QCD sum rules in a Bayesian approach

1. QCD sum rules in a Bayesian approach arXiv: 1005.2459 [hep-ph] YIPQS workshop on “Exotics from Heavy Ion Collisions” 19.5.2010 @ YITP Philipp Gubler (TokyoTech) Collaborator: Makoto Oka (TokyoTech)

2. Contents • Basics of QCD sum rules • Basics of the Maximum Entropy Method (MEM) • A first application of the method to the ρmeson • Conclusions • Outlook (Possible further applications)

3. The basics of QCD sum rules In this method the properties of the two point correlation function is fully exploited: is calculated “perturbatively” spectral function of the operator χ After the Borel transformation:

4. With the help of the OPE, the non-local operator χ(x)χ(0) is expanded in a series of local operators On with their corresponding Wilson coefficients Cn: The theoretical (QCD) side: OPE As the vacuum expectation value of the local operators are considered, these must be Lorentz and Gauge invariant, for example:

5. ρ(s) s The phenomenological (hadronic) side: The imaginary part of Π(q2) is parametrized as the hadronic spectrum: This spectral function is often approximated as pole (ground state) plus continuum spectrum in QCD sum rules: Is this assumption always appropriate?

6. An example: the σ-meson channel: Spectrum with Breit-Wigner peak: Spectrum with ππscattering: T.Kojo and D. Jido, Phys. Rev. D 78, 114005 (2008).

7. ρ(s) s The phenomenological (hadronic) side: The imaginary part of Π(q2) is parametrized as the hadronic spectrum: This spectral function is approximated as pole (ground state) plus continuum spectrum in QCD sum rules: This assumption is not necessary when MEM is used!

8. Basics of the Maximum Entropy Method (1) A mathematical problem: given ? “Kernel” (but only incomplete and with error) This is an ill-posed problem. But, one may have additional information on ρ(ω), such as:

9. Basics of the Maximum Entropy Method (2) For example… - Lattice QCD: → M.Asakawa, T.Hatsuda and Y.Nakahara, Prog. Part. Nucl. Phys. 46, 459 (2001). Spectral function: Usually: - exponential fits, - variational method, …

10. Basics of the Maximum Entropy Method (3) or… - QCD sum rules: Usually: “pole + continuum”, …

11. Basics of the Maximum Entropy Method (4) How can one include this additional information and find the most probable image of ρ(ω)? → Bayes’ Theorem prior probability likelihood function

12. Basics of the Maximum Entropy Method (5) Likelihood function Gaussian distribution is assumed: Corresponds to ordinary χ2-fitting. Prior probability (Shannon-Jaynes entropy) “default model”

13. Basics of the Maximum Entropy Method (6) Summary Finding the most probable image of ρ(ω) corresponds to finding the maximum of αS[ρ] – L[ρ]. → Bryan’s method: R.K. Bryan, Eur. Biophys. J. 18, 165 (1990). • How is α determined? → The average is taken: determined using Bayes’ theorem - What about the default model m(ω)? → The dependence of the final result on the default model must be checked.

14. Application to the ρmeson channel One of the first and most successful application of QCD sum rules was the analysis of the ρ meson channel. The “pole + continuum” assumption works well in this case. Y. Kwon, M. Procura, and W. Weise, Phys. Rev. C 78, 055203 (2008). e+e-→ nπ (n: even) The experimental knowledge of the spectral function allows us generate realistic mock data.

15. Generating mock data: analyzed region Centred at Gmock(M), we generate gaussianly distributed values as an input of the analysis.

16. MEM artifacts, induced due to the sharply rising default model How is the default model chosen? Numerical results:

17. Why is it difficult to reproduce the width? Compared to mρ and Fρ, the width of the input spectral function is only poorly reproduced. The reason for this failure lies in the lack of sensitivity of Gmock(M) on the width. We conclude that the sum rule of the ρ-meson contains almost no information on the width, making it impossible to give any reliable prediction on its value.

18. Analysis of the OPE data: We use three parameter sets in our analysis: (from the Gell-Mann-Oakes-Renner relation)

19. 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 GOPE(M). D.B. Leinweber, Annals Phys. 322, 1949 (1996).

20. Experiment: mρ= 0.77 GeV Fρ= 0.141 GeV Results (1)

21. Results (2) The dependence of the ρ-meson properties on the values of the condensates:

22. Conclusions • We have shown that MEM can be applied to QCD sum rules • The “pole + continuum” ansatz is not a necessity • The properties of the experimentally observed ρ-meson peak are reproduced with a precision of 10%~30% (except width)

23. Outlook (Possible further applications) • Baryonic channels • Behavior of various hadrons at finite temperature or density • e.g. Charmonium • Tetraquarks • Pentaquarks scattering states ↔ resonances ?

24. Backup slides

25. What happens for a constant default model?

26. Dependence of the results on various parameters: on Mmax: on σ(M) and Mmin:

27. What happens in case of no input peak?

28. How is Fρ obtained?

29. Basics of the Maximum Entropy Method (4) Prior probability (1) Monkey argument: Probability of ni balls falling into position i: M balls Poisson distribution Probability of a certain image (n1, n2, …,nN): ni balls (probability: pi, expectation value: Mpi=λi)

30. Basics of the Maximum Entropy Method (5) Prior probability (2) To change the discrete image (n1, n2, …,nN) into a continuous function, one takes a small number q and defines: Then, the probability for the image A(ω) to be in Πi dAi becomes: (Shannon-Jaynes entropy) “default model”