Cross-Section Bootstrap: Unveiling the Froissart Amplitude

Alessandro Georgoudis; Andrea L. Guerrieri; Miguel Correia

REVIEW 4 major objections 6 minor 6 cited by

Cross-Section Bootstrap: Unveiling the Froissart Amplitude

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A finite-energy bound on the total cross-section follows from unitarity and crossing symmetry alone, and lands within a factor of ten of proton-proton data.

desk verdict The analytic bound is a real result; the exclusion plot is an extrapolation without a certificate, which is the one thing standing between this paper and a clean acceptance. read the letter →

arxiv 2506.04313 v1 pith:EHTCPYE3 submitted 2025-06-04 hep-th hep-phnucl-th

Miguel Correia , Alessandro Georgoudis , Andrea L. Guerrieri This is my paper · ORCID

classification hep-thhep-phnucl-th

keywords S-matrixbootstrapFroissartboundintegratedtotalcross-sectionReggetrajectoriesPomeroneikonalapproximationscalarscatteringfiniteenergybounds

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to show that the total scattering cross-section, integrated over energies up to a finite s, is bounded from above in any gapped scalar theory by a number that depends only on a single low-energy coefficient, with no assumption about the spin-2 scattering length. If the bound and its numerical evaluation are correct, the tightly constrained quantity sits within a factor of about ten of measured proton-proton and proton-antiproton cross-sections, making Froissart-type saturation quantitatively testable rather than asymptotic. The paper also identifies the amplitude that saturates the bound at high energies — the 'Froissart amplitude' — and claims it is a universal, strongly-coupled object with a rising cross-section, a shrinking diffractive cone, Regge trajectories including a Pomeron-like one, and an impact-parameter profile shaped like a growing ring rather than a disk. Such an object, if real, would give a first-principles handle on the soft, non-perturbative regime of QCD.

What carries the argument

The machinery is a two-step inequality. First, after splitting the partial-wave sum at a cutoff $L$, unitarity fixes the low-spin contribution by setting $\mathrm{Re}\, S_\ell = -1$, while the high-spin tail is bounded by relating it, through Eq. (11), to the positive dispersive coefficient $c_{2k}(t_0) = \frac{2^{1-k}}{\pi}\int_4^\infty ds \frac{\mathrm{Im}\,T(s,t_0)}{(s-2+t_0/2)^{2k+1}}$, defined by derivatives of the amplitude at the crossing-symmetric point. Optimizing over $L$ and $t_0$ yields Eq. (12), whose large-$s$ limit is the d-dimensional Froissart bound. The numerical part uses a multi-foliation ansatz (14) with unitarity imposed via positive semi-definite matrices and improved positivity constraints, solved with SDPB to obtain $\max c_2(t_0)$ and the exclusion bound.

What would settle it

Compute a rigorous dual upper bound on $c_2(t_0)$ at $t_0 = 4/3$ (or any fixed $t_0$) using the method of [46]; if that dual bound falls strictly below the extrapolated value plotted in Fig. 2, the exclusion bound in Fig. 1 would have to be revised, since no consistent amplitude could reach the extrapolated $c_2$. Alternatively, construct or numerically find a bootstrap-allowed amplitude whose integrated cross-section at some finite $s$ exceeds the extrapolated bound of Fig. 3.

Watch

Extended reading notes

Core claim

The central discovery is an inequality, Eq. (12), that bounds the integrated total cross-section for identical scalar scattering at any finite energy s in any spacetime dimension d, using only unitarity, crossing symmetry, and polynomial boundedness of the amplitude. The bound is expressed in terms of the dispersive coefficient $c_{2k}(t_0)$, which is positive by a dispersion relation and can be bounded from above numerically by the S-matrix bootstrap. Removing the finiteness assumption on the D-wave scattering length made by Yndurain, the bound is universal for gapped scalars; in the asymptotic limit it reduces to the d-dimensional Froissart bound with the correct $\log^{d-2} s$ growth. Numerically, the authors maximize the integrated cross-section directly and conjecture that the extremal amplitude coincides in the high-energy limit with the universal amplitude that maximizes all low-energy coefficients $c_{2k}(t_0)$; this 'Froissart amplitude' is then exhibited in detail, with its Regge spectrum, diffractive minima, and eikonal ring profile.

Load-bearing premise

The plotted 'optimal exclusion bound' in Fig. 1 uses a value of $\max c_2(t_0)$ obtained by extrapolating truncated numerical bootstrap solutions (up to $N=20$, with 814 parameters) to $N\to\infty$ with a power-law fit, and that extrapolation has no rigorous error certificate; if the true infinite-$N$ maximum is lower, the practical bound weakens, though equation (12) remains a valid bound for whatever $c_2$ one inserts.

Editorial extensions

If this is right

Any consistent gapped scalar theory in $d$ dimensions has an integrated total cross-section bounded at finite $s$ by Eq. (12); the bound is strict and requires no assumption on the spin-2 scattering length.
In four dimensions, with $m \approx 1\,\mathrm{GeV}$, the optimal exclusion bound lies within a factor of about ten of measured $pp$ and $\bar p p$ integrated cross-sections, making the bound phenomenologically relevant rather than purely asymptotic.
The amplitude that maximizes the integrated cross-section at high energies is conjectured to be the universal Froissart amplitude; it exhibits a rising total cross-section, a shrinking diffractive cone with multiple minima, and Regge trajectories, one with intercept near 1.15 resembling the phenomenological Pomeron.
The eikonal profile of the Froissart amplitude is a slowly expanding annulus, so Froissart growth is realised by a growing ring rather than a growing disk; a simple 'white ring' model reproduces the diffractive pattern and predicts a Regge cut with branch points at $\ell = 1 \pm r_0 \sqrt{t}$.
The Froissart amplitude carries a spin-2 bound state at threshold, which diverges in the $t_0 \to 4$ limit, explaining why the finite-energy optimal $t_0$ cannot be taken at threshold.

Reading between the lines

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

I infer that if the white ring picture is generic, the asymptotic elastic-to-total ratio need not approach the black disk value $1/2$; inelastic scattering would be increasingly peripheral, offering a bootstrap-derived analogue of the 'hollowness' effect discussed phenomenologically.
The conjectured identity between the Froissart amplitude and the $\min c_0/\max c_2$ cusp is testable within the bootstrap: any other objective that drives amplitudes to the same cusp (e.g., maximizing higher $c_{2k}$) should produce the same Regge spectrum, a check the authors have not performed.
I infer that extending the same bound to fermionic (proton-like) amplitudes would likely modify the intercept structure and potentially produce an Odderon trajectory, since Bose symmetry is what currently removes odd spins.
The bound's dependence on a single low-energy coefficient suggests a concrete test: measuring the Wilson coefficient $c_2(t_0)$ in a candidate theory fixes the maximal integrated cross-section, so a computed cross-section above the bound would falsify the S-matrix assumptions.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. First, the analytic result is real: Eq. (12) gives a strict finite-energy upper bound on the integrated total cross-section for gapped scalars in any d, using crossing, unitarity, and a single low-energy coefficient c2(t0). That removes Yndurain's spin-2 scattering-length assumption and recovers Froissart asymptotically. Second, the striking exclusion plot in Fig. 1 is not a proven universal bound, because it uses max c2(t0) estimated by power-law extrapolation of primal bootstrap solutions up to N=20. Primal solutions give lower bounds on the maximum, and the RHS of (12) grows with c2, so the plot is an estimate of where the true bound lies, not a certificate.

What the paper does well: the derivation in Section II is clean, and the appendix on the d-dimensional Froissart bound is a useful counterpart to the existing literature. The numerical work is substantial: 814 parameters, careful foliation, improved positivity, and a reproducible setup with SDPB. The authors are honest about convergence issues in the Chew–Frautschi plot and the slow approach to the asymptotic limit. The "white ring" eikonal model is a nice heuristic, and the connection to Kupsch and Oehme is well placed. The paper also generalizes the bound to arbitrary test functions in Appendix I.

The soft spots: (1) the main quantitative claim—the factor-of-ten comparison with pp data—depends on the extrapolation of max c2(t0). The paper cites 5% agreement with a dual bound at t0=4/3, but no dual certificate for the full curve is provided. This should be fixed or clearly labeled as an estimate. (2) The Froissart amplitude's Regge spectrum, especially the singular trajectories and the Pomeron intercept, is extracted from a truncated ansatz and is not fully converged; the authors say so, but the conjectural status should be prominent in the abstract. (3) The spin-2 threshold bound state is inferred from indirect evidence rather than constructed; that is fine as a conjecture, but it is not a theorem.

Overall: the analytic half of the paper is a solid contribution that deserves a serious referee. The numerical half is plausible and well executed but not yet rigorous. I would referee it, asking for dual bounds or softened claims about the optimal exclusion curve. The paper is worth discussing and citing.

Referee Report

4 major / 6 minor

Summary. The paper derives an upper bound on the integrated 2->2 total cross-section for gapped identical scalar theories in d dimensions, Eq. (12), valid at finite s for any t0 in (0,4), even L, and integer k, once the low-energy dispersive coefficient c_{2k}(t0) is known. Setting c_{2k}(t0) to its maximum over consistent amplitudes yields a theory-independent exclusion bound. In d=4 the asymptotic limit reduces to the Froissart bound. The authors then compute max c_2(t0) numerically with the S-matrix bootstrap using a truncated multi-foliation ansatz (14) with up to N=20 and 814 parameters, extrapolate to N->infinity, and use the result to plot an 'optimal exclusion bound' compared with pp and pbarp data, claiming agreement within a factor of about ten for m about 1 GeV. They identify the extremal amplitude at the (min c0, max c2) cusp as the conjectured 'Froissart amplitude' and study its total cross-section, Regge trajectories, diffractive minima, and eikonal profile, proposing a 'white ring' annulus picture.

Significance. Eq. (12) is a clean and potentially important finite-energy generalization of the Froissart bound: it converts the asymptotic log^2 growth into a finite-s inequality controlled by a single positive low-energy coefficient, and it is derived transparently from unitarity, crossing, and dispersion relations. The numerical exploration is state-of-the-art and the paper is rich in phenomenology. If the extrapolated max c_2(t0) were replaced by certified dual bounds, the exclusion plot would be a rigorous universal statement; as it stands, the analytic derivation is the lasting core, while the 'factor of ten' and the detailed properties of the Froissart amplitude are numerical and conditional evidence. The paper should be credited for being explicit about many limitations (e.g., footnotes on numerical convergence), but those limitations are in tension with some abstract-level claims.

major comments (4)

[Section III.A, Figs. 1-2] The input max c_2(t0) used in the exclusion bound is obtained by extrapolating finite-N primal bootstrap solutions. A primal feasible solution provides a lower bound on the true maximum, and since the right-hand side of Eq. (12) is increasing in c_2(t0), the red excluded region in Fig. 1 can lie below the true bound. The statement in Section III.A that the extrapolation agrees with a dual bound to 5% at t0=4/3 is a useful cross-check, but Fig. 1 is produced by minimizing Eq. (12) over t0, and no certified upper bounds for the full curve max c_2(t0) are given. The 'optimal exclusion bound' is therefore a numerical estimate, not a proven universal bound; the quantitative claim in the abstract and in Fig. 1 should be rephrased or supported by dual certificates.
[Section IV.B, Fig. 7, and abstract] The Regge trajectories, the Pomeron-like trajectory B with intercept alpha0 about 1.1, the daughter trajectories, and the 'singular' trajectories Z_i are obtained by analytic continuation of the truncated N=20 amplitude. The paper itself notes in footnote 18 that the continuation near ell~1 may be affected by systematic numerical errors, and in Fig. 19 that trajectory B is the least stable. Despite this, the abstract states as a result a 'surprisingly rich spectrum of resonances aligning along Regge trajectories, including Pomeron-like and daughter trajectories'. These are numerical indications for a conjectured amplitude, not established properties; the language should be qualified accordingly.
[Section IV.C, Figs. 8-9, Appendix D] The 'white ring' picture is an interpretation based on fitting the phenomenological model (20) to the differential cross-section of the numerical amplitude. The eikonal representation (D1)-(D4) is an approximation valid at large s, and the fitted radii R1 and R2 are not derived from the bootstrap. Therefore the claim in the abstract that 'the eikonal representation reveals that the scattering is localized within an annular region' is model-dependent evidence, not a first-principles consequence. This should be made explicit in the main text, not only in the figure captions.
[Section III.A and Appendix G] The existence of a spin-2 bound state at threshold is inferred from the growth of c_2(t0) as t0 approaches 4 and from the ell=2 phase-shift behavior at increasing N (Fig. 17), but the ansatz (14) contains no such singularity at s=4. The claim in Appendix G that 'we can confidently claim that the Froissart amplitude features a spin two threshold state' goes beyond what an extrapolation of finite-N data can establish. Since this divergence is also used to explain why the finite-s optimization avoids t0 approaching 4, the claim should be labelled as a numerical conjecture or supported by an explicit singular ansatz.

minor comments (6)

[Section I, Fig. 1] The m about 1 GeV assumption underlying the comparison with pp data should be stated as an external physical input, not a consequence of the bootstrap; varying m shifts the bound by powers of m.
[Section III.A, Figs. 2-3] In Fig. 2 and Fig. 3, the N-to-infinity extrapolation is shown without specifying the power-law fitting function, the fit range, or the systematic error; please provide these details or a reference.
[Section IV.B, Fig. 7] The caption of Fig. 7 says 'Blue dots denote the particles showing up at real values of the spin arranging into Regge trajectories'; this should be 'at non-integer values of the spin' to be consistent with the text and with Fig. 19.
[Section III.A] The dual bound [80,81] is cited as 'work in progress' in [81]; if the dual calculation is available, include the relevant numbers and error bars in the text so the 5% cross-check can be assessed.
[Section IV.B, Eq. (17)] Equation (17) should define Disc_t T(s,t) and specify the integration contour; as written, the Froissart-Gribov formula requires more explanation for the non-expert reader.
[Section II, Eq. (8)] The footnote 'for some finite arbitrary even L' should say 'positive even L at least 2', since the derivation uses (L-1)^{d-2} and the partial-wave split at ell=L.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the bound's low-energy input c2 is separately maximized, and the predicted cross-section is not used to fit it.

full rationale

The analytic bound (12) is a conditional inequality valid for any value of the low-energy coefficient c2k(t0); the paper then supplies max c2k(t0) by a separate primal S-matrix bootstrap maximization. No equation re-inserts the integrated cross-section, the pp data, or the Froissart amplitude into the computation of c2, so the output observable is not an input by construction. The Froissart amplitude is defined as the maximizer of c2, not of the integrated cross-section, and its rising total cross-section, diffractive minima, and Regge structure are derived properties of that amplitude rather than fitted targets. The power-law extrapolation of max c2(t0) from finite-N solutions is an uncertified numerical estimate, which weakens the rigor of the finite-energy exclusion curve in Fig. 1 but is a correctness concern, not circularity. The only notable self-citation is the 5% agreement with a dual bound at t0=4/3 attributed to [80,81]; that check is a heuristic validation, not the load-bearing source of the bound, and the underlying dual method [46] is published independent work. Consequently no circular step can be exhibited from the text.

Assumptions & free parameters 3 free parameters · 3 assumptions · 1 invented entities

The analytical bound officially has one physical input, max c2, plus standard S-matrix axioms. The practical numerical bound imports extrapolation and ansatz-completeness assumptions. The Froissart-amplitude picture adds fitted ring radii and conjectured spectral features, but these do not enter the central bound.

free parameters (3)

mass scale m for pp comparison = 1 GeV
The authors set m roughly 1 GeV as the relevant interaction range when comparing the scalar bound to pp and pbarp data in Fig. 1; this is a modelling choice, not derived.
white ring radii R1, R2 = R1 approx 2.3, R2 approx 3.5 at s=50 and 4.4 at s=200
Fitted to reproduce the diffractive cone of the Froissart amplitude in Figs. 8-9; used to support the annulus-growth model, not the central bound.
power-law extrapolation parameters for c0 and c2 = not reported numerically
The N to infinity extrapolation in Fig. 16 uses an ad hoc fit a + b x^c; this feeds the plotted max c2 used in the finite-energy exclusion bound.

assumptions (3)

domain assumption Polynomial boundedness |T(s,t)| < s^{2k} for t at most t0
Used to justify the Cauchy contour deformation and the positive dispersion relation for c2k in Eq. (10). The paper appeals to Regge boundedness, which Appendix C says has not been proven from axiomatic QFT.
domain assumption Maximal analyticity and Sommerfeld-Watson continuation of partial waves to complex spin
Used in Section IV.B and Appendix C to identify Regge trajectories; footnote 23 states this assumption has not been proven from axiomatic QFT.
ad hoc to paper The finite-N multi-foliation ansatz (14) converges to the full space of allowed amplitudes
The numerical bounds and the Froissart amplitude rely on truncating at N=20 and then extrapolating; no convergence theorem is provided beyond empirical checks.

invented entities (1)

Spin-2 threshold bound state
purpose: Explains the divergence of max c2(t0) as t0 approaches 4 and the special cusp structure at the boundary of the allowed region
Inferred from the growing threshold feature in the ell=2 phase shift and from the universality of max c2; no direct observable prediction outside the paper is provided.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Cross-Section Bootstrap: Unveiling the Froissart Amplitude." pith.science (2026). https://pith.science/paper/EHTCPYE3

@misc{pith2026250604313,
  author       = {Pith},
  title        = {Pith review of: Cross-Section Bootstrap: Unveiling the Froissart Amplitude},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EHTCPYE3}},
  note         = {Machine review of arXiv:2506.04313}
}

read the original abstract

We derive a universal bound on the integrated total scattering cross-section at \emph{finite} energies, expressed in terms of a single low-energy coefficient constrained by the non-perturbative S-matrix Bootstrap. At high energies, the bound is compared with proton-proton scattering data; at low energies, with numerical bootstrap results obtained by directly maximizing the cross-section. We conjecture that the amplitude saturating the cross-section at high energies lies at a strongly-coupled corner of the allowed space of low-energy parameters. This universal amplitude exhibits a rising total cross-section, a shrinking elastic differential cross-section with multiple diffractive minima, and a surprisingly rich spectrum of resonances aligning along Regge trajectories, including Pomeron-like and daughter trajectories, as well as unusual ``singular" trajectories in the forward limit which appear deeply interconnected with Froissart growth. Remarkably, the eikonal representation reveals that the scattering is localized within an annular region that slowly expands with energy, challenging the traditional ``disk" diffraction picture. Our results open the door to theoretical and phenomenological studies of \emph{soft} high-energy hadronic scattering via the S-matrix Bootstrap.

Figures

Figures reproduced from arXiv: 2506.04313 by the authors.

**Figure 2.** FIG. 2. Max [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** FIG. 4. Profile of 1 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Figures from the paper (14 more)

**Figure 6.** Figure 6: FIG. 6. Profile of the total cross-section of the Frois [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: FIG. 7. Chew-Frautschi plot of the Froissart amplitude. Dots colored in red, orange, and yellow denote even integer [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: FIG. 8. Diffractive cone of the Froissart amplitude. It [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: FIG. 9. Density plot of the eikonal profile of the Frois [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]

**Figure 10.** Figure 10: FIG. 10. Direct pointwise maximization of the total [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]

**Figure 11.** Figure 11: FIG. 11. Complex angular momentum plane. In blue, [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]

**Figure 12.** Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]

**Figure 14.** Figure 14: FIG. 14. Profile of [PITH_FULL_IMAGE:figures/full_fig_p018_14.png]

**Figure 16.** Figure 16: FIG. 16. Behavior of [PITH_FULL_IMAGE:figures/full_fig_p019_16.png]

**Figure 17.** Figure 17: FIG. 17. Phase shift for [PITH_FULL_IMAGE:figures/full_fig_p020_17.png]

**Figure 18.** Figure 18: FIG. 18. Projection of the partial waves via exp( [PITH_FULL_IMAGE:figures/full_fig_p021_18.png]

**Figure 19.** Figure 19: FIG. 19. Real and imaginary parts of the squared mass [PITH_FULL_IMAGE:figures/full_fig_p021_19.png]

**Figure 20.** Figure 20: FIG. 20. Effective growth exponent [PITH_FULL_IMAGE:figures/full_fig_p022_20.png]

**Figure 22.** Figure 22: FIG. 22. Partial waves and phase shifts of the Froissart Amplitude for [PITH_FULL_IMAGE:figures/full_fig_p023_22.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

(Super)$\,$Gravity from Positivity
hep-th 2025-07 conditional novelty 8.0 of 10

Positivity of scattering amplitudes forces weakly coupled EFTs of a massive spin-3/2 particle to include gravity, and for charged states a gauged photon with g=2 and WGC-saturating charge.
Unitary Dual-Resonance S-matrices
hep-th 2026-07 conditional novelty 7.0 of 10

Smearing Veneziano blocks over continuous Regge slopes yields local, analytic, crossing-symmetric S-matrices with full partial-wave unitarity and controllable second-sheet resonances.
Strings from Almost Nothing
hep-th 2025-08 conditional novelty 7.0 of 10

Ultrasoft, minimally structured scattering amplitudes are forced onto the Veneziano and Virasoro-Shapiro amplitudes of string theory.
S-matrix bootstrap bounds on self-interacting dark matter
hep-ph 2026-07 conditional novelty 6.0 of 10

Weakly coupled scalar self-interacting dark matter cannot be heavier than ~0.3 GeV (generic) or ~MeV (derivative-coupled pNGB), much tighter than the 12 GeV unitarity bound.
Factorization of elastic, single, and double diffractive $pp$ scattering
hep-ph 2026-07 conditional novelty 6.0 of 10

SCET with Glauber operators factorizes elastic/single/double diffractive pp scattering, proving non-universality of hadronic functions versus ep while rapidity anomalous dimensions remain universal.
Bootstrapping Gravity with Crossing Symmetric Dispersion Relations
hep-th 2025-06 conditional novelty 5.0 of 10

A crossing-symmetric dispersion-relation bootstrap reproduces known gravitational EFT bounds and gives new tree-level limits on the graviton coupling to a massive spin-4 state.

Reference graph

Works this paper leans on

139 extracted references · 24 canonical work pages · cited by 6 Pith papers

[1]

white ring

The observed shrinking of theppdiffractive cone at a rate faster than logsat √s= 13 TeV 11 as reported by the TOTEM collaboration [8, 9]. This is a feature of our Froissart amplitude and of the analytical “white ring” model with ex- plicit Froissart growth
[2]

A diffractive-like pattern with multiple minima that becomes more pronounced at higher en- ergies. While only one minimum is currently observed at the LHC, current fits, which use a combination of Regge-pole and eikonal mod- els, typically predict further structures beyond current experimental resolution [105–108]. It would be intriguing if, at higher ene...
[3]

The presence of a spin-2 resonance near thresh- old in our Froissart amplitude, as seen in Fig
[4]

Amaldi, Adv

U. Amaldi, Adv. Ser. Direct. High Energy Phys.23, 257 (2015)

2015
[5]

Froissart, Phys

M. Froissart, Phys. Rev.123, 1053 (1961). 31 Maximizing the cross-section at a point corresponds to choosing a delta-function kernel. Such a choice lies out- side the class of admissible test functions considered here and is not covered by the analytic bounds discussed in this work. In general we do not expect a bound in this case, see Appendix A

1961
[6]

hollowness

If this particle belongs to a Regge trajec- tory that is associated with universal growth, it should have vacuum quantum numbers. Cu- riously thef 2(1950) resonance [9] matches this description, which has been alternatively pro- posed as the first state on the Donnachie- Landshoff Pomeron trajectory [101, 109]. Notably, the Froissart amplitude we construc...

1950
[7]

white ring

From (C6) and since dσel dt >0 the weaker bound follows Z 0 −t0 dσel dt dt≤σ tot(s),(C8) for sufficiently small but finitet 0, that is−s≪ t0 <0. Plugging (C7) into the above, we see thatR 0 −s0 s2α′tdt∼1/logsin the limits→ ∞which in- deed provides some shrinking and makes the LHS of (C8) grow as log 2d−5s. However the RHS goes as logd−2sand the bound is v...
[8]

Antchevet al.(TOTEM), Eur

G. Antchevet al.(TOTEM), Eur. Phys. J. C79, 103 (2019), arXiv:1712.06153 [hep-ex]

arXiv 2019

Show all 139 references

[9]

Here, we display the Regge trajectories in the complexs-plane, parametrized by spinℓ

Regge trajectories in the complex plane Figure 19 presents a more detailed view of the imaginary-plane projection of the Chew–Frautschi plot shown in Figure 7. Here, we display the Regge trajectories in the complexs-plane, parametrized by spinℓ. The goal is to visualize the an...
[10]

Asymptotic growth of the F roissart Amplitude To probe the asymptotic growth of the Froissart amplitude, we study the effective exponent αeff 0 (s) =s d ds log |T(s,0)|,(G3) 21 ℓ=0 ℓ=2 ℓ=4 ℓ=6 Imρ 0.1 0.3 0.5 0.7 0.9 ℓ=8 Reρ FIG. 18. Projection of the partial waves via exp(−lo...
[11]

Near-forward zeros of the F roissart Amplitude A remarkable result by Auberson et al. [104] es- tablishes that amplitudes violating the Pomeranchuk theorem, such as those with cross-sections that grow with energy, must exhibit zeros of the amplitude T(s, t) near the negative r...
[12]

Dymarsky, JHEP07, 106 (2015), arXiv:1412.8642 [hep-ph]

A. Dymarsky, JHEP07, 106 (2015), arXiv:1412.8642 [hep-ph]. 25

2015 arXiv
[13]

Grosset al., Eur

F. Grosset al., Eur. Phys. J. C83, 1125 (2023), arXiv:2212.11107 [hep-ph]

2023 arXiv
[14]

G. F. Chew and S. C. Frautschi, Phys. Rev. Lett. 8, 41 (1962)

1962
[15]

Chew,The Analytic S Matrix: A Basis for Nu- clear Democracy(W

G. Chew,The Analytic S Matrix: A Basis for Nu- clear Democracy(W. A. Benjamin, 1966)

1966
[16]

Kharzeev and M

D. Kharzeev and M. Stephanov,Froissart bound and chiral limit in QCD,unpublished
[17]

Giordano and E

M. Giordano and E. Meggiolaro, JHEP03, 002 (2014), arXiv:1311.3133 [hep-ph]

2014 arXiv
[18]

Martin, Phys

A. Martin, Phys. Rev.129, 1432 (1963)

1963
[19]

J. R. Cudell, V. V. Ezhela, P. Gauron, K. Kang, Y. V. Kuyanov, S. B. Lugovsky, E. Martynov, B. Nicolescu, E. A. Razuvaev, and N. P. Tkachenko (COMPETE), Phys. Rev. Lett.89, 201801 (2002), arXiv:hep-ph/0206172

2002 arXiv
[20]

Martin and S

A. Martin and S. M. Roy, Phys. Rev. D89, 045015 (2014), arXiv:1306.5210 [hep-ph]

2014 arXiv
[21]

Navaset al.(Particle Data Group), Phys

S. Navaset al.(Particle Data Group), Phys. Rev. D110, 030001 (2024)

2024
[22]

A. Grau, S. Pacetti, G. Pancheri, and Y. N. Srivas- tava, Phys. Lett. B714, 70 (2012), arXiv:1206.1076 [hep-ph]

2012 arXiv
[23]

M. M. Block and F. Halzen, Phys. Rev. D86, 051504 (2012), arXiv:1208.4086 [hep-ph]

2012 arXiv
[24]

Y. He, A. Irrgang, and M. Kruczenski, JHEP11, 093 (2018), arXiv:1805.02812 [hep-th]

2018 arXiv
[25]

Ostapchenko, Phys

S. Ostapchenko, Phys. Rev. D81, 114028 (2010), arXiv:1003.0196 [hep-ph]

2010 arXiv
[26]

V. A. Khoze, A. D. Martin, and M. G. Ryskin, Int. J. Mod. Phys. A30, 1542004 (2015), arXiv:1402.2778 [hep-ph]

2015 arXiv
[27]

Anselm and V

A. Anselm and V. Gribov, Physics Letters B40, 487 (1972)

1972
[28]

Karateev, S

D. Karateev, S. Kuhn, and J. a. Penedones, JHEP 07, 035 (2020), arXiv:1912.08940 [hep-th]

2020 arXiv
[29]

Elias Mir´ o, A

J. Elias Mir´ o, A. L. Guerrieri, A. Hebbar, J. a. Penedones, and P. Vieira, Phys. Rev. Lett.123, 221602 (2019), arXiv:1906.08098 [hep-th]

2019 arXiv
[30]

Martin, AIP Conf

A. Martin, AIP Conf. Proc.1105, 258 (2009), arXiv:0812.0680 [hep-ph]

2009 arXiv
[31]

Kupsch, (2008), arXiv:0801.4871 [math-ph]

J. Kupsch, (2008), arXiv:0801.4871 [math-ph]

2008 arXiv
[32]

C´ ordova, Y

L. C´ ordova, Y. He, M. Kruczenski, and P. Vieira, JHEP04, 142 (2020), arXiv:1909.06495 [hep-th]

2020 arXiv
[33]

M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees, and P. Vieira, JHEP11, 143 (2017), arXiv:1607.06110 [hep-th]

2017 arXiv
[35]

Doroud and J

N. Doroud and J. Elias Mir´ o, JHEP09, 052 (2018), arXiv:1804.04376 [hep-th]

2018 arXiv
[36]

A. Bose, P. Haldar, A. Sinha, P. Sinha, and S. S. Ti- wari, SciPost Phys.9, 081 (2020), arXiv:2006.12213 [hep-th]

2020 arXiv
[37]

C´ ordova and P

L. C´ ordova and P. Vieira, JHEP12, 063 (2018), arXiv:1805.11143 [hep-th]

2018 arXiv
[38]

A. L. Guerrieri, J. Penedones, and P. Vieira, Phys. Rev. Lett.122, 241604 (2019), arXiv:1810.12849 [hep-th]

2019 arXiv
[39]

Homrich, J

A. Homrich, J. a. Penedones, J. Toledo, B. C. van Rees, and P. Vieira, JHEP11, 076 (2019), arXiv:1905.06905 [hep-th]

2019 arXiv
[40]

Guerrieri, J

A. Guerrieri, J. Penedones, and P. Vieira, Phys. Rev. Lett.127, 081601 (2021), arXiv:2102.02847 [hep-th]

2021 arXiv
[41]

Correia, A

M. Correia, A. Sever, and A. Zhiboedov, SciPost Phys.13, 062 (2022), arXiv:2111.12100 [hep-th]

2022 arXiv
[42]

M. F. Paulos and Z. Zheng, JHEP05, 145 (2020), arXiv:1805.11429 [hep-th]

2020 arXiv
[43]

Bercini, M

C. Bercini, M. Fabri, A. Homrich, and P. Vieira, Phys. Rev. D101, 045022 (2020), arXiv:1909.06453 [hep-th]

2020 arXiv
[44]

Elias Mir´ o and A

J. Elias Mir´ o and A. Guerrieri, JHEP10, 126 (2021), arXiv:2106.07957 [hep-th]

2021 arXiv
[45]

Kruczenski and H

M. Kruczenski and H. Murali, JHEP04, 097 (2021), arXiv:2012.15576 [hep-th]

2021 arXiv
[46]

A. L. Guerrieri, A. Homrich, and P. Vieira, JHEP 11, 084 (2020), arXiv:2008.02770 [hep-th]

2020 arXiv
[47]

A. Bose, A. Sinha, and S. S. Tiwari, SciPost Phys. 10, 122 (2021), arXiv:2011.07944 [hep-th]

2021 arXiv
[49]

A. L. Guerrieri, J. Penedones, and P. Vieira, JHEP 06, 088 (2021), arXiv:2011.02802 [hep-th]

2021 arXiv
[50]

Hebbar, D

A. Hebbar, D. Karateev, and J. Penedones, (2020), arXiv:2011.11708 [hep-th]

2020 arXiv
[51]

Sinha and A

A. Sinha and A. Zahed, (2020), arXiv:2012.04877 [hep-th]

2020 arXiv
[52]

Albert and L

J. Albert and L. Rastelli, JHEP08, 151 (2022), arXiv:2203.11950 [hep-th]

2022 arXiv
[53]

Correia, J

M. Correia, J. Penedones, and A. Vuignier, JHEP 08, 108 (2023), arXiv:2212.03917 [hep-th]

2023 arXiv
[54]

Tourkine and A

P. Tourkine and A. Zhiboedov, JHEP07, 228 (2021), arXiv:2101.05211 [hep-th]

2021 arXiv
[55]

Karateev, J

D. Karateev, J. Marucha, J. a. Penedones, and B. Sahoo, JHEP12, 136 (2022), arXiv:2204.01786 [hep-th]

2022 arXiv
[56]

Albert and L

J. Albert and L. Rastelli, JHEP09, 039 (2024), arXiv:2307.01246 [hep-th]

2024 arXiv
[57]

He and M

Y. He and M. Kruczenski, JHEP08, 125 (2021), arXiv:2103.11484 [hep-th]

2021 arXiv
[58]

Guerrieri and A

A. Guerrieri and A. Sever, Phys. Rev. Lett.127, 251601 (2021), arXiv:2106.10257 [hep-th]

2021 arXiv
[59]

H. Chen, A. L. Fitzpatrick, and D. Karateev, JHEP 12, 092 (2022), arXiv:2207.12448 [hep-th]

2022 arXiv
[60]

Note that it is also possible to have an accumula- tion of resonances in the physical region that makes the cross-section highly oscillatory (but still finite)

This universality further supports the notion that max- imizing the cross-section at a given energy requires the excitation of all spins at that scale: a goal achiev- able only via Reggeization of these resonances, which enhances the interaction at lower energies. Note that it...
[61]

J. E. Miro, A. Guerrieri, and M. A. Gumus, (2022), arXiv:2210.01502 [hep-th]

2022 arXiv
[62]

Guerrieri, H

A. Guerrieri, H. Murali, J. Penedones, and P. Vieira, JHEP06, 064 (2023), arXiv:2212.00151 [hep-th]

2023 arXiv
[63]

H¨ aring, A

K. H¨ aring, A. Hebbar, D. Karateev, M. Meineri, and J. a. Penedones, JHEP2410, 103 (2024), arXiv:2211.05795 [hep-th]

2024 arXiv
[64]

F. Bhat, D. Chowdhury, A. Sinha, S. Tiwari, and A. Zahed, JHEP03, 157 (2024), arXiv:2311.03451 [hep-th]

2024 arXiv
[65]

He and M

Y. He and M. Kruczenski, (2024), arXiv:2403.10772 [hep-th]

2024 arXiv
[66]

Correia, Phys

M. Correia, Phys. Rev. D110, 025012 (2024)

2024
[67]

Acanfora, A

F. Acanfora, A. Guerrieri, K. H¨ aring, and D. Kara- teev, JHEP03, 028 (2024), arXiv:2310.06027 [hep- th]

2024 arXiv
[68]

Albert, W

J. Albert, W. Knop, and L. Rastelli, JHEP02, 157 (2025), arXiv:2406.12959 [hep-th]

2025 arXiv
[69]

Antunes, M

A. Antunes, M. S. Costa, and J. Pereira, Phys. Lett. B846, 138225 (2023), arXiv:2301.13219 [hep- th]

2023 arXiv
[70]

Albert, J

J. Albert, J. Henriksson, L. Rastelli, and A. Vichi, JHEP09, 172 (2024), arXiv:2312.15013 [hep-th]

2024 arXiv
[71]

A. L. Guerrieri, A. Hebbar, and B. C. van Rees, (2023), arXiv:2312.00127 [hep-th]

2023 arXiv
[72]

Cordova, M

L. Cordova, M. Correia, A. Georgoudis, and A. Vuignier, JHEP01, 093 (2024), arXiv:2311.03031 [hep-th]

2024 arXiv
[73]

Elias Miro, A

J. Elias Miro, A. L. Guerrieri, and M. A. Gumus, Phys. Rev. D110, 016007 (2024), arXiv:2311.09283 26 [hep-ph]

2024 arXiv
[74]

Tourkine and A

P. Tourkine and A. Zhiboedov, JHEP11, 005 (2023), arXiv:2303.08839 [hep-th]

2023 arXiv
[75]

Gaikwad, V

A. Gaikwad, V. Gorbenko, and A. L. Guerrieri, JHEP01, 090 (2024), arXiv:2310.20698 [hep-th]

2024 arXiv
[76]

universal rise of the cross-section

for thepp(p¯p) cross-section: σ± tot(s) =Hlog 2 s sM +P+ R1 s sM −η1 ∓R 2 s sM −η2 (H1) whereHis the coefficient of the Froissart term and controls the growth at infinity.R i are the intensi- ties of the secondary Regge poles andPis called the Pomeranchuk constant. Finallys M ...
[77]

Adams, N

A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis, and R. Rattazzi, JHEP10, 014 (2006), arXiv:hep-th/0602178

2006 arXiv
[78]

Copetti, L

C. Copetti, L. Cordova, and S. Komatsu, Phys. Rev. Lett.133, 181601 (2024), arXiv:2403.04835 [hep-th]

2024 arXiv
[79]

Copetti, L

C. Copetti, L. Cordova, and S. Komatsu, JHEP 03, 204 (2025), arXiv:2408.13132 [hep-th]

2025 arXiv
[80]

M. A. Gumus,Mapping out EFTs with analytic S- matrix, Ph.D. thesis, SISSA, Italy, Trieste (2023)

2023
[81]

Guerrieri, A

A. Guerrieri, A. Homrich, and P. Vieira, (2024), arXiv:2404.10812 [hep-th]

2024 arXiv
[82]

Guerrieri, K

A. Guerrieri, K. H¨ aring, and N. Su, (2024), arXiv:2410.23333 [hep-th]

2024 arXiv
[83]

M. A. Gumus, D. Leflot, P. Tourkine, and A. Zhi- boedov, (2024), arXiv:2412.09610 [hep-th]

2024 arXiv
[84]

Cordova and R

L. Cordova and R. Rodrigues, (2025), arXiv:2504.20239 [hep-th]

2025 arXiv
[85]

He and M

Y. He and M. Kruczenski, (2025), arXiv:2505.19332 [hep-th]

2025
[86]

F. J. Yndurain, Rev. Mod. Phys.44, 645 (1972)

1972
[87]

M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees, and P. Vieira, JHEP12, 040 (2019), arXiv:1708.06765 [hep-th]

2019 arXiv
[88]

Patrignaniet al.(Particle Data Group), Chin

C. Patrignaniet al.(Particle Data Group), Chin. Phys. C40, 100001 (2016)

2016
[89]

J. H. Schwarz, Phys. Rev.167, 1342 (1968)

1968
[90]

A. K. Common, Nuovo Cim. A69, 115 (1970)

1970
[91]

Landry and D

W. Landry and D. Simmons-Duffin, (2019), arXiv:1909.09745 [hep-th]

2019 arXiv
[92]

V. N. Gribov, E. M. Levin, and A. A. Migdal, Yad. Fiz.12, 173 (1970)

1970
[93]

Elias Mir´ o, A

J. Elias Mir´ o, A. Guerrieri, and M. Gumus, Work in progress
[94]

Correia, A

M. Correia, A. Sever, and A. Zhiboedov, JHEP03, 013 (2021), arXiv:2006.08221 [hep-th]

2021 arXiv
[95]

Z. Bern, D. Kosmopoulos, and A. Zhiboedov, (2021), arXiv:2103.12728 [hep-th]

2021 arXiv
[96]

D. D. Coon, Phys. Lett. B29, 669 (1969)

1969
[97]

Figueroa and P

F. Figueroa and P. Tourkine, Phys. Rev. Lett.129, 121602 (2022), arXiv:2201.12331 [hep-th]

2022 arXiv
[98]

V. N. Gribov and A. A. Migdal, Yad. Fiz.8, 1002 (1968)

1968
[99]

R. J. M. Covolan, J. Montanha, and K. A. Gou- lianos, Phys. Lett. B389, 176 (1996)

1996
[100]

E. G. S. Luna and M. J. Menon, Phys. Lett. B565, 123 (2003), arXiv:hep-ph/0305280

2003 arXiv
[101]

Donnachie, G

S. Donnachie, G. Dosch, P. Landshoff, and O. Nachtmann,Pomeron Physics and QCD(Cam- bridge University Press, 2002)

2002
[102]

Sawada, Phys

T. Sawada, Phys. Rev.165, 1848 (1968)

1968
[103]

Oehme, Physics Letters B31, 573 (1970)

R. Oehme, Physics Letters B31, 573 (1970)

1970
[104]

Auberson, T

G. Auberson, T. Kinoshita, and A. Martin, Phys. Rev. D3, 3185 (1971)

1971
[105]

J. B. Bronzan, Phys. Rev. D4, 2569 (1971)

1971
[106]

J. B. Bronzan and C. S. Hui, Phys. Rev. D5, 964 (1972)

1972
[107]

P. D. B. Collins,An Introduction to Regge Theory and High Energy Physics(Cambridge University Press, 1977)

1977
[108]

Oehme, Phys

R. Oehme, Phys. Rev. D4, 1485 (1971)

1971
[109]

Oehme, Phys

R. Oehme, Phys. Rev. D9, 2695 (1974)

1974
[110]

Pakvasa and H

S. Pakvasa and H. Sugawara, Phys. Lett. B48, 333 (1974)

1974
[111]

M. S. Dubovikov and K. A. Ter-Martirosian, Nucl. Phys. B124, 163 (1977)

1977
[112]

Kupsch, Nuovo Cim

J. Kupsch, Nuovo Cim. A71, 85 (1982)

1982
[113]

Jenkovszky, I

L. Jenkovszky, I. Szanyi, and C.-I. Tan, Eur. Phys. J. A54, 116 (2018), arXiv:1710.10594 [hep-ph]

2018 arXiv
[114]

Born and E

M. Born and E. Wolf,Principles of Optics: Electro- magnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Pergamon Press, Ox- ford, 1980)

1980
[115]

Hecht,Optics(2017)

E. Hecht,Optics(2017)

2017
[116]

Ruiz Arriola and W

E. Ruiz Arriola and W. Broniowski, Few Body Syst. 57, 485 (2016), arXiv:1602.00288 [hep-ph]. 27

2016 arXiv
[117]

Fiore, L

R. Fiore, L. L. Jenkovszky, R. Orava, E. Predazzi, A. Prokudin, and O. Selyugin, Int. J. Mod. Phys. A24, 2551 (2009), arXiv:0810.2902 [hep-ph]

2009 arXiv
[118]

Martynov and B

E. Martynov and B. Nicolescu, Eur. Phys. J. C79, 461 (2019), arXiv:1808.08580 [hep-ph]

2019 arXiv
[119]

Jenkovszky, R

L. Jenkovszky, R. Schicker, and I. Szanyi, Int. J. Mod. Phys. E27, 1830005 (2018), arXiv:1902.05614 [hep-ph]

2018 arXiv
[120]

shrinking cone

and, so far, has not been ruled out by basic principles. Appendix B: F roissart bound ind-dimensions In this Appendix, we discuss the asymptotic be- haviour of the bound in (12). We will interpret this limit as thed-dimensional version of the Frois- sart bound. Similar derivat...
[121]

Ferreira, A

E. Ferreira, A. K. Kohara, and T. Kodama, Eur. Phys. J. C81, 290 (2021), arXiv:2011.13335 [hep- ph]

2021 arXiv
[122]

Donnachie and P

A. Donnachie and P. V. Landshoff, Nucl. Phys. B 267, 690 (1986)

1986
[123]

Lukaszuk and B

L. Lukaszuk and B. Nicolescu, Lett. Nuovo Cim.8, 405 (1973)

1973
[124]

Martynov and B

E. Martynov and B. Nicolescu, Phys. Lett. B778, 414 (2018), arXiv:1711.03288 [hep-ph]

2018 arXiv
[125]

Antchevet al.(TOTEM), Eur

G. Antchevet al.(TOTEM), Eur. Phys. J. C80, 91 (2020), arXiv:1812.08610 [hep-ex]

2020 arXiv
[126]

Antchevet al.(TOTEM), Nucl

G. Antchevet al.(TOTEM), Nucl. Phys. B899, 527 (2015), arXiv:1503.08111 [hep-ex]

2015 arXiv
[127]

M. A. Gumus, D. Leflot, P. Tourkine, and A. Zhi- boedov,upcoming
[128]

G. B. Arfken and H. J. Weber,Mathematical meth- ods for physicists; 4th ed.(Academic Press, San Diego, CA, 1995) Chap. 11

1995
[129]

M¨ antysaari, Rept

H. M¨ antysaari, Rept. Prog. Phys.83, 082201 (2020), arXiv:2001.10705 [hep-ph]

2020 arXiv
[130]

Broniowski, L

W. Broniowski, L. Jenkovszky, E. Ruiz Arriola, and I. Szanyi, Phys. Rev. D98, 074012 (2018), arXiv:1806.04756 [hep-ph]

2018 arXiv
[131]

S. M. Troshin and N. E. Tyurin, J. Phys. G49, 115001 (2022), arXiv:2201.00315 [hep-ph]

2022 arXiv
[132]

Martin,Inability of field theory to exploit the full unitarity condition, Tech

A. Martin,Inability of field theory to exploit the full unitarity condition, Tech. Rep. (CERN, Geneva, 1966)

1966
[133]

Zhiboedov,Notes on the analytic S-matrix, GGI Lectures on the Theory of Fundamental Interac- tions (2022)

A. Zhiboedov,Notes on the analytic S-matrix, GGI Lectures on the Theory of Fundamental Interac- tions (2022)

2022
[134]

Haldar and A

P. Haldar and A. Sinha, SciPost Phys.8, 095 (2020), arXiv:1911.05974 [hep-th]

2020 arXiv
[135]

V. N. Gribov,The theory of complex angular momenta: Gribov lectures on theoretical physics, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2007)

2007
[136]

Rising cross-sections,

R. Oehme, “Rising cross-sections,” inSpringer Tracts in Modern Physics, Volume 61: Ergeb- nisse der exakten Naturwissenschaften Volume 61 (Springer Berlin Heidelberg, Berlin, Heidelberg,
[137]

Bellazzini, G

B. Bellazzini, G. Isabella, and M. M. Riva, JHEP 04, 023 (2023), arXiv:2211.00085 [hep-th]

2023 arXiv
[138]

H¨ aring and A

K. H¨ aring and A. Zhiboedov, (2024), arXiv:2410.21499 [hep-th]

2024 arXiv
[139]

Abramowitz and I

M. Abramowitz and I. Stegun,Handbook of Math- ematical Functions: With Formulas, Graphs, and Mathematical Tables, Applied mathematics series (Dover Publications, 1965)

1965
[141]

Lopez and G

C. Lopez and G. Mennessier, Phys. Lett. B58, 437 (1975)

1975
[750]

qK/REddAjcJ5sxuLJbFzQq2TupA=

We have verified that decimating the grid (e.g., by retaining only one out of every four points) does not significantly affect the numerical results, and can be useful for accelerating the solution of the boot- strap optimization problem. Using a higher spin cutoffL >18 does n...

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

white ring

The observed shrinking of theppdiffractive cone at a rate faster than logsat √s= 13 TeV 11 as reported by the TOTEM collaboration [8, 9]. This is a feature of our Froissart amplitude and of the analytical “white ring” model with ex- plicit Froissart growth

[2] [2]

A diffractive-like pattern with multiple minima that becomes more pronounced at higher en- ergies. While only one minimum is currently observed at the LHC, current fits, which use a combination of Regge-pole and eikonal mod- els, typically predict further structures beyond current experimental resolution [105–108]. It would be intriguing if, at higher ene...

[3] [3]

The presence of a spin-2 resonance near thresh- old in our Froissart amplitude, as seen in Fig

[4] [4]

Amaldi, Adv

U. Amaldi, Adv. Ser. Direct. High Energy Phys.23, 257 (2015)

2015

[5] [5]

Froissart, Phys

M. Froissart, Phys. Rev.123, 1053 (1961). 31 Maximizing the cross-section at a point corresponds to choosing a delta-function kernel. Such a choice lies out- side the class of admissible test functions considered here and is not covered by the analytic bounds discussed in this work. In general we do not expect a bound in this case, see Appendix A

1961

[6] [6]

hollowness

If this particle belongs to a Regge trajec- tory that is associated with universal growth, it should have vacuum quantum numbers. Cu- riously thef 2(1950) resonance [9] matches this description, which has been alternatively pro- posed as the first state on the Donnachie- Landshoff Pomeron trajectory [101, 109]. Notably, the Froissart amplitude we construc...

1950

[7] [7]

white ring

From (C6) and since dσel dt >0 the weaker bound follows Z 0 −t0 dσel dt dt≤σ tot(s),(C8) for sufficiently small but finitet 0, that is−s≪ t0 <0. Plugging (C7) into the above, we see thatR 0 −s0 s2α′tdt∼1/logsin the limits→ ∞which in- deed provides some shrinking and makes the LHS of (C8) grow as log 2d−5s. However the RHS goes as logd−2sand the bound is v...

[8] [8]

Antchevet al.(TOTEM), Eur

G. Antchevet al.(TOTEM), Eur. Phys. J. C79, 103 (2019), arXiv:1712.06153 [hep-ex]

arXiv 2019