Interpreting operator insertions in the S-Matrix

Question

At several points throughout Weinberg QFT Volume I, Weinberg claims that the sum of all diagrams which have in states $\alpha$ and out states $\beta$ and one off-shell photon at position $x$ is given by \begin{equation} \langle \beta|\hat{J}^{\mu}(x)|\alpha\rangle \tag{1} \end{equation} where $J^{\mu}(x)=\frac{\delta L_{QED}}{\delta A^{\mu}(x)}$. For instance he says this at Eq's(10.4.19),(10.5.1) and (13.6.1). However, I am under the impression that the amplitude for single photon emission is rather: \begin{equation} \langle \beta|\hat{A}^{\mu}(x)|\alpha\rangle \tag{2} \end{equation} Below Eq.(10.5.1) he says Eq(1) is the case in theories such as QED where the interaction is linear in $A^{\mu}(x)$. I do not understand this comment as Eq(2) is the amplitude for single photon emission?

My second question: In QCD, does only Eq(2) work for describing single gluon emission, supposing that we insert the gluon operator in place of the photon field operator?

Answer 1

The equality relies on the Schwinger-Dyson equations. The sum of all Feynman diagrams in which the state $\alpha$ goes to $\beta$ accompanied by the emission/absorption of a not necessarily on-shell photon of momentum $q^{\mu}$ is \begin{equation} \int d^dx \,e^{iq\cdot x}\,\langle\beta|\hat{A}^{\mu}(x)|\alpha\rangle \end{equation} where this amplitude includes the photon propagator. We can get rid of this propagator factor by multiplying through by $\Box_x^{\mu\nu}=\Box g_{\mu\nu}-(1-\frac{1}{\xi})\partial_{\mu}\partial_{\nu}$. So the amplitude without the propagator is \begin{equation} \int d^dx \,e^{iq\cdot x}\,\,\Box_x^{\mu\nu}\langle\beta|\hat{A}_{\nu}(x)|\alpha\rangle=\int d^dx \,e^{iq\cdot x}\,\,\langle\beta|\hat{J}^{\mu}(x)|\alpha\rangle \end{equation} Where the above equality is a Schwinger-Dyson equation for QED. In the case of multiple photon emissions in which there are multiple photon field operators, contact terms emerge at this last equality, however, inspecting these terms, one finds that they correspond to the disconnected components of the S-matrix and hence can be neglected.

In the case of QCD, in which a single gluon is emitted, one cannot make such a correspondence as the Schwinger-Dyson equations give \begin{equation} \int d^dx \,e^{iq\cdot x}\,\,\Box_x^{\mu\nu}\langle\beta|\hat{A}_{\nu}^a(x)|\alpha\rangle=\int d^dx \,e^{iq\cdot x}\,\,\langle\beta|\hat{J}^{\mu}_a(x)+gf^{abc}\partial_{\nu}\big(\hat{A}^{\nu}_b(x)\hat{A}^{\mu}_c(x)\big)|\alpha\rangle \end{equation} where $J^{\mu}_a=-f_{abc}F_c^{\nu\mu}A_{b\mu}-i\frac{\delta \mathcal{L}_{Matter}}{\delta D_{\nu}\psi}t_a\psi$ is the associated conserved current. For further details see Schwartz's QFT textbook Section 14.8