Quarkonium at finite Temperature from QCD Sum Rules and the Maximum Entropy Method Seminar at the Komaba Nuclear Theory Tokyo University Philipp Gubler (TokyoTech) 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, (2011). K. Ohtani, P. Gubler and M. Oka, Eur. Phys. J. A 47, 114 (2011).

Contents Introduction, Motivation The method: QCD sum rules and the maximum entropy method Results of the analysis of charmonia at finite temperature First results for bottomonia Conclusions and outlook

Introduction: Quarkonia General Motivation: Understanding the behavior of matter at high T. - Phase transition: QGP (T>T c ) ↔ confining phase (T<T c ) - Currently investigated at RHIC and LHC - Heavy Quarkonium: clean probe for experiment

Why are quarkonia useful? Prediction of J/ψ suppression above T c due to Debye screening: T. Matsui and H. Satz, Phys. Lett. B 178, 416 (1986). T. Hashimoto et al., Phys. Rev. Lett. 57, 2123 (1986). Lighter quarkonia melt at low T, while heavier ones melt at higher T → Thermometer of the QGP

Some Experimental Results S. Chatrchyan et al. [CMS Collaboration], Phys. Rev. Lett. 107, (2011). Evidence for both J/ψ and and excited states of Y are being observed. L. Kluberg and H. Satz, arXiv: [hep-ph]. Taken from:

Results from lattice QCD (1) 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 (2004). S. Datta et al, Phys. Rev. D69, (2004). T. Umeda et al, Eur. Phys. J. C39, 9 (2004). A. Jakovác et al, Phys. Rev. D75, (2007). G. Aarts et al, Phys. Rev. D 76, (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, (2006). H. Ohno et al, Phys. Rev. D 84, (2011).

Results from lattice QCD (2) Recently, first dedicated studies trying to analyze the behavior of bottomonium at finite T have appearaed. - G. Aarts et al, Phys. Rev. Lett 106, (2011). G. Aarts et al, JHEP 1111, 103 (2011). Melting of excited states just above T c ?

Problems on the lattice Number of avilable data points is reduced as T increases  Reproducability, Resolution of MEM changes with T Finite Box → No continuum  Several Volumes are needed Behavior of constant contribution to the correlator (transport peak at low energy) may influence the results  Needs to be carefully subtracted

The basics of QCD sum rules Asymptotic freedom

As the coupling constant increases at small energy scales, the perturbative expansion using Feynman diagrams breaks down in this region and one has to look for other methods. One example of such a non-perturbative method is the QCD sum rule approach: In the region of Π(q) dominated by large energy scales such as It can be calculated by some perturbative method, the Operator product expansion (OPE), which will be explained later.

On the other hand, in the region of q 2 >0 Π(q 2 ) contains information about real hadrons that couple to the current χ(x): These two regions of Π(q) are connected with the help of a dispersion relation: After the Borel transormation:

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

The phenomenological (hadronic) side: The imaginary part of Π(q 2 ) 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! s ρ(s)

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

Basics of the Maximum Entropy Method (2) How can one include this additional information and find the most probable image of ρ(ω)? → Bayes’ Theorem likelihood function prior probability

Basics of the Maximum Entropy Method (3) Likelihood function Gaussian distribution is assumed: Prior probability (Shannon-Jaynes entropy) “default model” Corresponds to ordinary χ 2 -fitting.

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

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

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 Compared to lattice: No reduction of data points that can be used for the analysis, allowing a direct comparison of T=0 and T≠0 spectral functions. A.I. Bochkarev and M.E. Shaposhnikov, Nucl. Phys. B 268, 220 (1986).

＋ ・ 1st-term ・ 2nd-term （ α s correction ） OPE in Feynman diagrams (pert.)

OPE in Feynman diagrams (non-pert.) ・ 3rd-term and 4th-term （ gluon condensate ）

The T-dependence of the condensates taken from: K. Morita and S.H. Lee, Phys. Rev. D82, (2010). G. Boyd et al, Nucl. Phys. B 469, 419 (1996). O. Kaczmarek et al, Phys. Rev. D 70, (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, (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).

MEM Analysis at T=0 m η ｃ =3.02 GeV (Exp: 2.98 GeV) S-wave P-wave m J/ψ =3.06 GeV (Exp: 3.10 GeV) m χ0 =3.36 GeV (Exp: 3.41 GeV)m χ1 =3.50 GeV (Exp: 3.51 GeV) PG, K. Morita and M. Oka, Phys. Rev. Lett. 107, (2011).

The charmonium spectral function at finite T PG, K. Morita and M. Oka, Phys. Rev. Lett. 107, (2011).

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

Where might we have problems? Higher order gluon condensates?  Probably not a problem, but needs to be checked Higher orders is α s ?  Maybe. Can be tested for vector channel Division between high- and low-energy contributions in OPE?  Clould be a problem at high T. Needs to be investigated carefully.

First results for bottomonium Υηbηb χ b0 χ b1 S-wave P-wave Calculated by K. Suzuki Preliminary

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

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 + continuumFit using a Gaussian peak + continuum Preliminary 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 ?

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 charmonia using finite temperature QCD sum rules and MEM J/ψ, η c, χ c0, χ c1 melt between T ~ 1.0 T c and T ~ 1.2 T c, which is below the values obtained in lattice QCD As for bottomonium, we have found preliminary evidence for the melting of Y(2S) and Y(3S) between 1.5 T c and 2.5 T c, while Y(1S) survives until 3.0 T c or higher. Furthermore,η b melts at around 3.0 T c, while χ b0 and χ b1 melt at around 2.0 ~ 2.5 T c

Outlook Check possible problems of our method  α s, higher twist, division of scale Calculate higher order gluon condensates on the lattice Try a different kernel with better resolution (“Gaussian sum rules”) Extend the method to other channels

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