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

[wormwoodsilver101]

TL;DR 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.

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 homework problem and explain the math to me. Thank You.

Question:

Equations from lecture notes:

Weak Charged Currents:

Weak Neutral Currents:

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"

 

[LightHero]

What is the meaning of the terms in these equations?Answer:The equations you have provided are from the Electroweak Unified Theory, which is a theory that unifies the electromagnetic and weak forces. In the first equation, the weak charged current, we see the term 'Jμ^W+', which represents the weak charged current. This current is responsible for the decay of particles such as the W+ boson, which mediates the weak interaction. It is composed of the charged quark-lepton pairs - a left-handed electron (e−L) and a right-handed positron (e+R). In the second equation, the weak neutral current, we see the term 'Jμ^Z', which represents the weak neutral current. This current is responsible for the decay of particles such as the Z boson, which mediates the weak interaction. It is composed of the neutral quark-lepton pairs - a left-handed neutrino (νL) and a right-handed anti-neutrino (νR). Finally, in the last equation, the EM current, we see the terms 'Z_μ' and 'Aμ'. These terms represent the fields associated with the electromagnetic force. The 'Aμ' term represents the electromagnetic four-vector potential, while the 'Zμ' term represents the field associated with the weak neutral current. Both of these fields interact with the quark-lepton pairs to generate the electromagnetic force.