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Infrared-Finite Positivity Bounds in 4D Gravity

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Ziyu Dong

High Energy Physics - Theory · Submitted 13 Oct 2026 · 84 references

Pith's verdict: The half-space approach is a genuinely new idea for IR-finite positivity bounds in gravity, but the central dispersion relation rests on an unproven analyticity assumption that the paper does not resolve; the bound is conditional on that gap.

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arXiv ID2610.19149 (version 1)
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CitationZiyu Dong. "Infrared-Finite Positivity Bounds in 4D Gravity." arXiv:2610.19149 (2026). https://pith.science/paper/CKZDEGFM
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The paper

The paper claims

This paper claims that an infrared-finite positivity bound on local four-scalar Wilson coefficients in 4D gravity can be obtained by formulating S-matrix scattering in a flat half-space with a Dirichlet boundary, where boundary…

The reading

The paper addresses a long-standing obstruction: in four-dimensional quantum gravity, dispersive positivity bounds on effective field theory (EFT) coefficients are typically infrared divergent because long-range graviton exchange introduces logarithmic sensitivity to an infrared cutoff. The authors propose a geometric solution. They formulate S-matrix scattering in an exactly flat half-space (a Minkowski region with a planar Dirichlet wall at $y=0$), where the wall reflects long-range fields. Using localized Gaussian wave packets whose centers are held at a finite distance from the wall, the gravitational exchange becomes infrared finite: the direct and reflected graviton contributions cancel the logarithmic divergence and leave a finite piece proportional to Newton's constant $G_N$. The authors construct a positive semidefinite smearing functional that acts on the absorptive part of the half-space amplitude, prove that it satisfies unitarity and crossing, and test its positivity numerically for momentum-conserving and non-conserving absorptive kernels. They then evaluate the corresponding dispersion relation and obtain an explicit bound on the dimensionless coefficient $g_2$ of the four-scalar contact term $g_2 (s^2+t^2+u^2)/\Lambda_{\rm EFT}^4$. The bound includes a finite gravitational allowance of order $G_N\Lambda_{\rm EFT}^2$ that is independent of the EFT scale at fixed dimensionless external-state parameters. The framework does not require specifying a UV completion, imposing a spectral ansatz, or expanding the spectrum perturbatively; it only assumes standard analyticity, crossing, and unitarity for the half-space S-matrix, together with the positivity of the smearing functional in both crossed channels.

What carries the argument

The central object is the half-space S-matrix defined on a flat Minkowski half-space $y>0$ with a Dirichlet boundary at $y=0$, realised as the $R\to 0$ limit of a flat region joined to an AdS$_4$ exterior. The wall preserves ISO(1,2) along the $(t,x,z)$ directions and breaks translations in $y$. Scattering states are constructed by projecting flat-space plane waves onto the odd eigenfunctions $\sin(k_y y)$. The half-space Green function (4.15) contains a direct and an image term; at zero wall-parallel momentum transfer it reduces to $\min(y,y')$, which is finite. The paper constructs positive semidefinite smearing functionals by taking an equatorial restriction of the flat-space partial-wave kernel (5.23), applying normal-momentum shaking via a congruence (5.9), and then projecting each leg with the half-space projector $\Pi_h$ (5.6). The resulting functional (5.40) integrates over the wall-parallel transfer $q_\parallel$ with a profile $\phi(-q_\parallel^2)=(1-q_\parallel/q_{\rm max})^3$ and over the four normal momenta $k_i$ with a Gaussian weight $W(k_i)$ (5.11). The dispersion relation uses twice-subtracted sum rules in the variable $s_h = s_\parallel - k_s^2$.

Extended reading notes

Core claim

The paper discovers that by scattering correlated wave packets in a flat half-space with a Dirichlet boundary, the infrared divergence in the gravitational contribution to dispersive positivity bounds cancels identically. The cancellation follows from the image method: the massless Green function in the half-space is $G_h(x,x') = (1/4\pi^2)[1/\left(-(x-x')^2+i0\right) - 1/\left(-(x-R_y x')^2+i0\right)]$, where the second term is the reflected image. For wave packets localized at finite distance from the wall, the combined term is finite. The leading gravitational correction to the bound then equals $G_N$ times a finite numerical factor that depends only on the dimensionless ratios of the packet parameters (the wall-parallel transfer scale $q_{\rm max}$, the normal-momentum width $\sigma$, and the distance $y_0$ from the wall). Crucially, the coefficient of this gravitational correction is independent of the EFT scale $\Lambda_{\rm EFT}$ and requires neither an arbitrary soft resolution scale nor a model-dependent UV scale. The paper demonstrates through numerical tests that the same smearing functional gives positive responses for both $s$-channel and $u$-channel absorptive spectra, for a wide range of masses, angular momenta, and normal-profile shapes. This is the first derivation of an infrared-finite positivity bound in 4D gravity that includes a finite gravitational correction from the full UV absorptive spectrum without any perturbative or spectral-shape assumptions.

Load-bearing premise

The half-space S-matrix, defined through the Dirichlet boundary induced by a curved exterior in the $R\to0$ limit, satisfies the same analyticity, crossing, and unitarity properties required for a dispersion relation, and the specific smearing functional preserves positivity in both the $s$ and $u$ channels for general absorptive spectra (the paper provides numerical support for a subset of parameters in Section 5.4, but a fully rigorous proof covering all normal-profile shapes and all spectral energies is not given).

Editorial extensions

If this is right

  • If the bound (5.71) is correct, it shows that gravitational corrections to EFT positivity bounds are finite and of order $G_N \Lambda_{\rm EFT}^2$, providing a concrete test for whether a given UV completion can produce a low-energy four-scalar contact term with a given $g_2$.
  • The construction suggests that other long-range forces (e.g., electromagnetism) could be regulated by the same half-space method, yielding infrared-finite positivity bounds for photon EFT coefficients.
  • The explicit string example in Section 6 demonstrates that the same geometric boundary also regulates the collective contribution of string resonances to the absorptive part, connecting infrared control to the UV spectrum of quantum gravity.
  • The framework can be generalised to spinning external states using the canonical boosts and helicity conventions of Section 4.4, opening the way to positivity bounds on higher-spin EFT operators (e.g., graviton four-point interactions).
  • The finite Dirichlet–Neumann strip (Section 4.2) provides a geometric regulator for soft-graviton loops, controlling residual packet-spreading effects; this can be used to systematically include gravitational loop corrections in the bound.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the half-space positivity functional works for general absorptive kernels as the numerical tests suggest, it would imply that the infinite-dimensional space of half-space spectral data (parametrized by normal profiles $f(k_1,k_2)$) still yields positivity in both crossed channels, which is a nontrivial strengthening of the usual partial-wave positivity that only sees conserved angular momentum.
  • The bound's independence of the EFT scale at fixed dimensionless parameters suggests that the gravitational correction is a genuine universal contribution that cannot be absorbed into renormalisation of the contact terms; it therefore provides a sharp criterion for whether a low-energy effective theory can be UV-completed into a consistent theory of quantum gravity.
  • The eikonal analysis of Section 3 implies that any IR regulation scheme that removes the constant zero mode (e.g., compactification on a sphere with zero-mode subtraction) fails to give a positive smearing kernel in the eikonal limit, because the zero-average Green function takes negative values; the half-space geometry is singled out as the simplest geometry that removes the zero mode while keepi
  • The numerical results at small $\sigma/q_{\rm max}$ show that the finite-width responses approach the strictly positive flat-space limits (5.46), implying that the bound is stable against perturbative corrections in $\sigma^2/q_{\rm max}^2$ and that the small-width hierarchy (which is needed for IR finiteness) does not endanger positivity.
Original abstract

We formulate an S-matrix bootstrap in an exactly flat half-space, where boundary reflection provides geometric infrared control for four-dimensional gravity. Under the channel positivity conditions, we obtain an infrared-finite positivity bound on local four-scalar Wilson coefficients that includes a finite gravitational correction of order $G_N$ and is derived from the full UV absorptive spectrum without specifying a UV theory, imposing a spectral ansatz, requiring weak coupling or expanding the spectrum perturbatively. At fixed dimensionless external state parameters, the coefficient of the leading gravitational correction is independent of the EFT scale and requires neither an arbitrary soft resolution scale nor a model-dependent UV scale. Gaussian wave-packet scattering localizes interactions within the flat region and directly probes bulk Wilson coefficients while exponentially suppressing sensitivity to the boundary. Long-range gravitons remain sensitive to the boundary and give a finite contribution. The smearing functional passes numerical positivity tests with general half-space absorptive kernels in both crossed channels. An explicit string example shows how boundary reflection suppresses the collective contribution of string resonances to the absorptive part of the amplitude, connecting infrared control of low-energy scattering to the UV spectrum of quantum gravity in the same dispersion relation. The framework opens a route to infrared-finite constraints on effective interactions across particle physics and gravity.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-15: 1 request, 6,306 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.28 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.56 · compares with prior results 0.18 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We formulate an S-matrix bootstrap in an exactly flat half-space, where boundary reflection provides geometric infrared control for four-dimensional gravity. Under the channel positivity conditions, we obtain an infrared-finite positivity bound on local four-scalar Wilson coefficients that includes a finite gravitational correction of order $G_N$ and is derived from the full UV absorptive spectrum without specifying a UV theory, imposing a spectral ansatz, requiring weak coupling or expanding the spectrum perturbatively. At fixed dimensionless external state parameters, the coefficient of the leading gravitational correction is independent of the EFT scale and requires neither an arbitrary soft resolution scale nor a model-dependent UV scale. Gaussian wave-packet scattering localizes interactions within the flat region and directly probes bulk Wilson coefficients while exponentially suppressing sensitivity to the boundary. Long-range gravitons remain sensitive to the boundary and give a finite contribution. The smearing functional passes numerical positivity tests with general half-space absorptive kernels in both crossed channels. An explicit string example shows how boundary reflection suppresses the collective contribution of string resonances to the absorptive part of the amplitude, connecting infrared control of low-energy scattering to the UV spectrum of quantum gravity in the same dispersion relation. The framework opens a route to infrared-finite constraints on effective interactions across particle physics and gravity.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We formulate an S-matrix bootstrap in an exactly flat half-space, where boundary reflection provides geometric infrared control for four-dimensional gravity. contribution 0.64 0.92 0.12 asserted 1.00
s1 Under the channel positivity conditions, we obtain an infrared-finite positivity bound on local four-scalar Wilson coefficients that includes a finite gravitational correction of order $G_N$ and is derived from the full UV absorptive spectrum without specifying a UV theory, imposing a spectral ansatz, requiring weak coupling or expanding the spectrum perturbatively. result 0.94 0.93 0.75 asserted 1.00
s2 At fixed dimensionless external state parameters, the coefficient of the leading gravitational correction is independent of the EFT scale and requires neither an arbitrary soft resolution scale nor a model-dependent UV scale. result 0.96 0.86 0.21 asserted 1.00
s3 Gaussian wave-packet scattering localizes interactions within the flat region and directly probes bulk Wilson coefficients while exponentially suppressing sensitivity to the boundary. method 0.95 0.72 0.06 asserted 1.00
s4 Long-range gravitons remain sensitive to the boundary and give a finite contribution. result 0.88 0.69 0.06 asserted 1.00
s5 The smearing functional passes numerical positivity tests with general half-space absorptive kernels in both crossed channels. result 0.87 0.88 0.07 asserted 1.00
s6 An explicit string example shows how boundary reflection suppresses the collective contribution of string resonances to the absorptive part of the amplitude, connecting infrared control of low-energy scattering to the UV spectrum of quantum gravity in the same dispersion relation. result 0.43 0.82 0.14 asserted 1.00
s7 The framework opens a route to infrared-finite constraints on effective interactions across particle physics and gravity. interpretation 0.65 0.44 0.29 hedged possibility 0.73

What kind of paper the classifier reports this is: theoretical result 0.67 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Schematic spatial view of two counterpropagating beams. A disk represents the general transverse space Σ2. Legs 1 → 3 and 2 → 4 follow the two separated beams at leading eikonal order. The arrows show physical velocities. The s–u crossing exchanges labels 1 and 3 on the upper beam. In the all-incoming convention, the initial and final helicity differences are hi,s = h1 − h2, hf,s = h4 − h3, hi,u = h3 − h2, hf,u = h4 − h1, hi,s − hf,s = hi,u − hf,u = h1 + h3 − h2 − h4. (3.4) The physical u channel therefore gives the same Bessel transform with b ∼ 2J/√ u and the crossed external heli
Figure 2. The finite low-energy contour C0 and the large contour C∞, both oriented counterclockwise. The physical cuts continue inside C0 (thin orange). For GN s0 ≪ 1, their leading eikonal contribution comes from outside (thick orange). The negative inner arc and positive cut integrals require a nonzero negative large-contour contribution. An IR regulator that retains the zero mode can supply a sufficiently large positive constant to make the regulated Green function nonnegative throughout the transverse space. Removing this regulator dependence from exclusive scattering requires a…
Figure 3. A flat core with a hyperbolic exterior, shown in the auxiliary embedding R 1,2 . The left panel replaces the hyperbolic cap by a flat disk, retaining a sharp circular join. The right panel smooths a finite annulus while preserving a strictly flat interior and the hyperbolic end. Light blue, green and gray indicate the flat, transition and hyperbolic regions. The right panel retains a translucent copy of the sharp join beneath the concentric, rotationally symmetric transition annulus. The coordinate τ is an embedding coordinate, not physical time. The outer rim only truncates the…
Figure 4. (a) Wave packets spread during propagation towards t → ±∞. Unlike
Figure 5. Schematic illustration of normal shaking of the incoming and outgoing relative momenta. The black reference directions lie on the fixed zx great circle. Blue arrows show independent normal shaking, and the dashed blue arc joins the displaced directions. The common rotation about y remains free. Shaking of the incoming and outgoing center-of-mass momenta is not shown. The one-side on-shell map (5.9) shakes the yˆ components of the incoming pair while keeping Ps∥ and the oriented plane p sd 1 ∧ p sd 2 fixed. The outgoing pair is treated in the same way. For spinning states, the canoni
Figure 6. Relative changes of the two physical-channel responses under one smearing prescription. Columns show J = 0, 11, 50, with the s channel above and the u channel below. Symmetric￾logarithmic tick labels give the relative changes. 1.21 2 5 10 20 50 100 0.00 0.08 0.16 0.24 0.32 M 6 s/u ̂ λs/u/q 2max s channel, σ/qmax = 10 −2 1.21 2 5 10 20 50 100 u channel, σ/qmax = 10 −2 M 2 s/u /q 2 max J = 0 J = 1 J = 2 J = 3 J = 4 J = 5 J = 6 J = 7 J = 8 J = 9 J = 10 J = 50
Figure 7. Mass scans testing the positivity of the s- and u-channel responses λbs and λbu at fixed angular momenta
Figure 8. Spin scans testing the positivity of the s- and u-channel responses λbs and λbu at fixed resonance masses. A general half-space absorptive operator need only respect ISO(1, 2), rather than ISO(1, 3), and therefore also depends on the four external normal momenta. A PSD spectral decomposition expresses this dependence through two-particle profiles f(k1, k2), using the same profile on the outgoing side with complex conjugation. At fixed conserved timelike momentum Ps∥ in the wall-parallel (t, z, x) space, the relative directions transform under – 70 –
Figure 9. Angular-momentum scans at fixed wall-parallel spectral energies for the same normal profile. Both panels use logarithmic vertical axes. To compare with the large-mass behavior in
Figure 10. Wall-parallel spectral-energy scans at fixed angular momenta for the same normal profile. The gray dotted line is the positive large-energy leading limit. We additionally vary the shape of the normal profile f(k1, k2). For the Gaussian products in (5.50), we replace the centers ±σ/2 by ±kc and sample kc σ ∈ {0.4, 0.6, 0.8}, τ σ ∈ {0.025, 0.05, 0.1}, kc + 10τ < 10σ. (5.53) The last condition keeps the ten-width neighborhoods of both Gaussian centers inside the normal integration band [−10σ, 10σ], with the profile suppressed by e −50 at ten widths from each center. We also replace…
Figure 11. Real part of the interaction Green function from (6.5) at y1 = y2 = y0. The b/y0 and ℓdiff/y0 axes use logarithmic spacing. For a shock with fixed source momentum and a narrow probe packet of energy E, so – 84 –
Figure 12. Leading imaginary part of the interaction Green function from (6.5). The surface uses the same coordinates and projection as

Figures reproduced from arXiv:2610.19149 by the authors.

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