Nonlocal Condensate Model for QCD Sum Rules Ron-Chou Hsieh Collaborator: Hsiang-nan Li arxiv:

2 Outline Concepts Local and non-local condensates Summary

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

4 Concepts Basic idea : D escribing 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 s 0

5 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

6 And with PCAC, we obtain Here assuming that there exists a threshold value s 0 which can separate the matrix element to lowest resonance state and other higher states. with

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 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

9 The choice of s 0 in pion form factor for Q 2 =1.99GeV 2

10 Local and non-local condensate Where does nonperturbtive contribution come from?

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

12 B.L. Ioffe and A.V. Smilga, NPB216(1983) Local condensate result of pion form factor

13 The infrared divergence problem

14 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/ A.P. Bakulev, A.V. Pimikov and N.G. Stefanis, Other models It has been demonstrated that Λcan be interpreted as the constituent quark mass.---PLB.217, p218(1991); Z.Physics, C68, p451(1995)

15 Compare with the simplest gauge invariant non-local condensate : Must obey following constrain condition

16 Local condensate: Nonlocal condensate:

17 Free propagator and exact propagator An exact propagator : The Wick theorem : The normal ordering :

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

19 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:

20 The quark condensate contribution can be obtained by the normal ordering The weight functions are parameterized as

21 small s region large s region

22 The dressed propagator for the quark is then given by With the definitions Using constrain condition

23 Pion form factor in QSR

24 With the matrix element is given by PCAC: Again, insert the complete set of states and identity:

25

26 Numerical results

27

28 Pion decay constant in QSR

29 Summary The infrared divergence problem can be solved by our nonlocal condensate model. The applicable energy region can be extended to 10 GeV 2 in the calculation of pion form factor. The predicted value of pion decay constant is very well. Can we use the Källén-Lehmann representation to improve the QSR approach?