Disperon QED

TL;DR

Disperon QED introduces a dispersive framework that recasts hadronic input functions into a dispersive integral over a fictitious massive mediator (the disperon), enabling standard loop techniques to compute otherwise intractable hadronic-in-loop contributions. By combining dispersion relations, OpenLoops, a threshold subtraction, and a hybrid DET (disperon EFT) approach, the paper provides a practical path to include HVP and form-factor effects in NNLO and beyond for low-energy processes like $e^+e^-\to\pi\pi$. It develops a universal threshold counterterm to handle threshold singularities, demonstrates IR-consistent cancellations, and validates the method against full one-loop calculations, achieving speed-ups via DET for large disperson masses. The work delivers a flexible toolset for precise Monte Carlo predictions in processes with external hadrons and paves the way for applications to more complex final states and to lepton–hadron scattering, with potential resonance-sensitive improvements near thresholds.

Abstract

We present disperon QED, a method to deal with data input in loop processes in Monte Carlo codes. It relies on dispersion relations, automated tools such as OpenLoops, effective field theory methods and a threshold subtraction. We motivate this method and apply it to the process $ee\toππ$ in McMule to deal with hadronic vacuum polarisation insertions in two-loop contributions as well as the vector form factor of the pion within the form-factor scalar QED approximation. The generality of this method for more complicated processes is emphasised.

Figures (8)

• Figure 1: The three tree-level diagrams that need to be calculated for the matching to DET. The first two diagrams are associated with disperons arising from the FsQED approach. In the last one the disperon exclusively interacts with leptons which is due to the VP insertion.
• Figure 2: Single-dispersive contribution for fixed kinematics but varying the dispersion parameter $s_1$, evaluated using the DET at dimension 6 and dimension 8 as well as OpenLoops in dp and qp. The vertical black line represents the value of ${s_1} = s$ and hence the location of the threshold singularity. The other jumps are due to zero crossings. The vertical red line is the value of $s_{\rm cut}$. Exact refers to the calculation in Mathematica.
• Figure 3: Double-dispersive contribution for fixed kinematics, similar to Figure \ref{['fig:single-disp']}.
• Figure 4: Topologies contributing to the threshold singularity in $ee\to \pi \pi \gamma$.
• Figure 5: The top panel shows the distribution for $\theta_{\rm avg}$ defined in \ref{['eq:thav']} at LO and the best prediction within McMule. The other panels show the various $\delta$'s defined before. Note that the purple line corresponds to a NNNLO effect that results in a $10^{-6}$ effect relative to LO.