Griffith's Elementary Particles Section 9.7 Electroweak Unification

[Jdeloz828]

TL;DR Summary

Working my way through understanding Griffith's discussion of electroweak unification (section 9.7). I have a few questions on this section.

1. On pg. 343 Griffith's expresses the weak current in terms of left-handed doublets.

j_μ^± = $\bar χ_L$γ_μτ^±$χ_L$

$χ_L$ = $\begin{pmatrix} ν_e \\ e \end{pmatrix}_L$

$\tau^+$ = $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ , $\tau^-$ = $\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$
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Should I physically interpret the left-handed doublet as implying that an electron and its neutrino are only different in their state vectors left-handed doublet component? Do the particles that get paired in these doublets necessarily represent excitations in the same underlying field?

2. Also on pg.343, Griffith's claims that if there were a third current corresponding to $$\tau^3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, $$ then we could contemplate a full weak isospin symmetry. What exactly does he mean by there being a full weak isospin symmetry? Does it have something to do with the structure of the eigenstates of the interation? What would be an example of this symmetry being broken? Also, why does the existence of this symmetry necessitate a third weak current corrensponding to $\tau^3$?

3. Finally, on 345-346, the author introduces the vector bosons $W^\mu, A^\mu, Z^\mu$ which contract with the current terms in the expression for the amplitude. Why is it necessary that we introduce separate vectors for this, couldn't the currents contract with a single vector field?

I feel like I'm still not forming a deep understanding of this section. Much of it seems very arbitrary to me. I find that whenever Griffith's starts discussing symmetries and representations he's never very precise about what he means. If anyone could shed some light on these questions, or this topic in general, it would be much appreciated.

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[eljose79]

1. Yes, the left-handed doublet implies that the electron and its neutrino are different only in their state vectors. This is because in the Standard Model of particle physics, the electron and its neutrino are part of the same electroweak doublet, where the weak force is responsible for the interactions between them. The particles in these doublets do represent excitations in the same underlying field, which is the electroweak field.

2. A full weak isospin symmetry refers to the idea that the weak force would treat particles in different isospin states (such as up and down quarks) in the same way. This symmetry would also imply that the weak force would not distinguish between particles with different weak isospin values. However, this symmetry is broken in nature, as the weak force does distinguish between particles with different isospin values. For example, the weak interaction between an up quark and a down quark is different from the weak interaction between two up quarks. The existence of this symmetry necessitates a third weak current corresponding to τ3 because the weak force is a gauge theory and the symmetry is related to the gauge symmetry of the theory.

3. The separate vector bosons Wμ, Aμ, and Zμ are necessary because they correspond to the different types of weak interactions. The W bosons mediate the charged weak interactions, while the Z boson mediates the neutral weak interactions. The A boson is the photon and mediates the electromagnetic interactions. These separate vector bosons are necessary because the weak force is a gauge theory and has different interactions depending on the type of particle involved. A single vector field would not be able to account for the different types of weak interactions.