Reading Week Assignment1
MA444D, The Standard Model of Elementary Particle
Physics, 2015/16
Lecturer: Stefan Sint
Problem 1, Goldstone theorem
Consider the Lagrangian density for a real scalar 2-component field ϕ
~T =
(ϕ1 , ϕ2 )
1
1
1
~ T ∂µϕ
~ − µ2 ϕ
~T ϕ
~ − λ(~
ϕT ϕ
~ )2 .
(1)
L = ∂µ ϕ
2
2
4
1. Determine the global symmetries of the Lagrangian density.
2. Assume that µ2 < 0. Determine the field configurations which minimize
the potential. How are any two of these related to each other?
3. Choose one particular field configuration ϕ
~ min which minimizes the potential and parametrize the fluctuations around it,
~=ϕ
φ
~ −ϕ
~ min .
(2)
~ and find the masses
Then rewrite the lagrangian density in terms of φ
of the particles described by this lagrangian density.
4. Explain how this example relates to the general Goldstone theorem as
discussed in class.
Problem 2
Verify the following identities without using an explicit representation for the
γ-matrices:
1. Verify that {γ5 , γ µ } = 0 for all µ = 0, 1, 2, 3.
1
Due Wednesday 9 March 2016 in class. Please staple your solutions together and make
sure that your name is readable.
2. The trace of an odd number of γ-matrices vanishes. Check this explicitly for tr(γ µ γ ν γ ρ ).
Hint: insert γ5 γ5 = 1 and use that {γ5 , γ µ } = 0, as well as the cyclicity
of the trace. The proof then easily generalises to any odd number of
γ-matrices
3.
γµ γ µ = 4
(3)
γµ γ ρ γ µ = −2γ ρ
(4)
γµ γ ρ γ σ γ µ = 4g ρσ
(5)
γµ γ ν γ ρ γ σ γ µ = −2γ σ γ ρ γ ν
(6)
4.
5.
6.
Problem 3
Kinematics of a scattering process 1 + 2 → 3 + 4. The four 4-momentum
vectors pi , i = 1, 2, 3, 4 of the particles are related by energy momentum
conservation p1 + p2 = p3 + p4 . No assumptions are made concerning the
particle masses mi .
1. Show that there are exactly 2 independent Lorentz invariants which
can be built from scalar products among the four 4-momentum vectors.
(Note that in this counting the particle masses m2i = p2i are taken as
given, so p2i are not counted as independent invariants).
2. The Mandelstam variables for such a two-body scattering process are
given by
s = (p1 + p2 )2 ,
t = (p1 − p3 )2 ,
u = (p1 − p4 )2 ,
(7)
and define 3 Lorentz invariants. Since only 2 independent Lorentz
invariants exist there must be a relation f (s, t, u) = 0. Show that
indeed, the following relation holds:
s + t + u = m21 + m22 + m23 + m24
(8)
Problem 4
Given the Yang-Mills part of the Glashow Weinberg-Salam-Model,
1
1
L(x) = − tr {Wµν (x)W µν (x)} − Bµν (x)B µν (x) ,
(9)
2
4
where Wµ and Bµ are the SU(2) and U(1) gauge fields, respectively, with their
respective field tensors Wµν and Bµν , evaluate the Euler-Lagrange equations
for both the SU(2) and U(1) gauge fields. Then perform the rotation by
the Weinberg angle θW and rewrite this lagrangian density in terms of the
physical Wµ± , Zµ boson fields and the photon Aµ . Determine all interaction
terms in the lagrangian and draw corresponding Feynman vertices.
Problem 5, Goldstone theorem and Higgs mechanism
Consider a complex scalar 1-component field φ with the Lagrangian density
L(x) = ∂µ φ∗ (x)∂ µ φ(x) − V (φ),
(10)
where the potential is further specified by
V (φ) = λ 2φ∗ (x)φ(x) − v 2
2
.
1. Identify the global symmetry of the potential. How many Goldstone
bosons should one expect from a spontaneous breaking of this symmetry?
2. Introduce 2 real scalar fields by reparameterizing φ as follows:
1
(11)
φ(x) = √ ρ(x) exp(iχ(x)/v)
2
Identify the minima of the potential.and parametrize fluctuations around
one of them. Determine the particle spectrum and masses described
by this Lagrangian density.
3. Couple the Lagrangian density in Eq (10) to an Abelian gauge field Aµ
through the substitution ∂µ → Dµ = ∂µ + iqAµ and the addition of a
Maxwell term, so that
1
L(x) = (Dµ φ)∗ (x)Dµ φ(x) − V (φ) − Fµν (x)F µν (x)
4
Using again the reparameterization of Eq. (11) show that the field χ(x)
can be eliminated by a gauge transformation. What is the particle
content and what are the masses described by this Lagrangian density?