Subatomic Physics

13.3. The Electroweak Interaction 409
Vµ alone between the left-handed and the right-handed wavefunctions:
ψ ∗ eL VµψeL = 1
2 ψ∗ e(Vµ −Aµ)ψe, (13.14)
ψ ∗ eR VµψeR = 1
2 ψ∗ e(Vµ + Aµ)ψe. (13.15)
Both sides of Eqs. (13.14) and (13.15) are equivalent, but the left-hand side provides
a convenient description of the weak currents of leptons. The weak charged currents
of the leptons are purely left-handed. Equation (13.14) consequently permits us to
write this current of the electron and its neutrino as
j ch
µ,wk = ψ ∗ eL VµψvL + ψ ∗ vL VµψeL, (13.16)
These currents can be expressed more succinctly in terms of the weak isospin notation,
which also brings out their symmetry under this operation; we use matrices
for this description.
In terms of the isospin and matrix notation, the currents for the doublet EL,
Eq. (13.16), can be written as
jµ,em = ψ ∗ ELVµ �
I3 + Y
�
ψEL + ψ
2
∗ ERVµ Y
2 ψER
= ψ ∗ ELVµI3ψEL + ψ ∗ EVµ Y
2 ψE
(13.17)
j ch
µ,wk = ψ ∗ EL Vµ2I−ψEL + ψ ∗ EL Vµ2I+ψEL,
j nc
µ,wk = ψ ∗ EL 2I3ψEL
with ER = eR and E = EL +ER. InthisequationY is a weak hypercharge operator
with eigenvalues given by Yl, i.e. Y|l〉 = Yl|l〉, where|l〉 is a lepton. We have also
introduced the isospin raising and lowering operators and matrices I+ and I−, and
I3
I± = 1
2 (I1 ± iI2),
I+ = 1
� �
0 1
, I− =
2 0 0
1
2
I3 = 1
� �
1 0
.
2 0 −1
The properties of these raising and lowering operators are
I+|νL〉 =0, 2I+|eL〉 = |νL〉,
I−|eL〉 =0, 2I−|νL〉 = |eL〉.
� �
0 0
, (13.18)
1 0
(13.19)
The coefficients in front of the isospin and the hypercharge operators in Eq. (13.17)
are fixed by Eq. (13.3) and the properties of the electron (charge = −e) and neutrino
(charge = zero).