aainstitutetext: Departamento de Sistemas Físicos, Químicos y Naturales,
Universidad Pablo de Olavide,
Ctra. Utrera km 1, 41013, Spain.

# Restoring rotational invariance for lattice QCD propagators.

F. de Soto<EMAIL_ADDRESS>
###### Abstract

This note presents a method to reduce the discretization errors appearing when
solving a Quantum Field Theory in a hypercubic lattice in both position and
momentum-space. The method exploits the artifacts that break rotational
symmetry to recover rotationally invariant results for two-point Green
functions. We show that a combination of the results obtained in position and
momentum space can be useful to signal the presence of rotationally invariant
artifacts making use of their approximate Fourier transforms in the continuum.
The method will be introduced using a Klein-Gordon propagator, and a direct
application to gluon propagator in quenched lattice QCD will be given.

##  1 Introduction

The presence of discretization errors in lattice field theories has been
largely studied since the beginning of lattice QCD. Indeed, when space-time is
discretized in an homogeneous lattice of spacing $a$, any observable will –in
general– have errors of order $\mathcal{O}(a^{n})$, with $n=1,2,\cdots$
depending on the lattice action and operator chosen. In this line, Symanzik
designed a systematic method to reduce discretization errors where improved
operators and lattice actions Symanzik:1983dc ; Symanzik:1983gh ;
Luscher:1996sc ; Luscher:1996ug  are used to eliminate the dominant
counterparts, thus increasing the order, $n$, of the discretization errors
present. Discretization errors disappear once the continuum limit is taken,
i.e., by computing the desired observable at different lattice spacings and
extrapolating to the $a\to 0$ limit.

A particularly vexing effect of discretizing space-time in a square lattice is
rotational invariance breaking: while the continuum (Euclidean) space-time in
$d$-dimensions is invariant under the rotations group 111A reminiscent of the
Lorentz invariance in Minkowsky space. $SO(d)$, in the lattice this symmetry
is broken to the discrete isometry group $H(d)$. As a consequence, lattice
propagators in momentum space instead of being smooth functions of $k^{2}$
acquire a spurious (in the sense that it is not expected in the continuum)
dependence on higher order invariants of the lattice isometry group. This type
of discretization error is responsible for the known half-fishbone structure
observed in the gluon or quark propagators when one considers all the lattice
momenta Becirevic:1999hj ; Becirevic:1999uc ; Boucaud:2003dx ; Bonnet:2001uh
and, over the years, different strategies have been employed to mitigate the
impact of rotational invariance breaking over the lattice results. Among them,
choosing only some particular values of the momenta Leinweber:1998uu , using a
tree-level correction Bonnet:2001uh ; Constantinou_2009  or extrapolating to
zero the higher order invariants of the lattice group deSoto:2007ht ;
Catumba:2021hcx .

The quest for small pion masses and lattice spacings has reached a new era in
lattice-QCD in the last decade, with several collaborations achieving pion
masses close to the physical value and lattice spacings below $0.1$fm with
$N_{f}=2$, $2+1$ and $2+1+1$ dynamical quark flavours
FlavourLatticeAveragingGroup:2019iem . The impact of discretization errors is
mitigated by the use of small lattice spacings, but they are still relevant as
there is a competition between discretization and statistical errors. As the
number of configurations increases, the statistical errors are reduced and
discretization ones become more relevant. In any case, the quest for precision
measurements reinforces the necessity for a careful treatment of
discretization errors.

While the impact of symmetry breaking artifacts in momentum space has been
widely studied, for correlation functions in position space one encounters
that the effects of breaking the rotational invariance are difficult to deal
with. Part of the difficulty is due to the fact that discretization errors
more relevant for small distances than for large ones (while in momentum space
they are relevant at large momenta). This has been found, for example, in the
lattice determination of the quenched potential (See Fig. 4.3 in Bali:2000gf
), or the renormalization constants for the vector and axial-vector quark
bilinears, $Z_{V}$ and $Z_{A}$ Aoki_2004 . In general, symmetry breaking
artifacts in position space are not so well understood as in momentum space
and are difficult to eliminate.

Discretization artifacts associated to rotational invariance breaking are
particularly relevant for observables such as gluon or ghost propagators
Boucaud:2017ksi ; Boucaud:2018xup ; Aguilar:2019uob ; Aguilar:2021okw , where
statistical errors are well under control. Indeed, the presence of
uncontrolled discretization errors in the large-momentum region of the
renormalization scale may have an impact on the low-momentum domain. The
effect would be similar to the one detected when there is a systematic
deviation in the lattice scale setting, as discussed in Boucaud:2017ksi .
Although they are also present for the lattice estimate of three-point
functions such as the three-gluon vertex, the typically larger statistical
errors prevent appreciating the effects of rotational invariance breaking.

In this paper we propose an unified method to deal with hypercubic errors both
in position and momentum space based in a generalization of the one presented
in deSoto:2007ht . Moreover, we will introduce the use of an approximate
Fourier transform of the lattice data after appropriate treatment of the
hypercubic errors in both position and momentum-space that can be used to
detect the presence of uncorrected O4-invariant discretization or finite-
volume errors.

The structure of the paper is as follows: in section 2 we will use the
propagator of a free massive field to discuss the rotational symmetry breaking
artifacts in both momentum and position space. Based in this analysis, we will
introduce a method to cure them in section 3 and apply it to the free boson
case where the exact solution is known in this section. In section 4, we will
use a Fourier transform to check the rotational invariance of the extrapolated
data. Finally, we will apply this method in section 5 to the quenched QCD
gluon propagator in position and momentum space for a small lattice.

##  2 Analysis of Klein-Gordon propagator

In order to get a pedagogical introduction to the problem, we will start by
considering the propagator of a free massive boson in momentum space. It has
the advantage that it can be straightforwardly obtained in either a discrete
or continuum space and both in a finite box or in an infinite space. In this
sense, two considerations are in order: i) when working with discrete
positions one has to make an approximation on how derivatives are written in
the discrete space, and ii) dealing with finite volumes implies making some
assumption on the boundary conditions. For the cases illustrated here, we
considered the simplest cases of forward derivative

$\displaystyle\frac{\partial\psi(x)}{\partial
x}\to\frac{\psi(x+a)-\psi(x)}{a}\ ,$ (1)

and periodic boundary conditions

$\displaystyle\psi(x+L)=\psi(x)\ ,$ (2)

respectively. We will also focus in the four-dimensional case and assume
through the manuscript that both the lattice spacing, $a$, and the number of
lattice sites, $N$, are equal for all the four space-time directions. While
the former is a common choice in lattice QCD, the latter is not so widely used
in practice, where lattices tend to be more elongated in time direction.

For the purpose of illustrating discretization and finite-volume errors, we
have considered four cases:

  * i)

continuum ($x$ takes any real value)

  * ii)

a discrete lattice in position space with $x=an$ and $n$ any integer number
(note that this case considers an infinite volume),

  * iii)

a finite box of size $L$ with periodic boundary conditions with $x$ taking any
value in the range $(0,L)$, and

  * iv)

the lattice case, with $x$ discrete and belonging to a box with periodic
boundary conditions.

For latter convenience, it is important to remark that the possible values of
the momenta in each one of the cases mentioned above are determined by the
restrictions in the positions. In this sense, if each component of the
position takes the discrete form $x_{i}=an_{i}$, where $a$ is the lattice
spacing and $n_{i}$ an integer, the accessible momenta will be restricted to
the range $(0,2\pi/a)$, while if the positions are restricted to be in a range
$(0,L)$, the accessible momenta will be discrete and of the form
$\frac{2\pi}{L}n_{i}$. Those relations between positions and momenta have been
detailed in table 1

Case | position | momenta
---|---|---
i) Continuum, infinite volume | $x_{\mu}\in R$ | $k_{\mu}\in R$
ii) Discrete, infinite volume | $x_{\mu}=a\ n_{i}$ | $k_{\mu}\in\left(0,\frac{2\pi}{a}\right)$
| $n_{\mu}\in\text{Integers}$ | p.b.c.
iii) Continuum, finite volume | $x_{\mu}=\in(0,L)$ | $k_{\mu}=\frac{2\pi}{L}n_{\mu}$
| p.b.c. | $n_{\mu}\in\text{Integers}$
iv) Discrete, finite volume | $x_{\mu}=a\ n_{\mu}$ | $k_{\mu}=\frac{2\pi}{Na}n_{\mu}$
(lattice) | $n_{\mu}=0,1,\cdots,N-1$ | $n_{\mu}=0,1,\cdots,N-1$
Table 1:  Possible values of each component of positions and momenta for each
one of the four cases considered.

###  2.1 Klein-Gordon propagator

#### Momentum space.

The propagator of a free boson field of mass $\mu$ in a continuum, infinite
space (case i) in Tab. 1) is given by:

$\Delta_{\text{cont}}(k)=\frac{1}{k^{2}+\mu^{2}}\ .$ (3)

with $k^{2}=\sum_{\mu=1}^{4}k_{\mu}^{2}$. The same expression holds when
working in a finite box with periodic boundary conditions (case iii) in
Tab.1), with the particularity that in this case only discrete values of the
momenta are allowed.

If a discretization is introduced in position space so that the allowed
positions take the form $x_{i}=an_{i}$, the propagator has the form:

$\Delta(k)=\frac{1}{\hat{k}^{2}+\mu^{2}}\ ,$ (4)

where $\hat{k}^{2}=\sum_{\mu=1}^{4}\hat{k}_{\mu}^{2}$ and $\hat{k}_{\mu}$ are
the lattice momenta that, using the forward discretization of derivatives (1),
take the form: $\hat{k}_{\mu}=\frac{2}{a}\sin\frac{ak_{\mu}}{2}$. Note that
this expression holds for discrete positions, both in the case they span over
an infinite volume or are limited to a finite lattice (cases ii) and iv) in
Tab. 1 respectively). The only difference between those two cases is that in
the former $k_{i}$ can take any real value in the range $(0,2\pi/a)$ while in
the latter the allowed momenta are discrete with
$k_{\mu}=\frac{2\pi}{Na}n_{\mu}$ with $n_{\mu}=0,1,\cdots,N-1$.

It is noticeable that the presence of a finite volume does not affect the
continuum propagator (3). This means that a free boson propagator in a finite
volume will be identical –for the momenta where it can be evaluated– to the
infinite volume result. This is no longer true in an interacting theory, where
the interactions can introduce a volume dependence. In any case, the fact that
finite volume effects do not affect the tree-level propagator may be at the
origin of the subdominant role played by finite volume effects in gluon
propagator. It is now widely accepted that finite-volume effects are almost
negligible Boucaud:2011ug  above $\sim 0.5{\rm GeV}$ but they become an
important issue as the infrared limit is approached Oliveira:2005hg ;
Bogolubsky:2007bw ; Cucchieri:2008mv ; Oliveira:2009nn ; Bornyakov:2009ug .
The modern capability of producing very large volume lattice simulations with
a lattice size $L\gtrsim 10{\rm fm}$ significatively reduces even more the
impact of finite volume errors Boucaud:2018xup ; Aguilar:2019uob . For the
particular case of gluon propagator, taking the infinite volume limit for the
space of local gauge transformations Zwanziger:1993dh ; Cucchieri:2016qyc  has
proven to be an efficient method to significantly reduce finite volume errors
affecting the low momenta. This would mean that finite volume effects for the
deep IR of gluon propagator would be associated to the Landau gauge-fixing in
the lattice and can be therefore cured by replicating the finite lattice and
fixing the gauge over the extended configuration.

#### Position space.

The position space propagator can be obtained from the momentum space one via
Fourier transform. In the case of a four-dimensional continuum, infinite space
it is given by:

$\Delta_{\text{cont}}(x)=\int\
\frac{d^{4}p}{(2\pi)^{4}}e^{-ipx}\Delta_{\text{Cont}}(p)=\frac{1}{4\pi^{2}x}\int_{0}^{\infty}dp\
p^{2}\Delta_{\text{Cont}}(p)J_{1}(px)=\frac{\mu\ K_{1}(\mu\ x)}{4\pi^{2}x}\ ,$
(5)

where we used the rotational symmetry for integrating out the angular part and
$J_{n}(z)$ and $K_{n}(z)$ represent Bessel functions of the first kind and
modified second kind respectively.

In an box of finite volume, the position space propagator is given by:

$\Delta(x)=\frac{1}{L^{4}}\sum_{k}{\Delta}(k)e^{-ikx}$ (6)

where

  * •

for the continuum case (labeled iii) in Tab. 1) the sum is extended to an
infinite number of momenta in the four dimensions of space-time and the
${\Delta}(k)$ corresponds to Eq. (3).

  * •

for the discrete case (labeled iv) in Tab. 1) the sum is extended to the
limited set of accesible momenta, $k_{\mu}=\frac{2\pi}{Na}n_{\mu}$, and
${\Delta}(k)$ corresponds to Eq. (4).

Finally, for the infinite-volume discrete case (labeled ii) in Tab.1) the
values of the propagator in position-space are obtained through:

$\Delta(x)=\int_{k_{\mu}\in\left(0,\frac{2\pi}{a}\right)}d^{4}k\
\Delta(k)e^{-ikx}$ (7)

where $\Delta(k)$ is given by Eq.(4).

From their construction, the deviations of Eq. (6) for the continuum case with
respect to Eq. (5) will be finite-volume errors while the deviations of Eq.
(7) with respect to Eq. (5) will be discretization errors. The lattice
propagator obtained using the discrete momentum space propagator (4) in (6)
will include a mixture of both discretization and finite-volume errors and
constitutes the actual object of study of this paper.

###  2.2 Rotational symmetry breaking artifacts in Klein-Gordon propagator

#### Momentum space

In order to illustrate the role of lattice artifacts associated to the
rotational symmetry breaking, we have plotted in Fig.1(a) the ratio

$\displaystyle\frac{{\Delta}(k)}{{\Delta}_{\rm cont}(k)}$ (8)

between the momentum-space Klein-Gordon propagator in a lattice (4) and its
continuum counterpart (3) for a lattice of size $N=32$ and a boson mass
222From now on, all dimensional quantities are expressed in terms of $a$. The
value $a\mu=0.1$ chosen would correspond to a boson of mass $\mu\sim 0.4{\rm
GeV}$ for a lattice spacing $a\sim 0.05{\rm fm}$. Note that those values are
in the approximate range of the dynamically generated gluon mass
Aguilar:2019uob  and typical values of spacings in current lattice QCD
calculations. $a\mu=0.1$. As this ratio should be $1$ in the continuum limit,
deviations from $1$ will be discretization errors. In the plot, each momentum
$(k_{1},k_{2},k_{3},k_{4})$ is represented separately. One observes that the
plot not only deviates from $1$, but some values of the squared momenta,
$k^{2}$, there are multiple values of the propagator. This is due to the fact
that momenta not related by the discrete lattice group H(4), i.e., by
permutations among the components of $k$ or sign changes, give a different
result. We will refer to each set of momenta related by the H(4) symmetry
group as orbit. Thus, instead of obtaining a smooth function that only depends
on $k^{2}$ as in the continuum, one obtains a different value of the
propagator for each orbit. Indeed, it depends not only on $k^{2}$ but also on
the higher order invariants of the H(4) group:
$k^{[4]}=\sum_{\mu}k_{\mu}^{4}$, $k^{[6]}=\sum_{\mu}k_{\mu}^{6}$, and
$k^{[8]}=\sum_{\mu}k_{\mu}^{8}$. To illustrate this, we picked the invariant
$k^{[4]}$, which will have the leading role, and used the dimensionless ratio:

$\frac{k^{[4]}}{(k^{2})^{2}}$ (9)

to set the color scale of Fig.1(a). Note that $\frac{k^{[4]}}{(k^{2})^{2}}$
takes values ranging from $1/4$ for momenta distributed along the lattice
diagonal, i.e. as $(\frac{k}{2},\frac{k}{2},\frac{k}{2},\frac{k}{2})$ to $1$
for momenta along an axis, $(k,0,0,0)$.

At first sight, discretization errors are highly correlated with the value of
$\frac{k^{[4]}}{(k^{2})^{2}}$, being the orbits whose momentum is maximally
shared among the four directions (called democratic orbits) closest to the
continuum result and the orbits with the momentum along an axis the most
distant ones. The deviations from $1$ in Fig. 1(a) are responsible for the
well-known half-fishbone structures that tend to appear in momentum-space
propagators Becirevic:1999hj ; Boucaud:2011ug . For comparison, we have also
plotted the lines for the momentum-space propagator in Eq. (4) for an
infinite-volume (where each component of the momenta can take any real value),
for momenta with extreme values of the ratio in Eq. (9), i.e. along the
lattice diagonal and axis.

This picture is, of course, very well known, and has triggered different
methods to recover rotationally invariant results. The role of the H(4)
invariants in the appearance of these discretization errors can be visualized
from a Taylor series of the discrete propagator (4) in the lattice spacing
$a$,

$\Delta(k)=\frac{1}{\hat{k}^{2}+\mu^{2}}=\frac{1}{k^{2}+\mu^{2}}+\frac{a^{2}}{12}\left(\frac{1}{k^{2}+\mu^{2}}\right)^{2}k^{[4]}+...\
,$ (10)

whose first term is proportional to the H(4) invariant $k^{[4]}$ (see
deSoto:2007ht  for more details). The fact that momenta along the lattice
diagonal have the minimum 333Recall that democratic orbits
$(\frac{k}{2},\frac{k}{2},\frac{k}{2},\frac{k}{2})$ have the minimum $k^{[4]}$
among all the orbits with the same $k^{2}$. contribution to the leading
$a^{2}$ correction in (10) are the origin of the democratic methods
Leinweber:1998uu  where only orbits along or close to the diagonal are
considered (the rest is discarded). H4 extrapolation methods deSoto:2007ht  go
one step forward and use all the lattice orbits to extrapolate the higher
order invariants to zero. A different and very successful course of action
Bonnet:2001uh  is plotting the lattice data as a function of $\hat{k}^{2}$
instead of $k^{2}$ and use a correction to the lattice propagator based in the
lattice tree-level propagator, ${\Delta}_{\rm TL}(k)$:

$\displaystyle{\Delta}_{\rm Corr}(k)={\Delta}_{\rm Latt}(k)\frac{{\Delta}_{\rm
Cont}(k)}{{\Delta}_{\rm TL}(k)}\ .$ (11)

For the free case studied in this section, this method is exact, and the use
of any of those two conditions separately would provide the continuum result
444Using $\hat{k}$ already solves the problem as (4) is the same function than
(3) changing $k$ by $\hat{k}$. On the other hand, as the lattice result is
tree-level, the ratio in (11) equals the continuum result by definition..

|
---|---
(a) | (b)
Figure 1: Ratio of the lattice Klein-Gordon propagator for $N=32$ and
$a\mu=0.1$ and its continuum counterpart in momentum (a) and position (b)
space. In (a) full lines have been added for the two limiting cases of orbit
along an axis and along the diagonal in a continuum box. In (b), empty symbols
represent the discrete infinite volume case. In both plots, the color map
corresponds to the dimensionless ratio $n^{[4]}/(n^{2}\cdot n^{2})$.

#### Position space

In analogy with what has been done in momentum space, the ratio of the Klein-
Gordon propagator in position-space in a lattice (7) to the continuum one (5)
has been plotted in Fig. 1(b) for the same parameters. The color scale
indicates in this case the value of the dimensionless ratio:

$\frac{x^{[4]}}{(x^{2})^{2}}\ .$ (12)

The plot includes both the lattice propagator for a finite volume (full
symbols) and an infinite lattice (empty symbols). Both sets show a large
spreading of the data for small distances, while for large $x$ they tend to
concentrate along a curve (around one for the infinite volume case and away
from one for the finite-volume case). It is important to remark that the
difference between both sets is a finite-volume effect which is responsible
for the differences at large distances. For short distances, nevertheless, the
data are almost identical in the two sets signaling the presence of rotational
symmetry breaking discretization artifacts that are large for this region.

If one considers the infinite-volume data (to avoid mixing in our discussion
finite-volume and discretization errors) of Fig. 1(b), one can observe that
democratic orbits tend to be below $1$ in the plot, and therefore, the method
based in a cylindrical cut along the diagonal will not work in general in
position space. Indeed, a closer look to the data shows that in this case the
“best” orbits (in the sense they are closer to the continuum result) have
$x^{[4]}/(x^{2})^{2}\sim 1/2$.

Contrarily to the momentum-space case, discretization and finite-volume
artifacts are not well known in position-space and, up to the knowledge of the
author, the only method that has been applied in position space
PhysRevLett.125.242002  is a tree-level correction as the one sketched in Eq.
(11) for momentum-space. Among the reasons that may explain why discretization
errors are harder to eliminate in position space, one can signal the fact that
while in momentum space, discretization errors have a sizable effect for large
values of $k^{2}$, where there are many lattice orbits
$(k_{1},k_{2},k_{2},k_{4})$ with the same $k^{2}$, in position space
discretization errors affect mainly the small-x region, where few orbits are
available. This asymmetry explains why extrapolations in $k^{[4]}$ efficiently
remove discretization errors while in $x^{[4]}$ the results are rather poor.
Moreover, while in momentum space democratic orbits can be used to reduce (at
least partially) discretization errors, it is no longer true in position-
space, where democratic orbits for small-$x$, where it exhibits large
deviations from the continuum value (see Fig. 1(b)). In next section we will
propose a method to deal with discretization (and possibly finite-volume)
errors in an unified way for both momentum and position-space.

##  3 Generalized H4 method

The observation Becirevic:1999hj ; Becirevic:1999uc  of a linear trend in
$k^{[4]}$ in the lattice data for any fixed value of $k^{2}$ motivated the
introduction of the so-called H4 extrapolation methods, that use the different
orbits with the same $k^{2}$ to make an extrapolation of $k^{[4]}$ to zero.
This fit, according to Eq.(10), would remove the leading discretization errors
of order $\mathcal{O}(a^{2})$ for a free massive boson propagator. In this
paper we generalize and extend the method originally developed in
Becirevic:1999uc ; Becirevic:1999hj ; Boucaud:2003dx  and described in detail
in deSoto:2007ht .

H4-extrapolation methods start by assuming an expansion 555For clarity only
the leading term in the invariant $k^{[4]}$ has been included here, although
higher order terms can be included, as detailed in deSoto:2007ht . A more
recent application of the method that takes into account $k^{[6]}$ and
${k^{[4]}}\cdot{k^{[4]}}$ terms can be found in Oliveira:2018lln . inspired by
the Taylor series in (10:)

$F(k^{2})=F(k^{2},k^{[4]})+a^{2}\ C(k^{2})\ k^{[4]}$ (13)

where $C(k^{2})$ is some function of $k^{2}$ that is fitted either separately
for each $k^{2}$ or –assuming it is a smooth function of $k^{2}$– for a range
of values of $k^{2}$. This method has been successfully used over the last two
decades for eliminating H4 errors from lattice gluon and ghost propagators
Boucaud:2011ug ; Boucaud:2018xup ; Aguilar:2019uob . The role played by higher
order corrections (such as $k^{[6]}$, $(k^{[4]})^{2}$, etc) is particularly
relevant for large momenta and it is therefore dangerous to extent the
H4-extrapolation to very large momenta without adding more terms. In general,
we limited the range of momenta to
$k^{2}_{\max}<\left(\frac{2\pi}{Na}\right)^{2}n_{\max}^{2}$, with
$n_{\max}=N/4$. A noticeable effort in the extension of this method for the
understanding of the structure of lattice gluon propagator and its
discretization artifacts using a different point of view has been presented in
Catumba:2021hcx ; Catumba:2021pdu .

In order to generalize the method for both positions and momenta, let us re-
write this expression i) in terms of $n^{2}$, $n^{[4]}$ which is a notation
valid both for momentum-space where $k^{2}=(\frac{2\pi}{L})^{2}\ n^{2}$ and
position-space with $x^{2}=a^{2}\ n^{2}$, ii) using the dimensionless ratio in
Eqs. (9-12) –note that this ratio is dimensionless both in position and
momentum space and can be written as $n^{[4]}/{(n^{2})}^{2}$, iii) as a
multiplicative correction, which will allow a dimensional analysis exercise
and is flexible to be applied to a wide variety of functions, and iv) adding a
parameter $\alpha_{4}$ to account for the fact that the $n^{[4]}\to 0$ limit
does not necessarily recover the continuum results (as it is observed in
position space, remind Fig. 1b). With all, it results:

$F(n^{2})=F(n^{2},n^{[4]})\times\left[1+c(n^{2})\left(\frac{n^{[4]}}{(n^{2})^{2}}-\alpha_{4}\right)\right]$
(14)

Note that Eq. (13) is recovered from Eq. (14) with $\alpha_{4}=0$ and
$a^{2}C(p^{2})\propto F(n^{2})\times\frac{c(n^{2})}{(n^{2})^{2}}$. In this
work, instead of fitting $c(n^{2})$ for each $n^{2}$ as an independent
parameter, we will make a global fit for all the values of $n^{2}$, searching
the optimal functional form of $c(n^{2})$. It is important to remark that this
is still only the first correction, and therefore it should not be extended to
too large values of $n^{2}$ where higher order invariants such as $n^{[6]}$
may play an important role.

The addition of the $\alpha_{4}$ term in Eq. (14) is not an H4-breaking term
but rather it prevents the H4 extrapolation to introduce O4 errors, which
would be the case in position-space as can be observed in Fig. 1(b) and will
be discussed below. It is important to remark that eliminating H4 errors
produce smooth results (in contrast with the noisy fluctuations obtained
otherwise) but not necessarily eliminates discretization errors. For
illustration, consider democratic orbits in Fig. 1(a); they have a smooth
behavior in $k^{2}$ but it is clearly different from one, and indeed shows a
linear trend with $k^{2}$.

#### Application to momentum-space

Let us consider the data for the lattice boson propagator in momentum space
and apply Eq. (14). We take the lattice data $F(n^{2},n^{[4]})$ for
$n^{2}<(N/4)^{2}$ and fit the artifact-free value $F(n^{2})$ and the function
$c(n^{2})$ using $\alpha_{4}=0$. To avoid any a priori guess about $c(n^{2})$,
we made a fit 666For the purpose of these fits, we have assigned an arbitrary
statistical error of $0.1\%$ to all the exact lattice data used. Note that
this value is not unrealistic in current lattice simulations with a large
number of gauge field configurations. using $c(n^{2})=c_{m}(n^{2})^{m}$, with
$m$ an integer number and $c_{m}$ a free parameter. The result obtained is
that the minimum $\chi^{2}$ is obtained for $m=1$. Indeed, we could have
anticipated this result using a simple dimensional analysis, as the simplest
way of obtaining a $k^{2}$-dependent term of order $a^{2}$ is $a^{2}k^{2}$
which implies for the dimensionless function $c(n^{2})\propto n^{2}$.

Once we know the fit prefers positive powers of $n^{2}$, we finally use a more
general form such as:

$c(n^{2})=c_{0}+c_{1}n^{2}+c_{2}(n^{2})^{2}$ (15)

and the parameters $c_{0}$, $c_{1}$, $c_{2}$ as well as the extrapolated
values $F(n^{2})$ for each $n^{2}$ are the free parameters of the fit. Note
that the number of orbits used (i.e., the number of input values
$F_{L}(n^{2},n^{[4]})$) is much larger than the number of fitted quantities
($n^{2}+3$ in this case). As an example with the four dimensional lattice of
$N=32$ points in each direction used, and keeping only orbits up to
$n^{2}=(L/4)^{2}$ we have $157$ orbits for $64$ values of $n^{2}$ and the
three parameters $c_{0}$, $c_{1}$, and $c_{2}$. It is important to remark that
the number of orbits grows very fast with $n^{2}$, and faster for larger
space-time dimensions.

Fitting the exact propagator in Eq. (4) with the form (14) and using
$\alpha_{4}=0$ and the expansion (15), we obtain the coefficients:
$c_{0}=(0.5\pm 1.6)\cdot 10^{-3}$, $c_{1}=(-3.2\pm 0.1)\cdot 10^{-3}$ and
$c_{2}=(4\pm 1)\cdot 10^{-6}$. With the results of the fit we have plotted in
Fig. 2(a) the ratio of H4 extrapolated propagator to the continuum one. In the
plot one can observe that the deviation from the exact continuum propagator
after the H4 extrapolation is very small (below $1\%$), while using the
democratic method it would have a already a bias of $\sim 5\%$ for the end of
our fitting window.

Notice that the value obtained in the fit for the dominant contribution,
$c_{1}$, is compatible with the one that can be deduced from Eq.(11) of Ref.
deSoto:2007ht , which would be
$-\frac{1}{12}\left(\frac{2\pi}{N}\right)^{2}\approx-3.213\cdot 10^{-3}$ for
$N=32$.

#### Application to position-space

Let us now apply the method explained above to the position-space data
represented in Fig. 1(b). We already discussed above that breaking the
rotational symmetry has a larger impact for small distances (small $n^{2}$).
This affects the capabilities of the traditional H4-extrapolation methods for
eliminating the artifacts because in this region there is a small number of
orbits. An additional difficulty is that the “optimal” value of the ratio (12)
seems to be $1/2$ instead of $0$ in this case. Let us fix $\alpha_{4}=1/2$
based in this empirical observation (we will come back to this issue later)
and perform the same analysis we did in momentum-space. The minimum
$\chi^{2}$-fit is obtained in this case for $c(n^{2})\sim 1/n^{2}$, which
again can be inferred from dimensional analysis, as a $x^{2}$-dependent
artifact of order $(a^{2})$ is of the form $a^{2}/x^{2}\sim(n^{2})^{-1}$.

The result of the H4 extrapolation 777To maintain the name of the method, we
use here the term “extrapolation” although in this case the $\alpha_{4}=1/2$
lies in the middle of the lattice orbits. is represented in Fig. 2(b) using in
this case

$c(n^{2})=c_{0}+\frac{c_{-1}}{n^{2}}+\frac{c_{-2}}{(n^{2})^{2}}$ (16)

with the best-fit values $c_{0}=(3.1\pm 0.9)\cdot 10^{-3}$, $c_{-1}=-2.71\pm
0.03$, and $c_{-2}=2.8\pm 0.1$. We checked that adding more terms to the
development in (16) does not significantly increase the quality of the fit.

|
---|---
(a) | (b)
Figure 2: Ratio of Klein-Gordon propagator in a $32^{4}$ lattice with respect
to the continuum one in momentum space (a) and position-space (b). Grey
circles represent the original lattice data and green squares the result of
applying the generalized H4 method. For comparison, the results of a
democratic cut with an angle smaller than $30^{\circ}$ with diagonal have been
added as light blue stars.

Although the magnitude of the errors in the data is highly reduced in this
case as well, there are two features in Fig. 2(b) which contrast with the
situation in the momentum-space case of Fig. 2(a). First, there is a non-
vanishing slope in $x^{2}$ in the data after applying the H4 extrapolation.
This signals the presence of finite volume errors that grow with $(x/L)^{2}$
and that are not present in the infinite-volume discrete data of Fig. 1(b).
Second, there is still a noticeable noise in the data for small $x^{2}$
(mainly below $(x/a)^{2}\sim 15$). This problem seems to be related to the
fact that the region with larger discretization errors is also the one with a
smaller number of orbits, as discussed above.

In our application of the generalised H4 extrapolation method both in momentum
and position space we focused in minimizing $\chi^{2}$, i.e., obtaining the
best functional description of the data using Eq.(14). The method highly
reduces the noise associated to the symmetry breaking artifacts and one could
add an extra condition over the smoothness of the extrapolated results.
Although it could provide an useful guide, we preferred here not to impose
smoothness, as it might add uncontrolled biases to the results.

##  4 Continuous Fourier transforms and O4-invariant artifacts

Even if the noise associated with rotational breaking artifacts had been
completely removed, the smooth results obtained could still be affected by
O4-symmetric discretization errors. In some cases, the method used to recover
rotational invariance may introduce an O4-symmetric artifact. Nevertheless,
the propagators in the continuum infinite-volume limit in position and
momentum space (let us call them generically $F(x)$ and ${F}(k)$) must be one
the Fourier transform of the other, i.e.:

$\displaystyle{F}(k)$ $\displaystyle=$ $\displaystyle\int d^{4}x\
e^{-ikx}F(x)=\frac{4\pi^{2}}{k}\int_{0}^{\infty}dx\ x^{2}\
J_{1}(kx)F(x)\quad\Leftrightarrow$ (17) $\displaystyle F(x)$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}e^{ikx}{F}(k)=\frac{1}{4\pi^{2}x}\int_{0}^{\infty}dk\
k^{2}\ J_{1}(kx){F}(k)$ (18)

where in the second equality the angular part has been integrated out.

Once we have applied the generalized H4 method introduced in previous section,
we can use the expressions (17) and (18) to check whether the functions
obtained satisfy those equations. If we had correctly recovered the continuum-
infinite-volume results through the suppression of the spurious dependence in
the higher order invariants of the H4 group, the Fourier transform of the
extrapolated result in position space would coincide with the extrapolated
result in momentum-space and vice-versa.

FFT$F_{latt}(x)$${F}_{latt}(k)$H4
method$F_{H4}(x)$${F}_{H4}(k)$$\mathcal{F}^{-1}$$\mathcal{F}$$F_{FT}(x)$${F}_{FT}(k)$
Figure 3: Schematic representation of the definition of propagators via
Fourier transforms in the continuum of the H4-extrapolated data. If the method
removed all lattice artifacts (both discretization and finite volume ones),
$F_{FT}(x)$ and $F_{H4}(x)$ would coincide, and the same can be said about
momentum-space propagators.

Computing the Fourier transform of the propagators that we know only at some
discrete values of the distance or momentum (those corresponding to an integer
value of $n^{2}$) is not straightforward. In order to compute the integrals in
Eqs.(17-18), we need a continuous description of $F(x)$ for all values of $x$
between $0$ and $\infty$ (and respectively ${F}(k)$ for all values of $k$). As
we have limited information for integer values of $n$, we are forced to make
some approximations. In the present work we have done the following:

  * •

For small values of $n^{2}$ we use a Padé approximant

$\frac{a_{0}+a_{1}n^{2}+a_{2}(n^{2})^{2}}{1+b_{1}n^{2}+b_{2}(n^{2})^{2}}$ (19)

whose coefficients are by obtained from a fit $n^{2}<n^{2}_{\text{m}in}$ in
each case. This behavior is assumed to describe the continuous function
between $n^{2}=0$ and $n^{2}_{\text{m}in}$. We fixed $n^{2}_{\text{m}in}=6$
for this work.

  * •

Among any two consecutive values of $n^{2}$ we assume a linear interpolation
between the lattice points.

  * •

For values of $n^{2}$ larger than the maximum value of $n^{2}$ considered,
$n^{2}_{\text{m}ax}=(N/4)^{2}$, we extend the propagator with the functional
form:

$\frac{d_{1}}{n^{2}}+\frac{d_{2}}{(n^{2})^{2}}+\frac{d_{3}}{(n^{2})^{3}}$ (20)

where the coefficients $d_{i}$ are fitted over the largest values of $n^{2}$
accessible.

Once we have set a continuous description of the discrete data, we perform
numerically the integrals in Eqs.(17-18) to obtain an approximation to the
Fourier transforms in the continuum. Although there is an unavoidable bias due
to our lack of information (in particular for the limits $n^{2}\to 0$ and
$n^{2}\to\infty$), this procedure allows us to make a comparison between the
H4-extrapolated propagator $\Delta_{H4}(x)$ and the one obtained through
Fourier transform of ${\Delta}_{H4}(k)$, ${\Delta}_{FT}(x)$.

In order to show the results of the Fourier transform, we will use the ratio
of the H4-extrapolated propagator in momentum space, ${\Delta}_{H4}(k)$,
divided by the Fourier transform of the H4-extrapolated one in position space,

${\Delta}_{FT}(k)=\frac{4\pi^{2}}{k}\int_{0}^{\infty}dx\ x^{2}\
J_{1}(kx)\Delta_{H4}(x).$ (21)

This ratio would be $1$ if we had fully eliminated lattice errors and if
Fourier transform was exact. We have represented this ratio for the Klein-
Gordon case studied in the previous section in Fig. 4(left) for five different
values of the parameter $\alpha_{4}$ used in the position-space H4
extrapolation ranging from $\alpha_{4}=0$ to $\alpha_{4}=1$. The figure shows
that except in the case $\alpha_{4}=0.5$, the ratio deviates significantly
from $1$, providing a sound argument for the choice of $\alpha_{4}=0.5$ we
made. Let us remark that within this plot we did not use the exact solution in
the continuum infinite-volume for this case, but only the results of the H4
extrapolation of the lattice data in both position and momentum space and the
Fourier transform.

|
---|---
Figure 4: Ratios of the Klein-Gordon propagator obtained through the
generalized H4 method and the one coming from the approximate continuum
Fourier transform. Left plot corresponds to momentum-space and right on to
position-space. In both plots, each set corresponds to a different value of
the parameter $\alpha_{4}$.

We can play the same game in the opposite direction of the Fourier transform,
i.e., compute the ratio of the H4-extrapolated propagator in position space to
the Fourier transform of the H4-extrapolated one in momentum space,
$\Delta_{FT}(x)$. The results are plotted in Fig. 4(right), and show that
there is a large bias for small distances in this case except for
$\alpha_{4}=0.5$. In this case, we can also identify a non-negligible slope in
$x^{2}$ for large values of $x^{2}$. This may correspond to a finite volume
error in $\Delta_{H4}(x)$, although we cannot exclude a bias introduced by the
inverse Fourier transform of ${\Delta}_{H4}(k)$.

Both figures provide the same message: only using the value $\alpha_{4}=1/2$
in the H4 extrapolation in position space, we obtain coherent results (from
the point of view of the Fourier transforms) in position and momentum space.

##  5 Application to Landau-gauge gluon propagator

#### Lattice set-up

In order to show an application of the method proposed for a simple but not
trivial case, we present in this section lattice calculations of gluon
propagator in both position and momentum-space. One thousand SU(3) quenched
gauge field configurations have been produced using Wilson gauge action for
$\beta=6.0$ and $N=32$ lattice points in each space-time direction. Landau
gauge has been fixed in the lattice through a Fourier accelerated method.
Details on the lattice action and gauge fixing can be found in Ayala:2012pb ;
Boucaud:2018xup . In order to write dimensional quantities in physical units,
we will use the lattice-spacing provided by Necco:2001xg , $a^{-1}\approx
2.045{\rm GeV}$.

Once the gauge field configurations are in Landau gauge, the fields are
defined from the link variables $U_{\mu}(x)$ as:

$\displaystyle
A_{\mu}(x+\hat{\mu}/2)=\left.\frac{U_{\mu}(x)-U^{\dagger}_{\mu}(x)}{2iag_{0}}\right|_{traceless}$
(22)

and its momentum-space counterpart is obtained via a fast Fourier transform
(FFT) as:

$\displaystyle{A}^{a}_{\mu}(k)=\frac{1}{2}{\rm
Tr}\sum_{x}A_{\mu}(x+\hat{\mu}/2)\lambda^{a}e^{ik(x+\hat{\mu}/2)}\ .$ (23)

The scalar component of momentum-space gluon propagator is obtained as:

$\displaystyle{\Delta}(k)=\frac{1}{24}\langle\widetilde{A}^{a}_{\mu}(k)\widetilde{A}^{a}_{\mu}(-k)\rangle$
(24)

where $\langle\cdots\rangle$ stands for average over gauge field
configurations. We will obtain the diagonal contribution to the position-space
propagator analogously as:

$\displaystyle\Delta(x)$ $\displaystyle=$
$\displaystyle\frac{1}{24L^{4}}\sum_{y}\langle
A^{a}_{\mu}(y)A^{a}_{\mu}(y-x)\rangle=$ (25) $\displaystyle=$
$\displaystyle\frac{1}{24L^{4}}\sum_{y}\sum_{k,k^{\prime}}e^{-iky}e^{-ik^{\prime}(y-x)}\langle\widetilde{A}^{a}_{\mu}(k)\widetilde{A}^{a}_{\mu}(k^{\prime})\rangle=$
(26) $\displaystyle=$ $\displaystyle\sum_{k}e^{-ikx}{\Delta}(k)$ (27)

which is the discrete inverse Fourier transform of the gluon propagator in
momentum space. Via this discrete Fourier transform we will obtain gluon
propagator in position space for all the lattice sites which should be a
rotationally invariant function of $x^{2}$ in the continuum limit.

#### Gluon propagator in momentum space

Gluon propagator in momentum-space after applying H4 extrapolation to reduce
rotational symmetry breaking errors has been represented in Fig. 5. For
comparison, we have chosen $\mu=4.3$ GeV as the renormalization scale and
added the precise parametrization discussed in Appendix B of Aguilar:2021okw .
There is a nice agreement except for the first point which is known to be
affected by a finite volume error.

The inset shows the ratio of bare gluon propagators obtained from H4
extrapolation in momentum space and the one coming from the Fourier transform
of the x-space one. Each set corresponds to a different value of the parameter
$\alpha_{4}$ used in the position-space H4 extrapolation. As in the free-boson
case, the case $\alpha_{4}=1/2$ seems to be preferred with a deviation from
$1$ that is below $\sim 5\%$ for the whole accessible momentum window.

Figure 5: Gluon propagator in momentum space renormalized at $\mu=4.3{\rm
GeV}$ as a function of the momentum for the H4 extrapolated data in momentum
space (purple full squares) and from the Fourier transform of the position-
space ones (open green circles). For comparison, the parametrization proposed
in Aguilar:2021okw  has been represented as the full line. The inset shows the
ratio of the bare gluon propagators obtained from the application of H4
extrapolation in momentum space and the one coming from the Fourier transform
of x-space ones. Each set corresponds to a different value of $\alpha_{4}$ for
the extrapolations in position space (from top to bottom $\alpha_{4}=0$,
$\alpha_{4}=1/4$, $\alpha_{4}=1/2$, $\alpha_{4}=3/4$, and $\alpha_{4}=1$.)

#### Gluon propagator in position space

Our results in position space have been represented in Fig. 6. For large
distances ($x\gtrsim 0.5{\rm fm}$), the behavior of Landau gauge gluon
propagator shows an exponential fall-off or Yukawa-like behavior
Suganuma:2010mm  due to the dynamical gluon mass generation which is in marked
contrast with the $1/x^{2}$ trend expected for a massless boson. As in the
previous case, the inset in Fig. 6 shows the ratio of the H4-extrapolated
propagator in position space to the Fourier transform of the momentum-space
one. We can appreciate in this case that the choice $\alpha_{4}=1/2$ is
clearly preferred, with an overall deviation of this ratio from $1$ which is
below $\sim 1\%$ for the whole set of positions. The only exception is the
first point where the value provided by the H4 extrapolation method is far
from the Fourier transform one.

Figure 6: Gluon propagator in position-space for the H4 extrapolated data in
position (purple full squares) and from the Fourier transform of the momentum-
space ones (open green circles, they are superimposed to the purple set except
for the first point). The inset shows the ratio of the bare gluon propagators
obtained from the application of H4 extrapolation in momentum space and the
one coming from the Fourier transform of x-space ones. Each set corresponds to
a different value of $\alpha_{4}$ for the extrapolations in position space
(from top to bottom $\alpha_{4}=1$, $3/4$, $1/2$, $1/4$, and $0$.).

#### Comparison with other methods

Let us finish with a comparison of the noise in the result of applying the H4
extrapolation, democratic method and the tree-level correction discussed in
Section 2. For the purposes of making a direct comparison of the noise
exhibited by the lattice data after application of these methods, we have
fitted gluon propagator in momentum space to the functional form:

${D}(k)=\frac{ak^{2}+b}{k^{4}+ck^{2}+d}$ (28)

that appears, for example, in a Refined Gribov-Zwanziger framework
Dudal:2008sp ; Dudal:2019vvt  and reproduces qualitatively very well the
lattice data. In left panel of Fig. 7 the ratio $\Delta(k)/D(k)$ has been
plotted for each method (for each method the function $D(k)$ has been fitted
separately and used for the ratio). The fluctuations around unity in this plot
are due to the hypercubic errors that have not been fully eliminated by the
method employed. It shows how in momentum space, H4-method clearly performs
much better than the others, with noise at the level of $\sim 0.1\%$.

In position space, we have fitted the data for small distances to a function
inspired by the asymptotic behavior of Eq.(5):

${D}(x)=\frac{e^{-\xi x}}{x^{3/2}}\times(a+bx+cx^{2})$ (29)

and represented the ratio $\Delta(x)/D(x)$ in the right panel of Fig. 7. In
this case the noise is larger than in momentum space, but is kept below $2\%$
with the H4-method. The performance of the democratic method in this sense is
similar except that the number of data points that survive the cylindrical cut
is only a little fraction of the total number, while in the H4-method all the
values of $n^{2}$ are used. Although the democratic method provides smooth
data, they seem to be rather far from the continuum limit, at least it was the
case for Klein-Gordon propagator (remind Fig. 2).

|
---|---
Figure 7: Ratio of the lattice data after treatment of the hypercubic
artifacts and the continuum fitting functions in Eqs. (28,29) for gluon
propagator in momentum-space (left) and position space (right).

It is important to remind that the conclusions of this paragraph are
independent of the presence of O4-symmetric artifacts, that can only be
eliminated using different lattice spacings and volumes or judiciously
applying the Fourier transform method as sketched above.

#### Final comment

There is one more comment in order with respect to the lattice errors (either
discretization or finite volume). When analyzing the inset plots of Figs. 5
and 6 for $\alpha_{4}=1/2$, one cannot distinguish without further analyses
the errors coming from the H4-extrapolation results from those that have been
introduced when computing the approximate Fourier transform described in
section 4. From our experience in the free case, we can judge that the
dependence in $k$ shown by the ratio in the inset plot of Fig. 5 must have
been introduced by the Fourier transform from positions to momenta rather than
by the H4-extrapolation in momentum space. This judgment, presented here
without proof, is based in the fact that H4 extrapolation tends to work better
in momentum than in position-space, for reasons already discussed along the
manuscript, and in the small-x dependency of $\Delta(x)$, that diverges, what
makes difficult to evaluate the inverse Fourier transform contribution from
this region.

Finally, due to i) the better performance of the H4-extrapolation method in
momentum space and ii) the difficulties of computing the approximate inverse
Fourier transform of propagators that diverge at zero, the ratio presented in
the inset of Fig. 6 seems better suited than the one in Fig.5 to analyze the
possible presence of uncorrected O4 artifacts in the propagators. Indeed, one
may devise a more involved method for a relative callibration of the lattice
spacing with a generalization of the one presented in Boucaud:2018xup  by
combining both position and momentum-space propagators for different $\beta$’s
and fix simultaneously the ratios of lattice spacings and the remaining $O4$
artifacts.

##  6 Conclusions

We have analyzed lattice errors appearing in boson propagators as a
consequence of rotational invariance breaking. Contrarily to other lattice
errors, the ones that are caused by breaking the rotational invariance can be
efficiently removed using the spurious dependence of the propagators on higher
order invariants of the discrete isometry group H(4). In order to accomplish
this, we have proposed a generalization of the H4-extrapolation method
presented in deSoto:2007ht  that can be straightforwardly applied both in
momentum and position space. We have presented the method using the case of a
free massive boson where the exact solution is known and then applied it to
the Landau-gauge gluon propagator.

We have shown how the democratic orbits in position space are not the closest
ones to the continuum results for a free-boson propagator, contrarily to the
momentum-space case. This appears as the basic underlying reason for the
failure of democratic methods in position space. The method proposed has the
advantages that no a priori assumption is made over the $n^{2}$-dependence of
the artifacts, and it does not assume any particular value of the ratio
$n^{[4]}/(n^{2})^{2}$ as being optimal. We have furthermore introduced the
(approximate) continuum Fourier transforms as a method to check the remaining
discretization errors after applying the H4 extrapolation. From a more
phenomenological point of view, the combination of momentum and position-space
lattice gluon propagators opens a new possible way of studying the deeply non-
perturbative physics related to gluon mass generation in Yang-Mills theories.

###### Acknowledgements.

The author is indebted to J. Rodríguez-Quintero for carefully reading this
manuscript. The author acknowledges financial support from Spanish MICINN
grant PID2019-107844-GB-C2, and regional Andalusian project P18-FR-5057. The
author acknowledges, too, the use of the computer facilities of C3UPO at Pablo
de Olavide University.

## References

  * (1) K. Symanzik, _Continuum Limit and Improved Action in Lattice Theories. 1. Principles and phi**4 Theory_ , _Nucl. Phys. B_ 226 (1983) 187.
  * (2) K. Symanzik, _Continuum Limit and Improved Action in Lattice Theories. 2. O(N) Nonlinear Sigma Model in Perturbation Theory_ , _Nucl. Phys. B_ 226 (1983) 205.
  * (3) M. Luscher, S. Sint, R. Sommer and P. Weisz, _Chiral symmetry and O(a) improvement in lattice QCD_ , _Nucl. Phys. B_ 478 (1996) 365 [hep-lat/9605038].
  * (4) M. Luscher, S. Sint, R. Sommer, P. Weisz and U. Wolff, _Nonperturbative O(a) improvement of lattice QCD_ , _Nucl. Phys. B_ 491 (1997) 323 [hep-lat/9609035].
  * (5) D. Becirevic, P. Boucaud, J.P. Leroy, J. Micheli, O. Pene, J. Rodriguez-Quintero et al., _Asymptotic scaling of the gluon propagator on the lattice_ , _Phys. Rev. D_ 61 (2000) 114508 [hep-ph/9910204].
  * (6) D. Becirevic, P. Boucaud, J.P. Leroy, J. Micheli, O. Pene, J. Rodriguez-Quintero et al., _Asymptotic behavior of the gluon propagator from lattice QCD_ , _Phys. Rev. D_ 60 (1999) 094509 [hep-ph/9903364].
  * (7) P. Boucaud, F. de Soto, J.P. Leroy, A. Le Yaouanc, J. Micheli, H. Moutarde et al., _Quark propagator and vertex: Systematic corrections of hypercubic artifacts from lattice simulations_ , _Phys. Lett. B_ 575 (2003) 256 [hep-lat/0307026].
  * (8) F.D.R. Bonnet, P.O. Bowman, D.B. Leinweber, A.G. Williams and J.M. Zanotti, _Infinite volume and continuum limits of the Landau gauge gluon propagator_ , _Phys. Rev. D_ 64 (2001) 034501 [hep-lat/0101013].
  * (9) UKQCD collaboration, _Asymptotic scaling and infrared behavior of the gluon propagator_ , _Phys. Rev. D_ 60 (1999) 094507 [hep-lat/9811027].
  * (10) M. Constantinou, V. Lubicz, H. Panagopoulos and F. Stylianou, _$\mathcal{O}(a^{2})$ corrections to the one-loop propagator and bilinears of clover fermions with symanzik improved gluons_, _Journal of High Energy Physics_ 2009 (2009) 064–064.
  * (11) F. de Soto and C. Roiesnel, _On the reduction of hypercubic lattice artifacts_ , _JHEP_ 09 (2007) 007 [0705.3523].
  * (12) G.T.R. Catumba, O. Oliveira and P.J. Silva, _$H(4)$ tensor representations for the lattice Landau gauge gluon propagator and the estimation of lattice artefacts_, _Phys. Rev. D_ 103 (2021) 074501 [2101.04978].
  * (13) Flavour Lattice Averaging Group collaboration, _FLAG Review 2019: Flavour Lattice Averaging Group (FLAG)_ , _Eur. Phys. J. C_ 80 (2020) 113 [1902.08191].
  * (14) G.S. Bali, _QCD forces and heavy quark bound states_ , _Phys. Rept._ 343 (2001) 1 [hep-ph/0001312].
  * (15) S. Aoki, M. Fukugita, N. Ishizuka, Y. Iwasaki, K. Kanaya, T. Kaneko et al., _Nonperturbative calculation of zv and za in domain-wall qcd on a finite box_ , _Physical Review D_ 70 (2004) .
  * (16) P. Boucaud, F. De Soto, J. Rodríguez-Quintero and S. Zafeiropoulos, _Comment on “Lattice gluon and ghost propagators and the strong coupling in pure $SU(3)$ Yang-Mills theory: Finite lattice spacing and volume effects”_, _Phys. Rev. D_ 96 (2017) 098501 [1704.02053].
  * (17) P. Boucaud, F. De Soto, K. Raya, J. Rodríguez-Quintero and S. Zafeiropoulos, _Discretization effects on renormalized gauge-field Green’s functions, scale setting, and the gluon mass_ , _Phys. Rev. D_ 98 (2018) 114515 [1809.05776].
  * (18) A.C. Aguilar, F. De Soto, M.N. Ferreira, J. Papavassiliou, J. Rodríguez-Quintero and S. Zafeiropoulos, _Gluon propagator and three-gluon vertex with dynamical quarks_ , _Eur. Phys. J. C_ 80 (2020) 154 [1912.12086].
  * (19) A.C. Aguilar, C.O. Ambrósio, F. De Soto, M.N. Ferreira, B.M. Oliveira, J. Papavassiliou et al., _Ghost dynamics in the soft gluon limit_ , 2107.00768.
  * (20) P. Boucaud, J.P. Leroy, A.L. Yaouanc, J. Micheli, O. Pene and J. Rodriguez-Quintero, _The Infrared Behaviour of the Pure Yang-Mills Green Functions_ , _Few Body Syst._ 53 (2012) 387 [1109.1936].
  * (21) O. Oliveira and P.J. Silva, _Finite volume effects in the gluon propagator_ , _PoS_ LAT2005 (2006) 287 [hep-lat/0509037].
  * (22) I.L. Bogolubsky, V.G. Bornyakov, G. Burgio, E.M. Ilgenfritz, M. Muller-Preussker and V.K. Mitrjushkin, _Improved Landau gauge fixing and the suppression of finite-volume effects of the lattice gluon propagator_ , _Phys. Rev. D_ 77 (2008) 014504 [0707.3611].
  * (23) A. Cucchieri and T. Mendes, _Infrared behavior and infinite-volume limit of gluon and ghost propagators in Yang-Mills theories_ , _PoS_ CONFINEMENT8 (2008) 040 [0812.3261].
  * (24) O. Oliveira and P.J. Silva, _The lattice infrared Landau gauge gluon propagator: from finite volume to the infinite volume_ , _PoS_ QCD-TNT09 (2009) 033 [0911.1643].
  * (25) V.G. Bornyakov, V.K. Mitrjushkin and M. Muller-Preussker, _SU(2) lattice gluon propagator: Continuum limit, finite-volume effects and infrared mass scale m(IR)_ , _Phys. Rev. D_ 81 (2010) 054503 [0912.4475].
  * (26) D. Zwanziger, _Fundamental modular region, Boltzmann factor and area law in lattice gauge theory_ , _Nucl. Phys. B_ 412 (1994) 657.
  * (27) A. Cucchieri and T. Mendes, _Bloch Waves in Minimal Landau Gauge and the Infinite-Volume Limit of Lattice Gauge Theory_ , _Phys. Rev. Lett._ 118 (2017) 192002 [1612.01279].
  * (28) S. Calì, K. Cichy, P. Korcyl and J. Simeth, _Running coupling constant from position-space current-current correlation functions in three-flavor lattice qcd_ , _Phys. Rev. Lett._ 125 (2020) 242002.
  * (29) O. Oliveira, P.J. Silva, J.-I. Skullerud and A. Sternbeck, _Quark propagator with two flavors of O(a)-improved Wilson fermions_ , _Phys. Rev. D_ 99 (2019) 094506 [1809.02541].
  * (30) G. Catumba, O. Oliveira and P.J. Silva, _Lattice artefacts on the Landau gauge gluon propagator from hypercubic tensor representations_ , in _38th International Symposium on Lattice Field Theory_ , 11, 2021 [2111.15022].
  * (31) A. Ayala, A. Bashir, D. Binosi, M. Cristoforetti and J. Rodriguez-Quintero, _Quark flavour effects on gluon and ghost propagators_ , _Phys. Rev. D_ 86 (2012) 074512 [1208.0795].
  * (32) S. Necco and R. Sommer, _The N(f) = 0 heavy quark potential from short to intermediate distances_ , _Nucl. Phys. B_ 622 (2002) 328 [hep-lat/0108008].
  * (33) H. Suganuma, T. Iritani, A. Yamamoto and H. Iida, _Lattice QCD Study for Gluon Propagator and Gluon Spectral Function_ , _PoS_ LATTICE2010 (2010) 289 [1011.0007].
  * (34) D. Dudal, J.A. Gracey, S.P. Sorella, N. Vandersickel and H. Verschelde, _A Refinement of the Gribov-Zwanziger approach in the Landau gauge: Infrared propagators in harmony with the lattice results_ , _Phys. Rev. D_ 78 (2008) 065047 [0806.4348].
  * (35) D. Dudal, O. Oliveira and P.J. Silva, _High Precision Statistical Landau Gauge Lattice Gluon Propagator Computation_ , _PoS_ Confinement2018 (2019) 265.