Propagator of the SU ( 2 ) gauge boson on a 3-dimensional lattice in - PowerPoint PPT Presentation

Propagator of the SU ( 2 ) gauge boson on a 3-dimensional lattice in the Landau gauge V. G. Bornyakov 2 , 3 , V. K. Mitrjushkin 1 , 3 , and R. N. Rogalyov 2 1. Joint Insitute of Nuclear Research 2. Institute for High-Energy Physics 3. Insitute of Theoretical and Experimental Physics, 23.09.2010

◮ Motivation ◮ Center symmetry ◮ Absolute and minimal Landau gauges ◮ Gauge fixing algorithm and technical details ◮ An approach to the absolute gauge ◮ Momentum dependence of the propagator ◮ The role of Z 3 2 sectors ◮ The effect of Gribov copies ◮ The effects of finite volume

Infrared behavior of the gluon propagator in the Landau gauge is of interest because ◮ Propagator is needed for calculation of physical quantities; ◮ The Kugo-Ojima and Gribov-Zwanziger confinement criteria are formulated in terms of propagator behavior in the Euclidean domain. If the Osterwalder-Schrader reflection positivity is violated for the gluon fields, one cannot construct the respective Hilbert space with positive metric. The gluon fields are not associated with asymptotic states. = ⇒ gluons are confined

◮ It is of interest to compare lattice and continuum results for the propagator ◮ Gauge fixing on a lattice is also of intererst because the respective continuum gauge theory is defined only in a particular gauge.

The gluon propagator in the Landau gauge: � δ µν − p µ p ν � D ab µν ( p ) = δ ab D ( p ) p 2 The Functional Renormalization Group (FRG) and the Schwinger–Dyson Equations (SDE) imply at p → 0 [Fischer, Pawlowsky, 2006; Alkofer etc]: ◮ scaling solution: D ( p ) ≃ ( p 2 ) 2 κ + ( 2 − D ) / 2 D Gh ( p ) ≃ ( p 2 ) − 1 − κ , (1) ◮ massive solution Z D ( p ) ≃ const D Gh ( p ) ≃ ( p 2 ) , (2)

� � S = 4 1 − 1 � 2T r U P g 2 P = x ,µ,ν where U x ,µ → Λ † Λ : x U x ,µ Λ x +ˆ µ , µ,ν U † ν,µ U † U P = U x ,µ U x +ˆ x ,ν x +ˆ We fix the absolute Landau U x ,µ ∈ SU ( 2 ) , D = 3 gauge by finding the global maximum of the functional 3 � F [ U ] = r U x ,µ , T (5) � U x ,µ = u 0 + i u a σ a , (3) x ,µ a = 1 Stationarity condition: µ = − 2 u a µ A a ga , (4) ∂ ν A a ν = 0 .

Gribov copies: residual gauge orbit R ( U ) = {U m |U m = U g m , δ F [ U m ] = 0 } . ◮ Minimal Landau gauge: to select any element ∈ R ◮ Absolute Landau gauge: to select the element with the maximal value of F [ U m ] . D ( p ) � = D ( p )!!! Problem of degenerate maxima.

Center symmetry: Z 2 : U x ,µ → − U x ,µ Z L ( x 1 , x 2 ) → − L ( x 1 , x 2 ) x 1 , x 2 N τ � � � � U ( x + j ˆ A c µ ( z )Γ c dz L ( x 1 , x 2 ) = Tr 3 , 3 ) = P exp iga . j = 1

We extend the gauge group Z 3 G − → G E = G × Z 2 , (6) where G = { Ω( x ) } , Ω( x ) ∈ SU ( 2 ) : U x ,µ → Ω † x U x ,µ Ω x +ˆ µ , . (7) Z 3 The configuration space {U} is divided into 8 Z 2 sectors, according to the signs of La La � � L ( x µ , x ν ) x µ = a x ν = a

Gauge fixing algorithm ◮ We generate a configuration U 0 using the heat bath method, ◮ perform Z Z 3 2 transformations and obtain U 1 , ..., U 7 associated with U 0 . All of them have the same Wilson action, however, they cannot be transformed into each other by a proper gauge transformation. Nevertheless, we consider them as Gribov copies corresponding to the extended gauge group.

◮ In the s th sector, we produce N B elements ¯ V sk of the gauge orbit, associated with U s . ◮ F sk ( g ) ≡ F ( V g sk ) (8) is the functional on G . Its maxima provide Gribov copies. ◮ we begin with the “Simulated Annealing” (SA) method and then proceed to the overrelaxation (OR) algorithm. SA is used for preliminary maximization of F sk ( g ) , the OR algorithm is more efficient at the final stage.

The SA algorithm generates gauge transformations g ( x ) by MC iterations with a statistical weight proportional to exp ( 4 V F sk [ G ] / T ) . T is an auxiliary parameter which is gradually decreased to maximize F sk [ g ] . [Bogolubsky et al., 2007; Schemel et al., 2006]: T init = 1 . 3 , T final = 0 . 01 After each quasi-equilibrium sweep, including both heatbath and microcanonical updates, T is decreased by equal intervals. The final SA temperature is fixed such that the quantity 3 � � � � � A a µ/ 2 ; µ − A a max (9) � � x +ˆ x − ˆ µ/ 2 ; µ x , a � � µ = 1 decreases monotonously during OR for the majority of gauge fixing trials. The number of the SA steps is set equal to 3000.

We use the standard Los-Alamos type overrelaxation with the parameter value ω = 1 . 7. The number of iterations: 500 ÷ 700 at L = 32 1500 ÷ 3000 at L = 80; in few cases, several times greater. The precision of gauge fixing: 3 � � � � � A a µ/ 2 ; µ − A a � < 10 − 7 max (10) � � x +ˆ x − ˆ µ/ 2 ; µ x , a � µ = 1 The configuration ¯ V sk with the greatest value of F sk [ g ] is referred to as “the k th Gribov copy in the s th sector”.

◮ We put the configurations ¯ V sk in the linear order: ¯ V sk → ¯ V r . There are two natural arrangements: V ( 1 ) ¯ = V sk , where r = N copy ( s − 1 ) + k ; (11) r V ( 2 ) ¯ = V sk ( j ) , where r = 8 ( k − 1 ) + s ; (12) r r runs from 1 to N tot copy = 8 N copy . ◮ Now we can take a part P n of the residual gauge orbit R ( U 0 ) consisting of n elements, 1 ≤ n ≤ N tot copy . ◮ Let F [¯ V r ] approaches its maximum on P n at ¯ V ¯ r .

◮ We evaluate (measure) the value of the propagator using ¯ V ¯ r . ◮ We repeat this procedure N meas times; the initial configuration U 0 ( j ) for each measurement being separated by 200 sweeps from the previous one in order to be considered as statistically independent. ◮ Then we take an average over the measurements.

L N meas N copy aL [Fm] F max 32 800 16 5.38 0 . 9192939 ± 0 . 0000173 40 400 20 6.73 0 . 9193018 ± 0 . 0000177 48 905 20 8.08 0 . 9193386 ± 0 . 0000091 56 788 20 9.43 0 . 9193515 ± 0 . 0000080 64 474 20 10.8 0 . 9193404 ± 0 . 0000078 72 578 35 12.1 0 . 9193656 ± 0 . 0000065 80 557 20 13.5 0 . 9193527 ± 0 . 0000055 Table: a √ σ ≈ 0 . 567, √ σ = 440 MeV; a = 0 . 168 Fm ∼ ( 1 . 17 GeV ) − 1 ; 1 GeV − 1 ≃ 0 . 197 Fm.

0.91932 0.9193 0.91928 F(n copy ) 0.91926 0.91924 0.91922 L=40: F(n copy ) 0.9192 0 20 40 60 80 100 120 140 160 n copy

0.919375 0.91937 0.919365 0.91936 F(n copy ) 0.919355 0.91935 0.919345 0.91934 0.919335 L=72: F(n copy ) 0.91933 0.919325 0 50 100 150 200 250 300 n copy

1 D(0) versus n c 0.95 L=40: D(p min ) versus n c D(2 p min ) versus n c 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0 25 50 75 100 125 150 n c An approach to the absolute Landau gauge: first we run Z 3 2 sectors

1 D(0) versus n c 0.95 L=80: D(p min ) versus n c D(2 p min ) versus n c 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0 25 50 75 100 125 150 n c An approach to the absolute Landau gauge: first we run Z 3 2 sectors

1 D(0) versus n c 0.95 L=40: D(p min ) versus n c D(2 p min ) versus n c 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0 25 50 75 100 125 150 n c An approach to the absolute Landau gauge: first we run within a single sector

1 D(0) versus n c 0.95 L=80: D(p min ) versus n c D(2 p min ) versus n c 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0 25 50 75 100 125 150 n c An approach to the absolute Landau gauge: first we run within a single sector

0.9 0.8 L = 80 0.7 0.6 D(p) 0.5 0.4 0.3 0.2 D(p) = (c 1 + c 2 p 2 c 3 ) / (c 4 + (c 5 + p 2 ) 2 ) 0.1 0 0 0.5 1 1.5 2 p 2 Fit parameters: c 1 = 0 . 122 ± 0 . 004 , c 2 = 0 . 668 ± 0 . 011 , c 3 = 0 . 563 ± 0 . 012 , c 4 = 0 . 184 ± 0 . 007 , c 5 = 0 . 335 ± 0 . 011, χ 2 = 1 . 33 N dof

0.9 0.8 L = 64 0.7 0.6 D(p) 0.5 0.4 0.3 2 + (p-c 3 ) 2 ) 0.2 D(p) = c 1 / (c 2 0.1 0 0 0.2 0.4 0.6 0.8 1 p Conventional parametrization by mass does not work

However, to study the infrared asymptotics more precisely, we should consider the infinite-volume limit. To take an example, D ( 0 ) versus L = Na Taking Gribov copies into account results in a substantial decrease of D ( 0 ) , D ( p min ) , D ( 2 p min ) . An analysis performed on a finite lattice with the neglect of such decrease may lead to erroneous conclusions on infrared behavior of the gluon propagator.

0.35 G 80 (p 2 ) L = 80: 0.3 zero 0.25 G 80 (p 2 ) 0.2 0.15 0.1 0.05 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 p 2 The effect of Gribov copies G L ( p ) = D ( first ) ( p ) − D ( best ) ( p ) (13) D ( best ) ( p )

0.35 G 56 (p 2 ) 0.3 L = 56: zero 0.25 G 56 (p 2 ) 0.2 0.15 0.1 0.05 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 p 2 G L ( p ) = D ( first ) ( p ) − D ( best ) ( p ) (14) D ( best ) ( p ) Maas [0808.3047]: G 56 ( 0 ) ≃ 0 . 1 , β = 4 . 24 In our study, the effect is 3 times greater.

0.4 0.35 0.3 0.25 G L (0) 0.2 0.15 0.1 0.05 0 0 0.005 0.01 0.015 0.02 0.025 0.03 1/L The effect of Gribov copies on the zero-momentum propagator as a function of volume