JAERI-Conf 99-008 JP0050035

23. Landau^- vfe^QCD > ^ a. V - v a > £ fflt&tb * iJ - X A The Lattice Landau Gauge QCD Simulation and the Confinement Mechanism

Hideo Nakajima1 Department of Information science, Utsunomiya University Sadataka Furui 2 School of Science and Engineering, Teikyo University

Abstract Numerical results of the gluon propagator and the ghost propagator in quenched lattice Landau gauge QCD simulation are reported. We observe that the gluon propa- gator is infrared finite and the ghost propagator is infrared divergent. The divergence of the ghost propagator is less singular than k~4.

1 The lattice Landau gauge QCD simulation

The Landau gauge fixing, the restriction on the gauge field such that dA = 0, specifies a region of the configuration space of gauge fields. The region such that the Faddeev-Popov determinant — dD^A) is positive is called the Gribov regionfl]. The Gribov region 0 is denned from the variation of the square norm of the gauge field [1, 2, 3, 4] :

A\\Ag\\2 = -(dA\t) + (t\ - dV\e)/2 + ••• (1) where g = ee . (2) The configuration is however not unique and, on the lattice, the minimal Landau gauge, i.e. restriction to the fundamental modular region is attempted[6, 7]. In the minimal Landau gauge[2], the fundamental modular region A is defined as the global minimum of ||J43||2 for each gauge orbit (A9) Gribov and Zwanziger[l, 5] showed that the restriction yields the gluon propagator which is free from infrared divergence and speculated that the singularity of the ghost propagator 1 Y' is enhanced from — to —. k2 k4 A solution of the Dyson-Schwinger equation in the Landau gauge[8] suggests that the gluon propagator is infrared finite and the ghost propagator is infrared divergent Dc{k) oc 1 (£2)1+0.92 ' We perform the lattice QCD simulation in /3 = 5 and 5.5 on 83 x 16 lattice in quenched Axi>x approximation, using the new definition of the gauge field : e = Ux

U £ 517(3), A£M = — AriM, Tr Ax

The restriction to the fundamental modular region was attempted by the smeared gauge fixing method[6] which was applied after the Landau gauge fixing in our particular method. [email protected] 2Speaker at the workshop, e-mail [email protected]

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We observed that the smeared gauge transformation appreciably reduces the norm in about 15% of the samples. Although there are about 8% exceptional samples where the norm is increased, the differences in the norms are less than 1% in all the 100 samples. We fixed remaining global gauge such that the zero mode of the gauge field component in the (1,1,1,1) direction is diagonalized and ordered, and observed that the expectation value of the gauge field is small but significantly nonvanishing.[7] The Polyakov loop after the global gauge fixing has the diagonal part expressed as U = eA Aa+j4 Ag. The expectation values of the diagonal components after the global gauge fixing is in the form diag(eia, 1, e~'a). It is a consequence of the diagonalization of the zero-mode of the gauge field which was suggested by the idea of the Abelian projection: i.e. if the U(l) part of the compact group SU(N) or SO(3) is left unbroken, one can minimise the off-diagonal component of the gauge fields and identify the magnetic charge. When the global gauge transformation is not done, the expectation values of {A3), (A8) are consistent to 0. t-direction z-direction diagonal 0.022(23) 0.028(26) 0.102(27) 3 1 A , A* \ -0.003(21) -0.003(23) 0.003(24) -0.019(20) -0.026(25) -0.105(28)

Table 1: The diagonal part of the Polyakov loop SU(3) algebra, after the smeared and global gauge fixing.

2 The gluon propagator

We measured the connected part of the gluon propagator :

D(c(nk)) = I (3)

In the case of (3 — 5.5, we show the results without the subtraction of the zero-mode and after the smeared and global gauge fixing and with subtraction. The scale is denned from the one loop renormalization group equation, i _I^ 2 2 2 2 ahiattice — e A>9 ((30g ) "°, where A/a«ice = AJV/5/28.81 for nj = 0. We take AMS = lOOMeV but for the data of (3 = 6[10], we take AMs = 0.128GeV according to their de- termination. Using the parameterisation of [11] based on the truncated Dyson-Schwinger equation, we obtained the complex masses of the gluon as in Table 2.

3 The ghost propagator

The ghost propagator is denned as

(4)

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Figure 1: The gluon propagator without Figure 2: The gluon propagator with global subtraction. (3 = 5.5, 83 x 16. and smeared gauge fixing and subtraction. (3 = 5.5, 83 x 16.

size Mi a-VGeV MJGeV ref /? = 5,83 x 16 0.75(15)a"1 0.48 0.36 this work /? = 5.5,83 xl6 0.71(15)a-1 0.84 0.59 this work /? = 6,323 x 64 0.32a"1 1.882 0.60 (Leinweber et al.)

Table 2: Complex mass of the gluon. a 1 is defined either from = OAGeV or 0.12SGeV. Fit of the off-axis diagonal propagator data. where a, b specify the color. The Faddeev-Popov operator is

M[U] = -(d • D[A]) = -(D[A] • d). (5)

X x The ghost propagator M. ^J\ — (Mo — Mi) is calculated perturbatively by using the Green function of the Poisson equation MQ1 = (—d2)'1 whose zero-mode is eliminated.

f (6) where Ad is the lattice adjoint operation[7]. The matrix elements between color eigenstates 6 a and |A x0)

ab -xo) dx = (7) are measured and obserbed that G°£(k) fitting by (ii) is better than (i) 1 . The simulation data shows that the off-diagonal element is consistent to zero but in the infrared region there are significant fluctuations.

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(3 = 5 0 = 5.5 (3 = 5 0 = 5.5 a b a b z K z K A 0.392(3) 1.2026(3) 0.773(9) 1.08(1) A 1.655(2) 0.1184(5) 1.582(5) 0.1293(12) A9 0.333(3) 1.2060(3) 0.683(6) 1.09(1) A3 1.597(2) 0.1045(5) 1.579(4) 0.1289(10)

Table 3: Parameters of the ghost propagator using A without global gauge fixing and with global gauge fixing A9. The left table is for the fit option i and the right table is for the fit option ii.

Figure 3: The ghost propagator after the Figure 4: The ghost propagator after the 3 3 global gauge fixing Gfg{k) (3 = 5.5, 8 x 16. global gauge nxmg GfJk) (3 = 5.5, 8 x 16. The fitted curve is 1.619/A;2*11137.

4 Discussion and Outlook

We observed that at /? = 5 the gluon propagator in the infrared region is suppressed[7]. At (3 = 5.5 the suppression is less significant, but it is infrared finite and the effective mass of the gluon is about 600MeV, consistent to the Mandula and Ogilvie[9], which are manifestations of the confinement. These results are qualitatively the same as in the SU(2) lattice simulation[12]. The ghost propagator is infrared divergent, but its divergence is less singular than k~4. We observe in the infrared region, significant fluctuations in the color space, which can be regarded as another manifestation of the confinement in the Landau gauge. Concerning the expectation value of the gauge field, which we observed in the lattice Landau gauge, there is an interesting conjecture on the quark confinement[13, 14]. The Hamiltonian of QCD describing the interaction of the light u and d quarks with the gauge fields contains the mass term M and the potential

V = -iguj^A^u - (8)

When (M) is small, (M + V) < 0 can be realized and a pair of u,d quarks from the condensate will convert qq pairs created by a collision into a meson pairs. A lattice simulation is necessary. The color confinement problem in general was extensively analyzed in the BRS formula- tion of the continuum gauge theory[15]. A sufficient condition of the color confinement given

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x c)b(y)\0)dx = (g^ - ^)<(p2) (9) / satisfies ul = —S%. The meaurement of w£ in the lattice Landau gauge is under way. We have encouraging results for the study of the color confinement mechanism. The authors thank Dr. Kikukawa and Dr. Hatsuda for suggestion of checking the Kugo- Ojima criterion, and S.F. thanks organizers of the workshop for stimulating discussion. This work was supported by High Energy Accelerator Research Organization, as KEK Supercomputer Project (Project No.98-34).

References

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[2] G. Dell'Antonio, D. Zwanziger, Commun. Math. Phys. 138 (1991) 291. [3] T. Maskawa, H. Nakajima, Prog. Theor. Phys. 60 (1978) 1526, Prog. Theor. Phys. 63 (1980) 641. [4] M. A. Semenov-Tyan-Shanskii, V. A. Franke, Zapiski Nauchnykh Seminarov, Leningrad- skage Otdelleniya Mathemat. Instituta im Steklov ANSSSR 120 (1982) 159. [5] D. Zwanziger, Nucl. Phys. B364 (1991) 127, Nucl. Phys. B412 (1994) 657.

[6] J.E. Hetrick and P.H. de Forcrand, Nucl. Phys B (Proc. Suppl.)63A-C,(1998) 838.

[7] H.Nakajima and S. Furui, Lattice 98 contribution, Bouldar(1998), hep- lat/9809080,9809081; Confinement III proceedings, June 1998, Jefferson Lab, NewPort. hep-lat/ 98090 78

[8] von Smekal, A. Hauck, R. Alkofer, Ann. Phys.267(1998) 1, hep-ph/9707327.

[9] J.E. Mandula and M. Ogilvie, Phys. Lett. B185 (1987) 127.

[10] D.B. Leinweber, J.I Skullerud, A.G. Williams and C. Parrinello, Phys. Rev.D58 031501, hep-lat/9803015; D.B. Leinweber, J.I Skullerud and A.G. Williams, hep-lat/9811027;

[11] M. Stingl, Z. Phys. A353 (1996) 423. [12] A. Cucchieri, Phys. Lett. B422 (1998) 233, hep-lat/9709015. [13] V.N. Gribov, Physica Scripta T15 (1987) 164. [14] K. Cahill and G. Herling, hep-lat/9809149

[15] T. Kugo and I. Ojima, Prog. Theor. Phys. Supp. 66 (1979) 1.

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