Correlation of the ghost and the quark in the lattice landau gauge QCD

Abstract

Effects of the quark field on the ghost propagator of the lattice Landau gauge are investigated by using the unquenched SU(3) configurations produced by the MILC collaboration and compared with quenched gauge configurations of SU(2) first copy of the overrelaxation gauge fixing, the parallel tempering (PT) gauge fixing and quenched SU(3) 56(4) configurations. We measure the color symmetric and the color antisymmetric ghost propagator and the Binder cumulant of the l¹ norm and the l² norm of color antisymmetric ghost propagators and investigate deviation from those of Gaussian distributions. In the first copy samples of quenched SU(2) we observe a large fluctuation in the Binder cumulant at the lowest momentum point. This fluctuation is reduced in the PT gauge fixed samples. The color anti-symmetric ghost propagator of quenched SU(3) configurations depends on the lattice size and is small as compared to the symmetric one in the large lattice of 56(4). The Binder cumulants of the quenched SU(2) and the Nf = 2 + 1 unquenched SU(3) are almost consistent with 3-d and 8-d Gaussian distribution, respectively. A comparison of the SU(3) unquenched configurations and quenched configurations indicates that the dynamical quarks have the effect of making color antisymmetric ghost propagator closer to the Gaussian distribution and the Kugo-Ojima color confinement parameter c closer to 1.

Ghost propagator; Quark propagator; Binder Cumulant; Dynamical mass

Correlation of the ghost and the quark in the lattice landau gauge QCD

Sadataka Furui^I; Hideo Nakajima^II

^ISchool of Science and Engineering, Teikyo University, 320-8551 Utsunomiya Japan

^IIDepartment of Information Science, Utsunomiya University, 321-8585 Utsunomiya Japan

ABSTRACT

Effects of the quark field on the ghost propagator of the lattice Landau gauge are investigated by using the unquenched SU(3) configurations produced by the MILC collaboration and compared with quenched gauge configurations of SU(2) first copy of the overrelaxation gauge fixing, the parallel tempering (PT) gauge fixing and quenched SU(3) 56⁴ configurations. We measure the color symmetric and the color antisymmetric ghost propagator and the Binder cumulant of the l¹ norm and the l² norm of color antisymmetric ghost propagators and investigate deviation from those of Gaussian distributions. In the first copy samples of quenched SU(2) we observe a large fluctuation in the Binder cumulant at the lowest momentum point. This fluctuation is reduced in the PT gauge fixed samples. The color anti-symmetric ghost propagator of quenched SU(3) configurations depends on the lattice size and is small as compared to the symmetric one in the large lattice of 56⁴. The Binder cumulants of the quenched SU(2) and the N_f = 2 + 1 unquenched SU(3) are almost consistent with 3-d and 8-d Gaussian distribution, respectively. A comparison of the SU(3) unquenched configurations and quenched configurations indicates that the dynamical quarks have the effect of making color antisymmetric ghost propagator closer to the Gaussian distribution and the Kugo-Ojima color confinement parameter c closer to 1.

Keywords: Ghost propagator; Quark propagator; Binder Cumulant; Dynamical mass

I. INTRODUCTION

In this presentation, we would like to show our study of color confinement and dynamical chiral symmetry breaking via unquenched lattice Landau gauge simulation using configurations given by the MILC collaboration in which Asqtad Kogut-Susskind (KS) fermions are used[1].

In our study of color confinement mechanism, we measure 1) the Kugo-Ojima confiment parameter c, 2) the A² condensate in running coupling, gluon propagator and quark propagator and 3) the ghost condensate parameter v and the Binder cumulant of the color anti-symmetric ghost propagator.

We are interested in the effect of quark fields on these variables since the confinement and the chiral symmetry breaking could be related with each other. We show our result of measurement of mass function M(q) of MILC_c(20³×64 lattice) and MILC_f(28³×96 lattice).

In the Kugo-Ojima theory[2], unphysical longitudinal gluons and ghosts do not appear in the physical spectrum by the BRST quartet mechanisme. Using arrows as the BRST transformations and B as the Nakanishi-Lautrup auxiliary field, the BRST transformation in the quark sector can be expressed as

Inclusion of fermion gives more restriction on the degrees of freedom of the ghost and it may change the fluctuation of the ghost propagator.

The Landau gauge adopted in this theory suffers from the Gribov problem[3], i.e. the uniqueness of the gauge field is not guaranteed in the simple gauge fixing procedure and in general there are gauge copies. In the Lagrangian field theory, presence of gauge copies is ignored, but in the lattice simulation of the infrared region, importance of the selection of a unique gauge is qualified[4, 6, 7]. The complete algorithm is however not known, despite various proposals (smearing [8], parallel tempering (PT)[6, 7] etc).

This paper is organized as follows. In sect.II, we review theories of the color confinement in infrared QCD. Results of Kugo-Ojima confinement parameter in lattice simulation are shown in sect.III. Simulation results of the ghost propagator, the gluon propagator, the QCD running coupling and the quark propagator are shown in sect. IV, V, VI and VII, respectively. Summary and discussion are given in sect. VIII.

II. COLOR CONFINEMENT

Let us summarize developments of the study of color confinement which are related to our work. In 1979 Kugo and Ojima[2] proposed a color confinement criterion, which will be explained in the following. Gribov[3] and Zwanziger[4, 5] gave condition of the confinement on the infrared exponents of the gluon propagator and the ghost propagator. These exponents define that of the running coupling a_s(q) in scheme. A measurement of a_s(q) via triple gluon vertex in quenched SU(3) lattice simulation by the Orsay group suggested infrared suppression and a presence of mass-dimension 2, A² condensates[9].

The relation between the mass-dimension 2 condensates and the Zwanziger's horizon condition generated by the restriction of the gauge field in the fundamental modular region was pointed out by the Gent group[10].

Although A² is not BRST invariant, a mixed condensates with c is BRST invariant[11]. The Gent group suggested in the Local Composite Operator (LCO) approach that the c condensates would manifest itself in the color antisymmetric ghost propagator.

The lattice simulation of the color antisymmetric ghost propagator in SU(2) lattice Landau gauge was performed by the group of Cucchieri[12].

A. Kugo-Ojima confinement criterion

The Kugo-Ojima confinement criterion is expressed by the parameter c defined in the two-point function

as u^ab(0) = -d^abc becomes 1. We adopt the SU(3) color matrix L normalized as trLªL^b = d^ab.

The parameter c is related to the renormalization factor of the gluon sector, the ghost sector and the quark sector as

If the finiteness of

₁ is proved, divergence of ₃ is a sufficient condition. If Z₃ vanishes in the infrared, Z₁ should have higher order 0. If Z₂ is finite should vanish.

B. Zwanziger's horizon condition

Zwanziger proposed realization of the unique gauge by restricting the configuration in the fundamental modular region. The condition on his two-point function given below is equivalent to the Kugo-Ojima theory in a naive continuum limit.

Here e = áå_x,µtr(L^a†S(U_x,µ)L^a)ñ / {( - 1)V}, where S(U_x,µ) is the coefficient of ¶_m of the covariant derivative. The horizon condition reads G_mm(q) - e = 0, and the l.h.s. of the condition is () + (d - 1)c - e = (d - 1)h where h = c - and dimension d = 4, and it follows that h = 0 ® horizon condition, and thus the horizon condition coincides with Kugo-Ojima criterion provided the covariant derivative approaches the naive continuum limit, i.e., e/d = 1.

III. LATTICE DATA OF THE KUGO-OJIMA PARAMETER

In our lattice simulation, we adopt two types of the gauge field definitions: 1) logU type in which the link variable U and the gauge field are related as U_x,µ = and 2) U-linear type in which A_x,µ is a traceless part of the difference of U_xµ and multiplied by one half.

The optimizing function corresponding to the two definitions are 1) The l² norm of A^g and 2) tr(2-(U^g_x,µ + U^g_x,µ ^† )), respectively.

Under infinitesimal gauge transformation g^-1dg = e, its variation reads for either definition as

DF_U(g) = -2á¶A^g|eñ + áe| - ¶D(U^g)|eñ + ...,

where the covariant derivative D_m(U) for two options reads commonly as

D_m(U_{x, µ}) f = S(U_{x, µ})¶_mf + [A_{x, µ}, ],

where ¶_mf = f(x + µ) -f(x), and = The function S(U_x,µ) in the logU definition is S(U_xµ) = .[13]

Stationality of F_U(g) means the Landau gauge, the local minimum means the Gribov region and the global minimum means the fundamental modular region[4].

In the quenched Landau gauge QCD simulation, the Kugo-Ojima parameter saturated at about 0.8, as shown in TABLE I. The parameter e/d is closer to 1 in the logU definition of the gauge field.

We measured the corresponding values of the unquenched configurations of MILC_c and MILC_f[16]. The lattice specifications are shown in TABLE II.

As shown in TABLE III, the Kugo-Ojima parameters of MILC configurations are consistent with 1. The qualitative difference from quenched simulations would be due to the difference of the ghost propagator caused by the quark field.

IV. THE GHOST PROPAGATOR

The ghost propagator is defined as the Fourier transform (FT) of the matrix element of the inverse Faddeev-Popov operator.

where = -¶_mD_m, and {}_xy means the matrix value. We define the ghost dressing function G(q²) as q²D_G(q²) and its exponent at around 0.4GeV as a_G.

We calculate the overlap to get the color diagonal ghost propagator

and color anti-symmetric ghost propagator

where = 2 for SU(2) and 6 for SU(3). Here f_c^b(x) and f_s^b(x) are the solution of f^b(x) = r^b(x) with r^b(x) = L^b cosq·x and L^b sinq·x, respectively.

The ghost dressing function of MILC_fb_imp = 7.09 and that of the quenched b = 6.4 and 6.45, the color symmetric and anti-symmetric ghost propagator of MILC_c are shown in [15]. FIG.1 is a fit of the ghost dressing function of MILC_f by the perturbative QCD(pQCD) in scheme[16].

A. The ghost condensate

In the LCO approach, preference of a specific direction of áf^abc

^bc^cñ in the color space is regarded as a signal of the ghost condensate. We define a parameter v µ |áf^abc

^bc^cñ| and following [12] measure its value as follows. We take into account the finite size effect of the lattice through a parameter r, defined as

where L is the lattice size and = 0,1,...L. In our asymmetric (L_x < L_t) lattice and the momentum q near diagonal in the 4-d space, we consider the sine momentum

and choose L² in the numerator as the square root of the volume .

Using this parameter r, the absolute value of the color antisymmetric ghost propagator is parameterized as

We tried to fit |f(q)| and f(q)² of MILC_fb_imp = 7.09 and 7.11 and found a relatively good parameter fit of |f(q)| with use of r = 134 and v = 0.026GeV², as shown in FIG.2.

Extraction of v from lattice data is not so easy due to the finite size effect of the asymmetric lattice. Except MILC_fb_imp = 7.09 which had difficulty in the fitted value from (q)² is larger than that from |f(q)|.

B. Binder cumulant

Cucchieri et al[12] measured the Binder cumulant of the color antisymmetric ghost propagator

and found that it is about 0.44, which is compatible with the expectation value of the d-dimensional Gaussian distribution, with d = 3.

The Binder cumulant U(q) of the color anti-symmetric ghost propagator f²(q) of MILC_fb = 7.09 m₀ = 13.6MeV/68MeV and MILC_cm₀ = 11.5MeV/82MeV are stable but the data of b_imp = 7.11 is rather noisy. In the region q > 1GeV, the Binder cumulant U(q) is 0.52(1) but in the region q > 1 GeV it is 0.58(3), which is consistent with that of Gaussian distribution U(q) ~ 0.583. The anomalous momentum dependence occurs also in the mass function of the quark propagator, which is discussed in the sect. VII.

We compared the Binder cumulant of SU(2) color antisymmetric ghost propagator of samples which are gauge fixed by the PT method [6] and the first copy of the overrelaxation. The first copy shows large fluctuation at the lowest momentum point. The Binder cumulant of quenched SU(3) 56⁴ color anti-symmetric ghost propagator is small as compared to color diagonal and its Binder cumulant is noisy.

In [14] we showed the Binder cumulant of MILC_c (21 samples). We increased the statistics of MILC_c to b_imp = 6.76 and 6.83, 20 samples each and extended the measurement to MILC_fb_imp = 7.09 and 7.11. In MILC_cb = 6.76, we found an exceptional sample which we encountered also in the quenched SU(3) 56⁴ lattice[7]. The ghost propagator of this sample has large exponent a_G. In the case of quenched simulation such copy had larger ||A_µ||² than the average and the PT gauge fixing excludes such samples.

V. THE GLUON PROPAGATOR

The gluon propagator is defined as the two-point function

The gluon dressing function is defined as Z(q²) = q²D_A(q²).

The gluon propagator and the dressing function of MILC_fb_imp = 7.09 are calculated in [16]. The data suggest that the gluon propagator is infrared finite. The infrared exponent of the gluon dressing function is defined as a_D. The exponent k of the Dyson-Schwinger equation(DSE) approach is defined at the lower momentum point, inaccessible in the present lattice simulation. We observe the numerical value of a_G is about k/2, and the tendency that a_D decreases but a_D > -1 i.e. the gluon propagator is not infrared vanishing. The combination of a_D + 2a_G is supposed to approach 0 in the DSE approach, but on the lattice it is about -0.1 in the MILC_c and -0.2 in the MILC_f.

VI. THE QCD RUNNING COUPLING

In the DSE approach, the ghost-gluon coupling in the scheme is calculated by the gluon dressing function Z₃ and the ghost dressing function ₃ and the vertex renormalization factor ₁ as

Our lattice simulation[16] of the gluon propagator and the ghost propagator of MILC_c yields the running coupling shown in FIG.3. There are deviations from the pQCD (dash-dotted line) and the DSE approach with k = 0.5 (long dashed line). As was done by the Orsay group[9], we consider a correction including the A² condensates and obtained áA²ñ ~ a few GeV².

The running coupling in the infrared can be estimated from the quark-gluon coupling

where Z₂ is the quark dressing function and is the vertex renormalization factor. An evaluation of Z₂(µ²,q²) is given in the next section.

VII. THE QUARK PROPAGATOR

We extended the measurement of the quark propagator using Asqtad action of MILC_c [14] to MILC_f. In the case of MILC_c, we compared the Asqtad action and the Staple+Naik action.

Due to long computation time for the convergence of the conjugate gradient method, the number of samples is of the order of 10 for each b_imp and the bare quark mass m₀.

The quark propagator is defined as a statistical average over Landau gauge fixed samples

In this expression, the inversion, is performed via conjugate gradient method after preconditioning, and we obtain

The mass function M(q) reflects dynamical chiral symmetry breaking. In high momentum region, it is parameterized as

where d_M = 12/(33-2N_f) and m(µ²) is the running mass. In the infrared region, we adopt the monopole fit

The momentum dependence of M(q) and Z₂(q) of m₀ = 13.6MeV in the infrared region of Asqtad action is smoother than that of the Staple+Naik action. It could be attributed to the effect of the tadpole renormalization. The parameters and L in our fit of the mass function are given in TABLE V.

We showed the quark wave function renormalization Z_y(q²) = g₁(µ²)/Z₂(q²) of MILC_f b_imp = 7.11 using the staple+Naik action in [14], where Z₂(q²) is the bare lattice data and g₁(q²) is the coefficient of g_m of the vector current vertex that compensates artefacts in Z₂.

We adopt áA²ñ as a fitting parameter and calculate[9]

where a(q) are data calculated in the scheme using the same MILC_f gauge configuration[7].

Here N_f is chosen to be 2 but the data does not change much for 3. We choose L_QCD = 0.691GeV and áyñ_m = -(0.7L_QCD)³[17, 18].

Since g₁(q²) in the infrared is expected to be given by the running coupling, the absence of suppression of the quark wave function renormalization suggests that the infrared suppression of the running coupling obtained by the ghost-gluon coupling could be an artefact.

In [20] the Z₂(q) is normalized to 1 at q = 3GeV. In our simulation without this kind of renormalization, Z₂(q) at q = 3GeV is close to 1 and the results are consistent. Our mass function M(q) of b_imp = 7.09 are about 20% larger than those of [20], but if M(q) is renormalized to the theoretical value of [20] at q = 3GeV we reproduce their data. Our mass function is consistent with DSE analysis[22].

In the case of b_imp = 7.11, the mass function of m₀ = 68MeV is consistent with [20], but that of m₀ = 27.2MeV is suppressed in the infrared and differ from [20]. The momentum dependence is correlated with the momentum dependence of the Binder cumulant of color anti-symmetric ghost propagator. The disagreements with [20] may be due to Gribov copies.

In the case of b_imp = 7.09, the chiral symmetry breaking mass measured by using the Staple+Naik action and measured by using the Asqtad action (FIG. 5) are consistent within about 5%, and there is no anomaly. Whether the Asqtad action has a non-QCD like behavior in a certain parameter region is under investigation.

VIII. SUMMARY AND DISCUSSION

In this presentation, we showed the Kugo-Ojima confinement criterion and the Zwanziger's horizon condition of lattice Landau gauge QCD. The infrared exponent of the ghost propagator a_G of the lattice is about half of the value k of the DSE[5, 21]. The gluon propagator is infrared finite. The Kugo-Ojima parameter c was saturated at about 0.8 in the quenched SU(3) but became consistent with 1 in the unquenched SU(3).

In b = 6.4, 56⁴ quenched SU(3) configurations, we found a copy whose a_G = 0.272 v.s. a_G(average)=0.223, and whose gluon propagator has an axis along which the reflection positivity is manifestly violated. In the b_imp = 6.76, 20³×64 MILC_c configurations, we find also an exceptional sample. The exceptional samples of unquenched configuration cause fluctuation in the Binder cumulant of color antisymmetric ghost propagator in the infrared, similar to the SU(2) first copy.

In the measurement of the running coupling a_s(q²) by the ghost-gluon coupling, we observed infrared suppression. The quark wave function renormalization Z_y(q) in the infrared suggests that the infrared suppression is a lattice artefact and the running coupling in the continuum limit freezes to a finite value. In the analysis, the artefact inherent to the compactness of the manifold should also be taken into account[23].

In the ghost sector, the Kugo-Ojima confinement criterion = 0 is satisfied. In the gluon sector, the continuum limit of Z₃ in the infrared is unknown, but the finite lattice data suggests that it is finite and non-zero. The infrared limit of Z₁ is not known, but the behavior of the ghost propagator suggests that it could be indefinite (fluctuating).

A natural question, 'How do the fixed points of the QCD renormalization group flow behave in the infrared?' may arise. In 1971, Wilson argued that the fixed point of the renormalization group flow of QCD could have limit cycle structure rather than a fixed point[24]. The deviation of the ghost dressing function from 4-loop pQCD result in the infrared region shown in FIG. 1 suggests that the flow of the ghost wave function renormalization is not a trivial one and it is correlated with the presence of dynamical quarks. The theorem on no renormalization effect on the ghost propagator[25] does not guarantee the validity of the simple extension of the ghost wave function renormalization from ultraviolet to infrared.

An extension of the analysis to finite temperature is straightforward. A preliminary simulation using MILC finite temperature configurations[26] shows that the Binder cumulant at finite temperature becomes smaller than that of Gaussian distribution as temperature rises. The effect of quenching randomness[27] played by quarks in the zero temperature unquenched configurations becomes weak above the critical temperature T_c. To clarify the nature of the infrared fixed points, systematic studies of finite size effects and the Gribov copy effects[7, 28] are necessary.

We observed that differences of the color antisymmetric ghost propagator in the quenched/unquenched and zero/finite temperature configurations make differences in the Kugo-Ojima parameter. The results imply that the quark field and the ghost field are correlated as the BRST quartet mechanism suggests.

This work is supported by the High Energy Accelerator Research Organization(KEK) supercomputer project No. 05-128 using Hitachi-SR8000 and the large scale simulation program No.5(FY2006) using SR11000. Numerical calculation of the ghost propagator was carried out by NEC-SX5 at the CMC of Osaka University and by NEC-SX8 at the Yukawa Institute Computer Facility. H.N. is supported by the MEXT grant in aid of scientific research in priority area No.13135210. S.F. thanks Attilio Cucchieri, Tereza Mendes and Silvio Sorella for the hospitality in Rio.

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Publication Dates

• Publication in this collection
  11 May 2007
• Date of issue
  Mar 2007

History

• Received
  11 Sept 2006