QCD sum rules in a Bayesian
approach
arXiv: 1005.2459 [hep-ph]
YIPQS workshop on
“Exotics from Heavy Ion Collisions”
19.5.2010 @ YITP
Philipp Gubler (TokyoTech)
Collaborator: Makoto Oka (TokyoTech)
Contents
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Basics of QCD sum rules
Basics of the Maximum Entropy Method
(MEM)
A first application of the method to the
ρmeson
Conclusions
Outlook (Possible further applications)
The basics of QCD sum rules
In this method the properties of the two point correlation function is
fully exploited:
is calculated
“perturbatively”
After the Borel transformation:
spectral function
of the operator χ
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:
ρ(s)
s
This spectral function is often approximated as pole (ground state) plus
continuum spectrum in QCD sum rules:
Is this assumption always appropriate?
An example: the σ-meson channel:
Spectrum with
Breit-Wigner
peak:
Spectrum with
ππ scattering:
T.Kojo and D. Jido, Phys. Rev. D 78, 114005 (2008).
The phenomenological (hadronic) side:
The imaginary part of Π(q2) 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)
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)
or…
- QCD sum rules:
Usually:
“pole + continuum”, …
Basics of the Maximum Entropy Method (4)
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 (5)
Likelihood function
Gaussian distribution is assumed:
Corresponds to
ordinary χ2-fitting.
Prior probability
(Shannon-Jaynes entropy)
“default model”
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].
→ Bryan’s method: R.K. Bryan, Eur. Biophys. J. 18, 165 (1990).
- How is α determined?
→ The average is taken:
determined using Bayes’ theorem
- What about the default model m(ω)?
→ The dependence of the final result on the default model must be checked.
Application to the ρmeson channel
One of the first and most successful application of QCD sum rules was
the analysis of the ρ meson channel.
The “pole + continuum” assumption works well in this case.
e+e- → nπ (n: even)
Y. Kwon, M. Procura, and W. Weise,
Phys. Rev. C 78, 055203 (2008).
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
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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)
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Baryonic channels
Behavior of various hadrons at finite
temperature or density
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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:
Probability of ni balls falling into
position i:
Poisson distribution
M balls
Probability of a certain image (n1, n2, …,nN):
ni balls
(probability: pi,
expectation value: Mpi=λi)
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”