measured on $Q^{2}$ domains that cover the pQCD Landau pole region: no
divergences, discontinuities, or other unusual behaviors have been found.
Thus, the $\alpha^{\mathrm{pQCD}}_{s}(Q^{2})$ divergence is merely an artifact
of perturbation theory whose appearance signals that all perturbative
approximants to any given quantity become invalid at some
$Q^{2}>\Lambda_{s}^{2}$.

Naturally, the internal inconsistency of perturbation theory and, a fortiori,
the unphysical divergence in $\alpha^{\mathrm{pQCD}}_{s}(Q^{2})$ cannot be
cured by computing the coupling to higher orders. In fact, higher orders can
only make the problems worse because pQCD approximants are all examples of
asymptotic series (Poincaré expansions) [360]. Writing such an approximant for
a measurable quantity in the form
$O_{N}(\alpha_{s})\sim\sum_{n=0}^{N}a_{n}\alpha_{s}^{n}$, then the sequence of
associated partial sums begins to depart from the true result for
$N>N_{\alpha}$, where the value of $N_{\alpha}$ may be estimated from the
condition $\alpha_{s}\,a_{N_{\alpha}+1}={\rm O}(a_{N_{\alpha}})$. Adding terms
with $N>N_{\alpha}$ leads to partial sums that rapidly proceed to diverge.
(Additional discussion can be found elsewhere [361].) To illustrate, consider
$Q^{2}\simeq Q_{0}^{2}=1\,$GeV2, i.e., in a neighbourhood of the UV-IR
transition; then, $\alpha^{\mathrm{pQCD},\overline{\rm
MS}}_{s}(Q^{2})/\pi\simeq 0.2$ and the asymptotic character of pQCD series
begins to become apparent at order N∼4LO. Different RSs can delay or hasten
the onset of this problem, but it cannot be avoided because one may always
express a running coupling computed in one RS to that obtained using another
via a power series in the latter coupling, as discussed in connection with Eq.
(2.3). On the other hand, if QCD is truly a theory with a connection to strong
interactions in Nature, then including nonperturbative effects must eliminate
the Landau pole [12, 35, 362] because Nature does not exhibit such a defect.

As will be detailed below, studying $\alpha_{s}$ is much more challenging in
the IR than in the UV. This is not because nonperturbative calculations are
necessarily more difficult than those using perturbation theory; the
impediments are rather of a different character. Namely, at this stage in the
development of QCD theory, there is no single agreed prescription for defining
and calculating an IR completion of QCD’s running coupling. Further, prima
facie, it is possible that more than one coupling may be needed to fully
characterize strong QCD. Notwithstanding these issues, it is crucial to
deliver a nonperturbative solution of QCD, and calculating $\alpha_{s}^{\rm
IR}$ is a key piece in that puzzle. Indeed, with a single $\alpha_{s}$ known
at all spacelike momenta, one would have in hand that quantity which describes
the strength of QCD’s interaction at all momentum scales [363], thereby
completing a big step toward an economical and fundamental description of low-
energy hadronic phenomena.

There are many additional, complementary reasons for determining the IR
behavior of $\alpha_{s}$. For instance, given that some definitions of
$\alpha_{s}$ involve ratios of field and vertex renormalization constants –
see, e.g., Eqs. (3.26), (3.27), then knowing $\alpha_{s}^{\rm IR}$ can deliver
insights into the long-wavelength properties of (dressed) gluons and quarks,
and even ghosts, if the formulation involves them. Of course, their UV
properties are well known, but emergent, nonperturbative phenomena are
expected to vastly modify the nature of QCD’s parton fields in the IR. Hence,
any sound understanding of a nonperturbative extension of $\alpha_{s}$ into
the IR will deliver information on the quasiparticle degrees-of-freedom that
replace partons in descriptions of low-energy phenomena.

In contrast to studies at UV momenta, no one approach to the calculation of
$\alpha_{s}^{\rm IR}$ has yet acquired the status of “preferred”. This is
largely because a nonperturbative solution of QCD is still being sought; and
in this hunt, many complementary approaches currently seem viable. Indeed,
using an amalgam of methods might be the only way to solve QCD. Hereafter, we
will describe a number of widely used approaches to the study of
$\alpha_{s}^{\rm IR}$.

As noted previously [12], there is currently no consensus on what might be
called the “correct” scheme for the calculation of $\alpha_{s}^{\rm IR}$.
Furthermore, some may not even consider that $\alpha_{s}^{\rm IR}$ is
universal because there are four individual UV-divergent interaction vertices
in the perturbative treatment of QCD. So, there could be up-to four distinct
couplings at infrared (IR) momenta; and when considering $n\geq 3$ point
functions, one has an uncountable infinity of choices for the momentum with
which the coupling runs. In our view, such considerations are spurious because
if QCD is a theory, then BRST symmetry will be preserved nonperturbatively;
hence, a unique IR completion of $\alpha_{s}$ exists. As we subsequently
elucidate, this position is supported by developments in the past
quinquennium.

Today, definitions and determinations of $\alpha_{s}^{\rm IR}$ are available
that ensure its universality and deliver consistent results. A universal PI
charge, $\hat{\alpha}(Q^{2})$, has been defined [35] using continuum Schwinger
function methods (CSMs) [357, 358, 359] and recently computed using a
combination of continuum and LGT inputs [36]. For reasons that are understood
both mathematically and phenomenologically, the prediction for
$\hat{\alpha}(Q^{2})$ is pointwise almost identical to
$\alpha_{g_{1}}(Q^{2})$, the experimental coupling [364, 365, 117] derived
from the Bjorken sum rule [366, 367] using the method of effective charges
[326, 327, 328], and also practically indistinguishable from the effective
charge, $\alpha^{\rm HLF}_{g_{1}}(Q^{2})$, obtained using HLFQCD [368].

A characterizing feature of all three charges is that they “freeze” in the IR,
saturating to a value $\alpha_{s}(Q^{2}=0)\approx\pi$, i.e., the couplings
become $Q^{2}$-independent at IR momenta and express a $Q^{2}=0$ fixed point,
something which was long ago conjectured – see, e.g., Refs. [84, 369, 370].
The $Q^{2}=0$ value of the coupling now becomes “critical” in the sense that
it must be large enough to ensure DCSB. Allowing for self-reinforcement via
the gluon-quark vertex, $\alpha_{s}(Q^{2}=0)\approx\pi$ is sufficient [40].

The universal (PI) feature of $\hat{\alpha}$, the consistency between
effective charge measurements and CSM and HLFQCD predictions, and the fact
that $\alpha_{g_{1}}$, $\alpha^{\rm HLF}_{g_{1}}$, $\hat{\alpha}$ have all
been used to make sensible predictions for key low-energy hadron
observables111For instance: the QCD scale $\Lambda_{s}$ – Secs. 3.6.6, 3.6.7;
hadron wave functions and spectra [371, 372, 373, 374, 375, 376, 377, 378,
379]; elastic and transition form factors [380, 379], polarized and
unpolarized quark and gluon PDFs and GPDs for different hadrons [339, 340,
342, 323]; leptonic decay constants of Nambu-Goldstone bosons [374, 375, 376,
377, 378, 381]; dressed-quark mass functions [6, Fig. 2.5], [40]; and a
unified picture of the soft and hard pomeron [382].  form three pillars in a
compelling argument that supports the existence of a canonical QCD coupling.
This coupling not only describes the strength of QCD’s interaction at all
momentum scales but also serves as the basis for predictions of a prodigious
array of hadronic phenomena. The past few years have also seen important
progress toward an understanding of the physical mechanism which underlies the
IR behavior (running) of QCD’s effective charge, viz. a deep understanding of
how a Schwinger mechanism [383, 384] in QCD leads to the emergence of a gluon
mass-scale [7, 8, 385, 386].

###  4.2 $\mathbf{\alpha_{s}}$ and emergent phenomena

As we saw at the beginning of Ch. 3, in the UV context, the dressing of
classical tree-diagrams by short-distance quantum loops leads to the running
of $\alpha_{s}$. This is only one part of the picture, however. Infrared
completions of $\alpha_{s}$ may include additional phenomena, including those
which cannot be captured by any summation of a finite number of diagrams, such
as confinement, DCSB, EHM, etc. These inherently nonperturbative phenomena are
likely connected; even, perhaps, essentially indistinguishable in origin. It
follows that any discussion of $\alpha_{s}^{\rm IR}$ will probe into the very
core of the most fundamental questions in the Standard Model.

We have explained why the Landau pole is artificial and unphysical. As a
consequence, any suggestion that confinement – a physical phenomenon – owes to
an IR slavery stemming from the Landau pole is erroneous. Notwithstanding
this, it is widely held that $\alpha_{s}^{\rm IR}(Q^{2}\simeq 0)$ should be
large (exceeding unity) in order for confinement to be realized. Yet
confinement scenarios exist which do not require a large coupling, e.g., the
supercritical binding model discussed in Ref. [46], which realizes a form of
confinement with $\alpha_{s}^{\rm IR}(Q^{2}\simeq 0)\approx 0.4$. Some other
small-coupling confinement models are described elsewhere [12].

Given the almost certain connection between $\alpha_{s}^{\rm IR}$ and Standard
Model emergent phenomena, the following discussion will range over a variety
of approaches to low-energy strong-interaction observables. Much work has
focused on pure gauge QCD, largely because this eliminates Nambu-Goldstone
bosons from the analysis. However, the attendant simplifications, while useful
for some forms of analysis, probably come at too high a price because light-
quark physics is at the heart of Nature: the proton, absolutely stable owing
to real-world confinement, is defined by its valence light-quark content;
hence, no discussion of proton stability can be conclusive in pure gauge QCD.
In our view, despite the complications they bring, it is better to include
light quarks from the outset, especially if the method pretends to an
explanation of hadron observables.

One may readily highlight the manifold character of the approaches being used
in the pursuit of $\alpha_{s}^{\rm IR}$ by listing just some of the scales
employed to characterize nonperturbative effects. For instance, as we have
already often noted, $\Lambda_{s}$ is a popular choice; it is also reasonable,
so long as proponents admit the caveat that, being RS-dependent, it is not an
objective quantity. Other examples include the mass of a given hadron, with
the proton being a good choice because, since its stable, the proton mass-
squared is a real number; QCD string tension, $\sigma$; AdS/QCD scale,
$\kappa$; chiral condensate; gluon mass-scale; and Gribov horizon parameter.
If wielding a unifying tool, then some or all of these scales can be related.
For instance, Eq. (3.53) provides a link between $\Lambda_{\overline{\rm MS}}$
and $\kappa$; Eq. (4.12) connects $\sigma$ and $\kappa$; and Ref. [41] argues
that the gluon mass scale and Gribov horizon parameter are identical.
Additional discussions can be found elsewhere [12, 6, 7, 8, 9, 10, 11].

###  4.3 Effective charge method

####  4.3.1 Generalities

The effective charge scheme described by Grunberg [326, 327, 328] associates a
QCD coupling with the expansion of any given observable restricted to first
order of a perturbative expansion in $\alpha_{s}^{\rm pQCD}$. Such an
effective charge implicitly incorporates terms of arbitrarily high order,
$n>1$, in the perturbative coupling, $[\alpha_{s}^{\rm pQCD}]^{n}$, expressed
in gluon bremsstrahlung and gluon vertex corrections at the probe–quark vertex
and also pair creations, viz. all the contributions that lead to DGLAP
evolution at $n>1$. A related prescription – in spirit – is the ’t Hooft
scheme [387, 363], which defines the coupling from the QCD $\beta$-function
truncated at 2 loops, Eq. (3.38). Like the 1-loop form, the result is RS-
independent. However, this scheme does not eliminate the Landau pole, and
[363]: “the fake infrared problem – obviously limits the use of $\alpha_{s}$
so defined, to sufficiently large momentum scales.” Herein, therefore, we
focus on the Grunberg prescription, which, being defined by an observable, has
no such problem.

We illustrate the Grunberg scheme with an example based on the Bjorken sum
rule [366, 367], which connects the nucleon axial charge, $g_{A}$, to the
integral of the isovector part of the nucleon spin structure function
$g_{1}(x,Q^{2})$:

$\displaystyle\Gamma_{1}^{p-n}(Q^{2}):=\int_{0}^{1}dx\,\big{[}g_{1}(x,Q^{2})-g_{1}(x,Q^{2})\big{]}\stackrel{{\scriptstyle\Lambda_{s}^{2}/Q^{2}\simeq
0}}{{=}}\frac{g_{A}}{6}.$ (4.1)

where we have highlighted that this simple form is valid on
$\Lambda_{s}^{2}/Q^{2}\simeq 0$. At any value of $Q^{2}$ outside this
neighborhood, there are corrections, which lead to the pQCD series [388, 389,
390, 161]:

$\displaystyle\Gamma_{1}^{\rm p-n}$ $\displaystyle(Q^{2})=\frac{g_{\rm
A}}{6}\bigg{[}1-\frac{\alpha^{\rm
pQCD}_{{\rm{s}}}(Q^{2})}{\pi}-3.58\left(\frac{\alpha^{\rm
pQCD}_{{\rm{s}}}(Q^{2})}{\pi}\right)^{2}-20.21\left(\frac{\alpha^{\rm
pQCD}_{{\rm{s}}}(Q^{2})}{\pi}\right)^{3}$
$\displaystyle\qquad-175.7\left(\frac{\alpha^{\rm
pQCD}_{{\rm{s}}}(Q^{2})}{\pi}\right)^{4}+\sim-893.38\left(\frac{\alpha^{\rm
pQCD}_{{\rm{s}}}(Q^{2})}{\pi}\right)^{5}$
$\displaystyle\qquad+\mathcal{O}\left(\big{(}\alpha^{\rm
pQCD}_{\rm{s}}\big{)}^{6}\right)\bigg{]}+\sum_{n>1}\frac{\mu_{2n}(Q^{2})}{Q^{2n-2}}.$
(4.2)

Here, the series coefficients are calculated in the $\overline{\rm MS}$ RS.
This generalization of the Bjorken sum is derived in the DGLAP framework and
is thus a solid prediction of pQCD. HT corrections are provided by the
coefficients $\\{\mu_{2n}(Q^{2})\\}$. The $Q^{2}$-dependence of the
$\\{\mu_{2n}\\}$ also arises from pQCD radiative corrections. While the
$\mu_{2n}$ can be phenomenologically determined – see Refs. [335, 336, 337],
pQCD radiative corrections also make them dependent on $\alpha_{s}^{\rm
pQCD}$. Consequently, the extraction of $\alpha_{s}^{\rm pQCD}$ from Eq. (4.2)
is difficult on domains where HT contributions are not negligible. Finally,
since HT contributions are nonperturbative, their computations is challenging
[391]; and for the Bjorken sum, they are currently only estimated with models
[392, 393, 394, 395, 396]. All such issues are circumvented via the effective
charge definition.

Implementing the Grunberg scheme, Eq. (4.2) is written

$\displaystyle\Gamma_{1}^{p-n}(Q^{2})$
$\displaystyle=:\frac{g_{A}}{6}\bigg{(}1-\frac{\alpha_{g_{1}}(Q^{2})}{\pi}\bigg{)}\,,$
(4.3a) $\displaystyle\mbox{or, equivalently,}\quad\alpha_{g_{1}}(Q^{2})$
$\displaystyle=\pi\left(1-\frac{6}{g_{A}}\Gamma_{1}^{p-n}(Q^{2})\right),$
(4.3b)

where the $g_{1}$ subscript indicates the observable chosen for the effective
charge.

With this definition of the coupling, both the short distance pQCD effects –
those within the bracket in Eq. (4.2) and the $Q^{2}$-dependence of the
$\\{\mu_{2n}\\}$ terms, and the long distance effects – the
$\\{{\mu_{2n}}/{Q^{2n}}\\}$ terms – are incorporated into $\alpha_{g_{1}}$.
This is analogous to the procedure that transforms the coupling constant in a
classical Lagrangian into a running effective coupling during the
renormalization process – see Sec. 2, with the additional feature that long
distance effects are also folded into $\alpha_{s}$. It is their inclusion, as
well as other non-perturbative mechanisms that may be omitted in Eq. (4.2),
which suppresses the Landau pole – see Sec. 3.4.2). One may therefore
interpret the effective charge approach as a renormalization prescription,
wherewith $\alpha_{s}$ becomes an observable, with all attendant features.

Although Grunberg’s prescription was initially intended only for the pQCD
domain, effective charges, when evaluated from measurements, naturally extend
to cover the nonperturbative domain. This feature will be our primary focus
herein. (For a recent use of effective charges in the UV, see the discussion
of LGT calculations of $\alpha_{s}(M_{\rm Z}^{2})$ [21, Eq. (335)].) In
addition to being defined in the IR, effective charges also have other merits,
e.g.: they acquire the RS-independence of every first order pQCD approximant –
see the discussion of Eq. (2.3); on the perturbative domain, they
nonperturbatively complete the pQCD series; and in being defined by an
observable, they do not exhibit a Landau pole.

A seeming disadvantage of effective charges is a loss of predictability;
namely, since such a charge is defined via one particular observable, then
there are as many distinct effective charges as there are physical processes.
However, owing to the strengths of CSRs [329], predictive power is preserved
on the perturbative domain, at least, because a choice of process is not
qualitatively different from a choice of RS. Extension of CSRs to the IR is
nontrivial. One attempt is described in Ref. [329]; and Refs. [112, 331, 325]
describe a nonperturbative method that relates, one to another, different
effective charges or RS-dependent couplings.

####  4.3.2 The effective charge $\mathbf{\alpha}_{g_{1}}(Q^{2})$

The generalized Bjorken sum rule, Eq. (4.3), is especially well suited to
defining an effective charge for several reasons:

  1. (i)

The pQCD series for $\Gamma_{1}^{p-n}$ has no $x$-dependence and, compared to
many other observables, is easier to compute. It is currently known up N3LO
with a N4LO estimate [397] – see also Ref. [398].

  2. (ii)

$\Gamma_{1}^{p-n}$ is an isovector quantity; so some potential contributions,
the $\Delta$-resonances and, at amplitude level, disconnected diagrams, are
eliminated. This facilitates calculations of $\Gamma_{1}^{p-n}$, e.g., those
of chiral effective field Theory [399] and LGT. Moreover, $\Gamma_{1}^{p-n}$
is an $x^{0}$-moment (local operator); hence, directly computable using LGT,
as opposed to the structure functions themselves (nonlocal quantities).
Contemporary LGT analyses of $g_{A}$ report results with sub-percent precision
[21, Sec. 10.3.1].

  3. (iii)

$g_{A}$ is precisely measured from neutron $\beta$ decay: $g_{A}=1.2762(5)$
[5].

  4. (iv)

$\Gamma_{1}^{p-n}$ is measured over a wide $Q^{2}$ domain: $Q^{2}/{\rm
GeV}^{2}\in[0.02,10]$ [336, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409,
410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424].

  5. (v)

Robust sum rules allow extractions of $\alpha_{g_{1}}(Q^{2})$ on domains not
covered by measurements: on $Q^{2}\simeq 0$ and the pQCD domain,
$\alpha_{g_{1}}(Q^{2})$ can be computed from Gerasimov-Drell-Hearn (GDH) [425,
426] and Bjorken [366, 367] sum rules, respectively, see Eqs. (4.4), (4.5),
(4.8).

  6. (vi)

$\alpha_{g_{1}}$ appears to be interpretable as a standard coupling, even in
the nonperturbative domain [427]. Moreover, crucially, $\alpha_{g_{1}}(Q^{2})$
connects to several calculations of $\alpha_{s}$ in the IR, including those
from CSMs and AdS/QCD, see below.

Given both that cross-sections are finite quantities and $Q^{2}\to
0\Rightarrow x=Q^{2}/(2M_{N}\nu)\to 0$, where $M_{N}$ is the nucleon mass and
$\nu$ is the energy transferred to this target, then the support of the
integrand in Eq. (4.1) must also vanish in this limit; namely,

$\lim_{Q^{2}\to
0}\Gamma_{1}(Q^{2})=0\quad\mbox{and}\quad\alpha_{g_{1}}(0)=\pi\,.$ (4.4)

Furthermore, the GDH sum rule can be written as follows [428, 429, 100]:

$\frac{8}{Q^{2}}\int_{0}^{1^{-}}g_{1}\left(x,Q^{2}\right)dx~{}\xrightarrow[Q^{2}\to
0]{~{}}\frac{-\kappa_{N}^{2}}{M_{N}^{2}},$ (4.5)

where $\kappa_{N}$ is the nucleon anomalous magnetic moment and $1^{-}$
indicates that the elastic contribution ($x=1$) to the integral is excluded.
Hence, Eqs. (4.3b), (4.5) entail:

$\alpha_{g_{1}}\left(Q^{2}\right)\stackrel{{\scriptstyle Q^{2}/M_{N}^{2}\simeq
0}}{{\approx}}\pi+\frac{3\pi}{4g_{A}}\left(\frac{\kappa_{p}^{2}}{M_{p}^{2}}-\frac{\kappa_{n}^{2}}{M_{n}^{2}}\right)Q^{2}\,.$
(4.6)

Finally, the empirical facts $M_{p}\approx M_{n}$ and
$\kappa_{p}+\kappa_{n}\lesssim 0$ entail that, with increasing $Q^{2}$,
$\alpha_{g_{1}}\left(Q^{2}\right)$ falls slowly away from its maximum value at
$Q^{2}=0$:

$\beta(Q^{2})\propto\frac{d\alpha_{g_{1}}}{dQ^{2}}\stackrel{{\scriptstyle
Q^{2}/M_{N}^{2}\simeq
0}}{{\approx}}\frac{3\pi}{4g_{A}}\left(\frac{\kappa_{p}^{2}}{M_{p}^{2}}-\frac{\kappa_{n}^{2}}{M_{n}^{2}}\right)\lesssim
0\,.$ (4.7)

Evidently, therefore, the Bjorken sum rule effective charge has an IR stable
fixed point.

It is also worth noting that, on any domain for which pQCD is valid, the
generalized Bjorken sum rule, Eq. (4.2), together with the pQCD determination
of $\alpha^{\rm pQCD}_{s}(Q^{2})$ yield, in $\overline{\rm MS}$:

$\displaystyle\alpha_{g_{1}}(Q^{2})=\alpha^{\rm
pQCD}_{s}(Q^{2})+3.58\frac{({\scriptstyle\alpha^{\rm
pQCD}_{s}(Q^{2})})^{2}}{\pi}+20.21\frac{({\scriptstyle\alpha^{\rm
pQCD}_{s}(Q^{2})})^{3}}{\pi^{2}}+$ $\displaystyle
175.7\frac{({\scriptstyle\alpha^{\rm
pQCD}_{s}(Q^{2})})^{4}}{\pi^{3}}\sim-893.38\frac{({\scriptstyle\alpha^{\rm
pQCD}_{s}(Q^{2})})^{5}}{\pi^{4}}+\mathcal{O}\left((\alpha^{\rm
pQCD}_{s})^{6}\right).$ (4.8)

One may alternatively use the PMC (Sec. 3.6) to express
$\alpha_{g_{1}}(Q^{2})$ as a conformal series, where the argument at order $n$
evolves with a calculable PMC scale $Q^{2}_{n}$. The series is nearly
independent of the choice of renormalization scale. The PMC, a rigorous
procedure which generalizes the BLM method [124], eliminates the occurrences
of the $\beta_{n}$-terms in the series coefficients, evident in Eq. (3.37). By
definition, these terms are non-conformal and the origin of the renormalon
divergence problem. The PMC is the non-Abelian generalization of the standard
Gell-Mann-Low method used for fixing the renormalization scale in QED. It is
thus consistent with the grand unification of QCD and electroweak theory. It
also satisfies the analytic condition that pQCD and QED must match in the
limit $N_{C}\to 0$ [96] – see also the discussion on page 3.18.

Using the PMC, perturbative approximants are expressed with different scales,
$Q^{2}_{n}$, at each order, in contrast to the standard scale-setting with a
single scale, $Q^{2}$, for any order. When using conventional scale setting,
the renormalization scale is heuristically estimated as the momentum transfer
characterizing the process, with an arbitrary uncertainty $0.5Q^{2}-2Q^{2}$
assigned. In contrast, the PMC scales, $Q^{2}_{n}$, correspond to the
effective virtuality in the processes contributing at order $n$. As in QED,
the PMC renormalization scales only depend on the choice of renormalization
scheme by a single universal factor.

For $\alpha_{g_{1}}$ [430],

$\displaystyle\alpha_{g_{1}}(Q^{2})=\alpha^{\rm
pQCD}_{s}(Q^{2}_{1})+1.146\frac{({\scriptstyle\alpha^{\rm
pQCD}_{s}(Q^{2}_{2})})^{2}}{\pi}+0.144\frac{({\scriptstyle\alpha^{\rm
pQCD}_{s}(Q^{2}_{3})})^{3}}{\pi^{2}}+$ $\displaystyle
0.763\frac{({\scriptstyle\alpha^{\rm
pQCD}_{s}(Q^{2}_{4})})^{4}}{\pi^{3}}+\mathcal{O}\left((\alpha^{\rm
pQCD}_{s})^{5}\right),$ (4.9)

where the series coefficients and PMC scales are computed at 4-loop, for
$n_{f}=3$ using $\overline{\rm MS}$ as the intermediate RS for $\alpha^{\rm
pQCD}_{s}$. The scale values are $Q^{2}_{1}=0.338Q^{2}$,
$Q^{2}_{2}=0.047Q^{2}$ and $Q^{2}_{3}=1600Q^{2}$. The last scale is
undetermined: it can be set to $Q^{2}_{4}=Q^{2}$ or $Q^{2}_{4}=Q^{2}_{3}$ with
negligible differences. Comparing Eqs. (4.8) and (4.9) reveals that the PMC
series has significantly better convergence properties and exhibits no obvious
sign of a renormalon divergence.

On the other hand, the low PMC scales, notably $Q^{2}_{2}=0.047\,Q^{2}$,
restrict applicability of the PMC series to higher values of $Q^{2}$ than the
$\overline{\rm MS}$ series: in practice, one must take $Q^{2}>2.2\,$GeV2, at
least [430]. To address this issue, $\alpha_{g_{1}}$ has been expressed using
the PMC with a single scale $Q^{2}_{\ast}$ and the $V$-scheme as intermediate
renormalization prescription for $\alpha^{\rm pQCD}_{s}$ [431]:

$\displaystyle\alpha_{g_{1}}(Q^{2})=\alpha^{\rm
pQCD,V}_{s}(Q^{2}_{*})+3.15\frac{({\scriptstyle\alpha^{\rm
pQCD,V}_{s}(Q^{2}_{*})})^{2}}{\pi}+20.46\frac{({\scriptstyle\alpha^{\rm
pQCD,V}_{s}(Q^{2}_{*})})^{3}}{\pi^{2}}+$ $\displaystyle
51.36\frac{({\scriptstyle\alpha^{\rm
pQCD,V}_{s}(Q^{2}_{*})})^{4}}{\pi^{3}}+\mathcal{O}\left((\alpha^{\rm
pQCD,V}_{s})^{5}\right),$ (4.10)

with $Q^{2}_{*}=Q^{2}e^{0.58+2.06{\scriptscriptstyle\alpha^{\rm
pQCD,V}_{s}(Q^{2})}-7.41({\scriptscriptstyle\alpha^{\rm
pQCD,V}_{s}(Q^{2})})^{2}}\simeq 1.8Q^{2}$. The expression for $\alpha^{\rm
pQCD,V}_{s}$ can be found elsewhere [12, Eq. (3.42)]. Plainly, the convergence
of the series in Eq. (4.10) is not as good as that in Eq. (4.9), but it is
improved compared to Eq. (4.8) and can be applied over a similar $Q^{2}$
range.

The advantages of notes (iv), (v) on page iv make $\alpha_{g_{1}}$ known at
any $Q^{2}$, and note (vi) indicates that it may be interpreted in terms of
calculations and comparable therewith. The Bjorken sum rule effective charge,
extracted [364, 365, 117] from measurements of $\Gamma_{1}^{\rm p-n}(Q^{2})$
at the world’s accelerator facilities [336, 400, 401, 402, 403, 404, 405, 406,
407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421,
422, 423, 424], is shown in Fig. 4.1.

Figure 4.1: Effective charge $\alpha_{g_{1}}(Q)/\pi$. The most recent
extractions from data [117] are shown by the filled blue circles and red
stars. Their inner error bars give the statistical uncertainties and the outer
ones represent the quadratic sum of the systematic and statistical
uncertainties. The open symbols show earlier extractions [364, 365], with the
error bars being the quadratic sum of the systematic and statistical
uncertainties. Recent theoretical predictions are also shown: CSM [35, 36]
(magenta curve and shaded band); and HLFQCD [368] (red line, using
$\kappa=0.534$ GeV). The cyan band and associated dashed curve are computed
using the Bjorken and GDH sum rules, respectively.

To understand the meaning of $\alpha_{g_{1}}$, one needs to recall the origin
of each term in Eq. (4.2). On $\Lambda_{s}^{2}/Q^{2}\simeq 0$,
$g_{1}(x,Q^{2})$ represents the net polarization222 That is the net number
density in light-front longitudinal momentum-space of quarks with spin
pointing along the nucleon spin minus the net density of quarks with spin
pointing oppositely. of quarks with light-front momentum fraction $x$, making
clear the reason for the presence of $g_{A}$ in Eq.(4.1). The spin-dependent
part of the scattering amplitude stems from the matrix elements of the axial
current $\bar{\psi}\vec{\tau}\gamma^{\mu}\gamma^{5}\psi$, where $\psi$ is the
nucleon state and $\vec{\tau}$ are the isospin Pauli matrices. By Noether’s
theorem, the conserved axial current is associated with the chiral symmetry
$\psi\to e^{i\vec{\phi}\cdot\vec{\tau}\gamma^{5}}\psi$. In an elastic
reaction, the axial current generates $g_{A}(Q^{2})$, the nucleon axial form
factor, just like the electromagnetic current $\bar{\psi}\gamma^{\mu}\psi$
generates the nucleon electromagnetic form factors $G_{E}(Q^{2})$ and
$G_{M}(Q^{2})$. A Fourier transform of $G_{E}(Q^{2})$ gives the electric
charge spatial distribution and likewise a Fourier transform of $g_{A}(Q^{2})$
represents the nucleon spin spatial distributions, viz. the evolution from the
nucleon center to its boundary of the net valence-quark polarization [432].
Thus, $g_{A}$ is the net valence-quark polarization, without spatial
resolution, i.e., the spatial average, which directly connects to the average
valence-quark polarization in momentum-space, $\int dxg_{1}$.

Using $\overline{\rm MS}$, the leading-twist $\alpha_{s}^{\rm
pQCD}$-dependence originates from gluon bremsstrahlung and gluon vertex
correction at the probe-photon+struck-quark interaction vertex. HT terms
comprise kinematical and dynamical corrections. The former arise from the
nonzero nucleon mass, which provides a scale violating the UV conformality of
QCD, i.e., Bjorken scaling. The latter represent interactions, via one or
several hard gluons, of the struck quark with the rest of the nucleon made of
two valence-quarks, sea quarks and gluons (nearly-spectator partons). This
part of the nucleon is nonperturbative; hence, the $\\{\mu_{2n}\\}$ express
nonperturbative distributions. In this picture, some HT contributions may
implicitly be related to confinement forces [433, 434]. However, HT terms do
not include 3-nucleon coherent reactions (resonances).

With the origin of the terms in the generalized Bjorken sum rule thus
explicated, $\alpha_{g_{1}}(Q)$ can be interpreted. Short-distance quantum
effects, such as vertex correction and vacuum polarization, cause the running
of $\alpha^{\rm pQCD}_{s}$. In addition, the $\alpha_{g_{1}}$ effective charge
includes gluon bremsstrahlung; gluon exchange between the incoming and
outgoing active quark; and, at low-$Q^{2}$, HT terms and resonance effects,
not formalized by the OPE and therefore not written in Eq. (4.2).

On the nonperturbative domain, where pQCD radiative corrections and HT have
merged, possibly for $Q^{2}\lesssim 0.6\,$GeV2 [112], the meaning of
$\alpha_{g_{1}}(Q)$ is more speculative [427]. If the contribution to
$\Gamma_{1}^{\rm p-n}$ from nucleon resonances dominates, the connection
between $\alpha_{g_{1}}(Q)$ and the QCD coupling becomes unclear at low
$Q^{2}$ since in the former case at least three color sources are involved,
whilst a coupling is defined from a two-body interaction; but if the
contribution to $\Gamma_{1}^{\rm p-n}$ of incoherent reactions dominates, like
at large $Q^{2}$, $\alpha_{g_{1}}(Q)$ may, at least approximately, still be
interpreted as a coupling. Note (ii) on page ii makes this plausible because
the contribution from $\Delta$ resonances, which dominates for
$\Gamma_{1}^{\rm p}$ and $\Gamma_{1}^{\rm n}$ separately, vanishes in
$\Gamma_{1}^{\rm p-n}$ [399]. The issue is then whether this ensures that
incoherent reactions dominate over the remaining resonances in
$\Gamma_{1}^{\rm p-n}$.

A | B
---|---
|

C

Figure 4.2: Experimental data (symbols) and pQCD prediction at leading twist
(grey bands) for the first moment $\Gamma_{1}(Q^{2})$. Top left: proton
$\Gamma_{1}^{\rm p}$, Top right: neutron $\Gamma_{1}^{\rm n}$. Bottom:
isovector part $\Gamma_{1}^{\rm p-n}$.

That it may be the case can be seen by comparing $\Gamma_{1}^{\rm p}$ and
$\Gamma_{1}^{\rm n}$ to $\Gamma_{1}^{\rm p-n}$ – see Fig. 4.2.

With decreasing $Q^{2}$, the incoherent reaction contributions (DIS and, at
low $Q^{2}$, nonresonant background) drop, as predicted by pQCD (grey bands).
On the pQCD domain, $\Gamma_{1}^{\rm p-n}$ is large because the incoherent
reactions contribute to the proton and neutron with opposite sign, owing to
isospin symmetry and the fact that up and down quarks contribute to the
nucleon spin with opposite sign (the up quarks tending to (anti)align with the
proton(neutron) spin, whilst down quarks have the opposite tendency – see Ref.
[432, Table I]). In fact, $\Gamma_{1}^{\rm p}$ and $\Gamma_{1}^{\rm n}$
display opposite signs.

On the nonperturbative domain, down to $\sim 0.6$ GeV2, the nonresonant
contributions to $\Gamma_{1}^{\rm p}$, $\Gamma_{1}^{\rm n}$ and
$\Gamma_{1}^{\rm p-n}$ are still relatively large, with no sign of structure
that would signal a change of the mechanism driving their $Q^{2}$-evolution
and the importance of the resonance contribution. Such structure appears for
$\Gamma_{1}^{\rm p}$ and $\Gamma_{1}^{\rm n}$ for $0\leq Q^{2}\leq 0.5\,$GeV2,
with a negative peak for $\Gamma_{1}^{\rm n}$ centered around $Q^{2}\simeq
0.2\,$GeV2 and a $\Gamma_{1}^{\rm p}$ that becomes negative and then turns-
over presumably owing to the same negative resonance contribution as for the
neutron. Yet, $\Gamma_{1}^{\rm p-n}$ continues to decrease smoothly, with no
sign of this structure. These observations suggest that resonances have
similar effects on the proton and the neutron, but cancel in their difference,
as expected if the $\Delta$ resonance is the main mechanism driving the low
$Q^{2}$-evolution of $\Gamma_{1}^{\rm p}$ and $\Gamma_{1}^{\rm n}$ [399].

Below $Q^{2}=0.2\,$GeV2, the resonance contribution dominates in the
individual $\Gamma_{1}^{\rm p}$ and $\Gamma_{1}^{\rm n}$. They are both
negative and therefore partly cancel in $\Gamma_{1}^{\rm p-n}$. Consequently,
the nonresonant reactions may be the dominant contribution to $\Gamma_{1}^{\rm
p-n}$ at any $Q^{2}$, with the residual resonance effects further suppressed
by global quark-hadron duality [435].

Neglecting any residual coherent (resonance) effects, $\alpha_{g_{1}}$ can be
interpreted as in the large $Q^{2}$ domain, i.e., the usual coupling of an
interaction between two force sources [436]. The first source is the struck
quark and the second source is either undefined (quasi-real gluon
bremsstrahlung) or the residual hadronic structure for high and low $Q^{2}$,
respectively. These observations highlight that the interaction need not be
carried by a single gluon, e.g., $H$-graphs (Ref. [9, Fig. 4]) are permitted;
and that this does not forbid the definition of a coupling because there is no
requirement that it be associated with single boson exchange. An explicit
example is provided in Refs. [38, 39, 40, 35, 36], in which the PI running
coupling, owing to its derivation using the PT-BFM, cannot be associated with
single gluon exchange, if that exchange is viewed in the textbook sense.

In contrast, effective charges drawn from a coupling to more than two resolved
sources, thereby allowing for several vertices, or that involve coherence
effects, may not easily be interpreted as a force coupling.

All in all, $\alpha_{g_{1}}$ is defined to follow, on the perturbative domain,
the Gell-Mann Low coupling prescription [27]. In the IR, it may still be
argued to represent a force coupling. The latter is a crucial and nontrivial
property, which may be peculiar to $\alpha_{g_{1}}$ as it largely filters out
coherent scattering effects. At high $Q^{2}$, $\alpha_{g_{1}}$ acquires its
$Q^{2}$-dependence from small distance quantum loops (vacuum polarization;
quark self-energy; vertex corrections), just like $\alpha^{\rm pQCD}_{s}$, to
which are added multi-gluon quasi-real emissions and photon-quark vertex
corrections. At lower $Q^{2}$, but still upon a domain for which pQCD is
applicable, HT terms contribute. At the lowest $Q^{2}$ values, incoherent
reactions still appear to dominate, perhaps involving the full panoply of
nonperturbative gauge sector interactions. In the past quinquennium, the
viability of this picture of $\alpha_{g_{1}}$ has received material support
from its agreement with theoretical calculations using definitions consistent
with $\alpha_{g_{1}}$ – see Fig. 4.1 and Secs. 4.5, 4.6.4.

###  4.4 $\alpha_{s}$ and a confining linear potential

The idea of strings to describe hadronic structure emerged in the 1960s [437,
438, 439, 440, 441], following a suggestion that hadron mass spectra can be
organized as sets of Regge trajectories [442], viz. that the masses-squared
depend linearly on both the principal quantum number $n$ (which gives the
number of nodes possessed by a hadron wavefunction) and the internal orbital
angular momentum, $L$. Plainly, given that neither of these quantities is
Poincaré invariant, some caution ought to be used in going forward from here
[443]. Nevertheless, it was subsequently suggested that, for mesons, such
behavior might indicate that these states are constituted from two
“constituent-like” quarks (QCD connection unknown) interacting via a potential
that grows linearly with their separation, $r$. Further phenomenology yielded
a nonrelativistic, static potential of the form:

$V(r)=-\frac{C_{F}\alpha_{V}\left(r\right)}{r}+\sigma r,$ (4.11)

known as the Cornell potential [444, 445, 446]. Here,
$C_{F}=\nicefrac{{4}}{{3}}$ is a QCD color factor, $\sigma\approx(0.42\,{\rm
GeV})^{2}$ is the string tension parameter and $\alpha_{V}(r)$ is the three-
dimensional Fourier transform to coordinate space of $\alpha_{s}(Q^{2})$ in
the $V$ RS.

The first term in Eq. (4.11) dominates at short distances and is a Coulomb-
like static potential attributed to perturbative one-gluon exchange. It
affects the hadronic wavefunctions and the fine structure of the hadron mass
spectrum. Refs. [289, 287] provide the expression of the perturbative static
source+sink potential up to O$(\alpha_{s}^{3})$, including logarithmic
corrections and $\alpha_{V}$ to 4-loop. The second term in Eq. (4.11) acts
like a string potential between the static source+sink pair. The string was
historically interpreted as stemming from gluons being condensed into a flux
tube, although other pictures have emerged that also generate a linear static
potential at long distance, e.g., the harmonic oscillator LF potential
discussed in Sec. 4.5, and the center vortices and instantons seen in LGT
studies [447, 448, 449, 450, 451, 452, 453].

This second term serves to produce confinement for the nonrelativistic
constituent-like quarks: in practice, it provides the basis for a useful
phenomenology of heavy quarkonium systems, $Q\bar{Q}$, with
$m_{Q}\gg\Lambda_{s}$. It also determines the slopes and intercepts of the
Regge trajectories. These can then be straightforwardly interpreted within the
model: $L$ expresses how fast the $Q\bar{Q}$ rotates around the hadron center
of mass. The faster they orbit, the larger is $L$ and the tension in the
string that balances the centrifugal force. This produces a larger hadron
binding energy, i.e., a mass increase. Whether the trajectory is linear or not
depends on model details [454, 455, 456]. Nevertheless, something like the
widely-used Cornell potential is a feature of strong interactions between
static source+sink pairs and should be reproduced in any nonperturbative
approach that aims to describe such systems. It is seen in LGT studies of
static source+sink pairs [457, 458] – as, too, is string breaking when
dynamical fermions are introduced [459, 460]; and Regge trajectories are found
in anti-de Sitter/conformal field theory (AdS/CFT) models of high-energy
fixed-angle scattering of glueballs [461]. In HLFQCD – see Sec. 4.5, one
obtains a prediction for $\sigma$ [462]:

$\sigma_{AdS}=2\kappa^{2}/\pi=(0.42\,\mbox{GeV})^{2},$ (4.12)

matching phenomenology-based expectations.

Working within a framework defined by Eq. (4.11), one can reach the position
that the hadron spectrum is sensitive to the long distance behavior of
$\alpha_{s}$ when that spectrum is interpreted in terms of an effective
charge. It has been long recognized that this indicates that $\alpha_{V}$,
taken from a potential defined without the confinement term, $\sigma r$,
freezes in the IR; albeit, there was no agreement on the value of the IR-fixed
point – see Ref. [12] for an overview of different, early works on the
subject.

Following Refs. [274, 273, 275], the static quark-quark potential has been
employed extensively by LGT practitioners to determine $\alpha_{V}$ in the UV
[276, 277, 278, 279, 280, 260, 262, 281, 282, 283, 284, 285, 286, 287, 288,
289, 290, 291, 292] – see Sec. 3.6.5. Recent LGT determinations of $V(r)$ up
to $\sim 1$ fm are available in Refs. [289, 287, 463].

Alternatively, one may include $\sigma r$ in the coupling, so that the
potential becomes:

$V(r)=-\frac{4}{3}\frac{\alpha_{V}\left(r\right)}{r}+\sigma
r=:-\frac{4}{3}\frac{\alpha_{\rm Rich}\left(r\right)}{r}$ (4.13)

or, in momentum space:

$V(|\vec{Q}|)=\frac{1}{\left(2\pi\right)^{3}}\int
d^{3}re^{-ir.Q}V(r)=-\frac{2}{3\pi^{3}}\frac{\alpha_{V}\left(|\vec{Q}|^{2}\right)}{Q^{2}}-\frac{\sigma}{\pi^{3}|\vec{Q}|^{4}}=:-\frac{2}{3\pi^{3}}\frac{\alpha_{Rich}(|\vec{Q}|^{2})}{|\vec{Q}|^{2}},$
(4.14)

where the subscript refers to Richardson, who first proposed this type of
definition [464]. In the IR, $V\propto 1/|\vec{Q}|^{4}$; so, $\alpha_{\rm
Rich}$ diverges as $1/|\vec{Q}|^{2}$. One can understand why $\alpha_{\rm
Rich}$ does not freeze, despite the general argument that confinement
suppresses, at hadronic scales, the quantum effects causing $\alpha_{s}$ to
run [37, 465, 41]: in connection with static potentials, such as $\sigma r$,
there no well-defined finite, nonzero wavelengths. The wavelengths of the
force carriers, whatever they are, tend to infinity and static sources have
zero wavelength. Therefore, the long wavelength suppression mechanism [37,
465, 41], whereby $\alpha_{s}$ freezes, is obscured by the loss of an
explanation of the potential in terms of (even dressed) quanta. Regular IR
behaviour can be obtained by introducing a gluon mass scale as, e.g., in Ref.
[466], but this does not typically lead to improved phenomenological outcomes.
Little progress has been made with an effective charge based on Richardson-
like ideas in the past quinquennium; so, we refer to Ref. [12] for a fair
discussion of the status of this scheme.

###  4.5 Holographic Light-Front QCD

####  4.5.1 Background sketch

Holographic Light-Front QCD (HLFQCD) [118] is a nonperturbative approach to
QCD phenomena based on AdS/CFT and light-front (LF) quantization. The latter
means that canonical quantization is performed using light-front time,
$x^{+}\equiv x^{0}+x^{3}$, rather than ordinary time $x^{0}$ [467]. A general
introduction to LF Hamiltonian QCD is given in Ref. [468] and the LF
computational rules are given in Ref.[333]. LF quantization offers many
advantages, some of which we now list.

  1. (a)

It is effectively Poincaré invariant and therefore free of pseudo-dynamical
effects, akin to inertial forces, in contrast to the standard canonical
quantization based on $x^{0}$ [443].

  2. (b)

It is generally considered to lead to a trivial vacuum structure [443].

  3. (c)

LF wavefunctions have a clear and straightforward interpretation, which
provides for a rigorous derivation and interpretation of the parton model
[468].

  4. (d)

It offers a systematic method for solving nonperturbative bound-state problems
[468].

Regarding note (d), we remark that LF quantization provides a rigorous
formulation of QCD with hadron structure described by a relativistic
Schrödinger equation. In principle, the components of the equation can be
derived directly from the QCD Lagrangian. However, in practice, the confining
potential term, $U(\zeta^{2})$, where $\zeta$ is the transverse parton
separation in LF coordinates, has only been analytically computed in $(1+1)$
dimensions [469]. In (3+1) dimensions, a first-principles computation of
$U(\zeta^{2})$ has thus far proved too difficult. One method, currently being
explored, aims to capitalize on the strengths of LGT [470]. Another approach
follows the path of gauge-gravity duality [50, 371, 372], which has proven
fruitful for delivering nonperturbative information about the QCD coupling.

Gauge-gravity, or AdS/CFT, duality posits that a classical gravitational
theory in a $(d+1)$ dimensional space-time, slightly curved (or, in
interaction language, a weakly coupled gravitational theory) projects, on the
boundary of the $(d+1)$ space, into a strongly coupled CFT in $d$ dimensional
flat Minkowski spacetime. One crucial application of AdS/CFT duality is that
it provides a method for the nonperturbative solution of some QFTs. Such
programs are classified via two categories. The top-down approach starts with
a superstring theory, whose background is chosen so that the desired
properties of the CFT are reproduced, in our context, e.g., some form of
confinement for QCD. The bottom-up approach starts with the CFT and determines
the gravitational theory. Once the dual theories are matched, weakly coupled
gravitational calculations can be performed to obtain strongly coupled gauge
CFT results.

Herein, we will be concerned with the bottom-up method. Applied to QCD, it
yields an AdS/QCD correspondence, which stems from the duality between the
group of isometries of a 5-dimensional AdS spacetime and the $SO(4,2)$
conformal group that approximately describes QCD. In fact, the Lagrangian in
Eq. (3.1), indicates that classical chromodynamics is a CFT in the chiral
($m_{q}=0$) limit. Observation, however, reveals that a $\sim 1\,$GeV scale
emerges in QCD, which breaks the classical conformal invariance and chiral
symmetry. This scale can be expressed as, inter alia, the confinement scale,
$\Lambda_{s}$, a hadron (proton or $\rho$) mass, the chiral symmetry breaking
scale, $\chi_{B}$, the HLFQCD scale $\kappa$, or the string tension $\sigma$.

In contrast, away from this emergent scale, the CFT character of QCD holds
approximately. On $M_{N}^{2}/Q^{2}\simeq 0$, where short-distance quantum
fluctuations and nonzero hadron mass effects are immaterial, it manifests as
Bjorken scaling [125, 126]. Within the complementary domain,
$Q^{2}/M_{N}^{2}\lesssim 1$, the experimental or calculated behavior of
$\alpha_{s}$ suggests that it freezes, asymptotically approaching an IR-fixed
point at $Q^{2}=0$. This, together with AdS/CFT duality, offers a
nonperturbative method for QCD calculations.

Classical gravitational calculations in 5-dimensional AdS-space, projected
onto the 4-dimensional AdS-space boundary, which is identified with physical
Minkowski spacetime, may be linked with solutions of QCD equations in which
quantum (and quark mass) effects are absent. From this perspective, AdS/QCD
provides a semiclassical approximation to QCD that incorporates many key
aspects, among them the conformal invariance of the chromodynamics Lagrangian.
If QCD is quantized on the LF, then AdS/QCD yields the HLFQCD approach [372,
371].

Numerous methods for determining $U(\zeta^{2})$ in the LF QCD Schrödinger-like
equation all yield the same potential, e.g., the following schemes have been
considered.

  1. (A)

Require the approximate conformal symmetry of QCD [118].

  2. (B)

Impose the approximate chiral symmetry of QCD, in practice requiring that
$U(\zeta^{2})$ yields massless pions [471].

  3. (C)

Require that $U(\zeta^{2})$, after transformation to a framework based on
instant-time $x^{0}$ quantization, yields the Cornell potential for static
quark-quark systems [462].

  4. (D)

Apply the de Alfaro-Fubini-Furlan procedure (dAFF) [472, 473, 474], which
allows for a scale in a Lagrangian while its corresponding Action remains
conformally invariant.

  5. (E)

Use the RGE flow equation [475].

Note (C) can intuitively be understood from the fact that the
$\zeta^{2}$-dependence of $U(\zeta^{2})$ is independent of the space
dimension. In the classical picture, applicable to HLFQCD, the force from a
static source is given by the field flux crossing a small boundary element,
e.g., a surface in 3-dimensional space. That the $\zeta^{2}$-dependence of
$U(\zeta^{2})$ is independent of the space dimension implies that the flux
must be that of a field propagating freely in the minimal number of
dimensions, i.e., one dimension. Otherwise, if the flux were that of a field
propagating freely in $n$-dimensions, then one could have the same flux
arrangement, i.e., the same $\zeta^{2}$-dependence in $d>n$ dimensions, but
not for $d<n$. Since, a force propagating in one dimension is constant – the
flux has nowhere to spread, the instant-time potential is linear.

All methods indicated above lead to the result that $U(\zeta^{2})$ has the
form of a harmonic oscillator [476]:

$U(\zeta^{2})=\kappa^{4}\zeta^{2}+b,$ (4.15)

where $\kappa$ depends only on AdS space dimension and $b$ is determined by
the spin $J$ representations in AdS space. For a 2-body system in AdS5,
$b=2\kappa(J-1)$. With $m_{q}=0$ and Eq. (4.15), a wide range of phenomena are
reproduced and predicted, including the following.

  1. (i)

Nonperturbative running of $\alpha_{s}(Q^{2})$ with the prediction of freezing
in the IR.

  2. (ii)

Regge trajectory descriptions of hadron spectra [338, 477]. (Within HLFQCD,
this property might be linked with the IR-freezing of $\alpha_{s}(Q^{2})$
[475].)

  3. (iii)

A connection between the soft and hard pomerons [382].

  4. (iv)

Prediction of a symmetry between the masses of baryons, mesons, and
tetraquarks with universal Regge trajectories [338, 477].

  5. (v)

Predictions for hadron form factors [118, 380], PDFs [339, 340, 342], and GPDs
[341].

In regard to prediction (i), one may ask how freezing happens since a
classical instantaneous potential is employed, as for $\alpha_{\rm Rich}$ in
Sec. 4.4. The answer may simply be that, like $\alpha_{\rm Rich}$, the maximum
wavelength argument [37, 465, 41] does not apply; but this does not preclude a
freezing of $\alpha_{s}$. In fact, the reason for the coupling’s freezing in
HLFQCD is plain: it is a conformal theory except in the vicinity of its scale
$\kappa$ and, as a semi-classical theory, other scales cannot be generated by
quantum anomalies. Conformality manifests itself for the $Q^{2}\ll\kappa^{2}$
(and $Q^{2}\gg\kappa^{2}$) as the freezing of coupling. So, one may argue that
although maximum wavelength argument does not apply explicitly, LFHQCD
expresses enough of QCD to ensure that $\alpha^{HLF}_{s}$ freezes, effectively
realizing the wavelength cutoff. The manner by which this is physically
realized is a complex question that involves multi-gluon dynamics. There is a
priori no contradiction with the fact that, in the non-relativistic case,
LFHQCD reproduces the linear potential included in $\alpha_{\rm Rich}$,
because this necessitates the introduction of massive quarks that break the
conformality of QCD and therefore directly void the cause of
$\alpha^{HLF}_{s}$ freezing. If so, such violation of the freezing would be a
quark mass effect, which can be excluded in the definition of the running
coupling in order to preserve a universal behavior independent of a quark
specific mass.

Some of the other predictions above require several parameters in addition to
$\kappa$,333 In the natural unit system $\hbar=c=1$, one free parameter is the
minimum required for a model or theory to describe the strong force since
absolute measurements are expressed in the conventional, human-chosen, unit,
e.g., GeV. For pQCD, that parameter is $\Lambda_{s}$, while it is $\kappa$ for
HLFQCD. The relation between $\Lambda_{s}$ and $\kappa$ [112] is given in Sec.
3.6.7. introduced to: describe higher Fock states in the LF wavefunction
solutions of the Schrödinger-like equation; parameterize SU(3)-flavor symmetry
breaking; and characterize a universal function [339, 340, 342] from which the
PDFs and GPDs are derived. Yet, HLFQCD describes a great deal of strong force
phenomenology with remarkably few parameters. In fact, only $\kappa$ is needed
to predict $\alpha_{s}(Q^{2})$, as will be shown next.

####  4.5.2 HLFQCD computation of $\boldmath\alpha_{g_{1}}(Q^{2})$

The measured $\alpha_{g_{1}}$ – Fig. 4.1 – is nearly constant in the deep IR,
i.e., QCD is approximately conformal on $Q^{2}/M_{N}^{2}\simeq 0$. This can
also be deduced by using the relation between the Bjorken sum
$\Gamma_{1}^{p-n}(Q^{2})$ and the generalized GDH sum rule, as detailed below
Eq. (4.6). Such near conformality encourages the use of HLFQCD to derive
$\alpha_{g_{1}}(Q^{2})$ [368].

The five dimensional AdS5 Action has the same form as that of general
relativity:

$S_{AdS_{5}}=-\frac{1}{4}\int\sqrt{g}\frac{1}{\hat{a}^{2}_{5}}F^{2}~{}d^{5}x.$
(4.16)

Here, $F$ is the gauge field, $\hat{a}_{5}$ its self-coupling, and
$g=\det(g_{\mu\nu})$, with $g_{\mu\nu}$, the AdS space metric, yielding the
invariant interval:

$ds^{2}=\frac{R^{2}}{z^{2}}\big{(}\eta_{\mu\nu}dx^{\mu}dx^{\nu}-dz^{2}\big{)}\,,$
(4.17)

where $R$ is the AdS radius and $\eta_{\mu\nu}$ is the Minkowski metric. The
fifth (holographic) dimension, $z$, is associated with the scale at which the
hadron is probed, i.e., it corresponds to $1/Q$.

$S_{AdS_{5}}$ generates a CFT, devoid of any scale. To introduce a
characterizing scale, a factor $e^{+\kappa^{2}z^{2}}$ can be used to distort
the geometry of AdS space, thereby breaking the conformal invariance of the
theory so that Eq. (4.17) becomes

$ds^{2}=\frac{R^{2}}{z^{2}}e^{\kappa^{2}z^{2}}\big{(}\eta_{\mu\nu}dx^{\mu}dx^{\nu}-dz^{2}\big{)}.$
(4.18)

Now $\kappa$ provides the theory’s scale, like $\Lambda_{s}$ does for QCD. It
is worth remarking here that only an exponential term $e^{\pm\kappa^{2}z^{2}}$
can yield the harmonic oscillator LF potential in Eq. (4.15) [372, 371]. The
positive exponent is selected by requiring a consistent AdS-LFQCD mapping
[118]. Indeed, the derivation of the spin term in the LF potential depends
directly upon the positive sign of the profile. Moreover, as will become
apparent in connection with Eq. (4.20), the positive sign is also necessary to
ensure the expected decrease of $\alpha_{s}$ with $Q^{2}$: whilst one should
not be prejudiced about the behavior of $\alpha_{s}(Q^{2})$ at low $Q^{2}$,
consistency between pQCD and HLFQCD on the intermediate $Q^{2}$ domain
requires that $\alpha_{s}^{HLF}(Q^{2})$ decrease with $Q^{2}$.

In short, $e^{+\kappa^{2}z^{2}}$ is the potential dual in Minkowski space,
retains the conformal symmetry of the action, and yields the uniquely suited
harmonic oscillator LF potential. The action in the distorted AdS geometry
becomes:

$S_{HLF}=-\frac{1}{4}\int\sqrt{g}\frac{1}{{\hat{a}_{5}}^{2}}F^{2}~{}e^{\kappa^{2}z^{2}}d^{5}x.$
(4.19)

In the chiral limit theory discussed here, $\kappa$ is universal and can thus
be extracted from hadron masses, e.g. [118], $\kappa=M_{\rm p}/2$ or
$\kappa=M_{\rho}/\sqrt{2}$, where $M_{\rho}$ is the $\rho$-meson mass; or
connected with $\Lambda_{s}$ [325]; or from some dimension associated with
hadron form factors [118, 380], etc. The $\kappa$ values determined by these
means are consistent and combining them yields $\kappa=0.523\pm 0.024$ GeV
[478].

Recall now that effective charges generalize the concept of running coupling
by including, in addition to the short distance quantum loops, short distance
gluon radiative effects (higher order DGLAP corrections), long distance parton
distribution correlations (HT corrections), and EHM contributions.
Analogously, in the AdS coupling of Eq. (4.19), one can incorporate the
$e^{+\kappa^{2}z^{2}}$ term that is dual to the HLFQCD confinement potential:
${a_{5}}(z^{2})\equiv{\hat{a}_{5}}e^{-\kappa^{2}z^{2}/2}$. Transforming
${a_{5}}(z^{2})$ to Minkowski 4-momentum space yields $\alpha^{\rm
HLF}_{s}(Q^{2})=\alpha^{\rm HLF}_{s}(0)e^{-\frac{Q^{2}}{4\kappa^{2}}}$, with
$\alpha^{\rm HLF}_{s}(0)$ an undetermined parameter. It may be fixed by a
scheme choice, e.g., requiring agreement with Eq.(4.4) imposes the $g_{1}$
scheme [329], in which case

$\alpha^{\rm HLF}_{g_{1}}(Q^{2})=\pi{\rm e}^{-Q^{2}/[4\kappa^{2}]}.$ (4.20)

The value of $\alpha^{\rm HLF}_{s}(0)$ encompasses the observable-dependence
of an effective charge – see Sec. 4.3, analogous to the RS-dependence of
$\alpha^{\rm pQCD}_{s}$. The pointwise behavior of the prediction in Eq.
(4.20) agrees well with low $Q^{2}$ data, see Fig. 4.1, despite having no
adjustable parameters after the scheme choice is imposed.

###  4.6 Continuum Schwinger Function Methods

Implicit in many aspects of the preceding discussion of effective charges and
running couplings is an appreciation of the role that can be played by
calculating and understanding QCD’s Schwinger functions – loosely, Euclidean-
space Green functions [479, 480]. These are the primary quantities computed in
LGT; and, of course, results and insights have also long been forthcoming from
an array of continuum methods. The key point is that in any rigorously well-
defined Poincaré-invariant quantum theory, knowledge of all its Schwinger
functions is necessary and sufficient to complete a solution of the theory.
Naturally, this does not mean it is the only way to solve the problem, but it
is one rigorous approach. As discussed above, LF QCD offers another.

One of the merits of CSMs is that some of the simpler Schwinger functions
calculated are also accessible to LGT, e.g., two-point functions (propagators)
are generally accessible and so are certain projections of three-point
functions. This opens the door to useful synergies, with combined results from
both approaches leading to greater insights than are achievable with either
alone.

####  4.6.1 Defining $\boldmath\alpha_{s}$ via three-point functions

One of the most direct routes to $\alpha_{s}$ using CSMs is paved by
identities such as that in Eq. (3.27). An example of their use is provided by
the ghost-gluon vertex, which has long been a focus because, when evaluated in
Landau gauge at zero incoming ghost momentum, the renormalization constant of
the gluon-ghost vertex is unity [99]: $\tilde{Z}_{1}\equiv 1$. Capitalizing on
this, then one can define a strong running coupling from the ghost-gluon
vertex as follows:

$\alpha_{s}^{\rm gh}\left(Q^{2}\right)=\alpha_{s}^{\rm
gh}\left(\mu\right)G^{2}\left(Q^{2},\mu\right)Z\left(Q^{2},\mu\right),$ (4.21)

where $\mu$ is the renormalization scale. Here, $G(Q^{2},\mu)$ and
$Z(Q^{2},\mu)$ are the ghost and (transverse) gluon propagator dressing
functions, respectively, which are obtained from calculated ghost and gluon
two-point functions:

$\delta^{bc}G(Q^{2},\mu):=Q^{2}G^{bc}(Q^{2},\mu),\quad\delta^{bc}Z(Q^{2},\mu):=-Q^{2}[\delta_{\mu\nu}-Q_{\mu}Q_{\nu}/Q^{2}]D^{bc}_{\mu\nu}(Q^{2},\mu),$
(4.22)

where $b$ and $c$ are color indices. The effective charge in Eq. (4.21) is
known as the Taylor coupling [481, 482]; and, as discussed in Sec. 3.6.6, it
has been used to obtain a precise value of $\alpha_{s}(M_{Z}^{2})$.

Other vertices, combined with a choice of kinematics, may also be used to
define a strong running coupling, viz. the 3-gluon vertex – $\alpha_{s}^{\rm
3g}$, 4-gluon vertex – $\alpha_{s}^{\rm 4g}$, and gluon-quark vertex –
$\alpha_{s}^{\rm gq}$. There also exist 2-ghost–2-gluon and 4-ghost vertices;
but they cannot be used to define a strong coupling because there are no
graphs involving solely a bare vertex – see, e.g., Ref. [482]). This exhausts
the set of vertex-based definitions of $\alpha_{s}$.

A natural extension of the Gell-Mann–Low effective charge to QCD would work
with the gluon vacuum polarization. However, it is not possible to do this
directly owing to the non-Abelian character of QCD, expressed in the STIs.
Notwithstanding that, a simple attempt is discussed in connection with Eq.
(4.29) below. Furthermore, as explained in Sec. 4.6.4, one can improve upon
that by combining the pinch technique [28, 29, 30, 31, 32] and background
field method [33, 34] to arrive at a unique QCD analogue of the QED running
coupling, which is drawn in Fig. 4.1.

Regarding definition via the 3-gluon vertex, a symmetric kinematic
configuration is often chosen [483]; namely, with each momentum, $k_{1,2,3}$,
understood to be entering the vertex, then $k_{1,2,3}^{2}=-Q^{2}$,
$2k_{i}\cdot k_{j}=Q^{2}$, $i,j=1,2,3$. In this case,

$\alpha_{s}^{\rm 3g}\left(Q^{2}\right)=\alpha_{s}^{\rm
3g}\left(\mu\right)[\Gamma^{\rm
3g}(Q^{2},\mu)^{2}Z^{3}(Q^{2},\mu)/\left\\{[\Gamma^{\rm
3g}(\mu^{2},\mu)^{2}Z^{3}(\mu^{2},\mu)\right\\},$ (4.23)

where $\Gamma^{\rm 3g}$ is the function that dresses the leading term in the
$3$-gluon vertex, i.e., the term which exists even at tree-level. In many
renormalization schemes, the denominator in Eq. (4.23) is unity.

If the 4-gluon vertex is chosen, then there are many possible choices of
kinematics. With choice “$C$” in Ref. [484],

$\alpha_{s}^{\rm 4g}\left(Q^{2}\right)=\alpha_{s}^{\rm
4g}\left(\mu\right)[\Gamma^{\rm
4g}(Q^{2},\mu)^{2}Z^{2}(Q^{2},\mu)/\\{[\Gamma^{\rm
4g}(\mu^{2},\mu)^{2}Z^{2}(\mu^{2},\mu)\\},$ (4.24)

where $\Gamma^{\rm 4g}$ is the analogue of $\Gamma^{\rm 3g}$ in Eq. (4.23).

Using the gluon-quark vertex, working with the so-called “1-2-3” or “totally
asymmetric” kinematic configuration $Q^{2}=k_{1}^{2}=k_{2}^{2}/2=k_{3}^{2}/3$
[485], one has

$\alpha_{s}^{\rm gq}\left(Q^{2}\right)=\alpha_{s}^{\rm
gq}\left(\mu\right)Z(Q^{2},\mu)Z_{f}(Q^{2},\mu)[\Gamma^{\rm
gq}\left(Q^{2},\mu\right)Z_{f}(Q^{2},\mu)]^{2}Z\left(Q^{2},\mu\right),$ (4.25)

where $\Gamma^{\rm gq}$, $Z_{f}$ are, respectively, the analogues of
$\Gamma^{\rm 3g}$, $Z$ for the gluon-quark vertex and quark two-point
function.

At UV momenta and using any RS that is independent of current-quark mass, all
vertex choices yield the same running coupling owing to STIs like those in Eq.
(3.26). However, CSMs typically employ MOM-like RSs when completing
calculations, so this symmetry is broken in the IR. Furthermore, the value of
$\alpha_{s}^{\rm gq}$ can explicitly depend on the current-quark mass, e.g.,
Ref. [486] produced results for $\alpha_{s}^{\rm gq}$ that are strongly
dependent on quark current-mass – see Fig. 11 therein, and Ref. [487, Fig. 17]
displays values of $\alpha_{s}^{\rm qg}$ that are smaller for the $s$ quark
than for the lighter quarks. The dependence on quark current-mass diminishes
with increasing $Q^{2}$.

Comparing Eq. (4.21) with Eqs. (4.23) – (4.25), it becomes clear that the
Taylor coupling presents the simplest choice if one elects to define a running
coupling from a vertex. The simplicity is, in fact, even greater because the
number of diagrams that must be computed is also smallest in this case.

Color-carrying correlation (Schwinger) functions are not directly observable.
Further, they need not individually be renormalization group invariant or
independent of the gauge parameter. In most CSM analyses, the (Poincaré-
covariant) Landau gauge is used because it is itself a fixed point of the
renormalization group and its transverse-projector character assists in
reducing the number of independent tensor structures that contribute to a
given result.

For instance, in general, the 3-gluon vertex involves 14 independent tensors,
but 10 of them are purely longitudinal; so, in Landau gauge, it is sufficient
to retain only the 4 remaining (transverse) structures [486]. Moreover, the
longitudinal component of the gluon 2-point function is not dynamical and
decouples from the transverse piece. Finally, $\tilde{Z}_{1}\equiv 1$ in
Landau gauge [99], in consequence of which $\alpha_{s}^{\rm gh}$ in Eq. (4.21)
only involves two-point functions. Others choices have also been explored,
including axial, Coulomb, Feynman, light-cone, maximal Abelian, and general
linear covariant gauges. Some analyses of the gauge-dependence of
$\alpha_{s}^{\rm gh}$ claim a weak sensitivity for gauges interpolating
between the Landau and Coulomb gauges, with a variation of less than 30% of
the IR-fixed point around the Landau gauge result [12, 488]. However, the
quantitative reliability of such estimates is difficult to judge because the
results can depend on practitioner-dependent choices, made in formulating the
Schwinger function computations, whose magnitude cannot readily be determined.

####  4.6.2 Complementary continuum formulations

There are two commonly used approaches to the continuum calculation of
Schwinger functions: Dyson-Schwinger equations (DSEs) and the functional
renormalization group (FRG). Hereafter we provide a brief sketch of both
schemes.

##### Dyson-Schwinger equations

The DSEs can be interpreted as the quantum equations-of-motion for the theory
under consideration [489, 490, 491, 492, 493, 494, 495]. Typically formulated
in Euclidean metric – for reasons explained elsewhere [9, Sec. 1], their
solutions are the theory’s Schwinger functions: with all such Schwinger
functions known, then the theory itself is solved. A Poincaré-covariant
formulation is always possible in a Poincaré-invariant quantum gauge field
theory.

The DSEs form an infinite tower of coupled nonlinear integral equations, with
the equation for a given $n$-point Schwinger function coupled to kindred
equations for higher-$n$-point functions: in QCD, this coupling can reach to
$n+2$. The DSE tower includes, inter alia, the gap equations for gluons and
quarks (and ghosts), the Bethe-Salpeter equations for meson bound states, and
Faddeev-like equations for baryon bound states.

The basic strength of this continuum approach is that it allows for an
essentially nonperturbative treatment of the theory. No diagrammatic expansion
of the Schwinger functions need be assumed to exist because the DSEs can be
derived directly from the theory’s generating functional. In principle, the
DSEs and their solutions deliver an exact representation of the complete
information content of the QFT under consideration.

In practice, of course, the complexity of quantum gauge field theory requires
that “sacrifices” be made in order to define a tractable problem: the tower is
truncated. A carelessly implemented truncation will typically lead to
violations of the theory’s principal symmetries. This was a problem in the
last millennium. However, during the past vicennium, material progress has
been made with the development of systematic, symmetry-preserving truncation
schemes [496, 497, 498, 373, 38, 39, 40, 499, 500, 501, 502]. Thus, today, in
an array of cases, it is even possible to identify and estimate truncation-
dependent uncertainties; then, the choice of truncation specifies a level of
approximation. The resulting integral equations are typically solved
numerically, something which is straightforward with contemporaneous computer
resources.

A typical, viable truncation scheme will ensure, amongst other things,
unitarity; multiplicative renormalizability; the Euclidean space equivalent of
Poincaré-covariance; preservation of Ward-Green-Takahashi identities; and that
perturbative results are preserved – especially, so far as we are concerned
herein, the UV behavior of the running coupling. Exploiting these features,
one ensures that the basic qualities of the underlying theory are preserved,
e.g., the number and properties of dressing functions (form factors) for each
$n$-point function. In QCD, this latter means that for quarks, there are two
dynamical form factors; gluons, one, viz. the vacuum polarization; ghost
fields, one; and a limited, specified number of dynamical form factors for
each vertex. These dressing functions are the outputs of the DSE solution
procedure; and, as stressed above, the inputs to any DSE calculation of a
running coupling.

Another merit of the DSEs is that since they deliver QCD Schwinger functions,
which are also the subject of LGT computations, then synergies between these
two nonperturbative approaches can be identified and exploited. Insofar as
$\alpha_{s}$ is concerned, relevant examples are provided by the following
Schwinger functions [503, 504, 505, 386]: ghost and gluon propagators; and
ghost-gluon and 3-gluon vertices. DSE solutions for these Schwinger functions
can be used to isolate and understand the body of artefacts deriving from
lattice regularization in LGT analyses and, vice versa, LGT results can be
used to expose and reduce the effects of DSE truncations.

##### Functional Renormalisation Group

Complementing the DSE approach, the FRG provides another in-principle exact
scheme for computing a theory’s Schwinger functions. Its origins may be traced
to the momentum shell renormalization group method developed in Refs. [506,
507, 508]. By progressively integrating the fast (large momentum) components
of a field, the method sums the field’s quantum fluctuations into high-
momentum Wilson shells – see Ref. [509] for an elucidation of the momentum
shell renormalization group and Refs. [510, 511, 512, 513, 514] for the FRG.

The FRG begins by setting an IR cutoff, $k$, at a boundary above which the
theory is truly perturbative. The value of $k$ is then progressively lowered
toward and into the nonperturbative domain. Thus, $k$ serves as a regulator
term, added to the classical action, which effectively generates a momentum-
dependent mass for the associated field. That mass vanishes in the UV.444 That
the quantum field (the gluon field in the context of this discussion) becomes
effectively massive is not surprising. Indeed, the purpose of some of the
counter terms that are added to the classical bare action during the
renormalization procedure is to ensure that the symmetries characterizing the
classical action are not broken, e.g., by a momentum cut-off. Consequently, a
mass term may be necessary to counter the artifacts associated with such a
renormalization procedure. The effect of the running mass is to decouple the
slow/IR components of the field – those of momenta lower than $k$, thereby
suppressing quantum fluctuations below the scale $k$.

Working with the thus regularized generating functional for one-particle
irreducible Schwinger functions, $\Gamma_{k}[\phi]$, where $\phi$ is here some
generic field, and differentiating with-respect-to $k$, one arrives at the
basic flow equation [510, 511, 512, 513, 514]:

$\frac{\partial}{\partial k}\Gamma_{k}[\phi]=\tfrac{1}{2}{\rm
Tr}\,\frac{\frac{\partial}{\partial k}R_{k}}{\Gamma_{k}^{(2)}[\phi]+R_{k}},$
(4.26)

where $R_{k}$ is the regulator function used to suppress the (IR) modes below
$k$, and $\Gamma_{k}^{(2)}$ is the functional second derivative of the action,
i.e., the inverse of the $\phi$-propagator after its modification by $R_{k}$.
The trace, $\rm Tr$, sums over internal indices, momenta and fields.

Eq. (4.26) describes the $k$-dependence of the theory, viz, of all the
theory’s Schwinger functions. It provides an exact flow equation for
$\Gamma_{k}[\phi]$ with a one-loop structure, in contrast to the multi-loop
structure of generic perturbation theory. A system of flow equations for
$n$-point Schwinger functions is then constructed by successively
differentiating Eq. (4.26) with respect to the field or source terms in the
action.

Taking the $k\to 0$ limit makes the cut-off disappear into the far IR,
integrating out the slow modes, viz. the QFT’s IR quantum fluctuations. It
results in an analytic, nonperturbative method that allows one to study QFTs
in a strong-coupling regime. In particular, running couplings can be studied
because they may be defined via correlation functions – see Sec. 4.6.1. By
construction, the FRG ensures the RG evolution expected from the QFT. This was
verified for $\alpha_{s}(Q^{2})$ in the pure Yang–Mills case in the UV regime
[515]. In practice, the infinite system of flow equations must be truncated.
As with the DSEs, this introduces uncertainties whose magnitude can be
difficult to assess. Additionally, the choice of regulator is not unique and
may be an extra source of arbitrariness.

Similarities between the DSE and FRG schemes are now plain. Their principal
foci are a theory’s Schwinger functions: the DSEs obtain them from a theory’s
quantum equations-of-motion, whereas the FRG works with the flow equation for
the effective action. Each scheme is identified with an infinite tower of
coupled equations; and supposing a sound approach is employed to make the
problem tractable, they deliver the complete solution of the QFT. In
principle, the DSE and FRG approaches must yield the same results for each
Schwinger function. In practice, different truncation schemes can lead to
mismatches that are artifacts of the choices made. Nevertheless, adding the
FRG to the DSE+LGT mix, then one has three distinct nonperturbative tools
whose results can be used to mutually check and improve the results of any
one.

####  4.6.3 CSM results for vertex-based couplings.

Once results for the relevant Schwinger functions are in hand, then the
couplings described in Sec. 4.6.1 can be computed. This is an active field and
numerous studies exist [482, 6, 7, 8, 9, 10, 11]. Herein, we exploit those
related to vertex-based running couplings from the past quinquennium. Earlier
contributions are discussed elsewhere [12].

##### IR behavior of Schwinger functions

Following Refs. [516, 517, 518, 519, 520, 493, 521, 522], despite existing
evidence to the contrary from both continuum and LGT analyses [28, 523], it
was considered plausible by some that the ghost dressing function diverged as
some irrational power of $Q^{2}$ at IR momenta, viz. $G(Q^{2},\mu)\propto
1/Q^{2\kappa}$, $0<\kappa\in\mathbb{P}$; the gluon dressing function vanished
as $Q^{4\kappa}$; and the leading dressing function of the 3-gluon vertex
diverged as $1/Q^{6\kappa}$. This position came to be called the “scaling
scenario”.

Subsequent studies, involving many distinct collaborations from diverse
locations, carefully reexamined the scaling possibility – see, e.g., Refs.
[524, 525, 526, 527, 528, 529, 530, 481]. Those continuum and LGT analyses
revealed that the scaling scenario is not realized in QCD. Instead, the ghost
dressing function is IR finite and, matching the original work [28, 523], the
gluon dressing function is monotonically decreasing from a nonzero maximum
value at $Q^{2}=0$. This is the “decoupling scenario”, so named because IR
massive gluons decouple from interactions on $Q^{2}\lesssim\Lambda_{s}^{2}$.
These conclusions have been confirmed in many more recent analyses [499, 531,
47, 532, 533, 48]. Furthermore, the leading 3-gluon vertex dressing function
has been found to exhibit a zero crossing to negative values at $Q^{2}\approx
1\,$GeV2 and to diverge logarithmically on $Q^{2}\simeq 0$ [534, 535, 536,
537]. Notwithstanding these facts, we will sometimes include scaling scenario
results for the purpose of illustrating the sensitivity of vertex couplings to
the character of Schwinger functions. (Additional background is provided in
Ref. [12, Sec. 4.4.3].)

##### Discussion of vertex-based couplings

It has been known for fifty years that the Landau gauge ghost-gluon vertex
function is finite [99]. Hence, whether an IR extension of the QCD running
coupling can meaningfully be defined via Eq. (4.21) depends on the IR behavior
of the gluon 2-point function. This question is settled in the affirmative by
the modern studies which have demonstrated that $Z(Q^{2},\mu)$ decreases
monotonically from a maximum value at $Q^{2}=0$.555The effect of light quarks
is not completely settled, but existing analyses indicate that they do not
alter this result [39].

As first argued in Ref. [28], the underlying cause of the decoupling behavior
of $Z(Q^{2},\mu)$ and, therefore, the nonzero IR-finite value of
$\alpha_{s}^{\rm gh}$, is dynamical generation of a running gluon mass,
$m_{g}(Q^{2})$, which regularizes the gluon propagator at IR momenta whilst
preserving the STIs. This mass function eliminates the Landau pole and forces
gluons to decouple from all interaction processes at length-scales greater
than $1/m_{g}(0)$. As explained in a more general context elsewhere [37], the
coupling therefore loses its $Q^{2}$-dependence, i.e., it “freezes”. A running
gluon mass is also expressed in other analysis methods, as discussed below;
albeit, the interpretation of the gluon mass-scale is sometimes different.

The source of the gluon running-mass has long been debated. Naturally, it may
depend on the framework used to analyze strong-field QCD. It was originally
supposed [28] to connect with formation of a gluon condensate, and that
possibility is still explored [538]. However, given the elusiveness of the
gluon condensate [539, Sec. 4.5], it is more likely to be the result of
another, better established phenomenon, viz. the Schwinger mechanism of gauge
boson mass generation [383, 384]. Recent efforts [385, 386], combining
continuum and LGT results, have provided strong support for this position by
demonstrating with confidence that a dynamically-generated, massless, colour-
carrying, scalar gluon+gluon correlation emerges as a feature of the dressed
3-gluon vertex. This massless pole triggers the Schwinger mechanism in QCD.
Alternative explanations typically rely on a realization of the scaling
scenario; hence, are disfavored. In any event, should a source for the gluon
mass scale be proposed that is unconnected with the Schwinger mechanism, then,
without artifice, that scheme must also simultaneously explain and reproduce
the appearance of the massless pole in the 3-gluon vertex, which, as shown in
Ref. [386], is a feature of QCD.

In Ref. [28], Feynman diagrams are resummed using the pinch technique, which
was actually developed for this calculation in order to ensure that a gauge-
independent result was obtained for the gluon dressing function. It was then
found that the numerical solution of the gap equation is well described by the
following simple analytic form, which preserves QCD’s one-loop renormalization
group behavior:

$\alpha_{s}(\mu^{2})\frac{Z(Q^{2},\mu)}{Q^{2}}=\frac{4\pi}{\beta_{0}\ln[\big{(}Q^{2}+4m_{g}^{2}(Q^{2})\big{)}/\Lambda_{s}^{2}\bigg{]}}\,\frac{1}{Q^{2}+m_{g}(Q^{2})}\,,$
(4.27)

where the gauge-independent running gluon mass is

$m_{g}^{2}\left(Q^{2}\right)=\frac{m_{0}^{2}}{\left[\mbox{ln}\left(\frac{Q^{2}+4m_{0}^{2}}{\Lambda_{s}^{2}}\right)/\mbox{ln}\left(\frac{4m_{0}^{2}}{\Lambda_{s}^{2}}\right)\right]^{12/11}},$
(4.28)

with $m_{0}=0.5\pm 0.2\,$GeV. It is worth recalling that whilst adding a mass
term directly to the QCD Lagrangian would violate gauge invariance, this mass
contributes only to the transverse part of the gluon 2-point function; hence,
is consistent with the STIs and therefore, gauge invariance.

The running mass in Eq. (4.28) vanishes on $\Lambda_{s}^{2}/Q^{2}\simeq 0$,
ensuring compatibility with pQCD. Moreover, Eq. (4.27) suggests a
straightforward continuation of the one-loop running coupling into the IR:

$\alpha_{s}^{\Pi}(Q^{2})=\frac{4\pi}{\beta_{0}\ln[\big{(}Q^{2}+4m_{g}^{2}(Q^{2})/\Lambda_{s}^{2}\big{)}\bigg{]}}\,,$
(4.29)

where the superscript $\Pi$ indicates that the coupling behavior stems from
the gluon vacuum polarisation. This coupling freezes at a maximum value in the
IR and decreases monotonically with increasing $Q^{2}$.

The result in Eq. (4.29) is gauge-independent, having been obtained using the
PT; but details depend on the 1-loop analysis used in Ref. [28] and the form
chosen in Eq. (4.29). Different values of $m_{0}$ may be obtained if the pinch
technique is not used and a different gauge is chosen and/or another RS, going
beyond 1-loop, were to be employed. Nevertheless, using the quoted central
value of $m_{0}$ and $\Lambda_{s}=0.63$ GeV, so that $\alpha_{s}^{\Pi}$
matches $\alpha_{s}^{\rm pQCD}$ in the UV, one finds
$\alpha_{s}^{\Pi}(Q^{2}=0)/\pi\approx 0.4$. This is compatible with the value
of the IR fixed point from HLFQCD (Sec. 4.5) expressed in $\overline{\rm MS}$:
$\alpha^{\mathrm{HLF}}_{\overline{\rm MS}}(0)/\pi=0.388\pm 0.047$ [331].

Eqs. (4.27)-(4.28) have been much discussed in the literature – see, e.g.,
Ref. [12] for a review of the many studies; developments or alterations of
their forms; and also the generic constraints on $m_{g}(Q^{2})$, its proposed
values, and $Q^{2}$-dependences. Typically, determinations agree with the
picture drawn in Ref. [28].

Although Refs. [540, 541, 542, 543, 28] laid a foundation for discussions of
$m_{g}(Q^{2})$ and $\alpha_{s}^{\rm IR}$, delivering predictions that have
subsequently been refined but not truly replaced [6, 7, 8, 9, 10, 11], they
are peculiar in relying entirely on 1-loop expressions for the gluon 2-point
function and running coupling, and using a ghost-free gauge. As noted above,
many studies have instead focused on vertex-based definitions of
$\alpha_{s}^{\rm IR}$; and, generally, the couplings thus defined are
inequivalent: on $Q^{2}\lesssim m_{N}^{2}$,
$\alpha_{s}^{\Pi}\neq\alpha_{s}^{\rm gh}\neq\alpha_{s}^{\rm
qg}\neq\alpha_{s}^{\rm 3g}\neq\alpha_{s}^{\rm 4g}$.

Figure 4.3:  Representative DSE results for vertex-based couplings.
$\alpha^{\rm gh}_{s}$: blue curve from Ref. [484] and red curve from Ref.
[544]). $\alpha^{\rm qg}_{s}$: cyan curves – Ref. [486], decoupling solution;
Ref. [485], scaling. (Quark degrees-of-freedom are only included in this
coupling. Ref. [486] – $m_{q}=30\,$MeV, with results also provided up to
$m_{q}=3.6\,$GeV; and Ref. [485] – chiral limit.) $\alpha^{\rm 3g}_{s}$:
magenta curves [484]. $\alpha^{\rm 4g}_{s}$: black curves [484]; and green
curve (right panel) [544]). Calculations are performed in Landau gauge and
using the MOM RS. The decoupling scenario – left panel – is that favored in
Nature [524, 525, 526, 527, 528, 529, 530, 481, 499, 531, 47, 532, 533, 48].
The $Q^{2}$-location of the maxima of the various running couplings depends on
the particular value $G(0,\mu^{2})$ at which the ghost dressing function
saturates. The vanishing of $\alpha_{s}^{\rm 3g}$ at a nonzero $Q^{2}$ value
owes to the IR zero-crossing of $\Gamma^{\rm 3g}(Q^{2})$, the 3-gluon vertex
dressing function [534, 535, 545, 546, 536, 537]. This is an unavoidable
consequence of the emergence of a gluon mass, which entails that IR gluon
modes are screened and ghost fields dominate gauge sector dynamics at IR
momenta: finiteness of the gluon DSE requires a zero-crossing in some of the
dressing functions. IR fixed points in vertex-based couplings are an artifact
of the scaling scenario.

As displayed in Fig. 4.3, DSE results for the ghost–gluon coupling express
$\alpha_{s}^{\rm gh}(0)=0$ in the decoupling scenario. On the other hand,
$\alpha_{s}^{\rm gh}(0)\neq 0$ in the disfavored scaling case. Several
calculations of the quark–gluon coupling $\alpha_{s}^{\rm qg}$ are also
depicted: qualitatively, the results are similar to those for $\alpha_{s}^{\rm
gh}$. Notably, $\alpha_{s}^{\rm qg}>\alpha_{s}^{\rm 3g}$ on the entire IR
domain. A more recent calculation, building upon LGT results for the gluon
propagator ($n_{f}=2+1$, one-loop, Landau gauge, MOM RS), yields a similar
coupling [547]. Owing to DCSB, the impact of dynamical quark loops is small,
so long as the number of light quarks does not exceed some critical value
[548, 549, 550, 39].

Calculations of $\alpha_{s}^{\rm 3g}(Q^{2})$ and $\alpha_{s}^{\rm 4g}(Q^{2})$
are also drawn in Fig. 4.3. At far IR momenta, these couplings are markedly
suppressed when compared with $\alpha_{s}^{{\rm gh},{\rm qg}}$. On the other
hand, with increasing $Q^{2}$, they reach a maximum value $\alpha_{s}^{\rm
3g,4g}\sim 1$ [12, 551, 552, 553, 554, 555, 484, 534, 556, 531, 499, 535, 545,
546, 557] before falling away to reproduce pQCD behavior.

Actually, it is evident in Fig. 4.3 that all vertex-based couplings exhibit a
(local) maximum at some nonzero value of $Q^{2}$. In order for CSM and HLFQCD
to be consistent, we judge that, in all cases, the locations of these maxima
must lie below the perturbative-nonperturbative transition scale determined
using HLFQCD, viz. $Q_{0}^{2}=1.32$ GeV2 in the MOM RS and using Landau gauge
[331]. This condition is satisfied (albeit, only just for $\alpha_{s}^{\rm
3g}$), irrespective of the particular CSM calculation considered [551, 553,
554, 534, 499, 484, 556, 531, 545, 535, 546, 505].

The IR suppression of $\alpha_{s}^{\rm 3g}(Q^{2})$ and $\alpha_{s}^{\rm
4g}(Q^{2})$ in MOM-like RSs occurs because, unlike gluons, no dynamical mass
emerges for ghost fields. Consequently, in the calculation of these couplings,
ghost loop contributions dominate in the IR, delivering logarithmic IR
divergences. These ghost contributions to the leading three- and four-gluon
Schwinger functions are negative, whereas the UV dominant gluon contributions
are positive. Thus, a clear signal for entry into a ghost dominance domain is
a sign-change of the gluon vertex functions. This sign-change has been
verified explicitly in Landau gauge studies of QCD’s pure-gauge sector [551,
552, 558, 553, 554, 534, 555, 499, 484, 556, 531, 545, 535, 546, 559]:
$\Gamma^{\rm 3g}(Q^{2})$ crosses zero around $Q^{2}\sim 0.05$ GeV2. It leads
to $\alpha_{s}^{\rm 3g}=0$ at that point, as can be seen from Fig. 4.3.

As also found for $\alpha_{s}^{\rm qg}$, light quark loops do not seem to
qualitatively alter this picture, only modifying the location of the zero-
crossing. Those analyses that have provided information on this issue indicate
that the zero survives, but is shifted somewhat lower in $Q^{2}$ [499, 557,
560]. Efforts to elucidate the full impact of dynamical quark loops are
continuing [39, 561, 487, 547, 562].

In Landau gauge, with two flavors ($n_{f}=2$) of physical-mass light quarks,
the FRG has also been used [563] to compute the Schwinger functions needed to
arrive at results for $\alpha_{s}^{\rm gh}$, $\alpha_{s}^{\rm qg}$,
$\alpha_{s}^{\rm 3g}$ and $\alpha_{s}^{\rm 4g}$, Eqs. (4.21)–(4.24). The
couplings were subsequently calculated, albeit, assuming $\alpha_{s}^{\rm
3g}=\alpha_{s}^{\rm 4g}$. In this analysis, the input ghost and gluon
propagators were taken from earlier pure-gauge DSE/FRG calculations and the
transition from parton to hadron degrees-of-freedom was modeled using a
dynamical hadronization technique that involves the addition of a scale-
dependent effective potential to the effective action [564]. The study
delivers IR-vanishing couplings, with $\alpha_{s}^{\rm gh}>\alpha_{s}^{\rm
qg}\gg\alpha_{s}^{\rm 3g}$ in the deep IR, but $\alpha_{s}^{\rm
qg}>\alpha_{s}^{\rm gh}>\alpha_{s}^{\rm 3g}$ on the higher momentum part of
the IR domain, in agreement with DSE, LGT and Curci-Ferrari results. The
couplings vanish in the IR owing to the generation of a gluon mass scale,
which leads to increasing IR suppression as the number of gluon legs grows.
However, the result $\alpha_{s}^{\rm qg}>\alpha_{s}^{\rm gh}$ at moderate IR
momenta was identified as a truncation artefact [563], in the absence of which
one might expect $\alpha_{s}^{\rm gh}>\alpha_{s}^{\rm qg}$ everywhere.

An expanded body of FRG analyses may be found in Refs. [556, 531, 561]: the
former does not involve dynamical quarks and the latter works with $n_{f}=2$.
Once again, all vertex-based couplings are computed: $\alpha_{s}^{\rm gh}$,
$\alpha_{s}^{\rm 3g}$, $\alpha_{s}^{\rm 4g}$ and $\alpha_{s}^{\rm qg}$. The
results generally agree qualitatively and (semi-)quantitatively with those
obtained using the DSEs, drawn in Fig. 4.3, except that these FRG studies do
not find a zero in $\Gamma^{\rm 3g}(Q^{2})$. The results for the ghost and
gluon propagators from Ref. [556] are used in the FRG study of Ref. [538] in
an attempt to develop an interpretation of the gluon mass scale as the result
of gluon condensation via a Higgs-like mechanism. The calculation of the gluon
effective mass, performed using the MOM RS, agrees with LGT results.

With all vertex-based couplings vanishing in the far IR, one arrives at a
picture of long-wavelength QCD as a noninteracting theory. Were this behavior
really a statement about a running coupling that describes QCD’s effective
charge at all accessible momentum scales [363], then one would be faced with
many conundrums. For instance, DCSB – the origin of constituent-like quark
masses – may even be absent, because that phenomenon requires a sufficiently
large integrated interaction strength on $Q^{2}\lesssim M_{N}^{2}$ [354]. A
resolution of these puzzles is discussed in Sec. 4.6.4, where a connection
between $\alpha_{s}^{\rm gh}$ and QCD’s Gell-Mann–Low effective charge is
explained. The result is a freezing coupling whose behavior is (practically)
indistinguishable from that obtained using HLFQCD.

####  4.6.4 Process-independent, effective charge

It will readily be recalled that the archetypal running coupling is that
computed in quantum electrodynamics (QED) [27]. It is now known with great
accuracy [5] and the running has directly been observed [565, 566].
Significantly, this Gell-Mann–Low effective charge is a renormalization group
invariant and process-independent (PI) running coupling. It is obtained simply
by computing the photon vacuum polarisation. Such a definition is
straightforward because ghost fields decouple in Abelian theories; hence, one
has the Ward identity [43, 567], in consequence of which the electric-charge
renormalization constant is equivalent to that of the photon field.
Physically, this means that the impact of dressing the interaction vertices is
entirely absorbed into the vacuum polarisation.

In QCD, that is not usually true because ghost fields do not generally
decouple from interactions, so the renormalization constants associated with
the running coupling and the gluon vacuum polarisation are different.666 It is
worth recalling that QCD can be formulated in one of the few ghost-free gauges
[568]. In that case, however, the number of independent tensors required to
fully express any given $n$-point function is considerably multiplied; so, in
practice, one arrives at a problem of at least equal complexity. This fact
underlies the introduction and analysis of the vertex-based definitions of
$\alpha_{s}$ described above. The problems with such definitions are now
plain, however. Whilst they are equivalent in the UV, there is a huge array of
distinct vertex-based couplings whose IR momentum-dependence depends on the
choices of vertex and kinematic configuration. This presents the fundamental
question: With $\alpha_{s}^{\rm gh}\neq\alpha_{s}^{\rm qg}\neq\alpha_{s}^{\rm
3g}\neq\alpha_{s}^{\rm 4g}$ and the behavior of any such coupling potentially
depending on the arrangement of momenta flowing into the vertex, can any
coupling of this type be identified as the single $\alpha_{s}$ that describes
QCD’s interaction strength at all momentum scales [363]? Furthermore, there
are also issues with the IR suppression (zero crossing) exhibited by these
couplings, which makes DCSB problematic, and the probable dependence on gauge
choice. These problems are resolved by a notable advance that begins with
ideas developed using QCD’s DSEs and then exploits results from LGT.

The key point is that there is an approach to analyzing QCD’s Schwinger
functions that preserves some of QED’s simplicity; namely, the combination of
BFM [33, 34] and PT [28, 29, 30, 31]. This approach acts to make QCD “look”
somewhat Abelian by enabling a systematic rearrangement of classes of diagrams
and their sums in order to arrive at modified Schwinger functions that satisfy
linear Slavnov-Taylor identities [99, 98].

In the gauge sector, working with Landau gauge, this BFM+PT approach yields a
modified gluon dressing function, from which one can compute a unique QCD
running coupling [35, 36], denoted $\hat{\alpha}_{s}$ herein, i.e., the
associated polarization captures all required features of the renormalization
group – just as is the case in QED [27]. One may express the result concretely
as follows:

$\hat{\alpha}_{s}(Q^{2}){\mathpzc
D}(Q^{2})=\alpha_{0}\left[\frac{G(Q^{2};\mu^{2})/G(0;\mu^{2})}{1-L(Q^{2};\mu^{2})G(Q^{2};\mu^{2})}\right]^{2}{\mathsf{D}}(Q^{2})\,,$
(4.30)

where the hitherto undefined quantities are the RGI product
${\mathsf{D}}(Q^{2})=[Z(Q^{2},\mu)/Q^{2}]\times$
$m_{g}^{2}(Q^{2},\mu)/m_{0}^{2}$, with $m_{g}(Q^{2},\mu)$ the running gluon
mass, $m_{0}$ a RGI gluon mass scale, and ${\mathsf{D}}(0)=1/m_{0}^{2}$;
${\mathpzc D}(Q^{2})$ is a RGI function, which behaves as the free propagator
for a boson with mass $m_{0}$ in both the far-IR and -UV and satisfies
${\mathsf{D}}(Q^{2})={\mathpzc D}(Q^{2})$ on $Q^{2}\lesssim\mu^{2}$; and
$L(Q^{2};\mu^{2})$ is that longitudinal part of the gluon-ghost vacuum
polarisation which vanishes at $Q^{2}=0$ [320]. Evidently,
$\alpha_{0}=\hat{\alpha}_{s}(Q^{2})$ is RGI. Thus defined, $\hat{\alpha}_{s}$
is process independent; namely, the same result is obtained independent of the
scattering process considered, whether gluon+gluon$\,\to\,$gluon+gluon,
quark+quark$\,\to\,$quark+quark, etc. Furthermore, and importantly, it depends
only on the momentum flowing through the vacuum polarization; so, is free of
the kinematic momentum-flow ambiguities attached to vertex-based definitions
of a running coupling – see the discussion of $\alpha_{s}^{\rm gh}$,
$\alpha_{s}^{\rm 3g}$, $\alpha_{s}^{\rm 4g}$, or $\alpha_{s}^{\rm gq}$ in Sec.
4.6.1.

The clean connection between coupling and gluon vacuum polarization is
facilitated by another particular feature of QCD, already mentioned often
above, viz. in Landau gauge the renormalization constant of the gluon-ghost
vertex is unity [99], Consequently, the effective charge obtained from the
modified gluon vacuum polarisation is directly connected with
$\alpha_{s}^{gh}$, the coupling deduced from the gluon-ghost vertex [305, 64,
320], and usually called the “Taylor coupling,” $\alpha_{\rm T}$ [309, 310].

Studying Eq. (4.30) and reviewing the character of $L$, $G$, one recognizes
that it is the inclusion of an infinite series of ghost-gluon scattering
contributions which transmogrifies the standard gluon vacuum polarization into
a form suitable for defining a Gell-Mann–Low-like effective charge for QCD.
Importantly, each of the Schwinger functions in Eq. (4.30) can be computed
using both continuum and LGT methods.

It is worth remarking here that $\hat{\alpha}_{s}$ is PI and RGI in any gauge;
but it is sufficient to know the result in Landau gauge, which is a convenient
calculational choice for both continuum and LGT studies. This is because
$\hat{\alpha}_{s}$ is form invariant under gauge transformations [35, 36] and
gauge covariance ensures that any such transformations can be absorbed into
the Schwinger functions of the quasiparticles whose interactions are described
by $\hat{\alpha}_{s}$ [569].

To date, the most refined calculation of $\hat{\alpha}_{s}$ is described in
Ref. [36], which combined contemporary results from continuum analyses of
QCD’s gauge sector and LGT configurations generated with three domain-wall
fermions at the physical pion mass [570, 571, 572] to obtain a parameter-free
prediction of $\hat{\alpha}_{s}(k^{2})$, characterized by

$\alpha_{0}/\pi=0.97(4)\,,\quad m_{0}=0.43(1)\,{\rm GeV}.$ (4.31)

It is now plain that this coupling saturates to a large, nonzero, finite value
at $Q^{2}=0$, with the dynamical generation of a running gluon mass having
eliminated the Landau pole. Notably, the gluon mass scale falls within the
window estimated in Ref. [28] – see the discussion of Eq. (4.27). The
resulting charge is drawn as the magenta curve and associated uncertainty band
in Fig. 4.1. It has been shown to deliver a unified description of many
observables [6, 9].

The results drawn in Fig. 4.1 reveal a remarkable fact:

$\hat{\alpha}_{s}(Q^{2})\approx\alpha_{g_{1}}^{\rm
HLF}(Q^{2})\approx\alpha_{g_{1}}(Q^{2})\,.$ (4.32)

There are small quantitative differences between these charges; but to all
practical intents and purposes, they are equivalent. Crucially, within mutual
uncertainties, they all freeze to the same IR value. These effectively
identical couplings therefore provide the best available candidate for that
running coupling/effective charge which describes QCD’s interaction strength
at all momentum scales [363].

One recognizes in Eq. (4.30) the expression of a force,
$F$=(coupling$\times$propagator). This strengthens the correspondence between
$\hat{\alpha}(Q^{2})$, $\alpha_{g_{1}}(Q^{2})$ and $\alpha^{\rm
HLF}_{g_{1}}(Q^{2})$ because the last two are defined in just this way.
Moreover, it highlights that EHM effects, such as confinement, are included in
all three couplings.

###  4.7 Lattice Gauge Theory

As a nonperturbative method, in which the computation of Schwinger functions
is conceptually straightforward, LGT is another approach to studying the
behavior of $\alpha_{s}$ in the IR. Unsurprisingly, therefore, many LGT
calculations of $\alpha_{s}$ use the same formulae and definitions as those
employed in CSMs; namely, Eqs. (4.21)-(4.24), defining $\alpha_{s}^{\rm gh}$,
$\alpha_{s}^{\rm qg}$, $\alpha_{s}^{\rm 3g}$, $\alpha_{s}^{\rm 4g}$.
Nevertheless, there are other possibilities. For instance, the strong running
coupling is accessible via Wilson loops, which provide the long distance
static quark-quark potential; hence, in turn $\alpha_{V}$, with recent results
available elsewhere [289, 287]. Wilson-loop calculations of $\alpha_{s}$ are
gauge invariant.

The basics of LGT are summarized in Sec. 3.6.5. Since fermion fields are
difficult to handle using a lattice regularization,777How to include dynamical
quarks in LGT calculations is well studied. Today, they are included in many
LGT studies. However, it prohibitively increases the CPU costs of calculations
in which large lattices are necessary, as is the case for any study of
$\alpha_{s}$ in the IR. Therefore, when quarks are not crucial, e.g., to
compute $\alpha_{s}^{\rm gh}$, $\alpha_{s}^{\rm 3g,~{}4g}$ – in contrast to
$\alpha_{s}^{\rm qg}$, dynamical quark fields are often quenched out of the
computation. It is expected that, generally, qualitative features revealed by
pure gauge studies should hold if quarks were present, but this is not
guaranteed [562]. we will ignore them for now and first discuss $\alpha_{s}$
calculated in the pure gauge sector. Just as with CSMs (unless the pinch-
technique is used), gauge-dependence affects LGT results if the studies make
use of $n$-point Schwinger functions. Such calculations are typically
performed in the Landau gauge and one of the MOM-type RSs.

For the computation of $\alpha_{s}$ using dressing functions, discretized
fields must be defined. Such definitions, while not unique, are equivalent up
to a proportionality factor [573, 574]. On a lattice, the discretized gluon
field, $\mathcal{A}$, is given at the lattice site, $x$, by the difference
between a link $U_{\overrightarrow{\mu}}$ and its adjoint:

$\mathcal{A}_{\overrightarrow{\mu}}^{c}(x)\equiv\frac{1}{2i}\left(U_{\overrightarrow{\mu}}(x)-U_{\overrightarrow{\mu}}^{\dagger}(x)\right)\xrightarrow[a\to
0]{}2a\sqrt{\pi\alpha_{s}^{\rm bare}}A_{\mu}^{c}(x),$ (4.33)

where $c$ is the color index, $\overrightarrow{\mu}$ the link direction, $a$
the lattice spacing and $A_{\mu}^{c}$ the physical gluon field.

The gluon propagator in the functional-integral formalism is the 2-point
Schwinger function

$D_{\mu\nu}^{bc}(x-y)=\int
d\mu(A,\bar{\psi},\psi)\,A_{\mu}^{b}(x)A_{\nu}^{c}(y)e^{-S}\equiv\left\langle
A_{\mu}^{b}(x)A_{\nu}^{c}(y)\right\rangle,$ (4.34)

where $d\mu(A,\bar{\psi},\psi)$ is the integration measure and $S$ is the
Euclidean-space QCD action. On a lattice, Eq. (4.34) becomes

$\mathcal{D}_{\overrightarrow{\mu}\overrightarrow{\nu}}^{bc}\left(x-y\right)=\left\langle\mathcal{A}_{\overrightarrow{\mu}}^{b}\left(x\right)\mathcal{A}_{\overrightarrow{\nu}}^{c}\left(y\right)\right\rangle.$
(4.35)

Its Fourier transform yields the discretized gluon propagator dressing
function,
$\mathcal{Z}(Q^{2})\equiv\delta^{bc}\delta^{\mu\nu}\mathcal{D}^{bc}_{\mu\nu}(Q^{2})Q^{2}.$
The inverse of the Faddeev–Popov operator $(-\partial+A)\partial$ defines the
ghost propagator, $D^{\rm G}_{bc}$, which, after discretization and Fourier-
transformation to momentum space, yields the ghost propagator dressing
function: $\mathcal{G}=\delta^{bc}Q^{2}\mathcal{D}_{bc}^{\rm
G}\left(Q^{2}\right)$. Discretized dressing functions for vertices are
obtained from the 3- and 4-point correlation functions. With these dressing
functions, the couplings $\alpha_{s}^{gh}$, $\alpha_{s}^{\rm 3g}$ and
$\alpha_{s}^{\rm 4g}$ are obtained from Eqs. (4.21)–(4.24).

As with other methods, LGT calculations are verified by comparing the UV
running of the obtained $\alpha_{s}(Q^{2})$, regardless of its definition,
with the RGE expectation. For LGT however, the discretization limits the UV
reach and computation accuracy. Likewise, finite size effects limit the IR
reach. This latter issue makes it difficult to verify phenomena expected in
the deep-IR, such as the zero-crossing of the 3-gluon dressing function
responsible for the minima of $\alpha_{s}^{\rm 3g}$ around $Q=0.1\,$GeV;
nevertheless, as described below, recent analyses point toward confirmation
[535, 546, 504]. Typical lattice physical size used by current LGT studies
range from 2-20 fm, with spacing from 0.05-0.2 fm. As noted above, modern LGT
calculations of $\alpha_{s}$ from $n$-point correlation functions deliver
results that confirm the decoupling scenario.

Herein, we sketch only those LGT results pertaining to $\alpha_{s}$ that were
obtained in the past quinquennium. Earlier analyses may be traced from Ref.
[12]. Beside the results themselves, LGT outputs also play a crucial role in
checking and improving the Schwinger functions obtained using other methods,
such as CSMs. Thus, much of LGT’s impact is already discussed in those
sections, with the references provided therein. There is also a the large body
of LGT analyses that compute the Schwinger functions alone, without making the
step to a result for one of the vertex-based couplings. This body includes
work on the center vortex approach to confinement, which involves simulations
of the gluon propagator and/or the long-distance static quark-quark potential
[447, 448, 449, 575, 451, 576, 577, 562, 450, 453, 452]. These topics are
omitted from our discussion.

As already discussed, LGT results for Schwinger functions are a crucial
component in the parameter-free prediction of the PI charge – see Sec. 4.6.4.
Likewise, in Ref. [546], both LGT and DSEs are used to determine the behavior
of the 3-gluon vertex. Specifically, Refs. [535, 546] study $SU(3)$-color in
Landau gauge with the aim of locating a zero-crossing in the 3-gluon vertex,
using a realistic (but pure gauge sector) high-statistics LGT simulation,
wherein LGT artifacts are well controlled. The zero-crossing is expected to
occur at a scale of $\sim 0.01\,$GeV2, which is difficult to reach for LGT
simulations. Nevertheless, a zero-crossing is found $Q^{2}/{\rm
GeV}^{2}=0.026\pm 0.08(\rm{stat})$. The $Q^{2}$ dependence of $\alpha_{s}^{\rm
3g}$ and the related vertex dressing function support the DSE conclusion on
the origin of the zero-crossing. Namely, the crossing marks entry into the
domain whereupon loop diagrams from massless ghosts, which contribute
negatively to the gluon vacuum polarization, come to dominate over the
positively contributing gluon loops – recall that the effective masses of the
latter suppress their contribution in the IR. The consistency between LGT and
DSE for the 3-gluon vertex is also checked in Ref. [504].

Another calculation of (inter alia) $\alpha_{s}^{\rm 3g}$ [578] has been used
to test the effect of the “Wilson flow” smoothing technique. That Landau gauge
calculation, using the MOM scheme, with both quenched and
$n_{f}=(2+1+1)$-unquenched field configurations, agrees well with the older
LGT calculations of $\alpha_{s}^{\rm 3g}$. An interpretation of the IR
behavior of $\alpha_{s}^{\rm 3g}$ in term of quasi-classical instantons is
offered therein. This has some similarity in spirit to the semi-classical
analysis that drives the derivation of $\alpha_{g_{1}}^{\rm AdS}$ – see Sec.
4.5.

Possible biases arising from finite volume and finite lattice spacing were
studied in Refs. [579, 580, 581] for ghost and gluon propagators and for
$\alpha_{s}^{\rm gh}$ (Landau gauge, MOM RS, pure gauge sector). Finite
lattice spacing effects were found to be small in the UV ($Q^{2}>1\,$GeV2),
and finite volume effects to be small everywhere, at least for the lattice
sizes used in current calculations. For instance, a three-fold increase in the
lattice spacing ($a/{\rm fm}=0.06\to 0.18$) increased $\alpha_{s}^{\rm gh}$ by
about 10% at its maximum, located on $Q^{2}\simeq 1\,$GeV2 (Fig. 4.3), leaving
the IR values unchanged; and decreased the $Q^{2}=0$ value of the gluon
propagator by roughly 10%.

The origins of the effects seem to be different for the gluon and ghost
propagators [503]. The issue with the gluon propagator appears to arise from a
non-optimal lattice spacing calibration and can thus be fixed by a careful
scale-setting procedure. That with ghost propagator, on the other hand, seems
to follow from the breaking of the O(4) symmetry (the group of rotations in 4
dimensions). Extrapolation methods exist to avoid the artifacts arising from
such breaking. Applying both scale-setting and extrapolation procedures should
thus yield an $\alpha_{s}^{\rm gh}$ largely free of the lattice discretization
artifacts [503].

This approach was applied to the gluon propagator in Ref. [505], which then
used it as an input, along with a LGT determination of the 3-gluon vertex, to
solve the DSE for the ghost-gluon vertex and obtain the ghost propagator,
$\alpha_{s}^{\rm gh}$ and $\alpha_{s}^{\rm 3g}$, all essentially free of DSE
truncation and LGT artifacts. Both couplings vanish in the IR, as expected for
LGT results on $\alpha_{s}^{\rm gh}$ since LGT realizes solely the decoupling
scenario. At any $Q^{2}$ value, $\alpha_{s}^{\rm gh}>\alpha_{s}^{\rm 3g}$. The
difference vanishes in the UV. These results were obtained in the Landau
gauge, MOM RS and pure gauge sector.

###  4.8 Refined Gribov–Zwanziger formalism

A somewhat different approach to strong-coupling QCD may be traced to Ref.
[582] and its focus on the manner by which one might properly fix the gauge in
QCD. The issue identified therein relates to the question of how one can
nonperturbatively ensure that the gauge fixing procedure selects only unique
gauge-inequivalent field configurations from the function space of gauge
fields. (No gauge transformation can connect two gauge inequivalent
configurations.) It is possible, even probable, that standard schemes –
implemented by introducing Faddeev-Popov ghost fields [95] – fail in this
regard by admitting (perhaps infinitely) many equivalent gauge fields that
satisfy the gauge fixing constraint; and this happens for every gauge-
inequivalent trajectory in the space of gauge fields. Along any one
trajectory, each of these fields is a member of the set of “Gribov copies” of
the desired field configuration. All members of this set are related, one to
another, by a gauge transformation; so, the action has the same value for each
and they all contribute the same amount to the functional integral. Moreover,
such a set is attached to each inequivalent gauge field trajectory, further
compounding the problem. Proper gauge fixing would select only one member from
each such set to be used in computing the QCD functional integral. Plainly,
the possibility of infinitely many gauge-equivalent, hence, action equivalent,
Gribov copies remaining after gauge fixing, and infinitely many sets of such
copies, forces one to question whether the functional integral has any meaning
at all [583].

Such gauge copies may contaminate and destabilize the formulation of a wide
range of gauges in which ghosts fields play a dynamical role, e.g., linear
covariant gauges and Coulomb gauge. They do not affect ghost-free
formulations, like Laplacian gauge [584], which was introduced for that very
purpose, the axial and LC gauges, see below Eq. (3.10), or the Weyl gauge, in
which the Coulomb potential is set to zero.

Following Ref. [582], the issue of Gribov copies and potential solutions were
further explored in Refs. [585, 586, 587, 588, 589]. The problems were
partially solved. An overview of the program is provided in Ref. [590] and
subsequent developments sketched in Ref. [12].

Such analyses are usually performed in Landau gauge, because, e.g., it is a
RGI linear covariant gauge and readily implemented in LGT. Prominent amongst
the schemes is modification of the QCD action, including a “horizon term”:

$\int d^{4}x\,\gamma\,g^{2}f^{abc}A_{\mu}^{b}(x)[{\mathpzc
M}^{-1}]^{ad}(x,x)f^{dec}A_{\mu}^{e}(x)\,,$ (4.36)

where

${\mathpzc
M}^{ab}(x,y)=[-\partial^{2}\delta^{ab}+\partial_{\mu}f^{abc}A_{\mu}^{c}(x)]\delta^{4}(x-y)$
(4.37)

is the Landau gauge Faddeev-Popov operator. The scale $\gamma$ controls the
strength of the horizon constraint and is fixed by the spacetime dimension and
number of gluon fields. (Issues concerning BRST (a-)symmetry,
renormalisability, etc., of such an action are discussed in Refs. [591, 590].)

The original program [582, 585] relies on an IR divergent ghost dressing
function and a gluon propagator that vanishes as $Q^{2}\to 0$. As discussed
above, such behavior is not realized in QCD: the QCD ghost propagator remains
that of a standard massless excitation and the gluon is characterised by a
dynamically-generated IR mass-scale, ensuring that the gluon propagator is
nonzero at $Q^{2}=0$ [7, 8, 11]. Thus, the original scheme is not viable.

A modification of the approach in Refs. [582, 585, 586, 587, 588, 589] is
canvassed in Refs. [592, 593, 594, 595, 596, 597], delivering a framework in
harmony with modern continuum and LGT Schwinger function results. (Other
procedures are described elsewhere [598, 599, 600, 601, 602].) It admits the
possibility that the ghost fields used to fully fix the gauge develop a
nonzero dimension-two condensate, whose presence yields a Landau-gauge 2-point
Schwinger function for the gluon with the following form [592, 593]:

$D_{\mu\nu}^{ab}(Q)=\delta^{bc}\left[\delta_{\mu\nu}-\frac{Q_{\mu}Q_{\nu}}{Q^{2}}\right]\frac{Q^{2}+M^{2}}{Q^{4}+Q^{2}\mathpzc{m}^{2}+\lambda^{4}}\,,$
(4.38)

where $\lambda^{4}=6\gamma g^{2}-{\mathpzc a}^{2}M^{2}$, with
$m_{\gamma}^{4}=\gamma g^{2}$ expressing the Gribov horizon scale;
$\mathpzc{m}^{2}=M^{2}-{\mathpzc a}^{2}$, with ${\mathpzc a}^{2}\propto\langle
A_{\mu}^{a}A_{\mu}^{a}\rangle$, a dimension-two gluon condensate, and $M^{2}$
related to the ghost-field condensate, both computed within a hadronic medium
[465]. Using this scheme, which supports a nonzero value for the gluon
propagator in the far-IR, one can obtain fair agreement with LGT results for
the gluon two-point function [594, 603, 595, 604]. In fact, at IR momenta
($Q^{2}\lesssim 1\,$GeV2), a good fit to high-statistics LGT results can be
obtained using Eq. (4.38). The refined Gribov-Zwanziger approach has been
employed [605] to study, at one-loop, the ghost-gluon interaction vertex using
a fixed perturbative $\alpha_{s}$ value (chosen to lie between 0.15 and 0.42).

Consistency between Eq. (4.38) and robust LGT results [530, 606] for the
Landau-gauge gluon two-point Schwinger function is also explored in Ref. [41].
That analysis introduced a novel constraint, insisting that the values
obtained for the parameters in Eq. (4.38) should ensure that
$D_{\mu\nu}^{ab}(Q)$ expresses a minimal level of consistency with parton-like
behaviour on the UV domain. This condition leads to a constraint relation
between the two principal mass scales in Eq. (4.38): $m_{\gamma}$, the Gribov
horizon scale; and $m_{g}=\lambda^{2}/M$, the IR gluon mass.

In general, there are three possible outcomes, which were all canvassed in
Ref. [41].

  1. (i)

If fitting LGT results using Eq. (4.38), subject to UV partonic constraints,
were to yield $m_{\gamma}\gg m_{g}$, then the Gribov gauge-fixing horizon is
affecting UV modes of the gluon. However, the validity of standard
perturbation theory shows this is not the case. Consequently, $m_{\gamma}\gg
m_{g}$ is unrealistic: had it been favoured by LGT results, then it would have
been necessary to discard either or both those results and the Gribov gauge-
field horizon condition.

  2. (ii)

The converse, $m_{\gamma}\ll m_{g}$, would indicate that the gluon mass alone
is sufficient to screen IR gluon modes, in which case the Gribov gauge-fixing
ambiguity could have no physical impact and any horizon term is redundant.

  3. (iii)

The detailed numerical analysis in Ref. [41] indicates that QCD occupies a
middle ground: $m_{\gamma}\approx m_{g}$, with each of a size ($\sim
0.5\,$GeV) that one associates with EHM in the gauge-sector. In this scenario,
the gluon mass and horizon scale play a practically identical role in
screening long-wavelength gluon modes, thereby dynamically eliminating all
gauge-fixing ambiguities. It was further found that, together, they set a
confinement scale of roughly 1 fm. In consequence of these findings, it was
conjectured that the gluon mass and Gribov gauge-field horizon are equivalent
emergent phenomena.

A stronger conclusion requires unquenched LGT results with better sensitivity
to IR momenta. Meanwhile, Ref. [41] worked with the quenched LGT results [530]
and found that a high-quality fit is achievable with $m_{\gamma}=m_{g}$. In
this realization of Scenario (iii), the horizon scale, ${\mathpzc a}^{2}$ and
ghost-field condensate can all be absorbed into a single running gluon mass,
$m_{g}(k^{2})$, whose dynamical appearance alone is sufficient to eliminate
gauge-fixing ambiguities and complete the definition of QCD. This leads one
back to the PI charge discussed in Sec. 4.6.4.

###  4.9 Curci-Ferrari model

The Curci–Ferrari model [607] is a massive Yang-Mills theory in which the mass
force carrier (here the gluon) is an additional parameter of the theory. A
mass term $\frac{m_{g}^{2}}{2}A_{\mu}^{a}A^{\mu}_{a}$ is added to the QCD
Lagrangian after gauge fixing, viz. its Faddeev-Popov expression, Eq. (3.21).
This yields a model that contains longitudinal gluon modes, which persist in
the UV and spoil renormalizability. The model can be made to recover a Landau-
gauge Yang-Mills theory when an appropriate zero mass gluon limit is
engineered.

The rationale for employing the model to explore the IR-behavior of
$\alpha_{s}$ and associated Schwinger functions [608], is the hypothesis that
the Faddeev-Popov quantization method becomes inapplicable in the IR and
should be modified to account for the presence of Gribov copies. The gluon
mass is then interpreted as an effective mass arising from the breaking of
BRST symmetry owing to the Gribov ambiguity [582]. In fact, the gluon
propagator obtained in the model is the same as that arising in Landau-gauge
pQCD, but for a mass term that suppresses the Landau pole thereby,
potentially, allowing the use of perturbation theory. Ref. [609] reviews the
application of the Curci–Ferrari model to QCD and Yang-Mills theories.
(Notably, the foundations of this approach are undermined by the conclusions
suggested in Ref. [41] – discussed in the preceding section, and the existence
of a Schwinger mechanism in QCD [385, 386].)

The scheme has been used to model gluon, quark and ghost propagators, as well
as the quark–gluon vertex. A one-loop perturbative treatment without dynamical
quarks results in a ghost-gluon coupling, $\alpha_{s}^{\rm gh}$, that vanishes
in the IR and remains small enough to validate a posteriori the use of
perturbation theory – see Ref. [12, Sec. 4.15]. Although the formalism is
perturbative, nonperturbative phenomena are nevertheless modelled by inclusion
of the gluon mass parameter. Generally, the Schwinger functions obtained agree
with CSM and LGT predictions in the decoupling scenario, although a scaling
scenario can be engineered [610].

Calculations of $\alpha_{s}^{\rm qg}$, $\alpha_{s}^{\rm 3g}$ and
$\alpha_{s}^{\rm 4g}$ have been carried out in the Curci-Ferrari model [611,
612, 613, 614, 615, 560]. The results agree with those obtained using
continuum and lattice Schwinger function methods – see Fig. 4.3 (left panel).
However, the agreement is weakened by the inclusion of dynamical quarks. This
might be explained by the observation that although $\alpha_{s}^{\rm gh}$,
$\alpha_{s}^{\rm 3g}$ and $\alpha_{s}^{\rm 4g}$ remain small (in Landau gauge)
at all $Q^{2}$ – see Fig. 4.3, the quark-gluon coupling $\alpha_{s}^{\rm qg}$
is found to be two- or even three-times larger in the higher energy part of
the IR-domain when compared with $\alpha_{s}^{\rm gh}$, $\alpha_{s}^{\rm 3g}$,
respectively. A $\nicefrac{{1}}{{N_{c}}}$ expansion has been used to address
this issue. Two-loop results are also available for propagators, and for
$\alpha_{s}^{\rm qg}$ and $\alpha_{s}^{\rm 3g}$ [612, 613, 614, 615].

There is an appealing simplicity in the fact that fair agreement with the
results of continuum and lattice Schwinger function methods in QCD is obtained
at 1-loop in the Curci-Ferrari model. This is largely a consequence of
$\alpha_{s}^{\rm gh}$, $\alpha_{s}^{\rm 3g}$ and $\alpha_{s}^{\rm 4g}$
remaining small in the model. Notably, 1-loop agreement is not new. Early DSE
calculations, featuring an effective gluon mass, used a 1-loop expression of
$\alpha_{s}$ and the propagators – see, e.g., the discussion of Eq. (4.29) –
to obtain results that were qualitatively and semi-quantitatively confirmed by
later LGT calculations. Similarly, the effective charge approach (Sec. 4.3)
defines the coupling using the 1-loop approximation to one or another given
observable. This indicates that for some quantities, at least, nonperturbative
effects can be encapsulated in just a few well-chosen parameters, e.g., the
effective gluon mass for $\alpha_{s}$ or the gluon propagator.

###  4.10 Analytic coupling

As discussed in connection with the effective charge approach, Sec. 4.3,
defining $\alpha_{s}$ so that it remains an observable is advantageous in both
the UV and IR domains. A different method to preserve this “observability”
attribute of $\alpha_{s}$ is to demand its analyticity in the complex
$Q^{2}$-plane,888 Except on the real timelike axis where singularities may
exist, corresponding to the production of on-shell particles. that is,
physically speaking, for $\alpha_{s}$ to be causal. Such a requirement removes
the Landau pole that appears at $Q\simeq\Lambda_{s}$ in the perturbative
treatment: the pole violates causality because it would represent the
production of causality-violating tachyons – see the last four paragraphs of
Sec. 2.

This approach to solving the Landau pole problem has a long history. It was
introduced in 1955 [616], applied soon thereafter to study the QED Landau pole
[617, 618, 619], and then applied to QCD [620]. Since then, the approach has
been pursued in various guises by several authors – See Refs. [12, 17] for
overviews.

The procedure may be formulated [620] using a spectral density, $\rho$, whose
form is usually taken as that expected from standard pQCD. Then, the theory of
dispersion relations is employed to mold an analytical coupling, $\alpha_{\rm
an}$, free of the Landau pole. This method is also, therefore, called the
dispersive approach.

Specifically, the Källén–Lehman spectral relation [621, 622] is employed:

$\alpha_{\rm
an}(Q^{2})=\frac{1}{\pi}\int_{0}^{\infty}d\nu\frac{\rho\left(\nu\right)}{\nu+Q^{2}},$
(4.39)

with $\nu$ the integration variable. Here, $\rho(\nu)$ is the density in
momentum space of a noninteracting gluon propagator, with effective mass
$\sqrt{\nu}$, that mediates the interaction. Note that the original
Källén–Lehman identity relates the full, viz. interacting, propagator, rather
than $\alpha_{\rm an}$, to the weighted sum of massive free-propagators. The
integration over $\nu$ means that $\alpha_{\rm an}$ is explicitly independent
of the mass term appearing in the propagator. Further, $\nu$ identifies as
$Q^{2}$, rather than a squared mass, in the spectral function – see Eq. (4.40)
below. (A variation that explicitly includes a gluon mass has been developed
[623, 12].) The form of $\rho(\nu)$ is chosen so that $\alpha_{\rm an}$ has
the desired UV and IR properties.

To find that $\rho(\nu)$ which would yield $\alpha_{\rm an}$ from Eq. (4.39),
the Schwarz reflection principle [624] is used. This entails that
$\rho\left(\nu\right)=\imaginary(\alpha_{\rm an}(-\nu+i\epsilon)\big{)}$
[625]. The standard choice is [620]:

$\rho\left(\nu\right)=\imaginary(\alpha_{s}(-\nu+i\epsilon)\big{)},$ (4.40)

with $\alpha_{s}$ the usual pQCD coupling. Inserting Eq. (4.40) into Eq.
(4.39) yields a Kramers–Krönig type of relation [626, 627] for $\alpha_{\rm
an}$, which ensures that it is analytical and causal.

The procedure described above is applicable at any order in the $\beta$
series, with the same features as the standard perturbative solutions for
$\alpha^{(l)}_{s}$ at $l+1$-loop – see Secs. 3.2-3.3); namely, that analytical
solutions exist for the 1-loop $\beta_{0}$ and 2-loop $\beta_{1}$, but the
higher order solutions are either numerical or approximately analytic. At
$\beta_{0}$-level, the procedure gives:

$\displaystyle\rho_{0}\left(\nu\right)$
$\displaystyle=\frac{\pi}{\beta_{0}[\ln^{2}(\nu/\Lambda_{s}^{2})+\pi^{2}]},$
(4.41a) $\displaystyle\alpha_{\rm an}^{(0)}\left(Q^{2}\right)$
$\displaystyle=\frac{4\pi}{\beta_{0}}\left(\frac{1}{\mbox{ln}(Q^{2}/\Lambda_{s}^{2})}+\frac{\Lambda_{s}^{2}}{\Lambda_{s}^{2}-Q^{2}}\right).$
(4.41b)

This coupling agrees with the UV expectation of pQCD and is free of the Landau
pole in the IR. In fact, its value at the would-have-been Landau pole is
$\alpha_{\rm an}^{(0)}(\Lambda_{s}^{2})/\pi=2/\beta_{0}\approx 0.2$. As
$Q^{2}\to 0$, all dependence on $\Lambda_{s}$ disappears and $\alpha_{\rm
an}(0)/\pi=4/\beta_{0}\simeq 0.41$, a result holding at all orders [625].
Albeit finite, $\alpha_{\rm an}(Q^{2})$ does not freeze in the IR, but
increases monotonically until reaching the $4\pi/\beta_{0}$ value.

The last term in Eq. (4.41b) does not depend on the higher-order terms in the
$\beta$ series and does not involve any $\ln(Q^{2}/\Lambda_{s}^{2})$ term.
Thus, it is a nonperturbative contribution, which expands at high $Q^{2}$ as
$(\Lambda_{s}/Q)^{2n}$, i.e., may be identified with the nonperturbative
higher-twist terms of the OPE formalism.

The couplings calculated at $l$-loop order, $\alpha_{\rm an}^{(l)}(Q^{2})$,
are numerically close to each other, viz. their values do not much depend on
the loop order. This was to be expected because all $\alpha_{\rm an}^{(l)}$
share the same IR and UV limits.

The small UV and IR values of $\alpha_{\rm an}^{(0)}$, and the fact that
$\alpha_{\rm an}^{(l)}\simeq\alpha_{\rm an}^{(0)}$ suggests that the
perturbative series associated with a quantity expanded in $\alpha_{\rm
an}^{(l)}$ should be valid on both the UV and IR domains, providing that all
nonperturbative effects are encompassed by $\alpha_{\rm an}$. To consistently
include the $\alpha^{(l)}_{\rm an}$ at each order $n$ in the pQCD series, an
$n$-th order spectral function is defined as
$\rho_{n}(\nu)=\imaginary(\alpha_{s}^{n}(-\nu)\big{)}$. This provides the
$n$-th order coupling,

$\alpha^{(l)}_{{\rm
an},(n)}=\frac{1}{\pi}\int_{0}^{\infty}d\nu\frac{\rho_{n}\left(\nu\right)}{\nu+Q^{2}},$
(4.42)

and a standard pQCD series,
$1+\Sigma_{n}a_{n}\big{[}\alpha_{s}^{(l)}(Q^{2})\big{]}^{n}$, is re-expressed
as $1+\Sigma_{n}a_{n}\alpha^{(l)}_{{\rm an},(n)}(Q^{2})$. Notice that the
series is unusual because it lacks a unique expansion parameter $\alpha_{s}$.
Instead, it uses parameters $\alpha_{{\rm an},(n)}$ in which the power of $n$
is implicit rather than explicit. As a consequence, $\alpha_{{\rm
an},(m+n)}\neq\alpha_{{\rm an},(m)}^{n}\neq\alpha_{{\rm an},(m)}\alpha_{{\rm
an},(n)}$; and, in particular, $\alpha_{{\rm an},(n)}\neq\alpha_{{\rm
an}}^{n}$. This approach is called analytical perturbation theory (APT) [620].
Its extension to non-integer values of $n$ is known as fractional APT (FAPT)
[628], and is motivated by the presence in QCD of operators with nonzero
anomalous dimensions.

In the past quinquennium, several authors have pursued the
analytical/dispersive approach. A method to extend the FAPT perturbative
series to higher orders using an $1/\ln(\nicefrac{{Q^{2}}}{{\Lambda_{s}}})$
expansion is proposed in Ref. [629] and tested on the Bjorken sum rule, Eq.
(4.2). The FAPT approach is used elsewhere [630, 631, 632] to compute
$\alpha_{s}^{\rm gh}$ for $N_{f}=3$ in the MiniMOM RS and Landau gauge on both
the IR and UV domains. The aim was to provide an analytical form for
$\alpha_{s}^{\rm gh}$ that qualitatively reproduces the decoupling solution
found numerically using LGT. To that end, $\rho$ was modeled, with ultimately
two free parameters adjusted so that the correct $\tau$-decay phenomenology,
at the edge of the pQCD domain, is recovered and that the maximum of
$\alpha_{s}^{\rm gh}$ in the IR matches LGT findings. In Ref. [633], a SU(3),
pure-gauge, $\beta$-function, inspired by the Novikov-Shifman-Vainshtein-
Zakharov $\beta$-function of supersymmetric gauge theories [634, 635, 636], is
treated with the analytic procedure to eliminate the Landau pole. The
resulting analytically-improved $\beta$-function provides an IR-safe
$\alpha_{s}$ that freezes to the IR-fixed point $\alpha_{s}=0.31$. Finally, in
Ref. [637], $\alpha_{\rm an}$ is studied at the 2-loop level, with an exact
analytical solution provided, albeit in an integral form that must be either
numerically estimated or approximated.

###  4.11 Screened massive expansion

The final approach to determining $\alpha_{s}$ in the IR that has seen
developments in the past quinquennium is the screened massive expansion
method. Its rationale is the smallness of vertex-based $\alpha_{s}(Q^{2})$
forms in the decoupling scenario – Fig. 4.3 (left panel), where
$\alpha_{s}(Q^{2})\xrightarrow[Q^{2}\to 0]{}0$, irrespective of which vertex
coupling is considered. This intimates that QCD in the IR might be treatable
perturbatively, with nonperturbative effects possibly encapsulated in
effective mass parameters [638, 639, 640, 641, 642]. Working from this
position, Refs. [638, 639, 640, 641, 642] propose a double perturbative
expansion for chiral QCD (massless current quarks) in the IR which, in
particular, enables computation of the ghost–gluon vertex coupling
$\alpha^{\rm gh}_{s}(Q^{2})$, Eq. (4.21), as well as propagators. The coupling
is first assumed to be small enough to permit the double perturbative
expansion and then calculated. The result is a numerically small coupling,
consistent with the initial assumption.

The method is discussed generally for a SU(N) Yang-Mills theory with massless
fermions in Refs. [639, 640]. Starting from the Faddeev–Popov Lagrangian of
that theory, trial massive propagators are used for the internal gluon lines
of the Feynman diagrams. While the mass parameter assigned to the propagators
eventually cancels in the renormalization process, distinct dynamical
effective masses for gluons and constituent quarks emerge from the loop
diagrams. The disappearance of the initial mass parameters makes the method
compatible with BRST symmetry, in contrast to other massive extensions of
SU(N).999Methods displaying emerging dynamical masses, e.g., DSE studies, also
respect the BRST symmetry, as noted above. The coupling and propagators are
obtained in Landau gauge using the $\overline{\rm MS}$ RS. They are consistent
with other nonperturbative methods, including LGT; and the smallness of
$\alpha^{\rm gh}_{s}(Q^{2})$ in the deep-IR means $\alpha^{\rm gh}_{s}$ is
dominated by its 1-loop solution. In particular, it consistently displays the
expected decoupling solution. $\alpha^{\rm gh}_{s}(Q^{2})$ peaks at
$Q^{2}\simeq 0.4$ GeV2, with values of 1.2–1.5, but this is a priori not
inconsistent with a perturbative expansion since pQCD series are developed in
the ratio $\alpha_{s}/\pi$.

The screened massive expansion can also be engineered [643] to produce
$\alpha_{s}^{\rm gh}$ in line with a scaling scenario – Fig. 4.3 (right
panel). To achieve such an outcome, one is required to employ a “screened MOM”
RS, i.e., force the renormalized gluon 2-point function to match a massive
free-propagator instead of the standard massless propagator at the momentum
subtraction point. The usual MOM RS always generates the decoupling scenario.

###  4.12 Other approaches

Other methods that have been used to model $\alpha_{s}$ include the Stochastic
Quantization formalism, where the role of ghosts is played by longitudinal
gluons; the background field method; the Banks-Zaks approach; and instanton
vacuum models. However, so far as we are aware, no new developments that are
pertinent to this review have been made with these methods during the past
quinquennium. Their current status, therefore, remains as described elsewhere
[12]. Additionally, the Bern-Kosower formalism [644] has recently been used to
compute SU(3) vertex functions, but without yet providing the corresponding
couplings [645].

##  Chapter 5 Summary and Perspective

The coupling $\alpha_{s}(Q^{2})$ is central to QCD and to the Standard Model
of particle physics. Ever since the dawn of the QCD era, vigorous efforts have