arXiv:hep-lat/0509033v2 24 Nov 2005
Hideo Nakajima∗ and Sadataka Furui† ∗
Dept. of Infor. Sci., Utsunomiya Univ., Utsunomiya, 320-8585 Japan E-mail:
[email protected] † School of Sci.& Engr., Teikyo Univ., Utsunomiya, 320-8551 Japan E-mail:
[email protected]
The Kugo-Ojima confinement criterion is verified in the unquenched Landau gauge QCD simulation. The valence quark propagator of the Kogut-Susskind fermion with use of the fermion action including the Naik term and the staple contribution is calculated on MILC Asqtad unquenched gauge configurations, and it shows infrared suppression of the quark propagator.
XXIIIrd International Symposium on Lattice Field Theory 25-30 July 2005 Trinity College, Dublin, Ireland ∗ Speaker.
c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence.
http://pos.sissa.it/
PoS(LAT2005)302
Color Confinement in lattice Landau gauge with unquenched Wilson and KS fermions
Color Confinement in lattice Landau gauge
Hideo Nakajima
1. Introduction
Under infinitesimal gauge transformation g−1 δ g = ε , its variation reads for either defintion as ∆FU (g) = −2h∂ Ag |ε i + hε | − ∂ D(U g)|ε i + · · ·,
where the covariant derivativative Dµ (U ) for two options reads commonly as Dµ (Ux,µ )φ = S(Ux,µ )∂µ φ + [Ax,µ , φ¯ ]
φ (x + µ ) + φ (x) where ∂µ φ = φ (x + µ ) − φ (x), and φ¯ = Stationality of the optimizing function 2 implies Landau gauge, the local minimum implies Gribov Region[2] and the global minimum implies Fundamental modular(FM) region[3]. We performed simulations on quenched configurations tabulated in Table 1[9, 10] and unquenched configurations of JLQCD[13], CP-PACS[14], Columbia University[15] and MILC[16] tabulated in Table 2[11]. β 6
1/a(GeV) 1.97
6.4
3.66
6.45
3.87
L 16 24 32 32 48 56 56
aL( f m) 1.60 2.40 3.20 1.72 2.59 3.02 2.86
definition of A U-linear/ logU U-linear/ logU U-linear/ logU U-linear/ logU U-linear/ logU U-linear/ logU logU
Table 1: Configurations used in the quenched QCD simulation
2. The Kugo-Ojima theory The Kugo-Ojima confinement criterion[1] is given by the fact that the parameter c defined as = −δ ab c in the eq(2.1) becomes 1 qµ qν ab 2 1 1 b −ip(x−y) a† (2.1) htr Λ Dµ (δµν − 2 )u (q ) = ∑ e [Aν , Λ ] i. q V x,y −∂ D xy
uab (0)
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In the lattice Landau gauge QCD simulation, we adopt two types of the gauge field definitions[7]: logU type: Ux,µ = eAx,µ , A†x,µ = −Ax,µ , 1 U linear type: Ax,µ = (Ux,µ −Ux,µ † )|trl p. , 2 Λa where trl p. implies traceless part. ( Aµ (x) = i ∑ Aµ a (x) √ , trΛa Λb = δ ab ) 2 a The corresponding optimizing functions are logU type: FU (g) = ||Ag ||2 = ∑x,µ tr Ag x,µ † Agx,µ , † U linear type: FU (g) = ∑x,µ tr 2 − ( Ux,g µ +Ux,g µ ) .
Color Confinement in lattice Landau gauge
JLQCD CP-PACS CU MILCc
Ksea 0.1340 0.1355 0.1357 0.1382
VW I I amVW ud /ams 0.134 0.093 0.087 0.020 0.025 0.010 0.040/0.050 0.007/0.050 0.0124/0.031 0.0062/0.031
Nf 2 2 2 2 2 2 2+1 2+1 2+1 2+1
1/a(GeV) 2.221 2.221 1.834 1.834 1.140 2.1 1.64 1.64 2.19 2.19
Ls 20 20 24 24 16 16 20 20 28 28
Lt 48 48 48 48 32 32 64 64 96 96
aLs (fm) 1.78 1.78 2.58 2.58 2.77 1.50 2.41 2.41 2.52 2.52
Table 2: Configurations used in the unquenched QCD simulation
The Zwanziger horizon condition[3] coincides with Kugo-Ojima criterion provided the covariant derivative approaches the naive continuum limit, i.e., e/d = 1[8]. We observe that in the quenched simulation c saturated at about 0.8, while in the unquenched simulation it is consistent with 1.
JLQCD CP-PACS CU MILCc MILC f
Ksea or β Ksea =0.1340 Ksea =0.1355 Ksea =0.1357 Ksea =0.1382 β =5.415 β =5.7 β =6.76 β =6.83 β =7.09 β =7.11
cx 0.89(9) 1.01(22) 0.86(6) 0.89(9) 0.84(7) 0.95(26) 1.04(11) 0.99(14) 1.06(13) 1.05(13)
ct 0.72(4) 0.67(5) 0.76(4) 0.72(4) 0.74(4) 0.58(6) 0.74(3) 0.75(3) 0.76(3) 0.76(3)
c 0.85(11) 0.92(24) 0.84(7) 0.85(11) 0.81(8) 0.86(28) 0.97(16) 0.93(16) 0.99(17) 0.98(17)
e/d 0.9296(2) 0.9340(1) 0.9388(1) 0.9409(1) 0.9242(3) 0.9414(2) 0.9325(1) 0.9339(1) 0.9409(1) 0.9412(1)
h -0.08(11) -0.01(24) -0.10(6) -0.05(9) -0.11(8) -0.08(28) 0.03(16) -0.00(16) 0.04(17) 0.04(17)
Table 3: The Kugo-Ojima parameter for the polarization along the spacial directions cx and that along the time direction ct and the average c, trace divided by the dimension e/d, horizon function deviation h of the unquenched Wilson fermion(JLQCD, CP-PACS), and KS fermion (MILCc ,CU,MILC f ). The logU definition of the gauge field is adopted.
3. Quark propagator In the unquenched lattice simulation with the improved KS fermion action, the MILC collaboration has replaced the link variables by fattening[5] Uµ (x) → c1Uµ (x) + ∑ w3 Sµν (x) + · · · (3)
ν
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PoS(LAT2005)302
MILC f
β 5.2 5.2 2.1 2.1 5.415 5.7 6.83(βimp ) 6.76(βimp ) 7.11(βimp ) 7.09(βimp )
Hideo Nakajima
Color Confinement in lattice Landau gauge
Hideo Nakajima
(3)
where Sµν is the staple contribution (3)
Sµν (x) = Uν (x)Uµ (x + νˆ )Uν† (x + µˆ )
D(U / )x,y =
1 4 (3) ηµ (x)sign(µ )[(c1 Uµ (x) + w3 ∑ Sµν (x))δy,x+µˆ ∑ 2 µ =−4 ν 6= µ
+c3Uµ (x)Uµ (x + µˆ )Uµ (x + 2µˆ )δy,x+3µˆ ]
where w3 = 9/64, c1 = 9/32 and c3 = −1/24, and ηµ (x) is given as ( ( µ ) 1 ν
