Quarkonium at finite Temperature from QCD Sum Rules
- Slides: 32
Quarkonium at finite Temperature from QCD Sum Rules and the Maximum Entropy Method 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). Seminar at the Komaba Nuclear Theory Group @ Tokyo University 7. 12. 2011 Philipp Gubler (Tokyo. Tech) Collaborators: Makoto Oka (Tokyo. Tech), Kenji Morita (YITP), Keisuke Ohtani (Tokyo. Tech), Kei Suzuki (Tokyo. Tech)
Contents n n n 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>Tc) ↔ confining phase (T<Tc) - Currently investigated at RHIC and LHC - Heavy Quarkonium: clean probe for experiment
Why are quarkonia useful? Prediction of J/ψ suppression above Tc 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 Evidence for both J/ψ and excited states of Y are being observed. S. Chatrchyan et al. [CMS Collaboration], Phys. Rev. Lett. 107, 052302 (2011). Taken from: L. Kluberg and H. Satz, ar. Xiv: 0901. 3831 [hep-ph].
Results from lattice QCD (1) M. Asakawa and T. Hatsuda, Phys. Rev. Lett. 92 012001 (2004). S. Datta et al, Phys. Rev. D 69, 094507 (2004). T. Umeda et al, Eur. Phys. J. C 39, 9 (2004). (schematic) - During the last 10 years, a picture has emerged from studies using lattice QCD (and MEM), where J/ψ survives above Tc, but dissolves below 2 Tc. A. Jakovác et al, Phys. Rev. D 75, 014506 (2007). G. Aarts et al, Phys. Rev. D 76, 094513 (2007). H. -T. Ding et al, Po. S LAT 2010, 180 (2010). - However, there also indications that J/ψ survives up to 2 Tc or higher. H. Iida et al, Phys. Rev. D 74, 074502 (2006). H. Ohno et al, Phys. Rev. D 84, 094504 (2011). Taken from H. Satz, Nucl. Part. Phys. 32, 25 (2006).
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, 061602 (2011). G. Aarts et al, JHEP 1111, 103 (2011). Melting of excited states just above Tc?
Problems on the lattice n Number of avilable data points is reduced as T increases q n Finite Box → No continuum q n Reproducability, Resolution of MEM changes with T Several Volumes are needed Behavior of constant contribution to the correlator (transport peak at low energy) may influence the results q 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 On with their corresponding Wilson coefficients Cn: 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: ρ(s) s 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!
Basics of the Maximum Entropy Method (1) A mathematical problem: given (but only incomplete and with error) “Kernel” ? This is an ill-posed problem. But, one may have additional information on ρ(ω), such as:
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 Likelihood function (3) Gaussian distribution is assumed: Corresponds to ordinary χ2 -fitting. Prior probability (Shannon-Jaynes entropy) “default model”
First applications in the light quark sector ρ-meson channel Experiment: mρ= 0. 77 Ge. V Fρ= 0. 141 Ge. V PG and M. Oka, Prog. Theor. Phys. 124, 995 (2010). Nucleon channel Experiment: m. N= 0. 94 Ge. V 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: 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 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.
OPE in Feynman diagrams (pert. ) ・ 1 st-term ・ 2 nd-term (αs correction) +
OPE in Feynman diagrams (nonpert. ) ・ 3 rd-term and 4 th-term (gluon condensate)
The T-dependence of the condensates 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). 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, 074505 (2004). taken from: K. Morita and S. H. Lee, Phys. Rev. D 82, 054008 (2010).
MEM Analysis at T=0 S-wave m. J/ψ=3. 06 Ge. V (Exp: 3. 10 Ge. V) mηc=3. 02 Ge. V (Exp: 2. 98 Ge. V) P-wave mχ0=3. 36 Ge. V (Exp: 3. 41 Ge. V) PG, K. Morita and M. Oka, Phys. Rev. Lett. 107, 092003 (2011). mχ1=3. 50 Ge. V (Exp: 3. 51 Ge. V)
The charmonium spectral function at finite T PG, K. Morita and M. Oka, Phys. Rev. Lett. 107, 092003 (2011).
What is going on behind the scenes ? The OPE data in the Vector channel at various T: T=1. 0 Tc T=0 T=1. 1 T=1. 2 cancellation between αs and condensate contributions
Where might we have problems? n Higher order gluon condensates? q n Higher orders is αs? q n Probably not a problem, but needs to be checked Maybe. Can be tested for vector channel Division between high- and low-energy contributions in OPE? q Clould be a problem at high T. Needs to be investigated carefully.
Pre li First results for bottomonium min Υ ηb S-wave χb 0 P-wave Calculated by K. Suzuki χb 1 ary
What about the excited states? Exciting results from LHC! Is it possible to reproduce this result with our method? Our resolution might not be good enough. S. Chatrchyan et al. [CMS Collaboration], Phys. Rev. Lett. 107, 052302 (2011).
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 Pre li min ary A clear reduction of the residue independent on the details of the fit is observed. Fit using a Gaussian peak + continuum Pre lim ina ry Consistent with melting of Y(3 S) and Y(2 S) states ?
Conclusions n n 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, ηc, χc 0, χc 1 melt between T ~ 1. 0 Tc and T ~ 1. 2 Tc, which is below the values obtained in lattice QCD As for bottomonium, we have found preliminary evidence for the melting of Y(2 S) and Y(3 S) between 1. 5 Tc and 2. 5 Tc, while Y(1 S) survives until 3. 0 Tc or higher. Furthermore, ηb melts at around 3. 0 Tc, while χb 0 and χb 1 melt at around 2. 0 ~ 2. 5 Tc
Outlook n Check possible problems of our method q n n n αs, 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
Backup slides
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