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
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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
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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)
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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
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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
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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)
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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
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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
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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)
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11. Peak quickly dissolves above Tc
Charmonia
Peak quickly dissolves above Tc
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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
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13. Bottomonia (1P)
Dissolved at 2-2.5Tc
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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
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15. Backup
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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
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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)
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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
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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
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20. Full QCD : Bazavov et al., ‘09
Gluon condensates
Smoother but same amount of change in Full QCD
Full QCD : Bazavov et al., ‘09
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21. Borel Transformation in QCDSR
Large Q2 limit + Probing resonance (large n)
Suppression of high energy part of r(s)
Solve
Dispersion relation =
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22. Mathematical aspects (Bertlmann, ’82)
Limit of hypergeometric func.
Whittaker function
Results
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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
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24. Results : S-wave states
Tonset = 1.07Tc
Tonset = 1.04Tc
hc starts to broaden earlier?
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25. Combined with 2nd Order Stark effect
ALL quantities change abruptly around Tc!!!
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26. A Lesson from (P+C) model
Varying only one parameter
m2, G s0
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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
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28. G’/G’rec : J/y Lattice data not available; but expected to be unity
Sensitive to spectral parameter : Strong constraint
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29. G’/G’rec : cc1 Lattice data not available; but expected to be unity
even above Tonset
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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
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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)
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