Desigualdades para o Propagador de Fantasmas

4.3 Desigualdades para o Propagador de Fantasmas

Logo após a conclusão do artigo [148] sobre as desigualdades para o propagador de

glúons, descrito na seção anterior, pudemos obter desigualdades simples também para o

propagador de fantasmas [147] (ver artigo anexado no m desta seção). De fato, usando

a relação (4.2) e as desigualdades

7

0 < λ

min

< λ

2

< λ

3

. . . ,

(4.20)

ca claro que G(p

2

₎

_{deve satisfazer às relações}

1

N

2 c

− 1

1

λmin

X

b

| eψ

min

(b, p)|

2

≤ G(p

2

)

(4.21)

e

G(p

2

)

_≤

1

N

2 c

− 1

1

λmin

X

i:λi6=0

X

b

| eψ

i

(b, p)|

2

.

(4.22)

Assim, após somar e subtrair na Eq. (4.22) os termos devidos às autofunções que perten-

cem ao kernel de M(b, x; c, y) e usando a relação de completeza

X

i

ψ

i

(b, x) ψ

i∗

(c, y) = δ

bc

δ

xy

,

(4.23)

onde

∗

_{indica conjugação complexa, é obtido o resultado}

G(p

2

)

_≤

1

λ

min



1

−

1

N

2 c

− 1

X

j:λj=0

X

b

| eψ

j

(b, p)|

2





.

(4.24)

No caso especíco do gauge de Landau, os autovetores com autovalor nulo são represen-

tados por modos constantes. Assim, para p 6= 0 obtemos as desigualdades

1

N

2 c

− 1

1

λ

min

X

b

| eψmin(b, p)_|

2

_{≤ G(p}

2

)

_≤

1

λ

min

.

(4.25)

É importante ressaltar que o limite superior na Eq. (4.25) torna-se uma igualdade se e

somente se os autovalores da matriz de Faddeev-Popov forem todos degenerados e iguais

a λmin.

7_{Para simplicar a notação consideramos autovalores (não-nulos) não-degenerados. Esta condição é}

Os resultados descritos acima permitem relacionar o comportamento infravermelho do

propagador de fantasmas G(p

2

_{)com a extrapolação do autovalor λmin}

_{no limite de volume}

innito. De fato, fazendo a hipótese de um comportamento do tipo p

−2−2κ

_{para G(p}

2

₎

no limite de momentos p pequenos, supondo λmin

∼ L

−2−α

_{para grandes lados de rede}

L

e lembrando que o menor momento não-nulo é dado na rede por pmin

= 2 sin(aπ/L), é

claro que a desigualdade (4.25) requer 2κ ≤ α. Isto implica que α > 0 é uma condição

necessária para obter um propagador de fantasmas G(p

2

₎

_{fortalecido (κ > 0) no limite}

infravermelho. Ao mesmo tempo, o limite inferior da equação (4.25) implica que α > 0 e

| eψ

min

(a, p)|

2

6= 0 são condições sucientes para obter κ > 0.

Devido ao grande esforço computacional necessário para o cálculo do propagador de

fantasmas G(p

2

₎

_{e do menor autovalor λmin}

_{da matriz de Faddeev-Popov, a vericação}

numérica das desigualdades (4.25) foi feita na Ref. [147] usando principalmente dados já

publicados, obtidos em geral para redes relativamente pequenas. Alem disso, foi vericado

somente o limite superior da equação (4.25). Apesar dessas limitações, o estudo evidenciou

que, no limite de grandes volumes de rede, o propagador de fantasmas calculado para o

menor momento pmin

é razoavelmente bem aproximado pelo valor 1/λmin

(ver a Figura

1 na mesma referência). Uma vericação mais cuidadosa

8

_{das desigualdades (4.25) foi}

recentemente completada [149] (ver quadro abaixo) usando no caso SU(2) redes de volume

até 64

4

_{para β = 2.2. Ao mesmo tempo, foi mostrado como melhorar de forma sistemática}

as desigualdades demonstradas acima. De fato, considerando também o autovalor λ2

e o

correspondente autovetor eψ

2

(b, p), é simples vericar que

1

N

2 c

− 1

X

b



1

λ

min

| e

ψ

min

(b, p)|

2

+

1

λ

2

| e

ψ

2

(b, p)|

2



≤ G(p

2

)

(4.26)

e que, para momentos p não nulos,

G(p

2

)

_≤



1

λ

min

−

1

λ

2



1

N

2 c

− 1

X

b

| eψ

min

(b, p)|

2

!

+

1

λ

2

.

(4.27)

8_{Isto foi possível graças ao acesso ao super-computador IBM Blue Gene/P, instalado em Abril de 2012}

na Rice University nos EUA, cujo uso é compartilhado entre pesquisadores da USP e da Universidade Rice.

4.3 Desigualdades para o Propagador de Fantasmas

87

Também é simples vericar que o novo limite superior (4.27) é em geral uma melhoria sobre

o limite superior da Eq. (4.25) e que os dois limites superiores coincidem para λmin

=

λ

2. Na Ref. [149] as velhas e as novas desigualdades para o propagador de fantasmas

foram vericadas numericamente considerando também diferentes cópias de Gribov. Os

resultados demonstraram claramente que (no limite de volume innito) λmin

≈ λ

2, como

esperado, e que G(p

2

min

)

≈ 1/λ

min

(ver Fig. 4.2), conrmando os dados apresentados no

trabalho original [147]. Como explicado logo abaixo da Eq. (4.25), este resultado implica

que λi

≈ λ

min

e representa uma vericação numérica do resultado all horizons are one

horizon, discutido na Seção 3 da Ref. [66]. Ao mesmo tempo, sendo G(p

2

min

)

muito

maior do que os limites inferiores, ca claro que o autovetor ψmin

deve ser em geral muito

diferente das ondas planas correspondentes ao autovalor p

2

min

do laplaciano discretizado.

De fato, foi obtido que a projeção

9

ψ

_min

(pmin) =

1

N

2 c

− 1

X

b

1

d

X

µ









e

ψmin



b,

2π

N

eµ

 







2

(4.28)

9_{Aqui e}

µ indica um vetor unitário na direção µ. Relembramos que o momento pmin é d-vezes degene-

1

10

100

1000

0.02

0.04

0.06

1/N

fc

Figura 4.2: O propagador de fantasmas G(p2

min) (triângulos cheios), os limites inferiores nas Eqs. (4.25) e (4.26) (círculos vazios e cheios) e os limites superiores nas Eqs. (4.25) e (4.27) (quadrados vazios e cheios) como função do inverso do lado de rede 1/N. Todas as grandezas são em unidades de rede. Para esta gura foi usado o conjunto de cópias de Gribov indicado na Ref. [149] como rst copy (fc). Observe a escala logarítmica no eixo y. As barras de erro (que não são visíveis) correspondem a um desvio padrão. (Na análise dos dados foram considerados somente os erros estatísticos.) Este gráco é apresentado na Figura 1 da Ref. [149].

4.3 Desigualdades para o Propagador de Fantasmas

89

é muito pequena para todas as congurações de gauge e cópias de Gribov consideradas

na Ref. [149]. Enm, foi vericado que, no limite de L grande, o menor autovalor λmin

da matriz de Faddeev-Popov vai a zero mais lentamente

10

_{do que 1/L}

2

_{, conrmando o}

comportamento massivo descrito no capítulo anterior para o caso quadridimensional.

10_{Este resultado está em desacordo com as conclusões apresentadas na Ref. [143], obtidas porém usando}

A. Cucchieri

Instituto de Fı´sica de Sa˜o Carlos, Universidade de Sa˜o Paulo, Caixa Postal 369, 13560-970 Sa˜o Carlos, SP, Brazil

T. Mendes*

DESY–Zeuthen, Platanenallee 6, 15738 Zeuthen, Germany (Received 17 April 2008; published 7 November 2008)

We present rigorous upper and lower bounds for the momentum-space ghost propagator GðpÞ of Yang- Mills theories in terms of the smallest nonzero eigenvalue (and of the corresponding eigenvector) of the Faddeev-Popov matrix. We apply our analysis to data from simulations of SU(2) lattice gauge theory in Landau gauge, using the largest lattice sizes to date. Our results suggest that, in three and in four space- time dimensions, the Landau gauge ghost propagator is not enhanced as compared to its tree-level behavior. This is also seen in plots and fits of the ghost dressing function. In the two-dimensional case, on the other hand, we find that GðpÞ diverges as p
2
2
with
 0:15, in agreement with A. Maas, Phys. Rev. D 75, 116004 (2007). We note that our discussion is general, although we make an application only to pure gauge theory in Landau gauge. Our simulations have been performed on the IBM supercomputer at the University of Sa˜o Paulo.

DOI:10.1103/PhysRevD.78.094503 PACS numbers: 11.15.Ha, 12.38.Aw, 14.70.Dj

I. INTRODUCTION

In the Gribov-Zwanziger scenario [1], confinement of quarks (in Landau and in Coulomb gauge) is related to a ghost propagator GðpÞ enhanced in the infrared (IR) limit when compared to the tree-level propagator 1=p2. An enhanced ghost propagator is also required by the Kugo- Ojima criterion [2] in order to explain confinement of color charge (in Landau gauge).

In order to investigate the origin of this IR enhancement we consider, following Ref. [3], a generic gauge condition F ½A ¼ 0, where A is the gauge field. The condition is imposed on the lattice by minimizing a functional E½U, where U is the (gauge) link variable. From the second variation of E½U one can define the Faddeev-Popov (FP) matrix Mða; x; b; yÞ. (Here we use a and b to indicate color indices, whereas x and y represent points of the lattice.) In the SUðN_cÞ case, M is an ðN_c2
1ÞV  ðN2_c
1ÞV matrix, where V ¼ Nd_{is the lattice volume, given by}

the number of points N along each side of the lattice and by the space-time dimension d. This matrix is real and sym- metric with respect to the exchange ða; xÞ $ ðb; yÞ (see Ref. [4] for a thorough discussion of the properties of the FP matrix in Landau gauge). At any (local) minimum of E½U, all the eigenvalues of M are positive (modulo trivial null eigenvalues). The set of all minima of E½U is the so- called Gribov region
. On the boundary of
—known as the first Gribov horizon @
—the smallest nontrivial ei- genvalue minof the FP matrix is null. Since the configu-

ration space has very large dimensionality we expect that,

in the limit of large lattice volumes V, entropy favors configurations near @
[5–7], i.e. min should go to zero

in this limit. This is indeed the case in 2d [8], 3d [9], and 4d Landau gauge [10], in 4d Coulomb gauge [7], and in 4d maximally Abelian gauge [11].

The ghost propagator is written in terms of the inverse of the FP matrix as GðpÞ ¼ 1 N2c
1 X x;y;a e
2ikðx
yÞ V M
1_{ða; x; a; yÞ; (1)}

where the lattice momentum pðkÞ has components pðkÞ ¼ 2 sinðk=NÞ with k¼ 0; 1; . . . ; N
1. Also,

here and in the formulas below, a path-integral average is understood. SinceMða; x; b; yÞ develops a null eigenvalue at the Gribov horizon @
, one might expect that the corresponding ghost propagator diverges at small momenta in the infinite-volume limit. This could in turn introduce a long-range effect in the theory, related to the color- confinement mechanism [5,7]. Indeed, in several numeri- cal studies (using relatively small lattice volumes), the ghost propagator GðpÞ seems to be IR-enhanced in 3d [9] and 4d Landau gauge [12,13] and in 4d Coulomb gauge [14]. On the other hand, in maximally Abelian gauge [11] one finds an IR-finite GðpÞ. At the same time, recent results in Landau gauge using very large lattice volumes [15–17] suggest an essentially tree-level-like IR ghost propagator GðpÞ in 3d and in 4d. (A similar result has been obtained in Refs. [18,19] using different analytic approaches.) Finally, in the 2d case, one finds [8] GðpÞ  p
2:3after extrapolat- ing data for the IR exponent
to infinite volume. Therefore, we have cases for which GðpÞ is not IR- enhanced in the infinite-volume limit and thus the argu- ment reported above cannot be valid in general [3]. The *Permanent address: Instituto de Fı´sica de Sa˜o Carlos,

Universidade de Sa˜o Paulo, C.P. 369, 13560-970 Sa˜o Carlos, SP, Brazil.

aim of this work is to understand under what conditions one should expect an IR-enhanced ghost propagator GðpÞ. This is done by investigating constraints on the behavior of GðpÞ and thus obtaining better control over its infinite- volume extrapolation. Clearly, since the only diverging quantity (besides the lattice volume) is 1=min, the IR

enhancement of GðpÞ may be closely related to this eigen- value and to the projection of its corresponding eigenvector cminða; xÞ on the plane waves e
2ikx=pffiffiffiffiVfor small mo-

menta pðkÞ.

II. LOWER AND UPPER BOUNDS FORGðpÞ In this section we obtain upper and lower bounds for the momentum-space ghost propagator GðpÞ in terms of the smallest nonzero eigenvalue min[and of the correspond-

ing eigenvectorcminða; xÞ] of the FP matrix Mða; x; b; yÞ.

To this end, let us first introduce our notation. As usual, we define the generic eigenvalue i ofMða; x; b; yÞ and the

corresponding eigenvector c_iða; xÞ by using the relation Mða; x; b; yÞciðb; yÞ ¼ iciða; xÞ, where the index i

takes values 1; 2; . . . ; ðNc2
1ÞV and the sum over repeated

indices is understood. As stressed in the Introduction, this matrix is positive (or semipositive) definite, whenever we use a minimizing condition in order to fix a gauge (minimal gauge). Also, since it is obtained from a second-order expansion, this matrix can always be written in a symmet- ric form. Then, the eigenvectors c_iða; xÞ can be assumed orthogonal to each other and normalized as P

a;xciða; xÞcjða; xÞ¼ ij, where  indicates complex

conjugation. Using this notation we can write M ða; x; b; yÞ ¼ X

i;i
0

iciða; xÞciðb; yÞ: (2)

Note that we are working in the space orthogonal to the kernel of the FP matrix. Then we can write the inverse of this matrix as [20]

M
1_{ða; x; b; yÞ ¼} X i;i
0


1i ciða; xÞciðb; yÞ: (3)

By using Eq. (1) we obtain

GðpÞ ¼ 1 N2c
1 X i;i
0 X a 
1i j ~ciða; pÞj2; (4)

where ~c_iða; pÞ ¼P_xc_iða; xÞe
2ikx=pffiffiffiffiV is the Fourier transform of the eigenvectorc_iða; xÞ. Since all the nonzero eigenvalues are positive and 0 < min i, we can write

the inequalities 1 Nc2
1 1 min X a j ~cminða; pÞj2 GðpÞ; (5) GðpÞ  1 Nc2
1 1 min X i;i
0 X a j ~ciða; pÞj2: (6)

(Note that we assume nondegenerate eigenvalues.

However, the second inequality applies also when minis

degenerate and the first one can be easily modified to take a degeneracy into account.) By summing and subtracting in Eq. (6) the contributions from the eigenvectors correspond- ing to a null eigenvalue and using the completeness rela- tionP_ic_iða; xÞc_iðb; yÞ ¼ _abxy, we find

GðpÞ  1 min  1
1 N2c
1 X j;j¼0 X a j ~cjða; pÞj2  : (7) Let us stress that the above results are simply a conse- quence of the positivity of the FP matrix, i.e. they apply to gauge-fixed configurations that belong to the interior of the first Gribov region.

III. BOUNDS IN LANDAU GAUGE

In the Landau case the eigenvectors corresponding to the zero eigenvalue are constant modes, i.e. they contribute only to the case p ¼ 0. Thus, for any nonzero momentum we have 1 N2c
1 1 min X a j ~cminða; pÞj2  GðpÞ  1 min: (8)

As said in the Introduction, in the infinite-volume limit the measure gets concentrated on the boundary of the (first) Gribov region, i.e. min goes to zero in this limit. In

particular, as for the Laplacian operator, one can expect to find min N
 for large N. Now, if we make the

hypothesis of a power-law behavior p
2
2
for GðpÞ in the IR limit and consider the smallest nonzero momentum on the lattice [i.e. ps¼ 2 sinð=NÞ], we have GðpsÞ 

N2þ2
in the limit of large N. Then from Eq. (8) we get 2 þ 2
 . Thus, assuming a power-law behavior for GðpÞ,  > 2 is a necessary condition to obtain a ghost propagator GðpÞ enhanced in the IR limit compared to the tree-level behavior 1=p2.

The lower bound in Eq. (8) clearly depends on the behavior of the quantity j ~cminða; psÞj2. Note that this

quantity cannot diverge in the infinite-volume limit. (Actually, since the eigenvectors are normalized, this quan- tity is always between 0 and 1.) Thus, if we furthermore make the assumption of a behavior at large N given by N
, with  0, we find the condition 
  2 þ 2
. As a check of the proposed bounds, we collect here results for the ghost propagator GðpsÞ at the smallest non-

zero momentum ps¼ 2 sinð=NÞ and for the smallest

nonzero eigenvalue minfrom Refs. [6,9,12,15] for the 3d

and 4d cases. These data are shown for the smaller lattices in TablesIandIIand for the larger lattices in Fig.1. We also present new data in the two-dimensional case. (These data were obtained together with the data for the gluon propagator reported in [21].) All data refer to the SU(2) case. Let us recall that recent studies [17,22] have verified the analytic prediction that Landau gauge gluon and ghost propagators in SU(2) and in SU(3) are rather similar. Thus,

A. CUCCHIERI AND T. MENDES PHYSICAL REVIEW D 78, 094503 (2008)

we expect that the analysis presented here should apply also to the SU(3) case. In all cases the quantity 1=minhas

been evaluated using the average value for minand propa-

gation of errors.

As one can see, the upper bound in Eq. (8) is always satisfied. (Of course, this is the case also when p
ps.)

The bound has also been verified using the data at  ¼ 0 from Refs. [6,12]. We have fitted the data for GðpsÞ and for

minas a function of L for the data shown in Fig.1using a

power-law Ansatz. It is interesting to note that (in 2d and in 4d) we actually find  smaller than 2 þ 2
. The bound is still satisfied, since the multiplicative constant in the fit of minis larger. It would be interesting to see how the upper

bound is realized when even larger lattice volumes are considered. Also, note that the upper bound in the 4d case seems to saturate as the volume increases. This would indicate that the contribution to GðpÞ from all the eigen- values i> minstays finite at large volume and that the

exponent , defined above, is zero. Let us note that this is not the case at smaller lattice volumes, since then one needs to consider the contribution to GðpÞ from the first 150–200 smaller eigenvalues in order to reproduce the behavior of the propagator at small momenta [10]. Finally, let us recall that finite-size effects for GðpÞ (at fixed p) are generally very small. This can be explained considering the spectral density of the FP matrix, as done in Ref. [3]. Thus, our results indicate that in 3d and in 4d the ghost propagator is likely not IR-enhanced, while in 2d

100 1000 10000 0 0.004 0.008 0.012 0.016 0.02 0.024 G(p s ), 1/ λmin 1/L 2D Results 10000 0.002 0.003 0.004 0.005 0.006 0.007 G(p s ) 1/L 3D Results 1000 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 G(p s ), 1/ λmin 1/L 4D Results

FIG. 1. The ghost propagator GðpsÞ for the smallest nonzero

momentum ps¼ 2 sinð=NÞ (in GeV
2) as a function of the

inverse lattice side 1=L (GeV) for the 2d case (top, at  ¼ 10 with volumes up to 3202), 3d case (center, at  ¼ 3, with volumes up to 3203), and the 4d case (bottom, at  ¼ 2:2 with volumes up to 1284). Data are taken from Ref. [15] for the 3d and the 4d cases. We also show (in 2d and in 4d) the inverse of the smallest nonzero eigenvalue min of the FP matrix (in

GeV
2_{). In these two cases one can verify the inequality}

1=min GðpsÞ. The fitting parameters are reported in

TableIII. Note that we did not evaluate minfor all our configu-

rations. TABLE II. As in TableIfor the 4d case. Data (in lattice units)

are taken from Refs. [6,12] using the average called first copy. We find similar results when considering data obtained using the average called absolute minimum.

N  GðpsÞ 1=min 8 0.8 8.94(8) 91(3) 12 0.8 22.1(6) 210(20) 16 0.8 40.6(7) 360(60) 8 1.6 7.2(1) 61(3) 16 1.6 32.1(3) 220(20) 8 2.7 3.4(1) 11.0(4) 12 2.7 7.1(3) 25(1) 16 2.7 12.9(6) 45(4)

volumes and  values in the 3d case. Data (in physical units) are taken from Ref. [9].

N  a
1[GeV] GðpsÞ [GeV
2] 1=min [GeV
2]

20 6.0 1.733 7.3(1) 32.5(7)

30 6.0 1.733 19.4(3) 94(2)

20 4.2 1.136 21.0(4) 107(3)

30 4.2 1.136 54.5(8) 282(6)

we obtain
 0:1. We note that our result in the 2d case is essentially in agreement with Ref. [8], where however the value of
is obtained by first fitting GðpÞ in an appropriate momentum range for each lattice volume and then per- forming the extrapolation to infinite volume by a four- parameter fit. The fits for minand GðpsÞ as a function of

L are usually simpler, as shown above.

These results are confirmed if one considers the dressing function p2Gðp2Þ at the largest lattice volumes for the three cases (see Fig.2). Indeed, the data in the 2d case can be fitted using a power-law Ansatz ap
2
(see TableIV), with
¼ 0:177ð2Þ, essentially in agreement with [8]. The fitted value for
decreases if one considers fewer points, drop- ping points with larger p, and becomes 0.152(7) when considering data with p22 ½0; 0:1. The same Ansatz does not work in 3d and in 4d. In these cases the data for the dressing function can be described by the Ansatz a
b logð1 þ cp2Þ, inspired by the logarithmic corrections suggested in Ref. [19]. In both cases one finds (approxi- mately) p2Gðp2Þ ! 4:5 in the limit p ! 0. This result supports the value
 0 obtained above, when consider- ing the bound using min. It is interesting that one can also

obtain a relatively good fit for the dressing function in the whole range of momenta by considering the fitting func- tion aðp
2kþ bp2eÞ=ð1 þ p2eÞ in the 2d case and a
b½logð1 þ cp2Þ þ dp2=ð1 þ p2Þ in 3d and in 4d. These fits and the corresponding parameters are reported in Fig.2 and in TableV.

IV. CONCLUSIONS

Our data suggest that the Landau gauge ghost propagator does not diverge faster than 1=p2 at small momenta in 3d and in 4d, while in 2d GðpÞ does not diverge faster than p
2
2
with
between 0.1 and 0.2, in agreement with

1 1.5 2 2.5 3 3.5 4 4.5 5 0.0001 0.001 0.01 0.1 1 10 p 2 G(p), a(p -2 k + b p 2 e )/(1+p 2 e ) p2 2D Results 1 1.5 2 2.5 3 3.5 4 4.5 5 0.0001 0.001 0.01 0.1 1 10 p 2 G(p), a-b[log(1+c p 2)+d p 2]/(1+p 2) p2 3D Results 1 1.5 2 2.5 3 3.5 4 4.5 0.001 0.01 0.1 1 10 100 p 2 G(p), a-b[log(1+c p 2)+d p 2]/(1+p 2) p2 4D Results

FIG. 2. The ghost dressing function p2Gðp2Þ as a function of p2(in GeV) for the 2d case (top, at  ¼ 10 with volume 3202), 3d case (center, at  ¼ 3, with volume 2403), and the 4d case (bottom, at  ¼ 2:2 with volume 804). The fitting functions and the corresponding fitting parameters are reported in TableV.

TABLE IV. Fits of p2Gðp2Þ using the Ansa¨tze ap
2
in the 2d case and a
b logð1 þ cp2Þ in the 3d and 4d cases. In the first case we used the data in the range p22 ½0; 0:5 for the fit. In the other two cases the data in the range p22 ½0; 1 have been considered.

d a
b c

2 1.134(7) 0.177(2)

3 4.7(1) 0.579(5) 320(20)

4 4.28(1) 0.69(2) 33(3)

TABLE III. Fits of GðpsÞ and of 1=min, respectively, using the

Ansa¨tze bL2þ2
and cL_{. In the 4d case the fit for 1=min has}

been done by considering only the three largest physical vol- umes. When considering the three smallest physical volumes we find  ¼ 0:9ð3Þ. In the other cases all data have been considered for the fit.

d b 2 þ 2
c 

2 0.026(1) 2.251(9) 0.12(3) 2.20(4)

3 0.11(3) 2.02(4)

4 0.086(3) 2.043(8) 1.2(1) 1.53(2)

TABLE V. Fits of p2Gðp2Þ using the Ansa¨tze aðp
2kþ bp2eÞ=ð1 þ p2eÞ in the 2d case and a
b½logð1 þ cp2Þ þ dp2=ð1 þ p2Þ in the 3d and 4d cases. In the three cases we used the whole range of momenta for the fit.

d a
b e c d

2 1.24(3) 0.16(2) 0.86(3) 0.75(15)

3 4.75(1) 0.491(5) 450(30) 7.1(1)

4 4.32(2) 0.38(1) 80(10) 8.2(3)

A. CUCCHIERI AND T. MENDES PHYSICAL REVIEW D 78, 094503 (2008)

and minas a function of L. In particular, we note that the

use of the upper bound to constrain GðpÞ is quite conve- nient, since mindoes not depend on p. It might thus be of

interest to optimize the evaluation of min and of

cminða; xÞ.

The results obtained here in the 3d and 4d cases, together with those reported in Ref. [21] for the gluon propagator, seem to contradict the Gribov-Zwanziger confinement scenario for Landau gauge. On the other hand, one should establish if the behavior of gluon and ghost propagators at very small momenta is really so essential for the explana- tion of confinement. After all, when dynamical quarks are considered [23], string breaking should manifest itself at a scale of about a fermi, i.e. for an energy scale of about 200 MeV. Thus, for the physics of hadrons the behavior of the propagators at intermediate momenta is probably more

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