Matrix elements of pseudoscalarQq¯mesons including relativistic and asymptotic-freedom effects

We present a detailed analysis of effects on the decay constants $f_P$ and mixing parameters $B$ of the pseudoscalar $Q\overline{q}$ mesons, coming from two fundamental dynamical properties of this system: relativistic dynamics and an asymptotically free Coulomb interaction. Our results derive entirely from the short-distance properties of a Dirac wave function for the light quark in a pure leading-log Coulomb potential. We obtain an analytical expression for the asymptotic behavior (r → 0) of this wave function. We also make use of a QCD effective action model for $Q\overline{q}$ mesons, in order to demonstrate that our analysis does not depend on any assumptions about the long-distance properties of this system. We show that the short-distance effects of the relativistic wave function with asymptotic freedom play an important role in the evolution of $f^P$ and $B$ with meson mass $M_P$. These corrections to $f_P$, which vanish as ${\alpha }_s\left(M_P\right)$ for $M_P\to \infty$, are to be superimposed on the leading short-distance correction due to Voloshin and Shifman and Politzer and Wise, and are found to be quantitatively more significant than the latter in the phenomenologically important region $M_D\lesssim M_P\lesssim M_B$.