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QCD sum rules for quarkonium at T>0

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Presentation on theme: "QCD sum rules for quarkonium at T>0"— Presentation transcript:

1 QCD sum rules for quarkonium at T>0
with Maximum Entropy Method Kenji Morita (Yukawa Institute for Theoretical Physics, Kyoto University) in Collaboration with Philipp Gubler, Kei Suzuki and Makoto Oka (Tokyo Institute of Technology) Ref : P.Gubler, KM, M.Oka, PRL107, (2011) P.Gubler, K.Suzuki, KM, M.Oka, in preparation

2 QCDSR Strategy for QCDSR Lattice QCD Current Correlation P(Q2)
Thermodynamic quantities / Coupling constant Lattice QCD Current Correlation P(Q2) Numerical Simulation Q2=-q2 ≪ 0 Gluon Condensates OPE QCDSR Imaginary Time Correlator +MEM Dispersion relation (II) with optimization This Work MEM Dispersion relation (I) Pole+Continuum Spectral function Goal 6 Oct, 2011 QWG

3 QCD sum rules q2=-Q2 OPE at large Q2 Matching to obtain m2 , s0 and G
Shifman-Vainshtein-Zakharov ‘79 “Hadronic” spectral function q2=-Q2 OPE at large Q2 m2 s0 Matching to obtain m2 , s0 and G at dim.d : Better convergence expected for heavier quark mass and Q2 >0 6 Oct, 2011 QWG

4 Requirement : Convergence Crude estimation by instanton liquid model
OPE w/ Optimization Borel transformed correlator Requirement : Convergence Dim.6 < 20% of OPE Crude estimation by instanton liquid model Gluon condensates in pure SU(3) LQCD (Boyd et al., ’96) 6 Oct, 2011 QWG

5 Relation to Spectral Function
Dispersion relation tells f m2, G s0 T↑ s0 m G →⇒↑ ↓⇒↑ → or ↑ If other quantities are fixed: m2 : decrease G : Increase s0 : decrease f : Increase Increase Conventional optimization procedure gives constraints among these parameters New proposal : Use MEM 6 Oct, 2011 QWG

6 Comparison with lattice
Input Approximation # of Sample Kernel T-dep Lattice G(t,T) Discretization O(10) (Nt=1/Ta) Nt QCDSR M(n) OPE T-dep condensate Continuous O(100) in practice G0, G2 6 Oct, 2011 QWG

7 Maximum Entropy Method
Maximize probability Shannon-Jaynes Entropy r(w) : most probable form and unique solution for given M and condition I Average over a for the final output r(w) Deviation from the best a for estimating error dr(w) Likelihood function “default model” matched to pQCD (w→infinity) 6 Oct, 2011 QWG

8 MEM for QCD sum rules Uncertainty in input MJ(n) Generate dMJ(n)
T-independent : mQ(mQ), as(MZ), G0(T=0) T-dependent : e, p (from lattice measurement) Generate dMJ(n) In practice, we use to remove constant contribution due to transport peak in V channel 6 Oct, 2011 QWG

9 MEM for QCD sum rules Uncertainty in input MJ(n) Generate dMJ(n)
T-independent : mQ(mQ), as(MZ), G0(T=0) T-dependent : e, p (from lattice measurement) Generate dMJ(n) In practice, we use to remove constant contribution due to transport peak in V channel M=3GeV/Result not sensitive Convergence of OPE Used in MEM 6 Oct, 2011 QWG

10 For technical details of QCDSR+MEM, Gubler and Oka, PTP124,995 (’10)
J/y from QCDSR+MEM P.Gubler, KM, M.Oka PRL’11 P.Gubler, K.Suzuki, KM, M.Oka, in preparation Resolution of the width not sufficient Peak contribution at T=0 consistent with exp. data Peak reduction is not due to increasing err at finite T For technical details of QCDSR+MEM, Gubler and Oka, PTP124,995 (’10) 6 Oct, 2011 QWG

11 Peak quickly dissolves above Tc
Charmonia Peak quickly dissolves above Tc 6 Oct, 2011 QWG

12 Bottomonia (1S) Dissolved at 2.5-3Tc Caveat : Peak = 1S+2S+3S
n range from as correction < 0.3 Condensates suppressed by (mc/mb)4 Caveat : Peak = 1S+2S+3S (Unlike charmonium) Dissolved at 2.5-3Tc 6 Oct, 2011 QWG

13 Bottomonia (1P) Dissolved at 2-2.5Tc 6 Oct, 2011 QWG

14 Summary and Outlook QCD sum rules+MEM
Spectral function without (pole+continuum) assumption Several advantages over lattice for MEM Width resolution is not sufficient Charmonium : sensitive to change near Tc Characterized by local operators (gluon condensate) Peak dissolved ~1.2Tc Bottomonium : modification at T>2Tc Only combined(1S+2S+3S) peak obtained Not sensitive to change near Tc 6 Oct, 2011 QWG

15 Backup 6 Oct, 2011 QWG

16 OPE for QCD sum rules Correlator in momentum space
Take spacelike momentum : q2 = -Q2 < 0 OPE and truncation valid for: Up to dim.4, rough estimation of dim.6 Temperature effect through condensates Meson at rest with respect to medium: q = (w,0) Hatsuda-Koike-Lee ‘93 6 Oct, 2011 QWG

17 From local operators to quarkonium : OPE
Treating as a short-distance process (Peskin ’79) × R~1/e One-gluon exchange × Info. on scale < e Ext. soft gluon k ≪ e info. on scale > e (binding energy) Leading order gives mass shift (2nd order Stark) 6 Oct, 2011 QWG

18 Optimization Technique
Moment of P(Q2) n and Q2 : free parameters Physical results should not depend on n and Q2 Need to choose suitable values Small Large n ○ Keep Cn small × Continuum dominated ○ Suppress high energy part of r(s) × Large Cn ○ Small n for resonance × Large Cn ○ Keep Cn small × Need large n to reach resonance Q2 6 Oct, 2011 QWG

19 Gluon condensates in medium
Appearance of the twist-2 gluon operator Relation to thermodynamic quantities Trace anomaly + traceless/symmetric term Energy-momentum tensor (caveat : pure gauge!) + Trace anomaly symmetric & traceless 6 Oct, 2011 QWG

20 Full QCD : Bazavov et al., ‘09
Gluon condensates Smoother but same amount of change in Full QCD Full QCD : Bazavov et al., ‘09 6 Oct, 2011 QWG

21 Borel Transformation in QCDSR
Large Q2 limit + Probing resonance (large n) Suppression of high energy part of r(s) Solve Dispersion relation = 6 Oct, 2011 QWG

22 Mathematical aspects (Bertlmann, ’82)
Limit of hypergeometric func. Whittaker function Results 6 Oct, 2011 QWG

23 OPE side : temperature dependence
T-dep Behavior of each OPE terms Scalar (bfb) Negative at low T G0 > 0 b < 0 Increase with T Twist-2 (cfc) Always positive Parameter 6 Oct, 2011 QWG

24 Results : S-wave states
Tonset = 1.07Tc Tonset = 1.04Tc hc starts to broaden earlier? 6 Oct, 2011 QWG

25 Combined with 2nd Order Stark effect
ALL quantities change abruptly around Tc!!! 6 Oct, 2011 QWG

26 A Lesson from (P+C) model
Varying only one parameter m2, G s0 6 Oct, 2011 QWG

27 Sensitivity to continuum
Too simple continuum? In QCDSR, continuum is well suppressed How about in the correlator? ITC is sensitive to what QCDSR is not. Note: we have imposed all higher states effects on continuum threshold. Explicit excited states (2s) enhances Grec in the model then reduce G/Grec 6 Oct, 2011 QWG

28 G’/G’rec : J/y Lattice data not available; but expected to be unity
Sensitive to spectral parameter : Strong constraint 6 Oct, 2011 QWG

29 G’/G’rec : cc1 Lattice data not available; but expected to be unity
even above Tonset 6 Oct, 2011 QWG

30 OPE in the heavy quark systems
Truncation at the leading order if , higher dimensional ones can be neglected Introducing temperature (Hatsuda-Koike-Lee,’93) Low enough / small change from vacuum value Temperature dependence only through operators = + q2 I G2 6 Oct, 2011 QWG

31 Temperature dependence from lattice QCD
Rapid change around Tc G2 quickly approaches to T 4 Max deviation from SB limit at 1.1Tc lattice data : Boyd et al, NPB469,419 (’96) 6 Oct, 2011 QWG


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