Neutrino Physics
Autumn semester 2013
Exercise 4
Drop into the box in the lobby by Wed 16.10. at 16.00.
1. It is shown in Lecture Notes that the object S * transforms under SU(2)
transformations similarly as  * , that is
(S * )i  (1  iaTa )ij (S * ) j ,
if
Ta*  S 1Ta S .
Show that in the case of the doublet presentation of SU(2), S  i 2 . What
would S be in the case of the triplet representation? Hint: Consult the
literature to find the generator matrices of the triplet (= adjoint)
representation of SU(2).
2. Explain why a conventional mass term of a charged lepton of the form
 T
L R + h.c. , where
L ) and R is a
L belongs to the doublet  L  ( L
singlet, is not invariant under the SU(2)  U(1) transformations, and show
that
 L R + h.c.
is, where  is a Higgs doublet with a suitable weak hypercharge. Calculate
the electric charges of the component fields of the Higgs doublet.
3. Show that tensors
𝑇 = 𝐿𝑇𝐿 𝐶𝑖𝜎2 𝐿𝐿 ,
𝑇𝑘 = 𝐿𝑇𝐿 𝐶𝑖𝜎2 𝜎𝑘 𝐿𝐿
transform according to the singlet (T) and triplet (Tk) representations
under the SU(2) transformations. Here LL transforms according to the
doublet representation, i.e.
𝐿𝐿 → (1 + 𝑖𝛼𝑖 𝜎𝑖 )𝐿𝐿 .