Thermodynamics for SU(2) gauge theory using gradient flow

Transcription

1 theory using Takehiro Hirakida, Etsuko Itou,2, Hiroaki Kouno 3 Kyushu U., RCNP, Kochi U. 2, Saga U. 3 NFQCD28, YITP, 5. June, 28 arxiv:85.76 [hep-lat] / 25

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4 SU(2) gauge theory SU(2) gauge theory (pure SU(2), 2-color QCD) has properties SU(3) gauge theory does. Asymptotic freedom Quark confinement Chiral symmetry breaking etc... Different point between SU(2) and SU(3)... the order of phase transition (pure SU(2): second, pure SU(N c 3): first) Numerical cost of lattice simulation: SU(2) < SU(3) SU(2) gauge theory is a good model for the methodological study on SU(3) gauge theory 4 / 25

5 Gradient flow Yang-Mills equation on lattice... } t V t (x, µ) = g { 2 x,µ S W V t (x, µ), V t (x, µ) = U(x, µ). t= g : bare coupling, t: flow-time, U(x, µ): link variable, S W : Wilson action. V t renormalized field. t typical energy scale of the renormalization. Composite operators with V t are UV finite in the positive flow-time (t > ). M. Luscher, JHEP 8 (2) 7. 5 / 25

6 Energy-momentum tensor Energy-momentum tensor (EMT)... Lattice regularization breaks the translational symmetry. Wave function renormalization is needed. We can directly calculate the renormalized EMT with 2 { Tµν R Uµν = lim + δ } µν [E E ], t α U 4α E U µν = G µρ G νρ δ µν 4 G ρσg ρσ, E = 4 G µνg µν. G µν : the field strength consisting of V t, α U, α E : coefficients calculated in the one-loop order of running coupling. EMT entropy density, trace anomaly, etc... 2 H. Suzuki, PTEP 23, 83B3 (23) 6 / 25

7 Our motivation. Previous study of SU(2) gauge theory... Thermodynamic quantities: measured using integral method only 3 carry out using in this work 2. Calculating the shear viscosity... Measuring of correlation function of the EMT is first step. A huge number of the configuration is needed (in previous work). Reduce using Gradient flow method(?) Prepare to calculate shear viscosity 3 P. Giudice and S. Piemonte, Eur. Phys. J. C 77 (27) no.2, 82 7 / 25

8 Target of this study Calculating thermodynamics for pure SU(2) using method. Scale setting (t scale) Compare with present scale settings. 2. Calculate EMT at finite temperature entropy density, trace anomaly, energy density, pressure eq. of state (EOS) Compare with Integral method Hard-Thermal-Loop (HTL) analysis SU(N c 3) theory 8 / 25

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10 Observable... Action density E = 4 G µνg µν, t 2 E(t) N 2 c. Reference scale: t 2 E(t) =. t=t a natural scaling-down of the SU(3) case 4. Configuration generation Wilson-plaquette action, N s = N t = 32 sweep = pseudo-heatbath + 2 over-relaxation sweep separation between measurements β # of Conf Gradient flow utilize Plaquette and Clover-leaf definition t/a 2 [., 32.], t/a 2 =. 4 M. Luscher, JHEP 8 (2) 7. / 25

11 t 2 <E(t)>..5 beta=2.5 plaquette beta=2.5 clover beta=2.7 plaquette beta=2.7 clover beta=2.85 plaquette beta=2.85 clover t/a 2 β t /a 2.83(2).839(3) 3.522() β t /a (36).96(2) 6.95(7) Best fit function (t /a 2.vs. β) ln(t /a 2 ) = (β 2.6).74(β 2.6) 2. / 25

12 Compare with other scale setting with r scale 5 (r c scale 6 ) : define via q q force 8t 8t =.62(86)(4), =.26(7)(7), r r c 8t =.3(43)(2)[fm]. r =.5[fm], r c =.26[fm]. with scale setting using the string tension 7 (a 2 σ) ln(a 2 /a 2 ) Eq.(3.) of JHEP 57 (25) 43 Our result take reference lattice spacing a at β = β both functions are available σ consistent β we cover T 2T c 5 R. Sommer, Nucl. Phys. B 4, 839 (994). 6 S. Necco and R. Sommer, Nucl. Phys. B 622, 328 (22). 7 M. Caselle, A. Nada and M. Panero, JHEP 57 (25) / 25

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14 Procedure and Simulation setup Steps to calculate renormalized EMT 8. Generate configuration at t = on Ns 3 N τ 2. Solve eq. in a 8t R 3. Construct renormalized EMT at each t 4. Carry out an extrapolation, first a, next t Simulation setup Wilson-plaquette action N s /N τ = 4, N τ = 6, 8,, 2 # of Conf. for each parameter: 2 sweep = pseudo-heatbath+n t over-relaxation sweep separation between measurements Gradient flow utilize Clover-leaf definition for action density t/a 2 [., 5.], t/a 2 =. 8 M. Asakawa et al., Phys. Rev. D 9, no., 5 (24) 4 / 25

15 Observables β.vs. T /T c for each N t 9 T /T c N τ = 6 N τ = 8 N τ = N τ = entropy density (s), trace anomaly ( ) st = ε + P = T R T R 44, = ε 3P = 4 µ= T R µµ. ε: energy density, P: pressure Compute ε and P from s and. 9 the critical β on N t = 6 from [J. Engels, J. Fingberg and D. E. Miller, Nucl. Phys. B 387 (992) 5.] 5 / 25

16 Flow-time dependence of s and (ε-3p)/t a> sqrt(8t) for Nτ=2 for Nτ= for Nτ=8 for Nτ=6 β=2.46, Nτ=6 β=2.55, Nτ=8 β=2.62, Nτ= β=2.67, Nτ=2 2a> sqrt(8t) for Nτ=2 β=2.63, Nτ=2 β=2.67, Nτ=2 β=2.82, Nτ=2 (ε+p)/t (8t) /2 T (8t) /2 T left panel: T =.2T c right panel: N t = 2 Fiducial window: 2a 8t N t a/2 lower: Using clover-leaf operator upper: Considering the effect of smearing by 6 / 25

17 a limit (ε-3p)/t (8t) /2 T=.25 T=.98Tc T=.8Tc T=.76Tc (8t) /2 T=.35 2 (ε+p)/t /Nτ /Nτ 2 8tT [.25,.4], δ( 8tT ) =. Each data is adopted closest to the fixed 8tT Constant extrapolation: to calculate the central value Linear extrapolation: to estimate the systematic error with constant extrapolation Constant.vs. Linear... 2 σ consistent 7 / 25

18 t limit (ε - 3P)/T T/Tc =.98 T/Tc =.8 T/Tc =.76 (ε + P)/T tT tT 2 8tT [.25,.4], δ( 8tT ) =. Carry out both constant- and linear-extrapolation We take the central result which is the better χ 2 /d.o.f Constant.vs. Linear... 2 σ consistent 8 / 25

19 Result 2.5 /T 4 s/t 3 /Τ 4 (EPJ C77 (27) 82, Nτ=5) ε/t ε=3p SB limit 2 T/Tc P/T 4 Left panel: s/t 4 (black symbol), /T 4 (red symbol) /T 4... consistent with the integral method in T > T c Right panel: ε/t 4.vs. P/T 4 (EOS) in T > T c Toward to SB limit (P/T 4, ε/t 4 ) = (π 2 /5, π 2 /5) 7 8% of SB limit for T 2T c NOT describe two-color QGP around T 2T c P. Giudice and S. Piemonte, Eur. Phys. J. C 77 (27) no.2, 82 9 / 25

20 Compare with HTL analysis ε(t)/ε SB HTL (NNLO with Λ error) HTL (NNLO with µ/2πt=.5,2.) Our results 2 3 T/Tc (T/Tc) 2 (Τ)/Τ 4. EPJ C77 (27) 82 (Nτ=5) 2 4 T/Tc Hard-Thermal-Loop (HTL) analysis... 2-color case in NNLO Left panel: ε/ε SB Our result is consistent with HTL in T > T c Right panel: (T /T c ) 2 /T 4 plateau and approaches to HTL result in.5t c T J. O. Andersen et al., JHEP 8 (2) 3. 2 / 25

21 Compare with SU(N c 3) theory ( /T 4 )/(SB limit of p SB /T 4 ) 2.5 Our result SU(2) (EPJ C77 (27) 82) SU(3) SU(4) SU(5) (NPB 564 (2) 33) SU(6) SU(8).5.5 (Tc/T) 2 Dashed lines: trace anomaly for SU(N c 3) theory 2 In high T : leading correction of /T 4 is /T 2 SU(N c 3): linear behavior until (T c /T ) 2.9 SU(2): Different behavior from the difference of order of phase transition 2 M. Panero, Phys. Rev. Lett. 3 (29) / 25

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23 and Future works We investigate the thermodynamics of the pure SU(2) gauge theory. Scale setting t 2 E(t) =. for SU(2) t=t 2. Obtaining s/t 3, /T 4, EOS Consistent with integral method and HTL analysis Future works Calculation of shear viscosities for pure SU(2) systematic study on T - and N c -dependence Constructing effective model of two-color QCD analyze the physics in 2-color chemical potential 23 / 25

24 Back Up: Topological Charge using 4 2 Q configuration number Figure: Result at β = 2.85, t/a 2 = 32 Topological charge Q Q = 32π 2 d 4 x ϵ µνρσ TrG µν G ρσ Q takes an almost integer-value autocorrelation can be negligible in our data sets 24 / 25

25 Back Up: Renormalized Polyakov loop It is believed that universality class of pure SU(2) is same as that of 3-D Ising model Renormalization condition 3 L R (T =.76Tc) = L R (T) Nτ=6 Nτ=8 Nτ= Nτ= T/T c Critical exponent (.3265(3) in 3-D Ising model) N τ = :.59(3) Nτ = 2:.242(3) 3 S. Borsanyi et al., Phys. Lett. B 73 (22) / 25