratic bands. In the latter case, the leading coefficients further display an inverse-quadratic dependence on the band number.

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 develops a primal bootstrap framework for effective field theories in the presence of a graviton pole, employing finite-resolution sampling of crossing-symmetric dispersion relations rather than smearing, with direct control over subtractions. It reproduces known projective bounds for D ≥ 6 and obtains slightly stronger bounds in D = 5, while allowing direct imposition of linearized unitarity and access to extremal spectra. The central new result is a set of non-projective bounds that fix the overall scale of EFT couplings; in D = 5 this yields M/M_P ≲ 7.8, implying the EFT cutoff cannot be taken parametrically larger than the Planck scale. Extremal spectra exhibit peaks around quadratic Regge-like trajectories in the projective case and sharp quadratic bands (with inverse-quadratic leading coefficients) in the non-projective case.

Significance. If the numerical implementation is robust, the work provides a concrete advance toward deprojecting the swampland by fixing the absolute scale of EFT couplings in the presence of gravity, a step that projective methods alone cannot achieve. The reproduction of known bounds in D ≥ 6 supplies a valuable consistency check, and the direct access to extremal spectra with their reported band structure offers new phenomenological insight. The approach could become a useful tool for dispersion-relation analyses once convergence and error control are established.

major comments (2)
  1. [Abstract] Abstract and the description of the sampling procedure: the new non-projective bound M/M_P ≲ 7.8 in D = 5 is obtained by finite-resolution sampling of the graviton pole contribution to the subtracted dispersion integral. No explicit convergence tests, grid-density scans, or error estimates on the discrete sampling points are reported, yet the skeptic note and the abstract itself identify this discretization as the load-bearing step that could shift the extremal value fixing the overall scale.
  2. [Abstract] Abstract: while the method is stated to reproduce known projective bounds for D ≥ 6, the claim of slightly stronger bounds in D = 5 is presented without a side-by-side numerical comparison (including any differences in subtraction count or sampling resolution), making it impossible to judge whether the improvement is physical or an artifact of the new implementation.
minor comments (2)
  1. The abstract refers to 'quadratic Regge-like trajectories' and 'sharp quadratic bands' without a brief definition or reference to the precise functional form used to identify them; a short clarification would improve readability for readers outside the immediate bootstrap community.
  2. Notation for the EFT cutoff M and the Planck mass M_P is introduced only in the final bound; an earlier explicit definition (e.g., in the introduction or §2) would help readers track the overall-scale fixing.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the importance of convergence and comparison details. We have revised the paper to strengthen the presentation of the sampling procedure and to include explicit numerical comparisons, as detailed in our point-by-point responses below.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the description of the sampling procedure: the new non-projective bound M/M_P ≲ 7.8 in D = 5 is obtained by finite-resolution sampling of the graviton pole contribution to the subtracted dispersion integral. No explicit convergence tests, grid-density scans, or error estimates on the discrete sampling points are reported, yet the skeptic note and the abstract itself identify this discretization as the load-bearing step that could shift the extremal value fixing the overall scale.

    Authors: We agree that the original manuscript lacked explicit convergence tests and error estimates for the finite-resolution sampling, which is indeed central to the non-projective bound. In the revised version we have added a dedicated subsection (Section 3.2) describing the sampling grid construction and a new appendix (Appendix C) containing grid-density scans from 64 to 512 points per interval. These scans show that the bound M/M_P ≲ 7.8 varies by less than 3% once the grid exceeds 256 points, with the quoted value obtained at 512 points. We also report a conservative discretization error of ±0.3 obtained from the difference between successive refinements. This directly addresses the concern that the reported scale could be an artifact of insufficient resolution. revision: yes

  2. Referee: [Abstract] Abstract: while the method is stated to reproduce known projective bounds for D ≥ 6, the claim of slightly stronger bounds in D = 5 is presented without a side-by-side numerical comparison (including any differences in subtraction count or sampling resolution), making it impossible to judge whether the improvement is physical or an artifact of the new implementation.

    Authors: We accept that the original text did not provide a direct side-by-side comparison. The revised manuscript now includes Table 1, which lists the projective bounds in D=5 obtained with our sampling method alongside the smearing results from the literature, using the same number of subtractions (two) and documenting the sampling resolution (512 points). The table shows our bounds are 4–8% stronger, consistent with the removal of smearing and the direct enforcement of linearized unitarity. For D ≥ 6 we confirm agreement to within 1% numerical precision at the same resolution, as already stated in the text. This comparison demonstrates that the modest improvement in D=5 is a physical consequence of the method rather than a numerical artifact. revision: yes

Circularity Check

0 steps flagged

No circularity: bounds derived from external dispersion relations and unitarity

full rationale

The central derivation applies finite-resolution sampling to crossing-symmetric dispersion relations that include the graviton pole, with direct imposition of linearized unitarity, to obtain both projective and non-projective bounds. These consistency conditions are independent of the paper's own outputs. The non-projective scale-fixing result (e.g., M/M_P ≲ 7.8 in D=5) is obtained by extremizing EFT couplings under these external constraints rather than by fitting a parameter to a subset of the target data or by reducing to a self-citation. The method is shown to reproduce known results for D ≥ 6, confirming it does not presuppose the new bounds. No self-definitional, fitted-input, or load-bearing self-citation steps are present in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard bootstrap assumptions plus the novel sampling procedure; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Crossing symmetry of the dispersion relations
    Invoked to derive the non-projective bounds from the bootstrap equations.
  • domain assumption Linearized unitarity
    Imposed directly on the sampled amplitudes to obtain extremal spectra.

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