Inclusive rates from smeared spectral densities in the two-dimensional O(3) non-linear $\sigma$-model

This work employs the spectral reconstruction approach of Ref. [1] to determine an inclusive rate in the $1+1$ dimensional O(3) non-linear $\sigma$-model, analogous to the QCD part of ${e}^+{e}^- \rightarrow \rm {hadrons}$. The Euclidean two-point correlation function of the conserved current $j$ is computed using Monte Carlo lattice field theory simulations for a variety of spacetime volumes and lattice spacings. The spectral density of this correlator is related to the inclusive rate for $j \rightarrow {\rm X}$ in which all final states produced by the external current are summed. The ill-posed inverse problem of determining the spectral density from the correlation function is made tractable through the determination of smeared spectral densities in which the desired density is convolved with a set of known smearing kernels of finite width $\epsilon$. The smooth energy dependence of the underlying spectral density enables a controlled $\epsilon \to 0$ extrapolation in the inelastic region, yielding the real-time inclusive rate without reference to individual finite-volume energies or matrix elements. Systematic uncertainties due cutoff effects and residual finite-volume effects are estimated and taken into account in the final error budget. After taking the continuum limit, the results are consistent with the known analytic rate to within the combined statistical and systematic errors. Above energies where 20-particle states contribute, the overall precision is sufficient to discern the four-particle contribution to the spectral density.