Energy-momentum tensor correlators in hot Yang-Mills theory

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, 6.3.017 Mikko Laine, Mikko Vepsäläinen, AV, 1008.363, 1011.4439 Mikko Laine, AV, Yan Zhu, 1108.159 York Schröder, Mikko Vepsäläinen, AV, Yan Zhu, 1109.6548 AV, Yan Zhu, 11.3818, 150.0556 Yan Zhu, Ongoing work Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 1 / 40

Table of contents 1 Motivation Transport coefficients and correlators Perturbative input Correlators from perturbation theory Basics of thermal Green s functions Our setup Computational techniques 3 Results Operator Product Expansions Euclidean correlators Spectral densities 4 Conclusions and outlook Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 / 40

Motivation Table of contents 1 Motivation Transport coefficients and correlators Perturbative input Correlators from perturbation theory Basics of thermal Green s functions Our setup Computational techniques 3 Results Operator Product Expansions Euclidean correlators Spectral densities 4 Conclusions and outlook Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 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, 6.3.017 4 / 40

Motivation Transport coefficients from data Transport coefficients and correlators 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, 6.3.017 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 : 1 η = lim ω 0 ω Im ρ 1,1 (ω) DR 1,1(ω, k = 0) lim ω 0 ω π ζ = lim ω 0 9 ij 1 ω Im DR ii,jj (ω, k = 0) π 9 ij lim ω 0 ρ ii,jj (ω) ω Problem: Lattice can only measure Euclidean correlators Spectral density available only through inversion of G(τ) = 0 dω π (β τ)ω ρ(ω)cosh sinh βω To extract IR limit of ρ, need to understand its behavior also at ω πt perturbative input needed Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 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, 101.1994] Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 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 (0909.058): Lattice data for spatial correlators of Tr F 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, 6.3.017 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 (0909.058): UV behavior of Tr F µν and Tr F µν Fµν correlators on the lattice completely different even though leading order OPEs identical Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 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 1 Inspect Operator Product Expansions (OPEs) at finite temperature Compare behavior of perturbative time-averaged spatial correlators to lattice QCD and AdS/CFT 3 Use spectral densities at zero wave vector to aid the determination of transport coefficients from lattice data Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 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 1 Inspect Operator Product Expansions (OPEs) at finite temperature Compare behavior of perturbative time-averaged spatial correlators to lattice QCD and AdS/CFT 3 Use spectral densities at zero wave vector to aid the determination of transport coefficients from lattice data Concretely: Specialize to scalar, pseudoscalar and shear operators θ c θ g B F a µνf a µν, χ c χ g B F a µν F a µν, η c η T 1 = c η F a 1µ F a µ and proceed from 1 to 3 working at NLO. Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 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 1 Inspect Operator Product Expansions (OPEs) at finite temperature Compare behavior of perturbative time-averaged spatial correlators to lattice QCD and AdS/CFT 3 Use spectral densities at zero wave vector to aid the determination of transport coefficients from lattice data When can perturbation theory be expected to work? Λ x,t ( Λx ) + ( ΛT ) 1 + (πt ) x 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, 6.3.017 9 / 40

Correlators from perturbation theory Table of contents 1 Motivation Transport coefficients and correlators Perturbative input Correlators from perturbation theory Basics of thermal Green s functions Our setup Computational techniques 3 Results Operator Product Expansions Euclidean correlators Spectral densities 4 Conclusions and outlook Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 10 / 40

Correlators from perturbation theory Basics of thermal Green s functions Correlation functions: generalities Plenitude of different Minkowskian correlators: Π > αβ (K) e ik X ˆφα(X ) ˆφ β (0), X Π < αβ (K) e ik X ˆφ β (0) ˆφ α(x ), X ρ αβ (K) e ik X 1 [ ˆφ α(x ), ˆφ ] β X (0), αβ (K) e ik X 1 { ˆφ α(x ), ˆφ } β X (0), Π R αβ (K) i e ik X [ ˆφ α(x ), ˆφ ]θ(t) β (0), X Π A αβ (K) i e ik X [ ˆφα(X ), ˆφ ] β (0) θ( t), X Π T αβ (K) e ik X ˆφα(X ) ˆφ β (0) θ(t) + ˆφ β (0) ˆφ α(x ) θ( t) X One Euclidean correlator, computable on the lattice: Π E αβ (K ) e ik X ˆφ α(x) ˆφ β (0) X Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 11 / 40

Correlators from perturbation theory Basics of thermal Green s functions Correlation functions: generalities However, in thermal equilibrium all correlators related through ρ: Π < αβ (K) = n B(k 0 )ρ αβ (K), 0 Π > eβk αβ (K) = e βk 0 1 ρ αβ(k) = [1 + n B (k 0 )] ρ αβ (K), αβ (K) = 1 [ Π > αβ (K) + Π < αβ (K)] = [ 1 + n B (k 0 ) ] ρ αβ (K). Im Π R αβ (K) = ρ αβ(k), Im Π A αβ (K) = ρ αβ(k), Π T αβ (K) = iπr αβ (K) + Π< αβ (K), Π E αβ (K ) = dk 0 π ρ αβ (k 0, k) k 0 ik n Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 1 / 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: ρ αβ (K) = Im Π E αβ (k n i[k 0 + i0 + ], 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, 6.3.017 13 / 40

Correlators from perturbation theory Basics of thermal Green s functions Correlation functions: generalities Simplest example: bosonic operator in free field theory Π E (K ) = Π R (K) = ρ(k) = Π T (K) = = P 1 k n + E k, E k = 1 k + m, (k 0 + i0 + ) + Ek ( ) 1 (k 0 ) Ek + iπ [ ] δ(k 0 E k ) δ(k 0 + E k ), E k π [ ] δ(k 0 E k ) δ(k 0 + E k ), E k i K m + i0 + + π δ(k m ) n B ( k 0 ) Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 14 / 40

Correlators from perturbation theory Our setup Setting up the two-loop calculation Our program: work within finite-t SU(3) Yang-Mills theory S E = β 0 dτ d 3 ɛ x { } 1 4 F µνf a µν a, write down diagrammatic expansions for Euclidean correlators of energy-momentum tensor components (X (τ, x)) G θ (X) θ(x)θ(0) c, G χ (X) χ(x)χ(0), G η (X) η(x)η(0) c, β G α (P) e ip X G α (X), Ḡ α (x) dτg α (X), X ρ α (ω) Im G α (p 0 = i(ω + iɛ), p = 0), and evaluate the necessary integrals up to NLO. For correct IR behavior, perform HTL resummation when needed. 0 Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 15 / 40

Correlators from perturbation theory Our setup Setting up the two-loop calculation End up computing two-loop two-point diagrams in dimensional regularization, the black dots representing the operators: NB: At T 0, 4d integrals replaced by 3+1d sum-integrals T Q Q q 0 =πnt q Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 16 / 40

Correlators from perturbation theory Our setup Hard Thermal Loop resummation For ρ(ω, p = 0), three different energy regimes: ω πt : Ordinary weak coupling expansion expected to converge, no resummations needed g T /π ω πt : Weak coupling expansion breaks down, but can be resummed using HTL effective theory ω g T /π: All hell breaks loose For consistency, extend both bulk and shear results to ω gt via HTL treatment: ρ QCD resummed = ρ QCD resummed ρ HTL resummed + ρ HTL resummed ρ QCD naive ρ HTL naive + ρ HTL resummed In both cases, no HTL vertex functions necessary to match the IR behavior of the unresummed result. Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 17 / 40

Correlators from perturbation theory Computational techniques Computational methods I: Identifying the masters Step 1: Perform Wick contractions in Euclidean correlator and perform Lorentz algebra (typically with FORM) Result: Expansion in terms of scalar masters [ G θ (P) 4d A c θ = (D ) Ja + 1 ] g4 J b B ] ] + g B {(D Nc ) [ (D 1)Ia + (D 4)I b + (D ) [Ic I d ]} 7D (D 4) + I 3 f Ig + (D ) [ 3Ie + 3I h + I i I j, Ja P Q, J b P 4 Q (Q P), I a 1 Q R, I b P Q R (R P), Q Q Q,R Q,R I h P 4 Q R (Q R) (R P), I i (Q P) 4 Q R (Q R) (R P), Q,R Q,R I i 4(Q P) Q R (Q R) (R P), I j P 6 Q R (Q R) (Q P) (R P) Q,R Q,R Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 18 / 40

Correlators from perturbation theory Computational techniques Computational methods II: Evaluating the masters The optimal method for dealing with the master integrals depends on the particular problem: Operator Product Expansions Explicit evaluation of Matsubara sums via cutting methods Expansion of the 3d integrals in powers of T /P Time-averaged spatial correlators Set p 0 = 0 in mom. space correlator and perform 3d Fourier transf. Calculation most conveniently handled directly in coordinate space, where the Matsubara sum trivializes Spectral functions: ρ(ω) Im G(p 0 = i(ω + iɛ), p = 0) Matsubara sum via cutting rules, then take explicitly the imaginary part δ-function constraints for the 3-momenta Most complicated part of the calculation: Dealing with the remaining spatial momentum integrals Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 0 / 40

Correlators from perturbation theory Computational techniques Example: Integral j in the spectral density Most complicated master integral in the bulk channel: P 6 I j Q R (Q R) (Q P) (R P) Q,R Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 1 / 40

Correlators from perturbation theory Computational techniques Example: Integral j in the spectral density After performing the Matsubara sum and taking the imaginary part, I j contribution written in terms of a two-fold phase space integral (E qr q r ): Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 / 40

Correlators from perturbation theory Computational techniques Example: Integral j in the spectral density After quite some work: final result in terms of finite 1d and d integrals, amenable to numerical evaluation (4π) 3 ρ Ij (ω) ω 4 (1 + n ω ) = + + + + + ω [ ( 4 1 dq n q 0 q ω 1 q ω [ ( dq n q ω q ω 1 q 4 [ ( dq n q ω q ω q ω ω q ω 4 4 dq dr 0 0 q ω dq dr ω 0 q dq dr 0 0 ) ( ln 1 q ) ω ) ( ln 1 q ) ω ) ( q ) ln ω 1 ω (1 q(q + ω ) ln + q ) ] ω ( 1 ) n ω q nq+r (1 + n ω r ) qr nr ( 1 ) n q ω (1 + n q r )(n q n r+ ω ) qr n r n ω ( 1 ) (1 + n q+ ω )n q+r n r+ ω qr nr ω (1 q(q + ω ) ln + q ) 1 ( q ) ] ω q ω ln ω ω (1 q(q + ω ) ln + q ) ( 1 + ω q 1 ) ( q ) ] q ω ln ω. Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 3 / 40

Results Table of contents 1 Motivation Transport coefficients and correlators Perturbative input Correlators from perturbation theory Basics of thermal Green s functions Our setup Computational techniques 3 Results Operator Product Expansions Euclidean correlators Spectral densities 4 Conclusions and outlook Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 4 / 40

Results Summary of NLO results OPEs Coord. space Spectral density Scalar (θ) [1] [] [4] Pseudoscalar (χ) [1] [] [4] Shear (η) [3] [5,6] [1] Mikko Laine, Mikko Vepsäläinen, AV, 1008.363 [] Mikko Laine, Mikko Vepsäläinen, AV, 1011.4439 [3] York Schröder, Mikko Vepsäläinen, AV, Yan Zhu, 1109.6548 [4] Mikko Laine, AV, Yan Zhu, 1108.159 [5] Yan Zhu, AV, 11.3818 [6] AV, Yan Zhu, 150.0556 Note: Inclusion of fermions possible in all cases (ongoing work in shear channel by Yan Zhu). Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 5 / 40

Results Wilson coefficients for OPEs Operator Product Expansions For P T, perform large momentum expansion of Euclidean correlators to obtain T 0 corrections to the OPEs G θ (P) 4c θ g4 = G χ(p) 16c χ g4 = G η(p) 4c η ( 3 p P g b 0 ( 3 p P 3 p n )[ 1 + g ( N c µ (4π) ln 3 P + 03 )] (e + p)(t ) 18 [1 + g b 0 ln µ 3 p n ζ θ P )[ 1 + g N c (4π) [ + 1 + g b 0 ln µ g b 0 ζ χp = { 1 + 3 ( p P 3 p n + 4 3g b 0 ] ( (e 3p)(T ) + O g 4, ( 3 1 ) P µ ln P + 347 )] (e + p)(t ) 18 ] ( (e 3p)(T ) + O ) 1 3 ] [1 g ( b 0 lnζ η (e 3p)(T ) + O g 4, g 4 1 ), P g [ N c (4π) + 41 ( p )] } P 3 p n (e + p)(t ) 1 ) P Note the appearance of Lorentz non-invariant operator e + p. Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 6 / 40

Results Wilson coefficients for OPEs Operator Product Expansions Behavior of Wilson coefficients in coordinate space: Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 7 / 40

Results Euclidean correlators Time averaged spatial correlators To compare with lattice results for spatial correlators, and to assess validity of OPE, determine next time-averaged spatial correlators Ḡ θ (x) β 0 dτg θ(x), Ḡχ(x) β 0 dτg χ(x): Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 8 / 40

Results Euclidean correlators Time averaged spatial correlators Qualitatively, NLO results slightly closer to lattice than LO ones However: difference between θ and χ channels much more suppressed in perturbative results Caveat: lattice results for equal time, not time averaged correlator Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 31 / 40

Results Spectral densities Spectral densities Bulk channel spectral densities: ρ θ (ω) 4d A c θ ρ χ (ω) 16d A cχ = = πω 4 { ( ) 1 + n ω (4π) g 4 + g6 N c (4π) πω 4 { ( ) 1 + n ω (4π) g 4 + g6 N c (4π) [ 3 [ 3 µ ln ω + 73 3 µ ln ω + 97 3 ] + 8 φ T (ω) + g4 me 4 } ω 4 φ HTL θ (ω) + O(g 8 ) ] + 8 φ T (ω) + g4 me 4 } ω 4 φ HTL χ (ω) + O(g 8 ) Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 33 / 40

Results Spectral densities Spectral densities Bulk channel spectral densities: ρ θ (ω) 4d A c θ ρ χ (ω) 16d A cχ = = πω 4 { ( ) 1 + n ω (4π) g 4 + g6 N c (4π) πω 4 { ( ) 1 + n ω (4π) g 4 + g6 N c (4π) [ 3 [ 3 µ ln ω + 73 3 µ ln ω + 97 3 ] + 8 φ T (ω) + g4 me 4 } ω 4 φ HTL θ (ω) + O(g 8 ) ] + 8 φ T (ω) + g4 me 4 } ω 4 φ HTL χ (ω) + O(g 8 ) H. B. Meyer, 100.3343 Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 34 / 40

Results Spectral densities Spectral densities In the imaginary time correlator, theoretical uncertainties (dependence on renormalization scale) considerably suppressed G(τ) = 0 dω π (β τ)ω ρ(ω)cosh sinh βω Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 35 / 40

Results Spectral densities Spectral densities In the imaginary time correlator, impressive agreement with lattice results in the short distance limit; divergence absent in the difference G(τ) = 0 dω π (β τ)ω ρ(ω)cosh sinh βω Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 36 / 40

Spectral densities Results Spectral densities For the imaginary time correlator, comparison possible also to IHQCD G(τ) = 0 dω π (β τ)ω ρ(ω)cosh sinh βω Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 37 / 40

Spectral densities Results Spectral densities In the shear channel, spectral density and imaginary time correlator converge wonderfully, but no lattice or AdS results to compare with at the moment: ρ η (ω) 4d A = G η(τ) = ω 4 ( ) { 1 + n ω 1 4π 10 + g [ ] N c (4π) 9 + φt η (ω) + m4 } E ω 4 φhtl η (ω/t, m E /T ) + O(g 8 ), [( ) ] dω cosh β 0 π ρη(ω) τ ω sinh βω Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 38 / 40

Table of contents Conclusions and outlook 1 Motivation Transport coefficients and correlators Perturbative input Correlators from perturbation theory Basics of thermal Green s functions Our setup Computational techniques 3 Results Operator Product Expansions Euclidean correlators Spectral densities 4 Conclusions and outlook Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 39 / 40

Conclusions and outlook Conclusions Perturbative evaluation of energy-momentum tensor correlators in thermal QCD useful for disentangling properties of the QGP Spectral densities needed to extract transport coefficients from lattice data for Euclidean correlators Spatial correlators useful way to compare lattice, pqcd and holographic predictions NLO results derived for OPEs in the bulk and shear channels Time averaged spatial correlator in the bulk channel Spectral density in the bulk and shear channels For further progress, accuracy of lattice results for Euclidean correlators the bottle neck Aleksi Vuorinen (University of Helsinki) Thermal correlators from pqcd Oxford, 6.3.017 40 / 40