# Motivation for Weinberg angle in electroweak gauge interaction?

Suppose I have the following lagrangian

If we only focus on the neutral current in the lagrangian:

Where $$L$$ is defined as:

And $$Y_L$$,$$Y_{R}^{\nu}$$,$$Y_{R}^{e}$$, are the hypercharge values of the different matter fields with respect to the $$U(1)_Y$$ gauge group.

So far so everything is clear to me.

However the follwing procedure confuses me: Seems like what one usually does is that one first identifies $$Z^{\mu}$$ $$Z^{\mu} \propto g'Y_{L}B_{\mu}+gW^{3}_{\mu}$$ Why is this? My guess is that in experiment the neutrino does not couple to photon field, is this right?

Then, one defines the photon field by demanding that the photon field $$A^{\mu}$$ is orthorgonal to $$Z^{\mu}$$. What's the definition of orthogonality in QFT? Why do we want it to be orthogal to $$Z^{\mu}$$? What's the physical motivation for the definition of $$A^{\mu}$$?

• Run to your SM QFT text. You are looking at the wrong terms in the Lagrangian. The relevant terms are the ones bilinear in $B_\mu, W^3_\mu$, and this is the mass matrix you diagonalize by an orthogonal transformation, redefining the fields above to $Z_\mu, A_\mu$, now decoupled (orthogonal). Given that, you may only then inspect what fermion currents these gauge fields couple too. – Cosmas Zachos May 4 at 0:53
• Near duplicate. – Cosmas Zachos May 4 at 1:00