Bounding scattering of charged particles in 1+1 dimensions

TL;DR

The paper develops a framework to bound 2→2 scattering of charged particles in 1+1 dimensions with global symmetries, focusing on $O(N)$ and $U(N)$ vector representations. By combining analyticity, crossing, and unitarity, it formulates an optimization problem for S-matrix residues and solves it numerically, revealing bounds that are saturated by no-particle-production S-matrices and by known integrable models. It maps S-matrix data onto symmetry channels and analyzes both real and complex symmetry groups, connecting to sine-Gordon and Gross-Neveu theories in specific cases. The results provide a practical bootstrap-like blueprint for constraining couplings in 1+1D QFTs and offer insight into how integrable S-matrices emerge as extremal solutions, with implications for higher-dimensional generalizations.

Abstract

We obtain general bounds on scattering processes involving charged particles in 1+1 spacetime dimensions. After a general analysis we derive mostly numerical bounds on couplings in theories with $O(N)$ and $U(N)$ global symmetries. The bounds are consistently saturated by $S$-matrices without particle production, and in many cases by known integrable $S$-matrices. Our work provides a blueprint for a similar analysis in higher dimensions.

Figures (19)

• Figure 1: $y$ variable
• Figure 2: Bound of a coupling to lightest pseudoscalar particle in the presence of other scalar or pseudoscalar states, and comparison with the sine-Gordon model, which saturates the bound as soon as two particles are present. Further explanations in the main text. The x-axis is the pole of the lightest bound state, and the y-axis is the corresponding residue bounds. We find there are is a sharp turn at the point when new bound states are created in the Sine-Gordon model.
• Figure 3: Comparision of the numerical $S$-matrix with the analytical sine-Gordon matrix with $\frac{\pi}{\xi}=2.5$. We have subtracted the pole structure in $S(s)$ for a cleaner plot.
• Figure 4: Comparison of bounds when we maximize the pseudoscalar or scalar couplings. The orange line is the curve for the pseudoscalar coupling when we maximize the scalar coupling. The blue line is the curve for the pseudoscalar coupling when we maximize the pseudoscalar coupling. By this figure we can verify Eq. \ref{['constraint']}
• Figure 5: Numerical bounds and the Gross-Neveu model points. This is a bound for the coupling to an antisymmetric tensor in the presence of a scalar bound state of the same mass.