in $q$. Evidently, the term $C_{5}$ does not contribute to this order. Then,
the expansion leads to the appearance of derivatives of the gluon propagator,
in analogy to Eq. (63), but also of the ghost-gluon kernel. Specifically, we
obtain for the WI of the three-gluon vertex and its displacement the
expression

$\mathit{L}_{{sg}}(r)=F(0)\left\\{\widetilde{Z}_{1}\frac{d\Delta^{-1}(r)}{dr^{2}}+\frac{{\cal
W}(r)}{r^{2}}\Delta^{-1}(r)\right\\}-{\mathbb{C}}(r)\,.$ (106)

In the above equation, $\mathit{L}_{{sg}}(r)$ is the form factor of the three-
gluon vertex defined in Eq. (76) and with lattice results shown in Fig. 11,
while ${\cal W}(r)$ is a particular derivative of the ghost-gluon kernel,
namely Aguilar et al. (2020, 2022)

${\cal W}(r)=-\frac{1}{3r^{2}}P^{\mu\nu}(r)\left[\frac{\partial
H_{\nu\mu}(p,q,r)}{\partial q^{\alpha}}\right]_{q=0}\,.$ (107)

For the detailed derivation of Eq. (106), we refer to Aguilar et al. (2022);
Papavassiliou (2022).

In the following section, we will use Eq. (106) to determine the displacement
amplitude ${\mathbb{C}}(r)$ from lattice inputs. To this end, we must first
pass to Euclidean space, where we note that

${\mathbb{C}}_{{\scriptscriptstyle\rm E}}(r^{2}_{{\scriptscriptstyle\rm
E}})=-\left.{\mathbb{C}}(r)\right|_{r^{2}=-r_{{\scriptscriptstyle\rm
E}}^{2}}\,,$ (108)

with the extra sign originating from the fact that ${\mathbb{C}}$ is a
derivative [see Eq. (41)]. Then, suppressing the indices “${\rm E}$” and
solving for ${\mathbb{C}}(r^{2})$, we obtain the central relation

${\mathbb{C}}(r)=\mathit{L}_{{sg}}(r)-F(0)\left\\{\frac{{\cal
W}(r)}{r^{2}}\Delta^{-1}(r)+\widetilde{Z}_{1}\frac{d\Delta^{-1}(r)}{dr^{2}}\right\\}\,.$
(109)

For the determination of ${\mathbb{C}}(r)$, we use lattice inputs for all the
quantities that appear on the r.h.s. of Eq. (109), with the exception of the
function ${\cal W}(r)$, which will be computed from the SDE satisfied by the
ghost-gluon kernel, derived and analyzed in the next section.

##  11 The ghost-gluon kernel contribution to the Ward identity

In this section, we derive the SDE that determines the key function ${\cal
W}(r)$; the resulting SDE will be solved using lattice inputs for the various
quantities entering in it. In addition, the infrared behavior of ${\cal W}(r)$
will be analyzed in detail, following an analytic procedure.

Our discussion starts with the SDE of the ghost-gluon kernel,
$H_{\mu\nu}(r,q,p)$, shown diagrammatically in Fig. 19, from which ${\cal
W}(r)$ can be obtained using Eq. (107).

Note that the similarity between the diagrams shown in Fig. 19 and those in
the bottom panel of Fig. 12, depicting the SDE of the ghost-gluon vertex, is a
simple reflection of the fundamental STI relating the ghost-gluon kernel with
the ghost-gluon vertex,

$\Gamma_{\nu}(r,q,p)=r^{\mu}H_{\mu\nu}(r,q,p)\,.$ (110)

Specifically, Eq. (110) is preserved by the SDEs of $\Gamma_{\nu}(r,q,p)$ and
$H_{\mu\nu}(r,q,p)$; indeed, contraction of each diagram $(h_{i}^{\mu\nu})$ of
Fig. 19 by $r^{\mu}$ yields the corresponding diagram $(g_{i}^{\nu})$ of Fig.
12 (up to a shift of $k\to-k-r$ for $i=1$, introduced to simplify certain
expressions). Note that, in Fig. 19, the diagram corresponding to the
$(g_{3})$ of Fig. 12 has been omitted, for the reason explained in the item
(i) of Section 8.

Figure 19:  SDE for the ghost-gluon scattering kernel, $H_{\mu\nu}(r,q,p)$. We
omit a diagram containing a 1PI four-point function.

It is well known that, in the Landau gauge, the momentum $q$ of the ghost
field in $H_{\mu\nu}(r,q,p)$ factors out of its quantum corrections Marciano
and Pagels (1978), allowing us to write Aguilar et al. (2019, 2020, 2022)

$H_{\mu\nu}(r,q,p)=g_{\mu\nu}+q^{\alpha}K_{\mu\nu\alpha}(r,q,p)\,,$ (111)

where no particular assumptions are made about the structure of the function
$K_{\mu\nu\alpha}(r,q,p)$. Following Eq. (107), we differentiate the r.h.s. of
Eq. (111) with respect to $q$ and subsequently set $q=0$, to obtain

${\cal W}(r)=-\frac{1}{3}r^{\alpha}P^{\mu\nu}(r)K_{\mu\nu\alpha}(r,0,-r)\,.$
(112)

Lastly, the _finite_ renormalization of ${\cal W}$ proceeds through the use of
Eqs. (14) and (86), which leads to the appearance of an overall factor of
${\widetilde{Z}}_{1}$ in the equations.

The outcome of the above steps is that ${\cal W}(r)$ can be written as

${\cal W}(r)={\cal W}_{1}(r)+{\cal W}_{2}(r)\,,$ (113)

where the ${\cal W}_{i}(r)$ are the contributions originating from the
diagrams $(h_{i}^{\mu\nu})$ of Fig. 19, respectively, and read

$\displaystyle{\cal W}_{1}(r)$ $\displaystyle=$
$\displaystyle\frac{\lambda{\widetilde{Z}}_{1}}{3}\int_{k}\Delta(k)D(k)D(k+r)(r\cdot
k)f(k,r)B_{1}(k+r,-k,-r)B_{1}(k,0,-k)\,,$ $\displaystyle{\cal W}_{2}(r)$
$\displaystyle=$
$\displaystyle\frac{\lambda{\widetilde{Z}}_{1}}{3}\int_{k}\Delta(k)\Delta(k+r)D(k+r)B_{1}(k+r,0,-k-r){\cal
I}_{{\cal W}}(-r,-k,k+r)\,,$ (114)

where $f(k,r)$ is given by Eq. (80), and we define the specific contribution
of the three-gluon vertex to the kernel of ${\cal W}(r^{2})$ as

$\displaystyle{\cal I}_{{\cal W}}(q,r,p)$
$\displaystyle:=\frac{1}{2}(q-r)^{\nu}\overline{\Gamma}^{\alpha}_{\alpha\nu}(q,r,p)\,.$
(115)

Note that, from Eq. (115) and the Bose symmetry of the
$\overline{\Gamma}\vphantom{\Gamma}_{\alpha\mu\nu}(q,r,p)$ under the exchange
$\\{q,\alpha\\}\leftrightarrow\\{r,\mu\\}$, it follows that

${\cal I}_{{\cal W}}(q,r,p)={\cal I}_{{\cal W}}(r,q,p)\,.$ (116)

At this point, by capitalizing on the planar degeneracy of
$\overline{\Gamma}\vphantom{\Gamma}_{\alpha\mu\nu}(q,r,p)$ discussed in Sec.
7, we obtain a compact, and yet accurate, approximation for ${\cal I}_{{\cal
W}}$. Specifically, using Eq. (77), we find

$\displaystyle{\cal I}_{{\cal W}}(q,r,p)$ $\displaystyle\approx{\cal I}_{{\cal
W}}^{0}(q,r,p)\mathit{L}_{{sg}}(s)\,,$ (117)

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

${\cal I}_{{\cal
W}}^{0}(q,r,p):=\frac{2f(q,r)}{p^{2}}\left[2q^{2}r^{2}-(q^{2}+r^{2})(q\cdot
r)-(q\cdot r)^{2}\right]\,.$ (118)

We remark that the approximation given by Eq. (117) becomes exact in the limit
$p=0$.

Using the above approximation for ${\cal I}_{{\cal W}}$, the contribution
${\cal W}_{2}(r)$ reads

$\displaystyle{\cal W}_{2}(r)=$
$\displaystyle\frac{2\lambda{\widetilde{Z}}_{1}}{3}\int_{k}\Delta(k)\frac{\Delta(k+r)D(k+r)}{(k+r)^{2}}B_{1}(k+r,0,-k-r)f(k,r)$
$\displaystyle\times\left[2r^{2}k^{2}-(r^{2}+k^{2})(r\cdot k)-(r\cdot
k)^{2}\right]\mathit{L}_{{sg}}({\hat{s}})\,,$ (119)

where we now have ${\hat{s}}^{2}=r^{2}+k^{2}+(r\cdot k)$.

Lastly, we transform ${\cal W}_{1}$ of Eq. (114) and ${\cal W}_{2}$ of Eq.
(119) to Euclidean space to obtain the final expression to be used for the
numerical determination of ${\cal W}$,

$\displaystyle{\cal W}_{1}(r)$ $\displaystyle=-\frac{r\alpha_{s}C_{\rm
A}{\widetilde{Z}}_{1}}{12\pi^{2}}\\!\int_{0}^{\infty}\\!dk^{2}k\Delta(k)F(k)B_{1}(k^{2},k^{2},\pi)\\!\int_{0}^{\pi}\\!d\phi
s_{\phi}^{4}c_{\phi}\frac{F(\sqrt{z})}{z}B_{1}(z,r^{2},\chi)\,,$
$\displaystyle{\cal W}_{2}(r)$ $\displaystyle=-\frac{r\alpha_{s}C_{\rm
A}{\widetilde{Z}}_{1}}{6\pi^{2}}\\!\int_{0}^{\infty}\\!dk^{2}\,k^{3}\Delta(k)\\!\int_{0}^{\pi}\\!d\phi\,s_{\phi}^{4}\Delta(\sqrt{z})B_{1}(z,z,\pi)\frac{F(\sqrt{z})}{z^{2}}\left[kr(2+c_{\phi}^{2})-zc_{\phi}\right]$
$\displaystyle\quad\times\mathit{L}_{{sg}}\left(r^{2}+k^{2}+rkc_{\phi}\right)\,,$
(120)

where $z$ has been defined below Eq. (89) and

$\chi:=\cos^{-1}\left[-\frac{(r+kc_{\phi})}{\sqrt{z}}\right]\,.$ (121)

We emphasize that we have used into the SDEs of both $B_{1}$ and ${\cal W}$,
given by Eqs. (88) and (120), respectively, the same approximation for the
three-gluon vertex, namely Eq. (77). Therefore, our analyses of $B_{1}$ and
${\cal W}$ are self-consistent, in the sense that the STI in Eq. (110) is
strictly preserved.

Before embarking on the numerical determination of ${\cal W}(r)$ for the
entire range of Euclidean momenta, we discuss the infrared behavior of this
function, and demonstrate an important self-consistency proof involving
${\mathbb{C}}(r)$.

Specifically, as discussed in Sec. 9, the $\mathit{L}_{{sg}}(r)$ and
$d\Delta^{-1}(r)/dr^{2}$ that appear in Eq. (109) are infrared divergent, due
to massless ghost loops present in their SDEs. Nevertheless, the BSE solutions
for the amplitude ${\mathbb{C}}(r)$ are all found to be finite at $r=0$, (cf.
Fig. 7) Aguilar et al. (2012a); Binosi and Papavassiliou (2018); Aguilar et
al. (2018, 2022). Therefore, in order for the WI displacement of Eq. (109) to
be consistent with the finite ${\mathbb{C}}(0)$ obtained from BSE solutions,
the infrared divergences of the ingredients appearing in Eq. (109) must cancel
against each other.

Indeed, a careful analysis of diagram $(e_{2})$ of Fig. 16 yields

$\lim_{r\to 0}\mathit{L}_{{sg}}(r)=\left[\frac{\alpha_{s}C_{\rm
A}{\widetilde{Z}}_{1}^{3}F^{3}(0)}{96\pi}\right]\ln\left(\frac{r^{2}}{\mu^{2}}\right)\,,$
(122)

up to infrared finite terms666We note that results identical to Eqs. (97) and
(122) for the infrared divergences of $d\Delta^{-1}(r)/dr^{2}$ and
$\mathit{L}_{{sg}}(r)$, respectively, have been previously derived within the
Curci-Ferrari model Barrios et al. (2022).. Then, assuming that only
$\mathit{L}_{{sg}}(r)$ and $d\Delta^{-1}/dr^{2}$ diverge and using the
asymptotic form of $d\Delta(r)/dr^{2}$ given in Eq. (97) into Eq. (109), we
find that the divergences do not fully cancel. Therefore, the finiteness of
${\mathbb{C}}(0)$ demands that the term ${\cal W}(r)/r^{2}$ appearing in the
WI must be infrared divergent.

Now, it is evident from Eq. (120) that ${\cal W}(r)$ vanishes as $r\to 0$.
Nevertheless, the ratio ${\cal W}(r)/r^{2}$ is found to diverge at the origin.
Specifically, expanding Eq. (120) around $r=0$, it can be shown that ${\cal
W}(r)/r^{2}$ has the asymptotic behavior

$\lim_{r\to 0}\frac{{\cal W}(r)}{r^{2}}=-\left[\frac{\alpha_{s}C_{\rm
A}{\widetilde{Z}}_{1}^{3}\Delta(0)F^{2}(0)}{96\pi}\right]\ln\left(\frac{r^{2}}{\mu^{2}}\right)\,.$
(123)

Then, combining Eqs. (123), (122) and (123) we find that the infrared
divergences in Eq. (109) cancel out exactly, leaving a finite
${\mathbb{C}}(0)$, in full agreement with the BSE results.

Figure 20:  Diagrammatic representation of the cancellation of the infrared
divergences originating from massless ghost loops in Eq. (109) to yield a
finite ${\mathbb{C}}(0)$. The red cross indicates that the overall ghost
momentum is factored out before being set to zero.

We finish the discussion of the infrared finiteness of ${\mathbb{C}}(0)$ with
a remark. In the absence of the Schwinger mechanism, i.e., for an identically
zero ${\mathbb{C}}(r)$, the infared divergences of $\mathit{L}_{{sg}}(r)$,
${\cal W}(r)/r^{2}$ and $d\Delta^{-1}(r)/dr^{2}$ must also cancel in Eq.
(109). For instance, this cancellation can be explicitly verified at the one
loop level777In the perturbative realization of Eq. (109) $F(0)$ also
diverges, participating in the overall cancellation of infrared divergences.,
where, evidently, ${\mathbb{C}}(r)=0$. In that case, however, the gluon
propagator is also massless, causing the gluonic loops contributing to the
functions entering Eq. (109) to also diverge, such that the cancellation
occurs among _all_ radiative diagrams. In contrast, in the presence of a gluon
mass, the cancellation of the remaining infrared divergences takes place at
the level of the ghost loops only, as illustrated diagrammatically in Fig. 20.

Figure 21:  ${\cal W}(r)$ obtained using the approximation Eq. (117) based on
the observed planar degeneracy of the three-gluon vertex in its kernel (blue
solid curve) together with uncertainty estimate (blue band).

We now return to the numerical determination of ${\cal W}(r)$ from Eq. (120).
To this end, we employ the fits to lattice data of Aguilar et al. (2021a) for
$\Delta(q)$ and $\mathit{L}_{{sg}}(q)$ shown in Figs. 9 and 11, respectively,
and the SDE solution for $F(q)$ shown in the left panel of Fig. 13. All of the
fits employed are constructed so as to reproduce the correct ultraviolet
behavior of the respective Green’s functions. For the value of the coupling in
the asymmetric MOM scheme we employ $g^{2}=4\pi\alpha_{s}$, with
$\alpha_{s}(4.3\text{ GeV})=0.27$, as determined in the lattice study of
Boucaud et al. (2017). Lastly, for $B_{1}$ we use the SDE result of Sec. 8,
shown in the right panel of Fig. 13, which reproduces accurately the available
lattice data for the ghost-gluon vertex.

Using the above ingredients into Eq. (120) we obtain the ${\cal W}(r)$ shown
as the blue solid curve in Fig. 21. The blue band in Fig. 21 represents the
error estimate on our results; the procedures followed to obtain it are
described in detail in Aguilar et al. (2022).

##  12 Displacement function from lattice inputs

In this section we determine ${\mathbb{C}}(r)$ from the main relation given in
Eq. (109).

For ${\cal W}(r)$ we use the result shown in Fig. 21, together with the curves
for $\mathit{L}_{{sg}}(r)$ from Fig. 11, $\Delta(r)$ and
$d\Delta^{-1}(r)/dr^{2}$ from Figs. 9 and 17, respectively, and the $F(r)$ of
Fig. 13. The ${\mathbb{C}}(r)$ obtained is shown as a black solid curve in the
left panel of Fig. 22. In the same panel, we show as points the estimates of
${\mathbb{C}}(r)$ obtained by using into Eq. (109) the lattice data points of
Ref. Aguilar et al. (2021a) _directly_ , rather than a fit. To estimate the
uncertainty in the resulting ${\mathbb{C}}(r)$, we combine the error estimate
of ${\cal W}(r)$, represented by the blue band in Fig. 21, with the
statistical error of the lattice data points for $\mathit{L}_{{sg}}(r)$ of
Aguilar et al. (2021a), and neglect the error in the gluon propagator, which
is much smaller than the errors in $\mathit{L}_{{sg}}$ and ${\cal W}$. Then, a
conservative error propagation analysis produces the error bars shown in Fig.
22.

Figure 22:  Left: Result for ${\mathbb{C}}(r)$ (black continuous line)
obtained from Eq. (109) using the ${\cal W}(r)$ shown in Fig. 21, the fits to
lattice data for $\Delta(r)$ and $\mathit{L}_{{sg}}(r)$ shown in Figs. 9 and
17, respectively, and the SDE solution for $F(r)$ shown in Fig. 13. The points
are obtained using for $\mathit{L}_{{sg}}(r)$ the data in Ref. Aguilar et al.
(2021a), with error bars denoting the error propagated from
$\mathit{L}_{{sg}}$ and ${\cal W}$. The green band is obtained by fitting the
upper and lower bounds of the points and guide the eye to the typical error
associated with ${\mathbb{C}}(r)$. Right: The ${\mathbb{C}}(r)$ of the left
panel is compared to the BSE prediction ${\mathbb{C}}_{\star}(r)$ (purple dot-
dashed and error band) of Fig. 7.

At this point, we quantify the significance of the ${\mathbb{C}}(r)$ obtained
above, in comparison to the null hypothesis result; evidently, in the absence
of the Schwinger mechanism, this latter quantity, to be denoted by
${\mathbb{C}}_{0}$ in what follows, vanishes identically, namely
${\mathbb{C}}_{0}=0$. To this end, we first compute the $\chi^{2}$ of our
points as

$\chi^{2}=\sum_{i}\frac{\left[{\mathbb{C}}(r_{i})-{\mathbb{C}}_{0}(r_{i})\right]^{2}}{\epsilon_{{\scriptscriptstyle{\mathbb{C}}(r_{i})}}^{2}}\,,$
(124)

i.e., the null hypothesis is taken as the estimator for our data. The sum runs
over the $n_{r}=515$ indices $i$ such that $r_{i}\in[0.3,4.3]$ GeV, the
interval of momenta for which the systematic error in our calculation of
${\cal W}(r)$ is best known, and
$\epsilon_{{\scriptscriptstyle{\mathbb{C}}(r_{i})}}$ denotes the error
estimate of ${\mathbb{C}}(r_{i})$. Then we obtain $\chi^{2}=2\,630$,
corresponding to $\chi^{2}_{\rm d.o.f.}=5.11$. The probability
$P_{{\mathbb{C}}_{0}}$ that our result for ${\mathbb{C}}$ is consistent with
the null hypothesis is vanishingly small, given by the formula

$P_{{\mathbb{C}}_{0}}=\int_{\chi^{2}=2\,630}^{\infty}\chi_{\rm{\scriptscriptstyle
PDF}}^{2}(515,x)dx=\left.\frac{\Gamma(n_{r}/2,\chi^{2}/2)}{\Gamma(n_{r}/2)}\right|_{n_{r}=515}^{\chi^{2}=2\,630}=7.3\times
10^{-280}\,.$ (125)

In fact, even if the errors were $0.95\%$ larger, i.e., nearly doubled, we
could still discard ${\mathbb{C}}_{0}$ at the $5\sigma$ confidence level.

In the right panel of Fig. 22 we compare ${\mathbb{C}}(r)$ to the BSE
prediction, ${\mathbb{C}}_{\star}(r)$, of Fig. 7, shown as a purple dot-dashed
curve and corresponding error band. In that panel, we observe an excellent
qualitative agreement between the two results. The most noticeable
quantitative difference is in the position of the minimum. Specifically,
${\mathbb{C}}$ reaches the minimum value of $-0.36\pm 0.11$ at
$r=1.93\begin{subarray}{c}+0.09\\\ -0.06\end{subarray}$ GeV, while the minimum
of ${\mathbb{C}}_{\star}$ is $-0.341\pm 0.003$ at $r=1.5\pm 0.1$.

Nevertheless, it is clear in the right panel of Fig. 22 that the BSE
prediction lies within the error estimate of the lattice-derived
${\mathbb{C}}(r)$. In fact, defining a $\chi^{2}$ measure for the discrepancy
between ${\mathbb{C}}$ and ${\mathbb{C}}_{\star}$ as

$\chi^{2}_{\star}=\sum_{i}\frac{\left[{\mathbb{C}}(r_{i})-{\mathbb{C}}_{\star}(r_{i})\right]^{2}}{\epsilon_{{\scriptscriptstyle{\mathbb{C}}(r_{i})}}^{2}}\,,$
(126)

we obtain $\chi^{2}_{\star}=258.5$, which is smaller than the number of
degrees of freedom. Then, this value of $\chi^{2}_{\star}$ amounts to a
probability of

$P_{{\mathbb{C}}_{\star}}=\left.\frac{\Gamma(n_{r}/2,\chi^{2}_{\star}/2)}{\Gamma(n_{r}/2)}\right|_{n_{r}=515}^{\chi^{2}_{\star}=258.5}=1-2.0\times
10^{-23}\,,$ (127)

showing that ${\mathbb{C}}_{\star}$ is statistically compatible with the
lattice derived ${\mathbb{C}}$, with probability extremely near unit.

##  13 Conclusions

The gauge sector of QCD is host to a wide array of subtle mechanisms that are
of vital importance for the self-consistency and infrared stability of the
theory. In the present work, we have offered a comprehensive review of the
intricate dynamics that account for some of the most prominent infrared
phenomena, such as the generation of a gluon mass through the action of the
Schwinger mechanism, the nonperturbative masslessness of the ghost, and the
characteristic features induced by this particular mass pattern to the form
factors of the three-gluon vertex.

The SDEs, supplemented by the judicious use of certain key results from
lattice QCD, provide a robust continuum framework for carrying out such a
demanding investigation. In fact, the results obtained from the SDEs are
increasingly reliable, passing successfully all sorts of tests imposed on
them. A particularly impressive, and certainly not isolated, case of such a
success has been outlined in detail in Section 6.

Symmetry and dynamics are tightly interwoven; therefore, the information
encoded in the STIs and WIs of the theory is particularly decisive for
unraveling basic dynamical patterns. A striking manifestation of the profound
connection between symmetry and dynamics is provided by the dual role played
by the function ${\mathbb{C}}(r)$: it is both the BS amplitude of the massless
states composed by a pair of gluons, and the quantity that embodies the
displacement induced to the WIs by the presence of these states.

In our opinion, the determination of ${\mathbb{C}}(r)$ described in Section 12
represents a major success of the entire set of concepts and techniques
surrounding the generation of a gluon mass through the action of the Schwinger
mechanism. Thus, fifty years after the genesis of QCD, we seem to be closing
in on the mechanism that the theory uses for curing the infrared instabilities
endemic to perturbation theory. We hope to be able to report further progress
in this direction in the near future.

The authors are supported by the Spanish MICINN grant PID2020-113334GB-I00. M.
N. F. acknowledges financial support from Generalitat Valenciana through
contract CIAPOS/2021/74. J. P. also acknowledges funding from the regional
Prometeo/2019/087 from the Generalitat Valenciana.

Not applicable.

###### Acknowledgements.

The authors thank A.C. Aguilar, D. Binosi, D. Ibáñez, J. Pawlowski, C.D.
Roberts, and J. Rodríguez-Quintero for several collaborations. The authors
declare no conflict of interest. Abbreviations The following abbreviations are
used in this work:
BFM | background field method
---|---
BQI | background-quantum identity
BRST | Becchi-Rouet-Stora-Tyutin
BS | Bethe-Salpeter
BSE | Bethe-Salpeter equation
EHM | emergent hadron mass
MOM | momentum subtraction (renormalization schemes)
PT | pinch technique
QCD | Quantum Chromodynamics
QED | Quantum Electrodynamics
RGI | renormalization group invariant
SDE | Schwinger-Dyson equation
STI | Slavnov-Taylor identity
WI | Ward identity
yes

##  Appendix A BQIs for the BSE amplitudes

In this Appendix, we use two particular BQIs in order to relate the
displacement functions ${\mathcal{C}}$ and ${\mathbb{C}}$ with their BFM
counterparts ${\widetilde{\mathcal{C}}}$ and $\widetilde{\mathbb{C}}$,
respectively.

The ghost-gluon vertices ${\rm{I}}\\!\Gamma_{\mu}(r,p,q)$ and
$\widetilde{{\rm{I}}\\!\Gamma}\vphantom{{\rm{I}}\\!\Gamma}_{\mu}(r,p,q)$ are
related by a BQI Binosi and Papavassiliou (2009), which reads

$\displaystyle\widetilde{{\rm{I}}\\!\Gamma}\vphantom{{\rm{I}}\\!\Gamma}_{\mu}(r,p,q)$
$\displaystyle=$
$\displaystyle\left\\{\left[1+G(q)\right]g^{\nu}_{\mu}+L(q)\frac{q_{\mu}q^{\nu}}{q^{2}}\right\\}{\rm{I}}\\!\Gamma_{\nu}(r,p,q)$
(128)
$\displaystyle+F^{-1}(p)p^{\nu}K_{\mu\nu}(r,q,p)+r^{2}F^{-1}(r)K_{\mu}(r,q,p)\,,$

where $K_{\mu}$ and $K_{\mu\nu}$ are two auxiliary functions, shown
diagrammatically in Fig. 23, while $G(q)$ and $L(q)$ are the form factors of
$\Lambda_{\mu\nu}(q)$, defined in Eq. (12).

Figure 23:  The auxiliary functions $K_{\mu}(q,r,p)$ and $K_{\mu\nu}(q,r,p)$,
appearing in the BQI of Eq. (128).

Next, we decompose the
$\widetilde{{\rm{I}}\\!\Gamma}\vphantom{{\rm{I}}\\!\Gamma}_{\mu}(r,p,q)$ and
${\rm{I}}\\!\Gamma_{\mu}(r,p,q)$ in Eq. (128) into their regular and pole
parts, using Eqs. (33) and (52), respectively. Note that the second and third
terms in Eq. (128) do not contain poles in $q^{2}$; this is so because
$K_{\mu\nu}(r,q,p)$ can contain (longitudinally coupled) poles only in the
$p_{\nu}$ channel, whereas $K_{\mu}(r,q,p)$ has no external gluon legs, and
hence no poles.

Then, multiplying Eq. (128) by $q^{2}$ we obtain

$q_{\mu}{\widetilde{C}}(r,p,q)=q_{\mu}\left[1+G(q)+L(q)\right]C(r,p,q)+{\cal
O}(q^{2})\,.$ (129)

Setting $q=0$ in Eq. (129) and using Eq. (18), we find

$C(r,-r,0)=Z_{1}F(0){\widetilde{C}}(r,-r,0)\,.$ (130)

Hence, using Eq. (61), we obtain the result in Eq. (39).

Then, expanding Eq. (129) to first order in $q$, using Eq. (41) for $C(r,p,q)$
and Eq. (62) for ${\widetilde{C}}(r,p,q)$, entails

${\mathcal{C}}(r)=Z_{1}F(0){\widetilde{\mathcal{C}}}(r)\,,$ (131)

which is one of the main results of this Appendix.

A relation identical to Eq. (131) can be obtained for ${\mathbb{C}}(r)$ and
its BFM analog, $\widetilde{\mathbb{C}}(r)$. The starting point of the
derivation is the BQI Binosi and Papavassiliou (2009)

$\displaystyle\widetilde{{\rm{I}}\\!\Gamma}\vphantom{{\rm{I}}\\!\Gamma}_{\alpha\mu\nu}(q,r,p)$
$\displaystyle=$
$\displaystyle\left\\{\left[1+G(q)\right]g^{\rho}_{\alpha}+L(q)\frac{q_{\alpha}q^{\rho}}{q^{2}}\right\\}{\rm{I}}\\!\Gamma_{\rho\mu\nu}(q,r,p)$
$\displaystyle+K_{\rho\nu\alpha}(r,q,p)P^{\rho}_{\mu}(r)\Delta^{-1}(r)-K_{\rho\mu\alpha}(p,q,r)P^{\rho}_{\nu}(p)\Delta^{-1}(p)\,,$

where $K_{\mu\nu\alpha}(r,q,p)$ is the function defined in Eq. (111).

Then, we note that the only longitudinal poles at $q=0$ present in Eq. (A) are
those contained in the ${\rm{I}}\\!\Gamma_{\alpha\mu\nu}(q,r,p)$ and
$\widetilde{{\rm{I}}\\!\Gamma}\vphantom{{\rm{I}}\\!\Gamma}_{\alpha\mu\nu}(q,r,p)$
vertices, with the auxiliary functions $K_{\alpha\nu\rho}(q,p,r)$ having poles
only in the $r_{\mu}$ and $p_{\nu}$ channels. As such, isolating the
$q_{\alpha}g_{\mu\nu}/q^{2}$ pole and expanding around $q=0$, one eventually
finds

${\widetilde{C}}_{1}(0,r,-r)=Z_{1}^{-1}F^{-1}(0)C_{1}(0,r,-r)=0\,,$ (133)

and

${\mathbb{C}}(r)=Z_{1}F(0)\widetilde{\mathbb{C}}(r)\,,$ (134)

where ${\widetilde{C}}_{1}(q,r,p)$ and $\widetilde{\mathbb{C}}(r^{2})$ are
defined in analogy to the Eqs. (37) and (41), and we used Eq. (39).

References

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