Electroweak Mixing: Hypercharge, Isospin & Photon/Z0

[Zarathustra0]

In the GWS electroweak model, there are two fundamental charges: weak hypercharge and the third component of weak isospin (henceforth referred to as hypercharge and isospin respectively). The gauge boson of hypercharge is the B0, and those of isospin are the W+, W0, and W-. The B0 and W0 mix to make the photon and Z0. What I don't understand is what determines the charges acted upon by the photon and Z0. The Z0 acts on isospin just like the Ws, but the photon acts on electric charge, which, in the context of electroweak theory, is defined to be the isospin plus half the hypercharge. How is this derived?

 

[Bill_K]

Z0, You can't derive it, of course, since it's a theory, but you can explain how we were led to it, and why it seems reasonable. W⁺ and W^- always couple to particles in pairs: e and ν_e, μ and ν_μ, u quark and d quark. It's as if the particles formed doublets in a symmetry group, which was called weak isospin. The W⁺ and W^- acted like stepping operators of an SU(2) group, but the third component W⁰ was missing.

Neutral weak currents were discovered in 1973 in processes such as neutrino-electron scattering. The first idea was that they were mediated by the missing W⁰. However things did not fit. Although the charged weak currents are known to be purely left-handed, the neutral current was found to be predominantly left-handed with a smaller right-handed component. So it did not simply fit into a triplet. But, if you had a triplet W plus something else there would be a fourth degree of freedom.

Weinberg's idea was that the fourth degree of freedom was electromagnetism. If so, it's clear how electromagnetism must fit in. Particles in each weak multiplet differ in charge by one. Also the average charges of the multiplets are displaced. By analogy with the strong interactions, we describe this by introducing a weak hypercharge Y and the formula Q = J³ + Y/2. Thus e_L^- and v_e form a doublet with total charge -1 (hence Y = -1) while e_R^- forms a singlet with total charge -1, hence Y = -2.

Weinberg said the B and the W⁰ were mixed into two orthogonal states A_μ = B_μ cos θ_W + W_μ³ sin θ_W and Z_μ = -B_μ sin θ_W + W_μ³ cos θ_W where θ_W is a weak mixing angle.

Ok, here's the key point. When you write out the electroweak neutral current interaction in terms of these rotated states,
g J_μ³ W_μ³ + ½ g' J_μ^Y B_μ = (g sin θ_W J_μ³ + ½ g' cos θ_W J_μ^Y) A_μ + (...) Z_μ
the coefficient of A_μ must be the electromagnetic current J_μ^EM = e(J_μ³ + 1/2 J_μ^Y). (This implies g sin θ_W = g' cos θ_W = e.)

The field Z_μ is simply whatever is orthogonal to that. It turns out to be J_μ^NC = J_μ³ - sin² J_μ^EM

 

[jtbell]

I thought GWS theory came before the discovery of weak neutral currents (c. 1968 versus 1973), and predicted their existence. That was one of the reasons the observation of neutral currents was such a big deal: it was a big step in confirming GWS theory.

 

[Bill_K]

Thanks, I stand corrected.

 

[Vanadium 50]

Bill is right, but you are starting from a bad place.

1) The "fundamental charge" of weak isospin is not T₃. It's T. This is where the SU(2) comes in - it means that the amount of weak charge held by a particle cannot be represented as a simple number (that would be a U(1) theory) but must be represented by a matrix.

2) Before symmetry breaking, you have a w₁, w₂, w₃. You don't know what electric charge is at that point in the derivation, so it's premature to label them by their charge. When going through the derivation, you will discover that one linear combination of w's has positive charge, one has negative charge, one combination of the B and the w₃ has the same couplings as the photon (and is identified with it), and the orthogonal combination is the Z.