Relation of vector and scalar matrix elements for hadronic transitions

Question

Consider a matrix element $$ F_{\mu}(p_{h'}, p_{h}, \dots) = \langle h'(p_{h'})|\bar{q}_{i}\gamma_{\mu}q_{j}|p_{h}\rangle, $$ describing transition of some initial hadron $h$ that contains a quark $q_{j}$ to some final hadron $h'$ that contains a quark $q_{i}$. $\dots$ means the dependence on other quantities like polarization vectors of $h, h'$.

In the literature often I meet a statement that (see e.g. expressions $(1)-(3)$ here and $(29)-(30)$ here, illustrating the transition $B \to \pi$)

$$ G(p_{h}, p_{h'},\dots) \equiv \langle h'(p_{h'})|\partial^{\mu}(\bar{q}_{i}\gamma_{\mu}q_{j})|p_{h}\rangle = $$ $$ =(m_{q_{j}} - m_{q_{i}})\langle h'(p_{h'})|\bar{q}_{i}q_{j}|p_{h}\rangle = (p_{h} - p_{h'})^{\mu}F_{\mu}(p_{h'},p_{h},\dots) $$ What I do not understand is the last equality, which can be represented in a form $$ \langle h'(p_{h'})|\partial^{\mu}(\bar{q}_{i}\gamma_{\mu}q_{j})|p_{h}\rangle = (p_{h} - p_{h'})^{\mu}F_{\mu}(p_{h'},p_{h},\dots) $$ I do not understand how to translate the quark momenta to the hadron momenta. Could you please explain this?

Answer 1

The momentum operator $P_\mu$ is the generator of translations, so $$ \partial_\mu (\bar q\gamma_\nu q) \propto \big[P_\mu,\,\bar q\gamma_\nu q\big]. $$ Substitute this into $$ \langle h'(p_{h'})|\,\partial^\mu (\bar q\gamma_\mu q)\,|p_h\rangle $$ and use $$ P^\mu\,|p_h\rangle = (p_h)^\mu\,|p_h\rangle $$ (and similarlly for the other state-vector) to derive the last equation shown in the question.