Rational basis completion for higher sectors containing elliptic subsectors

Determine how to construct and control off-diagonal differential-equation blocks describing higher sectors that contain an elliptic sector among their subsectors, including the rational subsector shifts needed before introducing monodromy-dependent transformations.

Background

The analysis of off-diagonal differential-equation blocks focuses on the dependence of an elliptic sector on its proper subsectors. A distinct situation occurs when a higher sector depends on an elliptic sector that appears among its subsectors.

The paper explicitly excludes this latter configuration from its analysis. Extending the rational-basis construction and the restricted epsilon dependence to such higher-sector couplings is therefore left unresolved.

References

In this section, we consider only the former case and leave the latter for future work.

— How to choose a good rational basis for elliptic Feynman integrals?  (2609.08875 - Chaubey et al., 8 Sep 2026) in Section 4, opening paragraph of “Off-diagonal differential equation blocks”

Finding a better motivated criterion for fixing this freedom is left for future work.

— How to choose a good rational basis for elliptic Feynman integrals?  (2609.08875 - Chaubey et al., 8 Sep 2026) in Section 4, near the end of “Off-diagonal differential equation blocks”; Conclusions and outlook

Much remains to be understood. Most urgently, it is important to clarify to what extent the residual freedom in the rational off-diagonal blocks is significant and whether there is a more intrinsic principle for fixing it.

— How to choose a good rational basis for elliptic Feynman integrals?  (2609.08875 - Chaubey et al., 8 Sep 2026) in Section 5, “Conclusions and outlook”