# Explicit form of the R-ratio of electron-positron annihilation into hadrons

## TL;DR

This paper derives an explicit, all-order expression for the R-ratio in electron-positron annihilation into hadrons, accounting for complex effects in the transition from spacelike to timelike domains, aiding theoretical analyses.

## Contribution

It provides a comprehensive explicit formula for the R-ratio at arbitrary loop level, improving the accuracy of strong coupling evaluations in electron-positron annihilation studies.

## Key findings

- Derived an explicit expression for the R-ratio at all loop levels.
- Clarified the impact of higher-order pi2-terms on strong coupling calculations.
- Presented multiple methods to approximate the R-ratio and their implications.

## Abstract

The explicit expression for the R-ratio of electron-positron annihilation into hadrons, which properly accounts for all the effects due to continuation of the spacelike perturbative results into the timelike domain, is obtained at an arbitrary loop level. Several equivalent ways to derive a commonly employed approximation of the R-ratio are recapped and the impact of discarded in the latter higher-order pi2-terms on the evaluation of the strong running coupling is elucidated. The obtained results substantially facilitate the theoretical study of electron-positron annihilation into hadrons and the related strong interaction processes.