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Nonlocal Condensate Model for QCD Sum Rules

Nonlocal Condensate Model for QCD Sum Rules. Ron-Chou Hsieh Academia Sinica, Taipei, Taiwan Collaborator: Prof. Hsiang-nan Li Ref: arXiv:0909.4763 (PLB698:140-145,2011). Outline. Concepts Nonlocal condensates model Summary. Pion form factor.

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Nonlocal Condensate Model for QCD Sum Rules

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  1. Nonlocal Condensate Model for QCD Sum Rules Ron-Chou Hsieh Academia Sinica, Taipei, Taiwan Collaborator: Prof. Hsiang-nan Li Ref: arXiv:0909.4763 (PLB698:140-145,2011)

  2. Outline • Concepts • Nonlocal condensates model • Summary

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

  4. Concepts • 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

  5. 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.

  6. 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.

  7. Dispersion relation Since the polarization operator can be written as a sum of two independent functions: with We then obtain

  8. The Borel transformation Act on the duality relation we obtained above, then

  9. Pion decay constant in QSR

  10. Non-local condensate model Where does nonperturbtive contribution come from? We assume that the nonperturbative contribution within the vacuum can be absorbed into the quark propagator

  11. Free propagator and exact propagator An exact propagator : The Wick theorem : The normal ordering :

  12. The Källén-Lehmann representation The exact fermion’s propagator : Non-perturbative part (normal ordering) Renormalized perturbative part

  13. The K-L representation can be recast into: We set the normal ordering piece as: Here we have modified the lower bound as

  14. Then the dressed propagator for the quark can be given by With the definitions

  15. The weight functions are parameterized as How to determine unknown parameters, ?

  16. The quark condensate contribution can be obtained from the normal ordering term and because they can also be Taylor series expanded as

  17. We get the constraint condition with

  18. Data fitting The threshold mass m is expected take a value of order of the constituent quark mass and set to 0.4 ± 0.1 0.7 1.25

  19. Pion decay constant in QSR

  20. The prediction of pion form factor

  21. Summary • 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.

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