QCD energy momentum tensor at finite temperature using gradient - PowerPoint PPT Presentation

QCD energy momentum tensor at finite temperature using gradient flow Yusuke Taniguchi for WHOT QCD collaboration S.Ejiri, R.Iwami, K.Kanaya, M.Kitazawa, M.Shirogane, H.Suzuki, Y.T, T.Umeda, N.Wakabayashi

Introduction Nuclear Matter pressure momentum stress direct measurement of thermodynamic quantity specific heat, viscosity, … From Nuclear Matter to Quark Matter Quark-Gluon-Plasma Energy momentum tensor Color Super- conductivity Nucleus Neutronstar Early Evolution of the Universe Temperature Density energy hot topics in QGP Poincare symmetry T µ ν   T 00 T 01 T 02 T 03 ? ? T 10 T 11 T 12 T 13   ?   T 20 T 21 T 22 T 23   T 30 T 31 T 32 T 33 •If we have T µ ν •Fluctuations and correlations of T µ ν

How to calculate T μν on lattice? Karsch coefficients problems non universal (No Poincare symmetry) •depends on: lattice action, operator terms in QCD Lagrangian when trace is taken Well established for E and P Measure expectation values on lattice terms in QCD Lagrangian Renormalization ψ ( x ) ← → δ µ ν ¯ δ µ ν ¯ ρσ ( x ) F a ρσ ( x ) δ µ ν F a D ψ ( x ) / ψ ( x ) ψ ( x ) ← → ← → ⇣ ⌘ ¯ µ ρ ( x ) F a νρ ( x ) F a ψ ( x ) D ν + γ ν ψ ( x ) γ µ D µ additive correction for δ µ ν ¯ ψ ( x ) ψ ( x )

Easier method for renormalization? Gradient Flow within smear field heat kernel Solution A kind of diffusion equation 2 t: flow time, dim=[length ] Flow the gauge field How to calculate T μν on lattice? Narayanan-Neuberger(2006) Lüscher(2009–) ∂ t A µ ( t, x ) = − δ S YM A µ ( t = 0 , x ) = A µ ( x ) δ A µ ∂ t A µ ( t, x ) = D ν G ν µ Z d 4 yK t ( x − y ) A µ ( y ) + interactions A µ ( t, x ) = K t ( x ) = e − x 2 / 4 t (4 π t ) D/ 2 √ 8 t

Gradient Flow as a renormalization scheme lattice operator MS scheme universal absorbed differences in lattice operator lattice action NP renormalized operator scale: operators are renormalized does not have UV divergence does not have contact term singularity How to calculate T μν on lattice? A great view point: flow + a→0 Re＜1ー ＞ Narayanan-Neuberger(2006), Lüscher(2010), Lüscher-Weisz(2011) Gauge operators with flowed field A µ ( t, x ) √ 8 t µ ν ( x, t ) F a µ ν F a . n e r e t i n fi

From flowed operator to proper operator NP renormalized operator which satisfies Need a matching factor like Wilson coefficient in flow scheme How to calculate T μν on lattice? WT identity at small t region Matching coefficients are calculable perturbatively 1 scale matching µ = √ 8 t H.Suzuki, PTEP 2013, 083B03 (2013) ⇢  � i� O 1 µ ν ( t, x ) − 1 h D E ˜ ˜ ˜ ˜ { T µ ν } WT ( x ) = lim c 1 ( t ) O 2 µ ν ( t, x ) + c 2 ( t ) O 2 µ ν ( t, x ) − O 2 µ ν ( t, x ) 4 t → 0 T =0 O 2 µ ν ( t, x ) = δ µ ν F a ρσ ( t, x ) O 1 µ ν ( t, x ) = F a νρ ( t, x ) ρσ F a µ ρ F a

Matching coefficients are calculable perturbatively at small t region How to calculate T μν on lattice? Matching coefficients at one loop From flowed operator to proper operator ✓ ◆ 1 1 9 γ − 18 ln 2 + 19 c 1 ( t ) = √ 8 t ) 2 − (4 π ) 2 4 g (1 / ¯ 1 33 c 2 ( t ) = (4 π ) 2 16 H.Suzuki, PTEP 2013, 083B03 (2013) ⇢  � i� O 1 µ ν ( t, x ) − 1 h D E ˜ ˜ ˜ ˜ { T µ ν } WT ( x ) = lim c 1 ( t ) O 2 µ ν ( t, x ) + c 2 ( t ) O 2 µ ν ( t, x ) − O 2 µ ν ( t, x ) 4 t → 0 T =0 O 2 µ ν ( t, x ) = δ µ ν F a ρσ ( t, x ) O 1 µ ν ( t, x ) = F a νρ ( t, x ) ρσ F a µ ρ F a

Three steps to calculate T μν appropriately defined on lattice 1. Flow the link variable 3. Multiply the coefficients and take t→0 limit How to calculate T μν on lattice? 2. Calculate expectation value of flowed operators ∂ t U µ ( t, x ) U † µ ( t, x ) = − g 2 0 ∂ x,µ S lat ( U ) O 2 µ ν ( t, x ) = δ µ ν F a ρσ ( t, x ) O 1 µ ν ( t, x ) = F a νρ ( t, x ) ρσ F a µ ρ F a ⇢  � i� O 1 µ ν ( t, x ) − 1 h D E ˜ ˜ ˜ ˜ { T µ ν } WT ( x ) = lim c 1 ( t ) O 2 µ ν ( t, x ) + c 2 ( t ) O 2 µ ν ( t, x ) − O 2 µ ν ( t, x ) 4 t → 0 T =0

Previous works and lessons quench flowed operator on lattice need to take a→0 limit the window region tamed at small t why we need t→0 +t 2 (dim8 operator) tamed at large t +t(dim6 operator) (dim4 operator) How to calculate T μν on lattice? BW12 T=1.66Tc (E+P)/T 4 ✓ a 2 ◆ O ( t ) O FlowQCD Collaboration t (2014-) t 2 � � O { T µ ν } ( x, t, a ) = { T µ ν } WT ( x ) + a 2 t + O ( a 2 T 2 , a 2 m 2 , a 2 Λ 2 QCD )

What’s new? Quarks included! No more renormalization is needed for composite op. Renormalization is needed for quark field flow the gauge field simultaneously Flow of quark field Lüscher, JHEP 1304, 123 (2013) ∂ t χ ( t, x ) = D µ D µ χ ( t, x ) χ ( t = 0 , x ) = ψ ( x ) χ ( t, x ) ← − ← − χ ( t = 0 , x ) = ¯ ¯ ψ ( x ) ∂ t ¯ χ ( t, x ) = ¯ D µ D µ χ R ( t, x ) = Z χ χ 0 ( t, x ) χ ( t, x ) χ ( t, x )) R = Z 2 (¯ χ (¯ χ ( t, x ) χ ( t, x )) 0

Three steps to calculate T μν 1. Flow the gauge and quark field 2. Calculate expectation value of flowed operators How to calculate T μν on lattice? 3. Multiply the coefficients and take t→0 limit ⇢ ⇣ D E ⌘ { T µ ν } q X ˜ 3 µ ν ( t, x ) − 2 ˜ ˜ 3 µ ν ( t, x ) − 2 ˜ O r O r O r O r WT ( x ) = lim c 3 ( t ) 4 µ ν ( t, x ) − 4 µ ν ( t, x ) t → 0 T =0 r = u,d,s ⌘� ⇣ D E ⌘ ⇣ D E X ˜ ˜ X ˜ ˜ O r O r c r O r O r + c 4 ( t ) 4 µ ν ( t, x ) − 4 µ ν ( t, x ) + 5 ( t ) 5 µ ν ( t, x ) − 5 µ ν ( t, x ) T =0 T =0 r = u,d,s r = u,d,s

How to calculate T μν on lattice? 3. Multiply the coefficients and take t→0 limit wave function renormalization VEV sub. ⇢ ⇣ D E ⌘ { T µ ν } q X ˜ 3 µ ν ( t, x ) − 2 ˜ ˜ 3 µ ν ( t, x ) − 2 ˜ O r O r O r O r WT ( x ) = lim c 3 ( t ) 4 µ ν ( t, x ) − 4 µ ν ( t, x ) t → 0 T =0 r = u,d,s ⌘� ⇣ D E ⌘ ⇣ D E ˜ ˜ ˜ ˜ X X O r O r c r O r O r + c 4 ( t ) 4 µ ν ( t, x ) − 4 µ ν ( t, x ) + 5 ( t ) 5 µ ν ( t, x ) − 5 µ ν ( t, x ) T =0 T =0 r = u,d,s r = u,d,s ← → ← → ⇣ ⌘ ˜ 3 µ ν ( t, x ) ≡ ϕ r ( t )¯ χ r ( t, x ) D ν + γ ν χ r ( t, x ) O r D µ γ µ χ r ( t, x ) ← → ˜ 4 µ ν ( t, x ) ≡ ϕ r ( t ) δ µ ν ¯ D χ r ( t, x ) / O r ˜ 5 µ ν ( t, x ) ≡ ϕ r ( t ) δ µ ν ¯ χ r ( t, x ) χ r ( t, x ) O r − 6 ϕ r ( t ) ≡ χ r ( t, x ) ← → D E (4 π ) 2 t 2 ¯ D χ r ( t, x ) / T =0 − 6 D E D E ˜ ˜ O r O r 2 3 µ ν ( t, x ) T =0 = 4 µ ν ( t, x ) T =0 = (4 π ) 2 t 2 δ µ ν

How to calculate T μν on lattice? 3. Multiply the coefficients and take t→0 limit ⇢ ⇣ D E ⌘ { T µ ν } q X ˜ 3 µ ν ( t, x ) − 2 ˜ ˜ 3 µ ν ( t, x ) − 2 ˜ O r O r O r O r WT ( x ) = lim c 3 ( t ) 4 µ ν ( t, x ) − 4 µ ν ( t, x ) t → 0 T =0 r = u,d,s ⌘� ⇣ D E ⌘ ⇣ D E ˜ ˜ ˜ ˜ X X O r O r c r O r O r + c 4 ( t ) 4 µ ν ( t, x ) − 4 µ ν ( t, x ) + 5 ( t ) 5 µ ν ( t, x ) − 5 µ ν ( t, x ) T =0 T =0 r = u,d,s r = u,d,s ( ◆) √ 8 t ) 2 ✓ c 3 ( t ) = 1 1 + ¯ g (1 / 2 + 4 3 ln(432) 4 (4 π ) 2 1 √ 8 t ) 2 c 4 ( t ) = (4 π ) 2 ¯ g (1 / √ ( ◆) 8 t ) 2 ✓ 1 + ¯ g (1 / 4 γ − 8 ln 2 + 14 3 + 4 √ c r 5 ( t ) = − ¯ m r (1 / 8 t ) 3 ln(432) (4 π ) 2 Makino-Suzuki, PTEP 2014, 063B02 (2014)

Numerical setups aT~1/N t artifact is small enough Iwasaki gauge action 3 28 x56 for T=0 3 32 xNt for T≠0 aT~1/N t artifact may be severe On an equal quark mass line NP improved Wilson fermion Nf=2+1 T=1/(aNt), Nt=16, 14, 12, 10, 8, 6, 4 Fixed scale method β=2.05 : a～0.07 [fm] β 8 12 10 16 14 6 m π /m ρ ~0.6 ( β =2.05) T[MeV] 100 200 300 400 500 T c m π m η ss ∼ 0 . 74 ∼ 0 . 6 m ρ m φ

T=279MeV T=174MeV t/a 2 t→0 limit by linear extrapolation t/a 2 t/a 2 T=199MeV T=465MeV T=233MeV p/T 4 T=349MeV 8 8 8 6 6 6 4 4 4 T ii /3T 4 T ii /3T 4 T ii /3T 4 2 2 2 0 0 0 -2 -2 -2 -4 -4 -4 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 t/a 2 t/a 2 t/a 2 8 8 8 6 6 6 4 4 4 T ii /3T 4 T ii /3T 4 T ii /3T 4 2 2 2 0 0 0 -2 -2 -2 -4 -4 -4 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 t/a 2 t/a 2 t/a 2

p/T 4 gradient flow as a function of T aT=1/Nt artifact is severe 10 gradient flow T-integration 8 6 p/T 4 4 2 WHOT-QCD, Phys. Rev. D 85, 094508 (2012) integration method 0 0 100 200 300 400 500 600 T (MeV)

e/T 4 t→0 limit by linear extrapolation t/a 2 t/a 2 T=174MeV T=199MeV T=233MeV T=349MeV T=465MeV T=279MeV t/a 2 0 0 0 -5 -5 -5 T 00 /T 4 T 00 /T 4 T 00 /T 4 -10 -10 -10 -15 -15 -15 -20 -20 -20 -25 -25 -25 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 t/a 2 t/a 2 t/a 2 0 0 0 -5 -5 -5 T 00 /T 4 T 00 /T 4 T 00 /T 4 -10 -10 -10 -15 -15 -15 -20 -20 -20 -25 -25 -25 0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 0.5 1 1.5 2 t/a 2 t/a 2 t/a 2

e/T 4 gradient flow as a function of T aT=1/Nt artifact is severe 25 gradient flow T-integration 20 15 e/T 4 10 WHOT-QCD, Phys. Rev. D 85, 094508 (2012) integration method 5 0 0 100 200 300 400 500 600 T (MeV)