1. Quarkonia at finite T from QCD sum rules and MEM
and light vector mesons at finite density
Quarkonia at finite T from QCD sum rules and MEM
P. Gubler and M. Oka, Prog. Theor. Phys. 124, 995 (2010).
P. Gubler, K. Morita and M. Oka, Phys. Rev. Lett. 107, (2011).
K. Ohtani, P. Gubler and M. Oka, Eur. Phys. J. A 47, 114 (2011).
K. Ohtani, P. Gubler and M. Oka, Phys. Rev. D 87, (2013).
K. Suzuki, P. Gubler、K. Morita and M. Oka, Nucl. Phys. A897, 28 (2013).
NFQCD
@ YITP, Kyoto University
Philipp Gubler (RIKEN, Nishina Center)
Collaborators:
Makoto Oka (TokyoTech), Kenji Morita (YITP), Keisuke Ohtani (TokyoTech),
Kei Suzuki (TokyoTech)

2. Contents Introduction, Motivation
The method: QCD sum rules and the maximum entropy method
Quarkonia at finite temperature
Light vector mesons in nuclear matter
Conclusions

3. Introduction: Hadrons in hot or dense matter
General Motivation: Understanding the behavior of matter at high T and μ.
QGP (T>Tc) ↔ confining phase (T<Tc)
Quarkonium suppression
Partial restoration of chiral symmetry in nuclear matter
Mass shift of light vector mesons ?
Broadening ?

4. 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:

5. More on the OPE in matter
non-perturbative condensates
perturbative Wilson coefficients
Change in hot or dense matter!

6. 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:

7. The Maximum Entropy Method
How can one include this additional information and find the most probable image of ρ(ω)?
→ Bayes’ Theorem
likelihood function
prior probability

8. 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).

9. A first test: mock data analysis
When only free c-quarks contribute to the spectral function, this should be reproduced in the MEM analysis.
Both J/ψ and ψ’ are included into the mock data, but we can only reproduce J/ψ.

10. First applications in the light quark sector
ρ-meson channel
Nucleon channel +
970 MeV -
1570 MeV
Experiment:
mρ= 0.77 GeV
Fρ= GeV
Experiment:
mN+= 0.94 GeV
mN- = 1.54 GeV
PG and M. Oka, Prog. Theor. Phys. 124, 995 (2010).
K. Ohtani, PG and M. Oka, Eur. Phys. J. A 47, 114 (2011). K. Ohtani, PG and M. Oka, Phys. Rev. D 87, (2013).

11. Quarkonium at finite T

12. The quarkonium sum rules at finite T
The application of QCD sum rules has been developed in:
A.I. Bochkarev and M.E. Shaposhnikov, Nucl. Phys. B 268, 220 (1986).
T. Hatsuda, Y. Koike and S.H. Lee, Nucl. Phys. B 394, 221 (1993).
depend on T

13. The T-dependence of the condensates
K. Morita and S.H. Lee, Phys. Rev. Lett. 100, (2008).
Considering the trace and the traceless part of the
energy momentum tensor, one can show that in tht quenched approximation, the T-dependent parts of the gluon condensates by thermodynamic quantities
such as energy density ε(T) and pressure p(T).
The values of ε(T) and p(T) 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).

14. Results for charmonium at T=0
S-wave
P-wave
mJ/ψ=3.06 GeV (Exp: 3.10 GeV)
mχ0=3.36 GeV (Exp: 3.41 GeV)
mηｃ=3.02 GeV (Exp: 2.98 GeV)
mχ1=3.50 GeV (Exp: 3.51 GeV)
PG, K. Morita and M. Oka, Phys. Rev. Lett. 107, (2011).

15. Results for charmonium at finite T
S-wave
P-wave
PG, K. Morita and M. Oka, Phys. Rev. Lett. 107, (2011).

16. Comparison with lattice results
Imaginary time correlator ratio:
Lattice data are taken from: A. Jakovác, P. Petreczky, K. Petrov and A. Velytsky, Phys. Rev. D 75, (2007).

17. Results for bottomonium at finite T
S-wave
P-wave
K. Suzuki, PG, K. Morita and M. Oka, Nucl. Phys. A897, 28 (2013).

18. The φ meson in dense matter

19. Light vector mesons at finite density
Important early study:
T. Hatsuda and S.H. Lee, Phys. Rev. C 46, R34 (1992).
Vector meson masses mainly drop due to changes of the quark condensates.
The most important condensates are:
for
for
Important assumption:
Might be wrong!

20. What has changed since 1992?
Vacuum
No fundamental changes since 1992

21. What has changed since 1992? Finite density effects
May have changed a bit
Finite density effects
Has changed a lot!!
Has changed a little
New term
Has changed a little

22. What has changed since 1992?
A.D. Martin, W.J. Stirling, R.S. Thorne and G. Watt, Eur. Phys. J. C 63, 189 (2009).
1992
2013
Some changes, but no big effect.
non-negligible

23. What has changed since 1992? The strangeness content of the nucleon:
Taken from M. Gong et al. (χQCD Collaboration), arXiv: [hep-ph].
y ~ 0.04
Value used by Hatsuda and Lee: y= Too big!!
The value of y has shrinked by a factor of about 5: a new analysis is necessary!

24. First results (vacuum)
Analysis of the sum rule is done using the maximum entropy method (MEM), which allows to extract the spectral function from the sum rules without any phenomenological ansatz.
mφ=1.05 GeV (Exp: 1.02 GeV)
mρ=0.75 GeV (Exp: 0.77 GeV)
Used input parameters:

25. φ meson at finite density
ruled out !?
The φ meson mass shift strongly depends on the strange sigma term.

26. What happened?
Let us examine the OPE at finite density more closely:
Measuring the φ meson mass shift in nuclear matter provides a strong constraint to the strange sigma term!

27. Relation between the φ meson mass shift and the strange sigma term

28. However…
Experiments seem to suggest something else:
Result of the E325 experiment at KEK
35 MeV mass reduction of the φ meson at nuclear matter density!
R. Muto et al, Phys. Rev. Lett. 98, (2007).

29. What could be wrong? 1. So far neglected condensates
Terms containing higher orders of ms and other so far neglected terms could have a non-negligible effect.
Effects of these terms are small
2. αs corrections
These corrections seem to be small
3. Underestimated density dependence of four-quark condensates
Does not seem to have a large effect.

30. Conclusions We have shown that MEM can be applied to QCD sum rules
We could observe the melting of the S-wave and P-wave quarkonia using finite temperature QCD sum rules and MEM
Using recent lattice results on the strange sigma term, we have shown that the φ-meson mass presumably experiences a smaller shift than thought before.

31. Backup slides

32. What about the excited states?
S. Chatrchyan et al. [CMS Collaboration],
Phys. Rev. Lett. 107, (2011).
Exciting results from LHC!
Is it possible to reproduce this result with our method?
Our resolution might not be good enough.

33. Extracting information on the excited states
However, we can at least investigate the behavior of the residue as a function of T.
Fit using a Breit-Wigner peak + continuum
Fit using a Gaussian peak + continuum
A clear reduction of the residue independent on the details of the fit is observed.
Consistent with melting of Y(3S) and Y(2S) states ?

34. What has changed since 1992? Value used by Hatsuda and Lee: 45 MeV
Taken from G.S. Bali et al., Nucl. Phys. B866, 1 (2013).
Value used by Hatsuda and Lee: 45 MeV
Recent lattice results are mostly consistent with the old values. The latest trend might point to a somewhat smaller value.

35. First results (ρ meson at finite density)
200 MeV
0.12 ~ 0.19
Consistent with the result of Hatsuda-Lee.
Used input parameters:
A. Semke and M.F.M. Lutz, Phys. Lett. B 717, 242 (2012).
M. Procura, T.R. Hemmert and W. Weise, Phys. Rev. D 69, (2004).