Two- and three-point Green's functions in two-dimensional Landau-gauge
  Yang-Mills theory
Two- and three-point Green’s functions in two-dimensional Landau-gauge Yang-Mills
theory
Axel Maas1, ∗
Department of Complex Physical Systems, Institute of Physics,
Slovak Academy of Sciences, Dúbravská cesta 9, SK-845 11 Bratislava, Slovakia
(Dated: November 4, 2018)
The ghost and gluon propagator and the ghost-gluon and three-gluon vertex of two-dimensional
SU(2) Yang-Mills theory in (minimal) Landau gauge are studied using lattice gauge theory. It is
found that the results are qualitatively similar to the ones in three and four dimensions. The propa-
gators and the Faddeev-Popov operator behave as expected from the Gribov-Zwanziger scenario. In
addition, finite volume effects affecting these Green’s functions are investigated systematically. The
critical infrared exponents of the propagators, as proposed in calculations using stochastic quanti-
zation and Dyson-Schwinger equations, are confirmed quantitatively. For this purpose lattices of
volume up to (42.7 fm)2 have been used.
PACS numbers: 11.10.Kk 11.15.-q 11.15.Ha 12.38.Aw
I. INTRODUCTION
Two-dimensional Yang-Mills theory turns out to be a
very fascinating topic. Quite a number of quantities, e.
g. the string tension [1], can be calculated exactly, al-
though not all quantities are (yet) known analytically.
In particular, up to now it was not possible to calcu-
late the Green’s functions in Landau gauge. However,
exactly these Green’s functions may contain interesting
information.
The reason for this is confinement. In two-dimensional
Yang-Mills theory, confinement in Landau gauge is al-
ready manifest in perturbation theory: All elementary
fields, the gluons and ghosts, form a BRST quartet, and
thus are confined according to the Kugo-Ojima mecha-
nism [2]. This can be extended non-perturbatively, pro-
vided that BRST symmetry is unbroken beyond pertur-
bation theory. This makes explicit the absence of propa-
gating degrees of freedom in two-dimensional Yang-Mills
theory. But even without propagating degrees of free-
dom, this permits to investigate the manifestation of the
quartet mechanism on the level of the Green’s functions.
In addition, the reasoning for the confinement scenario
of Gribov and Zwanziger [3, 4, 5, 6] is applicable to two
dimensions as well [6]. However, this scenario has no
direct manifestation on the perturbative level, as in the
case of the quartet mechanism. It is only manifest in
the infrared properties of correlation functions. In par-
ticular, the Gribov-Zwanziger scenario predicts that the
Faddeev-Popov operator Mab accumulates near-zero or
zero eigenvalues. As a consequence, the ghost propagator
DG, being the expectation value of the inverse Faddeev-
Popov operator, should be infrared diverging. Detailed
calculations using stochastic quantization [6] or Dyson-
Schwinger equations (DSEs) [7, 8] lead to a power-law
∗Electronic address: axel.maas@savba.sk
behavior in the far infrared in any dimension from two
to four,
DG(p) ∼p→0 p
−2−2κ. (1)
Furthermore, the gluon propagator is infrared vanishing,
and thereby explicitly positivity violating. Its scalar part
also behaves like a power-law in the far infrared,
D(p) =
(d− 1)
δµν −
Dµν(p) ∼p→0 p
−2−2t,
where d is the space-time dimension. The two exponents
are related by the sum rule
t+ 2κ+
= 0. (3)
Under the assumption of an infrared bare ghost-gluon
vertex, two possible values for κ are found, 0 and 1/5
[6, 8]. If physics is smooth as a function of dimensionality,
the non-zero exponent would be expected due to the re-
sults obtained in three and in four dimensions [6, 7, 8, 9].
Note that in calculations using the renormalization group
in the case of a bare ghost-gluon vertex the same equa-
tions as in DSE calculations are obtained, thus leading
to the same results for the infrared exponents in any di-
mension [10].
These two scenarios are two of the most discussed for
the confinement mechanism of gluons also in higher di-
mensions, see e. g. for four dimensions the reviews [11]
and in three dimensions [6, 8, 12]. A verification of their
predictions using lattice gauge theory in higher dimen-
sions has, however, turned out to be very complicated,
mainly due to finite volume effects. In three dimensions
only a qualitative agreement between the predictions of
the Gribov-Zwanziger scenario and functional calcula-
tions has been obtained [12, 13]. In four dimensions, the
lattice results are inconclusive (see e. g. [14, 15, 16, 17]).
Studies using Dyson-Schwinger equations in a finite vol-
ume support that these problems are, in fact, finite vol-
ume effects, and provide even a quantitative prediction
http://arxiv.org/abs/0704.0722v2
mailto:axel.maas@savba.sk
of these in four dimensions [18]. The latter are in ac-
ceptable agreement with the results obtained in lattice
calculations [18].
Here, for two dimensions, the accessible lattices permit
a quantitative test of the predictions. It will be shown
that the predictions, assumptions, and actually the value
of κ = 1/5, of the Gribov-Zwanziger scenario are found
in lattice calculations, and hence there is very strong ev-
idence for the Gribov-Zwanziger scenario to be at work.
In fact, it is possible to quantify the finite volume effects.
Hence, in the following a quantitative confirmation of
the predictions of the Gribov-Zwanziger scenario using
lattice gauge theory for two-dimensional SU(2) Yang-
Mills theory in (minimal) Landau gauge will be given.
Of course, with such results, one question immediately
arises when comparing the two-dimensional results to
those in higher dimensions: Why do they agree qualita-
tively on the level of two- and three-point Green’s func-
tions in the infrared? This points to a structural origin
of both, the Gribov-Zwanziger and the Kugo-Ojima sce-
nario, provided both are, in fact, correct. It is partic-
ularly tempting to then investigate the relation of both
scenarios in two dimensions. Also how the relation of
two-dimensional Yang-Mills theory to topological field
theory [19] comes then into play is immediately on one’s
mind. These, and similar questions arise when contem-
plating the results, and indicate that there are many in-
teresting opportunities still present in the study of two-
dimensional Yang-Mills theory. These are highly inter-
esting questions, and must be investigated in the future.
Within this work, however, as a first step just the
results from the lattice calculations will be collected.
The two-point functions, and as associated quantities
the Faddeev-Popov operator and the running coupling,
will be investigated in section II. The three-point func-
tions will afterwards be discussed in section III. A short
summary of the results will be given in section IV. The
technicalities of the lattice simulations can be found in
appendix A. Lattice artifacts other than finite volume ef-
fects will be discussed in appendix B. That the suppres-
sion of color indices in equations (1) and (2) is justified
will be shown in appendix C.
II. TWO-POINT FUNCTIONS
The definition and determination of the two-point
functions on the lattice, and the associated quantities,
have been repeatedly discussed in the literature (see, e.
g., [5, 12, 14, 15, 16, 17]). Here, the methods described
in [12] are employed. Furthermore, the appearance of β-
factors to obtain the correct scaling has been discussed
there, also in case of the three-point functions. Hence,
this will not be repeated here. To assign units to the
quantities, the exactly calculable string tension [1] has
been assigned the conventional value (440 MeV)2, as in
higher dimensions.
p [GeV]
0 0.2 0.4 0.6 0.8 1
Gluon propagator
p [GeV]
0 0.5 1 1.5 2 2.5 3
Gluon dressing function
FIG. 1: The top panel shows the gluon propagator at small
momenta for various volumes. The lower panel shows the
gluon dressing function over the whole accessible momentum
range. Open circles correspond to a volume of (42.7 fm)2, full
squares to (14.2 fm)2, full triangles to (7.11 fm)2, and upside-
down full triangles to (2.02 fm)2. The solid line in the top
panel is the function 4.5p4/5.
A. Gluon propagator
The gluon propagator is the most readily accessible
two-point correlation function. The results for the prop-
agator D(p), (2), and its associated dressing function
p2D(p) are shown in figure 1. A strongly infrared sup-
pressed gluon propagator is clearly visible. At the same
time, the infrared suppression increases with increasing
physical volume. In particular, while on a volume of (2.02
fm)2 the propagator appears to be infrared diverging, a
clear maximum appears already at a volume only a factor
1/L [GeV]
-210 -110
Gluon propagator at zero momentum
FIG. 2: The zero-momentum value D(0) of the gluon propa-
gator as a function of inverse edge length. The straight line is
the power-law fit 5.67L−0.79 to the 20 points at the smallest
volumes.
(2-3)2 larger. Only the point at the lowest non-vanishing
momentum and the point at zero are not consistent with
an infrared vanishing gluon propagator. This is, however,
expected [18]. The scaling of D(0) with volume, shown
in figure 2, makes it very likely that in the infinite volume
limit the gluon propagator vanishes at zero momentum,
as it vanishes like a power-law with inverse volume. In
fact, the exponent 0.79 of the determined power-law is
in very good agreement with the expectation [18] that it
should coincide with the exponent of the gluon propaga-
tor t = 4/5 of equation (2).
Furthermore, even the gluon dressing function does not
exhibit any qualitative difference to three dimensions.
In particular it also exhibits a (shallow) maximum. As
the propagator becomes ultraviolet constant, as a conse-
quence of asymptotic freedom, there is no intrinsic ne-
cessity for such a maximum, as in four dimensions. Its
presence in this theory without propagating degrees of
freedom is hence slightly surprising. However, in the con-
text of a DSE treatment, it is natural to expect such a
maximum due to the different signs of ghost and gluon
self-energy contributions [8].
The most interesting quantity is the far infrared be-
havior. It is clearly visible that the gluon propagator is
strongly infrared suppressed. The deviation at the very
lowest momenta points, however, shows a more massive
behavior, as expected from DSE-studies in finite volumes
[18]. However, the mass decreases rapidly with volume,
as discussed above, and a massive behavior is seen only
in a momentum window which rapidly decreases with in-
creasing volume. More interestingly, it is expected that
1/L [GeV]
0 0.1 0.2 0.3 0.4 0.5
Gluon infrared exponent
FIG. 3: The measured infrared exponent κZ obtained from
the gluon propagator. Two fits are given. The dashed line
corresponds to a fit of type (5) which is forced to go to the
predicted value κ = 1/5 at 1/L = 0, while the one given by
the solid line is not forced to do so. The fit parameters can
be found in table I.
in the regime1
≪ p ≪ ΛQCD (4)
the continuum behavior should prevail. In particular, the
gluon propagator should decrease even in a finite volume
in this domain like the power-law (2) [18]. Using the sum
rule (3), the exponent of the propagator itself should be
4κ. Such a power-law is shown in the top panel in figure
1, and agrees well with the data inside the domain (4).
To investigate this quantitatively, the effective expo-
nent κZ was determined. This was done by discarding
the two lowest non-vanishing momentum points. Then
the next five highest points in momentum were used to
fit a power-law. To obtain errors, the steepest and shal-
lowest curve consistent with a 1σ-confidence interval was
determined as well. That this is likely too optimistic is
shown by the scattering of the results below. If more
than one momentum representation for a given momen-
tum existed, the results were averaged over the various
representations, as the violation of rotational symmetry
is a minor effect that far in the infrared, see appendix B.
The results are shown in figure 3. While there are still
significant fluctuations at large volumina, the measured
exponents tend towards the continuum value.
The volume-dependence of the measured exponents
1 The characteristic scale ΛQCD is in two dimensions of course
proportional to the coupling constant g.
TABLE I: Fit parameters of formula (5). Fit 1 corresponds
to one with fixed a = κ = 1/5, fit 2 to one where a was fitted
as well.
Fit a b [fm] c [fm2] d [fm3]
1 1/5 0.130 -12.9 19.5
2 0.190 0.358 -14.0 20.9
can be fitted by the formula2
Z = a+
. (5)
Two fits have been done. In one case, a was fitted as
well, while in the second case a was set to the continuum
value κ = 1/5. However, even with a free, the result
is in reasonable agreement with 1/5. In particular, the
results are not consistent with an infrared finite gluon
propagator, which would be expected if κ = 0, the second
solution found in [6, 8]. The individual fit parameters are
given in table I.
Hence, the gluon propagator behaves quantitatively
exactly as predicted in the Gribov-Zwanziger scenario,
when finite volume effects are taken properly into ac-
count.
B. Ghost propagator
The ghost propagator has been determined along the
same line as in higher dimensions [5, 12]. However, more
interesting than the propagator itself is the dressing func-
tion p2DG(p). The propagator and the dressing function
are shown for different volumes in figure 4. It is clearly
visible that the dressing function is infrared diverging.
This already indicates that of the two possible exponents
κ = 0 and κ = 1/5 found [6, 8] only the latter one, if one
at all, is realized.
Compared to the case of the gluon propagator, finite
volume effects are hardly visible to the eye. It seems that
the propagator actually becomes less infrared diverging
with volume. From the quantitative evaluation below,
this is found to be not the case. What seems to be the
case is that the domain of closest approach to the origin is
affected by finite volume effects. Its modification leads to
the various changes in the infrared in a non-trivial man-
ner. If this is the case, the finite-volume effects would
be very hard to compare between lattice calculations
and functional calculations, as they would be dominated
by mid-momentum effects, which in functional methods
are usually most strongly affected by truncations [8, 11].
This would, on the other hand, explain why in four di-
mensions the finite volume effects in the ghost propaga-
tor have indeed been found to be at least to some extent
2 The cubic term is necessary to include all volumes.
p [GeV]
0 1 2 3 4 5
Ghost propagator
p [GeV]
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Ghost dressing function
FIG. 4: The top panel shows the ghost dressing function at
small momenta for various volumes. The lower panel shows
the ghost propagator over the whole accessible momentum
range. Open circles correspond to a volume of (42.7 fm)2,
full squares to (14.2 fm)2, full triangles to (7.11 fm)2, and
upside-down full triangles to (2.02 fm)2. The solid line is the
function 1.1p−2/5.
different in lattice and in Dyson-Schwinger calculations
[18]. In addition, Gribov-Singer effects [3, 20], which ac-
cording to the Gribov-Zwanziger scenario are irrelevant
in the infinite-volume limit [21], may still be relevant even
at volumes as large as those used here. This has not yet
been investigated in two dimensions in Landau gauge.
Even with the available volumes the effect is small. A
power-law with exponent κ = 1/5 already describes the
data quite well in the infrared, as shown in the top panel
of figure 4. Therefore, a more quantitative investigation
of the infrared behavior is required.
1/L [GeV]
0 0.1 0.2 0.3 0.4 0.5
Ghost infrared exponent
FIG. 5: The measured infrared exponent κ obtained from the
ghost propagator. Two fits of type (5) are given. The dashed
line corresponds to a fit which is forced to go to the predicted
value at 1/L = 0, while the one given by the solid line is not
forced to do so. The fit parameters can be found in table II.
TABLE II: Fit parameters for the ghost effective exponent
κG using formula (5). Fit 1 corresponds to one with fixed
a = κ = 1/5, fit 2 to one where a was fitted as well.
Fit a b [fm] c [fm2] d [fm3]
1 1/5 0.139 -0.711 0.173
2 0.150 1.28 -6.41 7.47
This is done by extracting the effective infrared ghost
exponent κG in the same way as in the case of the gluon
propagator. The results for κG are shown in figure 5.
While statistical errors are larger than in the case of
the gluon propagator, it is visible that all results clus-
ter around the predicted continuum value of κ = 1/5 at
large volumes. This is also seen from a fit of the type
(5). The corresponding fit parameters can be found in
table II. Due to statistical uncertainties it is not as clean
as for the gluon. However, it is visible that the exponent
does not vary strongly with volume. In fact, the effective
ghost exponent seems only to change by about a third
when changing the volume by almost two orders of mag-
nitude. Finally, even with the limited fit accuracy it is
not unreasonable that the results are, in fact, consistent
with the prediction of κ = 1/5.
Another possibility to check the continuum results is
to test the predicted sum rule (3). This is done by using
the effective measured exponents κZ and κG in figure 6.
Again, a fit of type (5) has been performed. The corre-
sponding fit parameters are given in table III. As already
anticipated from the individual results, the sum rule be-
comes better and better satisfied when approaching the
continuum limit. Hence, it seems very likely that the
1/L [GeV]
0 0.1 0.2 0.3 0.4 0.5
Infrared sum rule
FIG. 6: Test of the sum rule t+2κ+1 = 0, using the effective
ghost exponent κG, shown in figure 5, and the effective gluon
exponent tZ = −(1 + 2κZ), shown in figure 3. Two fits of
type (5) are given. The dashed line corresponds to a fit which
is forced to go to the predicted value at 1/L = 0, while the
one given by the solid line is not forced to do so. The fit
parameters are given in table III.
shiny relation (3) is, in fact, recovered in the continuum
limit.
TABLE III: Fit parameters for a formula of type (5) for the
sum rule. Fit 1 corresponds to one with fixed a = 0, fit 2 to
one where a was fitted as well.
Fit a b [fm] c [fm2] d [fm3]
1 0 0.0192 24.4 -38.6
2 -0.0806 1.85 15.2 -26.9
One of the particularly interesting results so far is that
the ghost exponent is only very weakly dependent on
the volume, compared to the one of the gluon. This
is in marked contrast to the case in four-dimensional
DSEs in a finite volume [18]. Furthermore, all attempts
to extract a ghost exponent in lattice calculations in
higher dimensions also yield a rather small, more or less
volume-independent exponent [17]. It is thus interesting
to compare the ghost propagator in various dimensions at
roughly the same volume. This is done in figure 7. Only
the momentum regime is shown which is accessible by all
of the lattices used. Furthermore, the propagators have
been normalized so that they coincide at a momentum of
2 GeV. For the momenta itself, the string tension was set
to the same value for all three different dimensionalities.
Aside from the question to which extent such a com-
parison is justified, the results behave as predicted: The
ghost propagator becomes more divergent with increasing
dimension. Also, it is in agreement with the predictions
p [GeV]
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
Ghost propagator
p [GeV]
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
Ghost dressing function
FIG. 7: Comparison of the ghost propagator in different di-
mensions. Circles are two dimensions, triangles are three
dimensions, and upside-down triangles are four dimensions.
The lattice volumes are (6.06 fm)2, (5.20 fm)3 [12], and (5.28
fm)4 [22], at β = 30, β = 4.2, and β = 2.3, respectively.
[6, 7, 8, 11] that the difference is more pronounced from
two to three dimensions than from three to four dimen-
sions: κ changes from 1/5 to ≈ 0.39 or 1/2 from two
to three dimensions. The four-dimensional exponent of
κ ≈ 0.59 is, on the other hand, rather close to the one in
three dimensions. This qualitative behavior, with all its
caveats, is another indication for the correctness of the
Gribov-Zwanziger scenario and the quantitative predic-
tions.
Furthermore, the result in two dimensions is in fact
confirming quantitatively the Gribov-Zwanziger scenario
in two dimensions. However, the very slow change in the
effective exponent over orders of magnitude in volume is
p [GeV]
0 0.5 1 1.5 2 2.5 3 3.5
Effective coupling
FIG. 8: The effective running coupling divided by p2 for var-
ious volumes. Open circles correspond to a volume of (21.3
fm)2, full squares to (14.2 fm)2, full triangles to (8.08 fm)2,
and upside-down full triangles to (2.02 fm)2.
indicative of what challenges have to be met in higher
dimensions to see the asymptotic ghost exponent.
Finally, the exact value of the exponent obtained in
DSE calculations depends on the projection of the ten-
sor equation for the ghost [8, 11]. The value of 1/5 is
obtained only in the case of a transverse projection [8].
This in turn implies automatically a certain structure of
the longitudinal (w. r. t. to the gluon momentum) tensor
structure of the ghost-gluon vertex, such that it leads for
arbitrary projections to the infrared exponent 1/5. This
then makes the determination of this tensor structure an
almost trivial exercise in the infrared limit. Furthermore,
this precisely prescribes how the Slavnov-Taylor identity
for the gluon propagator, and hence its transversality, is
recovered in the far infrared.
C. Running coupling
Although it is possibly a questionable concept in two-
dimensional Yang-Mills theory, it is possible to formally
define a running coupling. Analogous to higher dimen-
sions [11, 23], the quantity3 α(p) = p6D(p)DG(p)
then proportional to the coupling constant. In particu-
lar, as a consequence of the sum rule, the quantity α(p)
3 To improve the statistical behavior, the ghost dressing function
has been evaluated on a plane-wave source instead of a point
source, as in case of the propagator alone [12]. Hence only the
same volumes are accessible for the coupling constant as for the
ghost-gluon vertex below, where this is also necessary.
should behave in the infrared as p2. Hence α/p2 should
be constant.
From the results on the sum-rule, given in figure 6, it
is already clear that an infrared fixed point will hardly
be seen. However, the results, shown in figure 8, exhibit
such a fixed point at the largest volumes, provided the
lowest point at non-vanishing momentum is discarded4.
Note that the finite volume effects seem to make the run-
ning coupling diverging instead of vanishing, as in higher
dimensions [17, 18].
Thus at sufficiently large volumes, and taking finite
volume effects into account, it is in fact possible to ob-
serve a fixed point in the coupling in lattice gauge theory.
Note that there is a small, systematic overall factor
between the coupling obtained in the different volumes
shown in figure 8. This effect is not visible in the propaga-
tors themselves, but is increased here by taking effectively
the third power of the propagators. As this effect occurs
at all momenta, it is likely not simply a finite volume
effect. However, this can still be an O(a)-effect which is
caused, e. g. among other effects, by the fact that tadpole
corrections, which give overall-factors to the propagators,
have been neglected here [16, 24].
D. Faddeev-Popov operator
A last element in the analysis of the two-point cor-
relation functions are the properties of the Faddeev-
Popov operator, central to the Gribov-Zwanziger sce-
nario [3, 4, 5]. The results on the ghost propagator, which
is the expectation value of the inverse Faddeev-Popov op-
erator, already indicate the existence of an enhancement
of its eigenspectrum near zero eigenvalue. This enhance-
ment is the hallmark of the Gribov-Zwanziger scenario.
However, it is interesting to see the quantitative behav-
ior of the eigenspectrum. Hence the spectral properties
of the Faddeev-Popov operator have been determined as
well, using the technique described in [12].
The near-zero part of the eigenspectrum is shown for
various volumes in figure 9. The volume scaling of the
lowest eigenvalue is shown in figure 10. It is clearly visi-
ble that with increasing volume more and more eigenval-
ues are found near zero. This is the near-zero eigenvalue
enhancement, as predicted in the Gribov-Zwanziger sce-
nario5. In addition, the lowest eigenvalue vanishes in
the infinite-volume limit, and in fact vanishes faster than
4 For the coupling constant only edge momenta have been used,
in contrast to the propagators where also other momenta have
been included. Dismissing here only the lowest non-vanishing
momenta is thus equivalent to dismissing the two lowest non-
vanishing momenta in case of the propagators.
5 Note that the decrease towards larger eigenvalues seen in figure
9 is likely an artifact of the method to determine the eigenvalues
[12]. Furthermore, all eigenvalues are only found with multiplic-
ity 1.
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018
0.001
Spectrum near zero eigenvalue
FIG. 9: The near-zero part of the eigenvalue spectrum of
the Faddeev-Popov operator for volumes (2.02 fm)2 (dashed-
dotted line), (7.11 fm)2 (dotted line), (14.2 fm)2 (dashed line),
and (24.9 fm)2 (solid line). 1164228, 2261493, 3517400, and
2614098 eigenvalues have been enclosed in the full spectrum,
respectively.
1/L [GeV]
-210×3 -110 -110×2
Lowest eigenvalue of the Faddeev-Popov operator
FIG. 10: The volume-dependence of the lowest eigenvalue of
the Faddeev-Popov operator. The solid line is the function
0.314L−2.34 .
the lowest eigenvalue of the Laplacian. This is a prop-
erty which has also been observed in higher dimensions
[12, 22]. It has been argued that such a larger rate may be
necessary for the ghost propagator to develop an infrared
divergence [25]. It is therefore another direct evidence for
the validity of the Gribov-Zwanziger scenario. This van-
ishing of the lowest eigenvalue is in fact necessary for the
Gribov-Zwanziger mechanism to work: For infinite vol-
ume, an average configuration should be arbitrarily close
or on the common boundary of the fundamental mod-
ular region and the Gribov horizon, where by definition
the determinant of the Faddeev-Popov operator vanishes,
and thus must have at least one vanishing eigenvalue [5].
III. THREE-POINT FUNCTIONS
Investigating the vertices in two dimensions is a very
interesting task. On the one hand, the vertices do not
lend themselves easily to evaluation, since as three-point
functions they are much more strongly affected by statis-
tical fluctuations than two-point functions. Hence their
investigation has so far been limited to rather small
volumes in four [26, 27] and even in three dimensions
[12, 26]. On the other hand, the vertices describe inter-
action effects, and it is not a-priori clear how they should
behave in a theory without propagating degrees of free-
dom. In particular, the possibility that the three-gluon
vertex, or at least some of its tensor structures, could
change sign is a very interesting observation in higher di-
mensions [12, 26]. Whether this is also the case in two
dimensions, especially in large volumes, is thus also a
question of interest.
One drawback of investigating vertices in two dimen-
sions on a square lattice is the impossibility to construct
a momentum configuration such that all three momenta
are equal. This equal momentum configuration is the one
usually preferred in functional studies of the vertices [28],
as it permits to have only one external scale. However,
in higher dimensions it was found that the results do not
change qualitatively when instead two of the momenta
are taken to be orthogonal [12, 26]. This configuration
can be realized in two dimensions, and will thus be em-
ployed here.
In general, vertices have a significant amount of tensor
structures. To obtain a more simple function to measure
the interaction represented by a vertex, the quantity
Γtl,abcGabc
Γtl,abcDadDbeDcfΓtl,def
. (6)
will be evaluated instead. Here the indices a, . . . , f are
generic multi-indices, encompassing field-type, Lorentz
and color indices. Also, Dab are the propagators of the
fields, Gabc represent the full Green’s functions and Γtl,abc
are the corresponding tree-level vertices. This quantity
is defined such that it becomes equal to one if the full
and the tree-level vertex coincide. For a more detailed
discussion of this quantity and its properties, see [12].
There are two vertices in Landau-gauge Yang-Mills
theory. The first is the ghost-gluon vertex, which is
shown for four different volumes in figure 11. In this case
in fact the vertex is shown, as only one tensor structure,
the tree-level one, survives non-amputation [12].
As in the higher-dimensional cases [12, 26, 27], it ex-
hibits an essentially constant behavior, except for a pos-
sible small structure below roughly 1 GeV in ghost mo-
mentum. This structure is a maximum, with a drop to-
wards smaller momenta below the tree-level value. Fur-
thermore, the value at small ghost momenta and finite
gluon momenta is below 1, but finite. If a modification
away from a constant infrared behavior of this vertex
should exist, it must set in with an extremely small ef-
fective exponent to not be visible on these volumes.
These results are all in qualitative agreement with the
ghost-gluon vertex in higher dimensions [12, 26, 27]. In
particular, the results confirm the truncation scheme in
the far infrared used in two dimensions in stochastic
quantization and DSE calculations [6, 7, 8]. In that case
an infrared finite ghost-gluon vertex was assumed, de-
livering the critical infrared exponent κ = 1/5, which in
fact was observed in the previous section. This once more
nicely confirms the Gribov-Zwanziger scenario, which
leads directly to this type of approximation. Further-
more, in four dimensions the infrared critical exponent
of the ghost-gluon vertex is fixed, once the exponents of
the propagators are known [28]. This can be extended
to two dimensions and yields in fact an infrared constant
ghost-gluon vertex [29]. This is a very stringent test of
the scenario. The results found here in lattice calcula-
tions once more pass this test. Or, more aptly put, the
test passes the results.
The three-gluon vertex is much more troublesome to
calculate due to strong statistical fluctuations, in partic-
ular at large lattice (not physical) momenta. These are,
in fact, even more pronounced than in higher dimensions,
as was already observed when going from three to four di-
mensions [26]. Thus the uncertainty connected with this
vertex is quite large. Nonetheless, the results shown in
figure 12 are quite spectacular. At a point of about 300-
400 MeV, corresponding roughly to the position where
the plateau in the coupling constant develops or where
the gluon propagator starts to bend over, the quantity
changes sign. Thereafter, it diverges, likely like a
power-law, as can be seen from the bottom-left panel
in figure 12. Precisely such a divergence is expected in
higher dimensions [28]. This also compares very well to
lattice results in higher dimensions, which found the on-
set of such a negative divergence in three dimensions [12],
and at least an infrared suppression in four dimensions
[26]. Note, however that due to the contraction (6) not
necessarily one particular tensor structure of the vertex
changes sign. It is as well possible that two tensor struc-
tures have opposite sign throughout, but differ in mag-
nitude, and one is dominant in the infrared, while the
other dominates in the ultraviolet.
The infrared divergence of the three-gluon vertex when
one momentum vanishes is roughly in agreement with
a power-law with exponent −2.2 for the single external
scale, as can be seen in the bottom-left panel of figure 12.
Although this is not the momentum configuration used
in DSE calculations [28], there is again just one external
scale. It could be expected that the infrared behavior is
the same, if there is just one scale left. In that case, this
exponent of −2.2 is actually the one expected in DSE
q [GeV] (Ghost)
0.5 1
1.5 2
eV] (G
luon) 0
Ghost-gluon vertex, orthogonal momenta
FIG. 11: The ghost-gluon vertex for orthogonal momenta. The
top left panel shows the vertex for all possible orthogonal mo-
mentum configurations for a volume of (21.3 fm)2, with errors
suppressed. The ripple structure is an artifact of the method
[12], and vanishes with increasing statistics. The bottom left
and right panel show the vertex in two specific momentum con-
figurations. In one case the gluon momentum vanishes (left
panel), and in the other the gluon and ghost momenta are of
equal magnitude (right panel). In this case, various physical
volumes are compared. Open circles correspond to a volume
of (21.3 fm)2, full squares to (14.2 fm)2, full triangles to (8.08
fm)2, and upside-down full triangles to (2.02 fm)2.
q [GeV]
0 0.5 1 1.5 2 2.5 3 3.5
Ghost-gluon vertex, one momentum vanishing
q [GeV]
0 0.5 1 1.5 2 2.5 3 3.5
Ghost-gluon vertex, orthogonal momenta with two equal
calculations [29]. This statement applies as well to the
infrared constancy of the ghost-gluon vertex.
Taking this reasoning seriously would imply that all
two- and three-point functions exhibit exactly and quan-
titatively the infrared exponents predicted in DSE calcu-
lations, and are in agreement with the Gribov-Zwanziger
scenario. Therefore, this work here would represent the
first quantitative confirmation of these two frameworks
using lattice gauge theory.
It is of course tempting to also investigate higher n-
point functions. Unfortunately, this is currently out of
reach in the present approach. The reason is that only
non-amputated, full Green’s functions can be directly ob-
tained with the methods used here. Therefore, it would
be necessary to first subtract the not-connected part of
the amplitude, and then amputate the Green’s functions.
In general, the not-connected and the connected ampli-
tude have the same infrared behavior, at least in four
dimensions, if the predictions [28] are correct. There-
fore, it would be necessary to disentangle the sum of
two functions, which both have the same leading infrared
behavior. As the statistical fluctuations become larger
when increasing the number of external legs, the required
statistics become impractical at the current time. Hence
it would be necessary to reduce these fluctuations. It is
possible that e. g. including only results for the same sign
of the Polyakov loop6 would be helpful, as by this statis-
tical fluctuations, at least in case of the gluon propagator,
are reduced [30]. This has to be investigated further.
6 At finite volume, the value of the Polyakov loop is non-zero for
each individual configuration.
p [GeV]
0 0.5 1 1.5 2 2.5 3 3.5 4
Three-gluon vertex, one momentum vanishing
p [GeV]
0 0.5 1 1.5 2 2.5 3 3.5 4
Three-gluon vertex, orthogonal momenta with two equal
p [GeV]
-210×5 -110 -110×2 -110×3
Three-gluon vertex, one momentum vanishing
q [GeV
k [GeV]
Three-gluon vertex, orthogonal momenta
FIG. 12: The three-gluon vertex for orthogonal momenta. The top left and right panel show the vertex in two specific
momentum configurations. In one case one of the gluon momenta vanishes (left panel), and in the other two of the gluon
momenta are of equal magnitude (right panel). The bottom left panel shows a magnification of the low-momentum regime
for one momentum vanishing. In this case the absolute value of GA
is displayed. Various physical volumes are compared.
Open circles correspond to a volume of (21.3 fm)2, full squares to (14.2 fm)2, full triangles to (8.08 fm)2, and upside-down full
triangles to (2.02 fm)2. Finally, in the bottom right panel GA
is shown for the complete orthogonal momentum configuration
plane in case of the largest volume (21.3 fm)2. In case of the bottom left panel, results from all available volumes up to lattices
of size 1202 are shown, see appendix A. In addition to the previously used symbols, the remaining symbols correspond to (3.56
fm)2 (pluses), (4.04 fm)2 (open stars), (6.06 fm)2 (open crosses), (7.11 fm)2 (full stars), (10.1 fm)2 (open triangles), (10.7 fm)2
(diamonds), (12.1 fm)2 (full circles), and (17.8 fm)2 (open squares). The line is the function −0.17p−2.2.
IV. SUMMARY
The volumes accessible in two-dimensional Yang-Mills
theory permitted here to obtain the two-point and three-
point functions on very large lattices, up to (42.7 fm)2
and (21.3 fm)2, respectively. In particular, it was possible
to obtain quantitative results on the infrared behavior
with a precision which is unprecedented in the lattice
investigations of these quantities.
These results demonstrated that the gluon propaga-
tor is infrared vanishing, the ghost propagator is infrared
diverging, and the ’effective coupling constant’ also has
the expected qualitative infrared behavior. Moreover, it
was possible to make these statements quantitative. In-
cluding the effects of finite volume, it was possible to
determine the infinite-volume limit of the characteris-
tic infrared exponents for the two-point functions, and
demonstrate the validity of the sum-rule (3). In fact, the
value κ = 1/5 found coincides with one of the two pos-
sible values expected from stochastic quantization and
Dyson-Schwinger equations for an ’on-shell’, i. e. trans-
verse, gluon. Furthermore, the infrared behavior of the
vertices permit to close the system self-consistently in the
context of such equations. In particular, the ghost-gluon
vertex is infrared constant.
These results confirm the Gribov-Zwanziger scenario
in two dimensions. Without any dynamic, i. e. propa-
gating, degrees of freedom, all the infrared behavior is
still qualitative the same as in higher dimensions. This
implies that these effects in fact stem from the gauge-
fixing procedure, in essentially the way predicted by the
Gribov-Zwanziger scenario.
It will, of course, take some time before it is possible to
repeat the same in higher dimensions. One of the quan-
titative reasons is that the critical exponent in the gluon
observables decreases with increasing dimension [6, 8].
Hence the effects observed here will only be observable
for larger volumes in higher dimensions. Nonetheless,
the results are also in excellent qualitative agreement
with the predictions of DSE calculations for the finite
volume behavior of the propagators in four dimensions
[18]. Finally, the comparison of the ghost propagator for
different dimensions yields the pattern expected from the
Gribov-Zwanziger scenario.
However, these results should also be taken with care,
as two-dimensional Yang-Mills theory is different from its
higher-dimensional versions. And although there is little
evidence to the contrary, no rigorous implication exists
that the effects seen here translate themselves into higher
dimensions without changes. Hence a satisfactory state
of affairs in higher dimensions has to await equivalent
investigations in higher dimensions. Until then, these
results here are another piece of the puzzle, which seem
to indicate that the Gribov-Zwanziger scenario in Landau
gauge is valid also in higher dimensions.
These results are, beyond these questions, also inter-
esting on their own. It is very tempting to investigate
how these results relate to the host of exact results avail-
able in two-dimensional Yang-Mills theory, what is the
connection to the topological aspects of the theory, and,
last but not least, how and if an equivalence between
the Gribov-Zwanziger and the Kugo-Ojima confinement
scenario exists, at least in two dimensions.
Acknowledgments
The author is grateful to Attilio Cucchieri and Tereza
Mendes for many helpful and interesting discussions.
Furthermore he thanks all those (in particular, Markus
Huber) who always ask about two dimensions. This work
was supported by the DFG under grant MA 3935/1-2 and
in part by the Slovak Grant Agency for Science, Project
VEGA No. 2/6068/2006. The ROOT framework [31] has
been used in this project.
APPENDIX A: GENERATION OF
CONFIGURATIONS
The generation of configurations in two dimensions and
their gauge-fixing to Landau gauge can be and has been
done exactly as in higher dimensions [12]. In particular,
the confirmation of the Gribov-Zwanziger scenario in the
present work implies that the problem of Gribov-Singer
copies [3, 20] should also in two dimensions become ir-
relevant for Green’s functions in the infinite volume limit
[21]: Gribov-Singer effects should become smaller with
increasing volume. Hence they have been ignored here,
although, as discussed in section II, effects at finite vol-
ume cannot be excluded.
To give units to the momenta, the infinite volume limit
of the string tension for a given β, which can be deter-
mined analytically [1], is set to (440 MeV)2. The con-
figurations used are shown in table IV. The comparison
with the (also exactly known) infinite volume value of the
plaquette [1] shows that locally the continuum has been
reached. However, the discussion in section II shows that
this is not correct globally.
APPENDIX B: LATTICE ARTIFACTS OTHER
THAN FINITE VOLUME
As one of the main claims here is that the deviation
from the asymptotic continuum form in the infrared is a
pure finite-volume effect, it is necessary to check the in-
fluence of other lattice artifacts. In particular, discretiza-
tion effects and violation of rotational symmetry may be
relevant. The latter is known to be a significant effect
when comparing correlation functions measured along
different directions of the hypercube (see, e. g., [14]), in
the present case along an edge or along a diagonal. In
figure 13, these effects are explicitly checked. The results
are at roughly the same volume of about (10.3 fm)2 at
two different βs, 10 and 30, and results with momenta
along any possible direction are directly compared.
It is clearly visible that, despite a factor of nearly 2
in a, both results agree remarkably well over the whole
range of momenta. Thus discretization effects are nearly
negligible, at least for a volume of a few fm2 and mo-
mentum not too close to the maximum one. Treating
only the physical volume as an independent parameter
in the infrared throughout the main text is hence justi-
fied. Also no significant effect is seen of the violation of
rotational invariance, which is usually most pronounced
at intermediate momenta in the gluon dressing function.
For the current case a few tens of lattice sites along each
edge seem to be sufficient to have already a quite good
approximation of rotational invariance.
Furthermore, there is no distinct difference between
the gluon and the ghost dressing function in terms of
TABLE IV: Data of the configurations considered in the numerical simulations. The values for a are 1.108 GeV−1 for β = 10
and 1.951 GeV−1 for β = 30 [1]. The momenta p0, pi, and pf denote the lowest non-vanishing momentum and the beginning
and the end of the fit interval used in the determination of the effective exponents in section II, respectively. Note that for
N ≥ 140 not as many momentum configurations for the gluon propagator were available as for N ≤ 120. Ghost configurations
are the ones used to determine the ghost propagator, the properties of the Faddeev-Popov operator, the ghost-gluon vertex,
and the running coupling. Gluon configurations are the ones used to determine the gluon propagator and the three-gluon
vertex. As the autocorrelation time for the plaquette is less than one hybrid overrelaxation (HOR) sweep, all sweeps (after
thermalization) have been used for the plaquette measurement, given the number of plaquette configurations in the table. Note
that all ghost configurations are also included in the gluon configurations, the sets are not independent. In case of N ≥ 140,
only the propagators have been determined. Hence the number of both configurations coincide. The quantity < P > / < P∞ >
gives the ratio of the expectation value of the plaquette over the analytical infinite volume limit. The error is determined
according to [12]. Finally, p is the tuning parameter for the stochastic overrelaxation algorithm used for gauge-fixing [32], and
which has been obtained by linear self-adjustment [12]. Note that this quantity is not very precisely determined, and should be
used rather as an indication of the correct order. Sweeps is the number of HOR sweeps between two consecutive measurements
[12].
V [fm2] N =
V/a2 β p0 [MeV] pi [MeV] pf [MeV] Ghost config. Gluon config. Plaq. config. 1-< P > /P∞ p Sweeps
2.02 20 30 610 1206 1874 2430 11525 369211 -5(4) 10−6 0.83 30
3.56 20 10 347 685 1064 2102 12319 355257 1(1) 10−5 0.84 30
4.04 40 30 306 610 961 1964 10579 527689 1(2) 10−6 0.88 50
6.06 60 30 204 408 644 1723 7311 510688 0(1) 10−6 0.93 70
7.11 40 10 174 347 546 2161 10758 536786 -2(6) 10−6 0.87 50
8.08 80 30 153 306 484 1429 4898 438579 0(1) 10−6 0.90 90
10.1 100 30 123 245 387 747 1988 216391 -3(1) 10−6 0.96 110
10.7 60 10 116 232 366 1825 7108 496291 -2(4) 10−6 0.92 70
12.1 120 30 102 204 323 552 1754 225036 1(1) 10−6 0.95 130
14.1 140 30 87.6 175 371 368 368 53971 -2(2) 10−6 0.97 150
14.2 80 10 87.0 174 275 1582 6465 579900 -1(3) 10−6 0.92 90
16.2 160 30 76.6 153 325 291 291 48199 0(2) 10−6 0.98 170
17.8 100 10 69.6 139 220 1339 4337 478853 -2(3) 10−6 0.96 110
18.2 180 30 68.1 136 289 308 308 56724 -2(1) 10−6 0.96 190
20.2 200 30 61.3 123 260 199 199 40584 -2(1) 10−6 0.96 210
21.3 120 10 58.0 116 183 762 5236 678065 1(2) 10−6 0.93 130
22.2 220 30 55.7 111 236 232 232 51577 1(1) 10−6 0.99 230
24.2 240 30 51.1 102 217 232 232 55691 0(1) 10−6 0.98 250
24.9 140 10 49.7 99.4 211 517 517 76053 1(5) 10−6 0.96 150
28.4 160 10 43.5 87.0 184 455 455 75500 -8(4) 10−6 0.97 170
32.0 180 10 38.7 77.3 164 390 390 72034 -4(4) 10−6 0.97 190
35.6 200 10 34.8 69.6 148 328 328 66976 -4(4) 10−6 0.97 210
39.1 220 10 31.6 63.3 134 287 287 63703 3(3) 10−3 0.98 230
42.7 240 10 29.0 58.0 123 394 394 96075 0(2) 10−6 0.98 250
these artifacts. In case of the propagators these effects
would be even diminished, as the trivial factor p−2 helps
in the reduction of such artifacts. Hence the totally domi-
nant contribution for the artifacts in the correlation func-
tions in the infrared is clearly the finite physical volume.
Similar observations pertain to all quantities measured
here, and hence only the physical volumes are used as
explicit parameters in the main text, and no heed is paid
for the different β-values. The only exception observed
here is in the case of the running coupling in section II C,
where an overall scaling factor has been seen. This issue
has been discussed in detail in this section II C.
APPENDIX C: CONTRIBUTIONS IN OTHER
COLOR TENSOR STRUCTURES
There is no a-priori necessity for correlation functions
to carry only their tree-level color structure, although
such a color structure permits a consistent solution us-
ing functional methods in the infrared, at least in four
dimensions [18]. Therefore, this property should be ex-
plicitly checked. This is done for the ghost and the gluon
propagator in figure 14. All contributions are compatible
with zero. Furthermore, the average value decreases in all
cases with increasing statistics. So, within the statistics
available, there are no color-off-diagonal components in
the propagators. Due to the structure of the DSEs, it is
then very hard to imagine how the higher n-point Green’s
functions should have a color structure different from the
p [GeV]
0 1 2 3 4 5
Gluon dressing function
p [GeV]
0 1 2 3 4 5
Ghost dressing function
FIG. 13: Consequences of different discretizations and violation of rotational invariance in case of the gluon (left panel) and
ghost (right panel) dressing functions. Open circles correspond to a system at β = 30 and a volume of (10.1 fm)2, open stars
to a system at β = 10 and a volume of (10.7 fm)2. The different momentum directions have not been marked differently.
p [GeV]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
-0.04
-0.02
Off-diagonal gluon propagator
p [GeV]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
Off-diagonal ghost dressing function
FIG. 14: The color off-diagonal elements of the gluon propagator (left) and of the ghost propagator (right) on a (21.3 fm)2
volume.
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