# Weinberg mixing angle matrix for photon and $Z$ bosons

I'm currently studying the GWS-model and am a bit confused with the matrix (def. via the Weinberg mixing angle) "mixing" photon and $$Z$$ boson mass eigenstates. It is defined as

$$\begin{pmatrix} Z^0 \\ A \end{pmatrix} = \begin{pmatrix}\cos\theta_W & -\sin\theta_W\\ \sin\theta_W & \cos\theta_W\end{pmatrix} \begin{pmatrix} A^3 \\ B \end{pmatrix}$$

(Peskin and Schröder p.702)

So in the GWS-model the photon is massless, as it should be, but why is it then useful to mix the mass eigenstates of the massive $$Z^0$$ boson and the massless photon? Do they not also transform in different groups?

• If I remember correctly, it's in order to diagonalize the mass matrix and make the lagrangian density look nicer. Commented Sep 3, 2022 at 15:16

As your text (any text!) reminds you, you may change your basis from the theoretical, "clean" basis of gauge bosons $$(A^a,B)$$ living in different groups, SU(2) and hypercharge U(1), respectively, to the "physical", ugly/messy basis $$W^\pm, Z^0, A)$$ of mass eigenstates.