Asymptotic nature of the OPE/HQE expansions

Determine whether the operator product expansion for the e+e−→hadrons vacuum correlator Π(Q²) and the heavy-quark expansion for the inclusive semileptonic B→X_cℓν decay rate are only asymptotic (non-convergent) series whose coefficients eventually exhibit factorial growth, as conjectured in prior work.

Background

The paper reviews the operator product expansion (OPE) for Π(Q²) in e+e−→hadrons and the heavy-quark expansion (HQE) for inclusive semileptonic B decays. Motivated by instanton considerations and factorial growth patterns, the authors discuss the possibility that these expansions are asymptotic rather than convergent.

They reference a specific conjecture asserting asymptotic behavior and note the absence of a proof, highlighting a foundational uncertainty that directly impacts how quark–hadron duality violations should be modeled and interpreted.

References

It has been conjectured in that the expansions ee and b2csl are in fact only asymptotic expansions, meaning that starting at some power the series exhibits a factorial behaviour similar to what we have discussed in the toy example. Although we are not aware of a real proof of this assertion, the factorial growth of the number of independent matrix elements labelled by the index $k$ supports it.

— Quark-Hadron Duality Violations and Higher-Order $1/m_b$ Corrections in Inclusive Semileptonic $B$ Decays (2407.01473 - Mannel et al., 1 Jul 2024) in Section 2 (Definition of Duality Violations)