Screening mass of gluons in presence of external Abelian - PowerPoint PPT Presentation

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 − m k r k (1) m = 1 λ, λ – screening length (2) Example: QED el = 1 m 2 3 e 2 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 3 � F ( a ) F µν = µν t a , t a – generators of SU(N) group (5) a =1 // D. Gross, R. Pisarski, and L. Yaffe, Rev. Mod. Phys. 53 , 43 (1981) 1-loop calculations: el = 1 � N + N f � m 2 3 g 2 T 2 (6) 2 m 2 magn = 0 (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 µ 3 δ a 3 SU (2) High temperature: gH/T 2 << 1 Neutral gluon field: el ∼ g 2 T 2 � � m 2 � m 2 (9) 1 − C gH/T , magn = 0 // 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: el ∼ g 2 T 2 � � � m 2 m 2 magn ∼ g 2 T � 1 − C gH/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]] √ gH ∼ g 2 T m 2 magn ∼ g 4 T 2 ⇒ 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 neut = 0 m 2 ∼ g 4 T 2 m 2 ch ∼ g 2 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 � �O� ≈ 1 � D U O [ U ] e − S [ U ] − → O [ U k ] (11) Z K U k � � � D U e − S [ U ] , configurations U k are distributed with probability ∝ e − S [ Uk ] . � Z = D U = dU µ ( x ) , x,µ � � 1 − 1 � � Lattice Wilson action: S W = β N Re Tr U µν (12) µ>ν x 1 � a → 0 d 4 x F ( c ) µν ( x ) F ( c ) S W − → µν ( x ) (13) 4 g 2 µ ) U † ν ) U † U µν ( x ) = U µ ( x ) U ν ( x + ˆ µ ( x + ˆ 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 // T. A. DeGrand and D. Toussaint // M. Srednicki and L. Susskind, Nucl. Phys. B179 , 239 (1981) � � 1 − 1 � � S = β N Re Tr U µν ( n )Ξ µν ( n ) , n µ>ν Ξ µν ( n ) ∈ Z ( N ) Center of the SU(N) group: √ N Z ( N ) = { 1 · I } SU(2) case: Z (2) = { 1 · I, − 1 · I } 3 πi · I, 1 · I, e 3 πi · I } SU(3) case: Z (3) = { e − 2 2 Ξ µν ( n ) � = I if string ∩ U µν ( n ) Ξ µν ( n ) = − I if x = 0 , y = 0 , ∀ z, t N. V. Kolomoyets Screening mass of gluons in presence of external Abelian chromomagnetic field 10/28

Abelian Field Flux on the Lattice // S. Antropov et al. Plaquette: µ ) U † ν ) U † � U µν ( x ) = U µ ( x ) U ν ( x + ˆ µ ( x + ˆ ν ( x ) ⇒ U µν ( n ) = e ia 2 F µν ( n ) U µ ( n ) = e iaA µ ( n ) xy = e ia 2 ( H z + H ext ) = U xy e ia 2 H ext U ′ z z U ′ y (0 , n y , n z , n t ) = U y (0 , n y , n z , n t ) e iϕ ϕ = a 2 N x H ext z 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,      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 ) , � e iϕ 3 / 2    � 0  e iϕ = e iϕ 3 σ 3 / 2 = U µ ( n x , n y , n z , N t ) = U µ ( n x , n x , n z , 0) .  e − iϕ 3 / 2 0 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 i ( ϕ 3 + ϕ 8 / 3) / 2 0 0 √ e iϕ = e i ( ϕ 3 λ 3 + ϕ 8 λ 8 ) / 2 = e i ( − ϕ 3 + ϕ 8 / 3) / 2 0 0 (15)    √  e − iϕ 8 / 3 0 0 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 , 094506 (1999) [hep-lat/9903005] U µν ( n ) = e ia 2 F µν ( n ) U µ ( n ) = e iaA µ ( n ) a 2 F xy ( n ) → a 2 � F xy ( n ) + H ext � z A µ ( n ) → A µ ( n ) + H ext z xδ µ 2 t 3 / 2 U µ ( x + N x , y, z, t ) = U µ ( x, y, z, t ) ⇓ H ext z N x a = 2 πk, 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 ia 2 F µν ( n ) ≈ 1 + ia 2 F µν ( n ) , a → 0 (16) ∆ U µν ( n ) ≈ ia 2 ∆ F µν ( n ) (17) F ( ϕ ) = − T ln Z ( ϕ ) � D U e − S ( ϕ ) Z (0) , – free energy, Z ( ϕ ) = (18) ∂ n S ( ϕ ) � 1 ϕ n = S (0) + S ϕ ( ϕ ) � � S ( ϕ ) = S (0) + (19) � ∂ϕ n n ! � ϕ =0 n � Z ( ϕ ) D U e − S (0) e − S ϕ ( ϕ ) Z (0) = e − S ϕ ( ϕ ) Z ( ϕ ) = (20) f ( n ) ∼ ∂F ( n ) F ( ϕ ) = TS ϕ ( ϕ ) = T ( S ( ϕ ) − S (0)) (21) ∂β 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 β = 2 . 835 ( T ∼ 1 . 2 GeV) β = 3 . 020 ( T ∼ 1 . 9 GeV) External Abelian field flux ϕ = 0 . 08 ( ∼ 10 4 MeV 2 ) 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