1. Charmonium spectral functions at finite temperature from a Bayesian analysis of QCD sum rules
P. Gubler and M. Oka, Prog. Theor. Phys. 124, 995 (2010).
P. Gubler, K. Morita and M. Oka, in preparation.
Dense Strange Matter and Compressed Baryonic Matter
@ YITP, Kyoto University, Japan
Philipp Gubler (TokyoTech)
Collaborators: Makoto Oka (TokyoTech), Kenji Morita (YITP)

2. Contents QCD Sum Rules and the Maximum Entropy Method
Results of the analysis of charmonia at finite temperature
Conclusions and Outlook

3. QCD sum rules
M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147, 385 (1979); B147, 448 (1979).
In this method the properties of the two point correlation function is
fully exploited:
is calculated
“perturbatively”,
using OPE
spectral function
of the operator χ
After the Borel transformation:

4. The basic problem to be solved
given ?
“Kernel”
(but only incomplete and with error)
This is an ill-posed problem.
But, one may have additional information on ρ(ω), which can help to constrain the problem:
- Positivity:
- Asymptotic values:

5. The usual approach s
ρ(s)
The spectral function is usually assumed to be describable by a “pole + continuum” form (ground state + excited states):
This very crude ansatz often works surprisingly well, but…
This ansatz may not always be appropriate.
The dependence of the physical results on sth is often quite large.

6. The Maximum Entropy Method
→ Bayes’ Theorem
likelihood function
prior probability
(Shannon-Jaynes entropy)
Corresponds to ordinary χ2-fitting.
“default model”
M. Jarrel and J.E. Gubernatis, Phys. Rep. 269, 133 (1996).
M.Asakawa, T.Hatsuda and Y.Nakahara, Prog. Part. Nucl. Phys. 46, 459 (2001).

7. First applications in the light quark sector
ρ-meson channel
Nucleon channel
Experiment:
mρ= 0.77 GeV
Fρ= GeV
Experiment:
mN= 0.94 GeV
PG and M. Oka, Prog. Theor. Phys. 124, 995 (2010).
K. Ohtani, PG and M. Oka, in preparation.

8. Application to charmonium at finite temperature
- Prediction of “J/ψ Suppression by Quark-Gluon Plasma Formation”
T. Matsui and H. Satz, Phys. Lett. B 178, 416 (1986). … -
During the last 10 years, a picture has emerged from studies using lattice QCD (and MEM), where J/ψ survives above TC.
(schematic)
M. Asakawa and T. Hatsuda, Phys. Rev. Lett (2004).
S. Datta et al, Phys. Rev. D69, (2004).
T. Umeda et al, Eur. Phys. J. C39S1, 9 (2005). …
taken from
H. Satz, Nucl.Part.Phys. 32, 25 (2006).

9. The charmonium sum rules at T=0
The sum rule:
perturbative term including αs correction
Non-perturbative corrections including condensates up to dim 6
Developed and analyzed in:
M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147, 385 (1979); B147, 448 (1979).
L.J. Reinders, H.R. Rubinstein and S. Yazaki, Nucl. Phys. B 186, 109 (1981).
R.A. Bertlmann, Nucl. Phys. B 204, 387 (1982).
J. Marrow, J. Parker and G. Shaw, Z. Phys. C 37, 103 (1987).

10. MEM Analysis at T=0 mJ/ψ=3.06 GeV mηｃ=3.02 GeV
Here, the following values were used:
Experiment:
mJ/ψ= 3.10 GeV
mηc= 2.98 GeV

11. The charmonium sum rules at finite T
The application of QCD sum rules has been developed in:
T.Hatsuda, Y.Koike and S.H. Lee, Nucl. Phys. B 394, 221 (1993).
depend on T
A non-scalar twist-2 gluon condensates appears due to the non-existence of Lorentz invariance at finite temperature:
four-velocity of the medium

12. The T-dependence of the condensates
The energy-momentum tensor is considered:
After matching the trace part and the traceless part, one gets:
obtained from lattice QCD
These values are obtained from quenched lattice calculations:
G. Boyd et al, Nucl. Phys. B 469, 419 (1996).
O. Kaczmarek et al, Phys. Rev. D 70, (2004).
taken from:
K. Morita and S.H. Lee,
Phys. Rev. D82, (2010).
K. Morita and S.H. Lee, Phys. Rev. Lett. 100, (2008).
K. Morita and S.H. Lee, Phys. Rev. C 77, (2008).

13. The charmonium spectral function at finite T
Melting
Melting
negative shift of ~50 MeV,
consistent with the analysis of Morita and Lee.
negative shift of ~40 MeV,
consistent with the analysis of Morita and Lee.
Both J/ψ and ηc have melted completely.

14. Conclusions We have shown that MEM can be applied to QCD sum rules
The “pole + continuum” ansatz is not a necessity
We could observe the melting of the S-wave charmonia using finite temperature QCD sum rules and MEM
Both ηc and J/ψ seem to melt between T ~ 1.0 TC and T ~ 1.1 TC,, which is below the values obtained in lattice QCD

15. Outlook Analysis of P-wave states (χc0, χc1) Bottomium
Including higher orders (αs, twist)
Extending the method to investigations of other particles (D, …)