# Weak Isospin and the Weakly Interacting Bosons (Force Particles)

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We start with the prime weak interaction of the beta decay of the neutron to a proton plus electron plus anti-neutrino, in which we now understand that a down (d) quark (charge -1/3) in the neutron decays to an up (u) quark (charge +2/3) in the proton.  This is accomplished by the transitioning quarks emitting a virtual W¯ weak boson, which then realizes itself as an electron (charge -1) and a neutral anti-electron neutrino:

where t is the direction of time, not a particle.

By the way, Fred Reines (UC Irvine Nobel Prize winner) and Clyde Cowan discovered the electron neutrino in 1956 at Savannah River.  This picture is at Hanford in 1953, where in the picture, Fred is on the left and Clyde on the right.  I wrote an article describing this on a UCI Physics website.  Reines and Martin Perl (who discovered the tau lepton) shared the 1995 Nobel Prize in Physics.

In a previous post, we discussed the spin 1/2 fermions which are quarks and leptons.  The weak isospin interactions make a weird choice of what to interact with, and only choose the left handed quarks and leptons, where the spin is opposite the direction of the particle.  This results in a violation of parity symmetry, where left and right handed particles would be treated the same.  The theory of the left handed only interactions was called V-A theory, and was formulated by Robert Marshak and George Sudarshan, and independently by Richard Feynman and Murray Gell-Mann.  The V stands for a vector interaction postulated by Fermi just like the photon’s interaction, and the A is the parity opposite of that (axial vector) leading to parity violation.

For the left handed quarks and leptons form a set of weak doublets, I = 1/2, just like a spin 1/2 particle has two possible states of Sz = +1/2 or -1/2.  The up (u) and down (d) quarks are an I = 1/2 weak isospin doublet of weak isospin assignments Iz = +1/2 for the up quark and Iz = -1/2 for the down quark.  Similarly, the up type quark charm (c) and the down type quark strange (s) quark form another weak doublet, called the second generation.  Finally, the third generation weak doublet has the up type quark top (t) and the down type quark (b).  These are shown vertically as doublets, and horizontally as generations.

u    c    t

d    s    b

Left handed neutrinos are like up quarks and have weak isospin Iz = +1/2 and are ν_e (read nu-e), ν_μ, and v_τ.

Left handed negatively charged leptons are like down type quarks and have weak isospin Iz = -1/2 and are the electron, muon, and tau.

ν_e   ν_μ   ν_τ

e        μ        τ

Anti-particles such as anti-quarks, anti-neutrinos, and anti-leptons have the opposite Iz of their corresponding particles, but the same I = 1/2.

The basic W weak bosons W+, W¯, and Wº form a Weak Isospin I = 1 triplet, with Iz component projections: Iz = +1 for W+; Iz = 0 for Wº (W zero or nought); and Iz =  -1 for W¯.  This is an exact analogy to a Spin 1 particle, with Sz = +1, 0, and -1, z axis projections for Sz.

Just like there are zero spin particles with S = 0 and Sz = 0, there is a Weak Isospin I = 0 and Iz = 0 basic boson called the Bº (B-zero).

In weak interaction processes, weak isospin is conserved.  Looking at the neutron beta decay diagram at the start of this article, the decay starts with an down quark with Iz = -1/2. At the interaction vertex, it changes to an up quark with Iz = +1/2 and a W¯ with Iz = -1, which add up to Iz = -1/2, just what was started with in the down quark, so the interaction conserved Iz.  When the W¯ boson of Iz = -1 converts to an electron of Iz = -1/2 and an anti-electron neutrino of Iz = -1/2, the two particles have Iz = -1 just as the W¯ did, conserving Iz.  (Recall that the anti-neutrino has the opposite Iz of the neutrino).

Besides the charged W¯and W+ weak charged bosons, there is a neutral weak boson, the neutral Zº, which acts like the photon exchange in electromagnetism (q stands for any quark)

Now W+ and W¯ have the same couplings, which is also related to the gamma coupling and the Z° coupling.  This suggests a starting symmetry, where the up and down quarks couple to the interactions basically the same, as if they are two different states of the same underlying quark.  This is similar to the case of spin, where spin up and down of an electron are just states of the same underlying electron, and form a spin 1/2 doublet.  So the up and down quarks are taken as two states of the same weak interaction doublet.

For a weak boson to decay to two weak isospin 1/2 quarks, it could have their sum, or weak isospin 1, or their difference, which is weak isospin 0.  When we start formulating the ideal theory with weak isospin symmetry, the masses of the bosons and quarks must be zero.