Study on direct pion emission in decay $D^{*+} \to D^+ \pi$

The QCD multipole expansion (QCDME) is based on the quantum field theory, so should be more reliable. However, on another aspect, it refers to the non-perturbative QCD , so that has a certain application range. Even though it successfully explains the data of transition among members of the $\Upsilon$ ($\psi$) family, as Eichten indicates, beyond the production threshold of mediate states it fails to meet data by several orders. In this work, by studying a simple decay mode $D^*\to D+\pi^0$, where a pion may be emitted before $D^*$ transiting into $D$, we analyze the contribution of QCD multipole expansion. Whereas as the $D\pi$ portal is open, the dominant contribution is an OZI allowed process where a light quark-pair is excited out from vacuum and its contribution can be evaluated by the $^3P_0$ model. Since the direct pion emission is a process which is OZI suppressed and violates the isospin conservation, its contribution must be much smaller than the dominant one. By a careful calculation, we may quantitatively estimate how small the QCDME contribution should be and offer a quantitative interpretation for Eichten's statement.