# Experimental determination of the QCD effective charge $\alpha_{g_{1}}(Q)$

A. Deur,${}^{\njlab,\nuva}$ V. Burkert,${}^{\njlab}$ J.P. Chen,${}^{\njlab}$
W. Korsch${}^{\nuk}$ ${}^{\njlab}$Thomas Jefferson National Accelerator
Facility, Newport News, VA 23606. USA
${}^{\nuva}$ University of Virginia, Charlottesville, VA 22904. USA
${}^{\nuk}$University of Kentucky, Lexington, KY 40506. USA

###### Abstract

The QCD effective charge $\alpha_{g_{1}}(Q)$ is an observable that
characterizes the magnitude of the strong interaction. At high momentum $Q$,
it coincides with the QCD running coupling $\alpha_{\rm s}(Q)$. At low $Q$, it
offers a nonperturbative definition of the running coupling. We have extracted
$\alpha_{g_{1}}(Q)$ from measurements carried out at Jefferson Lab that span
the very low to moderately high $Q$ domain, $0.14\leq Q\leq 2.18$ GeV. The
precision of the new results is much improved over the previous extractions
and the reach in $Q$ at the lower end is significantly expanded. The data show
that $\alpha_{g_{1}}(Q)$ becomes $Q$-independent at very low $Q$. They compare
well with two recent predictions of the QCD effective charge based on Dyson-
Schwinger equations and on the AdS/CFT duality.

##  I Introduction

The behavior of Quantum Chromodynamics (QCD), the gauge theory of the strong
interaction, is determined by the magnitude of its coupling $\alpha_{\rm s}$.
It is large at low momentum, characterized here by $Q\equiv\sqrt{-q^{2}}$ with
$q^{2}$ the square of momentum transferred in the process of
electromagnetically probing a hadron. For $Q\ll 1$ GeV, $\alpha_{\rm
s}(Q)\gtrsim 1$ which is one of the crucial pieces leading to quark
confinement. For $Q\gg 1$ GeV, $\alpha_{\rm s}(Q)\lesssim 0.2$ which enables
the use of perturbative computational techniques (perturbative QCD, pQCD)
constituting an accurate analytical approximation of QCD. In this domain,
$\alpha^{\rm pQCD}_{\rm s}$ is well defined and known within an accuracy of 1%
at $Q=M_{Z^{0}}=91$ GeV, the $Z^{0}$ mass, and within a few percents at $Q$
values of a few GeV [1]. However, using pQCD at $Q\lesssim 1$ GeV produces a
diverging $\alpha^{\rm pQCD}_{\rm s}$ (Landau pole) that prohibits any
perturbative expansion in $\alpha^{\rm pQCD}_{\rm s}$ and signals the
breakdown of pQCD. In contrast, most nonperturbative methods, including
lattice QCD [2], the AdS/CFT (Anti-de-Sitter/Conformal Field Theory) duality
[3, 4] implemented using QCD’s light-front (LF) quantization [5] and a soft-
wall AdS potential (Holographic LF QCD, HLFQCD [6]) or solving the Dyson-
Schwinger equations (DSEs) [7] yield a finite $\alpha_{\rm s}$. In fact, many
theoretical approaches predict that $\alpha_{\rm s}$ “freezes” as $Q\to 0$,
viz, it loses its $Q$-dependence [8].

There are several possible definitions of $\alpha_{\rm s}$ in the
nonperturbative domain ($Q\lesssim 1$ GeV) [8]. We use here the effective
charge approach that defines $\alpha_{\rm s}$ from the perturbative series of
an observable truncated to its first order in $\alpha_{\rm s}$ [9]. Although
this definition can be applied for any $Q$ value, it was initially proposed
for the pQCD domain where it makes $\alpha_{\rm s}$ the equivalent of the
Gell-Mann Low coupling of Quantum Electrodynamics (QED), $\alpha$ [10]. With
this definition, $\alpha_{\rm s}$ can be evaluated at any $Q$ value, has no
low $Q$ divergence and is analytic around quark mass thresholds. Furthermore,
since the first order in $\alpha^{\rm pQCD}_{\rm s}$ of a pQCD approximant is
independent of the choice of renormalization scheme (RS), effective charges
are independent of RS and gauge choices. This promotes $\alpha_{\rm s}$ from a
parameter depending on chosen conventions to an observable, albeit with the
caveat that it becomes process-dependent since two observables produce
generally different effective charges. Yet, pQCD predictability is maintained
because effective charges are related without renormalization scale
ambiguities by Commensurate Scale Relations (CSR) [11]. CSR are known to hold
for pQCD and QED since the latter corresponds to the $N_{C}\to 0$ limit of
QCD, with $N_{C}$ the number of colors. For example, CSR explicitly relate
$\alpha_{g_{1}}$, $\alpha_{F_{3}}$, $\alpha_{\tau}$ and $\alpha_{R}$ defined
using the generalized Bjorken sum rule [12], the Gross-Llewellyn Smith sum
rule [13], and the perturbative approximant for the $\tau$-decay rate [14] and
$R_{e^{+}e-}$ [15], respectively. In fact, the choice of process to define an
effective charge is analogous to a RS choice for $\alpha^{\rm pQCD}_{\rm s}$
[16] and the procedure of extracting an effective charge, e.g., from
$\tau$-decay is denoted the $\tau$-scheme. Here, we discuss the effective
charge $\alpha_{g_{1}}(Q)$ ($g_{1}$-scheme) extracted using the generalized
Bjorken sum rule:

$\displaystyle\Gamma_{1}^{\rm p-n}(Q^{2})\equiv\int_{0}^{1^{-}}g_{1}^{\rm
p}(x,Q^{2})-g_{1}^{\rm n}(x,Q^{2})dx=\frac{g_{\rm
A}}{6}\bigg{[}1-\frac{\alpha^{\rm
pQCD}_{{\rm{s}}}(Q)}{\pi}-3.58\left(\frac{\alpha^{\rm
pQCD}_{{\rm{s}}}(Q)}{\pi}\right)^{2}$
$\displaystyle-20.21\left(\frac{\alpha^{\rm
pQCD}_{{\rm{s}}}(Q)}{\pi}\right)^{3}-175.7\left(\frac{\alpha^{\rm
pQCD}_{{\rm{s}}}(Q)}{\pi}\right)^{4}+\mathcal{O}\left(\big{(}\alpha^{\rm
pQCD}_{\rm{s}}\big{)}^{5}\right)...\bigg{]}+\sum_{n>1}\frac{\mu_{2n}}{Q^{2n-2}},$
(1)

where $x$ is the Bjorken scaling variable [17], $g_{\rm A}=1.2762(5)$ [2] is
the nucleon axial charge, $g_{1}^{\rm p(n)}$ is the longitudinal spin
structure function of the proton(neutron) obtained in polarized lepton-nucleon
scattering [18] and $\mu_{2n}$ are the Operator Product Expansion’s (OPE)
nonperturbative higher twist (HT) terms. The integral excludes the elastic
contribution at $x=1$. The series coefficients are computed for $n_{f}=3$ and
in the $\overline{\rm MS}$ RS for the $n>1$ $\alpha_{\rm{s}}^{n}$ terms [19].
They originate from the pQCD radiative corrections. Although the expansion (1)
is only applicable in the perturbative domain, i.e., at distance scales where
confinement effects are weak, the HT terms can be related to the latter [20]
and one may picture the terms of Eq. (1) as coherently merging together at low
$Q$ to produce confinement.

The effective charge $\alpha_{g_{1}}$ is defined from Eq. (1) expressed at
first order in coupling and twist:

$\displaystyle\Gamma_{1}^{\rm p-n}(Q^{2})\equiv\frac{g_{\rm
A}}{6}\left(1-\frac{\alpha_{g_{1}}(Q)}{\pi}\right)~{}\longrightarrow~{}\alpha_{g_{1}}(Q)\equiv\pi\left(1-\frac{6}{g_{\rm
A}}\Gamma_{1}^{\rm p-n}(Q)\right).$ (2)

Thus, in the domain where Eq. (2) applies, $\alpha_{g_{1}}$ can be interpreted
as a running coupling that not only includes short-distance effects such as
vertex correction and vacuum polarization, but all other effects, e.g., pQCD
radiative corrections and, in the lower-$Q$ domain of pQCD, HT terms and other
nonperturbative effects not formalized by OPE and therefore not included in
Eq. (2). The latter comes from coherent reactions of a hadron (resonances). In
the nonperturbative domain where pQCD radiative corrections and HT effects
have merged into global confinement effects, $\alpha_{g_{1}}$ may
approximately retain its interpretation as a coupling if the contribution to
$\Gamma_{1}^{\rm p-n}$ of nonresonant reactions continues to dominate, as they
do at large $Q$ [21].

There are several advantages to $\alpha_{g_{1}}$ [8]. First, rigorous sum
rules constrain $\alpha_{g_{1}}(Q)$ for $Q\to 0$ (the Gerasimov–Drell–Hearn
(GDH) sum rule [22]) and $Q\to\infty$ (the Bjorken sum rule). They provide
analytical expressions of $\alpha_{g_{1}}(Q)$ in these limits (blue dashed
line and cyan hatched band in Fig. 1). Furthermore, contributions from
$\Delta$ baryons are quenched in $\Gamma_{1}^{\rm p-n}$ [23], enhancing the
nonresonant reactions contribution to $\Gamma_{1}^{\rm p-n}$ relatively to the
resonance contribution, which helps toward interpreting $\alpha_{g_{1}}$ as a
coupling. If so, $\alpha_{g_{1}}$ would remain approximately equivalent to the
Gell-Mann Low coupling in the nonperturbative domain, a crucial property that
it is not obvious and may be specific to $\alpha_{g_{1}}$. Such property is
supported by the agreement between $\alpha_{g_{1}}$ and calculations of
couplings [24, 25] using a definition consistent with $\alpha_{g_{1}}$.

Former extractions of $\alpha_{g_{1}}$ [26] were obtained from experimental
data on $\Gamma_{1}^{\rm p-n}$ from CERN [27], DESY [28], Jefferson Lab (JLab)
[29] and SLAC [30], see Fig. 1. Since the results reported in Ref. [26],
progress occurred on both the experimental and theoretical fronts. Firstly,
when Ref. [26] was published, the meaning of $\alpha_{g_{1}}$ in the
nonperturbative region was unclear. Thus, the comparison in [26] of
$\alpha_{g_{1}}$ to theoretical predictions of the nonperturbative coupling
was tentative. This is now better understood: as just discussed,
$\alpha_{g_{1}}$ essentially retains its meaning of effective charge at low
$Q$ [21, 8]. Secondly, new data on $\Gamma_{1}^{\rm p-n}$ have become
available from CERN (COMPASS experiment) [31] and JLab (EG1dvcs experiment)
[32] at high $Q$, and from JLab (E97110, E03006 and E05111 experiments) [33]
at very low $Q$. Finally, new theoretical studies of the nonperturbative
behavior of $\alpha_{\rm s}$ were conducted, including the first use of the
AdS/CFT duality to describe the strong coupling in its nonperturbative domain
[24] and the identification of a process-independent (PI) effective charge
$\hat{\alpha}_{\rm PI}(Q)$ that unifies a large body of research from DSE and
lattice QCD to $\alpha_{s}$ [25, 34]. Connections between the nonperturbative
and perturbative effective charges were made [16, 35, 8], which permitted a
prediction of $\alpha_{\rm s}$ at the $Z_{0}$ pole, $\alpha_{\rm
s}^{\overline{\rm MS}}(M_{z}^{2})=0.1190\pm 0.0006$ at N3LO [36] that agrees
well with the 2021 Particle Data Group compilation, $\alpha_{\rm s}(M_{\rm
Z})=0.1179\pm 0.0009$ [2]. In addition to predicting quantities characterizing
hadronic structures [3, 25, 37], the effective charge helps establish
conformal behavior at low $Q$. Through AdS/CFT, this helps the investigation
of the physics beyond the standard model [4] or of the quark-gluon plasma [38]
in heavy ion collisions [39] and nuclear hydrodynamics [40] for the latter and
neutron stars [41].
Here, we report on new experimental data on $\alpha_{g_{1}}$ extracted from
[31, 32, 33] and how they compare with the latest theory predictions.

##  II Experimental extraction of $\alpha_{g_{1}}$

The new JLab data on $\Gamma_{1}^{\rm p-n}(Q)$ were taken by four experiments.
The first experiment, E97110 [42], occurred in the Hall A [43] of JLab. The
three others used the CLAS spectrometer [44] in JLab’s Hall B and were
experiments EG1dvcs [45], E03006 [46] and E05111 [47] (the two latter being
referred to as Experimental Group EG4). The four experiments occurred during
the 6 GeV era of JLab, before its 12 GeV upgrade. The experiments used a
polarized electron beam with energies ranging from 0.8 to 6 GeV. E97110
studied the spin structures of the neutron and 3He using the Hall A polarized
3He target with longitudinal and transverse polarization directions [48].
EG1dvcs, E03006, E05111 studied the proton, neutron and deuteron spin
structures using the Hall B longitudinally polarized ammonia (NH3 or ND3)
target [49]. The main purpose of EG1dvcs was high $Q$, up to 2.65 GeV
($Q^{2}=7$ GeV2), exclusive measurements of Deep Virtual Compton Scattering.
Therefore, it provided highly precise inclusive $\Gamma_{1}^{\rm p-n}$ data
compared to the older data in the same domain [27, 30, 28, 29]. E97110, E03006
and E05111 were dedicated to test Chiral Effective Field Theory predictions by
covering very low $Q$ domains: $0.19\leq Q\leq 0.49$, $0.11\leq Q\leq 0.92$
and $0.14\leq Q\leq 0.70$ GeV, respectively. To reach low $Q$ while covering
the large $x$ range necessary for the $\Gamma_{1}$ integral, high beam energy
(up to 4.4 GeV) was needed and the scattered electrons had to be detected at
small angles (down to about $5^{\circ}$). In Hall A, the low angles were
reached via a supplementary dipole magnet installed in front of the
spectrometer [50]. In Hall B, a Cherenkov Counter designed for high efficiency
at small angle was installed in one of the six sectors of CLAS [47] which
magnetic field was set to bent outward the scattered electrons. In addition,
both the Hall A and B targets were placed about 1 m upstream of their usual
positions.

The EG1dvcs data on proton and deuteron were combined to form $\Gamma_{1}^{\rm
p-n}$ over the range $0.78\leq Q\leq 2.18$ GeV [32]. The $\Gamma_{1}^{\rm
p-n}$ formed with the E97110 and EG4, data covers the $0.14\leq Q\leq 0.70$
GeV range [33]. The $\alpha_{g_{1}}$ data, obtained following Eq. (2), are
shown in Fig. 1 and given in Table 1. Also shown in the figure are the older
data presented in Ref. [29] including $\alpha_{F_{3}}$ extracted from the data
[51] and $\alpha_{g_{1}(\tau)}$ from the OPAL data on $\tau$-decay [14]. The
effective charge $\alpha_{F_{3}}$ is nearly identical to $\alpha_{g_{1}}$
[26], and $\alpha_{g_{1}(\tau)}$ was transformed from the $\tau$-scheme to the
$g_{1}$-scheme using the CSR [11]. Consequently, $\alpha_{F_{3}}$ and
$\alpha_{g_{1}(\tau)}$ are directly comparable to $\alpha_{g_{1}}$. We also
show in Fig. 1 the theory predictions from AdS/CFT [24] and DSE [25].
Remarkably, both predictions are parameter-free and gauge-invariant.

The AdS/CFT coupling $\alpha^{\rm HLF}_{g_{1}}$ is obtained in the HLFQCD
approach where QCD is quantized using LF coordinates [5]. The use of the
HLFQCD approach incorporates the underlying conformal (i.e., scale-invariant)
character of QCD at low and large $Q$. The deformation of the AdS5 space is
dual to a semiclassical potential that models quark confinement. This
potential can be determined with various methods that all lead to the same
harmonic oscillator form [3, 52, 53]. The effective charge $\alpha^{\rm
HLF}_{g_{1}}$ is dual to the product of the AdS5 coupling constant by the AdS5
space deformation term. Since the latter is dual to the CFT confinement force,
the meaning of $\alpha^{\rm HLF}_{g_{1}}$ is analogous to that of
$\alpha_{g_{1}}$ which at low $Q$ incorporates in $\alpha_{\rm s}$ confinement
effects. The $Q$-dependence of $\alpha^{\rm HLF}_{g_{1}}$ is controlled by a
single scale, e.g., the proton mass. The coupling is normalized to
$\alpha^{\rm HLF}_{g_{1}}(0)=\pi$ to obey the kinematic constraint that
$\Gamma_{1}^{\rm p-n}(0)=0$, i.e., $\alpha_{g_{1}}(0)=\pi$, see Eq. (2). This
normalization amounts to the RS choice of pQCD [16]. Thus, the $\alpha^{\rm
HLF}_{g_{1}}(Q)$ prediction is parameter-free. Above $Q\simeq 1$ GeV HLFQCD
stops to be valid because its semiclassical potential does not include, by
definition, the short distance quantum effects responsible for the running of
a coupling. This is palliated by matching HLFQCD and pQCD near $Q\simeq 1$ GeV
where both formalisms apply, thereby providing $\alpha^{\rm HLF}_{g_{1}}(Q)$
at all $Q$ [16].

The DSE effective charge $\hat{\alpha}_{\rm PI}$ [25] is obtained starting
with the Pinch Technique [54] and Background Field Method [55]. They allow us
to define a process-independent QCD coupling in terms of a mathematically
reconstructed gluon two-point function analogous to the Gell-Mann Low
effective charge of QED. The $\hat{\alpha}_{\rm PI}$ is then computed by
combining the solution of DSE compatible with lattice QCD results. The
definition of $\hat{\alpha}_{\rm PI}$ explicitly factors in a renormalization
group invariant interaction, thus making it, like $\alpha_{g_{1}}(Q)$ and
$\alpha^{\rm HLF}_{g_{1}}(Q)$, to incorporate confinement [56]. Like them,
$\hat{\alpha}_{\rm PI}(Q)$ freezes at low $Q$ with a predicted infrared fixed-
point of $\hat{\alpha}_{\rm PI}(0)=(0.97\pm 0.04)\pi$. The mechanism at the
origin of the freezing in the DSE framework is the emergence of a dynamical
gluon mass $m_{g}(Q)$ [54, 57] that (A) regulates the Landau pole and (B)
decouples the dynamics at scales $Q\lesssim m_{g}(0)$, thereby causing the
coupling to lose its $Q$-dependence [58]. Like $\alpha^{\rm HLF}_{g_{1}}$,
$\hat{\alpha}_{\rm PI}$ is parameter-free and gauge-invariant but, in contrast
to the former and $\alpha_{g_{1}}$, $\hat{\alpha}_{\rm PI}$ is also process-
independent. No parameter is varied to predict the infrared fixed-point
$\hat{\alpha}_{\rm PI}(0)$ since it is largely fixed by the value of
$m_{g}(0)$, nor a matching is necessary to ensure agreement with the
perturbative determination of $\alpha^{\rm pQCD}_{\rm{g_{1}}}$ from the
renormalization group equations and the Bjorken sum rule. Crucially, the
practical determination of $\hat{\alpha}_{\rm PI}(Q)$ consistently
incorporates the extensive information from Lattice QCD on the gluon and ghost
propagators, thereby connecting this technique to $\alpha_{g_{1}}$.

Figure 1: Effective charge $\alpha_{g_{1}}(Q)/\pi$ obtained from JLab
experiments E03006/E97110 [33] (solid stars), E03006/E05111 [33] (solid
circles) and EG1dvcs [32] (solid triangles) and from COMPASS [31] (solid
square). Inner error bars represent the statistical uncertainties and outer
ones the systematic and statistical uncertainties added quadratically. The
open symbols show the older world data [27, 30, 28, 29] with the error bars
the quadratic sum of the systematic and statistical uncertainties. Also shown
are the HLFQCD [24] (red line, using the HLFQCD scale $\kappa=0.534$ GeV [59])
and DSE [25] (magenta line and hatched band) parameter-free predictions of
effective charges. The dashed line and hatched cyan band are
$\alpha_{g_{1}}(Q)/\pi$ obtained from the GDH and Bjorken sum rules,
respectively.

The new data on $\alpha_{g_{1}}$ agree well with the older data and display a
much improved precision over the whole $Q$ range covered. In addition, the
data now reach clearly the freezing domain of QCD at very low $Q$. That
$\alpha_{g_{1}}$ freezes could be already inferred with the old data but only
by complementing them with the GDH sum rule or/and the $\alpha_{g_{1}}(0)=\pi$
constraint. For the first time, the onset of freezing is now visible with data
only. One notes that only three of the lowest $Q$ points agree with the GDH
expectation. This may signal a fast arising $Q$-dependence beyond the leading
behavior given by GDH. The data agree well with the $\alpha^{\rm HLF}_{g_{1}}$
and $\hat{\alpha}_{\rm PI}$ predictions. That such agreements would occur was
not obvious and is a significant finding. The possible tension between the
data and $\hat{\alpha}_{\rm PI}$ for the range $0.3\lesssim Q\lesssim 0.5$ GeV
may be because $\alpha_{g_{1}}$ and $\hat{\alpha}_{\rm PI}$ are not exactly
the same effective charges (e.g., at high $Q$,
$\nicefrac{{\alpha_{g_{1}}}}{{\hat{\alpha}_{\rm PI}}}\simeq 1+0.05\alpha^{\rm
pQCD}_{\rm{s}}\neq 1$), but it is noteworthy that it occurs only in the
moderately low $Q$ domain where the ghost-gluon vacuum effect as computed in
the Landau gauge contributes the most to $\hat{\alpha}_{\rm PI}$.

##  III Summary and conclusion

We used the new JLab data and COMPASS datum on the Bjorken sum to extract the
QCD effective charge $\alpha_{g_{1}}(Q)$ in the $Q$-range $0.14\leq Q\leq
2.18$ GeV. The new result displays a significantly higher precision compared
to the older extractions of $\alpha_{g_{1}}(Q)$, and improve the low $Q$ reach
by about a factor of 2.

The new data show that $\alpha_{g_{1}}(Q)$ “freezes”, viz, loses its
$Q$-dependence, at small $Q$, saturating at an infrared fixed-point
$\alpha_{g_{1}}(Q\simeq 0)\simeq\pi$. This was already apparent with the older
data when combined with the GDH sum rule expectation, but the new data
explicitly display the behavior without needing the sum rule and with
significantly higher precision. The freezing of $\alpha_{g_{1}}(Q)$ together
with the smallness of the light quark masses, makes QCD approximately
conformal at low $Q$. The conformal behavior vanishes when transiting from the
low-$Q$ effective degrees of freedom of QCD (hadrons) to the large-$Q$
fundamental ones (partons) where conformality is then restored (the long-known
Bjorken scaling [17]). This transition is revealed by the drastic change of
value of the effective charge. It occurs at a $Q$ value indicative of the
chiral symmetry breaking parameter, $\Lambda_{B}\simeq 1$ GeV. The breaking at
low $Q$ of chiral symmetry, one of the crucial properties of QCD, is believed
to cause the emergence of the global properties of hadrons.

The new data agree well with sum rule predictions and with the latest
predictions from DSE and from a AdS/CFT-based approach. It shows that a strong
coupling can be consistently defined in the nonperturbative domain of QCD,
namely as an effective charge analogous to the definition used in QED, and
that it can then be used to compute a large variety of hadronic quantities and
other phenomena in which the strong interaction plays a role.

Acknowledgments The authors thank D. Binosi, S. J. Brodsky, Z.-F. Cui, G. F.
de Téramond, J. Papavassiliou, C. D. Roberts and J. Rodríguez-Quintero for
their valuable comments on the manuscript. This work is supported by the U.S.
Department of Energy, Office of Science, Office of Nuclear Physics, contracts
DE-AC05-06OR23177 and DE-FG02-99ER41101.

$Q$ (GeV) | $\alpha_{g_{1}}\pm\rm stat.\pm syst.$
---|---
0.143 |  3.064 $\pm 0.043\pm 0.018$
0.156 |  3.129 $\pm 0.046\pm 0.019$
0.171 |  2.955 $\pm 0.046\pm 0.023$
0.187 |  3.083 $\pm 0.044\pm 0.024$
0.204 |  3.022 $\pm 0.049\pm 0.024$
0.223 |  3.002 $\pm 0.052\pm 0.027$
0.243 |  2.988 $\pm 0.055\pm 0.031$
0.266 |  2.947 $\pm 0.060\pm 0.035$
0.291 |  2.983 $\pm 0.065\pm 0.035$
0.317 |  2.961 $\pm 0.062\pm 0.038$
0.347 |  2.730 $\pm 0.070\pm 0.044$
0.379 |  2.853 $\pm 0.077\pm 0.040$
0.414 |  2.745 $\pm 0.076\pm 0.041$
0.452 |  2.779 $\pm 0.090\pm 0.043$
0.494 |  2.451 $\pm 0.094\pm 0.044$
0.540 |  2.397 $\pm 0.092\pm 0.039$
0.590 |  2.349 $\pm 0.101\pm 0.040$
0.645 |  2.431 $\pm 0.109\pm 0.043$
0.704 |  1.996 $\pm 0.131\pm 0.104$

$Q$ (GeV) | $\alpha_{g_{1}}\pm\rm stat.\pm syst.$
---|---
0.187 |  3.016 $\pm 0.009\pm 0.027$
0.239 |  2.973 $\pm 0.015\pm 0.035$
0.281 |  2.952 $\pm 0.021\pm 0.041$
0.316 |  2.929 $\pm 0.017\pm 0.048$
0.387 |  2.815 $\pm 0.021\pm 0.076$
0.447 |  2.704 $\pm 0.025\pm 0.086$
0.490 |  2.575 $\pm 0.031\pm 0.053$
0.775 |  1.743 $\pm 0.007\pm 0.071$
0.835 |  1.571 $\pm 0.007\pm 0.101$
0.917 |  1.419 $\pm 0.009\pm 0.132$
0.986 |  1.341 $\pm 0.010\pm 0.147$
1.088 |  1.272 $\pm 0.010\pm 0.156$
1.167 |  1.121 $\pm 0.013\pm 0.153$
1.261 |  0.955 $\pm 0.016\pm 0.146$
1.384 |  0.874 $\pm 0.016\pm 0.269$
1.522 |  0.730 $\pm 0.012\pm 0.280$
1.645 |  0.708 $\pm 0.009\pm 0.257$
1.795 |  0.617 $\pm 0.007\pm 0.254$
1.967 |  0.581 $\pm 0.006\pm 0.223$
2.177 |  0.636 $\pm 0.003\pm 0.187$

Table 1:  Data on $\alpha_{g_{1}}(Q)$ from JLab experiments EG4 (left),
EG4/E97110 (top right) and EG1dvcs (bottom right).

## References

  * [1] D. d’Enterria, et al. “The strong coupling constant: State of the art and the decade ahead,” [arXiv:2203.08271].
  * [2] P. A. Zyla et al. [Particle Data Group], “Review of Particle Physics,” PTEP 2020 (2020) 8, 083C01, and 2021 update.
  * [3] 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].
  * [4] A. Dobado and D. Espriu, “Strongly coupled theories beyond the Standard Model,” Prog. Part. Nucl. Phys. 115, 103813 (2020) [arXiv:1911.06844].
  * [5] P. A. M. Dirac, “Forms of relativistic dynamics,” Rev. Mod. Phys. 21, 392 (1949). For a recent review, see: S. J. Brodsky, A. Deur and C. D. Roberts, “Artificial Dynamical Effects in Quantum Field Theory,” [arXiv:2202.06051].
  * [6] S. J. Brodsky and G. F. de Téramond, “Light-front hadron dynamics and AdS/CFT correspondence,” Phys. Lett. B 582, 211-221 (2004) [arXiv:hep-th/0310227].
  * [7] P. Maris and C. D. Roberts, “Dyson-Schwinger equations: A Tool for hadron physics,” Int. J. Mod. Phys. E 12, 297-365 (2003) [nucl-th/0301049].
  * [8] A. Deur, S. J. Brodsky and G. F. de Téramond, “The QCD Running Coupling,” Prog. Part. Nucl. Phys. 90, 1 (2016) [arXiv:1604.08082].
  * [9] G. Grunberg, “Renormalization Group Improved Perturbative QCD,” Phys. Lett. 95B, 70 (1980) Erratum: [Phys. Lett. 110B, 501 (1982)]; “Renormalization Scheme Independent QCD and QED: The Method of effective charges,” Phys. Rev. D 29, 2315 (1984); “On Some Ambiguities in the Method of effective charges,” Phys. Rev. D 40, 680 (1989).
  * [10] M. Gell-Mann and F. E. Low, “Quantum electrodynamics at small distances,” Phys. Rev. 95, 1300 (1954).
  * [11] S. J. Brodsky and H. J. Lu, “Commensurate scale relations in quantum chromodynamics,” Phys. Rev. D 51, 3652 (1995) [hep-ph/9405218]. S. J. Brodsky, M. Mojaza and X. G. Wu, “Systematic Scale-Setting to All Orders: The Principle of Maximum Conformality and Commensurate Scale Relations,” Phys. Rev. D 89 (2014), 014027 [arXiv:1304.4631.
  * [12] J. D. Bjorken, “Applications of the Chiral $U(6)\times U(6)$ Algebra of Current Densities,” Phys. Rev. 148, 1467 (1966); “Inelastic Scattering of Polarized Leptons from Polarized Nucleons,” Phys. Rev. D 1, 1376 (1970).
  * [13] D. J. Gross and C. H. Llewellyn Smith, “High-energy neutrino-nucleon scattering, current algebra and partons,” Nucl. Phys. B 14, 337 (1969).
  * [14] 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) [hep-ph/0212078].
  * [15] S. G. Gorishnii, A. L. Kataev and S. A. Larin, “The $O(\alpha^{3}_{s})$-corrections to $\sigma_{tot}(e^{+}e^{-}\rightarrow hadrons)$ and $\Gamma(\tau^{-}\rightarrow\nu_{\tau}+hadrons)$ in QCD,” Phys. Lett. B 259 (1991), 144-150
  * [16] 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]; “On the interface between perturbative and nonperturbative QCD,” Phys. Lett. B 757, 275 (2016) [arXiv:1601.06568].
  * [17] J. D. Bjorken, ‘‘Asymptotic Sum Rules at Infinite Momentum,’’ Phys. Rev. 179 (1969), 1547-1553.
  * [18] A. Deur, S. J. Brodsky and G. F. de Téramond, ‘‘The Spin Structure of the Nucleon,’’ Rep. Prog. Phys., 82, 7 (2019) arXiv:1807.05250.
  * [19] A. L. Kataev, ‘‘The Ellis-Jaffe sum rule: The estimates of the next-to-next-to-leading order QCD corrections,’’ Phys. Rev. D 50, 5469 (1994) [hep-ph/9408248]; ‘‘Deep inelastic sum rules at the boundaries between perturbative and nonperturbative QCD,’’ Mod. Phys. Lett. A 20, 2007 (2005) [hep-ph/0505230]; P. A. Baikov, K. G. Chetyrkin and J. H. Kuhn, ‘‘Adler function, Bjorken sum rule, and the Crewther relation to order $\alpha_{\rm s}^{4}$ in a general gauge theory,’’ Phys. Rev. Lett. 104, 132004 (2010) [arXiv:1001.3606].
  * [20] M. Burkardt, ‘‘Transverse force on quarks in deep-inelastic scattering,’’ Phys. Rev. D 88, 114502 (2013) [arXiv:0810.3589]; M. Abdallah and M. Burkardt, ‘‘Transverse force on transversely polarized quarks in longitudinally polarized nucleons,’’ Phys. Rev. D 94, 094040 (2016) [arXiv:1610.01166].
  * [21] A. Deur, ‘‘Spin Sum Rules and the Strong Coupling Constant at large distance,’’ AIP Conf. Proc. 1155 (2009) 1, 112-121 [arXiv:0906.4285].
  * [22] S. B. Gerasimov, ‘‘A Sum rule for magnetic moments and the damping of the nucleon magnetic moment in nuclei,’’ Sov. J. Nucl. Phys. 2, 430 (1966) [Yad. Fiz. 2, 598 (1965)]; ‘‘S. D. Drell and A. C. Hearn, Exact Sum Rule for Nucleon Magnetic Moments,’’ Phys. Rev. Lett. 16, 908 (1966); M. Hosoda and K. Yamamoto ‘‘Sum Rule for the Magnetic Moment of the Dirac Particle,’’ Prog. Theor. Phys. 36 (2), 425 (1966).
  * [23] V. D. Burkert, ‘‘Comment on the generalized Gerasimov-Drell-Hearn sum rule in chiral perturbation theory,’’ Phys. Rev. D 63 (2001), 097904 [nucl-th/0004001.
  * [24] 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].
  * [25] D. Binosi, et al. ‘‘Process-independent strong running coupling,’’ Phys. Rev. D 96, 5, 054026 (2017) [arXiv:1612.04835]; Z. F. Cui, et al. ‘‘Effective charge from lattice QCD,’’ Chin. Phys. C 44, 8, 083102 (2020) [arXiv:1912.08232].
  * [26] A. Deur, V. Burkert, J. P. Chen and W. Korsch, ‘‘Experimental determination of the effective strong coupling constant,’’ Phys. Lett. B 650, 244 (2007); [hep-ph/0509113]; ‘‘Determination of the effective strong coupling constant $\alpha_{s,g_{1}}(Q^{2})$ from CLAS spin structure function data,’’ Phys. Lett. B 665, 349 (2008) [arXiv:0803.4119].
  * [27] B. Adeva et al. [SMC], ‘‘Measurement of the spin dependent structure function g1(x) of the deuteron,’’ Phys. Lett. B 302, 533 (1993); ‘‘The Spin dependent structure function g(1) (x) of the proton from polarized deep inelastic muon scattering,’’ Phys. Lett. B 412, 414 (1997); D. Adams et al. [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 (1994)] [hep-ph/9404270]; ‘‘A New measurement of the spin dependent structure function g1(x) of the deuteron,’’ Phys. Lett. B 357, 248 (1995); ‘‘The Spin dependent structure function g1(x) of the deuteron from polarized deep inelastic muon scattering,’’ Phys. Lett. B 396, 338 (1997); ‘‘Spin structure of the proton from polarized inclusive deep inelastic muon - proton scattering,’’ Phys. Rev. D 56, 5330 (1997) [hep-ex/9702005].
  * [28] 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) [hep-ex/0008037]; ‘‘The $Q^{2}$ dependence of the generalized Gerasimov-Drell-Hearn integral for the deuteron, proton and neutron,’’ Eur. Phys. J. C 26, 527 (2003) [hep-ex/0210047]; 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) [hep-ex/9809015].
  * [29] A. Deur et al., ‘‘Experimental determination of the evolution of the Bjorken integral at low $Q^{2}$,’’ Phys. Rev. Lett. 93, 212001 (2004) [hep-ex/0407007]; ‘‘Experimental study of isovector spin sum rules,’’ Phys. Rev. D 78, 032001 (2008) [arXiv:0802.3198]; K. Slifer et al. ‘‘Probing quark-gluon interactions with transverse polarized scattering,’’ Phys. Rev. Lett. 105, 101601 (2010) [arXiv:0812.0031].
  * [30] P. L. Anthony et al. [E142], ‘‘Deep inelastic scattering of polarized electrons by polarized He-3 and the study of the neutron spin structure,’’ Phys. Rev. D 54, 6620 (1996) [hep-ex/9610007]; K. Abe et al. [E143], ‘‘Precision measurement of the proton spin structure function g1(p),’’ Phys. Rev. Lett. 74, 346 (1995); ‘‘Precision measurement of the deuteron spin structure function g1(d),’’ Phys. Rev. Lett. 75, 25 (1995); ‘‘Measurements of the $Q^{2}$ dependence of the proton and deuteron spin structure functions g1(p) and g1(d),’’ Phys. Lett. B 364, 61 (1995) [hep-ex/9511015]; ‘‘Measurements of the proton and deuteron spin structure function g1 in the resonance region,’’ Phys. Rev. Lett. 78, 815 (1997) [hep-ex/9701004]; ‘‘Measurements of the proton and deuteron spin structure functions g(1) and g(2),’’ Phys. Rev. D 58, 112003 (1998) [hep-ph/9802357]; K. Abe et al. [E154], ‘‘Precision determination of the neutron spin structure function g1(n),’’ Phys. Rev. Lett. 79, 26 (1997) [hep-ex/9705012]; ‘‘Next-to-leading order QCD analysis of polarized deep inelastic scattering data,’’ Phys. Lett. B 405, 180 (1997) [hep-ph/9705344]; P. L. Anthony et al. [E155], ‘‘Measurement of the deuteron spin structure function g1(d)(x) for 1-(GeV/c)${}^{2}<Q^{2}<40$-(GeV/c)2,’’ Phys. Lett. B 463, 339 (1999) [hep-ex/9904002]; ‘‘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) [hep-ph/0007248].
  * [31] 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].
  * [32] A. Deur et al., ‘‘High precision determination of the $Q^{2}$ evolution of the Bjorken Sum,’’ Phys. Rev. D 90, 1, 012009 (2014) [arXiv:1405.7854].
  * [33] A. Deur, et al. ‘‘Experimental study of the behavior of the Bjorken sum at very low Q2,’’ Phys. Lett. B 825, 136878 (2022) [arXiv:2107.08133].
  * [34] J. Rodríguez-Quintero et al. ‘‘Process-independent effective coupling. From QCD Green’s functions to phenomenology,’’ Few Body Syst. 59, 6, 121 (2018) [arXiv:1801.10164].
  * [35] A. Deur et al. ‘‘Implications of the Principle of Maximum Conformality for the QCD Strong Coupling,’’ Phys. Lett. B 773, 98 (2017) [arXiv:1705.02384].
  * [36] A. Deur, S. J. Brodsky and G. F. de Téramond, ‘‘Determination of $\Lambda_{\bar{MS}}$ at five loops from holographic QCD,’’ J. Phys. G 44, 10, 105005 (2017) [arXiv:1608.04933].
  * [37] Z. F. Cui, et al., ‘‘Kaon and pion parton distributions,’’ Eur. Phys. J. C 80 (2020) no.11, 1064.
  * [38] R. A. Janik, ‘‘The Dynamics of Quark-Gluon Plasma and AdS/CFT,’’ Lect. Notes Phys. 828, 147-181 (2011) [arXiv:1003.3291].
  * [39] W. Busza, K. Rajagopal and W. van der Schee, ‘‘Heavy Ion Collisions: The Big Picture, and the Big Questions,’’ Ann. Rev. Nucl. Part. Sci. 68, 339-376 (2018) [arXiv:1802.04801].
  * [40] W. Florkowski, M. P. Heller and M. Spalinski, ‘‘New theories of relativistic hydrodynamics in the LHC era,’’ Rept. Prog. Phys. 81, no.4, 046001 (2018) [arXiv:1707.02282].
  * [41] N. Jokela, M. Järvinen and J. Remes, ‘‘Holographic QCD in the Veneziano limit and neutron stars,’’ JHEP 03, 041 (2019) [arXiv:1809.07770]; P. M. Chesler, N. Jokela, A. Loeb and A. Vuorinen, ‘‘Finite-temperature Equations of State for Neutron Star Mergers,’’ Phys. Rev. D 100, no.6, 066027 (2019) [arXiv:1906.08440]; C. Ecker, M. Järvinen, G. Nijs and W. van der Schee, ‘‘Gravitational waves from holographic neutron star mergers,’’ Phys. Rev. D 101, no.10, 103006 (2020) [arXiv:1908.03213].
  * [42] V. Sulkosky et al., ‘‘Measurement of the ${}^{3}He$ spin-structure functions and of neutron (3He) spin-dependent sum rules at $0.035\leq Q^{2}\leq 0.24$ GeV2,’’ Phys. Lett. B 805, 135428 (2020) [arXiv:1908.05709]; ‘‘Puzzle with the precession of the neutron spin,’’ Nature Phys. 17 687 (2021) [arXiv:2103.03333].
  * [43] J. Alcorn, et al. ‘‘Basic Instrumentation for Hall A at Jefferson Lab,’’ Nucl. Instrum. Meth. A 522, 294-346 (2004).
  * [44] B. A. Mecking et al. ‘‘The CEBAF Large Acceptance Spectrometer (CLAS),’’ Nucl. Instrum. Meth. A 503, 513-553 (2003).
  * [45] Y. Prok et al. ‘‘Precision measurements of $g_{1}$ of the proton and the deuteron with 6 GeV electrons,’’ Phys. Rev. C 90, 025212 (2014) [arXiv:1404.6231].
  * [46] X. Zheng et al., ‘‘Measurement of the proton spin structure at long distances,’’ Nature Phys. 17 736 (2021) (2021) [arXiv:2102.02658].
  * [47] K. P. Adhikari et al., ‘‘Measurement of the Q2-dependence of the deuteron spin structure function g1 and its moments at low Q2 with CLAS,’’ Phys. Rev. Lett. 120, 6, 062501 (2018) [arXiv:1711.01974].
  * [48] V. Sulkosky Ph.D. thesis, College of William & Mary (2007), Williamsburg, Va 23187 USA
  * [49] C. D. Keith et al., ‘‘A polarized target for the CLAS detector,’’ Nucl. Instrum. Meth. A 501, 327-339 (2003).
  * [50] F. Garibaldi et al., ‘‘High-resolution hypernuclear spectroscopy at Jefferson Lab, Hall A,’’ Phys. Rev. C 99, 054309 (2019) [arXiv:1807.09720].
  * [51] J. H. Kim et al., ‘‘A measurement of $\alpha_{\rm s}(Q^{2})$ from the Gross--Llewellyn Smith sum rule,’’ Phys. Rev. Lett. 81, 3595 (1998) [hep-ex/9808015].
  * [52] V. de Alfaro, S. Fubini and G. Furlan, ‘‘Conformal invariance in quantum mechanics,’’ Nuovo Cim. A 34, 569 (1976).
  * [53] A. P. Trawinski et al. ‘‘Effective confining potentials for QCD,’’ Phys. Rev. D 90, 074017 (2014) [arXiv:1403.5651].
  * [54] J. M. Cornwall, ‘‘Dynamical Mass Generation in Continuum QCD,’’ Phys. Rev. D 26, 1453 (1982); D. Binosi and J. Papavassiliou, ‘‘Pinch Technique: Theory and Applications,’’ Phys. Rept. 479, 1-152 (2009) [arXiv:0909.2536].
  * [55] L. F. Abbott, ‘‘The Background Field Method Beyond One Loop,’’ Nucl. Phys. B 185, 189-203 (1981).
  * [56] D. Binosi, L. Chang, J. Papavassiliou and C. D. Roberts, ‘‘Bridging a gap between continuum-QCD and ab initio predictions of hadron observables,’’ Phys. Lett. B 742 (2015), 183-188 [arXiv:1412.4782; F. Gao, S. X. Qin, C. D. Roberts and J. Rodriguez-Quintero, ‘‘Locating the Gribov horizon,’’ Phys. Rev. D 97 (2018) no.3, 034010 [arXiv:1706.04681].
  * [57] A. C. Aguilar, D. Binosi and J. Papavassiliou, ‘‘Gluon and ghost propagators in the Landau gauge: Deriving lattice results from Schwinger-Dyson equations,’’ Phys. Rev. D 78 (2008), 025010 [arXiv:0802.1870].
  * [58] S. J. Brodsky and R. Shrock, ‘‘Maximum wavelength of confined quarks and gluons and properties of quantum chromodynamics,’’ Phys. Lett. B 666, 95 (2008) [arXiv:0806.1535].
  * [59] R. S. Sufian et al., ‘‘Nonperturbative strange-quark sea from lattice QCD, light-front holography, and meson-baryon fluctuation models,’’ Phys. Rev. D 98 (2018) no.11, 114004 [arXiv:1809.04975].