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

We construct a QCD coupling $\mathcal{A}$ \( \mathcal{A} \) (Q²) in the Effective Charge (ECH) scheme of the canonical part d(Q²) of the (inelastic) polarised Bjorken Sum Rule (BSR) ${\overline{\mathrm{\Gamma }}}_1^{\mathrm{p}-\mathrm{n}}$ \( {\overline{\Gamma}}_1^{\textrm{p}-\textrm{n}} \) (Q²). In the perturbative domain, the coupling $\mathcal{A}$ \( \mathcal{A} \) (Q²) practically coincides with the perturbative coupling a(Q²) [≡ α_s(Q²)/π] in the four-loop ECH renormalisation scheme. In the deep infrared (IR) regime, $\mathcal{A}$ \( \mathcal{A} \) (Q²) behaves as suggested by the Holographic Light-Front QCD up to the second derivative. Furthermore, in contrast to its perturbative counterpart a(Q²), the coupling $\mathcal{A}$ \( \mathcal{A} \) (Q²) is holomorphic in the entire complex Q²-plane with the exception of the negative semiaxis, reflecting the holomorphic properties of the BSR observable d(Q²) [or: ${\overline{\mathrm{\Gamma }}}_1^{\mathrm{p}-\mathrm{n}}$ \( {\overline{\Gamma}}_1^{\textrm{p}-\textrm{n}} \) (Q²)] 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 ${\overline{\mathrm{\Gamma }}}_1^{\mathrm{p}-\mathrm{n}}$ \( {\overline{\Gamma}}_1^{\textrm{p}-\textrm{n}} \) (Q²) in the entire N_f = 3 regime 0 < Q² ≲ 5 GeV².