1. Recent results from QCD sum rule analyses based on the maximum entropy method International Symposium on Chiral Symmetry in Hadrons and Nuclei @Beihang University, Beijing, China 29.10.2013 Philipp Gubler (RIKEN, Nishina Center) Collaborators: Makoto Oka (TokyoTech), Kenji Morita (YITP), Keisuke Ohtani (TokyoTech), Kei Suzuki (TokyoTech) P. Gubler and M. Oka, Prog. Theor. Phys. 124, 995 (2010). P. Gubler, K. Morita and M. Oka, Phys. Rev. Lett. 107, 092003 (2011). K. Ohtani, P. Gubler and M. Oka, Eur. Phys. J. A 47, 114 (2011). K. Suzuki, P. Gubler 、 K. Morita and M. Oka, Nucl. Phys. A897, 28 (2013). K. Ohtani, P. Gubler and M. Oka, Phys. Rev. D 87, 034027 (2013).

2. Contents Introduction, Motivation  Basics of QCD sum rules  Why is the maximum entropy method (MEM) useful for QCD sum rules? Results for various systems  Quarkonium at finite temperature  Light vector mesons at finite density Conclusions and Outlook

3. QCD sum rules 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: M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147, 385 (1979); B147, 448 (1979).

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

5. This spectral function is approximated by a “pole + continuum” ansatz: s ρ(s) The traditional analysis method: s th Even though this ansatz is very crude, it works quite well in cases for which it is phenomenologically known to be close to reality. e.g.- charmonium (J/ψ) -ρ-meson OPE result

6. This ansatz can, however, not always work! T=0T>0 For instance, for: A more general analysis method is needed MEM analysis of QCD sum rules could be useful!

7. The basic problem to be solved given (but only incomplete and with error) ? “Kernel” 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:

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

9. likelihood functionprior probability M.Asakawa, T.Hatsuda and Y.Nakahara, Prog. Part. Nucl. Phys. 46, 459 (2001). M. Jarrel and J.E. Gubernatis, Phys. Rep. 269, 133 (1996). Corresponds to ordinary χ 2 -fitting. (Shannon-Jaynes entropy) “default model”

10. First applications in the light quark sector Experiment: m ρ = 0.77 GeV F ρ = 0.141 GeV PG and M. Oka, Prog. Theor. Phys. 124, 995 (2010). ρ-meson channel Nucleon channel Experiment: m N = 0.94 GeV K. Ohtani, PG and M. Oka, Eur. Phys. J. A 47, 114 (2011).

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.I. Bochkarev and M.E. Shaposhnikov, Nucl. Phys. B 268, 220 (1986). G. Boyd et al, Nucl. Phys. B 469, 419 (1996). O. Kaczmarek et al, Phys. Rev. D 70, 074505 (2004). For ε(T) and p(T), quenched lattice data are used:

12. The charmonium spectral function at finite T PG, K. Morita and M. Oka, Phys. Rev. Lett. 107, 092003 (2011). S-waveP-wave

13. Results for bottomonium K. Suzuki, PG, K. Morita and M. Oka, Nucl. Phys. A897, 28 (2013). S-waveP-wave

14. Introduction: Vector mesons at finite density Understanding the behavior of matter under extreme conditions - To be investigated at J-PARC - Vector mesons: clean probe for experiment - Firm theoretical understanding is necessary for interpreting the experimental results! Basic Motivation: Understanding the origin of mass and its relation to chiral symmetry of QCD

15. Vector mesons at finite density Vacuum

16. Density effects on the condensates

17. The strangeness content of the nucleon: recent developments Taken from M. Gong et al. (χQCD Collaboration), arXiv:1304.1194 [hep-ph]. Value used by Hatsuda and Lee: y=0.2 Too big!! y ~ 0.04 The value of y has shrinked by a factor of about 5: a new analysis is necessary!

18. φ meson at finite density 0.0 ~ 0.008 ruled out !? Measuring the φ meson mass shift in nuclear matter provides a strong constraint to the strangeness content of the nucleon.

19. Conclusions We have shown that MEM can be applied to QCD sum rules The “pole + continuum” is not a necessity We could observe the melting of the S-wave and P-wave quarkonia and estimated the corresponding melting temperatures 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.

20. Backup slides

21. Estimation of the error of G(M) Gaussianly distributed values for the various parameters are randomly generated. The error is extracted from the resulting distribution of G OPE (M). D.B. Leinweber, Annals Phys. 322, 1949 (1996). PG, M. Oka, Prog. Theor. Phys. 124, 995 (2010).

22. Results (2) The dependence of the ρ-meson properties on the values of the condensates: PG, M. Oka, Prog. Theor. Phys. 124, 995 (2010).

23. Results from lattice QCD During the last 10 years, a picture has emerged from studies using lattice QCD (and MEM), where J/ψ survives above T c, but dissolves below 2 T c. - (schematic) Taken from H. Satz, Nucl.Part.Phys. 32, 25 (2006). M. Asakawa and T. Hatsuda, Phys. Rev. Lett. 92 012001 (2004). S. Datta et al, Phys. Rev. D69, 094507 (2004). T. Umeda et al, Eur. Phys. J. C39, 9 (2004). A. Jakovác et al, Phys. Rev. D75, 014506 (2007). G. Aarts et al, Phys. Rev. D 76, 094513 (2007). H.-T. Ding et al, PoS LAT2010, 180 (2010). However, there are also indications that J/ψ survives up to 2 T c or higher. - H. Iida et al, Phys. Rev. D 74, 074502 (2006). H. Ohno et al, PoS LAT2008, 203 (2008).

24. The T-dependence of the condensates taken from: K. Morita and S.H. Lee, Phys. Rev. D82, 054008 (2010). G. Boyd et al, Nucl. Phys. B 469, 419 (1996). O. Kaczmarek et al, Phys. Rev. D 70, 074505 (2004). The values of ε(T) and p(T) are obtained from quenched lattice calculations: K. Morita and S.H. Lee, Phys. Rev. Lett. 100, 022301 (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).

25. What is going on behind the scenes ? T=0T=1.0 T c T=1.1 T c T=1.2 T c The OPE data in the Vector channel at various T: cancellation between α s and condensate contributions

26. First results (ρ meson at finite density) 200 MeV 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, 034505 (2004).