Nonperturbative Anomalous Thresholds

TL;DR

This work addresses anomalous thresholds—nonperturbative branch points that appear below normal thresholds in heavy-particle scattering—by deriving a nonperturbative framework grounded in extended unitarity and mass analyticity. In $d=2$, it yields explicit results: a simple pole at $s=a$ with a residue tied to the nonperturbative $m\,m\rightarrow m\,m$ amplitude, and a three-step construction that recovers the triangle diagram and passes a stringent check against the $E_8$ integrable model. The analysis is generalized to $d>2$, where the anomalous threshold manifests as a left-hand cut whose discontinuity matches Mandelstam's results (e.g., in $d=4$, $J=0$), and a nonperturbative double-pole formula for $M M \to M M$ is obtained in two dimensions. A nonperturbative test with the $E_8$ model reveals that overlapping singularities (such as a box diagram) are essential to reproduce the full double-pole structure. Overall, the paper provides a principled, nonperturbative derivation of Coleman-Thun-type poles and connects S-matrix bootstrap concepts with exact integrable-model data.

Abstract

Feynman diagrams (notably the triangle diagram) involving heavy enough particles contain branch cuts on the physical sheet - anomalous thresholds - which, unlike normal thresholds and bound-state poles, do not correspond to any asymptotic $n$-particle state. ``Who ordered that?" We show that anomalous thresholds arise as a consequence of established S-matrix principles and two reasonable assumptions: unitarity below the physical region and analyticity in the mass. We find explicit nonperturbative formulas for the anomalous threshold singularity and test them against the Coleman-Thun poles of the exactly solvable $E_8$ integrable model.

Figures (7)

• Figure 1: Complex $s$-plane for $MM \to MM$. Physical scattering occurs for $s \geq 4M^2$ where unitarity applies. Typically, unitarity is extended below to find further normal thresholds and simple poles, but not anomalous thresholds boyling1964hermitian.
• Figure 2: Left: Triangle diagram with anomalous threshold given by eq. \ref{['eq:triangle']}. Right: Representation of eq. \ref{['eq:unisolB3']}.
• Figure 3: Left: Representation of eq. \ref{['eq:Cdp']}. Right: Diagram with anomalous threshold given by eq. \ref{['eq:aboxdp']} in $d = 2$.
• Figure 4: $s'$ complex plane. In black: integration contour. In green: singularities of the integrand. Note in particular the presence of a branch cut of $\rho(s')$ for $s' \leq 4m^2$. In red: trajectory of $a$ as $M$ is increased. At $M = \sqrt{2}$, $a = 4m^2$ pinches the integration contour and forces a deformation.
• Figure 5: Graphical representation of solution to unitarity across the $2m$ cut, eqs. (7) and (21) in the main text. The sub-graphs $\alpha$, $\beta$ and $\sigma$ are $2m$-irreducible, i.e. do not contain any internal exchange of $2m$ (in the $s$-channel) meaning that $\mathrm{Disc} \,\alpha = \mathrm{Disc} \,\beta = \mathrm{Disc}\, \sigma = 0$ across this cut.