Yang-Mills propagators in linear covariant gauges from Nielsen identities

Abstract

We calculate gluon and ghost propagators in Yang-Mills theory in linear covariant gauges. To that end, we utilize Nielsen identities with Landau gauge propagators and vertices as the starting point. We present and discuss numerical results for the gluon and ghost propagators for values of the gauge parameter 0 < ξ ≤ 5. Extrapolating the propagators to ξ → ∞, we find the expected qualitative behavior. We provide arguments that our results are quantitatively reliable at least for values ξ ≲ 1/2 of the gauge-fixing parameter. It is shown that the correlation functions, and, in particular, the ghost propagator, change significantly with increasing gauge parameter. In turn, the ghost-gluon running coupling as well as the position of the zero crossing of the Schwinger function of the gluon propagator remain within the uncertainties of our calculation unchanged.

Full text

Yang-Mills propagators in linear covariant gauges from Nielsen identities
Martin Napetschnig ,1,* Reinhard Alkofer ,1,†Markus Q. Huber ,2,‡and Jan M. Pawlowski 3,4,§
1Institute of Physics, University of Graz, NAWI Graz, Universitätsplatz 5, 8010 Graz, Austria
2Institut für Theoretische Physik, Justus-Liebig-Universität Giessen, 35392 Giessen, Germany
3Institute for Theoretical Physics, Universität Heidelberg, Philosophenweg 12, D-69120 Germany
4ExtreMe Matter Institute EMMI, GSI, Planckstrasse 1, D-64291 Darmstadt, Germany
(Received 28 June 2021; accepted 5 August 2021; published 2 September 2021)
We calculate gluon and ghost propagators in Yang-Mills theory in linear covariant gauges. To that end,
we utilize Nielsen identities with Landau gauge propagators and vertices as the starting point. We present
and discuss numerical results for the gluon and ghost propagators for values of the gauge parameter
0<ξ≤5. Extrapolating the propagators to ξ→∞, we find the expected qualitative behavior. We provide
arguments that our results are quantitatively reliable at least for values ξ≲1=2of the gauge-fixing
parameter. It is shown that the correlation functions, and, in particular, the ghost propagator, change
significantly with increasing gauge parameter. In turn, the ghost-gluon running coupling as well as the
position of the zero crossing of the Schwinger function of the gluon propagator remain within the
uncertainties of our calculation unchanged.
DOI: 10.1103/PhysRevD.104.054003
I. INTRODUCTION
In the past decades, functional approaches such as
Dyson-Schwinger equations (DSEs) or functional renorm-
alization group (FRG) equations have very successfully
contributed to understanding many phenomena in quantum
chromodynamics (QCD), ranging from the hadron reso-
nance spectrum to the phase structure of QCD at non-
vanishing temperatures and densities. The majority of the
respective investigations have been carried out in the
Landau gauge due to the technical as well as conceptual
advantages this gauge provides.
Evidently, obtaining via such approaches results for
physical observables requires truncations to the full hier-
archy of coupled functional equations, typically chosen to
be of a given order in a systematic approximation scheme
such as the vertex expansion. This calls for checks of the
systematic errors of the respective results. For example, the
gauge independence of the computed observables would be
a very powerful self-consistency check. Although demon-
strating generic gauge independence is likely beyond reach
within functional approaches, the test of a reasonably
accurate independence of gauge-invariant quantities when
varying the gauge parameter within a given class of gauges
would provide a convincing (but also costly) verification of
the employed truncations.
In a first step toward such a self-consistency check, we
study in this work the propagators of elementary Yang-
Mills fields in the linear covariant gauges, thus extending
Ref. [1]. In addition, our investigation may serve for
corroborating the current state of the art of functional
studies in the Landau gauge; for the respective recent
Landau gauge DSE results for propagators and vertex
functions, see, e.g., Refs. [2,3] and for recent quantitative
FRG results Refs. [4,5]. These Yang-Mills results within
the Landau gauge, in particular, for the ghost and gluon
propagators, match quantitatively the respective available
lattice results; see, e.g., [6–9]. Detailed discussions of
Yang-Mills correlation functions in the Landau gauge as
well as more results on them can be found, for instance, in
Refs. [2–5,10–21], and references therein.
Herein, we extend previous studies [1,22] and compute
ghost and gluon propagators from Nielsen identities (NIs).
Further results on linear covariant gauges from functional
methods can be found, e.g., in Refs. [1,22–28], from the
(refined) Gribov-Zwanziger framework in Refs. [29–32],
from variational methods in Refs. [33–36], and from lattice
methods in Refs. [37–41]; see also the respective part in the
recent review [3] and references therein.
The NIs describe the dependence of correlation functions
on the gauge-fixing parameter in terms of a differential
equation of the effective action with respect to the gauge-
fixing parameter; see, e.g., [22,23,28]. The resulting
*martin.napetschnig@edu.uni-graz.at
†reinhard.alkofer@uni-graz.at
‡markus.huber@physik.jlug.de
§j.pawlowski@thphys.uni-heidelberg.de
Published by the American Physical Society under the terms of
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article’s title, journal citation,
and DOI. Funded by SCOAP3.
PHYSICAL REVIEW D 104, 054003 (2021)
2470-0010=2021=104(5)=054003(14) 054003-1 Published by the American Physical Society

equations for the correlations functions can be integrated
from the Landau gauge to any linear covariant gauge, and
we are going to report on an investigation in which we used
the quantitative DSE results for Landau gauge correlation
functions from Ref. [2] as a starting point for such an
integration of a set of coupled differential equations.
Moreover, NIs have also been derived for other families
of gauges. Often, these families go under the name
interpolating gauges. Examples include interpolating
gauges between the Landau gauge and the Coulomb gauge
[42,43], the Landau gauge and the maximally Abelian
gauge [26,44,45], the linear covariant gauges, the Coulomb
gauge and the maximally Abelian gauge [46,47], and the
linear covariant gauges, the maximally Abelian gauge, and
the Curci-Ferrari gauge [48].
This article is structured as follows. In the next section,
we introduce the correlation functions and their functional
equations. In Sec. III, the setup including truncations is
presented and discussed. Section IV contains the results,
and we conclude in Sec. Vwith a summary. Several
appendixes contain technicalities including discussions
of the RG and UV properties of the equations and the
model parameter dependence of the solution. The results
for the propagators can be downloaded from https://github
.com/markusqh/YM_data_LinCov.
II. CORRELATION FUNCTIONS AND THEIR
NIELSEN IDENTITIES
As usual in functional approaches, it is assumed that a
Wick rotation to Euclidean space has been performed. The
Lagrangian density of Yang-Mills theory in linear covariant
gauges is then given by
L¼LYM þLgf ;ð1Þ
with
LYM ¼1
4Fa
μνFa
μν;
Fa
μν ¼∂μAa
ν−∂νAa
μ−gfabcAb
μAc
ν;
Lgf ¼s¯
ca∂μAa
μ−iξ
2¯
caba:ð2Þ
The fields are the gluon field Aa
μ, the ghost field ca, the
antighost field ¯
ca, and the Nakanishi-Lautrup field ba.
The Nakanishi-Lautrup fields are introduced via the
Becchi-Rouet-Stora-Tyutin (BRST) transformation [49,50],
denoted by s, and given by
sAa
μ¼−Dab
μcb;
sca¼−1
2gfabccbcc;
s¯
ca¼iba;
sba¼0;ð3Þ
where
Dab
μ¼δab∂μþgfabcAc
μð4Þ
is the covariant derivative in the adjoint representa-
tion. Then, the gauge-fixing part of the Lagrangian reads
explicitly
Lgf ¼iba∂μAa
μ−¯
ca∂μð−Dab
μcbÞ−i
2χ¯
caba;ð5Þ
where χ¼sξwas introduced as the BRST transformation of
the gauge-fixing parameter ξ.
The BRST transformations are nonlinear for the gluon
and ghost fields. We are going to work with the Batalin-
Vilkovsky (BV) or antifield formalism (see, e.g., [51–54])
and introduce sources, called antifields, for them:
LBV ¼−Aa
μsAa
μ−casca
¼Aa
μDab
μcbþg
2fabccacbcc:ð6Þ
In LBV, additional vertices appear that contain an antifield
Aor cand are, hence, called antifield vertices.
In the Landau gauge, it is sufficient to consider the
completely transverse part of the dressing functions, as they
form a closed system [5,12], and we split the gluon
propagator into a transverse and longitudinal part written as
Dab
μν ðpÞ¼δabDμν ðpÞ¼δabðDT
μνðpÞþDL
μνðpÞÞ;
DT
μνðpÞ¼Pμν ðpÞZðp2Þ
p2;
DL
μνðpÞ¼pμpν
p2
ZLðp2Þ
p2;ð7Þ
where PμνðpÞ¼gμν −pμpν=p2is the transverse projection
operator. The Slavnov-Taylor identity (STI) for the gluon
propagator enforces its longitudinal dressing function to be
constant: ZLðp2Þ¼ξ; i.e., all quantum corrections are
transverse. Correspondingly, in the Landau gauge ξ¼0,
the gluon propagator is proportional to a transverse
projection operator. For ξ>0, this is no longer true, and
also the trivial longitudinal part enters.
The ghost has only one dressing function:
Dab
GðpÞ¼−δab Gðp2Þ
p2:ð8Þ
The ghost-gluon vertex can be conveniently split into a
transverse and a longitudinal part:
ΓAa
μ¯
cbccðk;p; qÞ¼−igfabcðDA¯
cc;T ðk;p; qÞPμνðkÞpν
þDA¯cc;Lðk;p; qÞkμÞ:ð9Þ
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We use a compact notation where the subscripts denote
the fields with indices corresponding to the momentum
arguments. The tree-level expression is, respectively,
DA¯
cc;T ðk;p; qÞ¼1and DA¯
cc;Lðk;p; qÞ¼p·k=k2.For
the three-gluon vertex, we use only a dressed tree-
level tensor, thereby neglecting components which are
subleading [55]:
ΓAa
μAb
νAc
ρðp; q; rÞ
¼igfabcCAAAðp; q; rÞ
×½ðp−qÞρgμν þðq−rÞμgνρ þðr−pÞνgμρ:ð10Þ
The NI encodes the dependence of the effective action Γon
the gauge-fixing parameter ξand reads
∂Γ
∂ξχ¼ZdxδΓ
δAa
μ
δΓ
δAa
μ
þδΓ
δca
δΓ
δcaþiδΓ
δ¯
caba:ð11Þ
The right-hand side of Eq. (11) follows from the BRST
invariance of the effective action. For the final form, we
differentiate Eq. (11) with respect to χand set χ¼0. This
leads to a master equation for the NIs:
∂Γ
∂ξ


χ¼0
¼Zdx∂δΓ
∂χδAa
μ
δΓ
δAa
μ
þδΓ
δAa
μ
∂δΓ
∂χδAa
μ
þ∂δΓ
∂χδca
δΓ
δca−δΓ
δca
∂δΓ
∂χδcaþiba∂δΓ
∂χδ¯
ca


χ¼0
:
ð12Þ
NIs for correlation functions can now be obtained by
applying further field derivatives to Eq. (12). For related
work on Nielsen identities, see [56,57].
For the propagators, the NIs in the form (12) used here
were derived in Ref. [23], where also a perturbative one-
loop analysis was done, and we refer to this reference for
further details. The NIs for the ghost and gluon propagators
read in Euclidean momentum space:
∂ξΓ¯
ccðp2Þ¼
ipμ
p2Γ¯
cχAμðp;0;−pÞþΓcχcðp;0;−pÞΓ¯
ccðp2Þ;
ð13aÞ
∂ξΓAμAνðp2Þ¼2ΓA
ρχAμðp; 0;−pÞΓAνAρðp2Þ;ð13bÞ
where we have suppressed the color indices. For the
numerical solution of Eqs. (13), we use approximations
for the vertices Γ¯
cχA,Γcχc, and ΓAχAwhich will be
discussed in detail in Sec. III.
For the ghost-gluon, the three-gluon, and the four-gluon
vertices, the NIs read, respectively,
∂ξΓAa
μ¯cbcc¼−ipρ
p2Γχ¯cbAd
ρAa
μ
Γ¯cdcbþΓχ¯cbAd
ρ
ΓccAd
ρAa
μ
þΓccχcdΓAa
μ¯
cbcdþΓ¯
cbcdΓAa
μccχcd
þΓAa
μAd
ρ
Γ¯
cbccχAd
ρþΓAd
ρ¯
cbccΓχAa
μAd
ρ;
∂ξΓAa
μAb
νAc
ρ¼ΓχAd
σAa
μAb
ν
ΓAc
ρAd
σþΓχAd
σAa
μ
ΓAb
νAc
ρAd
σ
þpermutations;
∂ξΓAa
μAb
νAc
ρAd
σ¼ΓχAe
τAa
μAb
νAc
ρ
ΓAd
σAe
τþΓχAe
τAa
μAb
ν
ΓAc
ρAd
σAe
τ
þΓχAe
τAa
μΓAb
νAc
ρAd
σAe
τþpermutations:ð14Þ
III. TRUNCATION AND INPUT
The NIs are exact functional equations, and we cannot
solve them without approximations. For the propagator
equations, we need Γ¯
cχAμðp; 0;−pÞ,Γcχcðp; 0;−pÞ, and
ΓAμχA
νðp; 0;−pÞ. We follow Ref. [22] and calculate them
from the first order in a skeleton expansion, also called
dressed-loop expansion, shown in Fig. 1. An additional
(a)
(c)
(b)
(d) (e)
FIG. 1. Skeleton expansions for Γ¯cχAμ,Γcχc, and ΓAχA(top to
bottom). Wiggly lines denote gluons and dotted ones ghosts.
Wiggly-dashed ones represent a mixed gluon-Nakanishi-Lautrup
field propagator. The antifields Aand cas well as χare
indicated explicitly. Propagators are all dressed; small (large) dots
represent bare (dressed) vertices.
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approximation in Ref. [22] was the use of bare vertices. In
this work, however, we keep the vertices dressed. The
resulting expressions KðiÞarising on the right-hand side of
Eq. (13) correspond to the loop diagrams i¼a,b,c,d,ein
Fig. 1and are provided in Appendix A.
Using these expressions and switching from the two-
point functions ΓAA and Γ¯cc to the dressing function Zand
G, respectively, the propagator NIs can be written as
∂ξln Gðp2Þ¼KðaÞþKðbÞþKðcÞ;
∂ξln Zðp2Þ¼KðdÞþKðe1ÞþKðe2Þ;ð15Þ
where the i¼econtribution has been split into two terms
for convenience; cf. Eqs. (A1).
From the loop diagrams, we can directly confirm that
the NIs transform correctly under a change of the renorm-
alization scale μ1→μ2. The NIs depend only on renor-
malized correlation functions and couplings Oiðp; μÞthat
schematically change as
Oiðp; μ1Þ→
ZOiðΛ;μ1Þ
ZOiðΛ;μ2ÞOiðp; μ2Þ;ð16Þ
where ZOiðΛ;μÞis the renormalization factor of the
respective correlation function or coupling Oi. The explicit
forms are deferred to Appendix B.
We exemplify the consistency of the left- and right-hand
sides of the NIs under a change of the renormalization
scale with diagram (a). Under a change of the renormal-
ization scale μ→ν, diagram (a) transforms, upon using
Eq. (B2),as
ZgðΛ;νÞ2˜
Z3ðΛ;νÞ2˜
Z1ðΛ;μÞ2
ZgðΛ;μÞ2˜
Z3ðΛ;μÞ2˜
Z1ðΛ;νÞ2¼Z3ðΛ;μÞ
Z3ðΛ;νÞ:ð17Þ
This is exactly as the ξderivative on the left-hand side of
Eq. (15) transforms. For all other diagrams, this analysis
can be repeated and leads to the same result.
As the starting value for integrating the NIs, we use
the Landau gauge ξ¼0, for which we have results for the
propagators and vertices. However, we also the need the
vertices for ξ>0. Based on the fact that the ξdependence
for the propagators found on the lattice [37–41] is small, we
adopt as working assumption that the ghost-gluon and
three-gluon vertices deviate only little as well and use
results for ξ¼0for all ξ. Only in the UV do we
accommodate the correct ξdependence by modifying the
anomalous running accordingly. In addition, we approxi-
mate their longitudinal parts with the transversely projected
ones due to the lack of concrete results for the former.
A final approximation consists in taking only a single
kinematic configuration for each vertex. For the three-
gluon vertex, this is a good approximation due to its small
angular dependence [2,55,58,59]. The ghost-gluon vertex
shows more angular dependence, which we neglect here,
though. This is justified by the overall modest variation of
the ghost-gluon vertex with respect to momenta.
The Landau gauge results [2,60] used as initial values for
solving the NIs are shown in Figs. 2and 3in comparison to
lattice results. Specifically, we use a self-contained solution
that possess several advantageous properties. Among them
are manifest gauge covariance expressed by the good
agreement of different couplings in the perturbative regime
and a unique treatment of quadratic divergences; for details,
we refer to Ref. [2]. We also fix the overall scale from these
results which was obtained by matching the maximum of
the gluon dressing function to lattice results.
In the Landau gauge, it is well studied that a family of
different solutions can be obtained from functional equa-
tions which differ in their IR behavior [12,68–74]. These
solutions are different only in the region below 2 GeV. Most
notably, the maximum in the gluon propagator changes.
FIG. 2. Ghost and gluon dressing functions (left) and gluon propagator (right) from lattice [61] and DSE calculations [2] in the Landau
gauge.
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To explore the existence of such solutions also beyond the
Landau gauge, we choose two different sets of solutions.
One of the solutions is the one that agrees best with lattice
results and has only a shallow, hardly visible maximum in
the gluon propagator. As a second choice, we take a
solution with a more pronounced maximum; see Fig. 2.
It remains to specify the models used for the vertices
ΓAAc and Γccc. When using bare vertices, we found that in
the infrared (IR) individual loop diagrams can qualitatively
modify the IR solution for the gluon propagator. This is
either resolved by cancellations between individual dia-
grams or by the IR behavior of the antifield vertices. We
explore the second option, as it is currently not clear if the
first one can be realized in the truncation we use.
As Ansatz for the vertices, we multiply their tree-level
tensors with products of the ghost and gluon dressing
functions with appropriate powers:
ΓAa
μAb
νccðp; q; kÞ¼gfabcHð¯
p2Þgμν;
Γcacbccðp; q; kÞ¼−gfabcHð¯
p2Þ;
HðxÞ¼GðxÞαZðxÞβ
GðsÞαZðsÞβ;ð18Þ
with ¯
p2¼ðp2þq2þk2Þ=2. The denominator ensures
that the vertex models are unity at ¯
p2¼s. The antifield
vertices run logarithmically like the ghost-gluon vertex, as
can be checked by a perturbative one-loop analysis. The
exponents are determined such that they respect this UV
behavior. As a second condition, the integrals in the NIs
should be IR finite. We make the simple Ansätze
α¼α0þα1ξ;β¼β0þβ1ξ;ð19Þ
where αiand βiare ξ-independent parameters. Enforcing
the conditions above, we obtain
α0¼−26β0
9;α1¼−9−4β0
6;β1¼9−4β0
12 :ð20Þ
β0>0is a free parameter for which we choose for
convenience β0¼1. A test of the sensitivity of our results
on this choice as well as a discussion of the function HðxÞ
is provided in Appendix D. It can be seen that the parameter
β0, if chosen within a reasonable range, influences only the
IR and this in a quantitatively mild way.
Both the ghost and the gluon NI have the form
∂ln M
∂ξ¼K: ð21Þ
Kdenotes the integrals from the skeleton expansions of the
vertices. The formal solution to this equation is
MðξÞ¼Mðξ0ÞeRξ
ξ0
dξK;ð22Þ
where ξ0denotes the gauge-fixing parameter for a known
solution. The quantity Kis obtained by numerically
calculating the integrals which are standard one-loop
two-point integrals. As a computational framework, we
use C
RASY
DSE [75]. The integrals are logarithmically
divergent and need to be renormalized. We do so by
momentum subtraction and require that the dressing
functions stay the same at the highest calculated momen-
tum point. It should be noted that no quadratic divergences
[76] are present. This situation has to be contrasted with the
DSE or flow equation for the gluon propagator for which
spurious quadratic divergences can arise; cf. Refs. [2,4,77].
IV. RESULTS
For the full nonperturbative solution, we solve the
differential equations (13) up to ξ¼5starting from the
Landau gauge. The check of the self-consistency of the UV
limit is deferred to Appendix C.
The results for the gluon dressing functions and the
gluon propagators are shown in Fig. 4. With increasing ξ,
FIG. 3. Ghost-gluon (left) and three-gluon vertices (right) in Landau gauge from lattice [62–64] and DSE calculations [2] in the
Landau gauge. For more lattice results on the three-gluon vertex, see, e.g., [65–67].
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the gluon propagator decreases. This can be seen in both the
maximum of the dressing function and the IR behavior of
the propagator. The difference between Landau gauge and
ξ¼0.5is not drastic and compatible with lattice results
[39]. We find at 0 and 1 GeV that the gluon propagator goes
down by 8% and 7%, respectively, for ξ¼0.5. For the
lattice results, these ratios are given in Ref. [39] as
approximately 10% and 5%, respectively, with errors of
a few percentage points each. It should be noted, though,
that the IR behavior of the gluon propagator depends on the
employed models for the antifield vertices. All individual
diagrams in the gluon NI are IR finite as shown in Fig. 12 in
Appendix A. This comes from the IR behavior of the
antifield vertices. If we used bare vertices, IR divergences
would arise that would qualitatively change the IR behavior
of the gluon propagator.
The ghost propagator shows for ξ>0the already known
logarithmic IR suppression [1,22]; see Fig. 5. This behavior
results from diagram (a); see Fig. 12. For small ξ, the
deviation from the Landau gauge is not very large. This
agrees with lattice results which do not see a change in the
ghost propagator up to ξ¼0.3and above approximately
500 MeV, which was the lowest accessible momentum
value [40,41]. In the continuum results displayed here,
evaluated down to 0.01 MeV, we do, however, see devia-
tions from the Landau gauge behavior below 500 MeV also
for small values of the gauge parameter ξ. Sizable devia-
tions appear for higher values of the gauge-fixing parameter
for which also the effect on the UV behavior becomes
visible.
Given the comparatively simple approximation
employed for the NIs, the agreement with lattice results
is very good. On the other hand, the method is very stable,
and we calculated up to ξ¼5without encountering any
problems. Indeed, we can easily check that for ξ¼3and
ξ¼13=3the correct one-loop anomalous dimensions are
produced, as for these values the ghost and the gluon
anomalous dimensions vanish, respectively. This is illus-
trated in Fig. 6, where the corresponding dressing functions
FIG. 4. The gluon dressing function (left) and propagator (right) for various ξincluding the starting point ξ¼0.
FIG. 5. The ghost dressing function for various ξincluding the
starting point ξ¼0.
FIG. 6. The UV behavior of the ghost and gluon dressing
functions for ξ¼3and ξ¼13=3, respectively.
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are compared to the Landau gauge ones in the momentum
region from 1 to 105GeV2.
The ghost-gluon coupling is defined via the relation
αðp2Þ¼αðμ2ÞG2ðp2ÞZðp2Þ½DA¯
ccðp2Þ2;ð23Þ
where the ghost-gluon vertex dressing DA¯
cc is evaluated at
the symmetric point and αðμ2Þ¼g2=ð4πÞ. One-loop uni-
versality entails that any dependence of the coupling on the
gauge-fixing parameter is suppressed at high momenta.
Beyond one loop, a dependence on ξcan appear; see, e.g.,
Refs. [78–80]. However, we can still assess the effect of the
truncation by comparing the couplings in the perturbative
regime above a few GeV. For the coupling, the correct
running of all quantities is important. We thus use the one-
loop resummed expression for the ghost-gluon vertex:
DA¯
cc;1lðp2Þ¼DA¯
ccðsÞ1þωln p2
s−3ξ=22
;ð24Þ
where ω¼11NcαðsÞ=ð12πÞG2ðsÞZðsÞ½DA¯
ccðsÞ2.We
show the couplings for various values of ξin Fig. 7.
Hereby, the scale sis chosen as 105GeV2; phrased
otherwise, the couplings are fixed at this value. As can
be seen, down to ≲10 GeV, they agree even for higher
values of the gauge-fixing parameters. Up to ξ¼0.5, the
agreement is good down to approximately 2 GeV. To
appreciate this agreement, we also show the coupling
without the ghost-gluon vertex dressing in Fig. 7. The
agreement is worse then, and the order of magnitudes is
different with ξ¼0having the largest coupling. We
additionally show the one-loop vertex dressing in the inset
to highlight that the vertex dressing becomes sizable
already for low ξ. It would be interesting to test the ξ
dependence of related quantities like the effective charge
defined in Ref. [81].
Another interesting quantity is the Schwinger function
ΔðtÞof the gluon propagator, defined as the Fourier
transformation of the momentum space propagator for
vanishing spatial momentum. If the propagator violates
positivity, this is reflected in the Schwinger function.
Figure 8shows the Schwinger function for various values
of the gauge-fixing parameter. Up to approximately ξ¼1,
the Schwinger function barely changes. In particular, the
position of the zero crossing does not move. Only for
higher values of ξdoes it move slightly. This stability is in
marked contrast to the situation for the family of solutions
in the Landau gauge, where the position of the zero
crossing moves [2]. This can be understood as the existence
of the zero crossing is related to the maximum of the gluon
propagator. For linear covariant gauges, we find that the
position of this maximum is basically constant in ξ.
Different members of the family of solutions in the
Landau gauge, on the other hand, exhibit different positions
for the maxima [2,4], and, hence, the Schwinger function
also changes.
Finally, to explore the fate of the family of different
solutions for correlation functions in the Landau gauge
[12,17,73,82–84], we also solved the NIs using a second
FIG. 7. Left: the coupling for various values of ξ. Right: the coupling without the vertex dressing. The inset shows the one-loop
expressions for the ghost-gluon vertex [Eq. (24)] used for the couplings.
FIG. 8. The absolute value of the Schwinger function for
various values of the gauge-fixing parameter. The Schwinger
function is negative for t≳1.3fm.
YANG-MILLS PROPAGATORS IN LINEAR COVARIANT GAUGES …PHYS. REV. D 104, 054003 (2021)
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Landau gauge solution. In this context, it should be
mentioned that these different solutions may correspond
to different nonperturbative infrared gauge completions
of the perturbative Landau gauge as discussed in
Refs. [12,83]. If this conjecture is correct, this would
correspond to a second gauge-fixing direction in addition
to ξ. Indeed, all observables in Yang-Mills theory and QCD
computed so far within this potential family of infrared
completions of the Landau gauge agree within the respec-
tive error bars. A specifically relevant example in the
present context of Yang-Mills theories is provided by the
glueball masses; see [85]. In line with the conjecture
discussed above, the obtained masses did not show any
deviations within errors.
In all plots in this section, we used up to here the solution
that is closest to lattice results. It is characterized by a very
flat maximum of the gluon propagator and a ghost dressing
function that is relatively small in the deep infrared. The
different Landau gauge solution used next as a starting
point for the NIs is shown in Fig. 2in comparison to the
previously used solution. The second solution has a
pronounced maximum in the gluon propagator and shows
a clear increase of the ghost dressing function at low
momenta. The results for a selection of values of ξare
shown in Fig. 9. The typical features of the Landau gauge
solution type are inherited by the ξ>0ones. In particular,
the gluon propagator has a maximum from which it follows
immediately that it violates positivity. Note that such a
property also leads to a spectral dimension of one in the
deep IR [86]. Correspondingly, if the maximum vanished,
this would imply a qualitative change of the type of
solution, and it is reassuring that we do not observe that.
Since a nonzero gauge-fixing parameter washes out the
gauge-fixing condition of the Landau gauge and, thus, the
differences between the two solutions, it is interesting to
check if the two solutions approach each other for high
values of ξ. To assess that, we plot the ghost dressing
function and the gluon propagator at fixed momenta as a
function of ξin Fig. 10. For the ghost dressing function, we
see that the two solutions come closer to each other for
higher values. For the gluon propagator, on the other hand,
this effect is not observed. In both cases, it seems plausible
that for ξ→∞the functions vanish. This limit corresponds
to removing the gauge fixing which, in fact, necessarily will
eventually lead to a vanishing gluon propagator. Phrased
otherwise, we see the expected behavior based on the
general properties of the linear covariant gauges. This
provides some confidence that the overall qualitative
behavior of the propagators is correct for all allowable
values of the gauge parameter 0<ξ<∞.
The distinct relative behavior of the two solutions is
clearly visible in Fig. 11, which shows the ratio between the
propagators at a fixed momentum point in the IR. The ratio
stays constant for the gluon propagator but depends
FIG. 9. Ghost dressing function (left) and gluon propagator (right) for two different decoupling solutions as starting points at various
values of ξ.
FIG. 10. The gauge parameter dependence of the ghost dressing
function (dashed line) and gluon propagator (solid line) at fixed
momenta. The lowest calculated value is for 1=ξ¼1=5.
NAPETSCHNIG, ALKOFER, HUBER, and PAWLOWSKI PHYS. REV. D 104, 054003 (2021)
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strongly on the gauge-fixing parameter for the ghost
propagator. Since the plotted ratio is G1=G2, this means
that the ghost propagator with higher values in the IR for
the Landau gauge decreases faster than the one with lower
values.
V. SUMMARY
We have calculated the ghost and gluon propagators of
Yang-Mills theory (respectively, quenched QCD) in the
linear covariant gauges for values of the gauge-fixing
parameter 0<ξ≤5. The starting point has been results
in the Landau gauge, ξ¼0, which were obtained in a self-
contained DSE calculation. As external input, we employed
the nonperturbative parts of the ghost-gluon and three-
gluon vertices of Landau gauge for all values of ξ.In
addition, we used Ansätze for the antifield vertices which
contain one free parameter. We found that the solutions are
not very sensitive to variations of this parameter.
In the IR, we recover the logarithmic suppression of the
ghost dressing function, predicted by earlier investigations,
and find an IR finite gluon propagator. The latter result,
however, happens by construction based on the antifield
vertex model. All results agree well, even quantitatively,
with available lattice results.
Compared to other methods, our setup is quite stable,
even at values of ξbeyond the Feynman gauge. In
particular, we recover the correct UV behavior for the
propagators most convincingly seen by vanishing anoma-
lous dimensions for the ghost and gluon dressing functions
at ξ¼3(Yennie gauge) and ξ¼13=3, respectively. We did
not encounter any signs of instability up to the highest
calculated value, ξ¼5.
While the changes of propagators and vertices are
sizable, observables are ξindependent. This calls for
respective studies of, e.g., glueball masses as done in
Ref. [85] for the Landau gauge. Such a study was beyond
the scope of the present work. Instead, as a first step in this
direction, we have discussed the ξdependence of the ghost-
gluon coupling (Fig. 7) and the zero crossing of the
Schwinger function (Fig. 8). We have shown that the ξ
dependence of both the coupling as well as the Schwinger
function zero crossing is very small up to ξ¼0.5, which is
highly nontrivial. Beyond ξ¼0.5, the reliability of the
current approximation is successively getting worse,
because we do not consider the backcoupling of the ξ
dependence in the vertices. Nevertheless, the observed
deviations are still quite small.
We also have explored the potential family of non-
perturbative infrared completions, as discussed in the
Landau gauge. We have tested two different starting points
and obtained two corresponding sets of solutions for ξ>0.
The qualitative features, in particular, violation of positiv-
ity, remain intact at least up to ξ¼5. In the limit of infinite
ξ, both propagators are in agreement with the expectation
that they vanish in this limit.
In the present work, we have restricted ourselves
to pure Yang-Mills theory. However, the inclusion
of dynamical quarks is straightforward, as there are no
direct quark contributions in the gluon and ghost
Nielsen identities [23]. Moreover, the Nielsen identity
for the quark propagator has a similar structure as those
for the other propagators and could be solved within a
skeleton expansion. It would be also interesting to extend
the current study to other covariant gauges like the
maximally Abelian gauge. There, direct calculations are
complicated due to its IR dominant two-loop dia-
grams [26,87].
ACKNOWLEDGMENTS
Support by the FWF (Austrian Science Fund) under
Contract No. P27380-N27 is gratefully acknowledged. This
work is supported by EMMI (ExtreMe Matter Institute, GSI)
and the BMBF (Federal Ministry of Education and Research,
Germany) Grant No. 05P18VHFCA. It is part of and
supported by the DFG (German Research Foundation)
Collaborative Research Centre SFB 1225 (ISOQUANT)
and the DFG under Germany’s Excellence Strategy EXC—
2181/1—390900948 (the Heidelberg Excellence Cluster
STRUCTURES).
APPENDIX A: DIAGRAMS IN THE NIELSEN
IDENTITIES OF THE PROPAGATORS
The loops displayed in Fig. 1lead to the following
expressions:
KðaÞ¼−Ncg2
2Zd4q
ð2πÞ4
GðqÞGðpþqÞ
q2ðpþqÞ2
×DA¯
cc
Lð−p;pþq;−qÞDA¯
cc
Lðq;p;−p−qÞ;ðA1aÞ
FIG. 11. The ratio of the propagators for two solutions. For the
ghost propagator, the lowest calculated momentum is used, and
for the gluon propagator, zero momentum.
YANG-MILLS PROPAGATORS IN LINEAR COVARIANT GAUGES …PHYS. REV. D 104, 054003 (2021)
054003-9

KðbÞ¼−Ncg2
2Zd4q
ð2πÞ4
GðqÞZðpþqÞ
p2q4ðpþqÞ2DA¯
cc
Tð−p−q;p; qÞðp2q2−ðp·qÞ2ÞÞCAAAð−p; p þq; −qÞ;ðA1bÞ
KðcÞ¼−Ncg2
2Zd4q
ð2πÞ4
GðqÞGðpþqÞ
q2ðpþqÞ2DA¯
cc
Lðq;−p−q; pÞΓcccð−p; p þq; −qÞ;ðA1cÞ
KðdÞ¼−Ncg2
3Zd4q
ð2πÞ4
GðqÞGðpþqÞ
p2q4ðpþqÞ2ðp2q2−ðp·qÞ2ÞÞDA¯
cc
Tð−p;pþq; qÞΓAAcð−q; −p; p þqÞ;ðA1dÞ
Kðe1Þ¼Ncg2
3Zd4q
ð2πÞ4
ZðpþqÞGðqÞ
p2q4ðpþqÞ4ðq2þ2p·qÞð3p4þðp·qÞ2þ2p2ðq2þ3p·qÞÞ
×CAAAð−p; p þq; −qÞΓAAcðpþq; −p; −qÞ;ðA1eÞ
Kðe2Þ¼Ncg2
3ξZd4q
ð2πÞ4
GðqÞ
q4ðpþqÞ4ðp2q2−ðp·qÞ2ÞCAAAð−p; p þq; −qÞΓAAcðpþq; −p; −qÞ:ðA1fÞ
The last integral was split into two parts to disentangle
the contribution from the transverse and longitudinal parts
of the gluon propagator. We used pμqνrρΓAAA
μνρ ðp; q; rÞ¼0
in several places to simplify the expressions. For example,
due to this, only the transversely projected part of the ghost-
gluon vertex appears in the second diagram. The results for
the individual KðiÞare shown in Fig. 12 for two different
values of ξ.
APPENDIX B: RENORMALIZATION GROUP
PROPERTIES OF CORRELATION FUNCTIONS
Here we provide the RG rescalings of the correlation
functions and couplings Oiused in the NIs, schematically
provided in Eq. (16).
Since all quantities in the identities are renormalized
ones, they behave under a change of the renormalization
group scale μ→νas follows:
Dμνðp2;μ2Þ¼Z3ðΛ;νÞ
Z3ðΛ;μÞDμνðp2;ν2Þ;
DGðp2;μ2Þ¼
˜
Z3ðΛ;νÞ
˜
Z3ðΛ;μÞDGðp2;ν2Þ;
DAAAðp2
i;μ2Þ¼Z1ðΛ;μÞ
Z1ðΛ;νÞDAAAðp2
i;ν2Þ;
DA¯
ccðp2
i;μ2Þ¼
˜
Z1ðΛ;μÞ
˜
Z1ðΛ;νÞDA¯
ccðp2
i;ν2Þ;
Dcccðp2
i;μ2Þ¼
˜
Z1ðΛ;μÞ
˜
Z1ðΛ;νÞDcccðp2
i;ν2Þ;
DAAcðp2
i;μ2Þ¼
˜
Z1ðΛ;μÞ
˜
Z1ðΛ;νÞDAAcðp2
i;ν2Þ;
gðμÞ¼ZgðΛ;νÞ
ZgðΛ;μÞgðνÞ;
ξðμÞ¼Z3ðΛ;νÞ
Z3ðΛ;μÞξðνÞ;ðB1Þ
FIG. 12. Contributions of individual diagrams at two values of ξfor the ghost (left) and the gluon (right) NIs.
NAPETSCHNIG, ALKOFER, HUBER, and PAWLOWSKI PHYS. REV. D 104, 054003 (2021)
054003-10

where Z3,
˜
Z3,Z1,
˜
Z1, and Zgare the renormalization
constants for the gluon propagator, the ghost propagator,
the three-gluon vertex, the ghost-gluon vertex, and the
coupling, respectively, which are related by the STIs
Z2
1¼Z2
gZ3
3;
˜
Z2
1¼Z2
gZ3
˜
Z2
3:ðB2Þ
APPENDIX C: CONSISTENT UV LIMIT OF
CORRELATION FUNCTIONS
Here we discuss the self-consistency of the UV behavior
of the NIs. The employed approximation is exact at the
perturbative one-loop level. Consequently, on the right-
hand side the correct ξ-dependent part of the anomalous
dimension must emerge. This can be seen as follows.
Consider the loop integrals with bare dressing functions for
large loop momenta q. The angle integrals can then be
performed and lead to
ðaÞ⟶
UV 3ω
88 lnp2
μ2;
ðbÞ⟶
UV −9ω
88 lnp2
μ2;
ðcÞ⟶
UV 3ω
22 lnp2
μ2;
ðdÞ⟶
UV −3ω
44 lnp2
μ2;
ðe1Þ⟶
UV 9ω
44 lnp2
μ2;
ðe2Þ⟶
UV 0:ðC1Þ
Only the logarithmic parts were kept, and μis a renorm-
alization scale. Summing up the corresponding coefficients
leads to
∂ξδ¼3
44 ;∂ξγ¼3
22 ;ðC2Þ
as can be checked with Table I. This is consistent to one-
loop order with the left-hand side of the equation.
TABLE I. The one-loop anomalous dimensions of the propa-
gators and vertices.
Anomalous dimension
Ghost propagator δ¼−9−3ξ
44
Gluon propagator γ¼−13−3ξ
22
Ghost-gluon vertex γghg ¼−3ξ
22
Three-gluon vertex γ3g¼17−9ξ
44
FIG. 13. The model employed for the antifield vertices for two
values of the parameter β0at different values for ξ.
FIG. 14. Ghost dressing function (top) and gluon propagator
(bottom) for β0¼0.5and 1 at various values of ξ.
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APPENDIX D: DEPENDENCE ON THE
PARAMETER β0
The model we employ for the antifield vertices depends
on one parameter, β0. For the results shown in the main
part, we used β0¼1. We tested the influence of this
parameter by calculating the propagators also with
β0¼0.5. The model function Hð¯
p2Þfor these two choices
is shown in Fig. 13. In the quantitatively relevant regime
around 1 GeV, the two parameter values lead only to small
differences for all ξ. The rise in the UV for larger ξcomes
directly from the anomalous dimension of the antifield
vertices.
The propagators obtained from β0¼0.5are compared to
the ones from β0¼1in Fig. 14. We can see that changing
β0affects basically only the IR. Only for ξ>4small effects
in the ghost dressing function are seen also in the mid-
momentum regime. We, thus, conclude that within
the present approximation scheme the dependence on the
model for the antifield vertices is of minor importance and
seen quantitatively only for low momenta.
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