Towards unifying perturbative and Holographic Light-Front QCD via holomorphic coupling

We construct a QCD coupling $\mathcal{A}(Q^2)$ in the Effective Charge (ECH) scheme of the canonical part $d(Q^2)$ of the (inelastic) polarised Bjorken Sum Rule (BSR) ${\bar{\mathrm{\Gamma }}}_1^{\mathrm{p}-\mathrm{n}}(Q^2)$. In the perturbative domain, the coupling $\mathcal{A}(Q^2)$ practically coincides with the perturbative coupling $a(Q^2)$ [$\equiv {\alpha }_s(Q^2)\mathrm{/}\pi$] in the four-loop ECH renormalisation scheme. In the deep infrared (IR) regime, $\mathcal{A}(Q^2)$ behaves as suggested by the Holographic Light-Front QCD up to the second derivative. Furthermore, in contrast to its perturbative counterpart $a(Q^2)$, the coupling $\mathcal{A}(Q^2)$ is holomorphic in the entire complex $Q^2$-plane with the exception of the negative semiaxis, reflecting the holomorphic properties of the BSR observable $d(Q^2)$ [or: ${\bar{\mathrm{\Gamma }}}_1^{\mathrm{p}-\mathrm{n}}(Q^2)$] as dictated by the general principles of the Quantum Field Theory. It turns out that the obtained coupling, used as ECH, reproduces quite well the experimental data for ${\bar{\mathrm{\Gamma }}}_1^{\mathrm{p}-\mathrm{n}}(Q^2)$ in the entire $N_f=3$ regime $0<Q^2\lesssim 5{\mathrm{G}\mathrm{e}\mathrm{V}}^2$.