This plot is only meant to be representative, rather than exhaustive, of the
recent and most accurate measurements of $\alpha_{s}$. Fitting this choice of
individual determinations (excluding one of the two DIS highly correlated
determinations) yields a $\chi^{2}/ndf$ of 4.3, significantly larger than 1.
This demonstrates the tensions between the different determinations. Using the
Unbiased Estimate, the average value of the coupling is
$\alpha_{s}(M_{Z}^{2})=0.1171\pm 0.0006$. However, since we used only an
arbitrarily selected sample of the available $\alpha_{s}$ determinations, this
number should not be taken as a world average or compared to such averages.
The $\chi^{2}/ndf$ values from other compilations before applying the Unbiased
Estimate procedure also underline these tensions. In particular, there are
clearly two classes of results differing by about 4%. The lower class of
values comes from DIS measurements, $\tau-$decay and thrust measurements. The
Unbiased Estimate is certainly not a flawless compilation procedure since it
does not account for possible correlations between results. While the question
of how to best combine all the data remains difficult, it is likely that the
large $\chi^{2}/ndf$ values are due to a tendency to underestimate
uncertainties rather than inadequate compilation procedures, indications of
physics beyond the Standard Model, or of a failing of QCD. Indeed, the
consistency within a few percent between all determinations is a crucial
verification of the validity of QCD.

##  Chapter 4 The strong coupling in the nonperturbative domain

As mentioned in the Preamble, knowing the strong coupling in the
nonperturbative domain is necessary to understand both high energy and
hadronic phenomena. An example is the calculation of the Sivers function [140]
encountered in single-spin pseudo-T-odd semi-inclusive DIS. In addition, the
_renormalization scale_ at arbitrarily small momenta $q^{2}\sim v^{2}S$ is
required to evaluate heavy quark production as one approaches threshold [141].
Finally, knowing the IR-behavior of $\alpha_{s}$ is important to understand
the mechanisms for dynamical chiral symmetry breaking [142]. The following
references provide other examples of phenomena for which knowing the IR-
behavior of $\alpha_{s}$ is required: [143, 144, 145, 146, 147, 148, 149]. In
addition, the review [151, 150] provides a concise description of the impact
of $\alpha_{s}$ on our understanding of hadron dynamics. Additional reviews on
$\alpha_{s}$ in the IR regime are given in [2, 152, 153].

Studying $\alpha_{s}$ in the IR is more challenging than in the UV domain
since the perturbative formalism involving the fundamental QCD fields cannot
be used. Consequently, a number of approaches have been explored. Each
approach has its benefits, justifications and limitations. In this second part
of this review, we will explore the insights brought by the various
approaches. Since there is still no consensus on how $\alpha_{s}$ should be
defined, such discussion is necessary for several reasons: 1) the different
definitions, and consequently meanings, of $\alpha_{s}$ need to be understood;
2) the approximations used and the arbitrary choices made (e.g., the RS and
gauge) need to be noted; 3) the connections between the different approaches
need to be identified; 4) it illustrates how this challenging problem can be
approached by a variety of methods; 5) it acknowledges the theoretical
contributions, early or recent, toward solving the problem of the IR-behavior
of $\alpha_{s}$; 6) finally, the variety of approaches to $\alpha_{s}$ in the
IR domain can allow a cross-check of the conclusions.

Although there are important results and constraints obtained from
experiments, most investigations of the IR-behavior of $\alpha_{s}$ have been
theoretical. Some of the insightful theoretical approaches address theories
which are closely related to QCD, including pure gluonic Yang–Mills theory
(without quark fields), spaces of different Minkowsky or Euclidean dimensions,
as well as SU($N_{C}$) with various numbers of colors $N_{C}$. In that last
context, it is useful to note that one recovers Abelian theory in the limit
$N_{C}\to 0$ at fixed $\alpha_{s}C_{F}=\alpha$ where $C_{F}={N^{2}_{C}-1\over
2N_{C}}$ [155].

Perturbative QCD, via Eq. (3.24), evidently predicts that $\alpha_{s}(Q^{2})$
diverges at the _Landau pole_ , when $Q^{2}\rightarrow\Lambda^{2}$. However,
this is not a meaningful prediction since it occurs, by definition, outside
the domain of pQCD’s validity. As we will discuss in Section 5.3, the absence
of nonperturbative terms in the series (3.24) is responsible for this
unphysical prediction. Consequently, the Landau singularity cannot be cured by
simply adding more perturbative terms to Eq. (3.24). In fact, they can worsen
the situation; for example, a typical quantum field theory perturbative
approximant is a _Poincaré series_ (_asymptotic series_) which diverges beyond
the order given approximately by the inverse of the expansion coefficient
[156]. At $Q^{2}=1$ GeV2, this is typically $\alpha_{s}/\pi\simeq 0.2$. Thus,
the perturbative approximant of an observable will start diverging after
$n\simeq 5$. Since the $\beta-$series is expanded in powers of $\hbar$, this
discussion does not directly concern its convergence, except that one
traditionally chooses the $\beta_{i}$ order in Eq. (3.24) to be the same as
the $\alpha_{s}$ order in the approximant series. Thus, the $\beta_{i}$ and
$\alpha_{s}$ orders are linked.

It is often heard that the diverging behavior of the perturbatively calculated
$\alpha_{s}$ is responsible for quark confinement. This ignores the fact that
the _Landau pole_ is unphysical and that Eq. (3.24) does not include important
diagrams which connect quarks via multi-gluon exchange such as the “H”
diagrams, see Fig. 2.2(d). Such diagrams must be included for large distances
calculations. Nevertheless, one can consider the possibility that $\alpha_{s}$
diverges as $1/Q^{2}$ in the $Q^{2}\rightarrow 0$ limit, or one can argue that
a large value for the coupling is an ingredient for confinement, but
confinement does not require $\alpha_{s}$ to diverge. For example, according
to lattice gauge theory simulations, the static
$\mbox{Q--}\overline{\mbox{Q}}$ potential continues to rise linearly well
after the coupling has reached a maximum value, typically at distances larger
than 0.4 fm. The predicted QCD potential continues to rise linearly for
$\mbox{Q--}\overline{\mbox{Q}}$ separations of order 1.3 fm [158] or even
larger distances for _quenched_ calculations [159]. Furthermore, the IR value
of
$\left\langle\alpha_{s}\right\rangle=\frac{1}{E}\int_{0}^{E}\,dQ\alpha_{s}(Q^{2})$
obtained from analyzing jet shape observables in $e^{+}e^{-}$ annihilation and
DIS is finite and typically modest:
$\left\langle\alpha_{s}\right\rangle=0.47\pm$0.07 for $E=2$ GeV [160]. In the
stochastic vacuum model approach to high-energy scattering [161], it is found
that $\alpha_{s}\simeq 0.81$ in the IR. In another approach, Gribov showed
that light quarks are super-critically bound if the averaged coupling has a
moderate value of 0.43 at large distances [162].

Most definitions of $\alpha_{s}$ in the IR attempt to generalize the pQCD
definition to include QCD’s confining effects. Thus, the question of the IR-
behavior of $\alpha_{s}$ is intimately linked to the topic of confinement.
Consequently, confinement needs also to be addressed here. In the approaches
that we will discuss, $\alpha_{s}$ is defined from the behavior of quark,
gluon, and/or ghost propagators which are believed to be directly relevant to
confinement. Such definitions of $\alpha_{s}$ are used in the Schwinger–Dyson
framework (Section 4.4), the lattice technique (Sections 3.9.4 and 4.5), the
Gribov–Zwanziger approach (Section 4.7), the Functional Renormalization Group
framework (Section 4.6) and Stochastic Quantization (Section 4.8).

A popular picture of confinement, first put forth by Gribov [163] and then
developed by Zwanziger (Section 4.7), is supported by the different framework
just mentioned. It is useful to have such confinement picture in mind in order
to understand the implications of the IR-behavior of $\alpha_{s}$. In the
Gribov–Zwanziger scenario, the gluon propagator is IR-suppressed compared to
an extrapolation of the $1/Q^{2}$ UV-behavior. Meanwhile, the ghost propagator
is IR-enhanced. Consequently, in this picture, confinement results from the
long-distance propagations of ghosts. Thus, the behavior of the ghost and
gluon propagators dictate the IR-evolution of $\alpha_{s}$.

Since ghosts are unphysical artifacts reflecting gauge choices and introduced
to force the gluon propagator to be transverse (see Eqs. (3.28) and (3.29) and
the discussion in between), a more intuitive but equivalent view is provided
by the Stochastic Quantization framework in which there are no ghosts. Their
role is played by longitudinal gluons in this formalism (see Section 4.8). At
distances close to a color charge where pQCD rules and where spherical
symmetry is not yet disturbed by the presence of another color charge, short
wavelength gluons can be represented by plane waves (on-shell; i.e.,
transversely polarized gluons). At distances relevant to the large distance
separation of the two color charges, the gluon field becomes highly asymmetric
and is postulated to collapse into a flux tube. Consequently, the plane waves
are fully distorted and longitudinally-polarized off-shell gluons dominate.
Furthermore, it was shown that the ghost propagator (a manifestation of
longitudinal gluons) in Landau gauge, $\partial_{\mu}A^{\mu}=0$, is related to
the instantaneous Coulomb propagator in the Coulomb gauge,
$\nabla\cdot\mathbf{A}=0$. The Coulomb propagator is thus enhanced at large
distance [164], which leads to a linear $\mbox{Q--}\overline{\mbox{Q}}$
potential [165, 166, 167]. A dominating longitudinal gluon propagator
indicates off-shell gluons, and thus the concept of effectively massive gluons
becomes relevant (see Section 4.4.4).

This picture of confinement implies a specific IR-behavior of $\alpha_{s}$,
namely that $\alpha_{s}$ loses its $Q^{2}-$dependence in the IR regime. In
other words, it effectively has a _conformal behavior_ , _freezing_ at a given
value: an _infrared fixed point_. Such behavior appears to be supported by
measurements and calculations using different techniques as we will see in the
rest of the review.

However, the Gribov–Zwanziger scenario described above has been challenged, as
will be further discussed in Section 4.4.3. In fact, as we already said, there
is no agreement on the IR-behavior of $\alpha_{s}$. The only certitude is that
the _Landau pole_ is unphysical. Rather than validating the _conformal_
scenario, some studies indicate that $\alpha_{s}(Q^{2})$ reaches its maximum
in the IR-UV transition region and then vanishes in the deep IR. Other
investigations point to a divergence of $\alpha_{s}(Q^{2})$ when
$Q^{2}\rightarrow 0$, where this divergence is unrelated to the _Landau pole_.

There are several reasons for these different conclusions. First and foremost,
multiple definitions of $\alpha_{s}(Q^{2})$ in the IR exist. Second,
approximations are not always under control, and thus they may lead to
unphysical artifacts. Third, there is no agreement on the more suitable form
(instant form or front form) [129], gauge, and RS to use. Finally, since some
of the results are model-dependent, some assumptions on which the model rests
may not be systematically valid.

In the next sections, we will review the different approaches used or
developed to study the low IR-behavior of $\alpha_{s}$, as listed by the
theoretical techniques. Then we will compare the various results and discuss
their differences.

First, we will briefly expand the discussion on the possible types of IR-
behaviors for $\alpha_{s}$. The possibility that $\alpha_{s}(Q^{2})$ loses its
$Q^{2}$-dependence in the IR; i.e., $\beta(Q^{2})\rightarrow 0$ in the IR
domain, was pointed out in the early days of QCD [16, 168, 169]. A general
argument, given in Refs. [5, 6], provides a physical explanation: color
confinement implies that long wavelengths of partons in hadrons are cut off.
Consequently, at this maximum wavelength corresponding to the typical hadron
size, the effects of loops in propagators and vertex corrections disappear.
Since these quantum effects are at the origin of the running of $\alpha_{s}$,
it should _freeze_ at the typical hadronic scale, provided that no other
phenomena are included in the IR-definition of $\alpha_{s}$: an infrared fixed
point becomes a natural consequence of confinement. In fact, a number of
theories and models that include confinement effects produce this feature, but
their predictions of $\alpha_{s}$ do not _freeze_ at a same value, and/or at
the same momentum scale. The concept that $\alpha_{s}$ _freezes_ is important
since it allows one to compute quantities involving integrals over the IR
domain of the coupling constant. It also permits the use of _conformal_ field
theory for nonperturbative QCD calculations. This connects to the AdS/CFT
approach (Sections 3.9.6 and 4.2), including the extension of the _CSR_
(Section 3.7.1 and Refs. [170] and [171]). In fact, the success of the AdS/CFT
predictions may be an indication that $\alpha_{s}$, with a definition relevant
to IR phenomenology, _freezes_ in the IR. Experimental measurements of
_effective charges_ , results from lattice gauge theory, the Schwinger–Dyson
formalism, the phenomenology of the hadron mass spectrum, the Gribov–Zwanziger
confinement scenario and other approaches, also support this behavior.

Alternatively, the QCD running coupling may diverge as $1/Q^{2}$, as suggested
by the behavior of the $\mbox{Q--}\overline{\mbox{Q}}$ static potential at
large distances. Yet another possibility is a monotonic increase of
$\alpha_{s}(Q^{2})$ as $Q^{2}$ becomes small, but without any divergence; i.e.
$\beta(Q^{2})$ remains non-zero and significantly negative. This is
exemplified by the “ analytic coupling”, see Section 4.9.1. Finally,
$\alpha_{s}$ may vanish in the IR. Several models, see e.g.  Refs. [160, 172,
173] and some Schwinger–Dyson and Lattice results indicate such behavior. Some
experimental results [174] are also suggestive of this behavior.

In the remaining part of this review, we will use $\alpha_{s}$ as a generic
designation for the strong coupling, and $\alpha_{***}$ for specific
couplings, where “∗∗∗” indicates the method, authors or schemes, defining
$\alpha_{s}$. We will start by discussing effective charges and related
definitions such as the coupling from holographic QCD and Sudakov charges.
Then we will discuss the coupling from the static
$\mbox{Q--}\overline{\mbox{Q}}$ potential and the information that the hadron
spectrum provides on the IR-behavior of $\alpha_{s}$. The Schwinger–Dyson,
lattice, functional renormalization group, Gribov–Zwanziger and stochastic
quantization approaches will be discussed thereafter. These approaches use the
same definition of the coupling. Finally, we will describe other frameworks,
the most developed being the analytic/dispersive approaches.

Although the different approaches are organized in different sections for the
sake of clarity, there are of course interrelations. For example, the
effective coupling of Dokshitzer _et al._ (Section 4.9.2) is related to the
_effective charge_ approach of Grunberg (Section 4.1) and to the Shirkov _et
al._ analytic approach (Section 4.9.1). Dokshitzer’s coupling can be
interpreted in terms of effectively massive gluon fields, which ties it to the
Schwinger–Dyson framework (Section 4.4), the lattice results (Section 4.5) and
other approaches addressing the IR-behavior of $\alpha_{s}$, since IR-
regularizations are often associated with the emergence of a mass scale. The
relevant mass scale can be interpreted as:

  * •

the mass scale characterizing the harmonic oscillator potential in the light-
front Schrödinger equation underlying quark bound states. Equivalently, it is
also the factor distorting the AdS space in AdS/QCD (Section 4.2);

  * •

the QCD string tension $\sigma$ (Section 4.3);

  * •

an effective gluon mass (Sections 4.4.4, 4.5 and 4.9.1);

  * •

a dispersive variable (“dispersive mass”) (Section 4.9.2);

  * •

the regulator which is introduced in the functional renormalization group
method (Section 4.6);

  * •

the Gribov mass (Section 4.7);

  * •

a glueball mass (Sections 4.9.1 and 4.10).

These scales are thus all related and typically take values of order 0.5
GeV$\simeq 1.5~{}\Lambda_{\overline{MS}}$. In addition, all of these
approaches have to connect to the phenomenological
$\mbox{Q--}\overline{\mbox{Q}}$ linear potential. While the existence of such
relations is physically suggestive, the explicit connection between the
various strong couplings $\alpha_{s}$ in the infrared is however often
unknown.

###  4.1 Effective charges

As we have noted, Eq. (3.24) implies that $\alpha_{s}(Q^{2})$, as derived from
pQCD, diverges when $Q^{2}\rightarrow\Lambda^{2}$. However, observables
measured across the domain extending from $Q^{2}\gg\Lambda^{2}$ to
$Q^{2}<\Lambda^{2}$ display no sign of discontinuity or unusual behavior. This
is expected since the _Landau pole_ is unphysical and $\Lambda$ is an
arbitrary quantity which depends on the choice of RS, see Table 3.1. In
contrast, observables must be RS-independent. This continuity of the
observables, along with Grunberg’s _effective charge_ approach, see Section
3.7.1, provides a definition of an effective coupling that behaves as
$\alpha_{pQCD}$ at large $Q^{2}$ but stays finite at small values of $Q^{2}$.

_Effective charges_ $\alpha^{eff}_{s}(Q^{2})$ are defined directly from
observables which are calculable in the pQCD domain. A prominent example,
which we will discuss in detail in the following sections, is the _effective
charge_ $\alpha_{g_{1}}(Q^{2})$ defined from the Bjorken sum rule [37]. This
_effective charge_ definition of the running coupling incorporates QCD
contributions inherent to deep inelastic scattering, QCD quantum corrections,
and also the $Q^{2}$ -dependent perturbative effects responsible for the
higher order terms of the observable’s perturbative series, e.g., gluon
bremsstrahlung.

Since it extends to low $Q^{2}$, an effective charge $\alpha^{eff}_{s}(Q^{2})$
also incorporates nonperturbative QCD contributions. Those can be organized as
power-law corrections and thus are often referred to as “ _higher-twist_ ”
contributions as classified by the operator-product expansion. For example,
the _effective charge_ defined from the integral in the Bjorken sum rule,
$\sum_{i=2,3...}^{\infty}\mu_{2i}^{p-n}(Q^{2})/Q^{2i-2}$, incorporates the
leading _twist_ series on the rhs of Eq. (3.40). The $\mu_{2i}^{p-n}(Q^{2})$
terms are themselves sums of kinematical or dynamical _higher-twists_ of
_twist_ order up to $2i$. These terms have the usual pQCD $Q^{2}$-logarithmic
dependence associated with asymptotic freedom and also can be interpreted in
terms of the confining force acting on the quarks [175]. Thus the loop effects
responsible for the UV running, the pQCD effects responsible for the higher
order DGLAP evolution [72], and the nonperturbative confining forces are all
incorporated into the effective QCD charge $\alpha^{eff}_{s}(Q^{2})$. In some
cases $\alpha^{eff}_{s}(Q^{2})$ can take values in the IR domain that would be
unphysical for a quantity measuring the absolute strength of a force. It is
e.g. the case for the $\alpha_{D}\left(Q=0\right)=-\pi$ where $D(Q^{2})$ is
the Adler function [39]. This is obviously not a problem for an effective
quantity.

A natural choice for an effective charge, close to the standard Gell-Mann–Low
definition of the QED running coupling [36] is the potential scheme
($V$-scheme), where $\alpha_{V}(Q^{2})$ is interpreted as the running
coefficient of the $1/Q^{2}$ potential acting between heavy quarks [7, 41].
However, as discussed in Section 3.7.1, $\alpha_{V}$ is affected by infrared
divergences at three loops and higher which prevent it from being calculated
perturbatively, even at high $Q^{2}$.

A related approach has been proposed by Dokshitzer _et al._ [176, 177]. As in
the case of _effective charges_ , Dokshitzer’s approach also generalizes the
QED Gell-Mann–Low coupling. It will be discussed in detail in Section 4.9.2.
Dokshitzer’s charge is intended to measure the effective interaction strength
in the IR in order to extend the use of the fundamental QCD fields outside the
UV domain. We will see that Grunberg’s _effective charge_ defined from the
Bjorken sum rule, $\alpha_{g_{1}}$, may be interpreted in a similar manner. In
distinction to the standard effective charge, Dokshitzer’s effective coupling
is built by imposing dispersion relation constraints on the coupling. The
effective charge $\alpha_{g_{1}}$ is also constrained by these restrictions
since dispersion relations applied to photoproduction cross-sections are the
basis for deriving the Gerasimov–Drell–Hearn sum rule [178], and this in turn
constrains the IR-behavior of $\alpha_{g_{1}}$. Thus, although the respective
formalisms of $\alpha_{g_{1}}$ and Dokshitzer’s effective coupling seem to be
different, the two couplings have some commonalities.

####  4.1.1 Measurement of the effective charge from the Bjorken sum rule.

The _effective charge_ $\alpha_{g_{1}}(Q^{2})$, defined from the Bjorken sum
rule, has been discussed for large $Q^{2}$ in Sections 3.7.1 and 3.9.1, where
it was shown that $\alpha_{g_{1}}(Q^{2})$ has a simple perturbative series
which is known to high orders in the $\overline{MS}$ scheme. An important
advantage of $\alpha_{g_{1}}(Q^{2})$ is that data exist at low, intermediate,
and high Q2 [75, 179, 180, 181, 182, 183, 184, 185, 186]. In addition,
rigorous sum rules dictate the behavior of $\alpha_{g_{1}}$ in the unmeasured
$Q^{2}\rightarrow 0$ and $Q^{2}\rightarrow\infty$ regions. A third advantage
is that the Bjorken sum is a non-singlet quantity, implying that some
resonance contributions to the sum, such as the $\Delta$(1232) resonance,
cancel out. This has important consequences. For example, disconnected
diagrams which are not easily computed on the lattice, do not contribute, thus
leading to more reliable lattice gauge theory estimates. Similarly, the
cancellation of the $\Delta$(1232) makes chiral perturbation calculations more
robust [187]. Finally, $\alpha_{g_{1}}$ is easier to interpret than other
effective charges due to similarities between the $g_{1}$ and $V-$schemes; see
Fig. 3.4 and the associated discussion. In effect, the _effective charge_
$\alpha_{g_{1}}$ can be evaluated at any $Q^{2}$, and its intuitive
interpretation makes it easy to compare with theoretical expectations.

The similarities between the $g_{1}$ and $V-$scheme have a physical origin:
First, $g_{1}$ is extracted from inclusive reactions, and the implicit sum
over all final states simplifies the theoretical expression for the cross-
section [188]. Second, the partial suppression of resonance contributions
enhances the non-resonant background contribution. This allows DIS-like
reactions to dominate the Bjorken integral even at low $Q^{2}$. In addition,
the integral over $x_{Bj}$ and global parton–hadron duality [132] amplify the
dominance of non-resonant reactions. In contrast, coherent state contributions
are not easily interpreted in terms of a QCD coupling: Indeed, if an elastic
contribution would be added to the definition of $\alpha_{g_{1}}$, it would
become significantly negative because the kinematic constraint
$\int_{0}^{1}dx_{Bj}~{}g_{1}(x_{Bj},Q^{2})\rightarrow 0$ as $Q^{2}\rightarrow
0$ (see Eq. (4.2) below) would then not hold; this would yield
$\alpha_{g_{1}+el}(Q^{2}=0)=-17.6$, which includes the elastic contribution,
rather than $\alpha_{g_{1}}(Q^{2}=0)=\pi$.

Two rigorous sum rules constrain $\alpha_{g_{1}}(Q^{2})$ in the limits
$Q^{2}\rightarrow 0$ and $Q^{2}\rightarrow\infty$: the Gerasimov–Drell–Hearn
(GDH) sum rule [178, 189] for $Q^{2}\rightarrow 0$, and the Bjorken sum rule
[37] for $Q^{2}\rightarrow\infty$. Let us first consider the latter. At large
$Q^{2}$, the rhs of Eq. (3.40) can be computed. Equating it to the rhs of Eq.
(3.41) yields the result:

$\alpha_{g_{1}}=\alpha_{\overline{MS}}+3.58\frac{\alpha_{\overline{MS}}^{2}}{\pi}+20.21\frac{\alpha_{\overline{MS}}^{3}}{\pi^{2}}+175.7\frac{\alpha_{\overline{MS}}^{4}}{\pi^{3}}+\mathcal{O}\left(\alpha_{\overline{MS}}^{5}\right),$
(4.1)

which is indicated by the blue band shown in Fig. 4.1. The width of the band
represents the uncertainties due to the value of $\Lambda_{\overline{MS}}$,
the truncation of the Bjorken series in Eq. (4.1), and the truncation of the
$\beta$ series used to compute $\alpha_{\overline{MS}}$ in Eq. (4.1). At the
smallest $Q^{2}$ typically considered for the applicability of pQCD,
$Q_{min}^{2}\gtrsim 1$ GeV2, the _asymptotic series_ (4.1) converges up to
order $n\sim\pi/\alpha_{\overline{MS}}(Q_{min}^{2})\simeq 4$ so one should
stop at this order, lest $175.7\alpha_{\overline{MS}}^{4}/\pi^{3}$ becomes
comparable to $\mathcal{O}\left(\alpha_{\overline{MS}}^{5}\right)$.

Figure 4.1:  Experimental data and sum rule constraints for the _effective
charges_ $\alpha_{g_{1}}(Q)/\pi$ and $\alpha_{F_{3}}(Q)/\pi$. The blue data
points are from Jlab [190], the green points are from Hermes [183], the black
points are from Fermilab [78], the red points are from CERN [186] and the
magenta points are from SLAC [179, 181, 182, 184].

As an alternative to Eq. (4.1), one can use the BLM/_PMC_ expression (see
Section 3.7.1 and Eq. (3.43)) for $\alpha_{g_{1}}(Q^{2})$ in terms of
$\alpha_{\overline{MS}}(Q^{2})$ [51]. In this case all nonzero $\beta$ terms
are shifted into the scales $Q^{*}$, $Q^{**}$, $\cdots$ of the
$\alpha_{\overline{MS}}$ coupling thus matching a _conformal_ expansion.
However, the domain of validity of this particular relation –a _CSR_ – may not
be useful due to the large decrease in the scales
$Q^{*}_{\overline{MS}}=0.37Q_{g_{1}}^{*}$ and
$Q^{***}_{\overline{MS}}=0.25Q_{g_{1}}^{***}$. This decrease renders the
computation of $\alpha_{\overline{MS}}$ problematic.

Let us consider now the GDH sum rule [178] which can be expressed as the
limit:

$\frac{8}{Q^{2}}\int_{0}^{1^{-}}dx_{Bj}~{}g_{1}\left(x_{Bj},Q^{2}\right)~{}\overrightarrow{{}_{Q^{2}\to
0}}~{}\frac{-\kappa_{N}^{2}}{M_{N}^{2}},$ (4.2)

where $\kappa_{N}$ is the anomalous magnetic moment of the target particle
(here a nucleon) and $M_{N}$ its mass. The elastic scattering contribution to
$g_{1}$, $x_{Bj}=1$ is excluded from the integral in Eq. (4.2). At $Q^{2}=0$,
there is no elastic reaction and at large $Q^{2}$ it is negligible, thus its
exclusion is unimportant. At intermediate $Q^{2}$, the elastic contribution
would be sizable if one chooses to include it to Eq. (3.41) (in that case, Eq.
(4.2) becomes invalid). However, as already discussed, this contribution
should be excluded for $\alpha_{g_{1}}$.

Eqs. (4.2) and (3.41) provide an important constraint on $\alpha_{g_{1}}$ at
$Q^{2}=0$:

$\frac{d\alpha_{g_{1}}\left(Q^{2}\right)}{dQ^{2}}|_{Q^{2}=0}=\frac{3\pi}{4g_{A}}\left(\frac{\kappa_{n}^{2}}{M_{n}^{2}}-\frac{\kappa_{p}^{2}}{M_{p}^{2}}\right).$
(4.3)

Furthermore, there is an additional kinematic constraint,
$\int_{0}^{1^{-}}dx_{Bj}g_{1}\left(x_{Bj},Q^{2}=0\right)=0$, because when
$Q^{2}$ is small, only the elastic scattering contribution is permitted.
However, the exclusive reaction is excluded from Eq. (3.41) leading to a
vanishing integral at $Q^{2}=0$. This result also follows from Eq. (4.2).
Consequently, the infrared fixed-point value

$\alpha_{g_{1}}\left(0\right)=\pi.$ (4.4)

follows from (3.41).

The two constraints given by Eqs. (4.3) and (4.4) are shown in Fig. 4.1 by the
dashed red line. The range between the low and high $Q^{2}$ constraints is
densely filled by experimental data [190], mostly coming from the Bjorken
integral data [75, 179, 180, 181, 182, 183, 184, 185, 186]. Other data come
from the CCFR measurement [78] of the Gross–Llewellyn Smith sum rule [74], Eq.
(3.47). At leading _twist_ , the GLS sum rule has the same $Q^{2}$-dependence
as the Bjorken sum rule, except for a small difference at order
$\alpha_{\overline{MS}}^{3}$ coming from the light-by-light contribution to
the GLS sum rule. Consequently, we expect in the perturbative domain that
$\alpha_{g_{1}}\left(Q^{2}\right)=\alpha_{F_{3}}\left(Q^{2}\right)$ up to
$\mathcal{O}(\alpha_{\overline{MS}}^{3})$, with $\alpha_{F_{3}}$ defined from
$\int^{1^{-}}_{0}dx~{}F_{3}(x,Q^{2})\equiv 3[1-\alpha_{F_{3}}/\pi]$. In
addition, the kinematic constraint leading to Eq. (4.4) also applies to the
GLS integral and thus
$\alpha_{F_{3}}\left(0\right)=\pi=\alpha_{g_{1}}\left(0\right)$. In addition,
as shown in Fig. 4.1, one can also use the value of $\alpha_{g_{1}(\tau)}$,
which is obtained using a _CSR_ , with the effective charge $\alpha_{\tau}$
defined from hadronic $\tau$ lepton decay, as reported in Ref. [40]. One also
notes that the JLab data at low-$Q^{2}$ tend toward $\alpha_{g_{1}}\simeq\pi$,
suggesting that $\alpha_{g_{1}}$ becomes nearly flat and thus “ _conformal_ ”
at $Q\lesssim 0.03$ GeV.

###### A prediction of QCD’s _conformality_ from the GDH sum rule

The approximate isospin symmetry between the proton and neutron implies that
$M_{p}\simeq M_{n}$. Although $\kappa_{p}$ and $\kappa_{n}$ differ in sign,
the squares are similar in magnitude:
$\kappa_{p}^{2}=3.21\simeq\kappa_{n}^{2}=3.66$. Consequently, the slope of
$\alpha_{g_{1}}\left(Q^{2}\right)$ near $Q^{2}=0$ as given by Eq. (4.3) is
suppressed. For example, in the deep IR region, $\alpha_{g_{1}}$ is expected
to decrease by only $\simeq 1\%$ over the $Q^{2}$-span of 0.1 GeV2, consistent
with near-_conformal_ behavior in the low-$Q^{2}$ region.

####  4.1.2 Measurement of the effective charge defined from $e^{+}e^{-}$
annihilation.

The ratio $R_{e^{+}e^{-}}(s)$ of cross sections for $e^{+}e^{-}$ annihilation
to hadrons divided by muon pair production, and the ratio of hadronic over
muonic $\tau-$decay widths, as well as the Adler function $D(Q^{2})$ [39],
have each been used to define various _effective charges_ [51, 53]. They are
closely related to each other by _CSRs_. For example, in the purely
perturbative domain, the couplings from $\tau$-decay and $R_{e^{+}e^{-}}$
ratios, $\alpha_{\tau}$ and $\alpha_{R}$ respectively, are predicted by the
CSRs to obey:

$\alpha_{\tau}(s)=2\int_{0}^{s}\frac{dt}{s}\left(1-\frac{t}{s}\right)^{2}\left(1+\frac{2t}{s}\right)\alpha_{R}(t),$
(4.5)

where $s$ and $t$ are the Mandelstam variables. The coupling
$\alpha_{\tau}(s)$ has been extracted from the OPAL experimental data [90] by
Brodsky and collaborators [40], with the result that $\alpha_{\tau}(0)\simeq
7.0$.

####  4.1.3 Sudakov effective charges

As is the case for the _effective charges_ just discussed above, the Sudakov
effective charges introduced by Gardi and Grunberg [191] essentially fold
perturbative corrections into the definition of the _effective coupling_. In
the large-$x_{Bj}$ limit of inclusive structure functions, collinear soft
multi-gluon emissions dominate the perturbative series. These effects induce
large logarithmic corrections (Sudakov double logarithms) which increase with
the order of the perturbative series. These corrections can be incorporated by
exponentiating (resumming) the Mellin transform of the structure function,
leading to an exponential $e^{-S}$ damping factor, where $S$ is the Sudakov
factor proportional to $\alpha_{s}\mbox{ln}^{2}$. In principle, this analysis
includes all the relevant perturbative contributions. Nonperturbative _power
corrections_ are not explicitly included in the Sudakov factor. However, $S$
arguably does depend on such corrections since they influence the perturbative
series through IR _renormalons_. The sum of the Sudakov anomalous dimensions,
which appear in the exponentiation, defines the Sudakov effective charge. Such
charges generalize the coupling definition of Ref. [60]. Most important, the
Sudakov _effective charges_ have a universal (scheme/observable-independent)
_freezing_ value.

The definition of the Sudakov effective charge is illustrated by an example
provided in Ref. [191]. The Mellin transform of the DIS structure function
$F_{2}(Q^{2},x_{Bj})$ is:

$\hat{F_{2}}\left(Q^{2},N\right)=\int_{0}^{\infty}dx_{Bj}\,x_{Bj}^{N-1}F_{2}\left(Q^{2},x_{Bj}\right),$
(4.6)

where $\hat{F_{2}}$ can be expressed, as usual, as a perturbative series in
powers of $\mbox{ln}(Q^{2})$ which can be exponentiated. The argument of the
exponentiated series is $\mbox{ln}(\hat{F_{2}})$. Differentiating it with
respect to $\mbox{ln}(Q^{2})$ yields:

$\frac{d~{}\mbox{ln}~{}(\hat{F_{2}})}{d~{}\mbox{ln}\left(Q^{2}\right)}=\frac{16}{3}\left[H\left(Q^{2}\right)+\int_{0}^{1}dz\,\frac{z^{N-1}-1}{1-z}\mathcal{A_{S}}\left[\left(1-z\right)Q^{2}\right]+\mathcal{O}\left(1/N\right)\right],$
(4.7)

where $H(Q^{2})$ and $\mathcal{A_{S}}(Q^{2})$ are given as perturbative series
in powers of $\alpha_{s}$. In particular,
$\mathcal{A_{S}}(Q^{2})=\frac{\alpha_{s}(Q^{2})}{4\pi}\left[1+A_{1}\alpha_{s}(Q^{2})+\cdots\right]$
defines the Sudakov effective charge. It is connected to the two Sudakov
anomalous dimensions $A$ and $B$:

$\mathcal{A_{S}}\left(Q^{2}\right)=\frac{3}{16}\left[A\left(\alpha_{s}\left(Q^{2}\right)\right)+\frac{dB\left(\alpha_{s}\left(Q^{2}\right)\right)}{d~{}\mbox{ln}\left(Q^{2}\right)}\right].$
(4.8)

In the large $Q^{2}$-limit, one has:

$\mathcal{A_{S}}\left(Q^{2}\right)=\alpha_{s}\left(Q^{2}\right)+\frac{3}{4}\beta_{0}\left(\alpha_{s}\left(Q^{2}\right)\right)^{2}+\left(\frac{5}{4}-\frac{\pi^{2}}{3}\right)\beta_{0}^{2}\left(\alpha_{s}\left(Q^{2}\right)\right)^{3}+\cdots.$
(4.9)

The IR limit is:

$\mathcal{A_{S}}\left(Q^{2}\right)=\frac{1}{2\beta_{0}}\left[1-\frac{2}{\pi}arctan\left(\frac{\mbox{ln}\left(Q^{2}/\Lambda^{2}\right)}{\pi}\right)+\frac{\Lambda^{2}}{Q^{2}}-\frac{\Lambda^{4}}{Q^{4}}\right],$
(4.10)

which has no Landau singularity, but it diverges toward negative values as
$Q^{2}\rightarrow 0$. Nonperturbative _power corrections_ may regularize this
unphysical behavior. An example of a simple ansatz for the nonperturbative
contributions, which can cancel divergences and lead to a $1/\beta_{0}$ _fixed
point_ , is given in [192]. $\mathcal{A_{S}}\left(Q^{2}\right)$ does not
_freeze_ in the IR, increasing from $1/\beta_{0}\simeq 0.10$ at $Q^{2}$=0
toward a maximum at $Q^{2}/\Lambda^{2}\simeq 0.84$.

It was already noticed in [193] that the _Landau pole_ can be suppressed by
the Sudakov damping factor $e^{-S}$, although this suppression was not
incorporated in the coupling discussed in Ref. [193] which retained the
standard pQCD definition.

###  4.2 AdS/CFT and Holographic QCD

The behavior of the QCD coupling in the IR can be obtained by enforcing the
underlying _conformal_ symmetry of QCD. In fact, conformal symmetry even
allows one to identify the form of the long-range color-confining potential by
incorporating a method due to de Alfaro, Fubini and Furlan (dAFF), as
discussed below. Its scale-dependence then leads to the running of the
coupling in the IR as well as a constraint in the pQCD domain.

As we discussed in Section 3.9.6, light-front holographic QCD is based on the
AdS/CFT correspondence [9] and the light-front quantization procedure based on
the frame-independent front form devised by Dirac [129]. In the front form,
the time evolution variable is $\tau=t+z/c$; i.e., the time along the light-
front (LF). The front-form results are independent of the observer’s Lorentz
frame.

In this framework, one can reduce the full QCD light-front Hamiltonian
equation to obtain a one–dimensional light-front Schrödinger Equation (LFSE)
acting on the valence Fock state. The eigenvalues of the LFSE determine the
hadron spectrum consisting of light quarks and the eigensolutions determine
their LF wavefunctions. The LFSE is analogous to the Schrödinger equation
describing hydrogenic atoms in QED [128] but it is relativistic and frame-
independent. The radial variable for the LFSE is identified as the invariant
distance between the $q$ and $\bar{q}$: $\zeta=b_{\perp}\sqrt{x(1-x)}$, where
$\zeta^{2}$ is conjugate to the light-front kinetic energy
$k^{2}_{\perp}/[x(1-x)]$. Here, $x$ is the light-front momentum fraction
$x=k^{+}/P^{+}=(k^{0}+k^{z})/(P^{0}+P^{z})$ and $b_{\perp}$ is the transverse
impact parameter.

If one requires the effective action which underlies the QCD Lagrangian to
remain _conformally_ invariant and extends the dAFF formalism to light-front
Hamiltonian theory [194, 195], the light-quark antiquark potential has the
unique form of a harmonic oscillator potential
$V(\zeta^{2})=\kappa^{4}\zeta^{2}$, and a mass gap arises. The dAFF mechanism,
originally derived in the context of $1+1$ quantum mechanics, enables the
emergence of a mass scale $\kappa$ while the action remains conformal. The
result is a nonperturbative relativistic light-front wave equation which
incorporates confinement and other essential spectroscopic and dynamical
features of hadron physics, including linear Regge trajectories with the same
slope in the radial quantum number $n$ and orbital angular momentum $L$.

Figure 4.2:  In QCD, hadronic dynamical supersymmetry is rooted in the
dynamics of color $SU(3)$ where $\bf\bar{3}\sim\bf 3\times\bf 3$.

It has been shown recently how the main features of nonperturbative QCD
dynamics are well captured in a semiclassical effective theory based on
superconformal quantum mechanics [196, 197] and its extension to light-front
physics [198, 199]. This new approach to hadron physics incorporates several
basic aspects one expects from QCD, namely confinement and the appearance of a
massless pion in the limit of zero quark masses. The specific breaking of
_conformal_ invariance uniquely determines the effective confinement
potential. The generalized supercharges connect the baryon and meson spectra
to each other in a remarkable manner. The partner mesons and baryons have the
same mass provided one identifies the relative orbital angular momentum
$L_{M}$ of the $q\bar{q}$ mesons with the relative internal orbital angular
momentum $L_{B}$ of the quark–diquark baryons in a cluster decomposition,
where $L_{M}=L_{B}+1$ [198, 199]. Furthermore, this framework gives remarkable
connections across the full light [200] and the full heavy-light hadron
spectra, where heavy quark masses break the conformal invariance, but the
underlying supersymmetry still holds [201]. It is important to recall that in
the context of hadronic physics the supersymmetric relations are not a
consequence of supersymmetric QCD, at the level of fundamental fields, but an
emergent dynamical supersymmetry from color $SU(3)_{C}$. This relies on the
fact that in $SU(3)_{C}$ a diquark can be in the same color representation as
an antiquark, namely a $\bf\bar{3}\sim\bf 3\times\bf 3$ [202, 203, 204] as
illustrated in Fig. 4.2.

The light-front harmonic oscillator potential corresponds to a linear
potential for bound states of nonrelativistic heavy quarks in the usually
employed instant-form of relativistic dynamics [205]. This links light-front
QCD to lattice gauge theory as well as heavy-quark effective theory. The mass
parameter $\kappa$ can be determined from a hadron mass; e.g.
$\kappa=M_{\rho}/\sqrt{2}$ or $\kappa=M_{p}/2$ [130], from the scale $\Lambda$
controlling pQCD evolution, as discussed in Section 3.9.6, or from other
experimental inputs such as the Bjorken sum $Q^{2}$-evolutions. Remarkably,
all of these determinations are in good agreement.

The underlying _conformal_ symmetry of QCD allows one to perform equivalent
calculations in a higher dimensional curved space-time where the dual theory
is weakly coupled and described by gravity. Mathematically, this dual
representations is based on the isomorphism of the $SO(4,2)$ group of
conformal transformations in physical space-time with the group of isometries
of AdS5 space. Because of this correspondence, one can identify the
fifth–dimension coordinate $z$ of AdS5 space with the light-front coordinate
$\zeta$. Light-front holography also implies that the potential
$V(\zeta^{2})=\kappa^{4}\zeta^{2}$ appearing in the meson LFSE can be obtained
by modifying AdS5 space using a specific “dilaton profile”
$e^{\pm\kappa^{2}z^{2}}$ [130, 206] (The plus sign dilaton profile is required
in light-front holographic QCD [130]). The quark–antiquark binding interaction
is thus related in AdS/QCD to the modification of the AdS space curvature
[195]. The same harmonic oscillator potential also emerges from the
application of the dAFF procedure to LF theory [194, 130].

Remarkably, one also finds that the same modification of the AdS5 action also
determines the IR-behavior of $\alpha_{s}\propto\exp{\left(-Q^{2}\over
4\kappa^{2}\right)}.$ Recall that in pQCD, the _effective coupling_
$\alpha_{pQCD}(Q^{2})$ is defined by folding short-distance quantum effects
into its evolution. Analogously, the $Q^{2}$-dependence of the AdS/QCD
effective coupling stems from the effects of the large-distance forces folded
into the coupling constant [170]. To obtain this result, one notes that the
AdS5 action is similar to the action of general relativity: $S_{GR}\propto\int
d^{4}x\sqrt{\mbox{det}(g_{\mu\nu})}R/G_{N}$, where $G_{N}$ is Newton’s
constant and $R$ is the Ricci scalar. In AdS/QCD, $\sqrt{R}$ is replaced by
the gluon field $F$, $\sqrt{G_{N}}$ is replaced by the gauge coupling
$g_{AdS}$, and the metric determinant becomes
$\sqrt{\mbox{det}(g_{AdS})}e^{\kappa^{2}z^{2}}$, which includes the
$e^{\kappa^{2}z^{2}}$ distortion of AdS space. This analysis yields the AdS
action:

$S_{AdS}=-\frac{1}{4}\intop
d^{4}x\,dz\sqrt{\mbox{det}(g_{AdS})}e^{\kappa^{2}z^{2}}\frac{1}{g_{AdS}^{2}}F^{2}.$
(4.11)

Just as $\alpha_{s}\equiv g^{2}/4\pi$ acquires its effective $Q^{2}$
dependence from short-distance QCD quantum effects, the initially constant AdS
coupling $\alpha_{AdS}\equiv g_{AdS}^{2}/4\pi$ is redefined to absorb the long
distance confining forces; i.e. the effects of the modification of the AdS
space curvature from _nonconformal_ confinement dynamics [170]:
$g_{AdS}^{2}\rightarrow g_{AdS}^{2}e^{-\kappa^{2}z^{2}}$. The five–dimensional
coupling $g_{AdS}(z)$ is mapped, modulo a constant, into the QCD coupling
$g_{s}$ of the confining theory in physical space-time using light-front
holography. Thus, by identifying $z$ with the invariant impact separation
variable $\zeta$, $g_{AdS}(z)\to g_{s}(\zeta)$, one predicts

$\alpha_{s}^{AdS}(\zeta)\equiv g_{s}^{2}(\zeta)/4\pi\propto
e^{-\kappa^{2}\zeta^{2}}.$ (4.12)

The physical coupling measured at the scale $Q$ is the light-front transverse
Fourier transform of the LF coupling $\alpha_{s}^{AdS}(\zeta)$ (4.12):

$\alpha_{s}^{AdS}(Q^{2})\sim\int^{\infty}_{0}\\!\zeta d\zeta\,J_{0}(\zeta
Q)\,\alpha_{s}^{AdS}(\zeta),$ (4.13)

in the $q^{+}=0$ light-front frame where
$Q^{2}=-q^{2}=-\mathbf{q}_{\perp}^{2}>0$ and $J_{0}$ is a Bessel function.
Using this ansatz we then have from Eq. (4.13)

$\alpha_{s}^{AdS}(Q^{2})=\alpha_{s}^{AdS}(0)\,e^{-Q^{2}/4\kappa^{2}},$ (4.14)

where $\alpha_{s}^{AdS}(0)=\pi$ in the $g_{1}$ scheme. The identification of
$\alpha_{s}^{AdS}(Q^{2})$ with the physical QCD running coupling in its
nonperturbative domain determines the space-time dependence at large distances
of the physical four–dimensional coupling $\alpha_{s}$.

The harmonic oscillator form of the LF potential
$V(\zeta^{2})=\kappa^{4}\zeta^{2}$, and thus the Gaussian form for the running
coupling, also follows from the dAFF procedure; i.e., the requirement that the
action remains _conformal_ [194, 130], even though a mass scale appears in the
confining potential of the light-front Hamiltonian.

Eq. (4.14) is valid only at the domain of $Q^{2}$ where QCD is a strongly
coupled theory and the AdS/CFT correspondence can be applied. Furthermore,
since the semi-classical approximation used in the light-front holographic
approach neglects quantum effects, vacuum polarization effects are not
incorporated in $\alpha_{AdS}$, and consequently it is not valid at large
$Q^{2}$. Nonetheless, $\alpha_{AdS}$ can be supplemented at large $Q^{2}$ by
either parameterizing the well-known pQCD effects at the origin of the large
$Q^{2}$ dependence, as done in [170], or by matching Eq. (4.14) to Eq. (3.24),
as done in [131] and discussed in Section 3.9.6. This matching procedure is
illustrated in Fig. 4.3.

Figure 4.3:  Unified coupling obtained from the analytic matching of
nonperturbative and perturbative QCD regimes. The procedure determines the
relation between $\Lambda_{\overline{MS}}$ and $\kappa$ or equivalently hadron
masses. The transition scale $Q_{0}$ between the large and short-distance
regimes of QCD is determined as well.

The Gaussian form $\alpha_{s}(Q^{2})=\alpha_{s}(0)e^{-Q^{2}/4\kappa^{2}}$
predicted by the light-front holographic approach describes the available
experimental data for the effective charge $\alpha_{g_{1}}(Q^{2})$ very well
over a large range of moment transfer; see Fig. 4.1. Conversely, a fit of the
experimental data shown in Fig. 4.1 with $\kappa$ as the free fit parameter
yields $\kappa=0.496\pm 0.007$, in good agreement with the determination from
the value $\kappa=0.50\pm 0.04$ obtained from the relation between $\kappa$
and hadron masses: $\kappa=M_{\rho}/\sqrt{2}$ or $\kappa=M_{p}/2$ [130]. It is
remarkable that the parameter $\kappa$ which fits the Bjorken sum rule data is
set independently by a hadron mass.

Additionally, it has been shown using the Schwinger–Dyson equations for the
quark propagator that if one assumes a Gaussian form for the QCD running
coupling, the quark propagator has poles above and below the real axis,
consistent with quark confinement [207]. Finally, the _scaling solution_ of
the Schwinger–Dyson equations yields a coupling in close agreement with
$\alpha_{AdS}$, with a _freezing_ value of 2.97 (Section 4.4).

###  4.3 The effective potential approach

####  4.3.1 The static $\mbox{Q--}\overline{\mbox{Q}}$ potential

A remarkable feature of hadron physics is that mesons and baryons composed of
light quarks all lie on Regge trajectories. The square of the hadron masses
are linear in both the internal orbital angular momentum $L$ and the principal
quantum number $n$ which is associated with the number of nodes in its
wavefunction. Moreover the slope in $L$ and $n$ are identical
phenomenologically. These universal features in fact follow from the
eigenvalues of LFSE with the harmonic oscillator potential
$V(\zeta^{2})=\kappa^{4}\zeta^{2}$ [130]. The lightest eigenvalue,
corresponding to the pion, has zero mass in the chiral limit $m_{q}\to 0.$

In the case of heavy quark $\mbox{Q--}\overline{\mbox{Q}}$ states, the
effective nonrelativistic static potential between two heavy quarks
($\mbox{Q--}\overline{\mbox{Q}}$) of mass $m_{Q}\gg\Lambda$ separated by a
distance $r$ is well described by:

$V(r)=-\frac{4}{3}\frac{\alpha_{V}\left(r\right)}{r}+\sigma r.$ (4.15)

The first term is a perturbative, Coulomb-type, one-gluon exchange
contribution. The hadronic wavefunctions and the fine structure of the hadron
spectrum are sensitive to this term. When $\alpha_{V}(r)$ is approximated as
an averaged coupling $\left\langle\alpha_{V}\right\rangle$, Eq. (4.15) is
known as the Cornell potential [208]. The mean value of the coupling
$\left\langle\alpha_{V}\right\rangle$ depends on the size of the hadrons
considered. Its value in GeV units spans from 0.19 to 0.4, with an optimal
value of 0.22 [208, 209, 210]. In the context of an IR-_freezing_ behavior of
$\alpha_{V}(Q^{2})$, this implies a _freezing_ value of $\alpha_{V}(0)=0.42\pm
0.03$ [208, 209, 210].

The second term in Eq. (4.15) is a linear confining potential which can be
interpreted as the string tension between the $\mbox{Q--}\overline{\mbox{Q}}$
pair, with $\sigma\sim 0.18$ GeV2. This term controls the slopes and
intercepts of the Regge trajectories. The QCD string picture was first
postulated by Nambu, Nielsen and Susskind to interpret the Regge trajectories
[211]. A common interpretation is that the string is formed by a
chromoelectric flux tube and is responsible for quark confinement. Regge
trajectories stem straightforwardly from the string picture: the faster a
hadron spins, the larger the string tension must be to compensate for the
centrifugal force, and hence the larger its mass [212].

In fact, the AdS/QCD holographic QCD framework which yields the harmonic
oscillator potential $V(\zeta^{2})=\kappa^{4}\zeta^{2}$ in the light-front
transverse coordinate $\zeta$ for light quark pairs, corresponds to a linear
potential for heavy quarks in the non-relativistic instant-form radial
coordinate $r$ for large distances [205]. The string tension for heavy quarks
can then be predicted from holographic QCD to be:

$\sigma_{AdS}=2\kappa^{2}/\pi\simeq 0.18~{}\mbox{GeV}^{2},$ (4.16)

which is remarkably consistent with fits to the quarkonium data.

A more accurate potential than that provided by Eq. (4.15) includes velocity-
dependent and spin-dependent corrections, see Refs. [213] and [49] for
reviews.

The validity of Eq. (4.15) is confirmed for heavy quarks by lattice QCD
simulations. However, in (2+1) lattice simulations, $\alpha_{V}(0)\sim 0.3$
[214], somewhat smaller than the value determined from the Cornell potential.
Earlier _quenched_ calculations resulted in an even smaller value of the
coupling, $\alpha_{V}(0)\sim 0.22$ [214].

In addition to lattice QCD, other nonperturbative approaches also suggest the
form of Eq. (4.15). In the Coulomb gauge, the instantaneous gluon propagator
leads to a linear static $\mbox{Q--}\overline{\mbox{Q}}$ potential [164, 165,
166, 167], as was advocated by Gribov and Zwanziger [163, 215]; see Section
4.7. Generally, it is expected that in the IR, the gluon propagator is
modified by nonperturbative effects and displays a
$1/\left(Q^{2}+f(Q^{2})\right)^{2}$ dipole form that would lead to the linear
IR potential [216]. (The function $f(Q^{2})$ is unimportant here. It is
typically related to an effective gluon mass or a glueball mass.) This is
studied in the Schwinger–Dyson formalism, see Section 4.4 and in particular
Refs. [217, 218], and in the related approach of the functional
renormalization group [219], see Section 4.6. A linear potential is also
recovered using background perturbation theory [220], see Section 4.10 for
details.

To summarize, Eq. (4.15) is well established. Nevertheless, there is no
agreement on the IR-behavior of $\alpha_{s}$, owing to its various possible
definitions. Still, within a specific definition, Eq. (4.15) -or more
generally hadron spectroscopy- does constrain $\alpha_{s}$ as we will now
discuss.

####  4.3.2 Constraints on the running coupling from the hadron spectrum

The hadron spectrum is sensitive to the IR-behavior of $\alpha_{s}$ when one
identifies the strong coupling as the _effective charge_ underlying the heavy
$\mbox{Q--}\overline{\mbox{Q}}$ potential. Hence, the heavy quark potential
can provide a useful information on $\alpha_{s}(Q^{2})$ in the IR. For example
in the Godfrey–Isgur quark/flux-tube model [221], $\alpha_{s}$ must _freeze_
in the IR at $\alpha_{s}(0)=0.6$. Other examples are the model of Zhang and
Koniuk [222], or the work of Badalian and collaborators using the framework of
background perturbation theory. Badalian _et al._ obtain $\alpha_{s}(0)=0.58$
from the analysis of the bottomonium, see Section 4.10 and Ref. [223].
Likewise, Andreev [224] uses several of the forms for $\alpha_{s}(Q^{2})$
suggested in the literature and constrains them with a Poincaré covariant
quark model to reproduce meson characteristics and experimental data,
including the Bjorken sum rule, Eq. (3.40), and the GLS sum rule, Eq. (3.47).
Andreev obtains a consistent set of behaviors by using $\alpha_{s}(Q^{2})$
defined in the $\overline{\mbox{MS}}$ scheme, but regularized with a constant
effective gluon mass using the massive analytic perturbation theory approach
(Section 4.9.1), or with Cornwall’s coupling (see Section 4.4.4) or with the
“synthetic coupling” (also discussed in Section 4.9.1). This procedure results
in a _freezing_ value $\alpha_{s}(0)\simeq 0.71$ in the $\overline{\mbox{MS}}$
renormalization scheme.

Baldicchi and Prosperi also tested several different expressions of
$\alpha_{s}(Q^{2})$ to check their compatibility with the meson spectrum
[225]. One of the expressions studied is a crude _freezing_ implementation:
below a given $Q_{0}$, $\alpha_{s}$ is constant, and above $Q_{0}$ it follows
the pQCD form. Another expression of $\alpha_{s}$ is given by the “analytic”
approach described in Section 4.9.1 and given by Eq. (4.54). This
$\alpha_{an}$ is finite and decreases monotonously with $Q^{2}$. A related
expression has been give by Dokshitzer _et al._ , and is given by Eq. (4.59).
Baldicchi and Prosperi conclude that the meson spectrum appears to be rather
insensitive to the type of coupling used [225].

Historically, these approaches follow from the pioneering works of Richardson
and of Buchmuller _et al._. These authors suggested that $\alpha_{s}$ should
be defined to incorporate a long range force in addition to the usual pQCD
effects responsible for the running of the coupling [226, 227]. In the non-
relativistic (static) approximation, Eq. (4.15) becomes in momentum space:

$V(Q)=\frac{1}{\left(2\pi\right)^{3}}\int
d^{3}re^{-ir.Q}V(r)=-\frac{1}{\pi^{3}}\frac{2}{3}\frac{\alpha_{V}\left(Q^{2}\right)}{Q^{2}}-\frac{\sigma}{\pi^{3}Q^{4}}.$
(4.17)

Incorporating in the coupling the long distance $\sigma$-term leads to the
definition of $\alpha_{Rich}(Q^{2})$:

$V(Q^{2})=-\frac{1}{\pi^{3}}\frac{2}{3}\frac{\alpha_{Rich}(Q^{2})}{Q^{2}}.$
(4.18)

Richardson [226] proposed that:

$\alpha_{Rich}\left(Q^{2}\right)=\frac{4\pi}{\beta_{0}\mbox{ln}\left(1+Q^{2}/\Lambda_{V}^{2}\right)},$
(4.19)

where $\Lambda_{V}$ should be used since $\alpha_{Rich}$ identifies to
$\alpha_{V}$ in the UV. The corresponding _$\beta$ -function_ is:

$\beta=-\beta_{0}\alpha_{Rich}^{2}\left(1-e^{-1/(\beta_{0}\alpha_{Rich})}\right).$
(4.20)

This expression anticipated results from the Schwinger–Dyson formalism; see
Eq. (4.28), in which the coupling is regulated by an effective dynamical gluon
mass $m_{g}(Q^{2})$. Using Richardson’s Eq. (4.19), the gluon mass is assumed
to be constant and equal to $\Lambda_{V}$, which is close to the $m_{g}(0)$
values suggested by Schwinger–Dyson and lattice studies. The $Q^{2}\rightarrow
0$ limit of Eq. (4.19) is:

$\alpha_{Rich}\left(Q^{2}\right)=\frac{4\pi\Lambda_{V}^{2}}{\beta_{0}Q^{2}},$
(4.21)

and the string tension

$\sigma=\frac{8\pi\Lambda_{V}^{2}}{3\beta_{0}}.$ (4.22)

Using the phenomenological value $\sigma=0.18$ GeV2, one obtains
$\Lambda_{V}=0.46$ GeV for $n_{f}=2$ in agreement with its high-$Q^{2}$ pQCD
determination $\Lambda_{V}=0.48$ GeV, see Table 3.1. Thus, if all the
mechanisms leading to the long-distance linear potential are included in the
definition of $\alpha_{V}$, then the $1/Q^{4}$ divergence of $V$ in the
instant-form coordinate system implies that $\alpha_{V}$ should diverge as
$3\sigma/4Q^{2}$ in the limit $Q^{2}\rightarrow 0$.

The fact that the linear $\mbox{Q--}\overline{\mbox{Q}}$ potential leads to
$\alpha_{V}\propto 1/Q^{2}$ has inspired several phenomenological models.
Belyakova and Nesterenko [228] have constructed a coupling with the
constraints that it follows $\alpha_{\overline{MS}}(Q^{2})$ in the UV and
$\alpha_{s}(Q^{2})\simeq\frac{4\pi}{\beta_{0}}\frac{\Lambda^{2}}{Q^{2}}$ in
the IR. The Belyakova–Nesterenko coupling is defined by the _$\beta$
-function_:

$\beta\left(\alpha_{BN}\right)=\beta_{\overline{MS}}^{(l)}\frac{1-e^{-8\pi/\alpha_{BN}\beta_{0}}\left(1-l^{2}\beta_{0}^{l}/\beta_{l}\right)}{1+l^{2}\left(\alpha_{BN}\beta_{0}/\left(4\pi\right)\right)^{l}}.$
(4.23)

It resembles the _$\beta$ -function_s proposed in Refs. [226, 209]. At leading
order, it yields:

$\alpha_{BN}\left(Q^{2}\right)=\frac{4\pi}{\beta_{0}}\frac{1}{W_{0}\left(Q^{2}/\Lambda^{2}\right)},$
(4.24)

where $W_{0}$ is the principal branch of the real-valued Lambert function; see
Section 3.2. The static $\mbox{Q--}\overline{\mbox{Q}}$ potential obtained for
$\Lambda^{(3)}=375\pm 40$ MeV agrees with both lattice QCD and the Cornell
potential, Eq. (4.15). The model provides the same relation between $\Lambda$
and $\sigma$ found by Richardson, see Eq. (4.22).

###  4.4 The Schwinger–Dyson formalism

Studies of the Schwinger–Dyson equation (SDE) for QCD have also provided
important insights into the IR-behavior of $\alpha_{s}(Q^{2})$. In effect, the
SDE (and the corresponding Bethe-Salpeter formalism for bound states)
incorporates an infinite set of coupled non-linear equations relating _Green’s
function_ s as well as other constraints from the QCD equations of motion. In
this formalism, the equation for the $n$-point function depends on the
equation for the $(n+1)$-point function. Although the SDE formalism can be
described using perturbative Feynman graphs, infinite sets of diagrams are
resummed into one-particle irreducible graphs, and thus the SDE can account
for the nonperturbative content of the theory. Alternatively, the SDE can be
derived using the generating functional of the _Green’s function_ s without
reference to perturbative graphs. The SDE formalism thus stems from the
fundamental equations of QCD, encompasses its nonperturbative effects, and is
able in principle to provide analytical results. As such, it provides a
powerful tool. Furthermore, unlike Lattice QCD, the SDE formalism is not
subject to unphysical artifacts resulting from space-time discretization, nor
unphysically large quark masses. These effects can complicate studies of
important hadronic phenomena.

However, the SDE framework has its own difficulties: the infinite system of
equations must be truncated to be amenable to solutions. In fact, any
truncation of the kernel conflicts with crossing symmetry. In the case of QED,
one must include an infinite number of crossed graph kernels in the Bethe-
Salpeter equation in order to derive the Dirac Coulomb equation for muonium
$\mu^{+}e^{-}$ in the limit of $m_{\mu}\to\infty.$ This implies that the SDE
approach contains approximations which may lead to effects that are not easily
understood and controlled. Also, the equations are difficult to solve and
analytical answers can be hard to obtain without further approximations.
Consequently, the SDE quantitative results are often obtained numerically. The
SDE truncations are typically chosen to ensure that cut off propagators are
positive (to retain unitarity), that the high-$Q^{2}$ behavior of the QCD
coupling matches RGE expectations, and that the _Green’s function_ s retain
their multiplicative renormalizability properties. The last property implies
that a full propagator is proportional to its corresponding bare propagator
via a “dressed function”.

Despite the difficulties inherent to the SDE, this approach provides an
important method for furthering our understanding of QCD in general and of
$\alpha_{s}$ in particular. It is especially powerful when studied alongside
other techniques which will be discussed in the next sections, noticeably
lattice QCD, so that these methods can supplement each other. In these
different approaches, couplings have the same definition and the RS is usually
the same (the MOM RS or derivatives). Consequently, comparing quantities in
SDE to other techniques is possible. As an example of the complementarity
between the SDE framework and lattice approaches, the SDE can be used to
understand the effects of space-time discretization and of choice of boundary
conditions for the lattice approach, whereas the lattice can be used to
understand the effect of truncating the infinite system of equations of the
Schwinger–Dyson formalism. For general reviews on SDE see Ref. [229]. Reviews
of SDE results discussing the IR-behavior of $\alpha_{s}$ can be found in
[230, 231].

####  4.4.1 The QCD coupling defined from Schwinger–Dyson equations

The behavior of the gluon and ghost Green’s functions is directly related to
the running of $\alpha_{s}$. The required propagators and vertices which
define these _Green’s function_ s have been extensively studied using the
Schwinger–Dyson formalism, on the lattice, and by using the Gribov–Zwanziger
approach [163, 215], as well as other methods. See [232, 233] for recent
reviews. For example, the strong coupling derived from the ghost–gluon vertex
$\alpha_{s}^{gh}(Q^{2})$ is given in the Landau gauge and the MOM RS by

$\alpha_{s}^{gh}\left(Q^{2}\right)=\alpha_{s}^{gh}\left(\mu\right)G^{2}\left(Q^{2},\mu\right)Z\left(Q^{2},\mu\right),$
(4.25)

where $\alpha^{gh}_{s}(\mu)$ is the coupling determined at the
_renormalization scale_ $\mu$, $Z(Q^{2},\mu)$ is the gluon propagator dressing
function, and $G(Q^{2},\mu)$ is the ghost propagator dressing function (the
ghost propagator being the inverse of the Fadeev–Popov operator). The RG
invariance of Eq. (4.25) is evident: the quantities forming the right-hand
side depend on $\mu$, but on the left hand side, $\alpha_{s}^{gh}$ does not.
The dressing functions are defined as
$\delta^{bc}G(Q^{2},\mu)/Q^{2}=G^{bc}(Q^{2},\mu)$ and
$-\delta^{bc}Z(Q^{2},\mu)(\delta_{\mu\nu}-q_{\mu}q_{\nu}/q^{2})/Q^{2}=D^{bc}_{\mu\nu}(Q^{2},\mu)$
where $b,c$ are color indices, $G^{bc}(Q^{2},\mu)$ is the ghost propagator,
and $D^{bc}_{\mu\nu}(Q^{2},\mu)$ is the gluon propagator. Since the dressing
functions relate the first-order (i.e. bare) _Green’s function_ s to their
fully (i.e. dressed) expressions, they are often called “form factors” in
analogy with the functions modifying the point-like expression for elastic
scattering due to hadron structure. There is no contribution from the vertex
dressing function in Eq. (4.25) owing to the _Slavnov–Taylor_ identity [33].

One sees from Eq. (4.25) that the analytic form of the propagators provides a
definition of the strong coupling. Other definitions are possible using the
three-gluon vertex (defining the coupling $\alpha_{s}^{3g}$), the four-gluon
vertex (defining $\alpha_{s}^{4g}$) or the quark–gluon vertex (defining
$\alpha_{s}^{qg}$). Since ghosts do not couple to themselves or to quarks, no
further coupling definitions are possible.

The $\alpha_{s}^{3g}$, $\alpha_{s}^{4g}$ and $\alpha_{s}^{qg}$ definitions
involve vertex dressing functions which are more difficult to study [22]. In
particular, there are a large number of contributing Feynman graphs. Thus
$\alpha_{s}^{gh}$ has been the focus of most of the initial SDE studies,
although several groups have calculated the other couplings, The quantities
$\alpha_{s}^{gh}$, $\alpha_{s}^{3g}$, $\alpha_{s}^{4g}$, $\alpha_{s}^{qg}$ are
the same in a massless RS, such as $\overline{MS}$, but not for a massive RS
such as the MOM RS [153]. Nevertheless, these couplings can be related to each
other [22, 25, 153, 154], and the differences vanish at large $Q^{2}$. Another
coupling used in the SDE is $\alpha_{gse}$, defined from the gluon self-energy
[234]. It has been shown to be closely related to $\alpha_{s}^{gh}$ in the
Landau gauge [235].

####  4.4.2 Gauge choices

_Green’s function_ s are not directly observable, and they become gauge-
dependent if one utilizes the MOM RS. This also makes the derived results for
$\alpha_{s}^{gh}$, $\alpha_{s}^{3g}$, $\alpha_{s}^{4g}$ and $\alpha_{s}^{qg}$
gauge dependent. An alternative procedure is to use Cornwall’s _Pinch_ scheme
to define a gauge-independent three-gluon coupling [54, 236].

Most SDE results relevant to $\alpha_{s}$ are obtained in the Landau gauge,
$\partial_{\mu}A^{\mu}=0$, since the relatively small numbers of fields and
vertices make computations of the SDE simpler relative to many other gauges.
It is also a Lorentz-covariant gauge, which, beside making the _Kugo–Ojima
confinement criterion_ applicable, further simplifies solving the SDE. For
example, of all the possible tensors (from bosons) or spinor (from fermions)
terms which form the 2-point _Green’s function_ , only two survive generic
symmetry requirements. They lead to a boson propagator that can be written as
$A(Q^{2})q^{2}\eta^{\mu\nu}-B(Q^{2})q^{\mu}q^{\nu}$. These two functions,
which are _a priori_ independent, become evidently equal in Lorentz-covariant
gauges such as the Landau gauge: $A(Q^{2})=B(Q^{2})$, yielding a propagator
that depends only on $Q^{2}$. In contrast, gauges which are not covariant have
propagators – and consequently predictions for $\alpha_{s}$ – which can depend
on another variable in addition to $Q^{2}$. For example, in the first
calculations of the gluon propagator done in the (non-covariant) axial gauge
[237], $B(Q^{2})$ was set to zero in order to force the gluon propagator to
only depend on $Q^{2}$, as is the case in the perturbative domain. The assumed
vanishing of $B(Q^{2})$ leads to a $1/Q^{4}$ behavior of the gluon propagator.
Since there are no ghosts in this gauge, one predicts a singular behavior
$\alpha_{s}\propto 1/Q^{2}$ in the IR. This behavior seems satisfactory, given
the expectation that it leads to a linear $\mbox{Q--}\overline{\mbox{Q}}$
potential; see Section 4.3. However, more complete SDE calculations in
covariant gauges and subsequent lattice calculations have shown this behavior
to be an artifact of the assumption $B(Q^{2})=0$.

Another advantage of the Landau gauge is that the equations involving the
transverse and longitudinal propagators decouple. The on-shell gluon
propagator should be transverse, and thus it can be studied independently. In
addition, in the Landau gauge the ghost–gluon vertex has a finite
renormalization constant [33, 238], which allows the coupling to be defined
and extracted from Eq. (4.25); using calculations involving only the two-point
_Green’s functions_ $G(Q^{2})$ and $Z(Q^{2})$. The Landau gauge is also
symmetric under the exchange of ghost and anti-ghost. The gauge-dependence of
the running coupling using this procedure has been argued to be weak for
couplings computed in gauges close to the Landau gauge. This was checked by
using various linear covariant gauges [231]. Notwithstanding the preeminence
of the Landau gauge, other gauges, such as the Coulomb, Feynman [239, 240,
241], axial [237], light-cone [234], maximal Abelian gauge, [242] and linear
covariant gauges (which interpolate between the Landau and the Feynman gauges)
have also been used [231, 243, 244].

####  4.4.3 Classes of solutions in the IR domain

There is presently a consensus that two types of solutions are possible for
the IR behavior of the gluon propagator, and thus for the IR behavior of
$\alpha_{s}$. The first solution, called “scaling scenario” (or also “
_conformal_ scenario”, “critical solution” or “ghost dominance scenario”), was
first obtained and studied using the SDE formalism. Many results obtained
using the SDE (see e.g. [245]-[253]), Lattice (see Section 4.5), variational
approach to the Yang–Mills Schrödinger equation of the vacuum [243] and the
exact renormalization group equations method (Section 4.6), show that in the
IR, $G(Q^{2},\mu^{2})$ and $Z(Q^{2},\mu^{2})$ scale with related power laws:
$G(Q^{2},\mu^{2})\propto(Q^{2})^{-\kappa^{\prime}}$ and
$Z(Q^{2},\mu^{2})\propto(Q^{2})^{2\kappa^{\prime}}$, where $\kappa^{\prime}$
is called the _IR exponent_. (This coefficient is generally labeled $\kappa$,
or sometimes $\alpha$, in the literature. We write it here $\kappa^{\prime}$
to differentiate it from the mass scale $\kappa$ used in holographic QCD in
Sections 3.9.6 and 4.2.). Such relations imply an IR _freezing_ of
$\alpha_{s}^{gh}$, see Eq. (4.25); thus the alternative name of “ _conformal_
scenario”. These relations also imply that ghosts dominate in the IR over
gluons. It was shown that the scaling scenario in the Landau gauge leads to a
well defined running of the coupling [249, 254, 255, 256]. In this gauge, the
_IR exponent_ was calculated to be $\kappa^{\prime}=(93-\sqrt{1201})/98\simeq
0.595$, using the SDE [243, 245, 249] or the functional renormalization group
method [257, 258]. This value leads to an IR _freezing_ value
$\alpha_{s}^{gh}\left(0\right)\simeq 2.972$.

It should be noted that the SDE computations of $\kappa^{\prime}$ can depend
on the approximations used to solve the SDE. This has resulted in a range of
values of $0.3\lesssim\kappa^{\prime}\lesssim 1$. Lattice calculations
indicate that $0.5\lesssim\kappa^{\prime}\lesssim 0.6$ with
$\kappa^{\prime}\simeq 0.5$ favored, see _e.g._ [259]. Lerche and von Smekal
[249], and then Alkofer and collaborators [260] have studied $\alpha_{s}^{gh}$
within this range using the Landau gauge and have obtained
$2.5\lesssim\alpha_{s}^{gh}\left(0\right)\lesssim 2.972$. The _scaling
solution_ also appears in other gauges. It has been studied in the Coulomb
gauge by Burgio _et al._ [261] using the lattice, and by Epple _et al._ [262]
using the SDE. The _IR exponent_ $\kappa^{\prime}$ is computed in the Landau,
interpolating and Coulomb gauges in Ref. [243], confirming
$\kappa^{\prime}\simeq 0.595$ in the Landau and interpolating gauges. In the
Coulomb gauge, two possible values are found: $\kappa^{\prime}\simeq 0.398$
and $\kappa^{\prime}=0.5$. These values are also found using the “functional
renormalization group” framework [256] – see Section 4.6 and by using a
variational approach in Refs. [218, 262]. However, a solution with
$\kappa^{\prime}=0.5$ seems not to be favored [256].

Fisher and collaborators have studied the effects of a finite volume in
lattice simulations and the corresponding effects of boundary conditions on
the value of $\kappa^{\prime}$ [263] by performing SDE calculations both on a
torus and for infinite space; one finds $\kappa^{\prime}\simeq 0.5$ and
$\kappa^{\prime}\simeq 0.6$, respectively. The coupling $\alpha_{s}^{gh}$
_freezes_ in the IR, irrespective of the value of $\kappa^{\prime}$. The work
of Fisher _et al._ also indicates an IR-vanishing gluon propagator in the case
of infinite space, whereas it stays finite in the finite-volume case. The
dependence of $\kappa^{\prime}$ with variable spatial dimensions was studied
in [264] for 2, 3 and 4 dimensions. The exponent $\kappa^{\prime}$ was found
to not vary dramatically, suggesting similar behavior of the underlying
_Green’s function_ s in these spaces.

A second possible solution for the IR behavior of $\alpha_{s}$, called the “
_decoupling solution_ ” (or “massive solution”, or “subcritical solution”),
was discovered by Boucaud and collaborators using lattice calculations [265].
In this case, the ghost propagator dressing function $G(Q^{2},\mu^{2})$ is
constant in the IR and the gluon propagator dressing function scales as
$Z(Q^{2},\mu^{2})\propto Q^{2}$. The predicted _IR exponents_ are different: 0
for the ghosts and 1 for the gluons. Thus according to Eq. (4.25) this
solution produces an IR-vanishing $\alpha_{s}^{gh}\propto Q^{2}$. The name
_decoupling solution_ originates from the vanishing of $Z(Q^{2},\mu^{2})$ as
$Q^{2}\rightarrow 0$, leading to free ghosts in the IR limit. That is, the
ghosts decouple from an effectively massive longitudinal gluon field. In the
decoupling scenario, the IR regime is still dominated by ghosts, although less
strongly than in the scaling scenario where they diverge.

The suppression of the gluon field in the _decoupling solution_ can be
interpreted as due to a dynamically generated gluon mass. The _decoupling
solution_ is thereby consistent with a Yukawa-like gluon propagator in the IR.
The massive solution is thus sometimes associated with massive scenarios, such
as Cornwall’s [234, 266] massive gluon which will be discussed in the next
section. However, Cornwall’s pioneering work is not in the Landau gauge; it
does not require ghosts, and it uses a different coupling, $\alpha_{gse}$
(although it has been shown to be similar to $\alpha_{s}^{gh}$ [235]).
Cornwall’s results produce a non-vanishing _freezing_ $\alpha_{gse}$.

Although it was first found on the lattice, the _decoupling solution_ is also
present in the SDE calculations [255, 265, 267, 268, 269], the _pinch_
technique background field method [270], an extended Gribov–Zwanziger approach
[271, 272], and from the stochastic quantization equations [273]. This last
work suggests, within the approximations used to truncate the equations, that
the _decoupling solution_ is a general solution of the theory. In contrast,
the _scaling solution_ would appear in specific gauges only, and represent a
particular solution stemming from the existence of a Gribov horizon (i.e., the
limit of the finite region of the field configuration space where the ghost
propagator remains positive and finite, see Section 4.7). This work also
indicates that the _decoupling solution_ has less effective action, which
explains why it seems favored by the lattice technique.

It is still unclear which solution is realized in the physical world.
Certainly, the description given above of the two solutions is gauge-
dependent, since some gauges, some relativistic form of dynamics, e.g., the
light-front, and some quantization methods do not require ghosts. In such a
case, the ghosts are replaced by other degrees of freedom, such as the
longitudinal component of the gluon propagator, see Section 4.8. Thus the two
solutions can still exist in a gauge without ghosts.

Some authors [263] have indicated that some decoupling-like results (that is
calculations leading to a vanishing coupling) may be due to a finite lattice-
size effect. However, an unphysical decoupling originating solely from finite
size effects seems to be now ruled out [274], and it would not explain why the
solution is also present in analytical methods. Fischer, Maas and Pawlowski
have argued that only the _scaling solution_ is compatible both with the
equations respecting the BRST symmetry [275], and with the color-confining
gluons [269]. Fischer and Pawlowski have also showed that the _scaling
solution_ is unique for the SDE and the exact renormalization group equations.
In particular, they showed that if the ghost and gluon external momenta
vanish, then one must realize the _scaling solution_ [254]. The study of
Alkofer, Huber and Schwenzer also disfavors the decoupling solution, at least
in the simplest realizations of QCD’s confinement [255]. Furthermore, the
_decoupling solution_ cannot satisfy the _Kugo–Ojima confinement criterion_
[276]. On the other hand, there are models supporting the _decoupling
solution_ : Boucaud _et al._ have interpreted this behavior in the context of
instantons [173]: the initial $\alpha_{s}(Q^{2})\propto Q^{4}$ IR-behavior is
predicted by the instanton liquid model, which represents the QCD vacuum as an
instanton liquid.

It has been shown in [225] and discussed in [151] that the two solutions lead
to sensibly similar descriptions of Nature in the IR domain, corresponding to
hadron mass/size scales. However, it is important to determine which solution
is being realized; for example, the _scaling solution_ justifies the approach
based on _conformal_ field theories, see Sections 3.9.6 and 4.2. In this
regard, the excellent description of QCD phenomenology by this approach and
the physical arguments for IR freezing of the QCD coupling given in Refs. [5,
6], would suggest that the _scaling solution_ is the one realized in Nature.

##### Typical forms of the gluon propagator for the two solutions

The _scaling solution_ , which leads to a freezing infrared fixed point value
for the QCD coupling, is consistent with the original Gribov–Zwanziger
approach of confinement (Section 4.7). In this case the gluon propagator
behaves as

$D_{\mu\nu}^{bc}(q)=\delta^{bc}\left(\eta_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{q^{2}}\right)\frac{q^{2}}{q^{4}+m_{gr}^{4}}.$
(4.26)

Here, $m_{gr}$ is the Gribov mass. In addition, if $Q^{2}\gg m^{2}_{gr}$ one
recovers the perturbative gluon propagator
$\delta^{bc}(\eta_{\mu\nu}-q_{\mu}q_{\nu}/q^{2})/q^{2}$.

The _decoupling solution_ , which leads to an IR-vanishing coupling when
$Q^{2}\to 0$, is itself consistent with an effective gluon propagator obeying
a Yukawa form [277],

$D_{\mu\nu}^{bc}(q)_{q\rightarrow
0}\propto\delta^{bc}\left(\eta_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{q^{2}}\right)\frac{1}{q^{2}-m_{g}^{2}},$
(4.27)

which can also be consistent with the generalized Gribov–Zwanziger scenario.
Nevertheless, in this refined Gribov–Zwanziger scenario, the _scaling
solution_ still seems preferred [278].

We will now discuss the topic of the effective gluon mass. As we have
mentioned earlier, the emergence of such a mass is equivalent to the
regularization of the behavior of $\alpha_{s}$ in the IR.

####  4.4.4 The massive gluon propagator

The first studies of the gluon propagator in the IR regime using the
Schwinger–Dyson formalism began in the 1970s [266, 279, 280]. Using the Landau
gauge, these early results indicated that $\alpha_{s}$ has no _Landau pole_ ,
and that in the IR, $\alpha_{s}(Q^{2})\propto 1/Q^{2}$ [279, 281]. A
calculation in the axial gauge [237] reached the same conclusion. The singular
dependence in the IR agrees naively with the expectation of a linear potential
in the static limit; see Section 4.3. However, in spite of this apparent
consistency, Cornwall [234] has argued that the $1/Q^{2}$ behavior is actually
erroneous due to the explicit exclusion of solutions which generate a
dynamical gluon mass that regulates the _Landau pole_. Earlier suggestions
that a gluon mass should suppress the unphysical Landau pole were made in
[266, 282, 283].

In Cornwall’s work an analytic form for the gluon propagator is proposed for
solving the SDE in the _pure gauge_ sector and in the light-cone gauge where
ghosts are unnecessary. Feynman graphs are resummed to form a gauge-
independent gluon propagator dressing function that modifies the free gluon
propagator which carries the gauge dependence. The resulting dynamically
generated gluon effective mass is $Q^{2}$-dependent and vanishes at
large-$Q^{2}$. This feature ensures compatibility with pQCD expectations, and
it accommodates the fact that a physical gluon mass is forbidden by gauge
invariance. Cornwall’s SDE solution to the gluon propagator suggests that the
perturbative form of $\alpha_{s}$, defined from the gluon self-energy, should
be modified in the IR as :

$\alpha_{gse}\left(Q^{2}\right)=\frac{4\pi}{\beta_{0}\mbox{ln}\left(\left[Q^{2}+\varsigma
m_{g}^{2}\left(Q^{2}\right)\right]/\Lambda^{2}\right)},$ (4.28)

with $\varsigma=4$ and $m_{g}(Q^{2})$ is an effective gluon mass:

$m_{g}^{2}\left(Q^{2}\right)=\frac{m^{2}}{\left[\mbox{ln}\left(\frac{Q^{2}+\xi
m^{2}}{\Lambda^{2}}\right)/\mbox{ln}\left(\frac{\varsigma
m^{2}}{\Lambda^{2}}\right)\right]^{12/11}},$ (4.29)

with $m=0.5\pm 0.2$ GeV and $\Lambda=0.26\pm 0.05$ GeV. The _freezing_ value,
$\alpha_{gse}(0)=0.42\pm 0.20$ agrees with the characteristic value of
$\alpha_{s}$ estimated using magnetic and color–magnetic spin–spin
interactions [284] and other determinations done in the context of hadron
spectroscopy, see Section 4.3. The limit $m=\Lambda$ is consistent with
Richardson’s form, Eq. (4.19). Renormalization group equations predict that at
large $Q^{2}$, $m_{g}^{2}(Q^{2})\propto\mbox{ln}(Q^{2})^{-12/11}$ where the
exponent is related to the strength of the static force in the _pure gauge_
sector; $4\pi(\beta_{0})^{-1}=12\pi/11$.

The gluon mass must satisfy the condition $m_{g}\geq\Lambda$ in order to
eliminate the _Landau pole_. Cornwall showed that using the _pinch_ technique
$m\simeq 1.2\,\Lambda$ [236] (a review of the _pinch_ technique, including its
application to the SDE and the computation of $\alpha_{s}$, can be found in
[285]). Values of $m$ and $\Lambda$ can also be obtained by assuming
Cornwall’s massive form for the gluon propagator and constraining it with
hadron phenomenology. For example, the isovector GDH sum rule for the nucleon
can be used to constrain $m_{g}$ by determining $\alpha_{\overline{MS}}$ at
$Q^{2}=0$. Solving Eqs. (4.1) and (4.4), one finds $m=0.350$ GeV and
$\Lambda=0.268$ GeV ($\alpha_{gse}(0)=0.59$). The ratio $m/\Lambda=1.31$
validates the pinch technique expectation that $m/\Lambda\simeq 1.2$ [236].
The GDH-based result is gauge independent, but this analysis depends on the
truncation order of Eq. (3.40). Since an _asymptotic series_ will reach its
optimal order at $n\sim(1/\mbox{coupling)}$, the value of
$\alpha_{s}(Q^{2}\simeq 1\mbox{ GeV}^{2})$ suggests that truncating Eq. (3.40)
at $n\sim 4$ is adequate.

Another way to use phenomenological information is to constrain Eqs. (4.28)
and (4.29) using data for hadron form factors. A good description of the
experimental pion and kaon form factors is obtained for $m=0.54$ GeV by Ji and
collaborators [286]. It yields $\alpha_{gse}(0)=0.4$ for $\Lambda=0.26$ GeV.
Brodsky and collaborators have combined the dimensional counting rules for the
form factors in the UV with Cornwall’s _freezing_ coupling for $m=0.44$ GeV
(i.e. $\alpha_{gse}(0)=0.47$ for $\Lambda=0.26$ GeV) to analyze the
photon–to–pion transition form factor and
$\gamma\gamma\rightarrow\pi^{+}\pi^{-}$. The data is well reproduced, but the
predicted normalization of the space-like pion form factor is higher than
measurements [287]. On the other hand, Aguilar and collaborators find that
$m=0.300$ GeV [288] describes the pion form factor data well, which implies a
_freezing_ value $\alpha_{gse}(0)=0.68$ ($\Lambda=0.26$ GeV).

In Ref. [289], the value of $m$ is constrained using measurements of the
proton–proton elastic cross section. This yields $m=0.370$ GeV
($\alpha_{gse}(0)=0.55$ for $\Lambda=0.26$ GeV) and in Refs. [290, 291] the
value $m=0.374$ GeV ($\alpha_{gse}(0)=0.54$ for $\Lambda=0.26$ GeV) is
obtained. The SDE are used in Ref. [292] to obtain $m=0.68$ GeV
($\alpha_{gse}(0)=0.35$ for $\Lambda=0.26$ GeV). In this analysis, the
predicted SDE coupling _freezes_ at $\alpha_{s}(0)=2.8$, a much larger value
than the $\alpha_{gse}(0)$ expected from Cornwall’s form. In Ref. [292], the
gluon mass and coupling are well parameterized in the UV and IR-UV transition
region $1<Q^{2}<18.5$ GeV2 by

$m_{g}^{2}\left(Q^{2}\right)=\frac{0.22+0.019Q^{2}}{1+1.76Q^{2}}$ (4.30)

and

$\alpha_{s}\left(Q^{2}\right)=12.9m_{g}^{2}\left(Q^{2}\right).$ (4.31)

All of these values of $m$ are within Cornwall’s proposed range. Lattice QCD
estimates of $m$ also fall within this range [293]. An analysis of the data
for inclusive radiative decays of the $J/\psi$ and $\Upsilon$ yields a
somewhat higher value, $m\sim 1$ GeV ($\alpha_{gse}(0)\sim 0.28$ for
$\Lambda=0.26$ GeV) [294].

Aguilar and Papavassiliou have summarized the studies of $m_{g}(Q^{2})$ in
Ref. [240]. These authors also show that a massive propagator effectively
describes the solution of the SDE in the gauge-independent _pinch_ technique
in the _pure gauge_ sector, but with two possible behaviors at large $Q^{2}$.
One solution is similar to Eq. (4.29), with
$m_{g}(Q^{2})\propto\mbox{ln}(Q^{2})^{-\gamma}$ (taking a typical value
$\gamma=0.86$), as expected from renormalization group equations. The second
solution falls with a power law:
$m_{g}(Q^{2})\propto\mbox{ln}(Q^{2})^{\gamma}/Q^{2}$ (with a typical value
$\gamma=1.12$), similarly to the power corrections obtained from the _OPE_.
The behavior of the gluon mass in Eq. (4.29) is consistent with the
logarithmic running of $m_{g}(Q^{2})$, as predicted by the first solution,
with the values $\varsigma=1.007$, $m=0.3$ GeV, $\Lambda=0.3$ GeV. This leads
to a _freezing_ value for $\alpha_{gse}(0)\simeq 1.2$. For the second power-
law solution for $m_{g}(Q^{2})$, the effective mass is well described by

$m_{g}^{2}\left(Q^{2}\right)=\frac{m^{4}}{\left[Q^{2}+m^{2}\right]\left[\mbox{ln}\left(\frac{Q^{2}+\epsilon
m^{2}}{\Lambda^{2}}\right)/\mbox{ln}\left(\frac{\epsilon
m^{2}}{\Lambda^{2}}\right)\right]^{\gamma}},$ (4.32)

with $\epsilon=1.046$, $m=0.5$ GeV, $\Lambda=0.3$ GeV and $\gamma=1.12$. This
leads to the _freezing_ value $\alpha_{gse}(0)\simeq 1.0$.

The results of Aguilar and Papavassiliou are gauge-invariant, and the
effective gluon mass can be interpreted as a gauge-invariant gluon vacuum
_condensate_ or nonperturbative correction of dimension four. In subsequent
work, [241], Aguilar and Papavassiliou used the Feynman gauge, the _pinch_
technique, and gauge invariant truncations of the gluon propagator to obtain
forms for $m_{g}(Q^{2})$ and $\alpha_{gse}(Q^{2})$. Effects from ghost loops
were neglected. Although the authors did not specify the _freezing_ value,
they provide a numerical example with a _freezing_ value of
$\alpha_{gse}\left(0\right)=0.91$ and a gluon mass normalization,
$m_{g}(0)=0.45$ GeV. An alternative nonperturbative approach to the gluon mass
has been developed by Aguilar and collaborators based on the SDE in the Landau
gauge [290] and the RGE [291]. The authors provide a numerical fit of their
results:

$m_{g}^{2}\left(Q^{2}\right)=\frac{m^{2}}{1+\left(Q^{2}/M^{2}\right)^{\gamma}},$
(4.33)

with $M=0.557$ GeV, $m=0.375$ GeV, and $\gamma=1.08$. Cornwall’s solution
(4.29), agrees well with these results. The gauge-dependence of $m_{g}(Q^{2})$
in these formulae has been investigated using linear covariant gauges in which
a gauge parameter $\xi$ continuously transitions between the Landau gauge
($\xi=0$) and the Feynman gauge ($\xi=1$). The resulting gluon mass
normalization is typically $m\simeq 0.37\pm 0.1$ GeV [244]. This implies
$\alpha_{gse}(0)=0.55\pm 0.03$ for $\Lambda=0.26$ GeV.

A closely related model has been recently proposed by Ayala _et al._ [295]. In
this analysis the _effective charge_ $\alpha_{V}$ has the form proposed by
Cornwall, Eq. (4.28), assuming the Aguilar _et al._ $Q^{2}$-dependence of the
effective gluon mass, Eq. (4.33), with $\gamma=1.15$ and a normalization
$m=\Lambda$. Thus Richardson’s limit, Eq. (4.19), of Cornwall’s coupling is
recovered at $Q^{2}=0$. Since the resulting coupling diverges as $1/Q^{2}$
when $Q^{2}\to 0$, this also leads to a linear potential for static quarks
consistent with quarkonium spectroscopy.

Cornwall’s coupling, Eq. (4.28), takes the form of the pQCD coupling at
$\beta_{0}$ order with an additional dynamical effective gluon mass term. In
effect, the nonperturbative effects which regularize $\alpha_{s}$ in the IR
are effectively incorporated into $m_{g}(Q^{2})$, together with higher order
pQCD loops. This is similar to the case of the _effective charges_ defined by
Grunberg, see Section 4.1: both nonperturbative and perturbative contributions
are incorporated into $\alpha_{s}(Q^{2})$. However, in Cornwall’s case,
observable-independent higher loops are folded in, whereas in Grunberg’s case,
observable-dependent gluonic bremsstrahlung is folded in. Nevertheless, the
two results can agree, once the scheme dependence is accounted for, see
Section 5.

Eqs. (4.28) and (4.29) have been augmented by further work by Cornwall and
Papavassiliou [296] using the _pinch_ technique and the SDE. One finds the
same form for the running coupling except that quark loops are now included:

$\alpha_{gse}\left(Q^{2}\right)=\frac{12\pi}{33\mbox{ln}\left[\left(Q^{2}+\epsilon
m_{g}^{2}\left(Q^{2}\right)\right)/\Lambda^{2}\right]-2n_{f}\mbox{ln}\left[\left(Q^{2}+\epsilon
M^{2}\right)/\Lambda^{2}\right]}.$ (4.34)

In this analysis, $\epsilon\simeq 4.8$, $M$ is identified with the string
tension: $M\simeq\sqrt{\sigma}\simeq 0.42$ GeV, and Eq. (4.29) is used for
$m_{g}(Q^{2})$. The screening role of the quark loops can be recognized in Eq.
(4.34). With $n_{f}=2$, $\Lambda=0.26$ GeV and $M=0.42$,
$\alpha_{gse}(0)=0.44$.

Other forms for the massive gluon propagator have been proposed using the
Gribov–Zwanziger approach; see Eq. (4.26). If one uses the SDE solution of
Stingl and of Habel _et al._ [297], the gluon propagator has the form

$D_{\mu\nu}^{bc}(q)=\delta^{bc}\left(\eta_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{q^{2}}\right)\frac{q^{2}}{q^{4}+2aq^{2}+m_{g}^{4}},$
(4.35)

where $a$ is a parameter constrained by phenomenology.

Lattice results can also be interpreted in terms of a massive gluon propagator
[298]. For example, the results of Marenzoni and collaborators [299] can be
fit by the form

$D(Q^{2})=\frac{1}{bQ^{2}\left(Q^{2}/\Lambda^{2}\right)^{\eta}+m_{g}^{2}},$
(4.36)

with $b=0.102(1)$, $\eta=0.532(12)$ and $m_{g}\simeq 0.4$ GeV. Similarly,
Oliveira and Bicudo have interpreted their lattice data in terms of a massive
gluon propagator. Their result for $Q^{2}\lesssim 0.25$ GeV2 can be described
assuming a constant IR gluon mass of about $m_{g}\simeq 0.7$ GeV [300]. Their
calculations are performed in the Landau gauge. A similar value is obtained in
Ref. [271]. The results are consistent with the _decoupling solution_ ,
yielding an IR-vanishing $\alpha_{s}$. In recent work [292], the SDE and
lattice data are also combined in the form of a massive gluon propagator. This
leads to an IR value of $\alpha_{s}(0)\simeq 0.9\,\pi$. Thus lattice studies
can in effect be interpreted as the emergence of an effective gluon mass.

Massive gluons have also been considered in approaches other than SDE and
lattice QCD; e.g., the Gribov–Zwanziger approach [301], the background
perturbation theory (Section 4.10), or the variational approach of Szczepaniak
and Swanson [165] which yields a constituent gluon mass of about 0.6 GeV.

Finally, we notice that an effective gluon mass can be used to generate
$1/Q^{2}$ _power corrections_ ; this is in spite of the fact that in the _OPE_
there is no _condensate_ of dimension-2 arising from a local gauge-invariant
operator [302]. However, in this case, the resulting effective gluon mass must
be imaginary (tachyonic gluon mass).

####  4.4.5 The coupling defined from the ghost–gluon vertex

The ghost–gluon coupling $\alpha_{s}^{gh}$ from Eq. (4.25) was first obtained
using the SDE formalism by von Smekal, Hauck and Alkofer [246]. Their Landau
gauge analysis is based on a self-consistent truncation scheme, where the
truncation implements the _Slavnov–Taylor_ identities [33] for the three-point
vertex function. The numerical results lead to a coupling agreeing with pQCD
evolution at large $Q^{2}$, while _freezing_ to a value of
$\alpha_{s}^{gh}\left(0\right)\simeq 9.5$. The following forms match well the
numerical solution [288]:

$\displaystyle\alpha_{s}^{gh}$ $\displaystyle=$ $\displaystyle
0.261+9.2621e^{-2\frac{\left(Q^{2}-0.0297\right)^{2}}{0.4689}};\mbox{ for
}Q^{2}<0.31\mbox{ GeV}^{2},$ $\displaystyle\alpha_{s}^{gh}$ $\displaystyle=$
$\displaystyle 1.4741+8.6072e^{-\frac{Q^{2}-0.1626}{0.3197}};\mbox{ for
}0.31<Q^{2}<1.3\mbox{ GeV}^{2},$ $\displaystyle\alpha_{s}^{gh}$
$\displaystyle=$ $\displaystyle\frac{1.4978}{\mbox{ln}(1.8488Q^{2})};\mbox{
for }Q^{2}>1.3\mbox{ GeV}^{2}.$ (4.37)

The numerical techniques have been subsequently improved, yielding smaller
_freezing_ values. Fischer _et al._ [250] obtained a numerical coupling
(Landau gauge, _pure gauge_ sector) which is well fitted by:

$\alpha_{s}^{gh}\left(Q\right)=\frac{2.972}{\mbox{ln}\left(e+5.292Q^{2.324}+0.034Q^{3.169}\right)},$
(4.38)

where the numerical coefficients are determined by matching to the value of
the pQCD coupling at the $Z^{0}$ mass:
$\alpha_{s}^{gh}(M_{Z}^{2})=\alpha_{s}^{pQCD}(M_{Z}^{2})$. This implies, in
the $MOM$ scheme, $\Lambda_{MOM}=0.715$ GeV and the _freezing_ value
$\alpha_{s}^{gh}\left(0\right)\simeq 2.972$; see Section 4.4.3.

An alternative truncation prescription was used by Bloch [303]. The result in
the _pure gauge_ theory is fitted by:

$\alpha_{s}^{gh}\left(Q^{2}\right)=\frac{1}{15+Q^{2}/\Lambda^{2}}\left(15\times
2.6+\frac{4\pi}{\beta_{0}}\left(\frac{1}{\mbox{ln}(Q^{2}/\Lambda^{2})}-\frac{1}{Q^{2}/\Lambda^{2}-1}\right)\frac{Q^{2}}{\Lambda^{2}}\right),$
(4.39)

where $\Lambda_{\overline{MS}}=0.33$ GeV in the $\overline{MS}$ scheme. In
this Landau gauge analysis, the _freezing_ value for $\alpha_{s}^{gh}$ is 2.6,
and the dynamically generated gluon mass is $m_{g}\simeq 0.35$ in the IR
domain. The form for the running ghost–gluon coupling is similar to the
behavior of $\alpha_{an}$ obtained in the analytic approach, Eq. (4.54), see
Section 4.9.1. The truncation prescription was subsequently improved in Ref.
[253], leading to numerical results parameterized as

$\alpha_{s}^{gh}\left(Q^{2}\right)=\frac{1}{1.16+Q^{4}/\Lambda^{4}}\biggl{(}3.49\left[1.16-0.070\left(Q^{2}/\Lambda^{2}\right)^{0.66}\right]\\\
+\left[Q^{4}/\Lambda^{4}\right]\left(Q^{2}/\Lambda^{2}+2\right)\alpha_{s}^{pQCD,\beta_{1}}\biggr{)},$
(4.40)

where $\alpha_{s}^{pQCD,\beta_{1}}$ is the perturbative coupling at order
$\beta_{1}$ obtained from Eq. (3.24) and $\Lambda=0.856$ GeV. This gives a
_freezing_ value of 3.49. Comparisons with the earlier results illustrate the
effects of modifying the truncation prescription as well as choosing a
different RS. To reinforce this point, we note that the earlier work by
Atkinson and Bloch with a bare truncation that leads to a _freezing_ value of
11.47 by averaging the SDE angular integrals, and to a _freezing_ at $4\pi/3$
with the angular integrals done exactly [247]. Bloch systematically studied
the effect of varying the truncation scheme and found that
$2\pi/3<\alpha_{s}^{gh}\left(0\right)<8\pi/3$ in SU(3) [248].

Aguilar and Natale [267] obtained an IR-vanishing coupling $\alpha_{s}^{gh}$.
These authors assumed a massless gluon propagator behaving as $1/Q^{2}$. The
calculations were done in the Landau gauge and pure Yang–Mills using the
renormalization prescription of [234, 296]. In subsequent works Aguilar _et
al._ employed a massive gluon propagator, rather than a massless one, leading
to a _freezing_ behavior of the coupling. Furthermore, in [235], Aguilar _et
al._ computed both $\alpha_{gse}$ and $\alpha_{s}^{gh}$ and found that the two
couplings have the same _freezing_ value of order 0.6 for $m_{g}(0)=0.5$ GeV.

Schleifenbaum and collaborators [243] have computed the gluon and ghost
propagators in both Landau and Coulomb gauges, finding that the gluon
propagator in the IR is suppressed, whereas the ghost propagator is singular.
These features yield an approximately linear potential and a ghost–gluon
coupling _freezing_ at $\alpha_{s}^{gh}\left(0\right)\simeq 3.65$ for the
Landau gauge. They also find two solutions in the Coulomb gauge
$\alpha_{s}^{gh}(0)\simeq 14.21/N_{C}=4.74$ or
$\alpha_{s}^{gh}(0)=16\pi/N_{C}\simeq 5.59$, with the value 4.74 favored
[256].

####  4.4.6 Couplings defined from the 3-gluon and 4-gluon vertices

In addition to the definition from the ghost–gluon vertex, Eq. (4.25),
couplings can also be defined from the 3-gluon and 4-gluon vertices:

$\alpha_{s}^{3g}\left(Q^{2}\right)=\alpha_{s}^{3g}\left(\mu\right)\left[\Gamma^{3g}(Q^{2},\mu)\right]^{2}Z^{3}(Q^{2},\mu),$
(4.41)
$\alpha_{s}^{4g}\left(Q^{2}\right)=\alpha_{s}^{4g}\left(\mu\right)\left[\Gamma^{4g}(Q^{2},\mu)\right]^{2}Z^{2}(Q^{2},\mu),$
(4.42)

where $\Gamma^{3g}$ and $\Gamma^{4g}$ are the dressing functions of the tree-
level 3-gluon and 4-gluon vertices. In contrast to the case of
$\alpha_{s}^{gh}$, dressing functions appear since there are no
_Slavnov–Taylor_ identities for these vertices. The three couplings
$\alpha_{s}^{gh}(Q^{2})$, $\alpha_{s}^{3g}(Q^{2})$ and
$\alpha_{s}^{4g}(Q^{2})$ are different in the MOM RS but can be related to
each other [22, 153].

The 3-gluon and 4-gluon vertices have been studied in the pure gauge sector by
Baker and Lee [304], Celmaster and Gonsalves [22], by Brandt and Frenkel
[305], as well as by Papavassiliou using the _pinch_ technique [306]. The
resulting couplings $\alpha_{s}^{3g}(Q^{2})$ and $\alpha_{s}^{4g}(Q^{2})$ were
first investigated by Alkofer, Fischer, and Llanes-Estrada [307]. They
concluded that these couplings must _freeze_ in the IR, but they did not
provide _freezing_ values. A recent perturbative analytical calculation of the
IR and UV behaviors of the 3-gluon vertex (Landau gauge and _pure gauge_
sector) is reported in [308].

Figure 4.4:  SDE computations of $\alpha_{s}$ defined using the 4-gluon vertex
(the green line is from Ref. [313] and black line is from Ref. [314]). The red
line, using the ghost–gluon vertex definition, is from Ref. [313] and the blue
line is from Ref. [314]. The magenta line for the 3-gluon vertex definition is
from Ref. [314]. Quark degrees of freedom are not included and the
calculations are done in the Landau gauge and MOM scheme.

The 3-gluon vertex was calculated nonperturbatively using the SDE by Eichmann
_et al._ [309] using the Landau gauge. Both _scaling_ and _decoupling
solutions_ were obtained. In the decoupling case, $\alpha_{s}^{gh}$ and
$\alpha_{s}^{3g}$ vanish in the IR, whereas in the scaling case
$\alpha_{s}^{gh}(0)=2.972$ and $\alpha_{s}^{3g}(0)=1.6\times 10^{-3}$; see
Fig. 4.4. A self-consistent SDE calculation of $\alpha_{s}^{3g}(Q^{2})$, also
in the Landau gauge, yields vanishing $\alpha_{s}^{gh}(Q^{2})$ and
$\alpha_{s}^{3g}(Q^{2})$ in the IR [310]. Huber, Campagnari, and Reinhardt
solved numerically, in the Coulomb gauge and _pure gauge_ sector, the
canonical recursive SDE for the three-gluon and ghost–gluon vertices. The
resulting ghost–gluon vertex is IR finite, whereas the three-gluon vertex is
IR singular [311].

The 4-gluon vertex was first studied nonperturbatively by Stingl, Driesen and
collaborators [312]. The calculation for the 4-gluon coupling
$\alpha_{s}^{4g}$ in the _pure gauge_ sector, assuming Landau gauge and the
MOM RS, by Kellermann and Fischer [313] yields a very small _freezing_ value,
$\alpha_{s}^{4g}(0)\simeq 2.77\times 10^{-3}$. The resulting behavior of
$\alpha_{s}^{4g}(Q^{2})$ follows the pQCD expectation at large $Q^{2}$ and
increases with decreasing $Q^{2}$ to about $Q^{2}=0.5$ GeV2. It then drops to
its _freezing_ value, which is reached at $Q^{2}\sim 10^{-2}$ GeV${}^{2};$ see
Fig. 4.4.

Cyrol, Huber and von Smekal [314] have recently computed self-consistent
values for the couplings controlling the 3-gluon and 4-gluon vertices. Again
this calculation is in the _pure gauge_ sector, assumes Landau gauge and the
MOM RS. In the _scaling solution_ , the 3-gluon and 4-gluon vertex couplings
_freeze_ at values that are very small compared to $\alpha_{s}^{gh}(0)=2.972$;
$\alpha_{s}^{3g}(0)=3.2\times 10^{-3}$ and $\alpha_{s}^{4g}(0)=4.2\times
10^{-4}$, confirming qualitatively the finding in Refs. [309, 313], namely
$\alpha_{s}^{3,4g}(0)\ll\alpha_{s}^{gh}(0)$. These running couplings are shown
in Fig. 4.4. In the _decoupling solution_ case, all three definitions of
$\alpha_{s}$ lead to IR-vanishing couplings.

####  4.4.7 Including quarks

Quark loops contribute to the running of $\alpha_{s}$ in the UV in the
opposite direction to the gluons, making $\alpha_{s}$ increase with $Q^{2}$.
Thus one expects that their inclusion will be to decrease $\alpha_{s}(0)$
compared to the results obtained in the _pure gauge_ sector. This comparison
is done with $\Lambda$ adjusted so that
$\alpha^{n_{f}=0}_{s}(M_{Z}^{2})=\alpha^{n_{f}=5}_{s}(M_{Z}^{2})$. _A
contrario_ , if the value of $\Lambda$ is kept the same for any $n_{f}$, the
quark loops would increase the value of $\alpha^{n_{f}>0}_{s}(0)$ compared to
$\alpha^{n_{f}=0}_{s}(0)$ since
$\alpha^{n_{f}>0}_{pQCD}(Q^{2})>\alpha^{n_{f}=0}_{pQCD}(Q^{2})$ and the
_Landau pole_ divergence of the perturbative coupling is steeper. In any case,
since effects other than vacuum polarization are included in the IR definition
of the coupling, the effect of quarks is not obvious, as illustrated by the
following example.

The influence of quark loops on the coupling has been investigated in ref.
[296]; see Eq. (4.34). The quark loops increase the _freezing_ value:
$\alpha^{n_{f}=2}_{gse}(0)=0.81$ in comparison with
$\alpha^{n_{f}=0}_{gse}(0)=0.60$. The values $\Lambda^{(5)}=0.09$ GeV and
$m=1.2\,\Lambda^{(5)}=0.11$ GeV are used for the $n_{f}=2$ calculation, and
$\Lambda^{(0)}=0.65$ GeV and $m=1.2\,\Lambda^{(0)}=0.78$ GeV are used for the
$n_{f}=0$ calculation –so that
$\alpha^{n_{f}=2}_{gse}(M_{Z}^{2})=\alpha^{n_{f}=0}_{gse}(M{{}_{Z}}^{2})=0.119$
at the $Z^{0}$ pole. The quark loop effect is thus non-negligible according to
[296]. However, Fisher and collaborators have recently reported that quark
loops are suppressed in the IR and thus have only a small effect for a
physical numbers of quark flavors, $n_{f}<6$ [315]. This result is apparent
from the function that fits their results (apart for a bump at $Q^{2}\simeq
0.5$ GeV2, which is believed to be unphysical and is omitted):

$\alpha_{s}^{gh}\left(Q^{2}\right)=\frac{1}{1+\frac{Q^{2}}{{\Lambda^{n_{f}}}^{2}}}\left(2.972+\frac{4\pi}{\beta_{0}}\left(\frac{1}{\mbox{ln}(\frac{Q^{2}}{{\Lambda^{n_{f}}}^{2}})}-\frac{1}{\frac{Q^{2}}{{\Lambda^{n_{f}}}^{2}}-1}\right)\frac{Q^{2}}{{\Lambda^{n_{f}}}^{2}}\right),$
(4.43)

with a _freezing_ value which is independent of $n_{f}$. The quark effects are
encoded in the _scale parameter_ : $\Lambda^{n_{f}=0}_{MOM}=0.71$ GeV and
$\Lambda^{n_{f}=3}_{MOM}=0.51$ GeV. (We note that the fit form is similar to
Eq. (4.39) and to the coupling found in the analytical approach, Eq. (4.54)).
Eq. (4.43) was used to study analytic properties of the gluon and quark
propagators in the Landau gauge [252].

Related to the inclusion of quarks, Alkofer, Fisher, and Llanes-Estrada have
studied the influence of chiral-symmetry breaking on the running coupling
$\alpha_{s}^{qg}(Q^{2})$, now defined by the quark–gluon vertex [316]:

$\alpha_{s}^{qg}\left(Q^{2}\right)=\alpha_{s}^{qg}\left(\mu\right)\xi^{2}_{1}\left(\mu,Q^{2}\right)P^{2}\left(\mu,Q^{2}\right)Z\left(\mu,Q^{2}\right),$
(4.44)

where $\xi_{1}$ is the vertex dressing function and $P$ the quark propagator
dressing function. It was found that the reduction of the number of linearly
independent Dirac tensors using chiral symmetry leads to different behavior of
the coupling. In the chiral symmetry case, which approximates well the light
quark sector, the coupling is IR-finite: $\alpha_{s}^{qg}(Q^{2})\simeq 2.5$.
In contrast, in the broken chiral symmetric case (as for the heavy quark
sector where quark loops are _quenched_), the additional tensors induce a
coupling diverging in the IR: $\alpha_{s}^{qg}(Q^{2})\propto 1/Q^{2}$. As
discussed in Section 4.3 the $1/Q^{2}$ behavior of the coupling gives rise to
a linear $\mbox{Q--}\overline{\mbox{Q}}$ potential for heavy static quarks.
This behavior can be compared to the expectation that QCD strings develop only
in the heavy-quark sector [317].

####  4.4.8 Gauge dependence

Many calculations in the SDE formalism and other frameworks yield a gauge-
dependent coupling $\alpha_{s}$. The question of the best choice of gauge and
its influence on the IR-behavior of $\alpha_{s}$ is open. (The gauge-
dependence subsides in the UV regime). Fisher has reported that couplings
computed in gauges close to the Landau gauge have a weak gauge-dependence
[231]. Fischer and Zwanziger [318] have studied the gauge-dependence of the
_scaling solution_ of $\alpha_{s}^{gh}$ using a class of transverse gauges
which interpolate between the Landau and Coulomb gauges. The IR value
$\alpha_{s}^{gh}\left(0\right)\simeq 2.972$, found for the _IR exponent_
$\kappa^{\prime}=0.595$, holds across the gauge range, except the Coulomb case
which has a singular limit. This particular case is studied in detail in Ref.
[319]. The _freezing_ value is found to be slightly lower:
$\alpha_{s}^{gh}\left(0\right)\simeq 2.333$. However, the already mentioned
calculations of the same quantity in Ref. [243], in both Landau and Coulomb
gauges, lead to the opposite trend: $\alpha_{s}^{gh}\left(0\right)\simeq 3.65$
in the Landau gauge and $\alpha_{s}^{gh}\left(0\right)\simeq 4.74$ in the
Coulomb gauge. (Two solutions are actually found in the Coulomb gauge:
$\alpha_{s}^{gh}\left(0\right)\simeq 4.74$ or
$\alpha_{s}^{gh}\left(0\right)\simeq 5.59$ for $\kappa^{\prime}\simeq 0.398$
or $\kappa^{\prime}=0.5$, respectively, with
$\alpha_{s}^{gh}\left(0\right)\simeq 4.74$ preferred). Recently, Huber has
numerically obtained the decoupling solutions of the SDE for linear covariant
gauges, varying its gauge fixing parameter between $0\leq\xi\leq 0.2$ [320].
In each case, the couplings extracted from the ghost–gluon and 3-gluon
vertices vanish as $Q^{2}\rightarrow 0$, albeit with a large gauge dependence
in the intermediate $Q^{2}$ domain of a few GeV2.

###  4.5 Lattice gauge theory

Lattice calculations provide another important method for studying the
behavior of the QCD coupling $\alpha_{s}$ in the IR regime. These studies
connect with the SDE framework. Both approaches have greatly benefited from
their complementarity. The basics of lattice QCD have been given in Section
3.9.4. Here, we will first describe the steps for typical lattice calculations
of $\alpha_{s}(Q^{2})$ in the IR. We will then discuss the results obtained by
several groups which use the numerical lattice approach to compute the QCD
coupling $\alpha_{s}$ in the nonperturbative infrared domain.

One method for obtaining $\alpha_{s}(Q^{2})$ in the IR is to compute the
static potential, as discussed in Sections 3.9.4 and 4.3. Another approach is
to compute the coupling from vertices. In this case, just as for the SDE
formalism, it amounts to determining correlation functions. Those were first
calculated on the lattice by Mandula and Ogilvie [293]. In the following, we
overview representative lattice methods to obtain $\alpha_{s}$ (valid in the
IR as well as the UV domains). The examples are restricted to the _pure gauge_
sector.

##### An example of the lattice determination of $\alpha^{gh}_{s}$

As discussed in Section 4.4, $\alpha_{s}(Q^{2})$ can be defined from the
ghost–gluon vertex, see Eq. (4.25). To compute the ghost and gluon propagator
dressing functions, discretized ghost and gluon fields need to be defined. The
discretized gluon field $\mathcal{A}_{\overrightarrow{\mu}}^{c}(x)$, where $c$
is the color index and $\overrightarrow{\mu}$ a link direction, may be defined
as the difference between a link and its adjoint:

$\mathcal{A}_{\overrightarrow{\mu}}^{c}(x)\equiv\frac{1}{2i}\left(U_{\overrightarrow{\mu}}(x)-U_{\overrightarrow{\mu}}^{\dagger}(x)\right)~{}\overrightarrow{{}_{a\to
0}}~{}2a\sqrt{\pi\alpha_{s}^{bare}}A_{\mu}^{c}(x).$ (4.45)

The continuum limit $a\rightarrow 0$ corresponds to the physical gluon field
$A_{\mu}^{c}$. The discretization of space-time induces a finite difference
$\mathcal{A}_{\overrightarrow{\mu}}^{c}(x)-2a\sqrt{\pi\alpha_{s}^{bare}}A_{\mu}^{c}(x)$.
This creates unphysical _tadpole_ contributions proportional to
$\alpha_{s}^{bare}$ or to $a^{3}$. Those can be eliminated by a redefinition
of the link variable or of the action, see Refs. [112] or [238]. Other
definitions of $\mathcal{A}_{\overrightarrow{\mu}}^{c}(x)$ have been used, and
are equivalent –up to a multiplicative renormalization factor [321]. The gluon
propagator from Eq. (3.53) is

$D_{\mu\nu}^{bc}(x-y)=\int
DxA_{\mu}^{b}(x)A_{\nu}^{c}(y)e^{-iS}\equiv\left\langle
A_{\mu}^{b}(x)A_{\nu}^{c}(y)\right\rangle.$ (4.46)

It gives on the lattice

$\mathcal{D}_{\overrightarrow{\mu}\overrightarrow{\nu}}^{bc}\left(x-y\right)=\left\langle\mathcal{A}_{\overrightarrow{\mu}}^{b}\left(x\right)\mathcal{A}_{\overrightarrow{\nu}}^{c}\left(y\right)\right\rangle.$
(4.47)

Then,
$\mathcal{D}_{\overrightarrow{\mu}\overrightarrow{\nu}}^{bc}\left(x-y\right)$
can be transformed to momentum space via a Fourier transform. This yields the
gluon propagator dressing function
$\mathcal{Z}(Q^{2})\equiv\delta^{bc}\delta^{\mu\nu}\mathcal{D}^{bc}_{\mu\nu}(Q^{2})Q^{2}$.
The lattice ghost propagator is obtained, by definition, from the inverse of
the Fadeev–Popov operator $(-\partial+A)A$; after discretization and
evaluation on the lattice, it is Fourier-transformed to momentum space. The
resulting ghost propagator dressing function is
$\mathcal{G}=\delta^{bc}Q^{2}\mathcal{D}_{bc}^{G}\left(Q^{2}\right)$. The
coupling $\alpha_{s}^{gh}(Q^{2})$ is then extracted from Eq. (4.25) after
applying the ghost and gluon propagator renormalization factors,
$\widetilde{Z}_{3}$ and $Z_{3}$, respectively; see Section 3.5. The validity
of the calculations can be verified by comparing the UV running of
$\alpha_{s}^{gh}(Q^{2})$ with the pQCD expectation.

The above procedure was applied by two different groups in [238] using
different lattice simulations in the _pure gauge_ sector. The results yield a
SU(2) running coupling which matches the pQCD running at $Q^{2}\gtrsim 4$ GeV2
and _freezes_ at $\alpha_{s}^{gh,SU(2)}(0)=5\pm 1$. It is close to the SDE
SU(2) result found in [253]: $\alpha_{s}^{gh,SU(2)}(0)=5.24$. This result is
extended to SU(3)-color QCD by scaling it by 2/3: it becomes
$\alpha_{s}^{gh}(0)=3.3\pm 0.7$. Since $\alpha_{s}^{gh}(\mu^{2})$ has been
determined in [238] by matching the lattice result to the 2-_loop_ pQCD
expectation, the prediction is scheme-independent, but it is dependent on the
order of truncation order of the perturbative $\beta$ series; it yields
$\Lambda_{2-loop}=1.1$ GeV in the UV.

##### An example of the determination of $\alpha^{3g}_{s}$

Another approach, first proposed in Ref. [322], is to measure the coupling by
calculating the 3- and 2-point gluon correlation functions,

$G_{3,\mu_{1}\mu_{2}\mu_{3}}(p_{1},p_{2},p_{3})\equiv\langle
A_{\mu_{1}}(p_{1})A_{\mu_{2}}(p_{2})A_{\mu_{3}}(p_{3})\rangle$

and $D(p)=Z(p_{2})/p_{2},$ respectively. The calculations are typically
carried out in the Landau gauge and MOM RS. The 3-gluon vertex $G_{3}$ can be
expressed in term of three gluon propagators $D$ and a 3-point vertex function
$\Gamma_{3}$ which is naturally identified with the coupling:

$G_{3,\mu_{1}\mu_{2}\mu_{3}}(p_{1},p_{2},p_{3})=\Gamma_{3,\nu_{1}\nu_{2}\nu_{3}}(p_{1},p_{2},p_{3})D_{\mu_{1}\nu_{1}}(p_{1})D_{\mu_{2}\nu_{2}}(p_{2})D_{\mu_{3}\nu_{3}}(p_{3}).$
(4.48)

The vertex function is then extracted as:

$\Gamma_{3,\nu_{1}\nu_{2}\nu_{3}}(p_{1},p_{2},p_{3})=G_{3,\mu_{1}\mu_{2}\mu_{3}}(p_{1},p_{2},p_{3})\prod_{n=1}^{3}D^{-1\mu_{n}}_{\nu_{n}}(p_{n}).$
(4.49)

The coupling is normally defined at the symmetric Euclidean point,
$\alpha_{s}(Q^{2})\equiv\Gamma_{3}(p_{1},p_{2},p_{3})$ for
$p_{1}^{2}=p_{2}^{2}=p_{3}^{2}=Q^{2}$ and $p_{1}+p_{2}+p_{3}=0$. Such
conditions correspond to the MOM scheme. In contrast, the minimal MOM scheme
$\widetilde{\mbox{MOM}}$ is preferred on the lattice, corresponding to the
condition $p_{1}=p_{2}+p_{3}=0$.

####  4.5.1 Lattice results for $\alpha_{s}$ in the IR

Lattice calculations for $\alpha_{V}$ defined by the static
$\mbox{Q--}\overline{\mbox{Q}}$ potential leads to an IR-_freezing_ coupling
[214]. However, for $\alpha_{s}$ defined using _Green’s function_ s, both IR-
vanishing and _freezing_ solutions exist, as for the SDE calculations.
Calculations by Bloch and collaborators [323], and by Furui and Nakajima [259]
result in the _scaling solution_. On the other hand, a majority of
calculations [277, 324, 325, 326, 327], indicate that in the Landau gauge and
MOM RS, $\alpha^{gh}_{s}$ vanishes when $Q^{2}\rightarrow 0$. Before
concluding that these results correspond to the _decoupling solution_
discussed in Section 4.4.3, one needs to explore more mundane possibilities.
One is that $G_{3}$ has been obtained from Eq. (4.49) using the perturbative
expression of the gluon propagator $D(Q^{2})$, which diverges as
$Q^{2}\rightarrow 0$. In this case, the IR-vanishing originates from an
artificial divergence originating from pQCD which is irrelevant in the IR
regime. Another possible reason is unphysical lattice finite-size effects
[263]. Finally, the IR-behavior obviously depends on the choice of definition
of the coupling, which can lead to diverging, _freezing_ or vanishing
$\alpha_{s}$. Nevertheless, it is now clear that the _decoupling solution_
seen in the SDE framework also genuinely exists on the lattice [328], where it
is dominant and where it was in fact first discovered [265].

Lattice data for the gluon and ghost propagators in SU(2) and SU(3) from Ref.
[329] have been used by Aguilar, Binosi and Papavassiliou [330] to form
$\alpha_{s}^{gh}$ from Eq. (4.25), and $\alpha_{gse}$ using the _pinch_
technique. As already discussed for other articles from these authors, a
massive gluon formula is assumed for the gluon propagator at tree-level,
$\Delta(Q^{2})\propto 1/[Q^{2}+m_{g}^{2}(Q^{2})]$, rather than a massless
propagator. This accounts for the _freezing_ of their couplings in the IR
rather than vanishing couplings. Assuming $m_{g}(0)=0.5$ GeV, the Landau gauge
coupling _freezes_ at $\alpha_{s}^{gh}(0)=4.45$; for $m_{g}(0)=0.6$ GeV, the
freezing value is $\alpha_{s}^{gh}(0)=6.40$. The gauge-invariant coupling
defined in the _pinch_ technique _freezes_ at $\alpha_{gse}(0)\simeq 0.5$ for
$m_{g}(0)=0.5$ GeV and $\alpha_{gse}(0)\simeq 0.4$ for $m_{g}(0)=0.6$ GeV.

Furui and Nakajima have used lattice gauge theory to calculate
$\alpha_{s}^{gh}(Q^{2})$ in the Landau gauge, $\widetilde{\mbox{MOM}}$ RS and
the _quenched_ approximation. Quarks are defined as domain wall fermions. The
resulting coupling _freezes_ at $\alpha_{s}^{gh}(0)=2.5\pm 0.5$ [259]. In
determining this value, the apparent IR-vanishing trend of $\alpha_{s}^{gh}$
was ignored since it was identified to be due to finite size effects [331].

Ayala and collaborators studied the influence of dynamical quarks on
$\alpha_{s}^{gh}$ by performing _quenched_ , and $n_{f}=2$ and 2+1+1
_unquenched_ lattice simulations [332]. These authors assume masses of order
20-50 MeV for the light quarks, 95 MeV for the $s$ quark mass, and 1.51 GeV
for the $c$ quark mass. The Landau gauge and MOM RS are used. There is no
direct influence of quarks on $\alpha_{s}^{gh}(Q^{2})$ for a fixed dynamical
gluon mass scale. However, as quarks change the value of $m_{g}$, they have an
indirect influence. The authors obtain a _freezing_ value of
$\alpha_{s}^{gh}(0)\simeq 3.2\pm 0.3$ for $m_{g}=0.5$ GeV, and
$\alpha_{s}^{gh}(0)\simeq 4.7\pm 0.4$ for $m_{g}=0.6$ GeV, both for $n_{f}=$
2+1+1.

Recently Maas and collaborators have studied systematic lattice effects on the
gluon and ghost propagators on $\alpha_{s}^{gh}(Q^{2})$ in the Landau gauge
[333]. The study was carried in 2, 3 and 4 dimensions and yields a solution
for $\alpha_{s}^{gh}$ which vanishes in the IR.

The gluon and ghost propagators and other correlation functions have been
evaluated on the lattice in many analyses, but without providing the IR-
behavior of $\alpha_{s}$. An example of such results is shown in Fig. 4.5.1.
In Ref. [334] the propagators have been obtained in the interpolating gauge in
order to test the gauge dependence of the correlation functions. The authors
obtained propagators compatible with the _Kugo–Ojima_ and the Gribov–Zwanziger
scenarios, regardless of the choice of gauge. In [335] it is shown that the
gluon propagator in 3D and 4D is non-zero and finite in the IR, leading to an
IR-finite coupling. The calculations are done in SU(2)-color but the
conclusion extends to SU(3)C. Indeed, a _quenched_ calculation done in the
Laplacian gauge for different numbers of colors $N_{C}$ (SU(2), SU(3) and
SU(4)) shows little dependence of the gluon propagator on $N_{C}$ [336]. This
is compatible with the expected $1/N_{C}$ dependence for the coupling because
$\alpha_{pQCD}$ factors the expressions of $\alpha_{s}$ defined from
propagators (Eqs. (4.25), (4.41) and (4.42)), and that in the _quenched_
approximation, $\alpha_{pQCD}\propto 1/\beta_{0}\propto N_{C}$.

The gluon dressing function $Z(Q^{2})=Q^{2}D(Q^{2})$, from _quenched_ and full
lattice calculations from Bowman _et al._ , Ref. [337].

Most of the propagator studies cited above are done in _quenched_ QCD and
Landau gauge. Calculations including dynamical quarks have also been carried
out, showing clear effects on the propagators from quark loops, but preserving
the qualitative features obtained in _quenched_ QCD [337]. Calculations in
other gauges also exist and display similar features as the Landau gauge
calculations, see e.g., Ref. [336] for calculations done in the Laplacian
gauge. Discretization, finite lattice size and _tadpole_ effects were studied
in Ref. [338].

The coupling $\alpha_{s}(Q^{2})$ can be also obtained from the static
potential. In recent results by Horsley and collaborators [339],
$\alpha_{V}(Q^{2})$ and its _$\beta$ -function_ are computed on a lattice
using numerical stochastic perturbation theory; see the reviews in Refs. [340,
341] and Section 4.8. The perturbative part of the static potential,
$4\alpha_{V}/3r$, is calculated from _Wilson loops_. In the _quenched_
results, $\alpha_{V}(Q^{2})$ diverges in the IR at $Q\simeq
0.7\Lambda_{\overline{MS}}$. When quarks are included, $\alpha_{V}(Q^{2})$
_freezes_ around $Q\simeq 0.8\,\Lambda_{\overline{MS}}$ with
$\alpha_{V}(0)\simeq 0.52$. The calculations are performed for SU(3)C and
include massless u and d quarks.

Finally, another method is to use the Schrödinger functional; see Section
3.9.4. It leads to a coupling in good agreement with pQCD down to $Q^{2}\simeq
0.4$ GeV2, but it diverges in the IR as $e^{m/Q}$ [342].

###  4.6 Functional renormalization group equations

The functional renormalization group (FRG) framework is similar to the SDE
approach. The central objects are _Green’s function_ s (although, only dressed
ones, while SDE must include both bare and dressed functions) and the method
provides an infinite system of “flow equations”, or “exact renormalization
group equations”, which fully describe the underlying quantum field theory in
continuous Euclidean space-time. Thus, the FRG equations and SDE must in
principle share the same solutions. In practice, the different approximations
needed to solve the FRG equations and the SDE can introduce differences. Those
can be used to assess the uncertainties on the solutions. Thereby, just like
SDE and Lattice benefited from their connection and complementarity, the FRG
approach provides yet another complementary framework.

The starting point of the FRG method is to define an IR cut-off $k$ above
which the theory is well understood and perturbatively calculable. Then the
value of $k$ is lowered to access the nonperturbative domain. The cut-off
takes the form of a regulator added to the classical action. In effect, it
adds a large, momentum-dependent, mass to the field, suppressing the low modes
of momenta below $k$ [258]. A flow equation for the _Green’s function_ s
expresses how those depend on $k$ [343]. This flow equation is obtained by
first partially deriving the generating functional of the quantum field theory
which describes its dependence with respect to $k$. This yields the flow
equation for the generating functional $\Gamma_{k}[\phi]$

$\frac{\partial}{\partial k}\Gamma_{k}[\phi]=\tfrac{1}{2}{\rm
Tr}\,\frac{\frac{\partial}{\partial k}R_{k}}{\Gamma_{k}^{(2)}[\phi]+R_{k}},$
(4.50)

where $R_{k}$ is the regulator which decouples the infrared modes with
momentum below the sliding scale $k$ and $\Gamma_{k}^{(2)}$, the second
functional derivative, is the full inverse propagator modified by the presence
of the regulator $R_{k}$. The trace stands for the sum over momenta, fields
and internal indices. The Wetterich equation (4.50) is an exact flow equation
for $\Gamma_{k}[\phi]$ which has a one-loop structure, in contrast with the
multi-loop diagram structure characteristic of perturbation theory.

The system of flow equations for the _Green’s function_ s are obtained by
successively deriving the flow equation of the generating functional with
respect to the source or field terms in the action. Once the $k\rightarrow 0$
limit is reached, the cut-off disappears and all the slow modes of the quantum
field theory are integrated out in the solution. One thus has an analytic,
nonperturbative method for investigating strongly interacting quantum field
theories. The FRG method is particularly relevant to the study of running
couplings since they can be defined from the _Green’s function_ s as in other
approaches.

The FRG equations thus provide the correct RG-scaling expected from the
theory. As verification, the pure Yang–Mills expectation for the UV running of
$\alpha_{s}(Q^{2})$ has been recovered within the FRG framework [344]. In
practice, the infinite system of equations must be truncated. As it is the
case of the SDE formalism, the truncation introduces uncertainties which can
be difficult to fully assess. Furthermore, multiple choices of regulator are
possible, which introduces some arbitrariness. For a review of the FRG method
and a discussion of its predictions for the running QCD coupling in the IR,
see e.g., Ref. [345].

Initial FRG studies which were performed down to the scale $\sim\Lambda$ found
an IR divergent solution of the gluon propagator, in agreement with the
scenario of IR slavery; see Refs. [219, 346]. However, since the $k\rightarrow
0$ limit was not reached, the nonperturbative effects were not fully accounted
for. More recent FRG investigations have now invalidated the IR slavery
conclusion. The ghost and gluon propagators have been studied within the FRG
formalism in the _pure field_ sector, using Landau gauge and 4D Euclidean
space [257, 258], assuming the _scaling solution_ as observed in the lattice
and SDE frameworks (Section 4.4.3). They yield a similar value of the _IR
exponent_ : in [257], it is found that $\kappa^{\prime}\simeq 0.52$, which is
compatible with the SDE value of 0.595, once the dependence of
$\kappa^{\prime}$ with the choice of cut-off implementation is accounted for.
The range $0.539\leq\kappa^{\prime}\leq 0.595$, with a strong preference for
the maximum value $\kappa^{\prime}=0.595$ was obtained in Ref. [258]. The
resulting _freezing_ value of the ghost–gluon coupling is
$\alpha_{s}^{gh}(0)=2.972$. All of these results are consistent with a
_freezing_ of the coupling due to IR ghost dominance and the _Kugo–Ojima
criterion_. The ghost and gluon propagators and the coupling
$\alpha_{s}^{gh}(Q^{2})$ have also been studied using the FGR approach in
[256]. These calculations have been also performed within the _pure field_
sector using Coulomb gauge and a focus on the _scaling solution_. The
resulting solution leads to a unique _freezing_ coupling, although its
_freezing_ value is not specified. The solution yields $\kappa^{\prime}=0.35$,
in agreement with Ref. [218] using a variational approach to the Yang–Mills
Schrödinger equation, and with the first solution obtained using the SDE in
[243].

The behavior of the running coupling is specifically addressed using FRG in
Ref. [344] within the _pure gauge_ field sector for both SU(2)C and SU(3)C.
The predicted coupling _freezes_ in the IR at the value of $7.7\pm 2$ for
SU(3)C and 11.3 for SU(2)C. The calculations were performed in the Landau
gauge. The inclusion of quarks is expected to lower the freezing values [344],
in agreement with the discussion in Section 4.4.7.

###  4.7 The Gribov–Zwanziger approach

Gribov was the first to point out the effects of gauge-fixing in a Yang–Mills
theory [163]. Those are directly relevant to the coupling when it is defined
using _Green’s functions_. The coupling is invariant under the local gauge
symmetry given by $\delta_{\omega}A_{\mu}^{a}=-D_{\mu}^{ab}\omega^{b}$ (with
$a,b$ color indices). However, the standard gauge-fixing relations, such as
the Coulomb, Feynman or Landau gauges, are not enough to uniquely specify the
gauge field. The vanishing eigenvalues of the Faddeev-Popov operator generate
copies of the fields; i.e., physically different fields which are related by
gauge transformations still exist, even after fixing the gauge. To avoid these
copies in the path-integral formalism, Gribov proposed to restrict the
Faddeev-Popov operator to an integration region where the operator is strictly
positive (the so-called “first Gribov” region). To implement such a
restriction, Gribov introduced a non-local operator. Even with this
prescription, copies still exist within the first Gribov region [347]. This
has prompted proposals to restrict the integration to an even smaller domain,
the “ fundamental modular region”, which is truly free of Gribov copies. It
was eventually found however, that the remaining copies appear to have no
influence on the solutions of the theory [245, 292, 348, 349], which is
consistent with the finding from lattice gauge theory, namely that the effect
of Gribov copies (“Gribov noise”) is negligible [321, 350].

The non-locality of Gribov’s operator limits the formalism to semi-classical
calculations. A local renormalizable Lagrangian restricting the integration
region to the first Gribov region was later proposed by Zwanziger [215],
enabling loop calculations. For a review of the Gribov–Zwanziger approach, see
Ref. [340].

Since the ghost propagator is the inverse of the Faddeev-Popov operator, the
ghost propagator has no pole in the Gribov approach, except on the horizon of
the Gribov region, which corresponds to the $Q^{2}\to 0$ limit. In this case,
the ghost propagator diverges, following a $1/Q^{4}$ dipole form. Gribov found
that when the integration of the generating functional, or any correlation
function, is restricted to the first Gribov region, the Gribov mass generated
by the copies suppresses the tree-level gluon propagator in the Landau gauge,
which should then take the IR form:

$D_{\mu\nu}^{bc}(q)=\delta^{bc}\left(\eta_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{q^{2}}\right)\frac{q^{2}}{q^{4}+m_{gr}^{4}},$
(4.51)

where $m_{gr}$ is the Gribov mass. With this form, the gluon propagator
vanishes in the IR, whereas as noted above, the ghost propagator diverges as
$1/Q^{4}$. This behavior is distinct from its $1/Q^{2}$ UV-behavior. This
scenario, the Gribov–Zwanziger approach, results in an IR _freezing_ of
$\alpha_{s}$ analogous to the _scaling solution_ seen in the SDE and lattice
studies; see Section 4.4.3 and Ref. [351]. However, the predicted _IR
exponent_ is $\kappa^{\prime}=1$ in this approach, differing from that of SDE
($\kappa^{\prime}\simeq 0.595$). The IR-_freezing_ value in this scenario was
calculated for the ghost–gluon coupling by Gracey to be
$\alpha_{s}^{gh}(0)=16/3\pi\simeq 1.70$ (Landau gauge, $\overline{MS}$, one
_loop_ calculation) [352].

The analytical properties of the calculations using the Gribov–Zwanziger
approach allows one to investigate the underlying reason for the _freezing_ of
the QCD coupling, at least at one loop: As one expects on general dimensional
considerations, the _freezing_ occurs because all dimensionful quantities
cancel each other in the IR, so the coupling becomes scale invariant. The
mechanism for such a cancellation has its origin in the interplay of the anti-
commutating Faddeev-Popov ghosts and the commutating “Zwanziger ghosts”. Those
arise from enforcing the Gribov region with the local Zwanziger Lagrangian.
More recent 2-_loop_ calculations of the gluon and ghost propagators are
available [216, 353]. Although the original Gribov–Zwanziger approach leads to
the _scaling solution_ , it should be noted that the decoupling scenario also
appears to be possible in an extended Gribov–Zwanziger approach, see Refs.
[271, 272].

The Gribov–Zwanziger scenario may appear gauge-dependent since gauge-fixing
procedures without Gribov copies exist, such as the axial or Laplacian gauges;
see the related discussion in Ref. [354]. The light-cone gauge $A^{+}=0$ is
also free of ghosts. In addition, there are no ghosts using the stochastic
gauge-fixing quantization method (see below) which, in fact, was introduced in
the QCD propagators/coupling context to circumvent the Gribov copy problem.
However, in spite of the apparent gauge dependence of the Gribov–Zwanziger
scenario, equivalent mechanisms are found in the ghost-less frameworks, as we
will see in the next section.

###  4.8 Stochastic quantization