Establish whether the scheme-dependent shift is fully explained by scale setting

Determine whether the observed shift in the central value of the long‑distance isovector window contribution to the hadronic vacuum polarization (computed when adopting the BMW20 scheme using w0 to set the scale instead of the fπ/fK scheme) is entirely explained by the different choice of scale‑setting quantity.

Background

The authors compare results obtained in their preferred fπ/fK scale‑setting scheme with those converted to the BMW20 scheme, which uses w0 determined via the Ω‑baryon mass. Using w0 leads to larger higher‑order cutoff effects and a shift in the central value of the long‑distance isovector contribution.

Understanding whether this shift is purely a scale‑setting effect is essential for consistent cross‑scheme comparisons and for combining results across collaborations employing different hadronic schemes.