2019 Standard Model Part II R.S.
Thorne
1. The matrix γ 5 ≡ iγ 0 γ 1 γ 2 γ 2 . Prove that (γ 5 )2 = 1 independent
of
representation.
0 0 1 i 0 σi
In the Weyl representation for the γ-matrices γ = ,γ = , where each entry
1 0 −σi 0
is a 2 × 2 sub-matrix and the σi are the Pauli spin matrices. Find the plane wave solutions of the
Dirac equation in this representation. Construct the helicity projection operators PR = (1 + γ 5 )/2
and PL = (1 − γ 5 )/2, and consider the solutions for case m = 0 and explain why this representation
is particularly useful for the massless limit.
2. Starting from the matrix element for the process
νµ + e− → µ− + νe
given by the current-current (four-fermion) theory of weak interactions,
√
M = (GF / 2)ū(µ)γ µ (1 − γ 5 )u(νµ )gµν ū(νe )γ ν (1 − γ 5 )u(e)
and using the result
dσ 1
= |M |2
dt 16πs2
where s = (p1 + p2 )2 and t = (p1 − p3 )2 , i.e. the usual Mandelstam variables, show that
σ = G2F s/π,
i.e. is inconsistent with perturbative unitarity. You will find the results tr(γ 5 γ · aγ · bγ · cγ · d) =
4iǫαβγδ aα bβ cγ dδ and ǫµναβ ǫµνγδ = −2(δαγ δβδ − δαδ δβγ ) useful in intermediate stages. Average over
initial electron spins and sum over final muon spins.
3. Consider an SU (2) gauge theory coupled to a complex two-component “doublet” scalar field φ
acting on which the SU (2) generators are represented by 12 τ , for τ = {τ1 , τ2 , τ3 } the usual Pauli
matrices, and
2
L = − 41 Fµν ·Fµν + (D µ φ)† Dµ φ − 21 λ φ† φ − 12 v 2 ,
where
Fµν = ∂µ Aν − ∂ν Aµ + g Aµ × Aν , Dµ φ = ∂µ φ − ig Aµ · 12 τ φ .
× denotes the 3-vector cross product, which
isequivalent to using the SU (2) structure constants.
1 0
Explain why we may choose φ = √2 (v + f ) and that the SU (2) gauge symmetry is completely
1
broken. What are the masses of the elementary particle states neglecting any quantum corrections?
4. The gauge part of the Lagrangian density for the Electro-Weak theory with gauge fields Aµ , Bµ ,
may be written as
L = − 14 Fµν ·Fµν − 14 B µν Bµν
where
Fµν = ∂µ Aν − ∂ν Aµ + gAµ × Aν , Bµν = ∂µ Bν − ∂ν Bµ .
1
Show that the coupling of the W to the electromagnetic field A is described by
LW,A = − 21 F W µν† F Wµν + ie W µ W ν† Fµν ,
F Wµν = dµ Wν − dν Wµ , dµ = ∂µ − ieAµ , Fµν = ∂µ Aν − ∂ν Aµ .
Hence determine the contribution of the W field to the electromagnetic current j µ (this is identified
from the term of the form j µ Aµ in LW,A ).
5. In order for symmetries present at the classical level to be preserved at the quantum level, and
hence to avoid for so-called anomalies destroying the renormalizability of a theory, the quantity
tr({TaR (R), TbR (R)}TcR (R)) − tr({TaL (R), TbL (R)}TcL (R))
must vanish. TaR (R) and TaL (R) are the generators for the weak charge and hypercharge in the
appropriate representation for each right-handed and left-handed particle respectively, the trace
includes the sum over all fermions, and {TaR (R), TbR (R)} denotes the anticommutator of the two
generators. Show that for a single family the standard model is indeed anomaly free, provided one
uses a piece of knowledge about quarks from outside the electroweak sector.
6. For any complex matrix M , show that M M † is positive and hermitian. Let V † M M † V = Λ,
for unitary V , and Λ diagonal with real positive eigenvalues. Writing Λ = Md Md† , where Md is
diagonal, show that remaining freedom to redefine V up to a diagonal matrix of phases can be used
to set the eigenvalues of Md to be real and positive. Define hermitian matrix H by
H ≡ V Md V †
and U ≡ H −1 M . Show that U is unitary and that
V † M W = Md
where W = U † V is a unitary matrix. (This is the bi-unitary transformation that is needed to
diagonalise quark mass matrices in general).
For two generations of quarks with masses md , ms , mu , mc , suppose that for
the Lagrangian density
1 1
Lm = − q + m+ 2 (1 + γ5 )q+ + q − m− 2 (1 + γ5 )q− + hermitian conjugate , the mass matrices take
the form ′ ′
u d 0 A 0 C
q+ = , q− = , m+ = , m− =
c′ s′ A∗ B C∗ D
where u′ , d′ , c′ , s′ are quark
fields (the prime indicates they diagonalize the weak charged current
cos θ sin θ
interaction). If R∓ (θ) = , define θ+ by
∓ sin θ ± cos θ
0 |A| −1 mu 0
R+ (θ+ ) R− (θ+ ) =
|A| B 0 mc
with θ− similarly defined in terms of ms , md . Show that after suitable rephasing of quark fields,
one can use R± to diagonalize m± . Hence, from the mixing matrix so generated in the quark weak
charged current interaction, show that the Cabbibo angle in this case is given by
r r
−1 md −1 mu
θC = θ− − θ+ = tan − tan .
ms mc
Check this relation against experimental data.
