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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−Aa

μsAa

μ−casca

¼Aa

μDab

μcbþg

2fabccacbcc:ð6Þ

In LBV, additional vertices appear that contain an antifield

Aor cand 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

NAPETSCHNIG, ALKOFER, HUBER, and PAWLOWSKI PHYS. REV. D 104, 054003 (2021)

054003-2

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

∂ξχ¼ZdxδΓ

δAa

μ

δΓ

δAa

μ

þδΓ

δca

δΓ

δcaþiδΓ

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

δAa

μ

þδΓ

δAa

μ

∂δΓ

∂χδAa

μ

þ∂δΓ

∂χδca

δΓ

δca−δΓ

δca

∂δΓ

∂χδcaþiba∂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;−pÞþΓcχcðp;0;−pÞΓ¯

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 ΓAχAwhich 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

ΓccAd

ρAa

μ

þΓccχcdΓAa

μ¯

cbcdþΓ¯

cbcdΓAa

μccχcd

þΓAa

μAd

ρ

Γ¯

cbccχAd

ρþΓAd

ρ¯

cbccΓχAa

μAd

ρ;

∂ξΓAa

μAb

νAc

ρ¼ΓχAd

σAa

μAb

ν

ΓAc

ρAd

σþΓχAd

σAa

μ

ΓAb

νAc

ρAd

σ

þpermutations;

∂ξΓAa

μAb

νAc

ρAd

σ¼ΓχAe

τAa

μAb

νAc

ρ

ΓAd

σAe

τþΓχAe

τAa

μAb

ν

ΓAc

ρAd

σAe

τ

þΓχAe

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,Γcχcðp; 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 Aand cas well as χare

indicated explicitly. Propagators are all dressed; small (large) dots

represent bare (dressed) vertices.

YANG-MILLS PROPAGATORS IN LINEAR COVARIANT GAUGES …PHYS. REV. D 104, 054003 (2021)

054003-3

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.

NAPETSCHNIG, ALKOFER, HUBER, and PAWLOWSKI PHYS. REV. D 104, 054003 (2021)

054003-4

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

ΓAAc and Γccc. 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:

ΓAa

μAb

νccðp; q; kÞ¼gfabcHð¯

p2Þgμν;

Γcacbccðp; 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].

YANG-MILLS PROPAGATORS IN LINEAR COVARIANT GAUGES …PHYS. REV. D 104, 054003 (2021)

054003-5

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.

NAPETSCHNIG, ALKOFER, HUBER, and PAWLOWSKI PHYS. REV. D 104, 054003 (2021)

054003-6

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¯

ccðp2Þ2;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¯

ccðsÞ1þωln p2

s−3ξ=22

;ð24Þ

where ω¼11NcαðsÞ=ð12πÞG2ðsÞZðsÞ½DA¯

ccðsÞ2.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)

054003-7

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)

054003-8

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; pÞΓcccð−p; p þq; −qÞ;ðA1cÞ

KðdÞ¼−Ncg2

3Zd4q

ð2πÞ4

GðqÞGðpþqÞ

p2q4ðpþqÞ2ðp2q2−ðp·qÞ2ÞÞDA¯

cc

Tð−p;pþq; qÞΓAAcð−q; −p; p þqÞ;ðA1dÞ

Kðe1Þ¼Ncg2

3Zd4q

ð2πÞ4

ZðpþqÞGðqÞ

p2q4ðpþqÞ4ðq2þ2p·qÞð3p4þðp·qÞ2þ2p2ðq2þ3p·qÞÞ

×CAAAð−p; p þq; −qÞΓAAcðpþq; −p; −qÞ;ðA1eÞ

Kðe2Þ¼Ncg2

3ξZd4q

ð2πÞ4

GðqÞ

q4ðpþqÞ4ðp2q2−ðp·qÞ2ÞCAAAð−p; p þq; −qÞΓAAcðpþq; −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;

Dcccðp2

i;μ2Þ¼

˜

Z1ðΛ;μÞ

˜

Z1ðΛ;νÞDcccðp2

i;ν2Þ;

DAAcðp2

i;μ2Þ¼

˜

Z1ðΛ;μÞ

˜

Z1ðΛ;νÞDAAcðp2

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 lnp2

μ2;

ðbÞ⟶

UV −9ω

88 lnp2

μ2;

ðcÞ⟶

UV 3ω

22 lnp2

μ2;

ðdÞ⟶

UV −3ω

44 lnp2

μ2;

ðe1Þ⟶

UV 9ω

44 lnp2

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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054003-11

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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