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A series of \emph{two-particle-recursively-reducible} (2PRR) graphs that impose two-particle unitarity. Each of the graphs can be reduced to a single vertex by performing two-particle cuts. Each graph comes with its crossing images. We call the amplitudes defined via this series \emph{2PRR scattering amplitudes}.

A series of \emph{two-particle-recursively-reducible} (2PRR) graphs that impose two-particle unitarity. Each of the graphs can be reduced to a single vertex by performing two-particle cuts. Each graph comes with its crossing images. We call the amplitudes defined via this series \emph{2PRR scattering amplitudes}.

We characterize the space of amplitudes using \emph{the multi-particle data}. In perturbation theory, it is defined by the sum of all two-particle-irreducible graphs.

Structure of the double discontinuity $\rho(s,t)$ in the $(s,t)$-plane. {\color{red}\textbf{Red}:} leading $s$-channel Landau curve (eq.\eqref{eq:LeadingLC-s}), {\color{blue} \textbf{Blue}:} leading $t$-channel Landau curve (Eq.~\eqref{eq:LeadingLC-t}) \textbf{Black}: conjectured leading Landau curve for the onset of the inelastic contribution $\rho_\MP$~\cite{Correia:2021etg} (see Eq.~\eqref{eq:PlanarCrossLandauCurve}). The purple region indicates the domain where both elastic contributions overlap. In the $(x=4/s,y=4/t)$-plane, the red and blue LC become straight lines, see e.g.\cite[Fig.~4]{Tourkine:2023xtu}.

The Mahoux--Martin region in the $(s,t)$ plane, shown as a hatched area. In this region the double spectral function $\rho(s,t)$ is strictly positive.

Integration domain of the Mandelstam equation and choice of pair $(\eta_1,\eta_2)$.

Example of a sparse, constant-$z$ grid $\mathcal{G}_\mathrm{DD}$, shown as blue dots. The grid consists of $26$ lines of constant complex scattering angle $z=\cos\theta$, each sampled at $26$ points (with an accumulation near $x=0$ and $y=0$, not visible by eye on the figure). The red curve denotes the $s$-channel–leaning Landau curve, which delimits the support of $\rho_\mathrm{NN}$ (orange and gray region). The gray region shows the domain where the multi-particle data $\rho_\mathrm{MP}$ is supported; it is bounded by the planar cross-graph Landau curve shown in black.

: Single spectral density $\rho(x)$ of the 2PRR amplitude at $c_0=40\pi$ as a function of $x=4/s$. Right panel zooms in the large energy region $s > 10^4$ which can be well fitted by the logarithmic decay given by $\frac{32\pi^3}{9} \frac{1}{\log(s)^2} \left( 1 + C \, \frac{\log(\log(s))}{\log(s)} \right)$ and the best-fit is given by $C \simeq 16.72$.

: Double spectral density of the 2PRR amplitude at $c_0 = 40\pi$ in the $(x,y)=(4/s,4/t)$ plane, shown both as a 3D surface plot (left) and a density plot (right). In the right panel, leading Landau curves~\eqref{eq:LeadingLC} delineate the support of the double discontinuity, and are indicated in gray.

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: Fixed-$t$ slices of the double discontinuity $\rho(s,t)$ of the 2PRR amplitude at $c_0=40\pi$ on log-linear scale. The right panel provides a zoomed-in view of the $y$-range of the left panel, highlighting the sign changes of the $t=20$ slice (red) around $s\sim10^5$. In the limit $s\to\infty$, all fixed-$t$ slices slowly decay.

: Fixed-$t$ slices of the elastic piece $\rho_{\mathrm{el}}(s,t)$ in the double discontinuity at $c_0=40\pi$, as defined in \eqref{eq:mandelstam-eqn}, on log-log scale. All slices exhibit a universal $1/s$ decay in the large-$s$ regime. The same $1/s$ decay is observed for all values of the coupling $c_0$ (for $c_0=40\pi$ the fit gives $C^\mathrm{st}=20$).

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Spin-zero partial wave for the 2PRR amplitude at $c_0=40\pi$. It exhibits particle production beyond four-particle threshold $x<0.25$ and is transparent as $x\to0$.

Spin-zero partial wave for the 2PRR amplitude at $c_0=40\pi$. It exhibits particle production beyond four-particle threshold $x<0.25$ and is transparent as $x\to0$.

\underline{Left:} Modulus squared of the spin-zero partial wave for the 2PRR amplitude at $c_0 = 40\pi$. \underline{Right:} Magnified plot: we estimate a posteriori the numerical error based on the size of the oscillations.

\underline{Left:} Modulus squared of the spin-zero partial wave for the 2PRR amplitude at $c_0 = 40\pi$. \underline{Right:} Magnified plot: we estimate a posteriori the numerical error based on the size of the oscillations.

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: \underline{Left:} Modulus squared of higher partial waves ($|S_J|^2$ with $J = 2, 4, \dots, 10$) of the 2PRR amplitude at $c_0 = 40\pi$. \underline{Right:} Maximal inelasticity for each spin $J$, i.e.\ $1 - \min_s |S_J|^2$, plotted in log-log scale as a function of $J$. The plot highlights the kinematic decay of inelasticity at large spin. : Rescaled inelasticities $\tilde{\eta}_J$ for $J = 2, \dots, 10$ (with $|S_J|^2 = 1 - \eta_J$) as a function of $x$ in log-linear scale, defined in Eq.~\eqref{eq:rescaled_J}, plotted alongside the spin-zero inelasticity as a function of $x = 4/s$.

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Regge behavior of the real (left panel) and imaginary (right panel) parts of spin-zero partial wave ($\im f_0$ and $\re f_0$) for different high-energy cutoffs, shown for the 2PRR amplitude at $c_0 = 40\pi$. Increasing the cutoff leads to a smooth extension of the slow Regge decay, indicating that the asymptotic regime has been reached. Using this criterion, we find asymptotic behavior for all amplitudes with $c_0 < 44\pi$.

Regge behavior of the real (left panel) and imaginary (right panel) parts of spin-zero partial wave ($\im f_0$ and $\re f_0$) for different high-energy cutoffs, shown for the 2PRR amplitude at $c_0 = 40\pi$. Increasing the cutoff leads to a smooth extension of the slow Regge decay, indicating that the asymptotic regime has been reached. Using this criterion, we find asymptotic behavior for all amplitudes with $c_0 < 44\pi$.

: Regge behavior of the 2PRR scattering amplitude $|T(s,-1)|$ at the coupling $c_0=40\pi$ as a function of $s$. The orange curve corresponds to a fitting function of the form $T_\infty + k/\log^{\gamma}(s)$. For $t_0=-1$, we find that $T_\infty \simeq 7.5$, $k \simeq 240$, and $\gamma \simeq 1.4$. We also probed that at a different coupling, $c_0=20\pi$ for instance, the constant at infinity becomes $T_\infty = 4.5$.

: Imaginary part of the amplitude for the 2PRR amplitude at $c_0=40\pi$. It falls off logarithmically which is well-fitted by the black dashed line.

: Fixed-angle high-energy behavior of the 2PRR scattering amplitude at $c_0=40\pi$. The modulus $|T(s,t(\theta))|$ is shown as a function $s$ for several fixed scattering angles~$\theta$. The black dashed curve corresponds to a fit of the large-$s$ behavior at $\theta=0.5\pi$ of the form~\eqref{eq:amp_log_decay}, with best-fit coefficient $C^\text{st} \simeq 63$. Note the transition from Regge to fixed-angle regime at energies $s \sim \theta^{-2}$.

: Single (left) and double (right) spectral densities at fixed-$t=100$ for 2PRR amplitudes at all available couplings $c_0$. Their sizes grow with increasing $c_0$. The growth rate as a function of the coupling can be fitted by a power-law $(c_0)^\gamma$ with an exponent $2 \leq \gamma \leq 3$. {The bump around $x=0.2$ coincide with the $t$-channel leading Landau curve. At $t=100$ both $\rho_{\rm el}(s,t)$ and $\rho_{\rm el}(s,t)$ contribute for $x\leq0.24$, but only $\rho_{\rm el}(s,t)$ remains for $x>0.24$.}

: Real part of the spin-zero partial wave coefficient, $\re f_0(s)$, shown in log–log scale for couplings $c_0$ ranging from $4\pi$ to $56\pi$. Reggeized solutions are displayed as solid curves and color-coded, while non-Reggeized amplitudes are shown in gray with different dashed styles. For the largest three couplings, Reggeization could not be achieved; these curves were therefore obtained using the lowest energy cutoff, $s_{\max}=10^8\,m^2$. Unlike the Reggeized solutions, the real part $\re f_0(s)$ for these non-Reggeized amplitudes becomes negative at large energies. The origin of this behavior, physical or numerical, cannot be firmly established at present.

Double discontinuity $\rho(s,t)$ at fixed momentum transfer $t$ as a function of $x=4/s$, shown for several values of the coupling (colored solid curves). The orange dashed line indicates the position of the leading $t$-channel Landau curve, while the purple dashed line marks the location of the subleading Landau curve at this value of $t$. A mild change in convexity of $\rho(s,t)$ is visible near the latter for $t=200,\,400$ and $500$, although the effect is subtle.

: $|S_0|$ and $|S_2|$ at different $c_0$ couplings ranging from $4\pi$ to $56\pi$. The inelasticity grows with the coupling, this observation remains valid for higher spin partial waves. The Reggeized amplitudes ($c_0\leq44\pi$) are displayed as solid curves, while non-Reggeized amplitudes are shown in gray-dashed curves.

: Maximum inelasticity $\max_s\eta_0(s)$ as a function of the coupling $c_0$, shown up to $c_0=56\pi$. The red dotted line indicates the maximal coupling~\cite{Paulos:2017fhb,Guerrieri:2021tak}. The blue curve corresponds to a power-law fit of the form $\alpha\left(\frac{c_0}{32\pi}\right)^p$, with fit parameters $\alpha=0.12$ and $p=2.5$. This fit suggests that $\max_s\eta_0(s)$ would reach unity at $\frac{c_0}{32\pi}=2.384\,$.

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: \underline{Left:} Quasi-elastic amplitude at $c_0 = 28\pi$, with $\eta_0 = 0$ and $\rho_{\mathrm{MP}} = 0$. \underline{Right:} Amplitude at $c_0 = 28\pi$ with $\eta_0 = 0$. $\rho_{\mathrm{MP}}$ is turned on, and the constraint $|S_2|^2 = 1$ is enforced during training through the loss function. The same vertical scale is used in both panels; a zoomed-in version of the second plot is also provided. :

: : \underline{Left:} 3D plot of the double discontinuity of the quasi-elastic amplitude at $c_0 = 28\pi$. \underline{Right:} Amplitude at $c_0=28\pi$ with $\eta_0=0$ where $\rho_\mathrm{MP}$ has been used to set $|S_2|^2=1$ as visible on the right plot of Fig.~\ref{fig:Aks_higher_PW}. The boundary of the support of $\rho_\mathrm{MP}$ is shown in black.

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Bounding constants for Aks screening.

Spin-zero partial wave at $c_0=40\pi$ from the SDPB solution fed with inelasticity data up to $J_\text{min}=10$ obtained from Fig.~\ref{fig:SJ_2PRR}. Shades with increasing opacities correspond to $N_\text{max}=8,9,10,11,12$. Both panels are to be compared with Fig.~\ref{fig:S0_2PRR} -- note the agreement up to $x\simeq10^{-3}$ (or $s\simeq 10^3$) and lack of convergence on the SDPB side for $s>10^3$. The right-most dip of $|S_0(x)|^2$ corresponds to the physical particle production in the $S$-wave, while the subsequent dips are transient.

Spin-zero partial wave at $c_0=40\pi$ from the SDPB solution fed with inelasticity data up to $J_\text{min}=10$ obtained from Fig.~\ref{fig:SJ_2PRR}. Shades with increasing opacities correspond to $N_\text{max}=8,9,10,11,12$. Both panels are to be compared with Fig.~\ref{fig:S0_2PRR} -- note the agreement up to $x\simeq10^{-3}$ (or $s\simeq 10^3$) and lack of convergence on the SDPB side for $s>10^3$. The right-most dip of $|S_0(x)|^2$ corresponds to the physical particle production in the $S$-wave, while the subsequent dips are transient.

: Comparison between the forward amplitudes at $c_0=40\pi$ given by SDPB (solid) and the NN (dashed) solutions as a function of $N_\mmax$ at fixed $J_\text{min}=10$. The farthest wavelet center in the ansatz are $\sigma \in \{ 1191, 1807, 2637, 3726, 5122\}$ respectively for each $N_\mmax$.

: Comparison between the forward amplitudes given by SDPB (solid) and the NN (dashed) solutions as a function of $J_\text{min}$ at fixed $N_\mmax=10$.

Iterated $s$-channel two-particle unitarity graph (the “Aks graph”) with a $2n$-particle cut in the $t$-channel.

Bose symmetry of the four-particle final state relates the 2PRR Aks graph to the MP non-planar box graph.

An example of an MP graph that follows from four-particle unitarity, where the 2PRR amplitude is a sub-process of a larger diagram.