# Relation of vector and scalar matrix elements for hadronic transitions

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?

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.