Exact-unitarity dual formulation of the 3+1-dimensional S-matrix bootstrap: closing the primal-dual duality gap and the Karplus double-discontinuity domain

The modern S-matrix bootstrap bounds the low-energy observables of gapped quantum field theories (theories with a mass gap, so the lightest particle has mass $m > 0$) in d = 3+1 dimensions from two sides: primal constructions build explicit amplitudes from an ansatz with maximal (Landau) analyticity and the full nonlinear two-particle unitarity, while rigorous dual bounds derive exclusion regions from fixed-$t$ dispersion relations, axiomatic analyticity, and unitarity. The two sides disagree: with linearized-unitarity duals, the exclusion bounds on the quartic-coupling/derivative-expansion plane miss the primal allowed region by roughly an order of magnitude in the effective-field-theory cutoff setting, and a residual gap persists even in duals that impose the exact nonlinear two-particle unitarity equation only up to a finite spin cutoff and below a finite energy. The open problem is to construct a fully rigorous dual formulation in d = 3+1 that enforces the exact nonlinear two-particle unitarity equation - bounding the real parts of the partial waves through the full unitarity relation rather than the linearized absorptive-part bounds - for all spins and all energies, optionally together with infrared spectral constraints at an EFT cutoff, and to determine whether the optimal dual bounds then converge to the primal bounds on the low-energy Wilson coefficients, or whether the residual duality gap is irreducible and reflects the difference between the rigorously proven (Martin-Lehmann) analyticity domain and the maximal (Landau) analyticity assumed by the primal ansatz. A companion question is the role of the correct Karplus-region support of the double discontinuity, and which observables are sensitive to it.

Background

Consider the elastic scattering of two identical massive scalar particles of mass $m$ in $d = 3+1$ spacetime dimensions. The amplitude $M(s,t,u)$, a function of the Mandelstam invariants $s$, $t$, $u = 4m^2 - s - t$, is constrained by the general principles of relativistic quantum field theory: crossing symmetry (one analytic function describes all channels), analyticity in the invariant variables (branch cuts starting at the two-particle thresholds $s, t, u = 4m^2$), and unitarity of the S-matrix. The numerical S-matrix bootstrap asks for the allowed values of observables - most commonly the coefficients $c_0$ (the quartic coupling) and $c_2$ (the leading higher-derivative correction) of the Taylor expansion of the amplitude about the crossing-symmetric point $s = t = u = 4m^2/3$ - over all amplitudes satisfying these principles.

In the primal approach one parameterizes the amplitude by a crossing-symmetric ansatz built from a conformal map $\rho(s)$ that sends the cut $s$-plane into the unit disk, imposes unitarity on a grid of energies, and maximizes linear functionals of the coefficients by semidefinite programming (the convex-optimization technique underlying the numerical conformal bootstrap). The extremal amplitudes so constructed populate, and bound from inside, the allowed region. In the dual approach one derives rigorous exclusion bounds instead: positive functionals that any consistent amplitude must satisfy, so that a negative value excludes a whole region of coefficient space.

Unitarity enters through the partial-wave expansion. Writing the amplitude as a sum over angular momentum $\ell$ with partial-wave coefficients $f_\ell(s)$, two-particle (elastic) unitarity below the first inelastic threshold reads $|S_\ell(s)| \le 1$ with $S_\ell(s) = 1 + i\rho^2(s) f_\ell(s)$, where $\rho^2(s) = \sqrt{s - 4m^2}/\sqrt{s}$ is the two-body phase-space factor; equivalently $2\,\mathrm{Im}\, f_\ell(s) \ge \rho^2(s)\,|f_\ell(s)|^2$. This is a nonlinear (quadratic) constraint coupling the real and imaginary parts of each partial wave. Linearized unitarity keeps only $0 \le \mathrm{Im}\, f_\ell \le 2/\rho^2$, discarding all information about the real parts; imposing the full inequality is what makes the bootstrap two-sided, since dispersion relations then bound the real parts in terms of the absorptive part. In the primal, the quadratic inequality is recast as a positive-semidefinite $2 \times 2$ matrix condition linear in the ansatz parameters, preserving convexity.

Dual bounds start from dispersion relations. A fixed-$t$ dispersion relation expresses the amplitude at any $s$ as an integral over its absorptive part plus subtraction constants, which are themselves the low-energy coefficients; projecting on partial waves gives the Roy equations, integral equations relating the real and imaginary parts of the partial waves. Applying Lagrange (weak) duality to the resulting optimization yields a dual functional whose minimum is a rigorous bound. The fixed-$t$ dual of Guerrieri and Sever relies only on analyticity domains that are rigorously proven (the Martin-Lehmann domain) and can impose the exact nonlinear unitarity equation, but only for partial waves below a finite spin cutoff and, because of the limited proven validity range of the Roy equations, only for $4m^2 \le s \le 12m^2$ in the standard implementation. The primal ansatz, by contrast, assumes maximal (Landau) analyticity - the strongest analytic structure compatible with the Mandelstam representation, in which the amplitude is reconstructed from a single spectral density and a double spectral density $\rho(s,t)$.

The source paper (Elias Miró, Guerrieri and Gümüş, JHEP 05 (2023) 001, arXiv:2210.01502) compared the two sides quantitatively in the effective-field-theory (EFT) cutoff setting, where an infrared cutoff $\Lambda$ is imposed together with extra constraints on the absorptive part below $\Lambda$ (bounding arcs of the amplitude, method M1, or the one-loop EFT imaginary part, method M2). Its primal bounds and its linearized-unitarity dual bounds disagree by roughly an order of magnitude: at $\Lambda^2 = 64m^2$, the dual gives $c_2 \Lambda^4 \le 138.8$ versus the primal $c_2 \Lambda^4 \lesssim 8$ under M2 (283.3 versus $\lesssim 45$ under M1). The authors state the expectation that the gap should close once the constraints on the real parts are included. The same paper studied the analytic domain of the double discontinuity: elastic unitarity forces the double spectral density to vanish below the Karplus curve (given, in units $m = 1$, by the hyperbolas $t = 4s/(s-16)$ and $t = 16s/(s-4)$), whereas bootstrap ansatze typically give the double discontinuity support on the whole square $[4m^2, \infty)^2$. A first rectangle approximation of the Karplus region produced no visible change in the $c_i$ bounds, leaving open which observables are sensitive to the precise support.

Subsequent work sharpened rather than settled the question. A toy four-dimensional model with the double discontinuity set to zero (so that only the spin-zero partial wave carries an imaginary part) showed perfect agreement between primal, dual and neural-network optimizers, but that setting is far from the full problem. Constructive approaches that iterate unitarity inside the Mandelstam representation (the Atkinson-Mandelstam approach) now produce four-dimensional amplitudes with exact two-particle unitarity and double-discontinuity support only above the leading Landau curves, and indicate that low-energy couplings are rather insensitive to the inelastic input while the high-energy (Regge) behavior is not. On the rigorous side, crossing-symmetric dispersion relations have extended Roy-like equations to all energies, removing one feasibility limitation of the dual; and the direct sequel of the source paper implemented the fixed-$t$ dual with nonlinear unitarity up to a finite spin cutoff alongside high-precision primal amplitudes, finding that a gap remains and interpreting it as the difference between axiomatic and Landau analyticity rather than as a defect of the unitarity treatment - an interpretation supported by numerics but not proven. Whether the gap can be closed, and which part of it is irreducible, is the open problem.

Problem Statement

For the elastic $2 \to 2$ scattering of identical massive scalars in $d = 3+1$ dimensions, construct a fully rigorous dual formulation of the S-matrix bootstrap - an exclusion-bound functional derived by weak duality from analyticity, crossing and unitarity, resting only on rigorously established analyticity - that enforces the exact nonlinear two-particle unitarity equation $2\,\mathrm{Im}\, f_\ell(s) \ge \rho^2(s)\,|f_\ell(s)|^2$ (with $\rho^2(s) = \sqrt{1 - 4m^2/s}$ the two-body phase-space factor; equivalently, bounds on the real parts of the partial waves through the full unitarity relation) for all partial waves $\ell$ and over the full range of physical energies $s \ge 4m^2$, rather than (i) only the linearized constraint $0 \le \mathrm{Im}\, f_\ell \le 2/\rho^2$, (ii) a finite spin cutoff, or (iii) the limited energy window $4m^2 \le s \le 12m^2$ of the standard fixed-$t$ Roy equations. The formulation may optionally also admit the EFT-cutoff variant of the source: dual bounds on $c_2$ at a cutoff $\Lambda$ combined with infrared constraints on the absorptive part (the M1 arc constraints or the M2 spectral constraints), where linearized-unitarity duals currently miss the primal bound by an order of magnitude ($c_2 \Lambda^4 \le 138.8$ versus $\lesssim 8$ at $\Lambda^2 = 64m^2$ under M2). Treating the EFT-cutoff variant is optional; if it is treated, either the M1 or the M2 constraint may be used.

Using this dual, determine whether the optimal dual exclusion bounds on the low-energy Wilson coefficients - the boundary of the $(c_0, c_2)$ island and, when the optional EFT-cutoff variant is treated, its EFT-cutoff slices - converge to the optimal primal bounds, or whether a residual gap survives at full spin and energy coverage; if a gap survives, establish its origin: is it an irreducible consequence of the dual's weaker (axiomatic, Martin-Lehmann) analyticity domain versus the primal's maximal (Landau) analyticity?

Two companion questions from the same source: incorporate the correct support of the double discontinuity - the Karplus region bounded, in $m = 1$ units, by the hyperbolas $t = 4s/(s-16)$ and $t = 16s/(s-4)$, in place of the full square $[4m^2, \infty)^2$ assumed by standard ansatze - into the analytic structure of either the dual or the primal, and determine which observables (low-energy Wilson coefficients, Regge intercepts - the exponents governing the amplitude's high-energy growth - and high-energy behavior, total cross sections, resonance spectra) are sensitive to the precise double-discontinuity support.

How these parts combine into a resolution is fixed as follows. The exact-unitarity dual construction and the convergence/origin determination are jointly required: constructing the dual without deciding whether its optimal bounds converge to the primal bounds on the low-energy Wilson coefficients, and without establishing the origin of any gap that survives, leaves the task unresolved. The two companion questions - incorporating the Karplus double-discontinuity support into the dual or the primal, and identifying the observables sensitive to that support - are also required parts of the resolution rather than optional side-tracks.

References

1. Joan Elias Miró, Andrea Guerrieri, Mehmet Asım Gümüş, "Bridging Positivity and S-Matrix Bootstrap Bounds", JHEP 05 (2023) 001, DOI 10.1007/JHEP05(2023)001, arXiv:2210.01502, https://arxiv.org/abs/2210.01502
2. Andrea Guerrieri, Amit Sever, "Rigorous Bounds on the Analytic S Matrix", Phys. Rev. Lett. 127 (2021) 251601, DOI 10.1103/PhysRevLett.127.251601, arXiv:2106.10257, https://arxiv.org/abs/2106.10257
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