Renormalon-based resummation of Bjorken polarised sum rule in holomorphic QCD

Abstract

Approximate knowledge of the renormalon structure of the Bjorken polarised sum rule (BSR) ${\bar{\Gamma }}_1^{p-n}\left(Q^2\right)$ leads to the corresponding BSR characteristic function that allows us to evaluate the leading-twist part of BSR. In our previous work [1], this evaluation (resummation) was performed using perturbative QCD (pQCD) coupling $a\left(Q^2\right)\equiv {\alpha }_s\left(Q^2\right)/\pi$ in specific renormalisation schemes. In the present paper, we continue this work, by using instead holomorphic couplings [$a\left(Q^2\right)\mapsto A\left(Q^2\right)$] that have no Landau singularities and thus require, in contrast to the pQCD case, no regularisation of the resummation formula. The $D=2$ and $D=4$ terms are included in the Operator Product Expansion (OPE) of inelastic BSR, and fits are performed to the available experimental data in a specific interval $(Q_{\min }^2,Q_{\max }^2)$ where $Q_{\max }^2=4.74{\mathrm{GeV}}^2$. We needed relatively high $Q_{\min }^2\approx 1.7{\mathrm{GeV}}^2$ in the pQCD case since the pQCD coupling $a\left(Q^2\right)$ has Landau singularities at $Q^2\lesssim 1{\mathrm{GeV}}^2$. Now, when holomorphic (AQCD) couplings $A\left(Q^2\right)$ are used, no such problems occur: for the 3δAQCD and 2δAQCD variants the preferred values are $Q_{\min }^2\approx 0.6{\mathrm{GeV}}^2$. The preferred values of ${\alpha }_s$ in general cannot be unambiguously extracted, due to large uncertainties of the experimental BSR data. At a fixed value of ${\alpha }_s^{\overline{\mathrm{MS}}}\left(M_Z^2\right)$, the values of the $D=2$ and $D=4$ residue parameters are determined in all cases, with the corresponding uncertainties.