Subatomic Physics

13.3. The Electroweak Interaction 411
with a coupling strength g ′ to the B field. If we multiply this equation on the left
by ψ∗ , we can write the expectation value of the interaction terms as
eR
Hint(eR) = −g′
2 ψ∗ eR (V0B0 − V · B)YψeR
= − g′
2 ψ∗ eR gµνVµBνYψeR, (13.23)
where a sum is implied over repeated indices. For the left-handed component of the
electron and neutrino we use the isospin doublet ψEL; it gets coupled to both the
W fields with strength g and the B field with strength g ′ , so that equation (13.21)
becomes
i�V0
�
∂ g′
− i
∂t � B0
Y g
− i
2 � � I· � �
W0 ψEL
�
= −i�cV · ∇ − i g′ Y g
B − i
�c 2 �c � I · � �
W ψEL. (13.24)
Again, we multiply on the left by ψ∗ EL and isolate the interaction terms, which, in
the shorthand notation of Eq. (13.23), are
Hint(EL) =−ψ ∗ ELgµνVµ � ′ g
2 BνY + g� I· � �
Wν ψEL. (13.25)
Table 13.1: Eigenvalues of the Weak Hypercharge.
The eigenvalues can be translated to
more massive families.
Particle or
Multiplet EL eR fL uR dR
Y −1 −2 1/3 4/3 −2/3
The hypercharge operator Y commutes with the isospin operator I, and has eigenvalues
Y given by Table 13.1. At this stage it appears that we have introduced two
new coupling constants, g and g ′ .
However, because we know the strength of the coupling of the electron to the
electromagnetic field, only one of them is unknown. To see the relationship between
the coupling constants g, g ′ and e, we write out the two interaction terms
Eqs. (13.23) and (13.25) in terms of the physical fields W ± ,Z 0 and A. The charged
current interaction part is
Hint(charged currents) = −g
√ 2 (ψ ∗ νL gµνVµWν (+) ψeL + ψ ∗ eL gµνVµWν (−) ψνL),
(13.26)