Nonlocal Condensate Model for QCD Sum Rules

Ron-Chou HsiehAcademia Sinica, Taipei, Taiwan

Collaborator: Prof. Hsiang-nan LiRef: arXiv:0909.4763 (PLB698:140-145,2011)

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Outline Concepts Nonlocal condensates model Summary

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The pion form factor can be written as the convolution of a hard-scattering amplitude and wave function

HT

xHT

Pion form factor

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ConceptsConcepts

Basic idea : Describing the nonperturbative contribution by a set of phenomenological effective Feynman rules ------- “quark-hadron duality”.

How to do it ?Dispersion relation : a phenomenological procedure which connect perturbative and non-perturbative corrections with the lowest-lying resonances in the corresponding channels by using of the Borel improved dispersion relations

Borel transformation : Giving a selection rule of s0

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Quark-hadron duality

A simple example: Pion decay constant

Firstly, consider a polarization operator which was defined as the vacuum average of the current product:

where the state is the exact vacuum which contains non-perturbative information inside.

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Now, we can insert a complete set of states and the following identity between two currents

then obtain

with

Here assuming that there exists a threshold value s0 which can separate the matrix element to lowest resonance state and other higher states.

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Dispersion relation

Since the polarization operator can be written as a sum of two independent functions:

with

We then obtain

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The Borel transformation

Act on the duality relation we obtained above, then

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Pion decay constant in QSR

1 .5 2 .0 2 .5 3 .0 3 .5 4 .0 4 .5 5 .0

0 .1 2 2 5

0 .1 2 3 0

0 .1 2 3 5

0 .1 2 4 0

0 .1 2 4 5

0 .1 2 5 0

M G e V

f

s0 0 .5 9

f 0.1307

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Non-local condensate Non-local condensate modelmodel

Where does nonperturbtive contribution come from?

We assume that the nonperturbative contribution within the vacuum can be absorbed into the quark propagator

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Free propagator and exact propagatorFree propagator and exact propagator

An exact propagator :

The Wick theorem :

The normal ordering :

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The Källén-Lehmann representation

The exact fermion’s propagator :

Non-perturbative part (normal ordering)

Renormalized perturbative part

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Here we have modified the lower bound as

The K-L representation can be recast into:

We set the normal ordering piece as:

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Then the dressed propagator for the quark can be given by

With the definitions

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The weight functions are parameterized as

How to determine unknown parameters, ?

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The quark condensate contribution can be obtained from the normal ordering term

and because they can also be Taylor series expanded as

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We get the constraint condition

with

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Data fitting

The threshold mass m is expected take a value of order of the constituent quark mass and set to

0.4 ± 0.10.7

1.25

Pion decay constant in QSRf 0.1307

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The prediction of pion form factor

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SummarySummary We developed a new model that nonperturbative contribution

can be calculated directly by using Feynman rule within the framework of QCD sum rules approach.

The predicted behavior of pion form factor is very well. The negative probability of the quark propagator could be the

explanation of quark confinement.