Scattering amplitudes from dispersive iterations of unitarity

TL;DR

The paper develops and numerically implements a dispersive, Mandelstam-representation-based method to construct relativistic 2→2 scattering amplitudes that satisfy analyticity, crossing and both elastic and inelastic unitarity in $d=3$ and $d=4$. By fixing a triad of inputs—the subtraction constant $\lambda$, the multi-particle production data $\eta_{MP}(s)$, and the multi-particle double spectral density $\rho_{MP}(s,t)$—the authors perform iterative unitarity updates to generate nonperturbative amplitudes at a fixed point, with amplitudes displaying features such as particle production and nontrivial Regge behavior. They classify amplitude constructions into toy-models (zero double discontinuity), 2QE (mostly elastic $J=0$), and 2PR (spin analyticity to $J=0$), and further enrich the framework with acnode-inspired multi-particle input to probe multi-particle effects. The study reveals a rich nonperturbative structure, including Landau curves, Regge behavior, and UV/IR sensitivity, while demonstrating convergence in a controlled regime and outlining open questions on extending to full multi-particle unitarity and gravity. The results offer a concrete nonperturbative realization of the S-matrix bootstrap in higher dimensions and provide a bridge between dispersive amplitudes and perturbative Feynman diagrams in scalar theories.

Abstract

We present and numerically implement a computational method to construct relativistic scattering amplitudes that obey analyticity, crossing, elastic and inelastic unitarity in three and four spacetime dimensions. The algorithm is based on the Mandelstam representation of the amplitude and iterations of unitarity. The input for the iterative procedure is given by the multi-particle double spectral density, the S-wave inelasticity, and the value of the amplitude at the crossing-symmetric point. The output, obtained at the fixed point of the iteration of unitarity, is a nonperturbative scattering amplitude. The amplitudes we obtain exhibit interesting features, such as non-zero particle production, intricate high-energy and near the two-particle threshold behavior. Scattering amplitudes obtained by initializing the iteration process with zero (or small) multi-particle input end up close to saturating the S-matrix bounds derived by other methods. There is a version of the iterative algorithm that is directly related to Feynman diagrams: it effectively re-sums infinitely many two-particle reducible planar Feynman graphs in the $φ^4$ theory, which remarkably produces a unitary nonperturbative scattering amplitude function. Finally, we discuss how the algorithm can be further refined by including multi-particle unitarity.

Figures (66)

• Figure 1: Diagrams that generate $2\to 2$ scattering amplitude in $\phi^4$ theory up to three loops. Diagrams inside a dashed frame are those for which the double spectral density is non-zero. In our iteration scheme the tree-level contact diagram (magenta) has the meaning of the fixed value of the amplitude at the crossing-symmetric point $\textcolor{violet}{\lambda} \equiv T({4 m^2 \over 3}, {4 m^2 \over 3})$. The black diagrams are then generated by iterations of two-particle unitarity and crossing-symmetric Mandelstam representation. The "frog" diagram (green) describes multi-particle production in the S-wave, which serves as an input for the algorithm $\textcolor{OliveGreen}{\eta_{MP}(s)}$. The "open envelope" diagram (blue) is the first one that contributes to the multi-particle double spectral density $\textcolor{blue}{\rho_{MP}(s,t)}$ which we also take as an input. In this way the iterative algorithm is specified by the triad $(\textcolor{violet}{\lambda}, \textcolor{OliveGreen}{\eta_{MP}(s)}, \textcolor{blue}{\rho_{MP}(s,t)})$. Given multi-particle data, the nonperturbative amplitude is effectively generated by iterations of two-particle unitarity gluing and dispersive crossing-symmetrization via the Mandelstam representation. Importantly, imposing multi-particle unitarity would lead to extra unitarity relations capturing both the frog and the open envelope diagrams.
• Figure 2: Domain of support of the double spectral density $\rho(s,t)$ in a theory with $\mathbb{Z}_2$ symmetry ($\phi^4$-type interaction) with $s_{\text{MP}}=16m^2$. The Landau curves which separate the region $\rho(s,t)=0$ from $\rho(s,t) \neq 0$ are given in \ref{['eq:LCleading']}.
• Figure 3: Typical integration domain of the Mandelstam equation \ref{['eq:mandelstam-eqn']}. The integration range for the complexified angles is $\infty > \eta',\eta" \geq 1+{4 m^2 \over s-4m^2}$ and $\eta_+ \leq z$, where recall that $\eta_+ = \eta' \eta" + \sqrt{(\eta'^2-1)(\eta"^2-1)}$.
• Figure 4: The region of the nontrivial support of the double spectral density in the $x,y$ variables. It is given in terms of the location of the leading Landau curves \ref{['eq:LC-def']}.
• Figure 5: Linear interpolation of a slowly decaying function, $-1/\log(x)$ in this example. The first grid point is at $10^{-9}$, and then the interpolating function sharply connects to zero. The dispersive integrals used in the iteration process have $1/s$ kernels which suppresses the integration at the cutoff scale and renders this effect innocuous, if the cutoff is high enough. Note also that the garland-shape of the piecewise-linear interpolating function is due to the log-log scale.