Physical realizability of two-particle quasi-elastic (2QE) amplitudes under full multi-particle unitarity

The dispersive iteration program of Tourkine and Zhiboedov (JHEP 11 (2023) 005) constructs nonperturbative two-to-two scattering amplitudes of identical massive scalars in $d=3,4$ as fixed points of an iteration of two-particle unitarity in the Mandelstam representation, initialized by the triad $(\lambda,\eta_{\rm MP}(s),\rho_{\rm MP}(s,t))$: the amplitude value at the crossing-symmetric point, the S-wave multi-particle inelasticity, and the multi-particle double spectral density. Initializing with $(\lambda,0,0)$ defines the two-particle quasi-elastic (2QE) amplitudes: a one-parameter family with a purely elastic S-wave ($|S_0(s)|=1$ for all $s\ge 4m^2$), no bound states, and small nonzero particle production in higher partial waves, lying very close to the boundary of the region allowed by modern S-matrix bootstrap bounds. By construction these amplitudes satisfy crossing, Mandelstam analyticity, and two-particle elastic and inelastic unitarity; what they do not satisfy by construction is multi-particle unitarity — the unitarity of the full S-matrix on states with three or more particles. The open question is whether the 2QE amplitudes are physically realizable: either they can be excluded by multi-particle unitarity (as the authors conjecture, e.g. because Bose symmetry of multi-particle final states forces graphs whose $J=0$ parts are projected out in the 2QE scheme), or a consistent multi-particle completion exists. Adjacent rigorous results — the Aks theorem that scattering implies production in $d>2$, the Landau-curve analysis of multi-particle singularities, and recent exponential lower bounds on the multi-particle data needed to screen low-spin inelasticity — do not decide the question in either direction.

Background

The analytic S-matrix program aims to determine relativistic scattering amplitudes directly from their structural principles — unitarity, crossing symmetry, and analyticity — rather than from a Lagrangian. Consider the two-to-two scattering amplitude $T(s,t)$ of identical scalar particles of mass $m$ in $d$ spacetime dimensions, where $s$ and $t$ are the Mandelstam invariants ($s$ is the squared center-of-mass energy, $t$ the squared momentum transfer). The Mandelstam representation writes $T(s,t)$ as a double dispersion relation in $s$ and $t$ with a single-discontinuity spectral density $\rho(s)$ and a double spectral density $\rho(s,t)$; Tourkine and Zhiboedov use a once-subtracted version whose subtraction constant is the amplitude value at the crossing-symmetric point, $\lambda=T(4m^2/3,4m^2/3)$. Decomposing the amplitude into partial waves (the angular-momentum expansion of the amplitude; $S_J(s)$ denotes the partial-wave S-matrix at spin $J$), unitarity of the two-to-two sector splits into elastic unitarity, which fixes the imaginary part of $T$ in terms of $|T|^2$ when only two-particle intermediate states contribute, and inelastic unitarity, $1-|S_J(s)|^2=\eta_J(s)\ge 0$, which measures particle production: the leakage of probability into final states with more than two particles. In this framework the double spectral density decomposes as $\rho(s,t)=\rho_{\rm el}(s,t)+\rho_{\rm el}(t,s)+\rho_{\rm MP}(s,t)$, where $\rho_{\rm el}$ is generated by elastic unitarity through the Mandelstam equation (a nonlinear integral equation expressing two-particle unitarity in all partial waves with $\mathrm{Re}\,J>0$), and the multi-particle double spectral density $\rho_{\rm MP}(s,t)$ encodes physics that is multi-particle in all channels; multi-particle unitarity forces its support to lie above the leading multi-particle Landau curve — the boundary, determined by the Landau equations of Feynman graphs such as the open-envelope graph, of the region of the $s$-$t$ plane where multi-particle singularities appear. The S-wave ($J=0$) inelasticity is separately parametrized by $\eta_{\rm MP}(s)$.

Reviving Atkinson's old idea of building nonperturbative amplitudes as fixed points of a dispersive iteration of two-particle unitarity, Tourkine and Zhiboedov implemented the iteration numerically in $d=3$ and $d=4$. The iteration is initialized by the triad $(\lambda,\eta_{\rm MP}(s),\rho_{\rm MP}(s,t))$: the multi-particle data are held fixed as inputs, and the remaining spectral densities are computed at the fixed point. Initializing with the triad $(\lambda,0,0)$ defines the two-particle quasi-elastic (2QE) amplitudes: the S-wave is purely elastic, $|S_0(s)|=1$ for all $s\ge 4m^2$, there are no bound-state poles below the two-particle threshold, and the amplitude exhibits small but nonzero particle production in $J>0$ partial waves. This evades the theorem of Aks (1965), which states that any nontrivial crossing-symmetric unitary amplitude in $d>2$ must exhibit particle production — fully elastic scattering is possible only in $d=2$ — because 2QE amplitudes do carry production in their higher partial waves. The 2QE amplitudes are the most extremal among the amplitudes constructed in that work and lie very close to the boundary of the region allowed by the modern numerical S-matrix bootstrap (the program, initiated by Paulos, Penedones, Toledo, van Rees and Vieira and refined by Chen, Fitzpatrick and Karateev and by Elias Miró, Guerrieri and Gümüş, that bounds low-energy couplings of unitary amplitudes via semidefinite programming). They therefore probe the physical status of the bootstrap boundary itself, whose extremal amplitudes appear nearly elastic.

Diagrammatically, setting $\eta_{\rm MP}=0$ projects out the $J=0$ part of every graph with a multi-particle cut (the Aks graph and the frog graph), which suggests an inherent non-locality; nevertheless this is compatible with all constraints imposed in that work — two-particle elastic and inelastic unitarity, crossing, and Mandelstam analyticity. What is not imposed is multi-particle unitarity: the unitarity of the full S-matrix on states with three or more particles. Its nonperturbative form couples the two-to-two amplitude analytically continued to its second sheet (through the elastic unitarity cut) to multi-particle phase-space integrals, schematically $\rho_{\rm MP}(s,t)\sim K\,(T^{(2)}_{2\to2})^2(T^{(2)\,*}_{2\to2})^2$, together with Bose-symmetry constraints on multi-particle final states. Whether the 2QE amplitudes survive these additional constraints is the open question.

Problem Statement

Let $T_\lambda(s,t)$ denote the two-particle quasi-elastic (2QE) amplitude of identical scalar particles of mass $m$ constructed by the dispersive iteration of two-particle unitarity of Tourkine–Zhiboedov with input triad $(\lambda,0,0)$: vanishing S-wave multi-particle inelasticity $\eta_{\rm MP}(s)=0$ and vanishing multi-particle double spectral density $\rho_{\rm MP}(s,t)=0$, with one subtraction fixed by $\lambda=T(4m^2/3,4m^2/3)$, no bound-state poles below the two-particle threshold, in $d=3$ and $d=4$ spacetime dimensions, over the range of couplings where the iteration converges. By construction these amplitudes satisfy crossing symmetry, Mandelstam analyticity with one subtraction, and two-particle elastic and inelastic unitarity for every partial wave, with a purely elastic S-wave $|S_0(s)|=1$ for all $s\ge 4m^2$ and nonzero particle production in $J>0$ partial waves.

Determine whether the 2QE amplitudes are physically realizable under full multi-particle unitarity. Concretely, settle the question in one of two ways:

(a) Prove exclusion: derive from the unitarity of the full multi-particle S-matrix — together with the standard analyticity and crossing assumptions the 2QE amplitudes already obey — a contradiction with the 2QE fixed point, identifying the concrete relations violated (for example, generalized unitarity relations forcing $\rho_{\rm MP}(s,t)\ne 0$ above the leading multi-particle Landau curve, Bose-symmetry constraints on multi-particle final states, or second-sheet relations of the type discussed in section 9.2 of Tourkine–Zhiboedov, $\rho_{\rm MP}(s,t)\sim K_4\,(T_{2\to2}^{(2)})^2\,(T_{2\to2}^{(2)\,*})^2$);

(b) Construct a consistent completion: with $T_\lambda(s,t)$ held fixed as the given 2QE amplitude (not re-obtained by re-choosing the inputs of the two-to-two iteration), specify multi-particle data $(\eta_{\rm MP}(s),\rho_{\rm MP}(s,t))$ and higher-point amplitudes satisfying all multi-particle unitarity relations — including the second-sheet relations that constrain the analytically continued two-to-two amplitude itself — whose two-to-two restriction reproduces $T_\lambda(s,t)$, thereby establishing that 2QE amplitudes are not excluded by unitarity-based constraints.

The question is posed for the class of 2QE amplitudes as constructed, in both $d=3$ and $d=4$ over the full range of $\lambda$ where they exist. In either branch the axiom set with respect to which physical realizability is asserted or refuted must be stated explicitly. Partial results — bounds on how multi-particle data can screen low-spin production, or diagrammatic expectations that certain graphs are forced — do not by themselves resolve the question.

References

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