Nonlocal Condensate Model for QCD Sum Rules RonChou

Nonlocal Condensate Model for QCD Sum Rules Ron-Chou Hsieh Academia Sinica, Taipei, Taiwan Collaborator: Prof. Hsiang-nan Li Ref: ar. Xiv: 0909. 4763 (PLB 698: 140 -145, 2011)

Outline n n n Concepts Nonlocal condensates model Summary 2

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

Concepts n Basic idea : Describing the nonperturbative contribution by a set of phenomenological effective Feynman rules ------- “quark-hadron duality”. n How to do it ? ØDispersion 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 ØBorel transformation : Giving a selection rule of s 0 4

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 nonperturbative information inside. 5

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 s 0 which can separate the matrix element to lowest resonance state and other higher states. 6

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

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

Pion decay constant in QSR 9

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 10

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

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

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

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

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

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

We get the constraint condition with 17

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 18

Pion decay constant in QSR

The prediction of pion form factor 20

Summary n n n 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. 21