# Feynman diagram and vertices in a hadron decay

This question is about Feynman diagram and vertices in a hadron decay, which comes from Problem 89.5 in Srednicki's textbook Quantum Field Theory. The reaction involved is $$\begin{equation} d\rightarrow u + e + \overline{\nu}_{e} \tag{1} \end{equation}$$

After integrating out the $$W^{\pm}$$ fields, we get an effective interaction between the hadron and lepton currents that includes $$\begin{equation} \mathcal{L}_{eff}=2\sqrt{2}Z_{C}C(\mathcal{\overline{E}}_{L}\gamma^{\mu}\mathcal{N}_{eL})(\mathcal{\overline{U}}_{L}\gamma_{\mu}\mathcal{D}_{L}), \tag{89.35} \end{equation}$$ where $$Z_{C}$$ is renormalizing factor, $$C$$ is constant, and $$\begin{equation} \mathcal{E} \equiv \left( \begin{array}{c} e \\ \overline{e}^{\dagger} \end{array} \right), \hspace{0.2in} \mathcal{N}_{e} \equiv \left( \begin{array}{c} \nu_{e} \\ \nu_{e}^{\dagger} \end{array} \right), \hspace{0.2in} \mathcal{U} \equiv \left( \begin{array}{c} u \\ \overline{u}^{\dagger} \end{array} \right), \hspace{0.2in} \mathcal{D} \equiv \left( \begin{array}{c} d \\ \overline{d}^{\dagger} \end{array} \right) \end{equation}$$ are Dirac fields for the electron, electron neutrino, up quark, and down quark respectively. Using Fierz identity, eq.(89.35) can be written as $$\begin{equation} \mathcal{L}_{eff}=2\sqrt{2}Z_{C}C(\mathcal{\overline{E}}_{L}\gamma^{\mu}\mathcal{D}_{L})(\mathcal{\overline{U}}_{L}\gamma_{\mu}\mathcal{N}_{eL}), \tag{89.36} \end{equation}$$

In the Feynman diagram for the interaction eq. (89.35), the two vertices are $$d-u-W^{-}$$ and $$e-\overline{\nu}_{e}-W^{-}$$. Does eq. (89.36) imply a different interaction that requires a different diagram? If so, what are the two vertices in this diagram? (they seem to be $$e-d-X$$ and $$u-\overline{\nu}_{e}-X$$; what is $$X$$?) Furthermore, does eq. (89.36) imply a quark process different from (1)?

If someone says that eq. (89.36) is just a mathematical transformation using Fierz identity and does not correspond to a physical interaction, this seems not the case, because by making analogy of the diagram of eq. (89.36) to the one-loop correction to the photon-fermion-fermion vertex (Fig. 62.3) in spinor electrodynamics (Section 62 in Srednicki's textbook), Problem 89.5 part b) asks us to prove the same result of Problem 62.2. This suggests that eq. (89.36) implies a diagram different from that of eq. (89.35) but the same as that of Fig. 62.3 in Problem 62.2.

• I think that Fierz identity doesn't apply to a product containing four different fermion fields. It may work for interaction with photons, because it doesn't transform one fermion into a different one, but the weak interaction does. Apr 26, 2019 at 9:54

My guess is that, in this problem, since the $$W^{\pm}$$ fields are integrated out, and eqs. (89.35) and (89.36) are effective (not exact) Lagrangian, so their Feynman diagram is just a 4-vertex of $$d-u-e-\overline{\nu}_{e}$$.