Is vertex-dressing-function analyticity forced? The phase constraint behind confinement from the QCD effective charge

Deur (2026) showed that the analytic structure of the observable QCD effective charge $\alpha_{g_1}$ — two complex-conjugate singularities at $Q^2=\pm i\Lambda_s^2$ — can be mapped onto the dressed quark and gluon propagators, whose long-distance behavior $e^{-\Lambda_s|x|/\sqrt{2}}|x|^{-5/2+d_a}$ then signals color confinement. The mapping hinges on how the branch-cut phases of the dressing functions $Z_1$ (quark-gluon vertex), $Z_2$ (quark propagator), $Z_3$ (gluon propagator), and $Z_{3g}$ (three-gluon vertex) are distributed: the reality of $\alpha_{g_1}$ on the spacelike axis forces the cross-sector phase relation $b_1+b_2=b_3+b_{3g}$ for all $Q^2$, and the existing derivation satisfies it by assuming that the vertex dressing functions are analytic. Whether a dynamical mechanism or self-consistency condition of QCD could enforce this relation with non-analytic vertices — or whether vertex analyticity is instead forced, locking in the confinement criterion — is open; the derivation additionally carries acknowledged caveats concerning its scheme and gauge dependence, the absence of ghosts in the light-front framework, and its restriction to a local expansion around the singularities.

Background

Quantum chromodynamics (QCD), the gauge theory of quarks and gluons, is governed by a running coupling whose strength depends on the momentum transfer $Q^2$. In perturbation theory this coupling is defined only up to a choice of renormalization scheme, and it develops an unphysical Landau pole at low $Q^2$. Grunberg's effective-charge method removes both defects by defining the coupling directly from an observable: a measured cross section is rearranged so that all QCD corrections are absorbed into one scheme-independent function of $Q^2$. The effective charge $\alpha_{g_1}$ is of this type: it is defined through the Bjorken sum rule, which relates the difference between the proton and neutron spin structure functions $g_1^p-g_1^n$, integrated over the spin-polarized parton distribution (partons are the quarks and gluons inside the nucleon), to $(1/12)|g_A|[1-3\alpha_{g_1}(Q^2)/\pi+\cdots]$, with $g_A$ the weak axial charge; polarized deep-inelastic scattering data, chiefly from Jefferson Lab, determine $\alpha_{g_1}$ over a wide $Q^2$ range including the nonperturbative infrared.

Light-front holographic QCD (LFHQCD) is a semiclassical first approximation to QCD that combines light-front quantization (a formulation of relativistic dynamics in which the state is evolved in a light-like coordinate rather than in time) with the AdS/CFT gauge-gravity correspondence, a duality relating a quantum field theory to a gravity theory in a higher-dimensional curved spacetime; it supplies a nonperturbative continuation of the coupling into the infrared. Requiring maximal analyticity — the coupling must be holomorphic in the complex $Q^2$ plane except at singularities forced by dynamics, with the perturbative Landau pole split into a pair of complex-conjugate singularities — de Téramond et al. (2024) fixed a unique analytic form of $\alpha_{g_1}$ consistent with renormalization-group evolution, LFHQCD, and the world data on $\alpha_s$. In the infrared this form has two complex-conjugate singularities at $Q^2=\pm i\Lambda_s^2$, with $\Lambda_s$ the QCD scale.

In QCD the full (dressed) Green's functions are written as bare ones multiplied by dressing functions $Z_i(Q^2)$. Extracted from the quark-gluon sector, $\alpha_{g_1}(Q^2)=\alpha_{g_1}(Q_0^2)Z_1^2(Q^2)Z_2^2(Q^2)Z_3(Q^2)$, while the three-gluon sector gives $\alpha_{g_1}(Q^2)=\alpha_{g_1}(Q_0^2)Z_3^3(Q^2)Z_{3g}^2(Q^2)$: $Z_1$ dresses the quark-gluon vertex, $Z_2$ the quark propagator, $Z_3$ the gluon propagator, and $Z_{3g}$ the three-gluon vertex. These functions are outputs of the Dyson-Schwinger equations (DSEs), the infinite tower of coupled integral equations QCD provides for its own Green's functions; each dressing function obeys its own equation, the different sectors being coupled only through loop integrations. Gauge theories also impose Ward identities (in the Abelian case) and Slavnov-Taylor identities (in QCD, involving Faddeev-Popov ghosts — the unphysical scalar fields that implement gauge fixing in covariant gauges), which constrain certain combinations of renormalization constants; in QED the Ward identity even forces the vertex and fermion-propagator renormalizations to coincide, but no known identity relates vertex and propagator phases across different sectors of QCD.

Deur (2026) connects the analytic structure of $\alpha_{g_1}$ to confinement. Near $Q^2=\pm i\Lambda_s^2$ the dressing functions develop branch cuts, so their logarithms are complex on the real spacelike axis: $\ln Z_i=a_i+ib_i$, where the $b_i(Q^2)$ are phases. Since $\alpha_{g_1}$ is an observable, it is real on that axis, so the phases of both factorizations must reproduce $\mathrm{Im}\,\ln[\alpha_{g_1}(Q^2)/\alpha_{g_1}(Q_0^2)]$: $2b_1+2b_2+b_3=3b_3+2b_{3g}$ for all $Q^2$, that is

$b_1(Q^2)+b_2(Q^2)=b_3(Q^2)+b_{3g}(Q^2).$

The derivation assumes that at least one vertex dressing function is analytic ($b_1=b_{3g}=0$; analyticity of $Z_1$ implies that of $Z_{3g}$ and vice versa) — the natural solution satisfying the constraint without fine-tuning between the quark-gluon and three-gluon sectors. It yields $b_2=b_3$: quark and gluon propagators share one singularity structure, and a steepest-descent (saddle-point) evaluation of their Fourier transforms gives a long-distance behavior $P(x)\propto e^{-\Lambda_s|x|/\sqrt{2}}|x|^{-5/2+d_a}$ with $d_a=4/(12+3\pi^2)\approx 0.1$, so parton propagation is exponentially suppressed beyond distances of order $\sqrt{2}/\Lambda_s$ — an intuitive criterion for color confinement. Numerically, $\Lambda_s/\sqrt{2}=0.44$ GeV in the momentum-subtraction (MOM) renormalization scheme is close to the emergent gluon mass $m_0=0.43(1)$ GeV found in lattice QCD and DSE studies. The Gribov gluon propagator $s/(s^2+\Lambda_s^4)$, the modified propagator of Gribov's original confinement scenario, appears as a special limiting case of the general form. The framework works in the $g_1$ renormalization scheme and light-front gauge, contains no ghost fields, and uses only a local expansion of the dressing functions near $Q^2=\pm i\Lambda_s^2$. What is not known is whether the analyticity assumption is forced — and that is the open problem.

Problem Statement

How is the analytic structure of the observable QCD effective charge $\alpha_{g_1}$ — its two complex-conjugate singularities at $Q^2=\pm i\Lambda_s^2$ — distributed among the dressing functions of the QCD Green's functions, and is the assumption-free answer compatible with confinement?

Core question: does there exist a dynamical mechanism or self-consistency condition of QCD — an identity, symmetry, or constraint emerging from the coupled Dyson-Schwinger tower — that enforces the cross-sector phase relation $b_1(Q^2)+b_2(Q^2)=b_3(Q^2)+b_{3g}(Q^2)$ for all spacelike $Q^2$ while the vertex dressing functions are non-analytic ($b_1\neq 0$ and/or $b_{3g}\neq 0$), maintaining the reality of $\alpha_{g_1}$ without fine-tuning between the quark-gluon and three-gluon sectors? Or can it be established that no such mechanism exists, so that the analyticity of at least one vertex dressing function ($Z_1$ or $Z_{3g}$) is forced, together with its consequences: equal sharing of the confinement structure between the quark and gluon propagators and the exponentially suppressed long-distance forms $P(x)\propto e^{-\Lambda_s|x|/\sqrt{2}}|x|^{-5/2+d_a}$? Here “without fine-tuning” means that the relation follows from a symmetry, identity, or dynamical self-consistency condition acting generically in $Q^2$, rather than being satisfied only through a special matching of the parameters of the separate quark-gluon and three-gluon sectors — each dressing function obeying its own equation within the coupled Dyson-Schwinger tower. What analytic structure do the propagators carry in each branch, and does suppression of propagation at distances beyond order $1/\Lambda_s$ — the confinement criterion — survive?

The same source acknowledges caveats that delimit the setting of this question: the derived propagator forms and the confinement scale are established so far only in the $g_1$ renormalization scheme and light-front gauge, a framework without Faddeev-Popov ghosts (which are dynamically active in other gauges and schemes), and the derivation uses only a local expansion of the dressing functions near $Q^2=\pm i\Lambda_s^2$, so the long-distance behavior it yields does not necessarily extend to shorter distances. These limitations describe the scope of the existing derivation; they are not themselves posed by the source as open questions.

References

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