Rigorous convergence of the two-dimensional unitarity iteration and the continuum fate of the CDD fractal

Tourkine and Zhiboedov (2021) implemented, for two-dimensional scattering of identical massive bosons, a dispersive iteration that reconstructs the scattering amplitude from prescribed particle-production data by iterating unitarity and crossing to a fixed point, and discovered a fractal, initialization-dependent basin structure connected to the Castillejo-Dalitz-Dyson (CDD) ambiguity. Two linked mathematical questions from that work remain open. First, no rigorous, continuum-level convergence criteria are known for the iteration: the analogue of Atkinson's four-dimensional existence proof fails in two dimensions because the unitarity kernel carries a $1/\sqrt{1-x}$ threshold factor that escapes control by Hölder norms, and only discretized, numerical convergence domains have been mapped. Second, the 'CDD fractal' - the erratic dependence of the Newton iteration's fixed point (its CDD sector, i.e. the number and positions of its CDD zeros) on the starting point when several sectors share the same inelasticity input - has not been characterized in the continuum limit: it is unknown whether basins of attraction retain finite size or shrink to zero, making the sector-selection map nowhere-continuous. This problem asks for a rigorous convergence theory of the two-dimensional unitarity iteration (contraction or Newton-Kantorovich criteria on explicit function spaces, delineating the admissible inelasticities and pole data) together with a mathematical classification of the continuum CDD-sector basin structure.

Background

Consider scattering of identical massive scalar bosons of mass $m$ in $1+1$ spacetime dimensions. The two-body kinematics then depends on a single invariant, the center-of-mass energy squared $s$ (the momentum transfer vanishes and $u=4m^2-s$), so the $S$-matrix is a single function $S(s)=1+iT(s)/\sqrt{s(s-4m^2)}$, the square-root Jacobian being specific to two dimensions. The axioms imposed are: Mandelstam analyticity, i.e. holomorphy of $S$ in the complex $s$-plane cut along the right (physical) cut $s\ge 4m^2$ and the left (crossed) cut $s\le 0$, with poles only at bound-state locations $0<s<4m^2$; real analyticity $T(s^*)=T(s)^*$; crossing symmetry $S(s)=S(4m^2-s)$; and unitarity, which on the cut reads $\operatorname{Im}T(s)=|T(s)|^2/(2\sqrt{s(s-4m^2)})+v_i(s)$, where the inelasticity $v_i(s)\ge 0$ (zero below the particle-production threshold) measures the fraction of probability flux lost to multi-particle final states. In 1968, D. Atkinson proved constructively, in four spacetime dimensions, that functions satisfying all these axioms exist: combining the dispersion relation (which reconstructs $T$ from its imaginary part through a crossing-symmetrized principal-value integral) with the unitarity equation defines a nonlinear map on a space of functions, and Atkinson showed that a ball in a Hölder norm is mapped into itself and contracted there, so the iteration converges to a fixed point that is the desired amplitude. This remains essentially the only constructive existence result of its type, and for over half a century it was never implemented numerically.

Tourkine and Zhiboedov (2021) implemented the dispersive iteration numerically in two dimensions, for the inverse problem: given the inelasticity $v_i(s)$, the bound-state pole positions, and the constant at infinity $c_\infty=\lim_{s\to\infty}T(s)$, construct the amplitude. Defining $\Phi(\operatorname{Im}T)(s)=|T(s)|^2/(2\sqrt{s(s-4m^2)})+v_i(s)$ with $T$ reconstructed from $\operatorname{Im}T$ by the dispersion relation (pole couplings, or $c_\infty$, being fixed at each step by the threshold condition $\operatorname{Re}T(4m^2)=0$), they studied the fixed-point iteration $\operatorname{Im}T_{n+1}=\Phi(\operatorname{Im}T_n)$ and the Newton-Kantorovich method (the functional analogue of Newton's method, applied to $\Psi=\mathrm{id}-\Phi$), on discretized grids. The explicit solution is known in this setting: purely elastic amplitudes ($v_i=0$) are products of CDD pole and CDD zero factors - the elementary factors, introduced by Castillejo, Dalitz and Dyson in 1956, that have unit modulus on the cuts and carry a single pole or zero - and with inelasticity the general solution is such a product times a universal exponential determined by $v_i$. In particular, distinct CDD sectors (amplitudes with different numbers and positions of CDD zeros, matched to the same $c_\infty$) share the same input data: this is one facet of the CDD ambiguity, the classic underdetermination of elastic amplitudes by unitarity data.

Two findings of that work frame the open questions. First, convergence was established only numerically. Adapting Atkinson's proof to two dimensions fails at a specific step: in the variable $x$ that maps the right cut to $x\in[0,1]$ (with $x\to 1$ the two-particle threshold $s\to 4m^2$), the two-dimensional unitarity kernel carries a factor $1/\sqrt{1-x}$, whereas in $d$ dimensions the corresponding threshold factor $(1-x)^{(d-3)/2}$ vanishes at $x=1$ for $d=4$. The divergent factor can make the Hölder norm $\|\rho\|=\sup_{x_1,x_2}|\rho(x_1)-\rho(x_2)|/|x_1-x_2|^{\mu}$ of the image exceed that of the input near threshold, so a plain Hölder ball is not mapped into itself; the authors also found divergent fixed-point iterations for amplitudes whose Hölder norms are comparable to convergent ones, indicating that Hölder boundedness alone cannot characterize the convergence domain. Second, because many CDD sectors share the same input, the Newton iteration's endpoint depends on the starting point: interpolating initial conditions between a one-zero and a three-zero CDD amplitude, the number and positions of the converged amplitude's zeros change erratically with the interpolation parameter on arbitrarily fine scales - a 'CDD fractal', reminiscent of the classical fractal basins (Julia sets) of Newton's method for polynomial roots. The authors did not characterize this structure, and posed its continuum fate explicitly as an open question: either all flat basins of attraction shrink to zero size and the sector-selection curve becomes nowhere-continuous, or finite-size basins persist at all scales.

Problem Statement

Setting. Identical massive scalar bosons of mass $m$ in $1+1$ dimensions; $S(s)=1+iT(s)/\sqrt{s(s-4m^2)}$ with $s$ the center-of-mass energy squared. The amplitude must satisfy Mandelstam analyticity (holomorphy in the cut $s$-plane, right cut $s\ge 4m^2$, left cut $s\le 0$, bound-state poles in $0<s<4m^2$), real analyticity, crossing $S(s)=S(4m^2-s)$, and unitarity in the form $\operatorname{Im}T(s)=|T(s)|^2/(2\sqrt{s(s-4m^2)})+v_i(s)$ for $s\ge 4m^2$, where the inelasticity $v_i(s)\ge 0$ is prescribed input together with the bound-state pole positions and the constant at infinity $c_\infty=\lim_{s\to\infty}T(s)$. The dispersive iteration is the map $\Phi$ defined by $\Phi(\operatorname{Im}T)(s)=|T(s)|^2/(2\sqrt{s(s-4m^2)})+v_i(s)$, where $T$ on the right is reconstructed from $\operatorname{Im}T$ via the crossing-symmetrized dispersion relation, with pole couplings (or $c_\infty$) fixed at each step by the threshold condition $\operatorname{Re}T(4m^2)=0$; the fixed-point iteration is $\operatorname{Im}T_{n+1}=\Phi(\operatorname{Im}T_n)$, and Newton's method is applied to $\Psi=\mathrm{id}-\Phi$. These are the continuous maps whose discretizations were implemented by Tourkine and Zhiboedov (2021).

(1) Rigorous convergence criteria. Establish continuum-level (functional, non-discretized) convergence criteria for this iteration: prove that $\Phi$ maps an explicitly specified function space into itself and contracts there in the spirit of Atkinson's four-dimensional proof, or that the Newton-Kantorovich hypotheses hold for $\Psi$ on an explicit domain, and thereby characterize the set of input data - inelasticity functions $v_i(s)$, pole positions, and $c_\infty$ - for which the iteration converges to a solution of the two-dimensional S-matrix axioms above. The analysis must resolve the role of the threshold factor $1/\sqrt{1-x}$ (in the variable $x$ mapping the right cut to $x\in[0,1]$, with $x\to 1$ the threshold $s\to 4m^2$), under which a ball in the Hölder norm $\sup_{x_1,x_2}|\rho(x_1)-\rho(x_2)|/|x_1-x_2|^{\mu}$ is not mapped into itself; a complete answer identifies the additional constraints (for example on derivatives or endpoint behavior) under which the argument goes through, and delineates the admissible data class for which that criterion applies.

(2) Continuum fate of the CDD fractal. Distinct CDD sectors - amplitudes differing in the number and positions of their CDD zeros but sharing the same inelasticity, pole data, and $c_\infty$ - are all consistent solutions for the same input, and the discretized Newton iteration reaches different sectors depending on the starting point, with fractal-like basin boundaries on the grids studied numerically. Determine the continuum-limit structure of these basins: prove which alternative holds - (a) all basins of attraction shrink to zero size, so that the map from initial data to the selected CDD sector is nowhere-continuous; (b) finite-size basins persist at all scales; or (c) a more regular structure emerges - and classify the basins mathematically (their measure-theoretic and topological structure, and their dependence on the discretization).

Deliverables (1) and (2) are two faces of one objective, a rigorous theory of the dispersive unitarity iteration in two dimensions: (1) establishes when the iteration defines an amplitude at all, and (2) establishes how it resolves the CDD ambiguity intrinsic to reconstructing an amplitude from production data; together they decide whether that reconstruction is a well-posed, algorithmically reproducible procedure modulo the CDD ambiguity.

References

1. Piotr Tourkine and Alexander Zhiboedov, "Scattering from production in 2d", JHEP 07 (2021) 228. DOI: 10.1007/JHEP07(2021)228, arXiv:2101.05211, https://arxiv.org/abs/2101.05211
2. Piotr Tourkine and Alexander Zhiboedov, "Scattering amplitudes from dispersive iterations of unitarity", JHEP 11 (2023) 005. DOI: 10.1007/JHEP11(2023)005, arXiv:2303.08839, https://arxiv.org/abs/2303.08839
3. Mehmet Asim Gumus, Damien Leflot, Piotr Tourkine and Alexander Zhiboedov, "The S-matrix bootstrap with neural optimizers. Part I. Zero double discontinuity", JHEP 07 (2025) 210. DOI: 10.1007/JHEP07(2025)210, arXiv:2412.09610, https://arxiv.org/abs/2412.09610
4. Mehmet Asim Gumus, Damien Leflot, Piotr Tourkine and Alexander Zhiboedov, "Neural S-matrix bootstrap II: solvable 4d amplitudes with particle production", arXiv:2601.22145, https://arxiv.org/abs/2601.22145
5. António Antunes, Miguel S. Costa and José Pereira, "Exploring inelasticity in the S-matrix Bootstrap", Phys. Lett. B 846 (2023) 138225. DOI: 10.1016/j.physletb.2023.138225, arXiv:2301.13219, https://arxiv.org/abs/2301.13219
6. Aurélien Dersy, Matthew D. Schwartz and Alexander Zhiboedov, "Reconstructing S-matrix Phases with Machine Learning", JHEP 05 (2024) 200. DOI: 10.1007/JHEP05(2024)200, arXiv:2308.09451, https://arxiv.org/abs/2308.09451
7. Joan Elias Miró, Andrea Guerrieri and Mehmet Asim Gumus, "The Phases of the Scalar S-Matrix Island", arXiv:2605.06613, https://arxiv.org/abs/2605.06613
8. Mehmet Asim Gumus, Simon Metayer and Piotr Tourkine, "Tracking S-matrix bounds across dimensions", arXiv:2512.24474, https://arxiv.org/abs/2512.24474
9. D. Atkinson, "A proof of the existence of functions that satisfy exactly both crossing and unitarity: I. Neutral pion-pion scattering. No subtractions", Nucl. Phys. B 7 (1968) 375-408. DOI: 10.1016/0550-3213(68)90091-6, https://doi.org/10.1016/0550-3213(68)90091-6
10. L. Castillejo, R. H. Dalitz and F. J. Dyson, "Low's Scattering Equation for the Charged and Neutral Scalar Theories", Phys. Rev. 101 (1956) 453-458. DOI: 10.1103/PhysRev.101.453, https://doi.org/10.1103/PhysRev.101.453