[1]Mauricio Narciso Ferreira

[1]Department of Theoretical Physics and IFIC, University of Valencia and
CSIC, E-46100, Valencia, Spain

# Evidence of the Schwinger mechanism from lattice QCD

<EMAIL_ADDRESS>*

###### Abstract

In quantum chromodynamics (QCD), gluons acquire a mass scale through the
action of the Schwinger mechanism. This mass emerges as a result of the
dynamical formation of massless bound-states of gluons which manifest as
longitudinally coupled poles in the vertices. In this contribution, we show
how the presence of these poles can be determined from lattice QCD results for
the propagators and vertices. The crucial observation that allows this
determination is that the Schwinger mechanism poles induce modifications,
called “displacements”, to the Ward identities (WIs) relating two- and three-
point functions. Importantly, the displacement functions correspond precisely
to the Bethe-Salpeter amplitudes of the massless bound-states. We apply this
idea to the case of the three-gluon vertex in pure Yang-Mills SU(3). Using
lattice results in the corresponding WI, we find an unequivocal displacement
and show that it is consistent with the prediction based on the Bethe-Salpeter
equation.

###### keywords:

Schwinger mechanism, gluon mass generation, lattice QCD, continuum Schwinger
function methods, emergence of hadron mass, non-perturbative quantum field
theory, quantum chromodynamics

##  1 Introduction

One of the most celebrated features of quantum chromodynamics (QCD) [1] is the
emergent hadron mass (EHM) [2, 3, 4, 5, 6, 7, 8, 9], i.e., the nonperturbative
generation of massive hadrons out of fundamental fields, gluons and quarks,
that are massless at the level of the Lagrangian. In this context, crucial
signals of EHM have been revealed in the infrared behavior of the QCD
propagators and vertices through the synergy between gauge-fixed lattice
simulations [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44,
45, 46, 47, 48, 49, 50, 51, 52, 53, 54] and continuum Schwinger function
methods (CSM) [55, 3, 56, 57, 58, 59, 8, 9], such as Schwinger–Dyson equations
(SDEs) [60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71] and the functional
renormalization group [72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82]. In
particular, it is now established that the gluon propagator saturates to a
finite value at the origin [17, 18, 19, 24, 25, 27, 22, 23, 26, 28, 31, 30,
32, 16, 29, 33, 43, 47, 83], which is an unequivocal signal of the dynamical
generation of a gluon mass scale proposed decades ago [84, 85, 86, 87, 88,
89].

Gluon mass generation has far-reaching implications. For instance, it prevents
QCD from developing a Landau pole, causes the effective decoupling of gluonic
modes beyond a maximum gluon wavelength [90], and suppresses Gribov copies
[91, 66, 92]. Moreover, it sets a scale for many other dimensionful
quantities, such as glueball masses [93, 94, 95, 96, 97]. The importance of
gluon mass generation has thus prompted an intense effort to elucidate the
mechanism behind its dynamical origin.

The notion that gauge bosons can acquire masses dynamically, without violating
gauge symmetry, originated with Schwinger in the sixties [98, 99] and has been
studied in various contexts since [100, 101, 102, 103, 104, 105, 86, 106, 107,
108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 7, 118]. In the particular
case of QCD, the activation of the Schwinger mechanism for gluon mass
generation hinges on the dynamical formation of massless, color-carrying,
bound-states of gluons [108, 110, 68, 113, 114, 116, 117, 7, 118]. Such
massless bound-states appear as poles in the interaction vertices, which, in
turn, lead to the saturation of the propagator.

A difficulty that arises in the quest to confirm the occurrence of the
Schwinger mechanism in QCD is that lattice simulations can only compute
transverse projections of the interaction vertices [12, 13, 20, 34, 35, 37,
47, 50, 52]. However, the Schwinger mechanism poles in the vertices are
strictly longitudinally coupled [107, 110, 68, 113, 114, 116, 117, 7, 118],
and thus cannot be directly seen in lattice results for the vertex functions.

Recently [116, 117], a method for confirming the existence of Schwinger
mechanism poles from lattice QCD has been put forth. This method is based on
the observation that the massless vertex poles induce crucial modifications to
the Ward identities (WIs) relating propagators and vertices. These
modifications, called “displacements”, consist of the appearance of the Bethe-
Salpeter (BS) amplitudes of the bound-states in the identities [68, 113, 7,
118], in addition to the propagators and pole-free vertex parts that are
present with or without the Schwinger mechanism. Hence, since the propagators
and pole-free vertex parts are accessible to lattice simulations, the
combination of lattice results for these quantities into the WIs allows us to
determine the BS amplitude. Then, if the latter is found to be nonzero, the
method allows us to confirm the occurrence of the Schwinger mechanism.

In the present contribution, we provide in Section 2 a brief overview of the
Schwinger mechanism and its realization in QCD through the formation of
massless poles in the vertices. For simplicity, we neglect the effect of
dynamical quarks, focusing instead on the pure Yang-Mills SU(3). Next, in
Section 3 we illustrate through the case of an Abelian vertex how such
massless poles displace the usual WIs. There we also present the WI
displacement for the three-gluon vertex, which will allow us to determine the
BS amplitude of the three-gluon vertex massless pole from lattice ingredients.
Then, in Section 4, we discuss the determination of the function ${\cal
W}(r)$, which is a special derivative of the ghost-gluon kernel and appears in
the three-gluon vertex WI displacement. In Section 5 we use the results of the
previous sections to determine the BS amplitude, analyzing the statistical
significance of the result and comparing it to the theoretical prediction
obtained directly from the Bethe-Salpeter equation (BSE). Finally, in Section
6 we present our conclusions.

##  2 Overview of the Schwinger mechanism

In the Landau gauge, which will be used throughout this work, the gluon
propagator can be written as
$\Delta^{ab}_{\mu\nu}(q)=-i\delta^{ab}\Delta(q)P_{\mu\nu}(q)$, where
$P_{\mu\nu}(q):=g_{\mu\nu}-q_{\mu}q_{\nu}/q^{2}$ is the transverse projector.

Figure 1: Diagrammatic representation of the gluon self-energy,
$\Pi_{\mu\nu}(q)$. In all our diagrams, wavy and dashed lines represent gluon
and ghost fields, respectively, while circles denote dressed propagators and
vertices. Feynman rules appropriate to our conventions are given in Appendix B
of [64].

The gluon propagator is determined in terms of the self-energy,
$\Pi_{\mu\nu}(q)$, given diagrammatically in Fig. 1. Gauge symmetry requires
that $\Pi_{\mu\nu}(q)=q^{2}{\bf\Pi}(q)P_{\mu\nu}(q)$, where ${\bf\Pi}(q)$
defines the dimensionless vacuum polarization. Then,

$\Delta^{-1}({q})=q^{2}[1+i{\bf\Pi}(q)]\,.$ (1)

The emergence of a gluon mass is signaled by the saturation of $\Delta(0)$ to
a finite value, illustrated in Fig. 2 with recent lattice data from Ref. [51].

Figure 2:  Lattice data (points) from Ref. [51] for the gluon propagator,
compared to a physically motivated fit given by Eq. (C11) of [116] (blue
solid).

The Schwinger mechanism is based on the observation that if the vacuum
polarization acquires a pole at zero momentum transfer $\Delta(0)$ will
saturate, even though no gluon mass term appears in the Lagrangian. Indeed, in
the presence of such a pole Eq. (1) has the limit

$\lim_{q\to 0}i{\bf\Pi}(q)=m^{2}/q^{2}\,\,\Longrightarrow\,\,\lim_{q\to
0}\,\Delta^{-1}(q)=\lim_{q\to
0}\,(q^{2}+m^{2})\,\,\Longrightarrow\,\,\Delta^{-1}(0)=m^{2}\,,$ (2)

written here in Euclidean space.

The mechanism leading to the emergence of a pole in ${\bf\Pi}(0)$ can vary for
different theories, see e.g., [100, 101]. For Yang-Mills theories, an elegant
nonperturbative mechanism has been put forward which is based on the formation
of a special kind of bound-state of gluons [103, 104, 86, 106, 107, 108, 119,
109, 110, 111, 68, 113, 114, 116, 7, 118]. This mechanism can be outlined
through the following sequence of ideas:

  1. (i)

First, it is assumed that the gluon self-interaction is strong enough to form
massless colored bound-states. These bound-states can be shown to not appear
in $S$-matrix elements, such that no new massless particle is introduced in
the spectrum of the theory [100, 101, 103, 104, 108, 7]. Nevertheless, the
bound-state propagator, $i/q^{2}$, induces a pole in a certain gluon-gluon
scattering kernel, illustrated diagrammatically in Fig. 3.

  2. (ii)

Consequently, the fundamental vertices of the theory acquire poles at zero
momentum transfer. This can be clearly seen in the case of the three-gluon
vertex by analyzing the SDE that governs its momentum evolution, shown in Fig.
4. Indeed, in that equation appears the aforementioned gluon-gluon scattering
kernel, which induces a pole in the vertex.

  3. (iii)

Finally, the massless poles in the vertices make their way naturally into the
vacuum polarization, thus activating the Schwinger mechanism.

Figure 3: Diagrammatic representation of the emergence of massless poles in
the gluon-gluon scattering kernel. This kernel is one-particle irreducible
with respect to vertical cuts. The first line shows some of the infinitely
many diagrams contributing to the kernel. This tower of interactions is
assumed to lead to the formation of a massless bound-state, with propagator
$i/q^{2}$, such that the kernel acquires a pole-free and a pole contribution.
Figure 4: Diagrammatic representation of the Schwinger-Dyson equation (SDE)
for the three-gluon vertex, where appears the gluon-gluon scattering kernel of
Fig. 3. The bracket defines the amplitude, $I_{\alpha}(q)$, for a gluon to
transition to a massless bound-state. Note that, by Bose symmetry, there must
also exist poles in the $r$ and $p$ channels, which are not shown. The
ellipsis denotes additional diagrams that are omitted for simplicity.

From now on we will focus on the three-gluon vertex, whose associated massless
bound-state pole is expected to be the leading contributor to gluon mass
generation [107, 120, 121, 116, 117]. We denote this vertex by
${\rm{I}}\\!\Gamma^{amn}_{\alpha\mu\nu}(q,r,p)=gf^{amn}{\rm{I}}\\!\Gamma_{\alpha\mu\nu}(q,r,p)$,
where $g$ is the gauge coupling and $f^{amn}$ are the SU(3) structure
constants.

The emergence of massless bound-state transitions in the three-gluon vertex
prompts us to split ${\rm{I}}\\!\Gamma_{\alpha\mu\nu}(q,r,p)$ into a pole-free
part, $\Gamma_{\alpha\mu\nu}(q,r,p)$, and a pole contribution,
$V_{\alpha\mu\nu}(q,r,p)$, i.e.,

${\rm{I}}\\!\Gamma_{\alpha\mu\nu}(q,r,p)=\Gamma_{\alpha\mu\nu}(q,r,p)+V_{\alpha\mu\nu}(q,r,p)\,.$
(3)

The dynamical origin of $V_{\alpha\mu\nu}(q,r,p)$ in the formation of massless
poles imposes a crucial constraint on its Lorentz structures. Specifically, by
Lorentz symmetry, the amplitude $I_{\alpha}(q)$ for a gluon to transition to a
massless bound-state, defined by the bracket in Fig. 4, must be of the form
$I_{\alpha}(q)=q_{\alpha}I(q)$, for some scalar $I(q)$. Hence, the $q=0$ pole
in the vertex must be associated with tensor structures longitudinal to the
leg carrying momentum $q$. Similar considerations show that the poles at $r=0$
and $p=0$ must be associated with tensors longitudinal to $r_{\mu}$ and
$p_{\nu}$, respectively. Therefore, the massless poles that trigger the
Schwinger mechanism must be strictly longitudinally coupled, such that [108,
110]

${P}_{\alpha^{\prime}}^{\alpha}(q){P}_{\mu^{\prime}}^{\mu}(r){P}_{\nu^{\prime}}^{\nu}(p)V_{\alpha\mu\nu}(q,r,p)=0\,.$
(4)

From Eq. (4), together with Bose symmetry of the vertex, we see that the pole
part $V_{\alpha\mu\nu}(q,r,p)$ can be written as

$V_{\alpha\mu\nu}(q,r,p)=\left(\frac{q_{\alpha}}{q^{2}}\right)C_{\mu\nu}(q,r,p)+\left(\frac{r_{\mu}}{r^{2}}\right)C_{\nu\alpha}(r,p,q)+\left(\frac{p_{\nu}}{p^{2}}\right)C_{\alpha\mu}(p,q,r)\,,$
(5)

with

$C_{\mu\nu}(q,r,p)=C_{1}g_{\mu\nu}+C_{2}r_{\mu}r_{\nu}+C_{3}p_{\mu}p_{\nu}+C_{4}r_{\mu}p_{\nu}+C_{5}p_{\mu}r_{\nu}\,,$
(6)

where $C_{i}\equiv C_{i}(q,r,p)$. Due to the transversality of the Landau
gauge gluon propagator, the form factors $C_{2,3,4}$ decouple in most
calculations. Moreover, in this gauge, the form factor $C_{5}$ can be shown to
not contribute to the gluon mass [108, 110, 116]. Hence, we will restrict our
discussion to $C_{1}$.

At this point, we emphasize that the massless bound-state that triggers the
Schwinger mechanism in QCD is not put in by hand, but emerges dynamically.
Indeed, as with any other bound-state, its formation is governed by a BSE
[108, 110, 121, 116, 7, 118], represented diagrammatically in the left panel
of Fig. 5.

The function that plays the role of BS amplitude in the BSE of Fig. 5 is
denoted by ${\mathbb{C}}(r)$ and is related to the form factor $C_{1}$ defined
in Eqs. (5) and (6). Specifically, note that Bose symmetry requires
$C_{1}(q,r,p)=-C_{1}(q,p,r)$, such that

$C_{1}(0,r,-r)=0\,.$ (7)

Then, in the vicinity of the $q=0$ pole,

$C_{1}(q,r,p)=2(q\cdot
r){\mathbb{C}}(r)\,,\qquad{\mathbb{C}}(r):=\left.\frac{\partial
C_{1}(q,r,p)}{\partial p^{2}}\right|_{q=0}\,.$ (8)

Figure 5:  Left: Bethe-Salpeter equation (BSE) governing the formation of the
massless bound-state that triggers the Schwinger mechanism. The ellipsis
denotes higher-order corrections to the gluon-gluon scattering kernel and
coupling to poles in vertices other than the three-gluon [108, 110, 121, 116,
7, 118]. Right: Bethe-Salpter (BS) amplitude, ${\mathbb{C}}(r)$, obtained in
[116], using the BSE of the left panel.

The vital first test of the Schwinger mechanism is the existence of nontrivial
solutions for ${\mathbb{C}}(r)$. Indeed, previous studies have shown that the
BSE of Fig. 5 admits nontrivial solutions, using lattice inputs for the
propagator and three-gluon vertex therein [108, 110, 121, 116, 7, 118]. The
most up-to-date solution was obtained in Ref. [116] and is shown in the right
panel of Fig. 5. For later convenience, this solution is denoted by
${\mathbb{C}}_{\star}(r)$, to distinguish it from the ${\mathbb{C}}(r)$ that
will be determined in Section 5 from the WI displacement. Note that, since the
BSE of Fig. 5 is a homogeneous equation, it only determines
${\mathbb{C}}_{\star}(r)$ up to a multiplicative constant; the particular
solution shown there has its scale set by matching it to the result obtained
in Section 5, as explained therein.

To conclude this section, we remark that coupling the poles of the three-gluon
and ghost-gluon vertices does not significantly affect the solution shown Fig.
5. Moreover, the pole associated with the ghost-gluon vertex is subleading in
comparison to ${\mathbb{C}}(r)$ [121, 116].

##  3 Ward identity displacement

It follows from the longitudinality property of $V_{\alpha\mu\nu}(q,r,p)$,
i.e., from Eq. (4), that the Schwinger mechanism massless poles cannot be
computed on the lattice by direct simulation of the three-gluon vertex.
Indeed, lattice QCD can only determine the transverse projections of the
vertex functions. In particular, for the three-gluon vertex, lattice
observables involve the projection
${\overline{\Gamma}}_{\alpha\mu\nu}(q,r,p)$, defined by [13, 20, 34, 35, 37,
47, 50, 52]

${\overline{\Gamma}}_{\alpha\mu\nu}(q,r,p):=P^{\alpha^{\prime}}_{\alpha}(q)P^{\mu^{\prime}}_{\mu}(r)P^{\nu^{\prime}}_{\nu}(p){\rm{I}}\\!\Gamma_{\alpha^{\prime}\mu^{\prime}\nu^{\prime}}(q,r,p)\,,$
(9)

rather than
${\rm{I}}\\!\Gamma_{\alpha^{\prime}\mu^{\prime}\nu^{\prime}}(q,r,p)$ itself.
Then, using Eqs. (3) and (4), we see that

${\overline{\Gamma}}_{\alpha\mu\nu}(q,r,p)=P^{\alpha^{\prime}}_{\alpha}(q)P^{\mu^{\prime}}_{\mu}(r)P^{\nu^{\prime}}_{\nu}(p)\Gamma_{\alpha^{\prime}\mu^{\prime}\nu^{\prime}}(q,r,p)\,,$
(10)

i.e., lattice simulations only have access to the pole-free part of the
vertex.

Nevertheless, a method for determining the BS amplitude, ${\mathbb{C}}(r)$,
from lattice results has recently been devised [116, 117]. The crucial
observation that enables this determination is that the BS amplitudes of the
massless vertex poles appear in the WI which relate two and three-point sector
functions [68, 113, 7, 118].

To fix the ideas, consider for simplicity the ghost-gluon vertex in the
background field method [122, 123, 124, 125, 126, 127, 128, 129, 130, 131,
132], denoted by ${\widetilde{\Gamma}}_{\mu}(q,r,p)$, where $q$, $r$ and $p$
stand for the gluon, antighost, and ghost momenta, respectively. This vertex
satisfies a Slavnov–Taylor identity (STI) [133, 134] identical in form to that
of the photon-scalar vertex of scalar QED. Specifically [135, 136, 64],

$q^{\mu}{\widetilde{\Gamma}}_{\mu}(q,r,p)=D^{-1}(p)-D^{-1}(r)\,,$ (11)

where $D^{ab}(q)=i\delta^{ab}D(q)$ denotes the ghost propagator. Note that, at
tree level ${\widetilde{\Gamma}}_{\mu}(q,r,p)=(r-p)_{\mu}$.

Now, let us assume that ${\widetilde{\Gamma}}_{\mu}(q,r,p)$ is a pole-free
function at $q=0$. From Eq. (11), we can derive the textbook WI by expanding
both sides to the first order in $q=0$ and equating coefficients of equal
orders. This procedure yields,

${\widetilde{\Gamma}}_{\mu}(0,r,-r)=2r_{\mu}\frac{\partial D^{-1}(r)}{\partial
r^{2}}\,.$ (12)

Equivalently, since Lorentz invariance implies
${\widetilde{\Gamma}}_{\mu}(0,r,-r)=r_{\mu}{\widetilde{\cal A}}(r)$, for some
scalar function ${\widetilde{\cal A}}(r)$, Eq. (12) can be recast as

${\widetilde{\cal A}}(r)=2\frac{\partial D^{-1}(r)}{\partial r^{2}}\,.$ (13)

Next, let us activate the Schwinger mechanism, such that the vertex acquires a
pole at $q=0$. By analogy to Eq. (3), we write

${\widetilde{\Gamma}}_{\mu}(q,r,p)\to{\widetilde{\rm{I}}\\!\Gamma}_{\mu}(q,r,p)={\widetilde{\Gamma}}_{\mu}(q,r,p)+\frac{q_{\mu}}{q^{2}}{\widetilde{C}}(q,r,p)\,,$
(14)

where ${\widetilde{\Gamma}}_{\mu}(q,r,p)$ now represents the pole-free part of
the vertex only, while ${\widetilde{C}}(q,r,p)$ is the residue of the
Schwinger mechanism pole.

Since the gauge symmetry is assumed to be unbroken, the STI of Eq. (11)
remains valid for the full vertex, i.e.,

$q^{\mu}{\widetilde{\rm{I}}\\!\Gamma}_{\mu}(q,r,p)=q^{\mu}{\widetilde{\Gamma}}_{\mu}(q,r,p)+{\widetilde{C}}(q,r,p)=D^{-1}(p)-D^{-1}(r)\,.$
(15)

Then we repeat the procedure of the derivation of the WI, expanding Eq. (15)
in a Taylor series around $q=0$. At zeroth order, Eq. (15) implies

${\widetilde{C}}(0,r,-r)=0\,,$ (16)

which is akin to the Eq. (7), derived for the three-gluon vertex from Bose
symmetry in Section 2.

Next, at first order we obtain

${\widetilde{\cal A}}(r)=2\left[\frac{\partial D^{-1}(r)}{\partial
r^{2}}-{\widetilde{\cal C}}(r)\right]\,,\qquad{\widetilde{\cal
C}}(r):=\left.\frac{\partial{\widetilde{C}}(q,r,p)}{\partial
p^{2}}\right|_{q=0}\,.$ (17)

Comparing Eqs. (13) and (17), we see that the WI for the form factor
${\widetilde{\cal A}}(r)$ gets modified, or “displaced”, by a derivative,
${\widetilde{\cal C}}(r)$, of the pole residue ${\widetilde{C}}(q,r,p)$. Note
that the above definition for ${\widetilde{\cal C}}(r)$ is completely
analogous to the BS amplitude ${\mathbb{C}}(r)$ of the three-gluon vertex,
defined in Eq. (8).

With Eq. (17) at hand, if the propagator $D(r)$ and the vertex form factor
${\widetilde{\cal A}}(r)$ are somehow known, we can compute ${\widetilde{\cal
C}}(r)$, thus determining if the vertex has a massless bound-state pole.

The same idea can be applied to the three-gluon vertex. The only fundamental
difference is the non-Abelian nature of its STI, which implies that the
relevant WI and its displacement have a more complicated form that mixes gluon
and ghost sector functions.

Specifically, the STI which relates the three-gluon vertex to the gluon
propagator is given by [1, 137, 138, 139, 140, 141]

$q^{\alpha}{\rm{I}}\\!\Gamma_{\alpha\mu\nu}(q,r,p)=F(q)\left[\Delta^{-1}(p)P_{\nu}^{\sigma}(p)H_{\sigma\mu}(p,q,r)-\Delta^{-1}(r)P_{\mu}^{\sigma}(r)H_{\sigma\nu}(r,q,p)\right]\,,$
(18)

where $F(q)$ is the ghost dressing function, defined by $D(q)=F(q)/q^{2}$, and
$H_{\nu\mu}(r,p,q)$ is the ghost-gluon kernel [142], which will be discussed
in the next section. We point out that the ghost propagator remains massless,
while its dressing function, $F(q)$, becomes finite at the origin [107, 143,
144, 145, 146, 28, 147, 148, 149, 150, 71, 142, 51, 14, 17, 19, 21, 24, 29,
39, 45], as shown in the left panel of Fig. 6.

Figure 6:  Left: Lattice data (points) from Refs. [39, 51] for the ghost
dressing function, $F(r)$. Right: Form factor $\mathit{L}_{{sg}}(r)$ of the
three-gluon vertex in the soft gluon limit obtained from lattice quantum
chromodynamics (QCD) in Ref. [50] (points). The blue solid curve in each panel
denotes a fit to the corresponding data, given by Eqs. (C6) and (C12) of
[116], for $F(r)$ and $\mathit{L}_{{sg}}(r)$, respectively.

The WI for the three-gluon vertex is obtained as a special case of the above
STI. To derive it, one expands Eq. (18) around $q=0$ and matches coefficients
of equal orders on each side of the resulting equation. Evidently, the zeroth-
order expansion leads again to Eq. (7). As for the first-order term, after a
suitable projection to isolate the classical tensor structure of the three-
gluon vertex, one obtains the relation (for detailed derivations see [116, 7])

${\mathbb{C}}(r)=\mathit{L}_{{sg}}(r)-F(0)\left\\{\frac{{\cal
W}(r)}{r^{2}}\Delta^{-1}(r)+\widetilde{Z}_{1}\frac{d\Delta^{-1}(r)}{dr^{2}}\right\\}\,.$
(19)

In the above equation, the displacement of the WI is _precisely_ the BS
amplitude ${\mathbb{C}}(r)$ of the Schwinger pole of the three-gluon vertex.
On the other hand, $\mathit{L}_{{sg}}(r)$ is the classical form factor of the
three-gluon vertex in the soft gluon limit, defined by [50]

$\mathit{L}_{{sg}}(r)=\left.\frac{\Gamma_{0}^{\alpha\mu\nu}(q,r,p)P_{\alpha\alpha^{\prime}}(q)P_{\mu\mu^{\prime}}(r)P_{\nu\nu^{\prime}}(p){\rm{I}}\\!\Gamma^{\alpha^{\prime}\mu^{\prime}\nu^{\prime}}(q,r,p)}{\Gamma_{0}^{\alpha\mu\nu}(q,r,p)P_{\alpha\alpha^{\prime}}(q)P_{\mu\mu^{\prime}}(r)P_{\nu\nu^{\prime}}(p)\Gamma_{0}^{\alpha^{\prime}\mu^{\prime}\nu^{\prime}}(q,r,p)}\right|_{q\to
0}\,,$ (20)

with $\Gamma_{0}^{\alpha\mu\nu}(q,r,p)$ denoting the tree-level form of the
vertex. By now, this form factor has been extensively studied on the lattice
[34, 35, 37, 47, 50, 52], such that its form is rather accurately known. In
the right panel of Fig. 6 we show the lattice results for
$\mathit{L}_{{sg}}(r)$ from [50] (points), together with a physically
motivated fit for it given by Eq. (C12) of [116] (blue continuous curve).

Lastly, ${\cal W}(r)$ is a particular derivative of the ghost-gluon kernel,
namely [151, 116]

${\cal W}(r)=-\frac{1}{3r^{2}}P^{\mu\nu}(r)\left[\frac{\partial
H_{\nu\mu}(p,q,r)}{\partial q^{\alpha}}\right]_{q=0}\,,$ (21)

while ${\widetilde{Z}}_{1}$ is the renormalization constant of the ghost-gluon
vertex. The latter is finite in the Landau gauge, by virtue of the well-known
Taylor theorem [133].

The Eq. (19) is the central relation that will enable us to determine the BS
amplitude, ${\mathbb{C}}(r)$, from lattice data for the gluon and ghost
propagators and the form factor $\mathit{L}_{{sg}}(r)$ of the three-gluon
vertex. To this end, we need first to determine the ghost-gluon kernel
derivative ${\cal W}(r)$ appearing in Eq. (19).

##  4 Ghost-gluon kernel contribution

Now we briefly describe our lattice-driven SDE determination of ${\cal W}(r)$.
The starting point of this analysis is the SDE that defines the ghost-gluon
kernel, shown diagrammatically in Fig. 7.

Figure 7:  SDE for the ghost–gluon scattering kernel, $H_{\mu\nu}(r,q,p)$. We
omit a diagram containing a 1PI four-point function, which has been shown to
contribute to the ghost-gluon vertex at the $2\%$ level only [152].

From that equation, the function ${\cal W}(r)$ can be isolated through Eq.
(21). The resulting expression for ${\cal W}(r)$ can be written as

${\cal W}(r)={\cal W}_{1}(r)+{\cal W}_{2}(r)\,,$ (22)

where the ${\cal W}_{i}(r)$ denote the contributions of the diagrams
$(h_{i}^{\mu\nu})$ in Fig. 7, respectively. These are given by

$\displaystyle{\cal W}_{1}(r)$ $\displaystyle=$
$\displaystyle{\widetilde{\lambda}}\int_{k}\Delta(k)D(k)D(k+r)(r\cdot
k)f(k,r)B_{1}(k+r,-k,-r)B_{1}(k,0,-k)\,,$ $\displaystyle{\cal W}_{2}(r)$
$\displaystyle=$
$\displaystyle{\widetilde{\lambda}}\int_{k}\Delta(k)\Delta(k+r)D(k+r)B_{1}(k+r,0,-k-r){\cal
I}_{{\cal W}}(-r,-k,k+r)\,,$ (23)

where ${\widetilde{\lambda}}:=ig^{2}C_{\rm A}{\widetilde{Z}}_{1}/6$,
$C_{\mathrm{A}}$ is the Casimir eigenvalue of the adjoint representation [$N$
for SU$(N)$], and

$f(k,r):=1-\frac{(r\cdot k)^{2}}{r^{2}k^{2}}\,.$ (24)

In addition to the gluon and ghost propagators, $\Delta(r)$ and $D(r)$,
respectively, Eq. (23) involves quantities that are related to the ghost-gluon
and three-gluon vertices, namely $B_{1}(r,p,q)$, ${\widetilde{Z}}_{1}$ and
${{\cal I}_{{\cal W}}}(q,r,p)$. Below we explain their meaning in detail.

  1. (i)

In Eq. (23), $B_{1}(r,p,q)$ denotes the classical form factor of the ghost-
gluon vertex, ${\rm{I}}\\!\Gamma_{\mu}(r,p,q)$, whose most general tensor
structure is given by

${\rm{I}}\\!\Gamma_{\mu}(r,p,q)=B_{1}(r,p,q)r_{\mu}+B_{2}(r,p,q)q_{\mu}\,.$
(25)

Hence, at tree level $B_{1}^{0}=1$ and $B_{2}^{0}=0$.

Note that the ghost-gluon vertex and kernel are related by the STI

${\rm{I}}\\!\Gamma_{\mu}(r,p,q)=r^{\nu}H_{\nu\mu}(q,r,p)\,.$ (26)

Thus, the general kinematics $B_{1}(r,p,q)$ can be determined through another
projection of the SDE of Fig. 7.

Such an SDE determination of the general kinematics $B_{1}(r,p,q)$ was
performed in Refs. [117, 118], also using lattice results as inputs for all
its ingredients. It is beyond the scope of the present work to describe this
analysis in detail. It suffices to mention that the results for $B_{1}(r,p,q)$
deviate only moderately from its tree-level value, in agreement with several
previous continuum studies [153, 154, 149, 150, 155, 142, 71, 156, 79, 157],
and reproduce the available lattice data from Ref. [14, 15]. As such, the
impact of the precise dressing of $B_{1}(r,p,q)$ on the ${\cal W}(r)$ computed
through Eq. (22) is under stringent control.

  2. (ii)

As previously mentioned, the ghost-gluon kernel, and hence ${\cal W}(r)$, is
finite in Landau gauge [133]. Nevertheless, multiplicative renormalization of
the theory leads to the appearance [151] of the ghost-gluon renormalization
constant, ${\widetilde{Z}}_{1}$, in Eq. (23). The _finite_ value of this
constant depends on the renormalization scheme adopted.

To take the most advantage of the lattice data for the propagators and the
three-gluon vertex, we adopt the scheme where $\Delta(r)$, $F(r)$ and
$\mathit{L}_{{sg}}(r)$ are most readily renormalized. Namely, the so-called
asymmetric MOM scheme [34, 37, 151, 50, 51, 118]. The latter is defined by the
prescriptions

$\Delta^{-1}(\mu^{2})=\mu^{2}\,,\qquad
F(\mu^{2})=1\,,\qquad\mathit{L}_{{sg}}(\mu^{2})=1\,,$ (27)

where we choose $\mu=4.3$ GeV as renormalization point. The corresponding
value for the coupling is $g^{2}=4\pi\alpha_{s}$, with $\alpha_{s}(4.3\text{
GeV})=0.27$, as determined in the lattice study of [37]. Within this
renormalization scheme, the same SDE analysis of Refs. [117, 118] used to
determine $B_{1}(r,p,q)$ also yields the value ${\widetilde{Z}}_{1}=0.9333$.

  3. (iii)

Finally, ${\cal I}_{{\cal W}}(q,r,p)$ is a particular transverse projection of
the three-gluon vertex, namely

$\displaystyle{\cal I}_{{\cal W}}(q,r,p)$
$\displaystyle:=\frac{1}{2}(q-r)^{\nu}{\overline{\Gamma}}^{\alpha}_{\alpha\mu}(q,r,p)\,,$
(28)

which encodes the total contribution of
${\overline{\Gamma}}^{\alpha}_{\alpha\mu}(q,r,p)$ to the SDE governing ${\cal
W}(r^{2})$. Note that the Bose symmetry of
${\overline{\Gamma}}^{\alpha}_{\alpha\mu}(q,r,p)$ implies

${\cal I}_{{\cal W}}(q,r,p)={\cal I}_{{\cal W}}(r,q,p)\,.$ (29)

In a series of previous works [151, 158, 116], the ingredient appearing in Eq.
(22) that represented the largest uncertainty was ${\cal I}_{{\cal
W}}(q,r,p)$. Since lattice results for the general kinematics three-gluon
vertex were not available then, in those references ${\cal W}(r)$ had been
approximated by various _Ansätze_ based on the Ball-Chiu construction of the
three-gluon vertex [137, 156]. Recently, general kinematics lattice data for
${\overline{\Gamma}}_{\alpha\mu\nu}(q,r,p)$ became available [52, 53, 54],
prompting a more accurate determination of ${\cal W}(r)$.

Remarkably the general kinematics lattice results of Refs. [52, 53, 54, 117],
as well as some continuum studies [159, 160, 161], revealed that a compact
expression provides a rather accurate approximation for the transversely
projected three-gluon vertex. Specifically,

${\overline{\Gamma}}_{\alpha\mu\nu}(q,r,p)\approx{\overline{\Gamma}}^{0}_{\alpha\mu\nu}(q,r,p)\mathit{L}_{{sg}}(s)\,,\qquad
s^{2}:=(q^{2}+r^{2}+p^{2})/2\,,$ (30)

where ${\overline{\Gamma}}^{0}_{\alpha\mu\nu}(q,r,p)$ denotes the tree-level
form of ${\overline{\Gamma}}_{\alpha\mu\nu}(q,r,p)$.

In Eq. (30) the sole dynamical ingredient is the soft gluon form factor,
$\mathit{L}_{{sg}}(r)$, of Fig. 6, which now appears evaluated at the Bose-
symmetric combination of momenta given by $s$. Note that general kinematics
form factors of the three-gluon vertex are expected to depend on three Lorentz
scalars. The fact that in Eq. (30) the form factor depends only on $s$, whose
values define planes in the coordinate system $(q^{2},r^{2},p^{2})$, has been
termed _planar degeneracy_ [52].

Using the planar degeneracy approximation of Eq. (30) into Eq. (28), we find a
similarly compact expression for ${\cal I}_{{\cal W}}(q,r,p)$, namely

${\cal I}_{{\cal W}}(q,r,p)\approx{\cal I}_{{\cal
W}}^{0}(q,r,p)\mathit{L}_{{sg}}(s)\,,$ (31)

where ${\cal I}_{{\cal W}}^{0}(q,r,p)$ is the tree-level value of ${\cal
I}_{{\cal W}}$, given by

${\cal I}_{{\cal
W}}^{0}(q,r,p):=\frac{2f(q,r)}{p^{2}}\left[2q^{2}r^{2}-(q^{2}+r^{2})(q\cdot
r)-(q\cdot r)^{2}\right]\,.$ (32)

The Eq. (31) provides us with a baseline for computing ${\cal W}(r)$
accurately and expeditiously.

In order to carry out the integrations over the whole momentum space in Eq.
(23), we employ fits for the lattice data for $\Delta(r)$ and $F(r)$ from [51]
and for the $\mathit{L}_{{sg}}(r)$ of [47] which are constructed to reproduce
the one-loop anomalous dimensions of these functions for large momenta. These
fits are given in Appendix C of [116] and are all renormalized consistently in
the asymmetric MOM scheme [151, 118].

Using the above ingredients, combined with the results for $B_{1}(r,p,q)$,
${\widetilde{Z}}_{1}$ and $\alpha_{s}$ mentioned in items $(i)$ and $(ii)$
above, we evaluate the Euclidean form of Eq. (23) to obtain ${\cal W}(r)$. The
result is shown as the blue solid curve in the left panel of Fig. 8.

Figure 8:  Left: ${\cal W}(r)$ obtained using the planar degeneracy
approximation of Eq. (31) for the three-gluon vertex (blue solid curve)
together with uncertainty estimate (blue band) obtained by using the Neural
Network predictor for ${\cal I}_{{\cal W}}$. Right: Lattice results (points)
for ${\cal I}_{{\cal W}}(q,r,p)$, when the momenta $q$ and $r$ are at an angle
of $2\pi/3$. The surface shows the result of the Neural Network predictor
trained on the data points. Another value for the angle can be seen in Fig. 4
of Ref. [117].

At this point, it is important to quantify the errors introduced in ${\cal
W}(r)$ by the use of the approximate form of the three-gluon vertex given in
Eq. (30). To that end, the projection ${\cal I}_{{\cal W}}(q,r,p)$ has been
computed _directly_ through lattice simulation in Ref. [117]. Results for
various lattice setups are shown as points in the right panel of Fig. 8. These
data correspond to momenta $q$ and $r$ at an angle of $2\pi/3$ and with
arbitrary magnitudes. For different angles, the ${\cal I}_{{\cal W}}(q,r,p)$
is qualitatively similar.

In order to employ the lattice results for ${\cal I}_{{\cal W}}(q,r,p)$ into
the SDE of ${\cal W}(r)$, we need a smooth interpolant. Since the data points
depend on three kinematic variables (the magnitudes of two momenta and the
angle between them), it is difficult to come up with a functional form that
fits them accurately. Moreover, since the data is noisy, standard interpolants
such as splines are unsuitable.

A reliable method to interpolate the general kinematics ${\cal I}_{{\cal
W}}(q,r,p)$ consists of training a Neural Network predictor on the lattice
data [117]. To this end, we randomly selected one-third of the 335 628 lattice
points for ${\cal I}_{{\cal W}}(q,r,p)$ as a training set. The data was then
fed into the Mathematica routine “Predict”, with the option “Neural Network”,
which outputs a smooth predictor function. The remaining 223 725 lattice data
points were then used to confirm the accuracy of the resulting interpolant, by
verifying that the predicted values were always within one standard deviation
of the actual lattice results [117].

In the right panel of Fig. 8, the Neural Network predictor for ${\cal
I}_{{\cal W}}(q,r,p)$ is represented by the color-mapped surface, which is
compared to the full set of lattice data for the angle between $q$ and $r$ set
at $2\pi/3$. In that figure, the accuracy and smoothness of the Neural Network
result are clearly seen.

The Neural Network predictor for ${\cal I}_{{\cal W}}(q,r,p)$ can then be used
directly into Eq. (23), as an alternative method for computing ${\cal W}(r)$.
Quite remarkably, the results for ${\cal W}(r)$ computed with this method and
those obtained from the planar degeneracy approximation of Eq. (31) differ by
only $2.5\%$ [117]. Combining this estimate of the systematic error with
propagated statistical error of $\mathit{L}_{{sg}}(r)$ [117] we obtain a total
error budget for ${\cal W}(r)$, which is represented as the blue band shown in
the left panel of Fig. 8.

##  5 Determination of the displacement amplitude from Lattice inputs

Now we are in position to determine ${\mathbb{C}}(r)$ from the WI
displacement, i.e., through Eq. (19).

Combining the blue curve for ${\cal W}(r)$ of Fig. 8 with the aforementioned
fits for $\Delta(r)$, $F(r)$ and $\mathit{L}_{{sg}}(r)$ into Eq. (19), we
obtain for ${\mathbb{C}}(r)$ the black solid curve in the left panel of Fig.
9. The points in the same panel show the result for ${\mathbb{C}}(r)$ obtained
by using _directly_ in Eq. (19) the lattice data points of [50] for
$\mathit{L}_{{sg}}(r)$, instead of a fit.

Figure 9:  Left: Result for ${\mathbb{C}}(r)$ obtained from the Ward identity
(WI) displacement, i.e., Eq. (19). The black continuous line results from
using the lattice-driven SDE result for ${\cal W}(r)$, shown in the right
panel of Fig. 8, and fits for $\Delta(r)$, $\mathit{L}_{{sg}}(r)$ and $F(r)$.
Using for $\mathit{L}_{{sg}}(r)$ the lattice data of Ref. [50] directly yields
the points. The green band emphasizes the typical size of the error estimate
for ${\mathbb{C}}(r)$ and is obtained by fitting the upper and lower bounds of
the error bars of the points. Right: Comparison of the ${\mathbb{C}}(r)$
(black line and green band) of the left panel to the BSE prediction,
${\mathbb{C}}_{\star}(r)$, (purple dot-dashed and error band) of Ref. [116].

The statistical significance of the above result for ${\mathbb{C}}(r)$ can be
quantified by comparing it to the null hypothesis, namely
${\mathbb{C}}={\mathbb{C}}_{0}=0$. To this end, we compute the $\chi^{2}$ of
our points for ${\mathbb{C}}(r)$, with the null hypothesis taken as the
estimator of the data, i.e.,

$\chi^{2}=\sum_{i=1}^{n_{r}}\frac{\left[{\mathbb{C}}(r_{i})-{\mathbb{C}}_{0}(r_{i})\right]^{2}}{\epsilon_{{\scriptscriptstyle{\mathbb{C}}(r_{i})}}^{2}}=2\,630\,.$
(33)

In the above equation, $\epsilon_{{\scriptscriptstyle{\mathbb{C}}(r_{i})}}$
denotes the error estimate of ${\mathbb{C}}(r_{i})$ (the error bars in Fig.
9). The sum is performed over the $n_{r}=515$ indices $i$ such that
$r_{i}\in[0.3,4.3]$ GeV.

From the result in Eq. (33), we can compute the probability,
$P_{{\mathbb{C}}_{0}}$, that our result for ${\mathbb{C}}(r)$ is consistent
with the null hypothesis. Denoting by $\chi_{\rm{\scriptscriptstyle
PDF}}^{2}(n_{r},x)$ the $\chi^{2}$ probability distribution function with
$n_{r}=515$ degrees of freedom, we obtain [117]

$P_{{\mathbb{C}}_{0}}=\int_{\chi^{2}=2\,630}^{\infty}\chi_{\rm{\scriptscriptstyle
PDF}}^{2}(515,x)dx=\left.\frac{\Gamma(n_{r}/2,\chi^{2}/2)}{\Gamma(n_{r}/2)}\right|_{n_{r}=515}^{\chi^{2}=2\,630}=7.3\times
10^{-280}\,.$ (34)

The vanishingly small probability obtained in Eq. (34) is to be understood as
meaning that, in the absence of additional uncertainties or correlations in
the data, the null hypothesis ${\mathbb{C}}_{0}$ is completely excluded.
Moreover, we point out that even if the error of _every data point_ for
${\mathbb{C}}(r)$ was $95\%$ larger we could still discard ${\mathbb{C}}_{0}$
at the $5\sigma$ confidence level.

The result for ${\mathbb{C}}(r)$ obtained in this way can then be compared to
the BSE prediction, ${\mathbb{C}}_{\star}(r)$, of Ref. [116], shown in the
right panel of Fig. 5. To this end, we first need to determine the overall
scale and sign of ${\mathbb{C}}_{\star}(r)$, which are left undetermined by
the homogeneous nature of the BSE.

Denoting by ${\mathbb{C}}_{{\scriptscriptstyle\rm BSE}}(r)$ a solution of the
homogeneous BSE, we first define

${\mathbb{C}}_{\star}(r)=b\,{\mathbb{C}}_{{\scriptscriptstyle\rm BSE}}(r)\,,$
(35)

with $b$ a constant. Then, we determine the multiplicative constant $b$ by
minimizing the $\chi^{2}$ measure for the discrepancy between ${\mathbb{C}}$
and ${\mathbb{C}}_{\star}$ as

$\chi^{2}_{\star}=\sum_{i}\frac{\left[{\mathbb{C}}(r_{i})-{\mathbb{C}}_{\star}(r_{i})\right]^{2}}{\epsilon_{{\scriptscriptstyle{\mathbb{C}}(r_{i})}}^{2}}\,.$
(36)

The result of this procedure is the ${\mathbb{C}}_{\star}(r)$ shown previously
in the right panel of Fig. 5.

Next, in the right panel of Fig. 9 we compare ${\mathbb{C}}(r)$ and
${\mathbb{C}}_{\star}(r)$ directly, finding a rather good agreement. The main
difference is in the position of the minimum, which is shifted from
$r=1.93\begin{subarray}{c}+0.09\\\ -0.06\end{subarray}$ GeV for
${\mathbb{C}}(r)$ to $r=1.5\pm 0.1$ for ${\mathbb{C}}_{\star}(r)$.

Finally, in addition to determining the scale and sign of
${\mathbb{C}}_{\star}(r)$, the $\chi^{2}_{\star}$ measure of Eq. (36) allows
us to perform a statistical analysis of the compatibility between the BSE
prediction and the lattice result. Specifically, after setting the scale of
${\mathbb{C}}_{\star}$ we obtain $\chi^{2}_{\star}=258.5$, which is smaller
than the number of degrees of freedom. Indeed, this value of
$\chi^{2}_{\star}$ translates to a near unit probability,

$P_{{\mathbb{C}}_{\star}}=\left.\frac{\Gamma(n_{r}/2,\chi^{2}_{\star}/2)}{\Gamma(n_{r}/2)}\right|_{n_{r}=515}^{\chi^{2}_{\star}=258.5}=1-2.0\times
10^{-23}\,,$ (37)

of the points ${\mathbb{C}}(r)$ being compatible with
${\mathbb{C}}_{\star}(r)$ [118].

##  6 Conclusion

The displacement of the WIs by the formation of massless vertex poles is a
distinctive feature of the Schwinger mechanism for gluon mass generation,
which allows its verification from lattice QCD results. In the present work,
we have used this framework to demonstrate that the three-gluon vertex in
Yang-Mills SU(3) has such a pole. Indeed, our analysis of the WI displacement
using lattice data unequivocally excludes the null hypothesis of a vanishing
BS amplitude, ${\mathbb{C}}=0$. Instead, our results reveal an excellent
agreement between the ${\mathbb{C}}(r)$ derived from the WI and the BSE
prediction, providing outstanding evidence for the occurrence of the Schwinger
mechanism in QCD.

It is important to emphasize that while the present analysis was carried out
in the simpler setting of pure Yang-Mills SU(3), the same ideas hold in the
presence of dynamical quarks. In particular, the WI displacement for the
three-gluon vertex retains exactly the same form as in Eq. (19) in the
unquenched case, for which lattice data for the propagators and vertices also
exist [162, 16, 29, 45, 47]. Indeed, a study is already underway to
investigate the Schwinger mechanism poles in the presence of quarks and should
be reported soon.

Finally, in the present work, we have focused entirely on the massless pole
content of the three-gluon vertex. However, once the Schwinger mechanism is
active, it is expected that massless poles appear in various vertices [113],
since the different vertices are connected to one another through the SDEs. As
such, other important signals of the Schwinger mechanism may be present in
functions such as the ghost-gluon kernel, and the quark-gluon and four-gluon
vertices, which are currently under investigation as well.

Acknowledgments

The author thanks A.C. Aguilar, J. Papavassiliou, C.D. Roberts, and J.
Rodríguez-Quintero for the collaborations.

## Declarations

Funding

M.N.F. is supported by the grant PID2020-113334GB-I00 and the contract
CIAPOS/2021/74, from the Spanish MICINN and the Generalitat Valenciana,
respectively.

Competing interests

The author has no relevant financial or non-financial interests to disclose.

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