Screening mass of gluons in presence of external Abelian chromomagnetic field

Screening mass of gluons in presence of external Abelian chromomagnetic field N. V. Kolomoyets, V. V. Skalozub Dnepropetrovsk National University Ukraine December 5, 2018 N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 1/28

Magnetic Mass Magnetic (electric) mass shows how fast magetic (electric) field decreases with distance in plasma. F µν = C(r) e mk r k (1) m = 1, λ screening length (2) λ Example: QED m 2 el = 1 3 e2 T 2 electric (Debye) mass (3) m 2 magn = 0 magnetic mass (4) Screened Long range N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 2/28

Magnetic Mass in SU(N) Gauge Theory F µν = 3 a=1 F (a) µν t a, t a generators of SU(N) group (5) 1-loop calculations: m 2 el = 1 ( 3 g2 T 2 N + N ) f 2 m 2 magn = 0 // D. Gross, R. Pisarski, and L. Yaffe, Rev. Mod. Phys. 53, 43 (1981) (6) (7) Higher orders, nonperturbative calculations: m 2 magn g 4 T 2 (8) m magn = 0 is not excluded N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 3/28

Hypothesis: not all color components of chromomagnetic field contribute to the magnetic mass N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 4/28

Magnetic Mass in the Presence of External Field External chromomagnetic field: H a µ = Hδ µ3 δ a3 High temperature: gh/t 2 << 1 SU(2) Neutral gluon field: m 2 el g2 T 2 ( 1 C gh/t ), m 2 magn = 0 (9) // M. Bordag and V. Skalozub, Phys. Rev. D 75, 125003 (2007) [hep-th/0611256] // S. Antropov, M. Bordag, V. Demchik and V. Skalozub, Int. J. Mod. Phys. A 26, 4831 (2011) [arxiv:1011.3147 [hep-ph]] Color-charged gluon fields: m 2 el g2 T 2 ( 1 C gh/t gh g 2 T m 2 magn g 4 T 2 ), m 2 magn g 2 T gh (10) // M. Bordag and V. Skalozub, Phys. Rev. D 77, 105013 (2008) [arxiv:0801.2306 [hep-th]] // M. Bordag and V. Skalozub, Phys. Rev. D 85, 065018 (2012) [arxiv:1201.1978 [hep-th]] Spontaneous field generation N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 5/28

Magnetic Mass on the Lattice SU(2) 1 T. A. DeGrand and D. Toussaint, The Behavior of Nonabelian Magnetic Fields at High Temperature, Phys. Rev. D 25, 526 (1982) Screening of the chromomagnetic field of the monopole-antimonopole string was shown Color structure could not be clarified by this method 2 S. Antropov, M. Bordag, V. Demchik and V. Skalozub, Long range chromomagnetic fields at high temperature, Int. J. Mod. Phys. A 26, 4831 (2011) [arxiv:1011.3147 [hep-ph]] Zero magnetic mass of the Abelian chromomagnetic field was shown Non-Abelian components of the chromomagnetic field were not investigated N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 6/28

Magetic Mass: Analytical Calculations vs Lattice m 0 m ch 0 m m 2 g 4 T 2 neut = 0 m 2 ch g2 T gh Perturb. On the lattice Perturb. On the lattice Perturb. The aim of this investigation: to show that m magn is produced by the charged components of the gluon field on the lattice N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 7/28

Quantum Gluodynamics on the Lattice Continuous Minkovsky space-time Euclidean 4D discrete lattice Continuous operators Discrete operators on the lattice Gluon fields SU(N) matrices at the links of the lattice Expectation value of a measured quantity O: O = 1 DU O[U]e S[U] O 1 O[U k ] (11) Z K U k Z = DU e S[U], DU = du µ(x), configurations U k are distributed with probability e S[U k ]. x,µ Lattice Wilson action: S W = β µ>ν a 0 S W 1 4g 2 x [ 1 1 N Re Tr U µν ] (12) d 4 x F (c) µν (x)f (c) µν (x) (13) U µν (x) = U µ (x)u ν (x + ˆµ)U µ(x + ˆν)U ν(x) plaquette. (14) N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 8/28

Idea of the investigation: Two external chromomagnetic fields are introduced on the lattice: field of monopole-antimonopole string; Abelan field flux. Screening of combination of that fields is investigated. N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 9/28

Monopole-Antimonopole String on the Lattice S = β n µ>ν Ξ µν (n) Z(N) // T. A. DeGrand and D. Toussaint // M. Srednicki and L. Susskind, Nucl. Phys. B179, 239 (1981) [ 1 1 ] N Re Tr U µν(n)ξ µν (n), Center of the SU(N) group: Z(N) = { N 1 I} SU(2) case: Z(2) = {1 I, 1 I} Ξ µν (n) I Ξ µν (n) = I if string U µν (n) if x = 0, y = 0, z, t SU(3) case: Z(3) = { e 2 3 πi I, 1 I, e 2 3 πi I} N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 10/28

Abelian Field Flux on the Lattice Plaquette: U µν (x) = U µ (x)u ν (x + ˆµ)U µ(x + ˆν)U ν(x) U µ (n) = e iaaµ(n) // S. Antropov et al. } U µν (n) = e ia2 F µν(n) U xy = e ia2 (H z+h ext z ) = U xy e ia2 H ext z U y(0, n y, n z, n t ) = U y (0, n y, n z, n t ) e iϕ Twisted boundary conditions: U y (N x, n y, n z, n t ) = U y (0, n y, n z, n t ) e iϕ, U µ (N x, n y, n z, n t ) = U µ (0, n y, n z, n t ), µ y, ϕ = a 2 N x H ext z U µ (n x, N y, n z, n t ) = U µ (n x, 0, n z, n t ), U µ (n x, n y, N z, n t ) = U µ (n x, n y, 0, n t ), U µ (n x, n y, n z, N t ) = U µ (n x, n x, n z, 0). e iϕ = e iϕ 3σ 3 /2 = ( e iϕ 3 /2 0 0 e iϕ 3/2 ) N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 11/28

Abelian Field Flux on the Lattice SU(3) e iϕ = e i(ϕ 3λ 3 +ϕ 8 λ 8 )/2 = e i(ϕ 3+ϕ 8 / 3)/2 0 0 0 e i( ϕ 3+ϕ 8 / 3)/2 0 0 0 e iϕ 8/ 3 (15) N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 12/28

Ext. Field through the Flux vs Ext. Field through the Strength TBC Cosmai & Cea // P. Cea and L. Cosmai, Phys. Rev. D 60, U µν (n) = e ia2 F µν(n) a 2 F xy (n) a 2 [ F xy (n) + Hz ext ] U µ (n) = e iaaµ(n) 094506 (1999) [hep-lat/9903005] A µ (n) A µ (n) + H ext z xδ µ2 t 3 /2 U µ (x + N x, y, z, t) = U µ (x, y, z, t) Hz ext N x a = 2πk, k Z 2 N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 13/28

Outline of the Investigation Lattices: N t N 3, N t = const Measured quantity: U = Re Tr U µν Investigated quantity: f(n) = U field U 0 U µν (n) = e ia2 F µν(n) 1 + ia 2 F µν (n), a 0 (16) U µν (n) ia 2 F µν (n) F (ϕ) = T ln Z(ϕ), free energy, Z(ϕ) = Z(0) S(ϕ) = S(0) + 1 n S(ϕ) n! ϕ n n Z(ϕ) = DU e S(0) e Sϕ(ϕ) F (ϕ) = T S ϕ (ϕ) = T (S(ϕ) S(0)) (17) DU e S(ϕ) (18) ϕ=0 ϕ n = S(0) + S ϕ (ϕ) (19) f(n) F (n) β Z(ϕ) Z(0) = e Sϕ(ϕ) (20) (21) N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 14/28

Outline of the Investigation Lattices: N t N 3, N t = const Measured quantity: U = Re Tr U µν Investigated quantity: f(n) = U field U 0 Possibilities for f: f 1/N 2 flux tubes, the flux is conserved; f 1/N 4 Coulombic behavior, flux spreads out over the available area; f e kn 2 screening of the field; k = m 2 magn; f 1/N spontaneous field generation, flux increases with distance. Simulations are performed in absence of external Abelian field flux ϕ; in presence of external Abelian field flux ϕ: ϕ is directed parallel to the monopole-antimonopole string. N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 15/28

Simulations Setup; SU(2) Lattices used: 4 N 3, N = 6, 8,..., 72 External Abelian field flux ϕ = 0.08 ( 10 4 MeV 2 ) β = 2.835 (T 1.2 GeV) β = 3.020 (T 1.9 GeV) β = 3.091 (T 2.3 GeV) Simulations are performed with the QCDGPU program (https://github.com/vadimdi/qcdgpu, V. Demchik, N. K., Comp. Sc. and Appl.,1, 1 (2014) [arxiv:hep-lat/1310.7087]) N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 16/28

χ 2 -analysis of the Data log f 7 8 9 10 11 12 13 6 8 10 12 14 N f i = U field U 0 i The data are fitted through minimization of χ 2 function: K χ 2 [y i log f(n i ; a)] 2 (a) =, (22) i=1 σ 2 i y i = log f i, f(n i ; a) = A N b e kn q. χ 2 min = χ2 (â) χ 2 ν; ν = K L; L = Length a Hypothesis testing: H 0 : f(n i ; a) describes the data; χ 2 min χ2 ν;0.05 H 1 : f(n i ; a) does not describe the data. χ 2 min > χ2 ν;0.05 Functions describing the data χ 2 = χ 2 (a) χ 2 min χ2 L χ 2 (a) χ 2 min + χ2 L;0.05 95% CIs for a CIs for screening parameters N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 17/28

SU(2) Results: Data at ϕ = 0 f(n) = U field U 0 0.0010 0.0008 ϕ = 0, β = 2. 835 ϕ = 0, β = 3. 020 ϕ = 0, β = 3. 091 7 8 ϕ = 0, β = 2. 835 ϕ = 0, β = 3. 020 ϕ = 0, β = 3. 091 0.0006 9 f log f 10 0.0004 11 0.0002 12 0.0000 6 8 10 12 14 N 13 6 8 10 12 14 N N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 18/28

SU(2) Results: Fitting at ϕ = 0 Function β = 2.835 χ 2 min χ2 ν;0.05 a / r ˆk ± 2σ CI 10 2 β = 3.020 χ 2 min χ2 ν;0.05 a / r ˆk ± 2σ CI 10 2 β = 3.091 χ 2 min χ2 ν;0.05 a / r ˆk ± 2σ CI 10 2 A/N 137 9.49 509 9.49 190 9.49 A/N 2 80.4 9.49 247 9.49 102 9.49 A/N 4 14.0 9.49 19.5 9.49 9.53 9.49 A e kn 0.40 7.81 63.9 ± 10.9 2.19 7.81 54.4 ± 4.5 0.98 7.81 56.8 ± 8.0 A e kn2 3.18 7.81 3.68 ± 0.63 11.8 7.81 3.33 ± 0.28 10.3 7.81 3.18 ± 0.45 (A/N) e kn 0.60 7.81 51.7 ± 10.9 4.14 7.81 41.5 ± 4.5 0.64 7.81 44.8 ± 8.0 (A/N) e kn2 1.49 7.81 2.99 ± 0.63 4.09 7.81 2.55 ± 0.28 5.10 7.81 2.52 ± 0.45 (A/N 2 ) e kn 0.99 7.81 39.5 ± 10.9 7.29 7.81 28.6 ± 4.5 0.77 7.81 32.7 ± 8.0 (A/N 2 ) e kn2 0.63 7.81 2.30 ± 0.63 2.16 7.81 1.78 ± 0.28 1.89 7.81 1.85 ± 0.45 (A/N 4 ) e kn 2.32 7.81 15.1 ± 10.9 17.2 7.81 2.80 ± 4.52 2.45 7.81 8.65 ± 7.96 (A/N 4 ) e kn2 1.36 7.81 0.91 ± 0.63 15.5 7.81 0.23 ± 0.28 1.58 7.81 0.52 ± 0.45 N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 19/28

SU(2) Results: Fitting at ϕ = 0 Function β = 2.835 χ 2 min χ2 ν;0.05 a / r ˆk ± 2σ CI 10 2 β = 3.020 χ 2 min χ2 ν;0.05 a / r ˆk ± 2σ CI 10 2 β = 3.091 χ 2 min χ2 ν;0.05 a / r ˆk ± 2σ CI 10 2 A/N 137 9.49 509 9.49 190 9.49 A/N 2 80.4 9.49 247 9.49 102 9.49 A/N 4 14.0 9.49 19.5 9.49 9.53 9.49 A e kn 0.40 7.81 63.9 ± 10.9 2.19 7.81 54.4 ± 4.5 0.98 7.81 56.8 ± 8.0 A e kn2 3.18 7.81 3.68 ± 0.63 11.8 7.81 3.33 ± 0.28 10.3 7.81 3.18 ± 0.45 (A/N) e kn 0.60 7.81 51.7 ± 10.9 4.14 7.81 41.5 ± 4.5 0.64 7.81 44.8 ± 8.0 (A/N) e kn2 1.49 7.81 2.99 ± 0.63 4.09 7.81 2.55 ± 0.28 5.10 7.81 2.52 ± 0.45 (A/N 2 ) e kn 0.99 7.81 39.5 ± 10.9 7.29 7.81 28.6 ± 4.5 0.77 7.81 32.7 ± 8.0 (A/N 2 ) e kn2 0.63 7.81 2.30 ± 0.63 2.16 7.81 1.78 ± 0.28 1.89 7.81 1.85 ± 0.45 (A/N 4 ) e kn 2.32 7.81 15.1 ± 10.9 17.2 7.81 2.80 ± 4.52 2.45 7.81 8.65 ± 7.96 (A/N 4 ) e kn2 1.36 7.81 0.91 ± 0.63 15.5 7.81 0.23 ± 0.28 1.58 7.81 0.52 ± 0.45 N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 20/28

SU(2) Results: Data at ϕ = 0.08 f(n) = U field U 0 0.0010 0.0008 0.0006 ϕ = 0, β = 2. 835 ϕ = 0, β = 3. 020 ϕ = 0, β = 3. 091 ϕ = 0. 08, β = 2. 835 ϕ = 0. 08, β = 3. 020 ϕ = 0. 08, β = 3. 091 7 8 9 ϕ = 0, β = 2. 835 ϕ = 0, β = 3. 020 ϕ = 0, β = 3. 091 ϕ = 0. 08, β = 2. 835 ϕ = 0. 08, β = 3. 020 ϕ = 0. 08, β = 3. 091 f log f 10 0.0004 11 0.0002 12 0.0000 10 20 30 40 50 60 70 N 13 10 20 30 40 50 60 70 N N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 21/28

SU(2) Results: Fitting at ϕ = 0.08 β = 2.835 β = 3.020 β = 3.091 Function χ 2 min χ2 ν;0.05 a /r χ 2 min χ2 ν;0.05 a /r χ 2 min χ2 ν;0.05 a /r A/N b 91.2 16.9 170 15.5 223 18.3 (A/N b ) e kn 30.0 15.5 44.9 14.1 69.0 16.9 (A/N b ) e kn2 47.2 15.5 73.7 14.1 118 16.9 A e B/N e kn 5.33 15.5 7.14 14.1 7.00 16.9 B = 20.3 ± 2.64 k = (1.09 ± 0.91) 10 2 B = 20.1 ± 1.84 k = (1.08 ± 0.75) 10 2 B = 21.3 ± 2.01 k = (6.90 ± 6.76) 10 3 18 0.045 16 0.040 0.035 14 0.030 B, GeV 1 12 10 k, GeV 0.025 0.020 0.015 8 0.010 0.005 6 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 T, GeV 0.000 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 T, GeV N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 22/28

Comparison of the results f(n) = A e B/N e kn χ 2 min β = 2.835 χ2 ν;0.05 a /r χ 2 min β = 3.020 χ2 ν;0.05 a /r χ 2 min β = 3.091 χ2 ν;0.05 a /r ϕ = 0 0.36 5.99 1.34 5.99 0.64 5.99 B = 2.81 ± 37.6 k = (6.83 ± 5.76) 10 1 B = 4.95 ± 14.2 k = (6.29 ± 2.46) 10 1 B = 5.04 ± 23.9 k = (4.92 ± 3.66) 10 1 ϕ = 0.08 5.33 15.5 7.14 14.1 7.00 16.9 B = 20.3 ± 2.64 k = (1.09 ± 0.91) 10 2 B = 20.1 ± 1.84 k = (1.08 ± 0.75) 10 2 B = 21.3 ± 2.01 k = (6.90 ± 6.76) 10 3 30 2.5 B, GeV 1 20 10 0 10 20 k, GeV 2.0 1.5 1.0 0.5 m 0 = 1.26 ± 0.41 GeV m 0.08 = (1.83 ± 0.87) 10 2 GeV at 95% CL 30 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 T, GeV 0.0 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 T, GeV N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 23/28

SU(3) Results: Data at ϕ = 0 4 N 3 lattices, N = 6,..., 88 f(l s ) β = 6.4 (T 580 MeV) - 4-6 - 8-10 0 20 40 60 80 L s χ 2 = 20.2 χ 2 ν;0.05 = 32.7 b = 2.00 ± 0.02 flux tubes χ 2 > χ 2 ν;0.05 rejection at 95% CL N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 24/28

SU(3) Results: Data at ϕ = 0 4 N 3 lattices, N = 6,..., 88 f(l s ) β = 6.4 (T 580 MeV) - 8.4-8.6-8.8-4 - 9.0-9.2-6 - 8 60 65 70 75 80 85 χ 2 = 3.05 χ 2 ν;0.05 = 12.6 b = 1.99 ± 0.23-10 0 20 40 60 80 L s - 8.4-8.6 χ 2 = 20.2 χ 2 ν;0.05 = 32.7 b = 2.00 ± 0.02 flux tubes χ 2 > χ 2 ν;0.05 rejection at 95% CL - 8.8-9.0-9.2 60 65 70 75 80 85 χ 2 = 2.24 χ 2 ν;0.05 = 12.6 b = (2.70 ± 0.32) 10 2 N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 25/28

Comparison with Literature SU(3) B. Grossman, S. Gupta, U. M. Heller and F. Karsch, Glueball - like screening masses in pure SU(3) at finite temperatures, Nucl. Phys. B 417, 289 (1994) [hep-lat/9309007]. External field sources are not introduced m el and m magn are measured through Plyakov loop correlators 2m magn = 5.8(4)T @ T = 1.5T c N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 26/28

Conclusions Both monopole-antimonopole string and external Abelian field flux are introduced on the lattice. Results of the previous investigations for SU(2) gauge group are reproduced. In SU(2) it is shown that adding of the Abelian field flux weakens the screening of the string field. This confirms that for the Abelian field m magn = 0; m magn of the monopole-antimonopole string field is produced by its non-abelian components. In SU(3) formation of flux tubes is obtained. N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 27/28

Conclusions Both monopole-antimonopole string and external Abelian field flux are introduced on the lattice. Results of the previous investigations for SU(2) gauge group are reproduced. In SU(2) it is shown that adding of the Abelian field flux weakens the screening of the string field. This confirms that for the Abelian field m magn = 0; m magn of the monopole-antimonopole string field is produced by its non-abelian components. In SU(3) formation of flux tubes is obtained. Thank you for your attention! N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 28/28