# Spectral dimension as a tool for analyzing nonpertubative propagators

Wolfgang Kern<EMAIL_ADDRESS>Institute of Physics, University
of Graz, NAWI Graz, Universitätsplatz 5, 8010 Graz, Austria  Markus Q. Huber
<EMAIL_ADDRESS>Institut für Theoretische Physik, Justus-
Liebig–Universität Giessen, 35392 Giessen, Germany  Reinhard Alkofer
<EMAIL_ADDRESS>Institute of Physics, University of Graz, NAWI
Graz, Universitätsplatz 5, 8010 Graz, Austria

###### Abstract

We derive general properties of the scale-dependent effective spectral
dimensions of nonperturbative gauge boson propagators as they appear as
solutions from different methods in Yang-Mills theories. In the ultraviolet
and for short timescales the anomalous dimensions of the propagators lead to a
slight decrease of the spectral dimension as compared to the one of a free
propagator. Lowering the momentum scale, the spectral dimension decreases
further. The class of propagators which display a maximum at Euclidean
momenta, and thus violate positivity, always approaches a spectral dimension
of one for large times. We also show that the longest time intervals are not
related to the deep infrared but to the momentum scale defined by the position
of the maximum.

correlation functions, Landau gauge, Dyson-Schwinger equations, Yang-Mills
theory, Gribov-Zwanziger framework

###### pacs:

12.38.Aw, 14.70.Dj, 12.38.Lg

##  I Introduction

The interest in the fully renormalized gluon propagator dates back into the
late seventies of the last century, see, e.g., Refs. Pagels:1976kq ;
Delbourgo:1978jw ; Oehme:1979ai , and the behavior of the gluon propagator in
the strongly interacting domain of quantum chromodynamics (QCD) is in the
focus of research until today, see, e.g., Alkofer:2000wg ; Maas:2011se ;
Aguilar:2015bud ; Huber:2018ned  and references therein. The question whether
the gluon is not only confined but also confining was raised already in the
first investigations of this topic Pagels:1976kq , and the thus motivated
assumption of an infrared singular gluon propagator received some support from
the observation that it might be responsible for the area law in the Wilson
loop West:1982bt  and thus quark confinement. Nowadays, however, it is
generally accepted that in the Landau gauge the gluon propagator is at small
momenta suppressed as compared to a free propagator, for recent reviews see,
e.g., Maas:2011se ; Aguilar:2015bud ; Huber:2018ned .111 This suppression
exists for the gauge boson propagator in Yang-Mills theory, i.e., the quenched
gluon propagator, and for the unquenched gluon propagator as long as the
number of light quarks is small. For a large number of flavors such that one
enters the conformal window a change in the behavior of the gluon propagator
is expected Hopfer:2014zna . Although it is evident that such a suppression
implies positivity violation Oehme:1979ai ; Oehme:1994pv , the relation to
gluon confinement stays elusive, and the understanding of the analytic
structure of the gluon propagator, despite some progress in this direction,
e.g., Alkofer:2003jj ; Strauss:2012dg ; Lowdon:2017uqe ; Lowdon:2018mbn ;
Cyrol:2018xeq ; Binosi:2019ecz , is still unsatisfactory.

Therefore the question arises whether the gluon propagator can be analyzed
with the help of other tools. Herein, we are going to employ in this respect
the concept of a spectral dimension. Introduced in Ref. Ambjorn:2005db  in the
context of causal dynamical triangulation studies in the research area of
quantum gravity, it has found wide-spread use in analyzing spacetimes
resulting from different approaches to quantum gravity, see, e.g.,
Carlip:2017eud ; Carlip:2019onx  for a recent compilation. Hereby, different
attempts toward quantum gravity typically see a dimensional reduction when
going from the infrared to the ultraviolet. Technically, in some of these
studies this is achieved by studying the spectral dimension $D_{S}$ via the
diffusion of a hypothetical scalar particle, and instead of taking the limit
of infinite diffusion times as in the mathematical definition of the spectral
dimension one studies a generalized effective spectral dimension $D_{S}(T)$ as
a function of the diffusion time $T$ Lauscher:2005qz . The quantity $D_{S}(T)$
is closely related to the two-point correlation function of the diffusing
particle. Therefore, given the propagators as resulting from a certain
approach to quantum gravity, one can perform investigations how the effective
dimension of spacetime changes, see Alkofer:2014raa  for an example of such a
study.

In the context of the here presented results it is important to note that the
spectral dimension can be interpreted as Haussdorff dimension of momentum
space Amelino-Camelia:2013gna ; Alkofer:2014raa ; Alkofer:2018zze . This makes
evident that the concept of a spectral dimension is not restricted to study
properties of a spacetime by using propagators but one can also employ it to
analyze propagators with respect to their nontrivial content due to
interactions. Local interactions which respect unitarity will always make the
spectrum as measured by the propagator “thinner”, i.e., we expect in a local
quantum field theory, and thus in Yang-Mills theory as well as QCD, a
reduction of the spectral dimension as compared to the topological dimension
of the underlying spacetime due to interactions. This reduction will be larger
for stronger interactions, and the hypothetical diffusion time will allow us
to draw conclusions of the scale-dependent impact of interactions onto the
propagator. As we will argue below, one important result of our investigation
is that for the class of propagators the gluon propagator belongs to, a simple
mapping of large diffusion times to small momenta is not correct. The maximum
displayed by such propagators is then not only indicative of positivity
violation but also responsible for breaking the “rule-of-thumb” that the
longest time intervals and largest distances relate to the smallest momenta.

This makes also plain that there is another technical distinction to quantum
gravity studies which are generically interested in sub-Planckian and
therefore the shortest distances. These are related, of course, to the
shortest diffusion times. As Yang-Mills theories are asymptotically free the
interactions are small for large momenta and short distances, and the expected
relation to short diffusion times is verified. In two and three dimensions we
find accordingly the topological dimension as value for the short-time
spectral dimension, while in four dimensions we find a reduction which we can
relate to the logarithmic corrections encoded in the anomalous dimension of
the gluon propagator. The most interesting momentum regime for our study is,
of course, the strongly interacting one. Somewhat surprisingly it turns out
that the largest diffusion times are not related to the deep-infrared behavior
of the propagator. Thus, the spectral dimension highlights different features
than other integral transformations, as, e.g., the Fourier transformation.

In the next section we discuss the effective spectral dimension and some of
its properties. Sec. III describes the input for which we calculate in Sec. IV
the spectral dimension as a function of the diffusion time. We summarize and
conclude in Sec. V. Some details of the input and proofs are deferred to
appendices.

##  II General considerations for the spectral dimension

###  II.1 The spectral dimension

We calculate the spectral dimension $D_{S}(T)$ from the return probability as
explained in Ref. Alkofer:2014raa . Diffusion is governed by the heat kernel
$K(x,x^{\prime},T)$ which describes quantitatively the diffusion from a start
position $x$ to a final position $x^{\prime}$ in some diffusion time $T$. The
heat kernel satisfies the heat kernel equation

$\displaystyle\left(\frac{\partial}{\partial
T}-\Delta_{x}\right)K(x,x^{\prime},T)=0$ (1)

with

$\displaystyle K(x,x^{\prime},0)=\delta(x-x^{\prime}).$ (2)

In this equation, $\Delta_{x}$ is the Laplacian operator. We can solve the
equation by Fourier transformation and obtain

$\displaystyle
K(x,x^{\prime},T)=\int\frac{d^{d}p}{(2\pi)^{d}}e^{ip(x-x^{\prime})}e^{-p^{2}T}.$
(3)

Generally, one can identify $K(x,x^{\prime},T)$ with matrix elements of the
operator $e^{\Delta_{x}T}$. The return probability $\mathcal{P}(T)$ is related
to $K(x,x^{\prime},T)$ via

$\displaystyle\mathcal{P}(T)=V^{-1}\int
d^{d}x\sqrt{g(x)}K(x,x,T)=V^{-1}\text{Tr}\left(e^{\Delta_{x}T}\right),$ (4)

viz., the average over the heat kernel gives the return probability. The total
volume is given by $V=\int d^{d}x\sqrt{g(x)}$ where $g(x)$ is the metric. In
flat space the return probability for a free propagators is

$\displaystyle\mathcal{P}(T)=(4\pi T)^{-d/2}.$ (5)

This leads to a spectral dimension, given by

$\displaystyle D_{S}(T)=-2\dfrac{\partial~{}\ln\mathcal{P}(T)}{\partial~{}\ln
T}=-2T\frac{\mathcal{P}^{\prime}(T)}{\mathcal{P}(T)}.$ (6)

of $D_{S}=d$ for any $T$.

Here we will use the Euclidean metric throughout. Anomalous behavior is then
introduced by higher order derivative terms, which correspond to deviations
from the classical behavior. Thus, we need to take into account quantum
fluctuations. Since the Laplace operator $\Delta_{x}$ corresponds to the
classical inverse propagator of a massless particle, we replace it now by the
dressed inverse propagator $F$. Eq. 1 changes then to

$\displaystyle\left(\partial_{T}+F(-\Delta_{x})\right)K_{g}(x,x^{\prime},T)=0.$
(7)

In an analogous way to above, we obtain the solution for this equation,

$\displaystyle
K(x,x^{\prime},T)=\int\frac{d^{d}p}{(2\pi)^{d}}e^{ip(x-x^{\prime})}e^{-F(p^{2})T},$
(8)

and for the return probability,

$\displaystyle\mathcal{P}(T)=\int\frac{d^{d}p}{(2\pi)^{d}}e^{-F(p^{2})T}.$ (9)

In the following we only consider a flat spacetime. The spectral dimension of
a free massless particle is then equal to the topological dimension of
spacetime, see Eq. (5). However, we expect that interactions between particles
lower the number of possible energy states which, in turn, lowers the
spacetime dimension felt by test particles. An intuitive way to see this is to
transform the integration variable as $k^{2}=F(p^{2})$ Amelino-Camelia:2013gna
; Amelino-Camelia:2013cfa  which leads to the following expression:

$\displaystyle\mathcal{P}(T)=\frac{\Omega_{d}}{(2\pi)^{d}}\int
dk\,\frac{k\,(F^{-1}(k^{2}))^{\frac{d-2}{2}}}{F^{\prime}(p^{2})}e^{-k^{2}\,T}.$
(10)

$\Omega_{d}$ is the angular integral. This can be interpreted as the return
probability of a free propagator in a nontrivial background determined by
$F(p^{2})$, viz., the quantum fluctuations of the propagator. A requirement
for this transformation is that there is a one-to-one correspondence between
$k$ and $p$.

The integral Eq. (9) can be interpreted as a nonlinear integral
transformation. A direct consequence of the function being in the exponent is
a different relation between $p^{2}$ and $T$ as compared to typical well-known
integral transformations. For example, low momenta are typically related to
large distances via a Fourier transformation. Here, however, low values of
$F(p^{2})$ are related to large $T$. If $F(p^{2})$ is monotonic and vanishes
at $p^{2}=0$, small $p^{2}$ and large $T$ are indeed related. However, if the
inverse propagator vanishes at $p_{0}\neq 0$, long diffusion times are related
to $F(p^{2})$ at $p^{2}\approx p_{0}^{2}$. The properties of the integral
transformation discussed above are inherited by the spectral dimension
$D_{S}(T)$ defined in Eq. (6).

###  II.2 Asymptotics of the spectral dimension

We will now discuss some general properties of the spectral dimension before
we turn to a full numerical analysis in Sec. IV. We will outline how some
specific problems of our input are overcome. We start the discussion with the
simple example of a massive propagator and will explain why using the same
method for Yang-Mills theory fails. We will then turn to two analytic examples
that correspond to the asymptotics of the input employed in Sec. IV.

For a massive particle the inverse propagator reads

$\displaystyle F(p^{2})=p^{2}+m^{2}.$ (11)

The return probability is222The prefactors cancel out in the spectral
dimension, so they are ignored here.

$\displaystyle\mathcal{P}(T)\propto\frac{e^{-m^{2}\,T}}{2T^{2}}$ (12)

from which the spectral dimension is calculated in four dimensions as

$\displaystyle D_{S}(T)=4+2\,m^{2}\,T.$ (13)

For $m=0$ the spectral dimension of the free massless propagator is obtained.
However, the mass leads to a contribution linear in $T$. This property remains
true for a more general inverse propagator where also a constant term is
separated:

$\displaystyle F(p^{2})=F^{(0)}(p^{2})+m^{2}.$ (14)

It was shown in Ref. Alkofer:2014raa  that this leads to a spectral dimension

$\displaystyle D_{S}(T)=2\,m^{2}\,T+D_{S}^{(0)}(T)$ (15)

where $D_{S}^{(0)}(T)$ is calculated from $F^{(0)}(p^{2})$. Thus, the authors
of Alkofer:2014raa  suggested to remove the trivial linear term by taking
$m^{2}=F(0)$ and considering $D_{S}^{(0)}(T)$ instead of $D_{S}(T)$.

However, this procedure runs into problems if the inverse propagator is not
monotonic and has a minimum. Then the subtracted inverse propagator
$F(p^{2})-F(0)$ is negative in some region which leads to a negative spectral
dimension for sufficiently large diffusion times, see Appendix B for a proof.

In principle, one can calculate the return probability including the linear
term and extract the coefficient from a fit. However, for large $T$ the return
probability is suppressed exponentially and the computation of the spectral
dimension is numerically not stable. Another method is to subtract the inverse
propagator at the minimum $p_{0}$:

$\displaystyle F(p^{2})\rightarrow F(p^{2})-F(p_{0}^{2}).$ (16)

This method turned out to be the most stable one and is used throughout this
work. However, Eq. (10) does no longer hold because the propagator is not a
bijective function. This could be circumvented by introducing an IR cutoff at
the minimum. We compared calculations with and without a cutoff at the minimum
and only found quantitative differences for intermediate diffusion times. The
asymptotic behavior is not affected, as the minimum itself dominates long
diffusion times for both cases. Thus all calculations shown here were done
without a cutoff.

We turn now to some general properties of the spectral dimension. In Ref.
Alkofer:2014raa  it was shown that for a polynomial inverse propagator the
behavior at short/long diffusion times depends on the highest/lowest exponent
of $p^{2}$. For the convenience of the reader the proof of this is given in
App. C. Here we are interested in nonpolynomial inverse propagators. At short
diffusion times the leading behavior is still determined by the highest
exponent $N_{\text{max}}$, which is $1$ here. Thus, we have

$\displaystyle\lim_{T\rightarrow 0}D_{S}(T)=d/N_{\text{max}}=d,$ (17)

where $d$ is the dimension of spacetime. However, in QCD the propagators
acquire an anomalous dimension and thus do not show a pure $1/p^{2}$ behavior.
This will be reflected in the spectral dimension which approaches $4$ very
slowly. Indeed, in simple test models containing a logarithm one can show that
the exponent of the logarithm is reflected in the spectral dimension for short
diffusion times. If spectral dimensions $D_{S,1}(T)$ and $D_{S,2}(T)$ with
exponents $\gamma_{1}$ and $\gamma_{2}$, respectively, are compared, one finds
that the ratio $(4-D_{S,1}(T))/(4-D_{S,2}(T))$ is equal to
$\gamma_{1}/\gamma_{2}$ for small $T$.

As mentioned above, the behavior of the spectral dimension for long diffusion
times is related to the behavior of $F(p^{2})\rightarrow 0$. Due to the
employed subtraction procedure, this happens at the minimum $p_{0}^{2}$. We
can approximate $F(p^{2})$ in this region by $a(p^{2}-p_{0}^{2})^{2}$, where
$a$ is an irrelevant constant. In that case, the spectral dimension is $1$ for
asymptotically large diffusion times. As shown in App. C, the spectral
dimension for an inverse propagator of the form $(p^{2}-p_{0}^{2})^{c}$ with
$c>0$ obeys

$\displaystyle\lim_{T\rightarrow\infty}D_{S}(T)=2/c.$ (18)

These analytic findings lead to an interesting connection to confinement. The
spectral representation of the propagator $D(p^{2})$ reads

$\displaystyle D(p^{2})=\int_{0}^{\infty}ds\frac{\rho(s)}{p^{2}+s}.$ (19)

This form assumes that there are no complex poles. However, they can be
included by taking into account the corresponding residues, see, e.g.,
Alkofer:2000wg ; Maas:2011se ; Huber:2018ned  for more details. Thus, Eq. (19)
encodes the complete analytic structure of the propagator. In particular, the
input in Sec. III was obtained with Euclidean metric, but Eq. (19) applies for
arbitrary complex $p^{2}$. In general, Eq. (19) holds (in the absence of
complex poles), but a necessary condition for a physical particle is that
$\rho(s)$ is not negative for any $s$. Thus, if one can show what $\rho(s)$ is
negative somewhere, the particle is removed from the physical state space. For
a propagator with a maximum it is straightforward to show that it violates
positivity. For this we consider the derivative of Eq. (19),

$\displaystyle\frac{dD(p^{2})}{dp^{2}}=-\int_{0}^{\infty}ds\frac{\rho(s)}{(p^{2}+s)^{2}}.$
(20)

If the propagator is not monotonic, for example, because it has a maximum, its
derivative is zero at this maximum. This, however, entails, that $\rho(s)$
must be negative in parts of the integration domain such that the integral can
vanish. While violation of positivity is sufficient to remove the gluon from
the physical spectrum, it is not a required condition. Thus, we can only
conclude that a gluon propagator with a maximum violates positivity and has a
spectral dimension of one for long diffusion times. This is one of the main
results of this work.

In Sec. IV we numerically confirm the behavior for short and long diffusion
times discussed above in all our calculations.

##  III Input

We summarize the propagators used for the calculations of spectral dimensions
in this section. For easier comparison the propagators are shown in Sec. IV
together with their spectral dimensions.

Figure 1: Gluon propagator. .

Figure 2: Spectral dimension. . Figure 3: Gluon propagator and its spectral
dimension for two-dimensional Yang-Mills theory.

Figure 4: Gluon propagators. .

Figure 5: Spectral dimensions. . Figure 6:  Gluon propagators and their
spectral dimensions for three-dimensional Yang-Mills theory obtained from two
different truncations.

Figure 7: Gluon propagators. .

Figure 8: Spectral dimensions. . Figure 9:  Gluon propagators and their
spectral dimensions for four-dimensional Yang-Mills theory obtained from one-
loop truncations of the decoupling type. The labels “a”, “b” and “c” indicate
three different models for the three-gluon vertex and “rc”, “as” and “gl”
three different subtraction methods for spurious divergences.

Figure 10: Gluon propagator. .

Figure 11: Spectral dimension. . Figure 12:  Gluon propagator and its spectral
dimension for four-dimensional Yang-Mills theory obtained from a one-loop
truncation of the scaling type.

###  III.1 Two dimensions

The gluon propagator for two dimensions was obtained from a calculation of the
gluon and ghost propagators using a bare ghost-gluon vertex and a model for
the three-gluon vertex. The data corresponds to the green lines in Fig. 15 of
Ref. Huber:2012zj . The gluon propagator is reproduced here in Fig. 3. In two
dimensions the coupling constant $g$ is a dimensionful quantity. In our
calculations we use the numeric value $1$ and plot dimensionful quantities in
units of $g$.

It should be noted that in three and four dimensions a family of solutions
exist for the propagators Boucaud:2008ji ; Fischer:2008uz ; Alkofer:2008jy .
The endpoint of this family with an IR vanishing gluon propagator and an IR
divergent ghost dressing function is called scaling solution vonSmekal:1997is
; vonSmekal:1997vx  where all dressings are characterized by power laws
Alkofer:2004it ; Alkofer:2008bs ; Fischer:2009tn . This solution necessarily
always has a maximum in the gluon propagator. For the other solutions both the
gluon propagator and the ghost dressing are IR finite, e.g., Cornwall:1981zr ;
Cucchieri:2007md ; Cucchieri:2008fc ; Sternbeck:2007ug ; Bogolubsky:2009dc ;
Dudal:2008sp ; Boucaud:2008ji ; Aguilar:2008xm ; Fischer:2008uz ;
Alkofer:2008jy ; Cyrol:2016tym ; Huber:2018ned ; Aguilar:2019kxz . In two
dimensions, however, only the scaling solution can be realized Dudal:2008xd ;
Huber:2012zj ; Zwanziger:2012xg .

###  III.2 Three dimensions

For three-dimensional Yang-Mills theory we used two different truncations of
Dyson-Schwinger equations. The first one consists of the one-loop truncated
propagator equations using a bare ghost-gluon vertex and the following model
for the tree-level dressing of the three-gluon vertex motivated by a similar
model in four dimensions Huber:2012kd :

$\displaystyle C^{AAA}(x,y,z)$
$\displaystyle=C^{AAA}_{\text{UV}}(x,y,z)+C^{AAA}_{\text{IR}}(x,y,z),$ (21)
$\displaystyle C^{AAA}_{\text{UV}}(x,y,z)$
$\displaystyle=\frac{1}{\sqrt{G(\overline{p}^{2})}},$ (22) $\displaystyle
C^{AAA}_{\text{IR}}(x,y,z)$
$\displaystyle=-G(\overline{p}^{2})^{3}\left(\frac{\Lambda^{2}_{\text{3g}}}{x+\Lambda^{2}_{\text{3g}}}\frac{\Lambda^{2}_{\text{3g}}}{y+\Lambda^{2}_{\text{3g}}}\frac{\Lambda^{2}_{\text{3g}}}{z+\Lambda^{2}_{\text{3g}}}\right)^{4},$
(23)

with $\overline{p}=(x+y+z)/2$ and $\Lambda_{\text{3g}}^{2}=10\,g^{4}$. The
second truncation consists of the full system of primitively divergent
correlation functions, viz., the propagators and the ghost-gluon, three-gluon,
and four-gluon vertices. Details can be found in Ref. Huber:2016tvc .

In three dimensions the coupling constant $g$ is a dimensionful quantity. In
our calculations we use the numeric value $1$. Due to ambiguities to set the
scale Huber:2016tvc  – which do not matter here – we continue with raw data,
so all dimensionful quantities should be considered as given in terms of $g$.

The spurious divergences related to the breaking of gauge covariance, see Ref.
Huber:2014tva  and references therein, are subtracted numerically by
determining the coefficients of the linear and logarithmic cutoff dependence
Huber:2016tvc .

The propagators for the two truncations are compared in Fig. 6.

###  III.3 Four dimensions

In four dimensions, we test a variety of different inputs which can be grouped
into four types. For the DSE results the scale was set by putting the gluon
dressing maximum to $0.94\,\text{GeV}$.

  * •

Solutions of a one-loop truncated gluon propagator DSE of decoupling type
(1LDec):
To test the effects of the three-gluon vertex and the method for the
subtraction of spurious divergences Huber:2014tva  we use combinations of
three different variants of each. One combination does not yield a solution,
so we have eight different setups. The three-gluon vertex models are called
“a”, “b”, and “c” and the subtraction methods as “gl”, “rc”, and “as”. Details
are explained in App. A.

  * •

Solutions of a one-loop truncated gluon propagator DSE of scaling type
(1LSca):
For testing a scaling type solution of the gluon propagator results from Fig.
4 (red line) in Ref. Huber:2014tva  are used. The subtraction of quadratic
divergences corresponds to “as” and the model for the three-gluon vertex is an
extended version of “a”.

  * •

Solutions of a two-loop truncated gluon propagator DSE of decoupling type
(2LDec):
We compare two different two-loop truncations. One uses models for the
propagators Huber:2017txg  and another one includes the vertices dynamically
Huber:2019wxx ; Huber:2019ip .

  * •

Refined Gribov-Zwanziger fits (RGZ):
The refined Gribov-Zwanziger framework, as an extension of the original work
by Gribov and Zwanziger Gribov:1977wm ; Zwanziger:1989mf ; Zwanziger:1992qr
taking into account condensates Dudal:2008sp ; Gracey:2010cg ; Dudal:2011gd ,
provides a closed form for the gluon propagator that can be used for fits to
lattice data. Here we use the parameters from Ref. Cucchieri:2011ig  and a
modified version that introduces a maximum into the propagator. The form we
employ is

$D_{\text{RGZ}}(p^{2})=C\frac{p^{2}+s}{p^{4}+u^{2}p^{2}+t^{2}}.$ (24)

For the parameters we use the following values Cucchieri:2011ig :

$\displaystyle C$ $\displaystyle=0.784,\quad$ $\displaystyle s$
$\displaystyle=2.508\,\text{GeV}^{2},$ (25) $\displaystyle t$
$\displaystyle=0.720\,\text{GeV}^{2},\quad$ $\displaystyle u$
$\displaystyle=0.768\,\text{GeV}.$ (26)

This gluon propagator does not exhibit a maximum, but we can modify the
parameters to introduce one. Specifically, we use $u=0.15\,\text{GeV}$ for
$D_{\text{RGZ,max}}(p^{2})$.

##  IV Results

The spectral dimension is calculated numerically from the second expression in
Eq. (6).

The results for two and three dimensions are shown in Figs. 3 and 6,
respectively. It clearly approaches the number of spacetime for short and one
for long diffusion times as expected from the discussion in Sec. II. In two
dimensions the transition is particularly smooth, which is most likely due to
the distinct maximum in the propagator. For three dimensions, we can also
compare different truncations and we find that indeed the behavior in the
intermediate regime varies strongly: The two-loop truncation shows a distinct
maximum while the one-loop truncation is monotonic and has almost no
structure. This is reflected in an increase of the spectral dimension for
intermediate diffusion times for the one-loop truncation.

In four dimensions we have a broader variety of setups. All calculations (with
the exception of the RGZ propagator) show that for long diffusion times the
asymptotic value of one is approached. For short diffusion times, we see that
the spectral dimension approaches four only very slowly as expected from the
discussion of logarithmic corrections in Sec. II. Details of that are
discussed below for each case. An exception are the RGZ propagators which have
no logarithmic corrections and thus approach four directly.

We start with the details of the one-loop truncations shown in Figs. 9 and 12.
The scaling case is almost featureless. However, comparing the different
solutions for the decoupling case, we see that this should be attributed to
the chosen three-gluon vertex model and the chosen subtraction scheme. The
setup for the scaling solution corresponds basically to “b as” with the
exception that the three-gluon vertex model includes a zero crossing. Changing
the three-gluon vertex and/or the subtraction scheme, this changes and a
plateau or even a maximum can appear.

In Sec. II we discussed that the ratio of the deviations of the spectral
dimensions from four for different anomalous dimensions is equal to the ratio
of anomalous dimensions. We can test that by using the spectral dimension of
the ghost propagator for comparison. However, the ghost propagator is negative
and the integral Eq. (6) does not converge. We thus define a pseudospectral
dimension $\widetilde{D}_{S}(T)$ that uses the negative ghost propagator, plot
the quantity

$\displaystyle R(T)=(4-D_{S}(T))/(4-\widetilde{D}_{S}(T))$ (27)

and compare it to the ratio $\gamma/\delta$ of the one-loop anomalous
dimensions of the gluon and ghost propagators, $\gamma=-13/22$ and
$\delta=-9/44$, respectively. This is shown in Fig. 13. For short diffusion
times, the ratios are indeed similar, with the degree of coincidence depending
on the truncation.

Figure 13: The ratio $R(T)$ for the one-loop truncated decoupling propagators
from Fig. 9.

We turn now to the two-loop truncations. The results for the spectral
dimensions are shown in Fig. 16. Both results show a similar behavior with a
plateau in the intermediate regime. An improvement compared to the one-loop
truncations can be seen for the ratio of anomalous dimensions, shown in Fig.
17, which approach the expected value very nicely to within $1\%$ for short
diffusion times. This could be related to the way the anomalous dimension is
realized in these truncations. For the one-loop truncations a renormalization
group improvement term needs to be included in the three-gluon vertex model
vonSmekal:1997vx ; Fischer:2002eq ; Huber:2012kd , but for the two-loop
truncations the anomalous dimension emerges correctly automatically
Huber:2018ned ; Huber:2019ip . Effects of such improvement terms were seen
previously Huber:2014tva  and also here by comparing different vertex models
for the one-loop truncations.

Figure 14: Gluon propagators. .

Figure 15: Spectral dimensions. . Figure 16:  Gluon propagators and their
spectral dimensions for four-dimensional Yang-Mills theory obtained from two-
loop truncations using either models for the vertices Huber:2017txg  or
dynamically calculated vertices Huber:2019wxx ; Huber:2019ip .  Figure 17: The
ratio $R(T)$ for the two-loop truncated propagators from Fig. 16.

As last example we consider the RGZ model. Since no logarithmic UV corrections
are contained, the spectral dimension approaches four for short diffusion
times very rapidly. The spectral dimensions for the two propagators with and
without maximum are shown in Fig. 20. For large diffusion times, we see large
differences between the two models which corroborate our discussion about the
effect of a maximum in the propagator. The standard RGZ propagator, which has
no maximum and (after subtraction) a leading term proportional to $p^{2}$,
turns back up for large diffusion times and approaches four, the canonical
dimension of spacetime. However, if we deform the propagator to contain a
maximum, the asymptotic value for long diffusion times is one as expected from
the general discussion in Sec. II.

Figure 18: Gluon propagators. .

Figure 19: Spectral dimensions. . Figure 20:  Gluon propagators and their
spectral dimensions for four-dimensional Yang-Mills theory from the refined
Gribov-Zwanziger framework Cucchieri:2011ig  and a modified version of it.

One general observation one can make from all these results is that the
distinctiveness of the maximum in the gluon propagator is connected to the
steepness of the spectral dimension in the intermediate regime. All the
propagators discussed have a maximum, except for the original RGZ propagator.
However, in some cases it is not very distinct, e.g., for the one-loop
truncation in three dimensions or the setup “c rc” in four dimensions, see
Figs. 6 and 9, respectively. In these cases, the spectral dimension goes up
again before it approaches the asymptotic value. In the three-dimensional
example, it even goes up as far as three. Given the analysis in Sec. II, we
know that there must be a jump in the asymptotic behavior of the spectral
dimension once the maximum reaches zero. This explains why for propagators
with a flat maximum the resulting value for the spectral dimension increases
in between before it decreases again and approaches eventually its lowest and
asymptotic value.

##  V Conclusions

We have calculated the spectral dimensions for Yang-Mills gluon propagators as
they are obtained from different truncations of Dyson-Schwinger equations or
from the refined Gribov-Zwanziger framework. The asymptotic behavior for short
diffusion times reflects the spacetime dimension, but in four dimensions
deviations from it are found which are a consequence of the propagator’s
perturbative (one-loop) anomalous dimension.

While short diffusion times are related to high energies, long diffusion times
are not necessarily related to low energies. We showed that for long diffusion
times the result depends on the existence of a maximum of the propagator: The
spectral dimension becomes then one. Interestingly, this type of propagators
is known to violate positivity. While a spectral dimension of one for high
diffusion times is not a necessary condition for positivity violation, it
nevertheless establishes an interesting link to confinement.

The behavior of the spectral dimension for intermediate diffusion times mainly
depends on the distinctiveness of the maximum in the gluon propagator.
Unfortunately, we could not identify dependencies on other aspects like the
level of sophistication of the truncation with which the gluon propagator was
calculated. It is interesting to note that the relation of the maximum with
the spectral dimension washed out any distinction between the decoupling and
scaling type of solutions. Largest distances, which clearly map to the longest
diffusion time, do not relate to the deep infrared but to the momentum regime
where the maximum is located. Thus, one may speculate that confinement is not
reflected in the infrared properties of correlation functions but in their
properties somewhat below one GeV. This would then have some analogy to the
phenomenon of dynamical chiral symmetry breaking, which in QCD is also driven
by the interactions at the sub-GeV scale and not by the deep infrared.

While the calculation of the spectral dimension as a function of the diffusion
time was done numerically, the main results we presented were obtained
analytically. Thus, they can be transferred directly to other cases, in
particular to unquenched propagators and other gauges like linear covariant,
Coulomb or maximally Abelian ones, for which results are available, e.g.,
Fischer:2003rp ; Huber:2009wh ; Alkofer:2011di ; Huber:2015ria ; Cyrol:2017ewj
; Reinhardt:2017pyr ; Aguilar:2012rz , thereby extending the here presented
study in order to elucidate the (in-)dependence of the results on the chosen
gauge.

## Acknowledgments

Funding by the FWF (Austrian Science Fund) under Contract No. P 27380-N27 and
the DFG (German Research Foundation) under Contract No. Fi970/11-1 are
gratefully acknowledged.

##  Appendix A Details on propagator solutions: One-loop truncated gluon
propagator DSE with different models and subtraction methods (1LDec)

Eight different combinations of three-gluon vertex models and subtraction
methods are used. The vertex models are the following ones:

$\displaystyle C_{\text{a}}^{AAA}(p,q,-p-q)$
$\displaystyle=\frac{1}{Z_{1}}G\left(\frac{x+y+z+\Lambda_{\text{IR}}^{2}}{2}\right)^{2\alpha}Z\left(\frac{x+y+z+\Lambda_{\text{IR}}^{2}}{2}\right)^{2\beta},$
(28) $\displaystyle C_{\text{b}}^{AAA}(p,q,-p-q)$
$\displaystyle=\frac{1}{Z_{1}}\left(\frac{G(\overline{p}^{2})}{Z(\overline{p}^{2})}\frac{\overline{p}^{2}}{\overline{p}^{2}+\Lambda_{s}^{2}}\right)^{2},$
(29) $\displaystyle C_{\text{c}}^{AAA}(p,q,-p-q)$
$\displaystyle=\frac{1}{Z_{1}}\frac{(G(y+\Lambda_{\text{IR}}^{2})G(z+\Lambda_{\text{IR}}^{2}))^{1-a/\delta-2a}}{(Z(y+\Lambda_{\text{IR}}^{2})Z(z+\Lambda_{\text{IR}}^{2}))^{1+a}},$
(30)

with $x=p^{2}$, $y=q^{2}$, $z=(p+q)^{2}$, and $\overline{p}^{2}=(x+y+z)/2$.
$G$ and $Z$ are the ghost and gluon dressing functions, $\delta$ is the
anomalous dimension of the ghost propagator and $Z_{1}$ the renormalization
constant of the three-gluon vertex. The models include terms for the
renormalization group improvement necessary for the one-loop truncated gluon
propagator DSE vonSmekal:1997vx ; Fischer:2002hn ; Huber:2012kd ;
Huber:2018ned . These models do not exhibit a zero crossing as seen with
different methods, e.g., Pelaez:2013cpa ; Aguilar:2013vaa ; Blum:2014gna ;
Huber:2014bba ; Eichmann:2014xya ; Vujinovic:2014fza ; Alkofer:2014taa ;
Williams:2015cvx ; Huber:2018ned ; Duarte:2016ieu ; Athenodorou:2016oyh ;
Sternbeck:2016ltn , but they are suppressed in the IR. The remaining
parameters are $\alpha=-2-6\delta$, $\beta=-1-3\delta$, and $a=3\delta$.
$\Lambda_{\text{IR}}$ is a small scale to suppress the IR divergence and
$\Lambda_{s}$ a scale of the order of $1\,\text{GeV}$ to tune the IR
suppression. The actual values are not important and vary in each calculation
due to the a posteriori determination of the physical scale. More on model “a”
can be found in Huber:2012kd , on model “b” in Huber:2017txg  and on model “c”
in Fischer:2002hn .

The three methods to subtract the spurious divergences in the gluon propagator
DSE are:

  * •

Analytic subtraction (as): The divergent part of the self-energy is subtracted
analytically Huber:2014tva .

  * •

Gluon loop subtraction (gl): The gluon loop is modified to subtract spurious
divergences Fischer:2002eq .

  * •

Second renormalization condition (rc): The spurious divergences are removed by
applying a second renormalization condition Collins:2008re ; Meyers:2014iwa ;
Huber:2017txg .

##  Appendix B Mass subtraction for nonmonotonic propagators leads to a
negative spectral dimension

We consider an inverse propagator with a minimum and subtract its value at
$p^{2}=0$ to perform the mass-term subtraction. We take this modified inverse
propagator as $F(p^{2})$. It is negative in some interval $(a,b)$. The related
return probability $\mathcal{P}_{a,b}(T)$ is positive because the integrand

$\displaystyle
p^{2}e^{-F(p^{2})T}=p^{2}e^{|F(p^{2})|T}=p^{2}\left(e^{|F(p^{2})|}\right)^{T}$
(31)

is positive in the complete interval. Additionally, the value of the integral
is increasing with T because of $e^{|F(p^{2})|}>1$. The derivative of the
return probability $\mathcal{P}^{\prime}_{a,b}(T)$ is positive too. The
derivative in (31) leads to

$\displaystyle\mathcal{P}^{\prime}_{a,b}(T)\propto\int\limits_{a}^{b}dp^{2}~{}p^{2}|F(p^{2})|\left(e^{|F(p^{2})|}\right)^{T}.$
(32)

Again, all factors of the integrand are positive and so is the integral
itself. It also increases with $T$.

We call the contribution of the return probability coming from the positive
part of $F(p^{2})$ $\mathcal{P}_{rest}(T)$. It is positive and decreases with
$T$. Its derivative $\mathcal{P}^{\prime}_{rest}(T)$, though, is negative due
to the factor $-F(p^{2})$ in the integrand. The spectral dimension reads

$\displaystyle
D_{S}(T)=-2\frac{\mathcal{P}^{\prime}(T)T}{\mathcal{P}(T)}=-2\frac{(\mathcal{P}^{\prime}_{a,b}(T)+\mathcal{P}^{\prime}_{rest}(T))T}{\mathcal{P}_{a,b}(T)+\mathcal{P}_{rest}(T)}.$
(33)

If $T$ is sufficiently large,
$|\mathcal{P}^{\prime}_{a,b}(T)|>|\mathcal{P}^{\prime}_{rest}(T)|$. Hence, the
numerator is positive in this case. The denominator, is also positive for any
$T$. Thus, the resulting spectral dimension is negative for large diffusion
times.

##  Appendix C Asymptotic behavior of the spectral dimension

We give the proofs for the two cases for the asymptotic behavior of the
spectral dimension considered in Sec. II. The first one is for a polynomial
form of the inverse propagator. This case was already considered in Ref.
Alkofer:2014raa  and we give the proof here for convenience of the reader.ß We
consider an inverse propagator of the form

$\displaystyle
F(p^{2})=(p^{2})^{n_{\text{max}}}+\ldots+(p^{2})^{n_{\text{min}}}.$ (34)

For the case $F(p^{2})=(p^{2})^{n}$, the integral for the return probability,
Eq. (9), can be done analytically. First, we perform a change of variable
$p\rightarrow(t/T)^{1/2n}$. The return probability is then proportional to

$\displaystyle\mathcal{P}(T)\propto
T^{-\frac{d}{2n}}\frac{1}{2n}\int_{0}^{\infty}dt\,t^{\frac{d}{2n}-1}e^{-t}=T^{-\frac{d}{2n}}\frac{1}{2n}\Gamma\left(\frac{d}{2n}\right),$
(35)

from which the spectral dimension is calculated as

$\displaystyle D_{S}(T)=-2\frac{d\,\ln P(T)}{d\,\ln T}=\frac{d}{n}.$ (36)

If the inverse propagator has the polynomiaßl form of Eq. (34), the power with
the largest/smallest $n$ will dominate for short/long diffusion times and we
obtain

$\displaystyle\lim_{T\rightarrow 0}D_{S}(T)$
$\displaystyle=\frac{d}{n_{\text{max}}},$ (37)
$\displaystyle\lim_{T\rightarrow\infty}D_{S}(T)$
$\displaystyle=\frac{d}{n_{\text{min}}}.$ (38)

The second case is for the inverse propagator of the form
$(p^{2}-p_{0}^{2})^{c}$ with $c>0$. Using Eq. (10) with
$F^{-1}(k^{2})=(k^{2})^{\frac{1}{c}}+p_{0}^{2}$ and
$F^{\prime}(p^{2})=c\,(k^{2})^{1-\frac{1}{c}}$, this leads to the following
integral:

$\displaystyle\mathcal{P}(T)\propto\int_{0}^{\infty}dk\,\frac{k^{\frac{2}{c}-1}\left(k^{\frac{2}{c}}+p_{0}^{2}\right)^{\frac{d}{2}-1}}{c}e^{-k^{2}\,T}.$
(39)

For $d=4$ this evaluates to

$\displaystyle\mathcal{P}(T)\propto\frac{T^{-\frac{2}{c}}\left(p_{0}^{2}\,T^{\frac{1}{c}}\,\Gamma\left(\frac{1}{c}\right)+\Gamma\left(\frac{2}{c}\right)\right)}{2c}$
(40)

so that the spectral dimension becomes

$\displaystyle
D_{S}(T)=\frac{2\,p_{0}^{2}\,T^{\frac{1}{c}}\Gamma\left(\frac{1}{c}\right)+4\,\Gamma\left(\frac{2}{c}\right)}{c\,p_{0}^{2}\,T^{\frac{1}{c}}\Gamma\left(\frac{1}{c}\right)+c\,\Gamma\left(\frac{2}{c}\right)}.$
(41)

In the limit of $T\rightarrow\infty$, this leads to $2/c$. For two dimensions
we obtain

$\displaystyle\mathcal{P}(T)\propto\frac{T^{-\frac{1}{c}}\Gamma\left(\frac{1}{c}\right)}{2c}$
(42)

and also $2/c$ for $T\rightarrow\infty$. We tested numerically that this is
also true for three dimensions.

## References

  * (1) H. Pagels, A Collective Model of the Hadrons, Phys. Rev. D14 (1976) 2747.
  * (2) R. Delbourgo, The Gluon Propagator, J. Phys. G5 (1979) 603.
  * (3) R. Oehme and W. Zimmermann, Quark and Gluon Propagators in Quantum Chromodynamics, Phys. Rev. D21 (1980) 471.
  * (4) R. Alkofer and L. von Smekal, The infrared behavior of QCD Green’s functions: Confinement, dynamical symmetry breaking, and hadrons as relativistic bound states, Phys. Rept. 353 (2001) 281, arXiv:hep-ph/0007355.
  * (5) A. Maas, Describing gauge bosons at zero and finite temperature, Phys.Rept. 524 (2013) 203–300, arXiv:1106.3942 [hep-ph].
  * (6) A. C. Aguilar, D. Binosi, and J. Papavassiliou, The Gluon Mass Generation Mechanism: A Concise Primer, Front. Phys.(Beijing) 11 no. 2, (2016) 111203, arXiv:1511.08361 [hep-ph].
  * (7) M. Q. Huber, Nonperturbative properties of Yang-Mills theories.  habilitation, Graz U., 2018.  arXiv:1808.05227 [hep-ph].
  * (8) G. B. West, Confinement, the Wilson Loop and the Gluon Propagator, Phys. Lett. 115B (1982) 468–472.
  * (9) M. Hopfer, C. S. Fischer, and R. Alkofer, Running coupling in the conformal window of large-Nf QCD, JHEP 11 (2014) 035, arXiv:1405.7031 [hep-ph].
  * (10) R. Oehme, Analytic structure of amplitudes in gauge theories with confinement, Int. J. Mod. Phys. A10 (1995) 1995–2014, arXiv:hep-th/9412040 [hep-th].
  * (11) R. Alkofer, W. Detmold, C. S. Fischer, and P. Maris, Analytic properties of the Landau gauge gluon and quark propagators, Phys. Rev. D70 (2004) 014014, arXiv:hep-ph/0309077.
  * (12) S. Strauss, C. S. Fischer, and C. Kellermann, Analytic structure of the Landau gauge gluon propagator, Phys.Rev.Lett. 109 (2012) 252001, arXiv:1208.6239 [hep-ph].
  * (13) P. Lowdon, Nonperturbative structure of the photon and gluon propagators, Phys. Rev. D96 no. 6, (2017) 065013, arXiv:1702.02954 [hep-th].
  * (14) P. Lowdon, Dyson–Schwinger equation constraints on the gluon propagator in BRST quantised QCD, Phys. Lett. B786 (2018) 399–402, arXiv:1801.09337 [hep-th].
  * (15) A. K. Cyrol, J. M. Pawlowski, A. Rothkopf, and N. Wink, Reconstructing the gluon, arXiv:1804.00945 [hep-ph].
  * (16) D. Binosi and R.-A. Tripolt, Spectral functions of confined particles, arXiv:1904.08172 [hep-ph].
  * (17) J. Ambjorn, J. Jurkiewicz, and R. Loll, Spectral dimension of the universe, Phys. Rev. Lett. 95 (2005) 171301, arXiv:hep-th/0505113 [hep-th].
  * (18) S. Carlip, Dimension and Dimensional Reduction in Quantum Gravity, Class. Quant. Grav. 34 no. 19, (2017) 193001, arXiv:1705.05417 [gr-qc].
  * (19) S. Carlip, Dimension and Dimensional Reduction in Quantum Gravity, Universe 5 (2019) 83, arXiv:1904.04379 [gr-qc].
  * (20) O. Lauscher and M. Reuter, Fractal spacetime structure in asymptotically safe gravity, JHEP 10 (2005) 050, arXiv:hep-th/0508202 [hep-th].
  * (21) N. Alkofer, F. Saueressig, and O. Zanusso, Spectral dimensions from the spectral action, Phys. Rev. D91 no. 2, (2015) 025025, arXiv:1410.7999 [hep-th].
  * (22) G. Amelino-Camelia, M. Arzano, G. Gubitosi, and J. Magueijo, Dimensional reduction in momentum space and scale-invariant cosmological fluctuations, Phys. Rev. D88 no. 10, (2013) 103524, arXiv:1309.3999 [gr-qc].
  * (23) N. Alkofer, Quantum Gravity from Fundamental Questions to Phenomenological Applications.  PhD thesis, Nijmegen U., 2018.  arXiv:1810.03132 [hep-th].
  * (24) G. Amelino-Camelia, M. Arzano, G. Gubitosi, and J. Magueijo, Planck-scale dimensional reduction without a preferred frame, Phys. Lett. B736 (2014) 317–320, arXiv:1311.3135 [gr-qc].
  * (25) M. Q. Huber, A. Maas, and L. von Smekal, Two- and three-point functions in two-dimensional Landau-gauge Yang-Mills theory: Continuum results, JHEP 1211 (2012) 035, arXiv:1207.0222 [hep-th].
  * (26) P. Boucaud et al., IR finiteness of the ghost dressing function from numerical resolution of the ghost SD equation, JHEP 06 (2008) 012, arXiv:0801.2721 [hep-ph].
  * (27) C. S. Fischer, A. Maas, and J. M. Pawlowski, On the infrared behavior of Landau gauge Yang-Mills theory, Annals Phys. 324 (2009) 2408–2437, arXiv:0810.1987 [hep-ph].
  * (28) R. Alkofer, M. Q. Huber, and K. Schwenzer, Infrared singularities in Landau gauge Yang-Mills theory, Phys. Rev. D81 (2010) 105010, arXiv:0801.2762 [hep-th].
  * (29) L. von Smekal, R. Alkofer, and A. Hauck, The infrared behavior of gluon and ghost propagators in Landau gauge QCD, Phys. Rev. Lett. 79 (1997) 3591–3594, arXiv:hep-ph/9705242.
  * (30) L. von Smekal, A. Hauck, and R. Alkofer, A solution to coupled Dyson-Schwinger equations for gluons and ghosts in Landau gauge, Ann. Phys. 267 (1998) 1, arXiv:hep-ph/9707327.
  * (31) R. Alkofer, C. S. Fischer, and F. J. Llanes-Estrada, Vertex functions and infrared fixed point in Landau gauge SU(N) Yang-Mills theory, Phys.Lett. B611 (2005) 279–288, arXiv:hep-th/0412330 [hep-th].
  * (32) R. Alkofer, C. S. Fischer, M. Q. Huber, F. J. Llanes-Estrada, and K. Schwenzer, Confinement and Green functions in Landau-gauge QCD, PoS CONFINEMENT8 (2008) 019, arXiv:0812.2896 [hep-ph].
  * (33) C. S. Fischer and J. M. Pawlowski, Uniqueness of infrared asymptotics in Landau gauge Yang- Mills theory II, Phys. Rev. D80 (2009) 025023, arXiv:0903.2193 [hep-th].
  * (34) J. M. Cornwall, Dynamical Mass Generation in Continuum QCD, Phys. Rev. D26 (1982) 1453.
  * (35) A. Cucchieri and T. Mendes, What’s up with IR gluon and ghost propagators in Landau gauge? A puzzling answer from huge lattices, PoS LAT2007 (2007) 297, arXiv:0710.0412 [hep-lat].
  * (36) A. Cucchieri and T. Mendes, Constraints on the IR behavior of the ghost propagator in Yang-Mills theories, Phys. Rev. D78 (2008) 094503, arXiv:0804.2371 [hep-lat].
  * (37) A. Sternbeck, L. von Smekal, D. Leinweber, and A. Williams, Comparing SU(2) to SU(3) gluodynamics on large lattices, PoS LAT2007 (2007) 340, arXiv:0710.1982 [hep-lat].
  * (38) I. L. Bogolubsky, E. M. Ilgenfritz, M. Müller-Preussker, and A. Sternbeck, Lattice gluodynamics computation of Landau gauge Green’s functions in the deep infrared, Phys. Lett. B676 (2009) 69–73, arXiv:0901.0736 [hep-lat].
  * (39) 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. D78 (2008) 065047, arXiv:0806.4348 [hep-th].
  * (40) A. Aguilar, D. Binosi, and J. Papavassiliou, Gluon and ghost propagators in the Landau gauge: Deriving lattice results from Schwinger-Dyson equations, Phys.Rev. D78 (2008) 025010, arXiv:0802.1870 [hep-ph].
  * (41) A. K. Cyrol, L. Fister, M. Mitter, J. M. Pawlowski, and N. Strodthoff, Landau gauge Yang-Mills correlation functions, Phys. Rev. D94 no. 5, (2016) 054005, arXiv:1605.01856 [hep-ph].
  * (42) A. C. Aguilar, M. N. Ferreira, C. T. Figueiredo, and J. Papavassiliou, Gluon mass scale through nonlinearities and vertex interplay, arXiv:1909.09826 [hep-ph].
  * (43) D. Dudal, S. P. Sorella, N. Vandersickel, and H. Verschelde, The effects of Gribov copies in 2D gauge theories, Phys. Lett. B680 (2009) 377–383, arXiv:0808.3379 [hep-th].
  * (44) D. Zwanziger, Some exact properties of the gluon propagator, arXiv:1209.1974 [hep-ph].
  * (45) M. Q. Huber and L. von Smekal, On the influence of three-point functions on the propagators of Landau gauge Yang-Mills theory, JHEP 1304 (2013) 149, arXiv:1211.6092 [hep-th].
  * (46) M. Q. Huber, Correlation functions of three-dimensional Yang-Mills theory from Dyson-Schwinger equations, Phys. Rev. D93 no. 8, (2016) 085033, arXiv:1602.02038 [hep-th].
  * (47) M. Q. Huber and L. von Smekal, Spurious divergences in Dyson-Schwinger equations, JHEP 1406 (2014) 015, arXiv:1404.3642 [hep-ph].
  * (48) M. Q. Huber, On non-primitively divergent vertices of Yang–Mills theory, Eur. Phys. J. C77 no. 11, (2017) 733, arXiv:1709.05848 [hep-ph].
  * (49) M. Q. Huber, Gluonic vertices and the gluon propagator in Landau gauge Yang-Mills theory 2019\.  arXiv:1909.10867 [hep-ph].
  * (50) M. Q. Huber, in preparation.
  * (51) V. Gribov, Quantization of Nonabelian Gauge Theories, Nucl.Phys. B139 (1978) 1.
  * (52) D. Zwanziger, Local and Renormalizable Action From the Gribov Horizon, Nucl. Phys. B323 (1989) 513–544.
  * (53) D. Zwanziger, Renormalizability of the critical limit of lattice gauge theory by BRS invariance, Nucl.Phys. B399 (1993) 477–513.
  * (54) J. Gracey, Alternative refined Gribov-Zwanziger Lagrangian, Phys.Rev. D82 (2010) 085032, arXiv:1009.3889 [hep-th].
  * (55) D. Dudal, S. Sorella, and N. Vandersickel, The dynamical origin of the refinement of the Gribov-Zwanziger theory, Phys.Rev. D84 (2011) 065039, arXiv:1105.3371 [hep-th].
  * (56) A. Cucchieri, D. Dudal, T. Mendes, and N. Vandersickel, Modeling the Gluon Propagator in Landau Gauge: Lattice Estimates of Pole Masses and Dimension-Two Condensates, arXiv:1111.2327 [hep-lat].
  * (57) C. S. Fischer, R. Alkofer, and H. Reinhardt, The elusiveness of infrared critical exponents in Landau gauge Yang-Mills theories, Phys. Rev. D65 (2002) 094008, arXiv:hep-ph/0202195.
  * (58) C. S. Fischer and R. Alkofer, Non-perturbative Propagators, Running Coupling and Dynamical Quark Mass of Landau gauge QCD, Phys. Rev. D67 (2003) 094020, arXiv:hep-ph/0301094.
  * (59) M. Q. Huber, K. Schwenzer, and R. Alkofer, On the infrared scaling solution of SU(N) Yang-Mills theories in the maximally Abelian gauge, Eur. Phys. J. C68 (2010) 581–600, arXiv:0904.1873 [hep-th].
  * (60) R. Alkofer, M. Q. Huber, V. Mader, and A. Windisch, On the Infrared Behaviour of QCD Green Functions in the Maximally Abelian Gauge, PoS QCD-TNT-II (2011) 003, arXiv:1112.6173 [hep-th].
  * (61) M. Q. Huber, Gluon and ghost propagators in linear covariant gauges, Phys. Rev. D91 no. 8, (2015) 085018, arXiv:1502.04057 [hep-ph].
  * (62) A. K. Cyrol, M. Mitter, J. M. Pawlowski, and N. Strodthoff, Nonperturbative quark, gluon, and meson correlators of unquenched QCD, Phys. Rev. D97 no. 5, (2018) 054006, arXiv:1706.06326 [hep-ph].
  * (63) H. Reinhardt, G. Burgio, D. Campagnari, E. Ebadati, J. Heffner, M. Quandt, P. Vastag, and H. Vogt, Hamiltonian approach to QCD in Coulomb gauge - a survey of recent results, Adv. High Energy Phys. 2018 (2018) 2312498, arXiv:1706.02702 [hep-th].
  * (64) A. C. Aguilar, D. Binosi, and J. Papavassiliou, Unquenching the gluon propagator with Schwinger-Dyson equations, Phys. Rev. D86 (2012) 014032, arXiv:1204.3868 [hep-ph].
  * (65) C. S. Fischer and R. Alkofer, Infrared exponents and running coupling of SU(N) Yang-Mills theories, Phys. Lett. B536 (2002) 177–184, arXiv:hep-ph/0202202.
  * (66) M. Pelaez, M. Tissier, and N. Wschebor, Three-point correlation functions in Yang-Mills theory, Phys.Rev. D88 (2013) 125003, arXiv:1310.2594 [hep-th].
  * (67) A. Aguilar, D. Binosi, D. Ibáñez, and J. Papavassiliou, Effects of divergent ghost loops on the Green’s functions of QCD, Phys.Rev. D89 (2014) 085008, arXiv:1312.1212 [hep-ph].
  * (68) A. Blum, M. Q. Huber, M. Mitter, and L. von Smekal, Gluonic three-point correlations in pure Landau gauge QCD, Phys. Rev. D 89 (2014) 061703(R), arXiv:1401.0713 [hep-ph].
  * (69) M. Q. Huber, A. L. Blum, M. Mitter, and L. von Smekal, On propagators and vertices of Landau gauge Yang-Mills theory, PoS QCD-TNT-III (2013) 018, arXiv:1401.5241 [hep-ph].
  * (70) G. Eichmann, R. Williams, R. Alkofer, and M. Vujinovic, The three-gluon vertex in Landau gauge, Phys.Rev. D89 (2014) 105014, arXiv:1402.1365 [hep-ph].
  * (71) M. Vujinovic, R. Alkofer, G. Eichmann, and R. Williams, Non-perturbative features of the three-gluon vertex in Landau gauge, Acta Phys. Polon. Supp. 7 no. 3, (2014) 607, arXiv:1404.4474 [hep-ph].
  * (72) R. Alkofer, G. Eichmann, C. S. Fischer, M. Hopfer, M. Vujinovic, R. Williams, and A. Windisch, On propagators and three-point functions in Landau gauge QCD and QCD-like theories, PoS QCD-TNT-III (2013) 003, arXiv:1405.7310 [hep-ph].
  * (73) R. Williams, C. S. Fischer, and W. Heupel, Light mesons in QCD and unquenching effects from the 3PI effective action, Phys. Rev. D93 no. 3, (2016) 034026, arXiv:1512.00455 [hep-ph].
  * (74) A. G. Duarte, O. Oliveira, and P. J. Silva, Further Evidence For Zero Crossing On The Three Gluon Vertex, Phys. Rev. D94 no. 7, (2016) 074502, arXiv:1607.03831 [hep-lat].
  * (75) A. Athenodorou, D. Binosi, P. Boucaud, F. De Soto, J. Papavassiliou, J. Rodriguez-Quintero, and S. Zafeiropoulos, On the zero crossing of the three-gluon vertex, Phys. Lett. B761 (2016) 444–449, arXiv:1607.01278 [hep-ph].
  * (76) A. Sternbeck, QCD propagators and vertices from lattice QCD (in memory of Michael Müller-Preußker), EPJ Web Conf. 137 (2017) 01020, arXiv:1612.06106 [hep-lat].
  * (77) J. Collins, Renormalization: An Introduction to Renormalization, the Renormalization Group and the Operator-product Expansion.  Cambridge University Press, 2008.
  * (78) J. Meyers and E. S. Swanson, The Gluon Propagator with Two-loop Schwinger-Dyson Equations, Phys.Rev. D90 (2014) 045037, arXiv:1403.4350 [hep-ph].