A Bayesian Analysis of QCD

Sum Rules

Baryons`10Osaka University8.12.2010Philipp Gubler (TokyoTech)

Collaborators: Makoto Oka (TokyoTech), Kenji Morita (GSI)

arXiv: 1005.2459 [hep-ph] (to be published in PTP)

Contents

QCD Sum Rules and the Maximum Entropy Method

Test of the method in the case of the ρmeson channel

First results of the analysis of charmonium at finite temperature

Conclusions

QCD sum rulesIn this method the properties of the two point correlation function isfully 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).

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:

The spectral function is usually assumed to be describable by a “pole + continuum” form (ground state + excited states):

This ansatz may not always be appropriate.

s

ρ(s)

The usual approach

This very crude ansatz often works surprisingly well, but…

The dependence of the physical results on sth is often quite large.

The Maximum Entropy Method→ Bayes’ Theorem

likelihood function prior 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 χ2-fitting.

(Shannon-Jaynes entropy)

“default model”

A first test case: the ρmeson channel

Experiment:

mρ= 0.77 GeV

Fρ= 0.141 GeVThe position and residue of the ρ-meson are reproduced, but not its width.

The sum rule:

Application to charmonium (J/ψ) at finite temperature- Prediction of “J/ψ Suppression by Quark-Gluon Plasma Formation”

T. Matsui and H. Satz, Phys. Lett. B 178, 416 (1986).…

During the last 10 years, a picture has emerged from studies using lattice QCD (and MEM), where J/ψ survives above 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. C39S1, 9 (2005).

…

The charmonium sum rules at T=0

Moment sum rules

Developed and analyzed in:

M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147, 385 (1979); B147, 448 (1979).

L.J. Reinders, H.R. Rubinstein and S. Yazaki, Nucl. Phys. B 186, 109 (1981).

Borel sum rules

R.A. Bertlmann, Nucl. Phys. B 204, 387 (1982).

We analyze two different sum rules:

Analysis at T=0

(Moments n=5~14 are used)

- The J/ψ peak is obtained close to the experimental value. - We also get a (nonsignificant) second peak.

→ Excited states?

→ MEM artifact?

Moment sum rules Borel sum rules

mJ/ψ=3.19 GeV mJ/ψ=3.11 GeV

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

A non-scalar twist-2 gluon condensates appears due to the non-existence of Lorentz invariance at finite temperature:

four-velocity of the medium

The T-dependence of the condensates

K. Morita and S.H. Lee, Phys. Rev. Lett. 100, 022301 (2008).

K. Morita and S.H. Lee, Phys. Rev. C. 77, 064904 (2008).

The energy-momentum tensor is considered:

After matching the trace part and the traceless part, one gets:

obtained from lattice QCD

taken from:

K. Morita and S.H. Lee, arXiv:0908.2856 [hep-ph].

These values 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).

The charmonium spectral function at finite T prelim

inary

(Moments n=5~14 are used)

Moment sum rules

melting of J/ψ!

Similar results are obtained with the Borel sum rules.

Conclusions

We have shown that MEM can be applied to QCD sum rules

The “pole + continuum” ansatz is not a necessity

We could observe the melting of J/ψ using finite temperature QCD sum rules and MEM

J/ψ seems to melt between T ~ 1.2 TC and T ~ 1.4 TC,, which is below the values obtained in lattice QCD