Quarkonium at finite Temperature from QCD Sum Rules and the Maximum Entropy Method

Seminar at the Komaba Nuclear Theory Group @ Tokyo University7.12.2011Philipp 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, 092003 (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 maximu

m entropy method Results of the analysis of charmonia at finite t

emperature First results for bottomonia Conclusions and outlook

Introduction: QuarkoniaGeneral Motivation: Understanding the behavior of matter at high T.

- Phase transition:

QGP (T>Tc) ↔ confining phase (T<T

c)

- Currently investigated

at RHIC and LHC

- Heavy Quarkonium: clean probefor 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

S. Chatrchyan et al. [CMS Collaboration], Phys. Rev. Lett. 107, 052302 (2011).

Evidence for both J/ψ and and excited states of Y are being observed.

L. Kluberg and H. Satz, arXiv: 0901.3831 [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 Tc, but dissolves below 2 Tc.

-

(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 Tc or higher.

-

H. Iida et al, Phys. Rev. D 74, 074502 (2006). H. Ohno et al, Phys. Rev. D 84, 094504 (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, 061602

(2011). G. Aarts et al, JHEP 1111, 103 (2011).

Melting of excited states just above Tc?

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 rulesAsymptotic 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 q2>0 Π(q2) 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(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!

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 proba

ble 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 χ2-fitting.

First applications in the light quark sector

Experiment:

mρ= 0.77 GeV

Fρ= 0.141 GeV

PG and M. Oka, Prog. Theor. Phys. 124, 995 (2010).

ρ-meson channel Nucleon channel

Experiment:

mN= 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 TThe 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 （ αs 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, 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 quantitiessuch as energy density ε(T) and pressure p(T).

MEM Analysis at T=0

mη ｃ =3.02 GeV (Exp: 2.98 GeV)

S-wave

P-wave

mJ/ψ=3.06 GeV (Exp: 3.10 GeV)

mχ0=3.36 GeV (Exp: 3.41 GeV) mχ1=3.50 GeV (Exp: 3.51 GeV)

PG, K. Morita and M. Oka, Phys. Rev. Lett. 107, 092003 (2011).

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 ?

T=0T=1.0 TcT=1.1 TcT=1.2 Tc

The OPE data in the Vector channel at various T:

cancellation between αs and condensate contributions

Where might we have problems?

Higher order gluon condensates? Probably not a problem, but needs to be checked

Higher orders is αs? 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 investi

gated carefully.

First results for bottomonium

Υ ηb

χb0

χb1

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, 052302 (2011).

Is it possible to reproduce this result with our method?

Our resolution might not be good enough.

Extracting information on the excited statesHowever, 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

Preliminary

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, ηc, χc0, χc1 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(2S) and Y(3S) between 1.5 Tc and 2.5 Tc, while Y(1S) survives until 3.0 Tc or higher. Furthermore,ηb melts at around 3.0 Tc, while χb0 and χb1 melt at around 2.0 ~ 2.5 Tc

Outlook

Check possible problems of our method 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