Introduction BetheSalpeter and Salpeter Eequation Wave Fuctions and

• • • Introduction Bethe-Salpeter and Salpeter Eequation Wave Fuctions and Decay Constants Numerical Results Summary

Introduction • • Leptonic and Nonleptonic Weak Decay CKM Matrix Element Neutral Mixing e. x. ,

Current Experimental Status

• CLEO (2005) • BES (2005)

Traditional Nonrelativistic Approach in QM • S Wave: the decay constant is related to the wave function at origin in coordinate space • P Wave : the decay constant is related to the differential of wave function at origin

Our Notation S Wave P Wave

This talk is based on the following letters: １，Phys. Lett. B 596: 84 -89, 2004 G. Cvetic, C. S. Kim, Guo-Li Wang, Wuk Namgung ２，Phys. Lett. B 633: 492 -496, 2006 Guo-Li Wang ３，Phys. Lett. B 650: 15 -21, 2007 Guo-Li Wang

Bethe-Salpeter and Salpeter Eequation • Bethe-Salpeter Equation: • Total momentum P and relative momentum q: • Normalization:

• There is difficulty about the kernel, the time-inspired interaction? • Reduced version is needed (instantaneous). • The momentum is divided into two parts: and the momenta : will become to the and

Salpeter Eequation • The reduced Bethe-Salpeter wave function: and the useful notations

Salpeter wave function include four parts And the Salpter equation can be written as: Salpeter equation (positive and negative energy wave function)

Normalization: We solve the full Salpeter equation will the simple Cornell potential: In momentum space:

Wave Fuctions and Decay Constants • The wave functions for S wave meson in a nonralativistic model are: • In our model the relativistic wave function for a pseudoscalar meson

• Since we have The four wave functions are not independent And the positive and negative wave function And the normalization

Decay Constant of meson

• Wave function for vector meson There are only four independent wave functions normalization

Decay Constant of meson

• Wave function for • Normalization: • Decay constant meson

• Wave function for • Normalization: • Decay constant meson

• Wave function for • Normalization: • Decay constant meson

Numerical Results • 1, Mass spectra of the mesons by fitting data and our best fit parameters

Decay constants of the mesons

2, Mass spectra of the mesons

Decay constants of the mesons

• Mass spectra of the P wave mesons

Decay constants of the P wave mesons

Summary • Relativistic corrections are important, especially for the P wave states. • The Decay Constants with successive radial excitation only fall slowly.

Thank you!