Plasmon mass scale in classical nonequilibrium gauge theory in two and three dimensions Jarkko Peuron(University of Jyväskylä) In collaboration with: T. Lappi (University of Jyväskylä) June 28 th 2018 Based on: Phys. Rev. D95 (2017) no.1, 014025, Phys.Rev. D97 (2018) no.3, 034017
Physics picture Figure : A. Kurkela, Nucl.Phys. A956 (2016) 136-143. Figure : E. Iancu 1105.0751 [hep-ph] Left: space-time evolution of an URHIC. Right: the validity ranges of classical and kinetic theory. We are interested in the quasiparticle interpretation of the classical theory. Simulate initial overoccupied gluon fields using classical Yang-Mills equations using real time lattice techniques. Jarkko Peuron Plasmon mass scale in classical nonequilibrium gauge theory in two and three dimensions 2 / 15
Quasiparticles in nonequilibrium plasma Solid state physics: plasmons, collective longitudinal excitations in electron gas. Here: strongly interacting gluon plasma, both transverse and longitudinal excitations. Quasiparticles are gluons, measure their mass using different methods. �� d 3 k 2 − 1 � (1) ε = 2 N c ( 2 π ) 3 ω ( k ) f ( k ) , � d 3 k � = 1 � 2 + | B i | 2 � �� � E i � . (2) ( 2 π ) 3 2 Keeping only quadratic terms in gauge potential we get | E C ( k ) | 2 � k 2 � 1 � 1 ω ( k ) | A C ( k ) | 2 f A + E ( k ) = + . (3) 2 − 1 � V ω ( k ) 4 N c Jarkko Peuron Plasmon mass scale in classical nonequilibrium gauge theory in two and three dimensions 3 / 15
Measuring plasmon mass Use 3 methods to determine their mass Effective dispersion relation (DR) at zero momentum �� � 2 � � � ˙ E ( k ) | E ( k ) | 2 � ≈ ω 2 (4) � Add uniform chromoelectric field, measure oscillations (UE , Kurkela & Moore Phys.Rev. D86 (2012) 056008). Perturbation theory, Hard Thermal Loops (HTL): � d 3 k pl = 4 f ( k ) ω 2 3 g 2 N c (5) ( 2 π ) 3 | k | Jarkko Peuron Plasmon mass scale in classical nonequilibrium gauge theory in two and three dimensions 4 / 15
Quasiparticle spectrum, 3D The initial quasiparticle spectrum satisfies � − k 2 � f ( k , t = 0 ) = n 0 k (6) ∆ exp . g 2 2 ∆ 2 n 0 = 1, � =0.3, 256 3 , t � =57 0.8 f(k), t � =57 0.7 0.6 f(k), t � =0 k/ � f(k) 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 k/ � Jarkko Peuron Plasmon mass scale in classical nonequilibrium gauge theory in two and three dimensions 5 / 15
Time dependence, 3D 128 3 128 3 1 n 0 =0.1 n 0 =2 (t Δ ) 2/7 ω pl2 /(n 0 Δ 2 ) 0.9 1 n 0 =0.5 n 0= 3 (t � ) 2/7 � pl2 /(n 0 � 2 ) n 0 =1 n 0 =5 0.8 0.7 0.6 0.5 0.5 0.4 HTL DR 3 0.3 0.25 UE DR 1 0.2 10 100 100 200 300 400 500 600 t Δ t � Left: All methods agree on the proposed power law ω 2 pl ∼ t − 2 / 7 (Kurkela & Moore Phys.Rev. D86 (2012) 056008) at late times. Left: DR method sensitive to fit cut off (3 and 1 different cutoffs in k / ∆ ). Right: More dense systems enter the scaling regime faster. Jarkko Peuron Plasmon mass scale in classical nonequilibrium gauge theory in two and three dimensions 6 / 15
Dependence on occupation number, 3D t � = 90, 128 3 0.35 � =0.3, HTL 0.3 � =0.3, UE � =0.3, DR � pl2 /(n 0 � 2 ) 0.25 0.2 0.15 0.1 0.05 0.1 0.2 0.4 0.8 1.6 3.2 n 0 Decreasing trend in n 0 explained by self similar scaling (Berges et. al. Phys.Rev. D89 (2014) no.11, 114007), which classical Yang-Mills theory exhibits self-similar evolution at late times. More dense systems enter this scaling regime faster and have spent larger fraction of their history on the scaling solution. Jarkko Peuron Plasmon mass scale in classical nonequilibrium gauge theory in two and three dimensions 7 / 15
Lattice cutoff dependence, 3D UV cutoff IR cutoff, two datasets for each method correspond to different UV cutoff. n0=1.0, t � =57 n0=1.0, t � =57 0.3 UE 0.28 UE 0.26 0.4 DR HTL 0.24 0.35 HTL HTL 0.22 � 2 / � 2 DR � 2 / � 2 0.3 UE 0.2 DR 0.25 0.18 0.2 0.16 0.14 0.15 0.12 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 50 100 150 200 250 � a s � L We see no IR cutoff dependence HTL method sensitive to UV cutoff. Agrees with UE in the continuum limit. Jarkko Peuron Plasmon mass scale in classical nonequilibrium gauge theory in two and three dimensions 8 / 15
Two-dimensional systems In the following consider 2D system in a fixed box. Mimic boost invariant expanding system. Use three dimensional lattice with N z = 1. The differences between different definitions of the occupation numbers are larger in 2D. Use two different estimates, f A and f EA . Due to gauge fixing ambiguities, can not compare directly with the initial scales. Use also autocorrelation to measure frequency in UE method. Jarkko Peuron Plasmon mass scale in classical nonequilibrium gauge theory in two and three dimensions 9 / 15
Occupation number and momentum scale in 2D Problem: After construction of links and gauge fixing the initial quasiparticle spectrum is deformed. n 0 = 1 , Qt = 0 , QL = 307 Qt = 0 , QL = 614 . 4 , Qa = 0 . 3 0 . 35 0 . 35 n 0 = 0 . 5 n 0 = 2 . 0 Q = 0 . 1 Q = 0 . 4 0 . 30 n 0 = 1 . 0 n 0 = 2 . 5 0 . 30 Q = 0 . 2 Q = 0 . 5 n 0 = 1 . 5 n 0 = 3 . 0 Q = 0 . 3 Q = 0 . 6 0 . 25 0 . 25 g 2 f E + A ( k ) g 2 f A + E ( k ) 0 . 20 0 . 20 n 0 0 . 15 0 . 15 0 . 10 0 . 10 0 . 05 0 . 05 0 . 00 0 . 00 0 1 2 3 4 5 0 1 2 3 4 5 k k Q Q Solution: Define Q and n 0 gauge invariantly so that they match to the initial condition for a dilute system. Jarkko Peuron Plasmon mass scale in classical nonequilibrium gauge theory in two and three dimensions 10 / 15
Gauge invariant observables Define � Tr ( D × B ) 2 � eff = 1 Q 2 � � B 2 �� 2 Tr π g 2 ε 2 d n eff 0 ≈ 2 − 1 ) Q 3 ( N c eff QL = 307 . 0 , Qt = 0 QL = 307 . 0 0 . 9 n 0 = 0 . 05 n 0 = 0 . 55 n 0 = 0 . 2 n 0 = 0 . 8 0 . 8 n 0 = 0 . 1 n 0 = 0 . 6 0 . 8 n 0 = 0 . 15 n 0 = 0 . 65 n 0 = 0 . 4 n 0 = 1 . 0 n 0 = 0 . 2 n 0 = 0 . 7 0 . 7 n 0 = 0 . 25 n 0 = 0 . 75 0 . 7 n 0 = 0 . 6 n 0 = 0 . 3 n 0 = 0 . 8 n 0 = 0 . 35 n 0 = 0 . 85 n 0 = 0 . 4 n 0 = 0 . 9 0 . 6 0 . 6 0 ( t = 0) Q eff ( t = 0) n 0 = 0 . 45 n 0 = 0 . 95 n 0 = 0 . 5 n 0 = 1 . 0 0 . 5 0 . 5 0 . 4 n eff 0 . 4 0 . 3 0 . 3 0 . 2 0 . 2 0 . 1 0 . 0 0 . 1 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 Q Q Jarkko Peuron Plasmon mass scale in classical nonequilibrium gauge theory in two and three dimensions 11 / 15
Time dependence, 2D Time dependence using all Time dependence at later methods times with HTL-A method n eff 0 = 0 . 22 , Q eff L = 321 . 0 , Q eff a s = 0 . 31 Q eff L = 264 . 0 , Q eff a s = 0 . 26 n eff 0 = 0 . 09 n eff 0 = 0 . 2 n eff 0 = 0 . 28 UE autocor. DR, T. n eff n eff − 1 0 = 0 . 12 0 = 0 . 24 t UE fit. DR, L 3 n eff n eff 0 = 0 . 16 0 = 0 . 27 − 1 HTL A. t 10 − 1 3 HTL EA. eff pl eff pl ω 2 Q 2 ω 2 Q 2 10 − 1 500 1000 1500 1800 10 2 Q eff t Q eff t − 1 / 3 power law at late Time-evolution of ω 2 pl consistent with t times. Jarkko Peuron Plasmon mass scale in classical nonequilibrium gauge theory in two and three dimensions 12 / 15
Dependence on occupation number, 2D ( Q eff t ) UE = 224 . 0 , Q eff L = 465 . 0 , Q eff a s = 0 . 26 1 . 6 HTL-A DR,L HTL-EA UE fit. 1 . 4 DR,T UE autocor. 1 . 2 eff n eff 0 1 . 0 pl ω 2 Q 2 0 . 8 0 . 6 0 . 4 0 . 05 0 . 10 0 . 15 0 . 20 0 . 25 0 . 30 n eff 0 ω 2 pl Observe similar trend as in 3D. Faster decrease in for Q 2 eff n eff 0 larger occupation number. Jarkko Peuron Plasmon mass scale in classical nonequilibrium gauge theory in two and three dimensions 13 / 15
Cutoff dependence, 2D IR cutoff UV cutoff n eff 0 = 0 . 32 , Q eff a s = 0 . 37 , ( Q eff t ) UE = 203 . 0 n eff 0 = 0 . 15 , Q eff L = 254 . 0 , ( Q eff t ) UE = 175 . 0 0 . 40 0 . 30 UE autocor. HTL EA. UE autocor. HTL EA. 0 . 35 UE fit. DR, T. UE fit. DR, T. 0 . 25 HTL A. DR, L HTL A. DR, L 0 . 30 0 . 20 0 . 25 eff eff ω 2 pl ω 2 pl Q 2 Q 2 0 . 15 0 . 20 0 . 10 0 . 15 0 . 05 0 . 10 0 . 05 0 . 00 0 100 200 300 400 500 600 700 800 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 Q eff a s Q eff L No IR cutoff dependence All results seem to increase when UV cutoff is taken to zero. The continuum limit does not seem to be divergent. Jarkko Peuron Plasmon mass scale in classical nonequilibrium gauge theory in two and three dimensions 14 / 15
Conclusions We have Measured quasiparticle mass in nonequilibrium gluon plasma Studied the time dependence of the mass. Results consistent pl ∼ t − 2 / 7 (3D) ω 2 pl ∼ t − 1 / 3 (2D). with ω 2 Studied the dependence of the mass on occupation number. ω 2 pl n 0 Q 2 falls faster for more dense systems. The UE and HTL seem to agree for 3D systems in the continuum limit. DR agrees with other methods within a factor of 2. In the future we would like to Measure the quasiparticle properties in anisotropic and expanding geometries using linear response analysis (See talk by K. Boguslavski). Jarkko Peuron Plasmon mass scale in classical nonequilibrium gauge theory in two and three dimensions 15 / 15
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