Nonlocal Condensate Model for QCD Sum Rules

Ron-Chou Hsieh

Collaborator: Hsiang-nan Liarxiv:0909.4763

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Outline

Concepts Local and non-local condensates 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 phenomenologically 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 : a) An improved expansion series

b) Give a selection rule of s0

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where the state is the exact vacuum which contain non-perturbative information and is axial vector current

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

Quark-hadron duality

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

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And with PCAC , we obtain

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

with

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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

The meaning of the Borel transformation becomes clear if we act on a particular term in the power expansion

Act on the Dispersion relation we obtained above, then

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The choice of s0 in pion form factor for Q2=1.99GeV2

1 .0 1 .5 2 .0 2 .5 3 .00 .1 7 0

0 .1 7 2

0 .1 7 4

0 .1 7 6

0 .1 7 8

M G e V

F

s0 0 .6 9 5

1 .0 1 .5 2 .0 2 .5 3 .00 .1 7 9

0 .1 8 0

0 .1 8 1

0 .1 8 2

0 .1 8 3

0 .1 8 4

0 .1 8 5

M G e V F

s0 0 .7 1 5

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Local and non-local Local and non-local condensatecondensate

Where does nonperturbtive contribution come from?

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Operator product expansion

In the QSR approach it is assumed that the confinement effects are sufficiently soft for the Taylor expansion:

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B.L. Ioffe and A.V. Smilga, NPB216(1983)373-407

Local condensate result of pion form factor

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The infrared divergence problem

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In 1986, S. V. Mikhailov and A. V. Radyushkin proposed:

with

Non-local condensate models

A.P. Bakulev, S.V. Mikhailov and N.G. Stefanis, hep-ph/0103119A.P. Bakulev, A.V. Pimikov and N.G. Stefanis, 0904.2304

Other modelsIt has been demonstrated that Λcan be interpreted as the constituent quark mass.---PLB.217, p218(1991); Z.Physics, C68, p451(1995)

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Compare with the simplest gauge invariant non-local condensate :

Must obey following constrain condition

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Local condensate:

Nonlocal condensate:

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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 means that for s larger than then the lower bound is cs otherwise is

Recast the equation into:

And set the nonperturbative piece as:

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The quark condensate contribution can be obtained by the normal ordering

The weight functions are parameterized as

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small s region large s region

m2

m2

s

2

2 scs

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The dressed propagator for the quark is then given by

With the definitions

Using constrain condition

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Pion form factor in QSR

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With the matrix element is given by PCAC:

Again, insert the complete set of states and identity:

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Numerical results

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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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SummarySummary The infrared divergence problem can be solved by our nonlocal con

densate model. The applicable energy region can be extended to 10 GeV2 in the calc

ulation of pion form factor. The predicted value of pion decay constant is very well. Can we use the Källén-Lehmann representation to improve th

e QSR approach?