Energy-momentum tensor correlators in hot Yang-Mills theory Aleksi Vuorinen University of Helsinki Micro-workshop on analytic properties of thermal correlators University of Oxford, 6.3.2017 Mikko Laine, Mikko Veps¨ al¨ ainen, AV, 1008.3263, 1011.4439 Mikko Laine, AV, Yan Zhu, 1108.1259 York Schr¨ oder, Mikko Veps¨ al¨ ainen, AV, Yan Zhu, 1109.6548 AV, Yan Zhu, 1212.3818, 1502.02556 Yan Zhu, Ongoing work Aleksi Vuorinen (University of Helsinki) Thermal correlators from pQCD Oxford, 6.3.2017 1 / 40
Table of contents Motivation 1 Transport coefficients and correlators Perturbative input Correlators from perturbation theory 2 Basics of thermal Green’s functions Our setup Computational techniques Results 3 Operator Product Expansions Euclidean correlators Spectral densities Conclusions and outlook 4 Aleksi Vuorinen (University of Helsinki) Thermal correlators from pQCD Oxford, 6.3.2017 2 / 40
Motivation Table of contents Motivation 1 Transport coefficients and correlators Perturbative input Correlators from perturbation theory 2 Basics of thermal Green’s functions Our setup Computational techniques Results 3 Operator Product Expansions Euclidean correlators Spectral densities Conclusions and outlook 4 Aleksi Vuorinen (University of Helsinki) Thermal correlators from pQCD Oxford, 6.3.2017 3 / 40
Motivation Transport coefficients and correlators Background: Heavy ion collisions Expansion of thermalizing plasma surprisingly well described in terms of a low energy effective theory — hydrodynamics UV physics encoded in transport coefficients: η , ζ ,.. Aleksi Vuorinen (University of Helsinki) Thermal correlators from pQCD Oxford, 6.3.2017 4 / 40
Motivation Transport coefficients and correlators Transport coefficients from data Observation: Hydro results particularly sensitive to shear viscosity RHIC data indicated extremely low viscosity; recently attempts towards extracting η ( T ) from RHIC+LHC data (Eskola et al.) Related general question: Can the QGP be characterized as strongly/weakly coupled at RHIC/LHC? Ultimate answers only from non-perturbative calculations in QCD Aleksi Vuorinen (University of Helsinki) Thermal correlators from pQCD Oxford, 6.3.2017 5 / 40
Motivation Transport coefficients and correlators Why pQCD I: Transport coefficients from lattice Kubo formulas: Transport coeffs. from IR limit of retarded Minkowski correlators — viscosities from those of energy momentum tensor T µν : ρ 12 , 12 ( ω ) 1 ω Im D R η = lim 12 , 12 ( ω, k = 0 ) ≡ lim ω ω → 0 ω → 0 � � ρ ii , jj ( ω ) π 1 π ω Im D R ζ = lim ii , jj ( ω, k = 0 ) ≡ lim 9 9 ω ω → 0 ω → 0 ij ij Problem: Lattice can only measure Euclidean correlators → Spectral density available only through inversion of � ∞ π ρ ( ω ) cosh ( β − 2 τ ) ω d ω 2 G ( τ ) = sinh βω 0 2 ∴ To extract IR limit of ρ , need to understand its behavior also at ω � π T — perturbative input needed Aleksi Vuorinen (University of Helsinki) Thermal correlators from pQCD Oxford, 6.3.2017 6 / 40
Motivation Transport coefficients and correlators Why pQCD I: Transport coefficients from lattice Analytic continuation from imaginary time correlator possible with precise lattice data and perturbative result Successful example: nonperturbative flavor current spectral density and flavor diffusion coefficient [Burnier, Laine, 1201.1994] Aleksi Vuorinen (University of Helsinki) Thermal correlators from pQCD Oxford, 6.3.2017 7 / 40
Motivation Transport coefficients and correlators Why pQCD II: Comparisons with lattice and AdS Euclidean correlators provide direct information about medium ⇒ Comparisons between lattice QCD, pQCD and AdS/CFT valuable Iqbal, Meyer (0909.0582): Lattice data for spatial correlators of Tr F 2 µν in agreement with strongly coupled N = 4 SYM, while leading order pQCD result completely off. How about NLO? Aleksi Vuorinen (University of Helsinki) Thermal correlators from pQCD Oxford, 6.3.2017 8 / 40
Motivation Transport coefficients and correlators Why pQCD II: Comparisons with lattice and AdS Euclidean correlators provide direct information about medium ⇒ Comparisons between lattice QCD, pQCD and AdS/CFT valuable Another curious result of Iqbal, Meyer (0909.0582): UV behavior of µν and − Tr F µν � Tr F 2 F µν correlators on the lattice completely different even though leading order OPEs identical Aleksi Vuorinen (University of Helsinki) Thermal correlators from pQCD Oxford, 6.3.2017 8 / 40
Motivation Perturbative input Challenge for perturbation theory Goal: Perturbatively evaluate Euclidean and Minkowskian correlators of T µν in hot Yang-Mills theory to Inspect Operator Product Expansions (OPEs) at finite temperature 1 Compare behavior of perturbative time-averaged spatial 2 correlators to lattice QCD and AdS/CFT Use spectral densities at zero wave vector to aid the determination 3 of transport coefficients from lattice data Aleksi Vuorinen (University of Helsinki) Thermal correlators from pQCD Oxford, 6.3.2017 9 / 40
Motivation Perturbative input Challenge for perturbation theory Goal: Perturbatively evaluate Euclidean and Minkowskian correlators of T µν in hot Yang-Mills theory to Inspect Operator Product Expansions (OPEs) at finite temperature 1 Compare behavior of perturbative time-averaged spatial 2 correlators to lattice QCD and AdS/CFT Use spectral densities at zero wave vector to aid the determination 3 of transport coefficients from lattice data Concretely: Specialize to scalar, pseudoscalar and shear operators θ ≡ c θ g 2 B F a µν F a χ ≡ c χ g 2 B F a µν � F a η ≡ 2 c η T 12 = − 2 c η F a 1 µ F a µν , µν , 2 µ and proceed from 1 to 3 working at NLO. Aleksi Vuorinen (University of Helsinki) Thermal correlators from pQCD Oxford, 6.3.2017 9 / 40
Motivation Perturbative input Challenge for perturbation theory Goal: Perturbatively evaluate Euclidean and Minkowskian correlators of T µν in hot Yang-Mills theory to Inspect Operator Product Expansions (OPEs) at finite temperature 1 Compare behavior of perturbative time-averaged spatial 2 correlators to lattice QCD and AdS/CFT Use spectral densities at zero wave vector to aid the determination 3 of transport coefficients from lattice data When can perturbation theory be expected to work? � �� ¯ � 2 + � ¯ � 2 1 ¯ Λ x , T ≃ Λ x Λ T ∼ x 2 + ( 2 π T ) 2 At least, if either x ≪ 1 / Λ QCD ( ω ≫ Λ QCD ) or T ≫ Λ QCD Aleksi Vuorinen (University of Helsinki) Thermal correlators from pQCD Oxford, 6.3.2017 9 / 40
Correlators from perturbation theory Table of contents Motivation 1 Transport coefficients and correlators Perturbative input Correlators from perturbation theory 2 Basics of thermal Green’s functions Our setup Computational techniques Results 3 Operator Product Expansions Euclidean correlators Spectral densities Conclusions and outlook 4 Aleksi Vuorinen (University of Helsinki) Thermal correlators from pQCD Oxford, 6.3.2017 10 / 40
Correlators from perturbation theory Basics of thermal Green’s functions Correlation functions: generalities Plenitude of different Minkowskian correlators: � e i K·X � � Π > φ † φ α ( X ) ˆ ˆ αβ ( K ) β ( 0 ) , ≡ X � e i K·X � � Π < φ † ˆ β ( 0 ) ˆ αβ ( K ) ≡ φ α ( X ) , X � e i K·X � 1 � �� φ α ( X ) , ˆ ˆ φ † ρ αβ ( K ) ≡ β ( 0 ) , 2 X � e i K·X � 1 � �� φ † φ α ( X ) , ˆ ˆ ∆ αβ ( K ) ≡ β ( 0 ) , 2 X � e i K·X �� � � Π R φ † φ α ( X ) , ˆ ˆ αβ ( K ) i β ( 0 ) θ ( t ) , ≡ X � e i K·X � � � � Π A φ † φ α ( X ) , ˆ ˆ αβ ( K ) ≡ i − β ( 0 ) θ ( − t ) , X � e i K·X � � Π T φ α ( X ) ˆ ˆ φ † β ( 0 ) θ ( t ) + ˆ φ † β ( 0 ) ˆ αβ ( K ) ≡ φ α ( X ) θ ( − t ) X One Euclidean correlator, computable on the lattice: � e iK · X � � Π E φ † φ α ( X ) ˆ ˆ αβ ( K ) β ( 0 ) ≡ X Aleksi Vuorinen (University of Helsinki) Thermal correlators from pQCD Oxford, 6.3.2017 11 / 40
Correlators from perturbation theory Basics of thermal Green’s functions Correlation functions: generalities However, in thermal equilibrium all correlators related through ρ : 2 n B ( k 0 ) ρ αβ ( K ) , Π < αβ ( K ) = e β k 0 e β k 0 − 1 ρ αβ ( K ) = 2 [ 1 + n B ( k 0 )] ρ αβ ( K ) , Π > αβ ( K ) = 2 � � � � 1 Π > αβ ( K ) + Π < 1 + 2 n B ( k 0 ) ∆ αβ ( K ) = αβ ( K ) = ρ αβ ( K ) . 2 Im Π R Im Π A αβ ( K ) ρ αβ ( K ) , αβ ( K ) = − ρ αβ ( K ) , = Π T − i Π R αβ ( K ) + Π < αβ ( K ) = αβ ( K ) , � ∞ d k 0 ρ αβ ( k 0 , k ) Π E αβ ( K ) = k 0 − ik n π −∞ Aleksi Vuorinen (University of Helsinki) Thermal correlators from pQCD Oxford, 6.3.2017 12 / 40
Correlators from perturbation theory Basics of thermal Green’s functions Correlation functions: generalities ...and the spectral function can in turn be given in terms of the Euclidean correlator: αβ ( k n → − i [ k 0 + i 0 + ] , k ) . Im Π E ρ αβ ( K ) = ∴ Analytic determination of Euclidean correlator, together with analytic continuation, enough to evaluate all Minkowskian Green’s functions! Surprising benefit: real-time quantities from the “simple” Feynman rules of the imaginary time formalism Aleksi Vuorinen (University of Helsinki) Thermal correlators from pQCD Oxford, 6.3.2017 13 / 40
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