QCD sum rules in a Bayesian approach

YIPQS workshop on

“Exotics from Heavy Ion Collisions”

19.5.2010 @ YITP

Philipp Gubler (TokyoTech)

Collaborator: Makoto Oka (TokyoTech)

arXiv: 1005.2459 [hep-ph]

Contents

Basics of QCD sum rules Basics of the Maximum Entropy Method (ME

M) A first application of the method to the ρmeso

n Conclusions Outlook (Possible further applications)

The basics of QCD sum rulesIn this method the properties of the two point correlation function isfully exploited:

is calculated “perturbatively” spectral function

of the operator χ

After the Borel transformation:

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 often approximated as pole (ground state) plus continuum spectrum in QCD sum rules:

Is this assumption always appropriate?

s

ρ(s)

An example: the σ-meson channel:

T.Kojo and D. Jido, Phys. Rev. D 78, 114005 (2008).

Spectrum with Breit-Wigner peak:

Spectrum with ππ scattering:

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)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, …

Basics of the Maximum Entropy Method (3)

- QCD sum rules:

or…

Usually: “pole + continuum”, …

Basics of the Maximum Entropy Method (4)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 (5)Likelihood function

Gaussian distribution is assumed:

Prior probability

(Shannon-Jaynes entropy)

“default model”

Corresponds to ordinary χ2-fitting.

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].

- How is α determined?

→ Bryan’s method: R.K. Bryan, Eur. Biophys. J. 18, 165 (1990).

determined using Bayes’ theorem

→ The average is taken:

- What about the default model m(ω)?

→ The dependence of the final result on the default model must be checked.

Application to the ρmeson channelOne of the first and most successful application of QCD sum rules was the analysis of the ρ meson channel.

Y. Kwon, M. Procura, and W. Weise, Phys. Rev. C 78, 055203 (2008).

e+e- → nπ (n: even)

The “pole + continuum” assumption works well in this case.

The experimental knowledge of the spectral function allows us generate realistic mock data.

Generating mock data:

analyzed region

Centred at Gmock(M), we generate gaussianly distributed values as an input of the analysis.

How is the default model chosen?

Numerical results:

MEM artifacts, induced due to the sharply rising default model

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.

Analysis of the OPE data:

We use three parameter sets in our analysis:

(from the Gell-Mann-Oakes-Renner relation)

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).

Results (1)

Experiment:

mρ= 0.77 GeV

Fρ= 0.141 GeV

Results (2)

The dependence of the ρ-meson properties on the values of the condensates:

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)

Outlook (Possible further applications) Baryonic channels Behavior of various hadrons at finite temperat

ure or density e.g. Charmonium

Tetraquarks Pentaquarks

scattering states ↔ resonances ?

Backup slides

What happens for a constant default model?

Dependence of the results on various parameters:

on Mmax:

on σ(M) and Mmin:

What happens in case of no input peak?

How is Fρ obtained?

Basics of the Maximum Entropy Method (4)Prior probability (1)

Monkey argument:

M balls

ni balls

(probability: pi, expectation value: Mpi=λi)

Probability of ni balls falling into position i:

Poisson distribution

Probability of a certain image (n1, n2, …,nN):

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”