# The strong coupling in the nonperturbative and near-perturbative regimes

Guy F. de Téramond<EMAIL_ADDRESS>Laboratorio de Física Teórica y
Computacional, Universidad de Costa Rica, 11501 San José, Costa Rica  Arpon
Paul<EMAIL_ADDRESS>School of Physics and Astronomy, University of
Minnesota, Minneapolis, Minnesota 55455, USA  Stanley J. Brodsky
<EMAIL_ADDRESS>SLAC National Accelerator Laboratory, Stanford
University, Stanford, CA 94309, USA  Alexandre Deur<EMAIL_ADDRESS>Thomas
Jefferson National Accelerator Facility Newport News, Virginia 23606, USA
Hans Günter Dosch<EMAIL_ADDRESS>Institut für Theoretische Physik der
Universität, D-69120 Heidelberg, Germany  Tianbo Liu<EMAIL_ADDRESS>Key
Laboratory of Particle Physics and Particle Irradiation (MOE), Institute of
Frontier and Interdisciplinary Science, Shandong University, Qingdao, Shandong
266237, China  Southern Center for Nuclear-Science Theory (SCNT), Institute of
Modern Physics, Chinese Academy of Sciences, Huizhou 516000, China  Raza
Sabbir Sufian<EMAIL_ADDRESS>RIKEN-BNL Research Center, Brookhaven
National Laboratory, Upton, New York 11973, USA  Physics Department,
Brookhaven National Laboratory, Upton, New York 11973, USA

###### Abstract

(HLFHS Collaboration)

We use analytic continuation to extend the gauge/gravity duality
nonperturbative description of the strong force coupling into the transition,
near-perturbative, regime where perturbative effects become important. By
excluding the unphysical region in coupling space from the flow of
singularities in the complex plane, we derive a specific relation between the
scales relevant at large and short distances; this relation is uniquely fixed
by requiring maximal analyticity. The unified effective coupling model gives
an accurate description of the data in the nonperturbative and the near-
perturbative regions.

††preprint: SLAC-PUB-17765

Introduction.– The running strong coupling $\alpha_{s}$ has been determined to
high orders in perturbation theory at large momentum transfer: It leads to
asymptotic freedom, a fundamental property of quantum chromodynamics (QCD) [1,
2]. However, a purely perturbative description of $\alpha_{s}$ is
inconsistent: It breaks down at the Landau pole at large distances [3].
Avoiding such unphysical singularities requires the introduction of an
infrared (IR) confinement scale which effectively suppresses the long
wavelength modes [4, 5], or some IR regulator by adding new terms to the
coupling [6, 7]: Analytic couplings often lead to an IR fixed-point [8]. A
unified and consistent treatment of $\alpha_{s}$ in both the perturbative and
nonperturbative domains should incorporate confinement dynamics, since it
implies the transition from the short distance partonic degrees of freedom to
the large distance hadronic degrees of freedom, the asymptotic states, where
perturbative QCD is not applicable. In practice an effective description is
adopted by the choice of an effective coupling subject to specific
constraints.

In quantum field theory (QFT) there is some freedom in the definition of the
running coupling [7, 9]. Several definitions restore the coupling to the
status of an observable quantity, notably the effective charge introduced by
Grunberg [10, 11]. It defines a QCD effective coupling $\alpha_{\rm eff}$ by
the approximant of any given observable truncated to first order in the
perturbative coupling $\alpha_{s}$. This makes the effective charge definition
of the QCD coupling analogous to that of QED [12]. An effective charge defined
from a specific observable can be used for any process using the relations
between the different effective charges [13], which guarantees the
predictability of the theory. Crucially, since the leading order approximant
of an observable is scheme- and gauge-independent, its associated coupling is
an observable as well, and therefore amenable to analytical continuation. An
important advantage of $\alpha_{\rm eff}$ is that confinement forces and
parton correlations are folded into its definition, thus both the IR and
ultraviolet (UV) domains are incorporated.

In Ref. [14], $\alpha_{\rm eff}$ was derived in the IR domain using HLFQCD,
and in Refs [15, 16, 17] we introduced a matching procedure between HLFQCD and
perturbative QCD. The procedure matches the two expressions for $\alpha_{s}$
at a single point $Q_{0}$ and yields an accurate prediction of $\alpha_{s}$ in
agreement with both the IR data [18, 19, 20] and the combined world average at
the $Z^{0}$ mass [21]. Yet, as we will discuss below, this prescription has
its limitations. In contrast, we use analytic continuation in this article to
extend the applicability of the nonperturbative holographic QCD description of
the strong coupling into the transition, near-perturbative, regime where
perturbative effects become important. Both the IR and the UV domains are
incorporated into a single analytic expression providing a continuous
transition between both domains for the $\beta$-function and any higher
derivative. In the present approach, the IR scale is required for the
suppression of the unphysical Landau pole. In fact, the flow of singularities
in the complex plane is responsible for the splitting of the Landau pole into
two complex conjugate singularities. Furthermore, by excluding the unphysical
region in coupling space, we obtain a specific relation between the mass scale
underlying color confinement and the QCD scale underlying the perturbative
interactions of quarks and gluons. The relation between the IR and UV scales
becomes unique and leads to a single mass scale by requiring maximal
analyticity in the flow of singularities. The resulting model is compared with
the experimental results for the effective charge in the $g_{1}$ scheme,
$\alpha_{g_{1}}$, defined by the Bjorken sum rule [22, 23], which is well-
measured in both the IR and UV domains.

Our approach has bearings on the analytic S-matrix bootstrap and duality
concepts and tools [24, 25, 26] which arise from general principles of QFT
such as symmetry, causality, unitarity and crossing, thus, rather
independently of the specific dynamics. Together with some large distance
input to limit the possible solutions, this approach can lead to bounds on
coupling constants [27]. Interestingly, in the pre-QCD period analytic
continuation provided the connection between perturbation theory and strong
interaction physics [28].

Interpolating effective coupling.– In nonperturbative models, a scale is
introduced to compute the hadron spectrum, whereas in a conformal gauge theory
a scale, denoted by $\Lambda$ for QCD, is introduced because of the necessity
of perturbative renormalization. A challenge is thus to understand how these
two scales are related depending on the symmetries of the underlying full
theory. As general constraints required to extend the holographic strong
coupling [14] beyond the IR, we demand the vanishing of the $\beta$-function
in the deep IR and UV to reflect the underlying conformal symmetry of QCD and
the analyticity of $\alpha_{\rm eff}$ for all possible physical solutions
consistent with the continuity of the physical observables.

The large $N_{C}$ limit [29] underlies the holographic approach to strongly
coupled QFT. In fact, the AdS/CFT duality, the correspondence between a
classical gravity theory in a five-dimensional anti-de Sitter (AdS) space and
a conformal field theory (CFT) in physical space-time, corresponds to the
large $N_{C}$ limit of a conformal theory in the UV asymptotic AdS boundary
[30, 31, 32]. This should remain true if we modify the IR region of AdS space
to introduce confinement. Our approach to holographic QCD in the light-front
(HLFQCD) is based on the embedding of Dirac’s relativistic front form of
dynamics [33] into AdS space. It leads to relativistic boost-invariant wave
equations in physical space-time, similar to the Schrödinger equation in
atomic physics [34, 35]. In HLFQCD, an IR mass scale can be introduced from an
emerging superconformal symmetry [36, 37, 38] by constructing a scale-deformed
supercharge operator, which is a superposition of supercharges within the
extended graded algebra [37]. This procedure determines uniquely the confining
interaction [39, 40, 41], as well as the corresponding modification of AdS
space in the IR domain [42, 43], while keeping the action conformal invariant
[36]: It also leads to hadronic supersymmetry between mesons, baryons and
tetraquarks [41].

To build a specific model, we demand that in the deep IR and UV domains the
$\beta$-function vanishes, thus reflecting the underlying conformal symmetry
of QCD. In fact, it is clear from a physical perspective, that in a confining
theory, gluons and quarks have a maximal wavelength, and thus all vacuum
polarization corrections to the gluon self-energy must decouple: An infrared
fixed point is a natural consequence of confinement [44].

Our expression for $\alpha_{\rm eff}$ incorporates the holographic results in
Ref. [14], compatible with superconformal symmetry, and it is consistent with
measurements of the strong coupling in the IR [20]. In the deep UV we will
recover the leading perturbative QCD logarithmic dependence. As for the
analytic continuation, we build a model, which leads to the splitting of the
Landau pole into two complex conjugate singularities, thus transforming the
unphysical singularities in the real axis into a singularity flow in the
complex $Q^{2}$-plane.

We express $\alpha_{\rm eff}$ in the space-like domain by

$\displaystyle\alpha_{\rm eff}(Q^{2})=\alpha_{\rm
eff}(0)\exp\left[-\bigintssss_{0}^{Q^{2}}\frac{du}{4\kappa^{2}+u\ln\left(\frac{u}{\Lambda^{2}}\right)}\right],$
(1)

which satisfies the constraints described above. Here, $Q^{2}=-q^{2}$ is the
virtuality scale, $\kappa$ is the IR mass scale in holographic QCD, $\Lambda$
is the QCD scale governing the logarithmic evolution of the coupling in the
UV, and $\alpha_{\rm eff}(0)$ is an IR fixed point value. In the $g_{1}$
scheme used here, it is fixed to $\pi$ by simple kinematical constraints.

Eq. (1) constrains both the deep IR and UV’s behavior:

$\displaystyle\alpha_{\rm eff}(Q^{2})\to
e^{-Q^{2}/4\kappa^{2}},\quad\mbox{for}\quad Q^{2}\ll 4\kappa^{2},$ (2)

the Gaussian expression found in [14], and

$\displaystyle\alpha_{\rm
eff}(Q^{2})\to\frac{1}{\ln\left(Q^{2}/\Lambda^{2}\right)},\quad\mbox{for}\quad
Q^{2}\gg\ 4\kappa^{2},$ (3)

the leading logarithmic $Q^{2}$-dependence. Thus, Eq. (1) provides a
continuous transition between the IR and the UV perturbative domains.

Figure 1:  Behavior of the effective coupling $\alpha_{\rm eff}$ from Eq. (1)
and its $\beta$-function (The strong coupling in the nonperturbative and near-
perturbative regimes) for the range of values of
$\Lambda^{2}=\frac{8}{\pi}\kappa^{2}$ to $\Lambda^{2}=2\kappa^{2}$. The
turning point of the $\beta$-function is located at $Q^{2}=2\kappa^{2}$. The
curves correspond to $\kappa\simeq$ 0.5 GeV [35], determined from the hadronic
mass spectrum, and the constraint $\alpha_{\rm eff}(0)=\pi$ in the $g_{1}$
scheme.

The $\beta$-function

$\displaystyle\beta(\alpha_{\rm eff})$
$\displaystyle=Q^{2}\,\frac{d\alpha_{\rm eff}(Q^{2})}{dQ^{2}}$
$\displaystyle=-\frac{Q^{2}}{4\kappa^{2}+Q^{2}\ln\left(\frac{Q^{2}}{\Lambda^{2}}\right)}\,\alpha_{\rm
eff}(Q^{2}),$ (4)

has a minimum at the turning point $Q^{2}=2\kappa^{2}$, independent of the
value of $\Lambda$, and vanishes in the deep IR and UV limits. This property
is illustrated in Fig. 1, where we choose as examples the values
$\Lambda^{2}=2\kappa^{2}$ (the inflection point) and
$\Lambda^{2}=\frac{8}{\pi}\kappa^{2}$ (the value for maximal analyticity found
below). The analytic expression (The strong coupling in the nonperturbative
and near-perturbative regimes) evolves continuously from the IR to the UV
functional forms of the $\beta$-function, namely from the nonperturbative
expression in holographic QCD [14] to the leading perturbative expression [9]

$\displaystyle\beta(\alpha_{\rm eff})$
$\displaystyle\to-\frac{Q^{2}}{4\kappa^{2}}\,\alpha_{\rm eff}(Q^{2}),\quad
Q^{2}\ll 4\kappa^{2},$ (5) $\displaystyle\beta(\alpha_{\rm eff})$
$\displaystyle\to-\alpha^{2}_{\rm eff}(Q^{2}),\quad Q^{2}\gg\ 4\kappa^{2}.$
(6)

The behavior of the effective coupling $\alpha_{\rm eff}$ and its all
$\beta$-function are shown in Fig. 1 for different ratios of the scales
$\kappa$ and $\Lambda$.

The large $N_{C}$ limit should remain valid when we extend $\alpha_{\rm eff}$
to the transition domain. This constraint implies no flavor dependence in Eq.
(1) since in this limit the ’t Hooft coupling $\lambda_{N}=g^{2}N_{C}$, for
fixed $\lambda_{N}$, becomes independent of the number of flavors [29]. The
effective coupling $\alpha_{\rm eff}$ in (1) is thus valid in the
nonperturbative and the near-perturbative domains. However, it remains an
approximation in the deep, fully perturbative, UV region, since there are no
quark mass thresholds in the conformal limit on which the present derivation
is based.

Singularity flows and confinement– General considerations of symmetry and
analyticity have guided us to the expression of $\alpha_{\rm eff}$ given by
Eq. (1). Analytic continuation of this expression leads us to study the flow
of singularities in the complex $Q^{2}$-plane which imposes strict constraints
on possible solutions. The actual flow follows from the equation

$\displaystyle 4\kappa^{2}+Q^{2}\ln\left(\frac{Q^{2}}{\Lambda^{2}}\right)=0,$
(7)

which determines the singularities of $\alpha_{\rm eff}(Q^{2})$ (1) and its
$\beta$-function (The strong coupling in the nonperturbative and near-
perturbative regimes) in the space-like domain $Q^{2}\geq 0$. Its solutions
are given by

$\displaystyle
Q_{u}^{2}({\kappa^{2}})=\Lambda^{2}\exp\left[W_{0}\left(-\frac{4\kappa^{2}}{\Lambda^{2}}\right)\right],$
(8)

and

$\displaystyle
Q_{l}^{2}({\kappa^{2}})=\Lambda^{2}\exp\left[W_{-1}\left(-\frac{4\kappa^{2}}{\Lambda^{2}}\right)\right],$
(9)

for fixed $\Lambda$, where $u$ and $l$ refer, respectively, to the solutions
in the upper and lower $Q^{2}$ half-planes. $W_{k}(z)$, the Lambert function,
is a multivalued function with a branch for each positive and negative integer
value of $k$. It is also useful to express $\kappa$ as a function of $Q^{2}$
along the locus of the solutions of (7)

$\displaystyle\kappa^{2}(Q^{2})=-\frac{1}{4}Q^{2}\ln\left(\frac{Q^{2}}{\Lambda^{2}}\right),$
(10)

with $Q^{2}\to Q_{u,l}^{2}$.

In the limit $\kappa\to 0$ we find the solutions

$\displaystyle Q_{u}^{2}(\kappa^{2}=0)=\Lambda^{2},\quad{\rm and}\quad
Q_{l}^{2}(\kappa^{2}=0)=0.$ (11)

The first solution, $Q_{u}$, which corresponds to the principal branch of the
Lambert function with $W_{0}(0)=0$, gives the Landau singularity at
$Q^{2}=\Lambda^{2}$, and the second solution, $Q_{l}$, which corresponds to a
lower branch, leads to an additional singularity at $Q^{2}=0$ since
$W_{-1}(0)=-\infty$.

Figure 2:  Flow of singularities in the complex $Q^{2}$-plane from the
solutions of (7) upon variation of $\kappa^{2}$ between
$0\leq\kappa^{2}\leq\frac{\pi}{8}\Lambda^{2}$ for fixed $\Lambda$: The flow
paths of $Q_{u}$ and $Q_{l}$ (Eqs. (8) and (9)) are marked blue and red,
respectively. The critical value $\kappa^{2}=\Lambda^{2}/4e$ corresponds to
the bifurcation point at $Q^{2}=\Lambda^{2}/e$. The effective coupling (1) is
not defined for $\kappa^{2}<\Lambda^{2}/4e$. For values of
$\kappa^{2}>\Lambda^{2}/4e$, the double pole at the critical bifurcation point
is split into two complex conjugate singularities, and the coupling (1) can be
computed for any value of $Q^{2}$. The upper limit for $\kappa$ leads to the
relation $\Lambda^{2}=\frac{8}{\pi}\kappa^{2}$ between the confinement and QCD
scales $\kappa$ and $\Lambda$. See also Table 1.

For small $\kappa>0$ the distance between the two singularities, located
initially for $\kappa=0$, at $Q^{2}=\Lambda^{2}$ and $Q^{2}=0$ starts to
shrink until they meet at a point determined by the maximal value of $\kappa$
in the interval $0<Q^{2}<\Lambda^{2}$ on the real axis, thus by the condition
$d\kappa^{2}(Q^{2})/dQ^{2}=0$ with the solution $Q^{2}=\frac{\Lambda^{2}}{e}$
as shown in Fig. 2. Also from (10) we find the critical value
$\kappa^{2}=\frac{\Lambda^{2}}{4e}$ (see Table 1), thus the two singularities
merge at the critical point

$\displaystyle
Q_{u}^{2}\left(\kappa^{2}=\frac{1}{4e}\Lambda^{2}\right)=\frac{1}{e}\Lambda^{2},\quad{\rm
and}\quad
Q_{l}^{2}\left(\kappa^{2}=\frac{1}{4e}\Lambda^{2}\right)=\frac{1}{e}\Lambda^{2},$
(12)

since $W_{0}\left(-\frac{1}{e}\right)=W_{-1}\left(-\frac{1}{e}\right)=-1$. For
$\kappa^{2}\leq\frac{\Lambda^{2}}{4e}$ the integral in Eq. (1) is not defined
since the singularities are located in the real axis where the integration is
performed.

$Q_{u}^{2}$ | $\kappa^{2}$ | $Q_{l}^{2}$ | $\kappa^{2}$
---|---|---|---
$\Lambda^{2}$ | 0 | 0 | 0
$~{}\Lambda^{2}/e~{}$ | $~{}\Lambda^{2}/4e~{}$ | $~{}\Lambda^{2}/e~{}$ | $~{}\Lambda^{2}/4e~{}$
$i\Lambda^{2}$ | $\pi\Lambda^{2}/8$ | $-i\Lambda^{2}$ | $\pi\Lambda^{2}/8$
Table 1:  Values for the solution of the flow equation (7), Eqs. (8) and (9),
for the specific points marked with black circles in Fig. 2, and the
corresponding values of $\kappa^{2}$ from (10).

One can combine the linear Cauchy-Riemann differential equations into the
second order equations

$\displaystyle\nabla^{2}{\rm
Re}\,\kappa^{2}=0,\quad\nabla^{2}\rm{Im}\,\kappa^{2}=0,$ (13)

which implies that a maximum (minimum) of $\rm{Re}\,\kappa^{2}$
($\rm{Im}\,\kappa^{2}$) along the real $Q^{2}$-axis corresponds to a minimum
(maximum) of $\rm{Re}\,\kappa^{2}$ ($\rm{Im}\,\kappa^{2}$) along the imaginary
direction, thus to a saddle point, as shown in Fig. 3 (left) for ${\rm
Re}\,\kappa^{2}$ at $\kappa^{2}=\Lambda^{2}/4e$. We also verify in Fig. 3
(right), by taking the intersection of ${\rm Im}\,\kappa^{2}$ with the plane
${\rm Im}\,\kappa^{2}=0$, that, as expected, $\kappa^{2}$ is real on the
singularity flow.

The gradient vectors $\vec{\nabla}{\rm Re}\,\kappa^{2}$ and $\vec{\nabla}{\rm
Im}\,\kappa^{2}$ are mutually orthogonal, $\vec{\nabla}{\rm
Re}\,\kappa^{2}\cdot\vec{\nabla}{\rm Im}\,\kappa^{2}$ = 0, and normal,
respectively, to the two curves ${\rm Re}\,\kappa^{2}$ = constant and ${\rm
Im}\,\kappa^{2}$ = constant. For both $\rm{Re}\,\kappa^{2}$ and
$\rm{Im}\,\kappa^{2}$, there is an intersection between two level curves at
the saddle point $Q^{2}=\Lambda^{2}/e$, as depicted in the bottom projection
planes in Fig. 3. The singularity flow for real $\kappa^{2}$ (i.e., at
$\rm{Im}\,\kappa^{2}=0$) follows the vector $\vec{\nabla}{\rm Re}\,\kappa^{2}$
(Fig. 3 (left)) which, because of the orthogonality of the gradient vectors
$\vec{\nabla}{\rm Re}\,\kappa^{2}$ and $\vec{\nabla}{\rm Im}\,\kappa^{2}$,
should follow the level curve for $\rm{Im}\,\kappa^{2}$. This is verified in
the projection plane in Fig. 3 (right) for the level curve which intersects
the saddle point. Hence, the singularity at the critical point at
$Q^{2}=\Lambda^{2}/e$ is a bifurcation point of the singularity flow from the
real axis into two complex conjugate singularities which flow into the lower
and upper half-planes of the complex plane. The bifurcation point thus
represents the critical value of the IR scale $\kappa^{2}$ above which the
coupling (1) and its $\beta$-function (The strong coupling in the
nonperturbative and near-perturbative regimes) – or any higher derivative, can
be computed for any value of $Q^{2}\geq 0$.

Figure 3:  Singularity flow for the real and imaginary components of
$\kappa^{2}$ from Eq. (7). The bifurcation point of the flow in the
$Q^{2}$-plane is a saddle point for ${\rm Re}\,\kappa^{2}$ (left). The
intersection of the surface ${\rm Im}\,\kappa^{2}$ with the plane ${\rm
Im}\,\kappa^{2}=0$, shown in shaded grey, illustrates that $\kappa^{2}$ is
real along the flow (right).

By further increasing $\kappa$, the complex conjugate singularities reach
their maximal separation at the points

$\displaystyle
Q_{u}^{2}\left(\kappa^{2}=\frac{\pi}{8}\Lambda^{2}\right)=i\Lambda^{2},\quad{\rm
and}\quad
Q_{l}^{2}\left(\kappa^{2}=\frac{\pi}{8}\Lambda^{2}\right)=-i\Lambda^{2},$ (14)

which are located along the imaginary axis (see Fig. 2) since
$W_{0}\left(-\frac{\pi}{2}\right)=i\frac{\pi}{2}$ and
$W_{1}\left(-\frac{\pi}{2}\right)=-i\frac{\pi}{2}$. From (10) we find

$\displaystyle\kappa^{2}=\frac{\pi}{8}\Lambda^{2},$ (15)

for the maximal separation along the imaginary axis. Although related by
analytic continuation, the space-like and time-like domains describe
physically different processes on different domains: The singularity flow from
the positive-real half-plane into the negative-real half-plane in Fig. 2 is
not possible. Eq. (15) represents an upper bound for the value of the
confinement scale $\kappa$ for a given value of $\Lambda$.

Maximal analyticity of the effective coupling in the spacelike domain leads to
a unique relation between the confinement scale $\kappa$ and the QCD scale
$\Lambda$: It implies that the effective coupling (1) is a holomorphic
function in the full $Q^{2}>0$ complex plane, except at its border $Q^{2}=0$.
This is compatible with general principles of QFT which require that spacelike
observables are analytic functions in the $Q^{2}$-Euclidean complex plane.

Comparison with experiment.– Imposing maximal analyticity also means that the
effective coupling has no free parameters once the scale $\kappa$ is fixed by
an observable quantity, for example by the $\rho$ mass. Furthermore, since
$\kappa$ is a physical quantity, it follows that $\Lambda$, in the present
nonperturbative approach, is also an observable quantity: The effective
coupling (1) is thus promoted from an effective interpolation formula to an
effective charge, namely to the status of an observable quantity [10, 11]. As
mentioned above in the limit of massless quarks there is no flavor dependence
of the coupling. As an approximation to this limit we take the effective
number of flavors to $n_{f}=3$. From the value
$\Lambda^{(n_{f}=3)}_{g1}=0.92\pm 0.05$ GeV in the $g_{1}$ scheme [16], one
obtains the value $\kappa=0.58\pm 0.03$ GeV in the maximal analyticity limit.
It is compatible with $\kappa=0.534\pm 0.025$ GeV from the light vector meson
spectrum in HLFQCD [45] and $\kappa=0.523\pm 0.024$ GeV from the light meson
and baryon spectroscopy including all radial and orbital excitations [46].

Figure 4:  Comparison of the effective strong coupling prediction from Eq. (1)
with $\alpha_{g1}$ extracted from the available data in Refs. [47, 48, 49, 50,
51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69,
70, 71]. The red band corresponds to $\kappa=0.534\pm 0.025$ GeV and maximal
quenching (15). Here, $\alpha_{\rm eff}$ is computed in the $g_{1}$ scheme,
which imposes $\alpha_{\rm eff}(0)=\pi$. The dashed red curve corresponds to
the point-matching procedure from Ref. [14] for $\kappa=0.534~{}{\rm GeV}$.
Similar behavior has been obtained from Dyson-Schwinger and lattice
computations in Refs. [72, 73], gray band. The blue band corresponds to the
perturbative 4-loop QCD prediction in the $g_{1}$ scheme [20].

We compare in Fig. 4 the predictions from Eq. (1), with available data,
including the most recent measurements from Ref. [20]. The data from the
Bjorken sum, listed by experiments, are from CERN [47, 48, 49, 50, 51, 52],
DESY [53, 54, 55], JLab [56, 57, 58, 59] and SLAC [60, 61, 62, 63, 64, 65, 66,
67, 68, 69]. The data without using the Bjorken sum are from CCFR (F3-SF) [70]
and OPAL [71]. We use the value of $\kappa=0.534\pm 0.025$ GeV from [45] and
the maximal quenching relation (15). The model describes accurately the
experimental results in the nonperturbative and transition domains, consistent
with a single scale governing the confinement dynamics of QCD.

Conclusion and outlook.–The results presented here provide a new framework for
extending the gauge/gravity results for the effective coupling [14] into the
transition regime between a strongly coupled regime and an asymptotically free
gauge theory. Instead of matching the IR and UV expressions at a given point
[15, 16, 17], we introduce a single analytic expression which leads to a
continuous transition from the IR to the UV expressions of the effective
strong coupling $\alpha_{\rm eff}$ and its $\beta$-function. The procedure,
based on analytic continuation, is compatible with the underlying conformal
symmetry properties of QCD and its large $N_{C}$ constraints. The study of the
analytic properties of the model determines the flow of singularities in the
complex plane leading to well-defined relations between the nonperturbative
and perturbative scales for the possible solutions of the strong coupling. The
Landau singularity in the real axis splits into two complex singularities
giving rise to the physical solutions. Imposing maximal analyticity in the
complex $Q^{2}$-plane provides a unique relation between the IR and UV scales
of the strong force, leading to an accurate description of the available data
in the nonperturbative and the perturbative transition region.

Extension of the present results to the full perturbative domain requires one
to account for the flavor thresholds at larger scales. In the framework of the
model developed here, this may be achieved by allowing for the $Q^{2}$
dependence of the confinement scale $\kappa$. Such a scale dependence is also
required to describe the spectroscopy of heavy quark systems in holographic
QCD [74, 42, 75]. The framework introduced in this article also naturally
lends itself to the analytic continuation of our results to the entire complex
plane including the timelike domain $Q^{2}<0$. These subjects are under
consideration.

Acknowledgments.– This work is supported in part by the Department of Energy
Contract No. DE-AC02-76SF00515. It is also supported in part by the U.S.
Department of Energy, Office of Science, Office of Nuclear Physics under
Contract No. DE-AC05-06OR23177. T.L. is supported by the National Natural
Science Foundation of China under Grants No. 12175117 and No. 12321005 and
Shandong Province Natural Science Foundation Grant No. ZFJH202303. R.S.S. is
supported by the Special Postdoctoral Researchers Program of RIKEN and RIKEN-
BNL Research Center.

## References

  * Gross and Wilczek [1973] D. J. Gross and F. Wilczek, Ultraviolet behavior of non-abelian gauge theories, Phys. Rev. Lett. 30, 1343 (1973).
  * Politzer [1973] H. D. Politzer, Reliable perturbative results for strong interactions?, Phys. Rev. Lett. 30, 1346 (1973).
  * Landau  _et al._ [1954] L. D. Landau, A. A. Abrikosov, and I. M. Khalatnikov, The removal of infinities in quantum electrodynamics, Dokl. Akad. Nauk SSSR 95, 10.1016/b978-0-08-010586-4.50083-3 (1954).
  * Gribov [1978] V. N. Gribov, Quantization of non-abelian gauge theories, Nucl. Phys. B 139, 1 (1978).
  * Cornwall [1982] J. M. Cornwall, Dynamical mass generation in continuum QCD, Phys. Rev. D 26, 1453 (1982).
  * Shirkov and Solovtsov [1996] D. V. Shirkov and I. L. Solovtsov, Analytic QCD running coupling with finite IR behavior and universal $\alpha_{s}(0)$ value, (1996), arXiv:hep-ph/9604363 .
  * Deur  _et al._ [2016a] A. Deur, S. J. Brodsky, and G. F. de Téramond, The QCD running coupling, Nucl. Phys. 90, 1 (2016a), arXiv:1604.08082 [hep-ph] .
  * Ayala and Cvetic [2015] C. Ayala and G. Cvetic, Mathematica and Fortran programs for various analytic QCD couplings, J. Phys. Conf. Ser. 608, 012064 (2015), arXiv:1411.1581 [hep-ph] .
  * Deur  _et al._ [2024] A. Deur, S. J. Brodsky, and C. D. Roberts, QCD running couplings and effective charges, Prog. Part. Nucl. Phys. 134, 104081 (2024), arXiv:2303.00723 [hep-ph] .
  * Grunberg [1980] G. Grunberg, Renormalization group improved perturbative QCD, Phys. Lett. B 95, 70 (1980), [Erratum: Phys.Lett.B 110, 501 (1982)].
  * Grunberg [1984] G. Grunberg, Renormalization-scheme-invariant QCD and QED: The method of effective charges, Phys. Rev. D 29, 2315 (1984).
  * Gell-Mann and Low [1954] M. Gell-Mann and F. E. Low, Quantum electrodynamics at small distances, Phys. Rev. 95, 1300 (1954).
  * Brodsky and Lu [1995] S. J. Brodsky and H. J. Lu, Commensurate scale relations in quantum chromodynamics, Phys. Rev. D 51, 3652 (1995), arXiv:hep-ph/9405218 .
  * Brodsky  _et al._ [2010] S. J. Brodsky, G. F. de Téramond, and A. Deur, Nonperturbative QCD coupling and its $\beta$-function from light-front holography, Phys. Rev. D 81, 096010 (2010), arXiv:1002.3948 [hep-ph] .
  * Deur  _et al._ [2015] A. Deur, S. J. Brodsky, and G. F. de Téramond, Connecting the hadron mass scale to the fundamental mass scale of quantum chromodynamics, Phys. Lett. B 750, 528 (2015), arXiv:1409.5488 [hep-ph] .
  * Deur  _et al._ [2016b] A. Deur, S. J. Brodsky, and G. F. de Téramond, On the interface between perturbative and nonperturbative QCD, Phys. Lett. B 757, 275 (2016b), arXiv:1601.06568 [hep-ph] .
  * Deur  _et al._ [2017] A. Deur, S. J. Brodsky, and G. F. de Téramond, Determination of $\Lambda_{\overline{MS}}$ at five loops from holographic QCD, J. Phys. G 44, 105005 (2017), arXiv:1608.04933 [hep-ph] .
  * Deur  _et al._ [2007] A. Deur, V. Burkert, J.-P. Chen, and W. Korsch, Experimental determination of the effective strong coupling constant, Phys. Lett. B 650, 244 (2007), arXiv:hep-ph/0509113 .
  * Deur  _et al._ [2008a] A. Deur, V. Burkert, J. P. Chen, and W. Korsch, Determination of the effective strong coupling constant $\alpha_{s,g_{1}}(Q^{2})$ from CLAS spin structure function data, Phys. Lett. B 665, 349 (2008a), arXiv:0803.4119 [hep-ph] .
  * Deur  _et al._ [2022a] A. Deur, V. Burkert, J. P. Chen, and W. Korsch, Experimental determination of the QCD effective charge $\alpha_{g_{1}}(Q)$, Particles 5, 171 (2022a), arXiv:2205.01169 [hep-ph] .
  * Workman  _et al._ [2022] R. L. Workman _et al._ (Particle Data Group), Review of Particle Physics, PTEP 2022, 083C01 (2022).
  * Bjorken [1966] J. D. Bjorken, Applications of the chiral $U(6)\otimes U(6)$ algebra of current densities, Phys. Rev. 148, 1467 (1966).
  * Bjorken [1970] J. D. Bjorken, Inelastic scattering of polarized leptons from polarized nucleons, Phys. Rev. D 1, 1376 (1970).
  * Chew and Frautschi [1961] G. F. Chew and S. C. Frautschi, Principle of equivalence for all strongly interacting particles within the $S$-matrix framework, Phys. Rev. Lett. 7, 394 (1961).
  * Eden  _et al._ [1966] R. J. Eden, P. V. Landshoff, D. I. Olive, and J. C. Polkinghorne,  _The analytic S-matrix_ (Cambridge Univ. Press, Cambridge, 1966).
  * Veneziano [1968] G. Veneziano, Construction of a crossing-symmetric, Regge-behaved amplitude for linearly rising trajectories, Nuovo Cim. A 57, 190 (1968).
  * Mizera [2024] S. Mizera, Physics of the analytic S-matrix, Phys. Rept. 1047, 1 (2024), arXiv:2306.05395 [hep-th] .
  * Mandelstam [1959] S. Mandelstam, Analytic properties of transition amplitudes in perturbation theory, Phys. Rev. 115, 1741 (1959).
  * ’t Hooft [1974] G. ’t Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B 72, 461 (1974).
  * Maldacena [1998] J. M. Maldacena, The large $N$ limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998), arXiv:hep-th/9711200 .
  * Gubser  _et al._ [1998] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428, 105 (1998), arXiv:hep-th/9802109 .
  * Witten [1998] E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2, 253 (1998), arXiv:hep-th/9802150 .
  * Dirac [1949] P. A. M. Dirac, Forms of relativistic dynamics, Rev. Mod. Phys. 21, 392 (1949).
  * de Téramond and Brodsky [2009] G. F. de Téramond and S. J. Brodsky, Light-front holography: A first approximation to QCD, Phys. Rev. Lett. 102, 081601 (2009), arXiv:0809.4899 [hep-ph] .
  * Brodsky  _et al._ [2015] S. J. Brodsky, G. F. de Téramond, H. G. Dosch, and J. Erlich, Light-front holographic QCD and emerging confinement, Phys. Rept. 584, 1 (2015), arXiv:1407.8131 [hep-ph] .
  * de Alfaro  _et al._ [1976] V. de Alfaro, S. Fubini, and G. Furlan, Conformal invariance in quantum mechanics, Nuovo Cim. A 34, 569 (1976).
  * Fubini and Rabinovici [1984] S. Fubini and E. Rabinovici, Superconformal quantum mechanics, Nucl. Phys. B 245, 17 (1984).
  * Akulov and Pashnev [1983] V. P. Akulov and A. I. Pashnev, Quantum superconformal model in (1,2) space, Theor. Math. Phys. 56, 862 (1983).
  * Brodsky  _et al._ [2014] S. J. Brodsky, G. F. de Téramond, and H. G. Dosch, Threefold complementary approach to holographic QCD, Phys. Lett. B 729, 3 (2014), arXiv:1302.4105 [hep-th] .
  * de Téramond  _et al._ [2015] G. F. de Téramond, H. G. Dosch, and S. J. Brodsky, Baryon spectrum from superconformal quantum mechanics and its light-front holographic embedding, Phys. Rev. D 91, 045040 (2015), arXiv:1411.5243 [hep-ph] .
  * Dosch  _et al._ [2015] H. G. Dosch, G. F. de Téramond, and S. J. Brodsky, Superconformal baryon-meson symmetry and light-front holographic QCD, Phys. Rev. D 91, 085016 (2015), arXiv:1501.00959 [hep-th] .
  * Dosch  _et al._ [2017] H. G. Dosch, G. F. de Téramond, and S. J. Brodsky, Supersymmetry Across the Light and Heavy-Light Hadronic Spectrum II, Phys. Rev. D 95, 034016 (2017), arXiv:1612.02370 [hep-ph] .
  * de Téramond and Brodsky [2023] G. F. de Téramond and S. J. Brodsky, Color symmetry and confinement as an underlying superconformal structure in holographic QCD (2023) arXiv:2308.09280 [hep-ph] .
  * Brodsky and de Téramond [2008] S. J. Brodsky and G. F. de Téramond, Light-front dynamics and AdS/QCD correspondence: The pion form factor in the space- and time-like regions, Phys. Rev. D 77, 056007 (2008), arXiv:0707.3859 [hep-ph] .
  * Sufian  _et al._ [2018] R. S. Sufian, T. Liu, G. F. de Téramond, H. G. Dosch, S. J. Brodsky, A. Deur, M. T. Islam, and B.-Q. Ma, Nonperturbative strange-quark sea from lattice QCD, light-front holography, and meson-baryon fluctuation models, Phys. Rev. D 98, 114004 (2018), arXiv:1809.04975 [hep-ph] .
  * Brodsky  _et al._ [2016] S. J. Brodsky, G. F. de Téramond, H. G. Dosch, and C. Lorcé, Universal effective hadron dynamics from superconformal algebra, Phys. Lett. B 759, 171 (2016), arXiv:1604.06746 [hep-ph] .
  * Adeva  _et al._ [1993] B. Adeva _et al._ (Spin Muon), Measurement of the spin-dependent structure function $g_{1}(x)$ of the deuteron, Phys. Lett. B 302, 533 (1993).
  * Adams  _et al._ [1994] D. Adams _et al._ (Spin Muon (SMC)), Measurement of the spin-dependent structure function $g_{1}(x)$ of the proton, Phys. Lett. B 329, 399 (1994), [Erratum: Phys.Lett.B 339, 332–333 (1994)], arXiv:hep-ph/9404270 .
  * Adeva  _et al._ [1997] B. Adeva _et al._ (Spin Muon (SMC)), The spin-dependent structure function $g_{1}(x)$ of the proton from polarized deep-inelastic muon scattering, Phys. Lett. B 412, 414 (1997).
  * Adams  _et al._ [1995] D. Adams _et al._ (Spin Muon), A new measurement of the spin dependent structure function $g_{1}(x)$ of the deuteron, Phys. Lett. B 357, 248 (1995).
  * Adams  _et al._ [1997] D. Adams _et al._ (Spin Muon (SMC)), Spin structure of the proton from polarized inclusive deep-inelastic muon-proton scattering, Phys. Rev. D 56, 5330 (1997), arXiv:hep-ex/9702005 .
  * Alekseev  _et al._ [2010] M. G. Alekseev _et al._ (COMPASS), The spin-dependent structure function of the proton $g_{1}^{p}$ and a test of the Bjorken sum rule, Phys. Lett. B 690, 466 (2010), arXiv:1001.4654 [hep-ex] .
  * Ackerstaff  _et al._ [1998] K. Ackerstaff _et al._ (HERMES), Determination of the deep inelastic contribution to the generalized Gerasimov-Drell-Hearn integral for the proton and neutron, Phys. Lett. B 444, 531 (1998), arXiv:hep-ex/9809015 .
  * Airapetian  _et al._ [2000] A. Airapetian _et al._ (HERMES), The $Q^{2}$-dependence of the generalized Gerasimov-Drell-Hearn integral for the proton, Phys. Lett. B 494, 1 (2000), arXiv:hep-ex/0008037 .
  * Airapetian  _et al._ [2003] A. Airapetian _et al._ (HERMES), The $Q^{2}$ dependence of the generalized Gerasimov-Drell-Hearn integral for the deuteron, proton and neutron, Eur. Phys. J. C 26, 527 (2003), arXiv:hep-ex/0210047 .
  * Deur  _et al._ [2004] A. Deur _et al._ , Experimental determination of the evolution of the Bjorken integral at low $Q^{2}$, Phys. Rev. Lett. 93, 212001 (2004), arXiv:hep-ex/0407007 .
  * Deur  _et al._ [2008b] A. Deur _et al._ , Experimental study of isovector spin sum rules, Phys. Rev. D 78, 032001 (2008b), arXiv:0802.3198 [nucl-ex] .
  * Deur  _et al._ [2014] A. Deur, Y. Prok, V. Burkert, D. Crabb, F. X. Girod, K. A. Griffioen, N. Guler, S. E. Kuhn, and N. Kvaltine, High precision determination of the $Q^{2}$ evolution of the Bjorken Sum, Phys. Rev. D 90, 012009 (2014), arXiv:1405.7854 [nucl-ex] .
  * Deur  _et al._ [2022b] A. Deur _et al._ , Experimental study of the behavior of the Bjorken sum at very low $Q^{2}$, Phys. Lett. B 825, 136878 (2022b), arXiv:2107.08133 [nucl-ex] .
  * Abe  _et al._ [1995a] K. Abe _et al._ (E143), Precision measurement of the proton spin structure function $g_{1}^{p}$, Phys. Rev. Lett. 74, 346 (1995a).
  * Abe  _et al._ [1995b] K. Abe _et al._ (E143), Precision measurement of the deuteron spin structure function $g_{1}^{d}$, Phys. Rev. Lett. 75, 25 (1995b).
  * Anthony  _et al._ [1996] P. L. Anthony _et al._ (E142), Deep inelastic scattering of polarized electrons by polarized 3He and the study of the neutron spin structure, Phys. Rev. D 54, 6620 (1996), arXiv:hep-ex/9610007 .
  * Abe  _et al._ [1995c] K. Abe _et al._ (E143), Measurements of the $Q^{2}$ dependence of the proton and deuteron spin structure functions $g_{1}^{p}$ and $g_{1}^{d}$, Phys. Lett. B 364, 61 (1995c), arXiv:hep-ex/9511015 .
  * Abe  _et al._ [1997a] K. Abe _et al._ (E143), Measurements of the proton and deuteron spin structure function $g_{1}$ in the resonance region, Phys. Rev. Lett. 78, 815 (1997a), arXiv:hep-ex/9701004 .
  * Abe  _et al._ [1997b] K. Abe _et al._ (E154), Precision determination of the neutron spin structure function $g_{1}^{n}$, Phys. Rev. Lett. 79, 26 (1997b), arXiv:hep-ex/9705012 .
  * Abe  _et al._ [1997c] K. Abe _et al._ (E154), Next-to-leading order QCD analysis of polarized deep inelastic scattering data, Phys. Lett. B 405, 180 (1997c), arXiv:hep-ph/9705344 .
  * Abe  _et al._ [1998] K. Abe _et al._ (E143), Measurements of the proton and deuteron spin structure functions $g_{1}$ and $g_{2}$, Phys. Rev. D 58, 112003 (1998), arXiv:hep-ph/9802357 .
  * Anthony  _et al._ [1999] P. L. Anthony _et al._ (E155), Measurement of the deuteron spin structure function $g_{1}^{d}(x)$ for $1-(GeV/c)^{2}<Q^{2}<40-(GeV/c)^{2}$, Phys. Lett. B 463, 339 (1999), arXiv:hep-ex/9904002 .
  * Anthony  _et al._ [2000] P. L. Anthony _et al._ (E155), Measurements of the $Q^{2}$ dependence of the proton and neutron spin structure functions $g_{1}^{p}$ and $g_{1}^{n}$, Phys. Lett. B 493, 19 (2000), arXiv:hep-ph/0007248 .
  * Kim  _et al._ [1998] J. H. Kim _et al._ , A Measurement of $\alpha_{s}(Q^{2})$ from the Gross-Llewellyn Smith sum rule, Phys. Rev. Lett. 81, 3595 (1998), arXiv:hep-ex/9808015 .
  * Brodsky  _et al._ [2003] S. J. Brodsky, S. Menke, C. Merino, and J. Rathsman, On the behavior of the effective QCD coupling $\alpha_{\tau}(s)$ at low scales, Phys. Rev. D 67, 055008 (2003), arXiv:hep-ph/0212078 .
  * Binosi  _et al._ [2017] D. Binosi, C. Mezrag, J. Papavassiliou, C. D. Roberts, and J. Rodriguez-Quintero, Process-independent strong running coupling, Phys. Rev. D 96, 054026 (2017), arXiv:1612.04835 [nucl-th] .
  * Cui  _et al._ [2020] Z.-F. Cui, J.-L. Zhang, D. Binosi, F. de Soto, C. Mezrag, J. Papavassiliou, C. D. Roberts, J. Rodríguez-Quintero, J. Segovia, and S. Zafeiropoulos, Effective charge from lattice QCD, Chin. Phys. C 44, 083102 (2020), arXiv:1912.08232 [hep-ph] .
  * Gutsche  _et al._ [2013] T. Gutsche, V. E. Lyubovitskij, I. Schmidt, and A. Vega, Chiral symmetry breaking and meson vave functions in soft-wall AdS/QCD, Phys. Rev. D 87, 056001 (2013), arXiv:1212.5196 [hep-ph] .
  * Nielsen  _et al._ [2018] M. Nielsen, S. J. Brodsky, G. F. de Téramond, H. G. Dosch, F. S. Navarra, and L. Zou, Supersymmetry in the double-heavy hadronic spectrum, Phys. Rev. D 98, 034002 (2018), arXiv:1805.11567 [hep-ph] .