Establish the regularity and scale behavior of parton distribution functions

Establish whether renormalized parton distribution functions are smooth, or at least sufficiently regular, in the momentum-fraction variable and determine the extent of their regularity in the momentum scale, including whether flavor-number-scheme matching induces genuine discontinuities in the scale dependence.

Background

The proposed discretization and reconstruction method assumes that proton parton distribution functions have sufficient regularity, particularly in the momentum-fraction variable, for numerical quadrature and inverse reconstruction. The paper formulates this assumption as an explicit conjecture rather than deriving it from the operator definition of the PDFs.

The appendix further raises the unresolved issue of whether discontinuities in the scale dependence at flavor-number-scheme matching points are physical or artifacts of fixed-order perturbation theory and scheme matching. Resolving this would clarify the appropriate functional space for the inverse problem.

References

We conjecture the following: Parton distribution functions $f_{q,h}(x,Q): [0,1] \times \mathbb{R}_+ \mapsto \mathbb{R}$ are smooth, or at least quite regular, functions of $x$, and perhaps also to some degree in $Q$. Sufficient regularity of the PDFs as functions of $x$, like smoothness or at least continuous differentiability, is required from the presumed solutions by the method constructed in Sec.~\ref{sec:discretization}.

— Mathematical inverse problem for the world data inference of the parton distribution functions of the proton  (2609.11896 - Hänninen, 10 Sep 2026) in Appendix, Section “Smoothness of the parton distribution functions”

Whether this type of a discontinuity of the PDFs would appear in an ``all orders consistent theory of QCD global analysis'' is not quite clear. This conjecture is left for future work.

— Mathematical inverse problem for the world data inference of the parton distribution functions of the proton  (2609.11896 - Hänninen, 10 Sep 2026) in Appendix, Section “Smoothness of the parton distribution functions”