Exploring Inelasticity in the S-Matrix Bootstrap

Abstract

The modern S-Matrix Bootstrap provides non-perturbative bounds on low-energy aspects of scattering amplitudes, leveraging the constraints of unitarity, analyticity and crossing. Typically, the solutions saturating such bounds also saturate the unitarity constraint as much as possible, meaning that they are almost exclusively elastic. This is expected to be unphysical in $d>2$ because of Aks' theorem. We explore this issue by adding inelasticity as an additional input, both using a primal approach in general dimensions which extends the usual ansatz, and establishing a dual formulation in the 2d case. We then measure the effects on the low-energy observables where we observe stronger bounds than in the standard setup.

Figures (14)

• Figure 1: The absolute value of $S_0$ for several $N_{max}$. $\phi$ is the argument of the complex $\rho$ variable defined below, such that $s$ correspondingly increases from $4$ to infinity.
• Figure 2: Direct comparison between profile $\beta^{(\textrm{s})}$ and the analytical results
• Figure 3: S-matrix components for the inelastic profile $\beta^{(\textrm{s})}$ with $\alpha=0.5$ and the analytical results in red dashes lines.
• Figure 4: Direct comparison between the profile $\beta^{(\textrm{s})}$ and the analytical results. The bound-state mass is fixed, $m_1^2 = 3$, $N_{\textrm{max}} = 5$ and $\bar{N}_{\textrm{max}} = 2$.
• Figure 5: S-matrix components for the inelastic profile $\beta^{(\textrm{s})}$ using the new Ansatz of (\ref{['eqn:newinelasticAnsatz']}). In red are the components of the analytical solution. Here, the parameters used were $m_1^2=3$, $N_{\textrm{max}} = 10$ and $\bar{N}_{\textrm{max}} = 10$ and $\alpha = 0.6$.