How to derive the currents of Z boson and the EM current?

Summary:

Starting from the gauge-invariant kinetic terms of the fermions of the first generation (Eq(7)) in the
lecture notes, find J^μ_W^3 and J^μ_B. Then using the definition for Z boson and photon (Eq(17) in the lecture
notes), reproduce the currents of the Z boson (Eq(18)) and the EM current (Eq(19)). You only need
to work on the terms involving the leptons of the first generation, i.e., e_L; e_R and v_e_L.

Main Question or Discussion Point

I am a student minor in physics and I am taking this course of particle physics. And I have been lost since the Non-Abelian Gauge Theory which a few lectures before Electroweak unified Theory lecture.

i am completely confused and overwhelmed by the math since the mid-term exam, since I was not equip with the mathematical background (such as group theory, symmetry, lie algebra, tensors...), and my lecturer didn't actually explain the mathematics needed, so it is very confusing for me and my classmates to understand and comprehend the course materials.

I have actually searched for other lecture notes on the internet to see if there is more information for me to patch up the missing math knowledge or other ways to explain Electroweak theory. However, it does not help, and some lecture notes uses a different set of notations, which confuses me even more, or they don't contain the equations that I need.

Some links for the notes that I have read:

https://webhome.weizmann.ac.il/home/yotams/notes/ep2.pdf

Lectures 21 - PHY357: Electroweak Unification

http://www.damtp.cam.ac.uk/user/ho/SM.pdf

So I would greatly appreciate it if anyone can help out with this hw problem and explain the math to me. Thank You.

Question:
capture-png.png

Equations from lecture notes:
1573372748809-png.png

1573372766189-png.png


1573372784724-png.png

1573372802334-png.png

1573372835258-png.png

Weak Charged Currents:
1573372886355-png.png

Weak Neutral Currents:
1573372925691-png.png

1573372950383-png.png

EM Current:
The terms of the field Z_μ and Aμ, the neutral-current part of the Lagrangian Eq can be written as "Hidden in lecture notes for homework"
1573372976894-png.png
 

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