been underway to understand it. As we have described herein, they can be
classified into two categories.

The first category we considered is that body which aims at a precise
determination of $\alpha_{s}$ at high energy, with the reference point set to
$M_{\rm Z}$, the $Z_{0}$ mass. There, the theoretical framework defining
$\alpha_{s}(Q^{2})$ is well understood and is based on the RGE [58, 27, 59]
and perturbation theory. The leading behavior of $\alpha_{s}(Q^{2})$ is a
logarithmic decrease with $Q^{2}$ [82, 83]. Accordingly, both $\alpha_{s}$ and
its $Q^{2}$-dependence vanish as $Q^{2}\to\infty$, which explains Bjorken
scaling [366] and enables the use of QCD perturbation theory for the
prediction of high-energy phenomena – arguably the keystone in establishing
QCD as the best existing candidate for a theory of the strong force.

Today, the main goal of high-energy studies of $\alpha_{s}$ is to determine
this coupling with an accuracy $\Delta\alpha_{s}/\alpha_{s}\ll 1\%$ [22], the
current global assessment being $\alpha_{s}(M_{\rm z})=0.1176(10)$ without LGT
($\Delta\alpha_{s}/\alpha_{s}=0.85\%$) and $\alpha_{s}(M_{\rm z})=0.1179(09)$
with LGT ($\Delta\alpha_{s}/\alpha_{s}=0.76\%$) [5].

High accuracy is evidently necessary for the understanding of QCD, but also
for Standard Model studies and searches for physics beyond the Standard Model.
For the former, it is because QCD corrections to Standard Model reactions
often dominate the uncertainties, since $\alpha_{s}$ is much larger than
$\alpha$ and $G_{F}$ and known with several orders-of-magnitude less accuracy.
An example of the latter is testing the universality of $\alpha_{s}(M_{\rm
z})$ by measuring it at a scale $Q^{2}\neq M^{2}_{\rm Z}$ and evolving it to
$M^{2}_{\rm Z}$.

Reaching $\Delta\alpha_{s}/\alpha_{s}\ll 1\%$ requires combining many
determinations of $\alpha_{s}(M_{\rm Z})$ obtained from distinct observables
whose approximant is well controlled. It also demands using either techniques
minimizing renormalization scale ambiguities [124, 119] or providing
calculations of pQCD approximants to at least past NLO. Several are currently
known to N3LO, while the $\beta$-series of QCD has been computed to five-loops
($\beta_{4}$) [104]. Nevertheless, when the common scheme of varying the
renormalization scale is employed [52] the uncertainty coming from the
truncation of pQCD series is generally a limiting factor in the extraction of
$\alpha_{s}(M_{\rm z})$ from measurement or LGT. In fact, LGT seems to be
emerging as a leading method for high precision determinations of
$\alpha_{s}(M_{\rm z})$ [21]; consequently, scrutinizing the systematics
associated with LGT has become crucial. Even so, experimental efforts on
$\alpha_{s}(M_{\rm z})$ remain imperative because LGT calculations do not
include physics beyond the Standard Model; thus, LGT can only contribute to
searches for such physics by providing a pure-QCD baseline.

The second category we discussed is the class of studies whose goal is to
understand $\alpha_{s}$ in the strong QCD regime. Like the precision efforts,
this is also challenging, albeit for different reasons. A first stumbling
block is placed by the many possible “obvious” definitions one can choose for
$\alpha_{s}$ in the IR: at first glance, no one choice is plainly superior to
another. In fact, it might be that several couplings with different values and
$Q^{2}$-dependence are necessary to characterize QCD: quark-gluon, ghost-
gluon, 3-gluon and 4-gluon vertices may couple distinctly. This could occur
when the Slavnov-Taylor identities – the QCD equivalent of the Ward-Green-
Takahashi identities in QED– fail to hold due to particular choices of gauge
and RS.

Compounding the number of distinct $\alpha_{s}$ behaviors is the enhancement
of the RS-dependence in the IR allied with the fact that the relation between
couplings calculated in different RSs is usually known only in the pQCD
framework. (See, however, the HLFQCD approach, Sec. 4.5 and Ref. [331], for an
exception.)

Finally, and not least amongst the reasons, nonperturbative calculations are
complex and difficult. This may lead to erroneous $\alpha_{s}$ behaviors owing
to artifacts from approximations, such as finite size effects in LGT or
truncation prescriptions for continuum Schwinger function methods.

Many prescriptions for $\alpha_{s}$ in IR have been studied, resulting in
$\alpha_{s}(Q^{2}\to 0)$ values ranging from 0 to $\infty$ [12]. While it
rapidly became clear that the Landau pole at $Q^{2}\simeq\Lambda^{2}_{s}$ was
a perturbative artifact and therefore could not represent the behavior of
$\alpha_{s}$ nor be the cause of color confinement, there was no consensus on
how $\alpha_{s}$ ought to behave in the IR.

A prominent proposal was that the coupling should possess an IR fixed point,
i.e., that it displays a plateau for $Q^{2}\ll\Lambda^{2}_{s}$, a behavior
called the (IR) freezing of $\alpha_{s}$. This was posited soon after the
advent of QCD [444, 84, 369, 28, 370, 646]. Other proposals were that
$\alpha_{s}$ should vanish as $Q^{2}\to 0$ [647, 648], or diverge as $1/Q^{2}$
[464] or that it should increase monotonically as $1/Q^{2}$, but without
diverging [620]. As explained above, many causes lie behind these widely
differing expectations.

Material progress in this area has been made in the past quinquennium: data
and theoretical calculations have now led to a coupling; namely, the effective
charge $\hat{\alpha}_{s}$ [35, 36] or the practically equivalent effective
charge $\alpha_{g_{1}}$ [364, 368], that is consistent and fruitful.
Consistent, because of the agreement between data [364, 365, 117] and the
parameter-free predictions from HLFQCD, DSE and LGT, which are in both cases
gauge-independent results by virtue of the nonperturbative technique for the
former and thanks to the pinch technique [28, 31] for the latter. As it
happens, this coupling also agrees with an earlier determination [28] after
correction for the difference arising from the different RS [331]. Consistent
also because the correlation functions underlying $\hat{\alpha}_{s}$ lead to
the expected IR fixed point. In fact, both the HLFQCD and the DSE/LGT
frameworks produce results with $\alpha_{s}(Q^{2}\ll\Lambda^{2}_{s})/\pi\simeq
1$.

Fruitful, because many hadronic quantities are calculable with such a coupling
and within the theoretical framework in which it is derived. These quantities
are, inter alia, hadron mass spectroscopy [112, 373], the polarized and
unpolarized PDFs and GPDs of hadrons [339, 340, 374, 375, 376, 377, 378],
elastic and transition form factors [380, 649, 650, 651], the quark mass
functions [652], the pion decay constant [373], the scale of QCD,
$\Lambda_{s}$ [325], and the connection between the soft and hard pomerons
[382].  The remarkable results for $\alpha_{s}(Q^{2})$ at all scales shown in
fig.4.1, combined with the elimination of scale and scheme ambiguities using
PMC scale setting [52], has opened a new range of predictions for physics
phenomena in both the perturbative and nonperturbative domains.  Therefore,
one now has in hand a coupling that can universally characterize QCD
interactions, that has the merit of being independent of the choice of gauge,
vertex, or process, and that can predict observables in the strong QCD domain.
This argues compellingly for the existence of a canonical QCD coupling, not
only in the UV region, but in the IR as well.

###### Acknowledgments.

This perspective is based on results obtained and insights developed through
collaborations with many people, to all of whom we are greatly indebted. This
work is supported by: U.S. Department of Energy, Office of Science, Office of
Nuclear Physics, contract DE-AC05-06OR23177 (AD), U.S. Department of Energy
contract DE-AC02-76SF00515 (SJB); and National Natural Science Foundation of
China grant no. 12135007 (CDR).

### Abbreviations

The following abbreviations are used in this manuscript:

AdS/CFT |  anti-de Sitter/conformal field theory
---|---
AdS/QCD |  anti-de Sitter/QCD
ALEPH |  Apparatus for LEP PHysics
APT |  analytical perturbation theory
ATLAS |  A Toroidal LHC Apparatus
BLM |  Brodsky-Lepage-Mackenzie
BRST |  Becchi-Rouet-Stora-Tyutin
CEBAF |  Continuous Electron Beam Accelerator Facility
CERN |  European Organization for Nuclear Research
CIPT |  contour-improved perturbation theory
CLAS |  CEBAF Large Acceptance spectrometer
CMS |  Compact Muon Solenoid (experiment)
CSR |  commensurate scale relations
CSMs |  continuum Schwinger function methods
dAFF |  de Alfaro-Fubini-Furlan
DCSB |  dynamical chiral symmetry breaking
DELPHI |  DEtector with Lepton, Photon and Hadron Identification
DESY |  Deutsches Elektronen-Synchrotron
DGLAP |  Dokshitzer–Gribov–Lipatov–Altarelli–Parisi
DIS |  deep inelastic scattering
DSE |  Dyson-Schwinger equation
EIC |  electron ion collider (at Brookhaven National Laboratory)
EicC |  electron ion collider in China
EHM |  emergent hadron mass
ERBL |  Efremov-Radyushkin-Brodsky-Lepage
fAPT |  fractional APT
FF |  (parton) fragmentation function
FLAG |  flavor lattice averaging group
FOPT |  fixed order perturbation theory
FRG |  functional renormalisation group
GPD |  generalized parton distribution
HERA |  Hadron-Electron Ring Accelerator
HLFQCD |  holographic light-front QCD
HRS |  High Resolution Spectrometer (experiment at PEP, or detector system at JLab)
HT |  higher-twist
IR |  infrared (physics/phenomena)
JLab |  Thomas Jefferson National Accelerator Facility
JADE |  JApan, Deutschland and England (experiment at PETRA)
L3 |  Third LEP experiment
LC |  Light Cone
LEP |  Large Electron-Positron collider
LF |  Light Front
LL |  leading logarithm
LGT |  lattice gauge theory
LHC |  large hadron collider (at CERN)
LO |  leading order
MS (and $\overline{\rm MS}$) |  Minimal Subtraction
NLL (and NiLL) |  next to … leading logarithm
NLO (and NiLO) |  next to … leading order
OPAL |  Omni-Purpose Apparatus for LEP
(P)DF |  (parton) distribution function
PDG |  Particle Data Group and associated publications
PEP |  Positron-Electron Project (facility at SLAC)
PETRA |  Positron-Electron Tandem Ring Accelerator
PI (charge) |  process-independent (charge)
PMC |  principle of maximum conformality
pQCD |  perturbative QCD
QCD |  quantum chromodynamics
QED |  quantum electrodynamics
QFT |  quantum field theory
RGE |  renormalization group equation
RGI |  renormalisation group invariant
RHIC |  relativistic heavy ion collider
RS |  Renormalization Scheme
SLAC |  Stanford Linear Accelerator Center
SLC |  Stanford Linear Collider
SLD |  SLAC Large Detector
SU(N) |  special unitary (group of degree n)
TASSO |  Two Arm Spectrometer SOlenoid
UV |  ultraviolet (physics/phenomena)

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