Constraining light-cone spectral functions with the lattice 1 Mikko - PowerPoint PPT Presentation

Constraining light-cone spectral functions with the lattice 1 Mikko Laine (University of Bern) 1 Supported by the SNF under grant 200020-168988. 1

Motivation 2

Production rate of photons from the quark-gluon plasma: k 2 d n γ � e ik ( t − z ) � J 1 em (0) J 1 em ( X ) � + O ( α 2 d t d ln k = em ) . 2 π 2 X µ + K q µ − ¯ q γ 3

ALICE/LHC data: 2 → γ PbPb X at s = 2.76 TeV with |y| < 0.75 NN × nPDF: CTEQ61 EPS09, FF: BFG set II 1 -1 Scaled with <T > = 13 mb AA ALICE 0-40% central ) -1 10 -2 γ γ γ = + (GeV prompt direct fragm γ fragm -2 10 T γ dp direct dN γ T using BFG set I y p prompt -3 10 ± ∆ A exp(-p /T) with T = 304 58 MeV T ev N 1 π -4 10 2 -5 10 KKKW 2013 -6 10 2 4 6 8 10 12 14 p (GeV) T 2 M. Klasen, C. Klein-B¨ osing, F. K¨ onig and J.P. Wessels, How robust is a thermal photon interpretation of the ALICE low- pT data?, 1307.7034. 4

A similar production rate applies for gravitational waves: 3 8 k 3 d ρ GW � e ik ( t − z ) � T 12 (0) T 12 ( X ) � . d t d ln k = πm 2 X Pl This constrains the highest temperature after Big Bang. 0 10 -10 10 m Pl Ω GW m Pl / T max Ω GW = y , T max -20 10 (T ) 0 Ω GW = y × T max . -30 hydrodynamics 10 m Pl leading log -40 10 -3 0 3 6 9 12 15 10 10 10 10 10 10 10 f / Hz 3 J. Ghiglieri, ML, Gravitational wave background from Standard Model physics: Qualitative features , 1504.02569. 5

Basic definitions 6

It is convenient to rewrite the photon production rate as 2 α em χ q d n γ n B ( k ) D eff ( k ) + O ( α 2 d t d 3 k = em ) . 3 π 2 Here n B ( k ) ≡ 1 / ( e βk − 1) is the Bose distribution, and χ q ∼ T 2 is a quark-number susceptibility which is easy to measure with lattice QCD / compute with pQCD. The relation applies to all orders in α s . µ + K q µ − ¯ q γ 7

The strong interactions are hidden in D eff ( k ) The “effective diffusion coefficient” is defined as  ρ V ( k, k ) , k > 0    2 χ q k  D eff ( k ) ≡ . ρ V ( ω, 0 ) lim , k = 0    3 χ q ω ω → 0+  Hydrodynamics shows that lim k → 0 D eff ( k ) = D (cf. later). Vector spectral function: e i ( ωt − k · x ) � 1 � 2[ V µ ( t, x ) , V µ (0)] � c , ρ V ( ω, k ) ≡ X V µ ψγ µ ψ , ¯ ≡ η = ( − + + +) . 8

General structure of ρ V 9

(i) Hydrodynamic regime For small k the general theory of statistical fluctuations applies, 4 and permits for a “hydrodynamic” prediction: 5 � ω 2 − k 2 ρ V ( ω, k ) � = ω 2 + D 2 k 4 + 2 χ q D . ω Here D ≡ lim k → 0 D eff ( k ) is the diffusion coefficient, and χ q is the quark number susceptibility, parametrizing the constant x � V 0 ( τ, x ) V 0 (0 , 0 ) � = χ q T . � correlator Note that ρ V can be negative in the space-like domain ω < k . 4 Cf. e.g. E.M. Lifshitz and L.P. Pitaevskii, Statistical Physics, Part 2 , § 88-89. 5 Cf. e.g. J. Hong and D. Teaney, Spectral densities for hot QCD plasmas in a leading log approximation, 1003.0699. 10

(i) pQCD √ ω 2 − k 2 � = 0 : 6 Leading order (LO) at M ≡ � cosh( ω + k ρ V ( ω, k ) = N c T M 2 4 T ) − ω θ ( k − ω ) � � � ln . cosh( ω − k 2 πk 2 T 4 T ) Leading-log order (LL) at M = 0 : 7 ρ V ( k, k ) = α s N c C F T 2 � 1 [1 − 2 n F ( k )] + O ( α s T 2 ) . � ln 4 α s 6 e.g. G. Aarts and J.M. Mart´ ınez Resco, Continuum and lattice meson spectral functions at nonzero momentum and high temperature, hep-lat/0507004. 7 J.I. Kapusta, P. Lichard and D. Seibert, High-energy photons from quark-gluon plasma versus hot hadronic gas, PRD 44 (1991) 2774; R. Baier, H. Nakkagawa, A. Ni´ egawa and K. Redlich, Production rate of hard thermal photons and screening of quark mass singularity, ZPC 53 (1992) 433. 11

Current status LO at M = 0 . 8 (only numerical result) NLO at M = 0 . 9 (only numerical result) NLO at M ∼ gT . 10 (only numerical result) NLO at M ∼ πT . 11 (only numerical result) N 4 LO at M ≫ πT . 12 (analytic result) 8 P.B. Arnold, G.D. Moore and L.G. Yaffe, Photon emission from ultrarelativistic plasmas, hep-ph/0109064; Photon emission from quark gluon plasma: Complete leading order results, hep-ph/0111107. 9 J. Ghiglieri et al , Next-to-leading order thermal photon production in a weakly coupled quark-gluon plasma, 1302.5970. 10 J. Ghiglieri and G.D. Moore, Low Mass Thermal Dilepton Production at NLO in a Weakly Coupled Quark-Gluon Plasma, 1410.4203. 11 ML, NLO thermal dilepton rate at non-zero momentum, 1310.0164. 12 S. Caron-Huot, Asymptotics of thermal spectral functions, 0903.3958; P.A. Baikov, uhn, Order α 4 K.G. Chetyrkin and J.H. K¨ s QCD Corrections to Z and τ Decays, 0801.1821. 12

(iii) AdS/CFT 13 In the IR the hydrodynamic prediction is reproduced, with the specific values D = 1 / (2 πT ) and χ q = N 2 c T 2 / 8 . One can also ask when hydrodynamics applies: the spectral function is close to hydrodynamics for k < ∼ 0 . 5 /D , and becomes negative at the smallest ω for k < ∼ 1 . 07 /D . 13 G. Policastro, D.T. Son and A.O. Starinets, From AdS / CFT correspondence to hydrodynamics, hep-th/0205052; S. Caron-Huot et al , Photon and dilepton production in supersymmetric Yang-Mills plasma, hep-th/0607237. 13

General comment on the real world N f = 0 : m 0++ ≫ 1 GeV ⇒ need to heat the system “a lot”. Concretely, T c / Λ MS ≃ 1 . 24 ⇒ α s (2 πT c ) = “small”. N f = 3 : m π ≪ 1 GeV ⇒ don’t need to heat a lot. Concretely, T c / Λ MS ≃ 0 . 45 ⇒ α s (2 πT c ) = “large”. So at least for the unquenched case, pQCD is not sufficient. 14

Non-perturbative approach: idea 1/2 15

What can we do with Euclidean lattice? � ∞ π ρ V ( ω, k ) cosh[ ω ( β 2 − τ )] d ω β ≡ 1 G V ( τ, k ) = , T . sinh[ ωβ 2 ] 0 In principle inversion is possible by the Cuniberti method, 14 if the perturbative UV tail ( τ ≪ β , ω ≫ πT ) is first subtracted. 15 In practice there is a “sign problem” in the inversion ⇒ fragile unless very high statistical precision available. 16 14 G. Cuniberti, E. De Micheli and G.A. Viano, Reconstructing the thermal Green functions at real times from those at imaginary times, cond-mat/0109175; F. Ferrari, The Analytic Renormalization Group, 1602.07355. 15 Y. Burnier, ML, Towards flavour diffusion coefficient and electrical conductivity without ultraviolet contamination , 1201.1994. 16 Y. Burnier, ML, L. Mether, A Test on analytic continuation of thermal imaginary-time data , 1101.5534. 16

Here: down-to-earth approach Trust UV from pQCD, fit an interpolating function in the IR. 17 Only a few coefficients can be fitted, so a “good” basis is needed. ω 4 ) O( α s trust O( α s ) ( g ≡ √ 4 πα s ): π T 1/2 ) O( α s fit g T O( α s 0 ) π T g T k 17 J. Ghiglieri, O. Kaczmarek, ML, F. Meyer, Lattice constraints on the thermal photon rate, 1604.07544. 17

Polynomial interpolation (assuming analyticity, V → ∞ ) Pick a point above which pQCD should apply, for instance � k 2 + ( πT ) 2 , ω 0 ≃ and use that to fix two coefficients: ρ V ( ω 0 , k ) ≡ β , ∂ ω ρ V ( ω 0 , k ) ≡ γ . Then the most general polynomial odd in ω takes the form n max ρ fit ≡ β ω 3 5 − 3 ω 2 − γ ω 3 1 − ω 2 δ n ω 1+2 n 1 − ω 2 � 2 . � � � � � � + ω 1+2 n 2 ω 3 ω 2 2 ω 2 ω 2 ω 2 0 0 0 0 0 0 n ≥ 0 18

How does the pQCD result look like? (“vacuum” ≡ LO+...) 18 T = 1.1T c , k = 4.189T 3 10 LPM / NLO / "vacuum" "vacuum" 2 10 1 10 2 ρ V / T 0 10 -1 lightcone 10 -2 10 -1 0 1 2 10 10 10 10 ω / T 18 3 T < ω < 10 T from J. Ghiglieri and G.D. Moore, Low Mass Thermal Dilepton Production at NLO in a Weakly Coupled Quark-Gluon Plasma, 1410.4203 ; ω > ∼ 10 T from I. Ghisoiu and ML, Interpolation of hard and soft dilepton rates, 1407.7955 ; ω ≫ 10 T from ML, NLO thermal dilepton rate at non-zero momentum, 1310.0164. The best available perturbative data, both for N f = 0 and N f = 3 , can be found at J. Ghiglieri and ML, web page http://www.laine.itp.unibe.ch/dilepton-lattice/ 19

Missing ingredient Below the light cone, ρ V is only known at LO. More information there could be an “inexpensive” way to constrain the fit (for now the whole IR domain ω ≤ ω 0 is fitted). T = 1.1T c , k = 4.189T 3 10 LPM / NLO / "vacuum" "vacuum" 2 10 1 10 2 ρ V / T 0 10 -1 lightcone 10 -2 10 -1 0 1 2 10 10 10 10 ω / T 20

Lattice details Imaginary-time observable: � e − i k · x � V i ( τ, x ) V i (0) − V 0 ( τ, x ) V 0 (0) � c . G V ( τ, k ) ≡ x Consider the full G V rather than G ii because this is relevant for dileptons and because much more is known within pQCD. Momenta are chosen along the lattice axes. With periodic boundary conditions this requires k = 2 πnT × N τ , N s where N s , N τ are the spatial and temporal lattice extents. 21

Ensemble N 3 β 0 s × N τ confs T/T c | t 0 k/T 96 3 × 32 7.192 314 1.12 2.094,4.189,6.283 144 3 × 48 7.544 358 1.14 192 3 × 64 7.793 242 1.15 96 3 × 28 7.192 232 1.28 1.833,3.665,5.498 144 3 × 42 7.544 417 1.31 192 3 × 56 7.793 273 1.31 With such large β 0 we are frozen to the trivial topological sector, 19 but do not expect this to affect the results dramatically. 19 S. Schaefer et al. [ALPHA Collaboration], Critical slowing down and error analysis in lattice QCD simulations, 1009.5228. 22