Crossing-symmetric dual S-matrix bootstrap and analytic cross-section bounds in $d>2$: can manifestly crossing-symmetric dispersion relations yield rigorous (possibly stronger-than-Froissart) bounds?

Sinha and Zahed (PRL 126, 181601, 2021) derived manifestly crossing-symmetric dispersion relations (CSDRs) for 2-to-2 scattering of identical scalars in four dimensions, built on the three-channel symmetric representation of Auberson and Khuri, and used them numerically to bound the total cross section at all energies. They posed two linked open questions that the follow-up literature has explicitly deferred rather than solved: (i) can these dispersion relations serve as the basis of a dual S-matrix bootstrap in spacetime dimension greater than two, upgrading the currently numerical positivity/bootstrap bounds to rigorous theorems; and (ii) can the method produce analytic bounds on the total cross section, and in particular settle whether imposing full crossing symmetry permits a bound stronger than the Froissart bound at high energies. This problem asks for the crossing-symmetric dispersive constraint system in d>2 to be completed into a rigorous dual bootstrap with certified analytic total-cross-section bounds, and for analytic bounds on the total cross section to be derived by this method, exploring the role of crossing symmetry -- unexamined in existing derivations of the Froissart bound -- in the high-energy behavior.

Background

Crossing symmetry states that the same analytic function of the Mandelstam variables $s$, $t$, $u$ (the Lorentz-invariant kinematic invariants with $s+t+u=4m^2$ for four identical scalars of mass $m$) describes the $s$-, $t$-, and $u$-channel processes of $2\to2$ scattering, continued into one another. The analytic S-matrix program extracts consequences of this symmetry via dispersion relations: Cauchy's integral formula applied to the amplitude along contours in the complex $s$ plane, with unitarity (the optical theorem, which relates the imaginary part of the forward amplitude to the total cross section) supplying the discontinuities. Classical fixed-$t$ dispersion relations treat the crossed channels asymmetrically, and the celebrated Froissart bound (Froissart 1961; Martin, with sharp constants fixed by Martin and Roy) states that under the standard axioms the total cross section cannot grow faster than $\pi/m_\pi^2 \log^2(s/s_0)$ at asymptotic energy $s$, where $m_\pi$ is the lightest exchanged mass (the pion mass in hadronic applications) and $s_0$ is a fixed reference energy; whether this $\log^2$ growth is optimal within the axiom class has never been settled. Auberson and Khuri (1972) constructed a rigorous parametric dispersion representation with manifest three-channel crossing symmetry, which Sinha and Zahed (2021) revisited and modernized: using a particular expansion of the amplitude about the crossing-symmetric point $s=t=u=4m^2/3$ (later given a systematic geometric meaning as an expansion around the crossing-symmetric Regge limit) together with the partial-wave expansion (the expansion of the amplitude in angular-momentum eigenfunctions, whose coefficients $a_\ell(s)$ obey the unitarity inequalities $|a_\ell(s)|\le1$ from probability conservation) and a lattice of sign constraints called null constraints (identities that vanish identically in the exact theory, so that any sign-definite manipulation of them must vanish on average), they obtained a numerical pointwise upper bound on the total cross section, valid for all $s\ge4m^2$, by maximizing the partial-wave spin sum $\sum_\ell(2\ell+1)a_\ell(s)$ subject to the null constraints and partial-wave unitarity. The S-matrix bootstrap is the program of mapping the space of amplitudes consistent with the axioms. The primal approach parameterizes candidate amplitudes and optimizes within them; the dual approach, realized in two dimensions by Guerrieri, Homrich and Vieira (2020) and in a fixed-$t$ form in $d\ge3$ by Guerrieri and Sever (2021) via Roy-type equations (Roy equations are fixed-$t$ dispersion relations for the partial-wave amplitudes, with crossing imposed as an explicit linear constraint), sets up a dual pair of optimization problems over infinite-dimensional positive semidefinite spaces (partial-wave unitarity) for which extrema certify rigorous bounds. Guerrieri and Sever's $d\ge3$ construction imposes elastic unitarity (probability conservation for the $2\to2$ channel alone, which strictly holds only below the first inelastic threshold) only below a capping energy and leaves the manifestly crossing-symmetric formulation explicitly to future work, writing that one may instead use the Auberson-Khuri-type symmetric relations, which they 'leave to a future exploration'. Sinha and Zahed's paper closes by asking for exactly this: a dual bootstrap in higher dimensions built on their crossing-symmetric dispersion relations, and analytic bounds on the total cross section from their method, including the question of whether crossing symmetry permits a bound stronger than Froissart's at high energies. Subsequent work (Bhat, Chowdhury, Sinha, Tiwari and Zahed 2024; Elias Miro, Guerrieri, Gumus and Zahed 2026) has improved the numerical bootstrap with locally crossing-symmetric Roy equations and an all-energy geometric formulation of the crossing-symmetric contour, while continuing to defer the dual problem and the high-energy behavior; Correia, Georgoudis and Guerrieri (2025) derived a universal analytic bound on the integrated total cross section in arbitrary dimension by classical fixed-$t$ methods, reducing to the Froissart-Martin result in four dimensions, and conjectured a saturating 'Froissart amplitude', but did not use the crossing-symmetric null-constraint method and left the factor-of-five gap to numerical bounds, the optimality of the $\log^2$ form, and the stronger-than-Froissart question open.

Problem Statement

Formulate and solve the dual S-matrix bootstrap in spacetime dimension $d>2$ on the basis of manifestly crossing-symmetric dispersion relations, and use it to derive analytic bounds on the total cross section, investigating the high-energy behavior. Concretely, for $2\to2$ scattering of identical scalar particles of mass $m$ subject to the standard analytic S-matrix axioms (analyticity in the Mandelstam variables with the crossing-symmetric domain of Martin, partial-wave unitarity, and polynomial boundedness), the problem comprises three jointly required results tied to one objective, turning manifest crossing-symmetric dispersive constraints into rigorous bounds: (1) Dual formulation: recast the crossing-symmetric dispersion relation of Sinha-Zahed (equivalently the Auberson-Khuri three-channel symmetric representation or its geometric all-energy contour reformulation) together with partial-wave unitarity as a well-posed dual pair of optimization problems over the partial-wave coefficients, with the crossing-symmetric variables as the natural coordinates; (2) Certified bounds: from the dual pair, produce rigorous (theorem-grade, not merely numerical) bounds on the total cross section $\sigma_{\mathrm{tot}}(s)$ for all $s\ge4m^2$; (3) Froissart question: derive analytic bounds on the total cross section from the crossing-symmetric constraint system and determine whether imposing full crossing symmetry permits a bound stronger than the Froissart bound at high energies, exploring the role of crossing symmetry, which existing derivations of the Froissart bound have not addressed.

References

1. A. Sinha and A. Zahed, 'Crossing Symmetric Dispersion Relations in Quantum Field Theories', Phys. Rev. Lett. 126, 181601 (2021), arXiv:2012.04877, https://arxiv.org/abs/2012.04877, DOI:10.1103/PhysRevLett.126.181601
2. G. Auberson and N. N. Khuri, 'Rigorous Parametric Dispersion Representation with Three-Channel Symmetry', Phys. Rev. D 6, 2953 (1972), DOI:10.1103/PhysRevD.6.2953, https://doi.org/10.1103/PhysRevD.6.2953
3. M. Froissart, 'Asymptotic Behavior and Subtractions in the Mandelstam Representation', Phys. Rev. 123, 1053 (1961), DOI:10.1103/PhysRev.123.1053, https://doi.org/10.1103/PhysRev.123.1053
4. A. Martin and S. M. Roy, 'Froissart bound on total cross section without unknown constants', Phys. Rev. D 89, 045015 (2014), arXiv:1306.5210, https://arxiv.org/abs/1306.5210, DOI:10.1103/PhysRevD.89.045015
5. A. L. Guerrieri, A. Homrich and P. Vieira, 'Dual S-matrix bootstrap. Part I. 2D theory', JHEP 11, 084 (2020), arXiv:2008.02770, https://arxiv.org/abs/2008.02770, DOI:10.1007/JHEP11(2020)084
6. A. Guerrieri and A. Sever, 'Rigorous Bounds on the Analytic S Matrix', Phys. Rev. Lett. 127, 251601 (2021), arXiv:2106.10257, https://arxiv.org/abs/2106.10257, DOI:10.1103/PhysRevLett.127.251601
7. R. Gopakumar, A. Sinha and A. Zahed, 'Crossing Symmetric Dispersion Relations for Mellin Amplitudes', Phys. Rev. Lett. 126, 211602 (2021), arXiv:2101.09017, https://arxiv.org/abs/2101.09017, DOI:10.1103/PhysRevLett.126.211602
8. N. Arkani-Hamed, T.-C. Huang and Y.-t. Huang, 'The EFT-Hedron', JHEP 05, 259 (2021), arXiv:2012.15849, https://arxiv.org/abs/2012.15849, DOI:10.1007/JHEP05(2021)259
9. S. D. Chowdhury, K. Ghosh, P. Haldar, P. Raman and A. Sinha, 'Crossing Symmetric Spinning S-matrix Bootstrap: EFT bounds', SciPost Phys. 13, 051 (2022), arXiv:2112.11755, https://arxiv.org/abs/2112.11755, DOI:10.21468/SciPostPhys.13.3.051
10. D. Chowdhury, P. Haldar and A. Zahed, 'Locality and Analyticity of the Crossing Symmetric Dispersion Relation', JHEP 10, 180 (2022), arXiv:2205.13762, https://arxiv.org/abs/2205.13762, DOI:10.1007/JHEP10(2022)180
11. C. Song, 'Crossing-Symmetric Dispersion Relations without Spurious Singularities', Phys. Rev. Lett. 131, 161602 (2023), arXiv:2305.03669, https://arxiv.org/abs/2305.03669, DOI:10.1103/PhysRevLett.131.161602
12. J. Elias Miro, A. Guerrieri and M. A. Gumus, 'Bridging Positivity and S-matrix Bootstrap Bounds', JHEP 05, 001 (2023), arXiv:2210.01502, https://arxiv.org/abs/2210.01502, DOI:10.1007/JHEP05(2023)001
13. F. Bhat, D. Chowdhury, A. Sinha, S. Tiwari and A. Zahed, 'Bootstrapping High-Energy Observables', JHEP 03, 157 (2024), arXiv:2311.03451, https://arxiv.org/abs/2311.03451, DOI:10.1007/JHEP03(2024)157
14. M. Correia, A. Georgoudis and A. L. Guerrieri, 'Cross-Section Bootstrap: Unveiling the Froissart Amplitude', arXiv:2506.04313 (2025), https://arxiv.org/abs/2506.04313
15. C. Pasiecznik, 'Bootstrapping Gravity with Crossing Symmetric Dispersion Relations', arXiv:2506.09884 (2025), https://arxiv.org/abs/2506.09884
16. F. Bhat, A. P. Saha and A. Sinha, 'Stringy Dispersion Relation for Field Theories', Phys. Rev. D 113, 066016 (2026), arXiv:2506.03862, https://arxiv.org/abs/2506.03862, DOI:10.1103/ljzx-q254
17. G. Peng, L.-Q. Shao, A. Tokareva and Y. Xu, 'Running EFT-hedron with null constraints at loop level', JHEP 03, 163 (2026), arXiv:2501.09717, https://arxiv.org/abs/2501.09717, DOI:10.1007/JHEP03(2026)163
18. C. de Rham, A. J. Tolley, Z.-H. Wang and S. Zhou, 'Primal S-matrix Bootstrap with Dispersion Relations', JHEP 01, 027 (2026), arXiv:2506.22546, https://arxiv.org/abs/2506.22546, DOI:10.1007/JHEP01(2026)027
19. J. Elias Miro, A. Guerrieri, M. A. Gumus and A. Zahed, 'A Geometric View on Crossing-Symmetric Dispersion Relations', JHEP 07, 255 (2026), arXiv:2509.14170, https://arxiv.org/abs/2509.14170, DOI:10.1007/JHEP07(2026)255