Heavy quarkonium physics from Euclidean correlators

Transcription

1 Heavy quarkonium physics from Euclidean correlators Yannis Burnier Albert Einstein Center for Fundamental Physics, Bern University Talk at ECT*, based on [YB, M. Laine, JHEP 1211 (2012) 086] [YB, A.Rothkopf, Phys. Rev. D 86 (2012) ] [YB, A. Rothkopf, in preparation] April 3, 2013

2 Outline 1 Introduction 2 Massive vector current correlator in thermal QCD at NLO Vector current correlators as observable NLO Calculation Results and comparison to the lattice 3 HTL Wilson loop to check the extraction of the static potential HTL and lattice Wilson loop Extraction of the potential from the Wilson loop spectrum Analytic continuation from the Euclidean correlator 4 Conclusion

3 Outline 1 Introduction 2 Massive vector current correlator in thermal QCD at NLO Vector current correlators as observable NLO Calculation Results and comparison to the lattice 3 HTL Wilson loop to check the extraction of the static potential HTL and lattice Wilson loop Extraction of the potential from the Wilson loop spectrum Analytic continuation from the Euclidean correlator 4 Conclusion

4 Some Euclidean observables for heavy quarkonium Vector (spatial) current correlator Definition: G V, ii (τ) = ( Ψγµ, i Ψ ) (τ, x) ( Ψγ µ, i Ψ) (τ, x) T x Physics: Transport peak, heavy quark diffusion (large τ) Quarkonium physics, bound state (intermediate τ) High energy scattering (small τ)

5 Some Euclidean observables for heavy quarkonium Vector (spatial) current correlator Definition: G V, ii (τ) = x ( Ψγµ, i Ψ ) (τ, x) ( Ψγ µ, i Ψ) (τ, x) T Physics: Transport peak, heavy quark diffusion (large τ) Quarkonium physics, bound state (intermediate τ) High energy scattering (small τ) Wilson loop Defined as an integral on a square path (r, τ): W = 1 N c Ptr[e ig dxµ A µ(x) ] T Physics: Heavy bound state physics, heavy quark potential.

6 How to extract the physics from Euclidean correlators? Physical quantities we are interested in are defined in Minkowski space. They can be fitted from the correlator s spectrum ρ(ω) = Disc[C E ( iω)], or C E (τ) = dωk(ω, τ)ρ(ω). Perturbation theory: Lattice: Handle Euclidean and Minkowski Not accurate for large coupling Hard to catch infrared physics Contains all the physics Bound to Euclidean ΡV Ω Ω Diffusion Coefficient Energy of bound state Need analytic continuation to Minkowski space... Can our understanding from perturbation theory help? Ω

7 Outline 1 Introduction 2 Massive vector current correlator in thermal QCD at NLO Vector current correlators as observable NLO Calculation Results and comparison to the lattice 3 HTL Wilson loop to check the extraction of the static potential HTL and lattice Wilson loop Extraction of the potential from the Wilson loop spectrum Analytic continuation from the Euclidean correlator 4 Conclusion

8 Vector current correlators as observable Get some intuition from LO perturbation theory G V (τ) = G 00 G ii = ( Ψγ µ Ψ ) (τ, x) ( Ψγ µ Ψ ) (τ, x) T G 00 (τ) is constant (current conservation) one defines the susceptibility χ = βg 00 At LO, G 00 = 4N c p Tn F(E p ) x G V and G ii have both a constant and a τ-dependent part: At LO, G const V G τ V = G τ ii = 2N c p = G 00 Gii const = 4N c p ( ) 2 + M2 E D p 2 Ep,Ep (τ) τ-dependence parametrized by the functions M 2 Tn F (Ep) E 2 p D E k+1 E 1 E k (τ) e(e1+ +E k )(β τ)+(e k+1)τ + e (E1+ +E k )τ+(e k+1)(β τ) [e βe1 ± 1] [e βen ± 1]

9 Vector current correlators as observable Get some intuition from LO perturbation theory G V (τ) = G 00 G ii = ( Ψγ µ Ψ ) (τ, x) ( Ψγ µ Ψ ) (τ, x) T G 00 (τ) is constant (current conservation) one defines the susceptibility χ = βg 00 At LO, G 00 = 4N c p Tn F(E p ) x G V and G ii have both a constant and a τ-dependent part: At LO, G const V G τ V = G τ ii = 2N c p = G 00 Gii const = 4N c p ( ) 2 + M2 E D p 2 Ep,Ep (τ) τ-dependence parametrized by the functions M 2 Tn F (Ep) E 2 p D E k+1 E 1 E k (τ) e(e1+ +E k )(β τ)+(e k+1)τ + e (E1+ +E k )τ+(e k+1)(β τ) [e βe1 ± 1] [e βen ± 1]

10 Leading order results G ii, G 00 and how they contribution to ρ ii 100 Gii T 3 G00 T 3 Χ T 2 G const ii T G Τ T 3 10 G Τ ii T 3 Ρii Ω T At τ = β/2 the constant part = time dependent part. In fact D Ep,Ep (β/2) = 2Tn F(Ep). D Ep,Ep (τ) = K(τ,2Ep) cosh(ω(τ β/2) 1+2n B (E p), with K(τ, ω) = sinh(ωβ/2)

11 Diagrams and renormalization We calculated the NLO vector correlator for any quark mass M: [YB, M.Laine, JHEP 1211 (2012) 086] The genuine 2-loop graphs for G V (ω) amount to + = 4g 2 { 2(D 2) 4(D 2) C A C F K 2 + K{P } P P Q P P K P Q + 4(D 2)M 2 (D 2) 2 Q 2 2(D 2) K 2 2 P + P Q K 2 4(D 2)M 2 (D 2) 2 Q 2 P K P Q 2 P P K P Q 16M 2 + 2(D 2) 2 K Q 4(D 2)Q 2 K 2 + 8M 4 2(D 4)M 2 Q 2 (D 2)Q 4 P P K P Q K 2 P P K P Q P K Q 2(D 4)M (D 2)(8 D)Q2 + (D 2)K M 4 4(D 2)Q 2 M 2 } P P K P Q P K Q K 2 2 P P K +... P Q The only parameter to renormalize at this order is the mass M: { = 4C A δm 2 D M 2 (D 2)Q 2 } {P } P P P Q 2 P P Q

12 Parametrization of the master integrals From the Fourier representation, G V (τ) = T ω n e iωnτ G V (ω n ) Calculate all master integrals separately Write the results into some common basis nf (E p ), D Ep,E p (τ), D ɛk E pe pk (τ), D ɛ k E pe pk (τ),... Sum the masters and simplify Remains 2 to 3 integrals to perform numerically τ dependence cancels in G 00 G 00, G V UV and IR finite after mass renormalization Some terms are IR sensitive, only the sum is finite

13 Results T χ NLO Tn [ F (E p) nb (ɛ k ) 4g 2 = C A C F p p 2 + n )] F(E k ) (1 M2 k ɛ k E k k 2 GV NLO { [ const. 4g 2 = Tn nb (ɛ k ) 1 F (E p) C A C F p k ɛ k p 2 3 ] Ep 2 + n [ F(E k ) 1 E k p 2 3 Ep 2 M2 p 2 k 2 M2 Ep 2E k 2 + M2 (4Ek 2 ]} M2 ) 2pkEp 2E k 2 ln p + k p k GV NLO τ-dep. D 2Ep (τ) )( 4g 2 = [(3 C A C F p 4π 2 + M2 Ep 2 1 p ln Ep + p ) ] E p E p p { D ɛk E + P pe (τ) [ pk p,k z ɛ k E pe pk (k 2 (E pk E p) 2 2Ep 2 ) + k M2 Ep E ] pk +... k + E p + E pk D ɛ k E pe (τ) [ ] pk + z ɛ k E pe pk (k 2 (E pk E p) 2 2Ep )

14 Comparison to the lattice To compare the curves, scale G ii and χ to the free, M=0 result G ii to the reconstructed correlator G NLO,rec (τ, T ) = ii G lattice,rec (τ, T, T ) = ii T = 1.45 T c, T c = 1.25 Λ_ MS dω 2π ( ρnlo,vac )K(τ, ω) V G(τ, T ) τ (T,T ) χ / χ free T = 1.45 T c, T c = 1.25 Λ_ MS LO NLO lattice M / T T = 1.45 T c, T c = 1.25 Λ_ MS M/T = LO NLO lattice free G ii / G ii LO NLO lattice M/T = τ T rec G ii / G ii M/T = 1 M/T = τ T [Ding, Francis, Kaczmarek, Karsch, Satz, Soeldner, ; Ding, Francis, Kaczmarek, Karsch, Laermann, Soeldner, ]

15 Addition of a transport peak The spectral function ρ ii /ω has a δ(ω) transport peak (LO, NLO) In the full result one expects a Lorentzian shape. Replace in G ii the constant part by a τ-dependent part from ρ (L) ii (ω 0) 3Dχ ωη2 ω 2 + η 2 We vary D tuning η to keep the area under ρ(ω)/ω constant T = 1.45 T c, T c = 1.25 Λ_ MS T = 1.45 T c, T c = 1.25 Λ_ MS M/T = 3.5, 2πT D = oo transport peak lattice 1.15 M/T = 3.5, 2πT D = oo transport peak lattice rec G ii / G ii πT D = 5 2πT D = 1 rec G ii / G ii πT D = 5 2πT D = τ T τ T

16 Addition of a bound state peak In the temperature range of interest at most one peak. Slightly to the left from the free quark-antiquark threshold Model this by a skewed Breit-Wigner shape added to NLO ρ (BW) ii (ω 2M) Set M 2γ, add to the thermal NLO result Add to vacuum result with A 5A, γ γ/5 _ T = 1.45 T c, T c = 1.25 Λ MS A ω 2 γ 2 (ω 2M + M) 2 + γ 2 T = 1.45 T c, T c = 1.25 Λ_ MS M/T = 3.5 added resonance lattice M/T = 3.5 added resonance lattice rec G ii / G ii A = 0.1 A = 0.5 rec G ii / G ii M / M = τ T M / M = τ T

17 Outline 1 Introduction 2 Massive vector current correlator in thermal QCD at NLO Vector current correlators as observable NLO Calculation Results and comparison to the lattice 3 HTL Wilson loop to check the extraction of the static potential HTL and lattice Wilson loop Extraction of the potential from the Wilson loop spectrum Analytic continuation from the Euclidean correlator 4 Conclusion

18 Wilson loop at LO in HTL resummed perturbation theory LO diagrams [Laine, Philipsen, Romatschke, Tassler, ] { d 3 q e iq3r + e iq3r 2 τ dq 0 W (τ, r) = 1 + C F (2π) 3 2 q 2 + Π L (0, q) + π n B(q 0 ) [ ( 1 h(τ, q 0 ) ρ L (q 0, q) q 2 1 ) ( 1 (q 0 ) 2 + ρ T (q 0, q) q3 2 1 ) ]} q 2, Form this relation we can get: The Euclidean correlator, performing the integrals numerically The real time correlator W (τ = it, r) The potential lim t t log(w (it, r)) = V (r) The spectral function ρ(r, ω) = 1 2π dt e iωt W (it, r) I avoid issues of renormalization introduce a lattice-like cutoff Λ

19 HTL and lattice Wilson loop Start by comparing the Euclidean correlators: 0.1 1e+00 1e-01 T=2.33T C Τ W (τ,r) 1e-02 β=7 1e-03 ξ=4 N x =20, N τ =32 1e τ HTL Euclidean correlator similar (more power in the UV) Resulting HTL potential known Use it to test the extraction of the potential

20 Wilson loop spectral function Analytic approximation for ω Re[V (r)] ρ Λ (r, ω) = 1 π Im[V (r)] cos δ (Re[V (r)] ω) sin δ (Im[V (r)]) 2 + (Re[V (r)] ω) 2 Skewed Lorenzian with δ = Im[V (r)] 2T

21 Wilson loop spectral function Analytic approximation for ω Re[V (r)] ρ Λ (r, ω) = 1 π Im[V (r)] cos δ (Re[V (r)] ω) sin δ (Im[V (r)]) 2 + (Re[V (r)] ω) 2 Im[V (r)] Skewed Lorenzian with δ = 2T Numerical results form integration of the real-time correlator: r=1/ r=7/ Ω

22 Spectral function fit and extracted value for the potential In principle the potential is: V (r) = lim t dω ω e iωt ρ(r,ω) dω e iωt ρ(r,ω) In the t limit, only the first peak ρ(r, ω) should contribute [YB, Rothkopf, ] ρ (r, ω) = 1 Im[V (r)] cos(re[σ (r)]) (Re[V (r)] ω)sin(re[σ (r)]) eim[σ (r)] π Im[V ](r) 2 + (Re[V ](r) ω) 2 +c 0 (r) + c 1 (r)t Q Q (Re[V ](r) ω) + c 2(r)t 2 Q Q (Re[V ](r) ω)2 + Parameters: e Im[σ (r)], Im[V (r)], Re[V (r)] ; Re[σ (r)] ; c 0 (r) ; c 1 (r) ; c 2 (r) The lowest peak contains all the information for the potential!

23 MEM to get the spectral function Can we get the spectral function? W Τ,r Ρ r,ω From N τ = 32 points, usual MEM (with 32 basis functions) fails given the large ω range needed Too few basis function to reconstruct the Euclidean correlator One might remove points at small/large τ to get the peak In any case not well controlled

24 Extended MEM to get the spectral function From N τ = 32 points, extended MEM (with 80 basis functions, β = 20, ω [ 10, 15], r = 1):

25 Extended MEM to get the spectral function From N τ = 32 points, extended MEM (with 80 basis functions, β = 20, ω [ 10, 15], r = 1): Ω Position of the peak Re[V ](r) OK Euclidean correlator reconstructed Low resolution, Numerically expensive, unstable results

26 Bayesian curve fitting to get the spectral function New entropy functionnal Basis: 800 bars

27 Bayesian curve fitting to get the spectral function New entropy functionnal Basis: 800 bars N τ = 32 Great result at small r! Re[V (r)] correct. Im[V (r)] OK at small r Ω PRELIMINARY 1 Re V r r r Ω

28 Outline 1 Introduction 2 Massive vector current correlator in thermal QCD at NLO Vector current correlators as observable NLO Calculation Results and comparison to the lattice 3 HTL Wilson loop to check the extraction of the static potential HTL and lattice Wilson loop Extraction of the potential from the Wilson loop spectrum Analytic continuation from the Euclidean correlator 4 Conclusion

29 Conclusion G V : Encodes a lot of information Hard to disentangle the different types of physics Require very accurate lattice data Next: ρ at NLO for any M physics Oher channels G S, G PS, G A, behave differently

30 Conclusion G V : Encodes a lot of information Hard to disentangle the different types of physics Require very accurate lattice data Next: ρ at NLO for any M physics Oher channels G S, G PS, G A, behave differently W : We are now confident that we can get Re[V (r)] The previous measurements of Im[V (r)] were MEM artifacts Next: What should be N τ, δ to get the correct Im[V(r)]?