Critical Behavior of heavy quarkonia in medium from QCD sum rules
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1 Critical Behavior of heavy quarkonia in medium from QCD sum rules Kenji Morita Institute of Physics and Applied Physics, Yonsei University in Collaboration with Su Houng Lee
2 Aim of this talk: 1. Practical procedure of QCDSR calculation 1. For introduction : S.H.Lee s talk 2. Borel transformation technique 3. How to extract spectral properties 2. Temperature dependence of spectral parameters 1. Change across the phase transition : how change of gluon condensates converts into quarkonia? 3. Construction of the imaginary time correlator 1. Reconstructed spectral function 2. Imaginary time correlator 3. Comparison with lattice QCD results Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 2
3 Strategy Current Correlation Π(Q 2 ) Thermodynamic quantities / Coupling constant Lattice QCD Q 2 =-q 2 0 Gluon Condensates Numerical Simulation OPE QCDSR Imaginary Time Correlator Dispersion relation (II) with optimization Dispersion relation (I) MEM Pole+Continuum Spectral function Goal Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 3
4 QCD Sum Rules for Heavy Quarkonium Current-current correlation function Take spacelike momentum : q 2 = -Q 2 < 0 OPE and truncation valid for: Temperature effect only through condensates Meson at rest with respect to medium: q = (ω,0)( Longitudinal and transverse polarizations are no longer independent Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 4
5 Gluon condensates Smoother but same amount of change in Full QCD Full QCD : Cheng et al., 08 Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 5
6 Borel Transformation in QCDSR Physical meaning Large Q 2 limit + Probing resonance (large n) Suppression of high energy part of ρ(s) Dispersion relation Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 6
7 OPE side : temperature dependence T-dep Behavior of each OPE terms Scalar (bφ( b ) Negative at low T G 0 >0 b<0 Increase with T Twist-2 2 (cφ( c ) Always positive Increase with T Parameter Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 7
8 Sum rule constraints From the OPE... f Increase T m 2, Γ s 0 If other quantities are fixed: m 2 : decrease Γ : Increase s 0 : decrease f : Increase Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 8
9 Borel Window : where QCDSR works well Borel Window : M 2 range such that... Criterion 1 OPE convergence in the Window Power correction is small enough M 2 min Criterion 2 Pole should dominate M 2 max Criterion 3 Mass should not depend on M 2, or must have local mininum/maximum Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 9
10 How to choose the best solution? Searching for (m,γ,s,s 0 ) giving the flattest Borel curve Caveats Solution is not unique! Many combination can give similarly flat curve! Need to fix either Γ or s 0 Changing s 0 M 2 max changes This method seems to work only when either M 2 min or M 2 max is fixed Useful for determine Γ and m at a fixed T and s 0 Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 10
11 Systematics of Borel curve modification T=0, Γ=0, larger s 0 T=0, Γ=0, moderate s 0 T=0, Γ>0, moderate s 0 Favored m M 2 min M 2 M 2 max M 2 max M 2 max Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 11
12 Systematics of Borel curve modification T=0, Γ=0, moderate s 0 T~T c, Γ=0, moderate s 0 T~T c, Γ>0, moderate s 0 T~T c, Γ=0, smaller s 0 favored m M 2 min M 2 min M 2 min M 2 min M 2 M 2 max M2 M 2 max max M 2 max Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 12
13 Systematics of Borel curve modification T~T c, Γ=0, moderate s 0 T>1.05T c, Γ=0, moderate s 0 T>1.05T c, Γ=0, smaller s 0 T>1.05T c, Γ>0, moderate s 0 Two favored. smaller χ 2 depends on the case m M 2 min M 2 min M 2 max M 2 M 2 max Minimum exists only if width or threshold changes Beyond T onset, width is necessary for obtaining minimum Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 13
14 Results for Charmonia Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 14
15 Results : S-wave S states : vacuum results T onset = 1.07T c η c starts to broaden earlier? T onset = 1.04T c Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 15
16 Results : P-wave P states T onset = 1.05T c T onset = 1.05T c Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 16
17 Indication from the results Constraints among m, Γ and s 0 One needs to fix m or s 0 (not Γ) Effect of temperature : as T increases, Mass should decrease around T c 2 nd s 0 nd order Stark effect (Lee( Caveat : Lee-Morita PRD75) T>T onset, higher dim. condensates will be important T>T onset m or Γ Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 17
18 Combined with 2 nd Order Stark effect ALL quantities change abruptly around T c!!! Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 18
19 Combined with 2 nd Order Stark effect Beyond this point, Stark effect mass shift becomes too large Note: Precisely applicable up to 1.1Tc ALL quantities change abruptly around T c!!! Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 19
20 Comparison with lattice QCD? MEM spectral density : too coarse, especially above T c Spectral modification from QCDSR : O(100MeV) More accurate quantity : Imaginary time correlator Dispersion relation We can put the phenomenological side used in QCDSR. Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 20
21 Modeled spectral density Parameters from the result combined with Stark effect Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 21
22 Imaginary time correlator : results Looking at medium modification J/ψ, T=0.87T c =257 MeV Deviation at τ > 0.2fm Below T c : zero-mode contribution should be different Lattice : Jakovac et al., PRD75, ( 07) Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 22
23 Imaginary time correlator : results Looking at medium modification J/ψ, T=0.87T c =257 MeV Neglecting zero mode Agree with lattice when continuum threshold does not change (no mass change in this case) Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 23
24 Imaginary time correlator : model study Varying only one parameter Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 24
25 Imaginary time correlator : J/ψ above T c J/ψ, T=1.07T c =316 MeV Agreement in lower s 0 cases at τ < 0.2 Zero mode contribution seems overestimated? Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 25
26 Imaginary time correlator : J/ψ above T c J/ψ, T=1.07T c =316 MeV Agreement in lower s 0 cases at τ < 0.2 Zero mode contribution seems overestimated? Also consistent with Stark effect w/o zero mode, G/G rec ~1 is possible with a mixture of pole and continuum modification Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 26
27 Imaginary time correlator : J/ψ above T c J/ψ, T=1.09T c =322 MeV lattice data : finest spacing Agreement becomes worse Zero mode contribution seems overestimated? Caveat : T=1.09T c is above the onset of the width Improvement needed Correct zero mode More modification needed Check zero mode through χ states Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 27
28 Imaginary time correlator : J/ψ above T c J/ψ, T=1.09T c =322 MeV lattice data : finest spacing Agreement becomes worse Zero mode contribution seems overestimated? Caveat : T=1.09T c is above the onset of the width Improvement needed Correct zero mode More modification needed Check zero mode through χ states Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 28
29 Imaginary time correlator : χ c above T c Zero mode contribution dominates Agree with lattice data but not sensitive to detailed spectral change Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 29
30 Imaginary time correlator : χ c above T c Zero mode contribution dominates; in free gas approximation, it increases at T increases Larger than lattice : change from 1.07T c? Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 30
31 Imaginary time correlator : η c above T c No zero mode in PS channel Larger than lattice : different from J/ψ Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 31
32 Discussion Too simple continuum? In QCDSR, continuum is well suppressed How about in the correlator? ITC might be sensitive to what QCDSR is not. Note: we have imposed all higher states effects on continuum threshold. Explicit excited states (2s) will enhance G rec in the model then reduce G/G rec Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 32
33 Relative contribution from each ρ i (ω) In G rec, temperature effect is small at T=1.07T c Continuum dominates the correlator Zero mode < 20% in V channel Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 33
34 Relative contribution from each ρ i (ω) In G rec, temperature effect is small at T=1.07T c Continuum dominates the correlator Zero mode contribution dominates at large τ Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 34
35 Relative contribution from each ρ i (ω) In G rec, temperature effect is small at T=1.07T c Continuum dominates the correlator at τ < 0.2fm Zero mode dominates at τ > 0.25fm Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 35
36 Results for Bottomonia: Condensates contribution to OPE is much smaller : (m b /m c ) 4 ~100 m b = 4.15 GeV, α s (8m 2 b )=0.12 M 2 min : 5% dim.4 contribution Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 36
37 Spectral modification above T c T onset = 2.09T c T onset = 1.96T c Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 37
38 Combined with 2 nd order Stark effect Both seem to exist at 2T c Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 38
39 Spectral modification above T c T onset = 1.52T c T onset = 1.46T c Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 39
40 Imaginary time correlator : Υ Agree with lattice data at T=1.15T c : δm = -25MeV, Γ=0, Deviation at T=1.54Tc Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 40
41 Imaginary time correlator : η b Agree with lattice data at T=1.15T c : δm = -18MeV, Γ=0, Deviation at T=1.54Tc Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 41
42 Imaginary time correlator : χ b0 Agree with coarser lattice, not with finer one Deviation at T=1.54Tc Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 42
43 Summary and Outlook QCDSR gives constraints on the spectral modification Instability of the Borel curve indicates onset of broadening QCDSR : translates change of condensates into pole Charmonium : sudden changes across T c Bottomonium : moderate changes above T c G/G rec : more sensitive to continuum QCDSR-constrained modification is not inconsistent with lattice G/G G/G rec Decreasing threshold, mass, and overlap constant / Increasing width Agreement might be accidental For clarification, we need more realistic continuum model including ing excited states and thermal effects, which do NOT affect QCDSR Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 43
44 A trial Including ψ at T=0 (using exp. data from PDG) QCDSR re-fit: only 0.3% change of mass and 10% increase of continuum threshold Imaginary time correlator at T=0: 3% change Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 44
45 Backup Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 45
46 OPE in the heavy quark systems Truncation at the leading order q 2 = I G 2 if, higher dimensional ones can n be neglected Introducing temperature (Hatsuda (Hatsuda-Koike-Lee, Lee, 93) Low enough / small change from vacuum value Temperature dependence only through operators Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 46
47 Mathematical aspects (Bertlmann, 82) Limit of hypergeometric func. Results Whittaker function Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 47
48 QCD sum rules Hadronic spectral density (what we want to know) OPE at large Q 2 m 2 s 0 Matching to obtain m 2 and s 0 R.H.S is just a SUM of spectral density, but there are many ways to optimize it! Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 48
49 Gluon condensates in medium Appearance of the twist-2 2 gluon operator + Relation to thermodynamic quantities Trace anomaly + traceless/symmetric term Energy-momentum tensor Trace anomaly symmetric & traceless Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 49
50 Temperature dependence from lattice QCD Rapid change around T c G 2 quickly approaches to T 4 Max deviation from SB limit at 1.1T c lattice data : Boyd et al, NPB469,419 ( 96) Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 50
51 Examples for χ 2 evaluation (T=0)( Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 51
52 Examples for χ 2 evaluation (T=1.0T( c ) Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 52
53 Examples for χ 2 evaluation (T=1.07T( c ) Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 53
54 Imaginary time correlator : Υ Less continuum dominant than J/ψ T=1.15T c is similar to T=0 when G/G rec ~1 Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 54
55 Imaginary time correlator : η b Agree with lattice data at T=1.15T c : δm = -18MeV, Γ=0, Deviation at T=1.54Tc Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 55
56 Imaginary time correlator : χ b0 Agree with coarse lattice but with finer one Deviation at T=1.54Tc Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 56
57 Effect of zero-mode Landau damping contributes the correlator Assuming free thermal quarks in medium We use Thermal distribution of heavy quark Caveat : continuum part may be too simple! We imposed all medium effects on the change of threshold Jun 18, 2009 Joint CATHIE-INT mini-program Quarkonium in Hot Media : from QCD to Experiment 57
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