Imperial College London
MSc EXAMINATION May 2012
This paper is also taken for the relevant Examination for the Associateship
ADVANCED QUANTUM FIELD THEORY
For Students in Quantum Fields and Fundamental Forces
Friday, 25 May 2012: 14:00 to 17:00
Answer 3 out of the following 4 questions.
Marks shown on this paper are indicative of those the Examiners anticipate assigning.
General Instructions
Complete the front cover of each of the THREE answer books provided.
If an electronic calculator is used, write its serial number at the top of the front cover of each answer
book.
USE ONE ANSWER BOOK FOR EACH QUESTION.
Enter the number of each question attempted in the box on the front cover of its corresponding
answer book.
Hand in THREE answer books even if they have not all been used.
You are reminded that Examiners attach great importance to legibility, accuracy and
clarity of expression.
c Imperial College London 2012
1
Go to the next page for questions
You may use the following results without proof:
• Loop integral in d dimensions (Minkowskian):
Z
d−2n
dd p
1
Γ(n − d/2)
2
n im
In (m ) ≡
=
(−1)
d
2
2
n
d/2
(2π) (p − m )
(4π)
Γ(n)
• Gamma functions: zΓ(z) = Γ(z + 1) and
!
m
2 X1
+
− γ + O()
p=1 p
(−1)m
=
Γ −m +
2
m!
for integer m ≥ 0
• Feynman parameters:
1
=
ab
1
Z
0
dx
[xa + (1 − x)b]2
• Gaussian integral
– Normal
R
1
T
dN q qi qj e− 2 q Mq
= M−1 ij ,
R
1 T
−
q
Mq
dN q e 2
where q = (q1 , . . . , qN ) and M is an N × N matrix.
– Grassmannian
R
†
dN c∗ dN c ci c∗j e−c Mc
−1
R
,
=
M
ij
dN c∗ dN c e−c† Mc
where c = (c1 , . . . , cN ) and M is an N × N matrix.
• Gaussian two-point function
– Normal
R
1
4
R
4
Dφ φ(x)φ(y)e− 2 d xd yφ(x)M (x,y)φ(y)
R
= M −1 (x, y).
R
− 12 d4 xd4 yφ(x)M (x,y)φ(y)
Dφ e
– Grassmannian
R 4 4
R
Dψ̄Dψ ψα (x)ψ̄β (y)e− d xd yψ̄α (x)Mαβ (x,y)ψβ (y)
−1
R
R
=
M
(x, y).
4
4
αβ
Dψ̄Dψ e− d xd yψ̄α (x)Mαβ (x,y)ψβ (y)
• Gamma matrices:
trγ µ = 0,
trγ 5 = 0,
{γ µ , γ ν } = 2g µν ,
5 µ
(γ 5 )† = γ 5 ,
γ , γ = 0.
2
Go to the next page for questions
1.
(i) Show that in 4 − dimensions
Z
d4− p
4πµ2
1
im2 2
µ
+ ln
=
+ 1 − γ + O() ,
(2π)4− p2 − m2
16π 2 m2
and
µ
Z
d4− p
1
i
=
4−
2
2
2
(2π) (p − m )
16π 2
2
4πµ2
+ ln
− γ + O() .
m2
[6 marks]
(ii) Consider the integral
I2µνρ
=µ
qµqν qρ
d4− q
.
(2π)4− ((k + q)2 − m2 )(q 2 − m2 )
Z
(a) Show that the divergent part is
i
k2
µνρ
µ νρ
ν µρ
ρ µν
2
µ ν ρ 2
I2 = −
(k g + k g + k g ) m −
+k k k
+ finite.
64π 2
6
[8 marks]
(b) Use the integrals
Z
Z
1
dx x ln(1 − 2x)2 = −1,
0
1
5
dx x2 ln(1 − 2x)2 = − ,
9
Z0 1
1
dx x3 ln(1 − 2x)2 = − ,
3
0
to show that for k 2 = 4m2 , the full integral, including finite parts, is
im2
2
4πµ2
5
µνρ
µ νρ
ν µρ
ρ µν
I2
= −
(k g + k g + k g )
+ ln
−γ+
192π 2
m2
3
2
2
im
2
4πµ
4
−
(k µ k ν k ρ )
+ ln
−γ+
.
2
2
64π
m
3
(1)
[6 marks]
[Total 20 marks]
3
Please go to the next page
2. The Lagrangian of scalar electrodynamics (theory of an electrically charged scalar field) is
1
λ
1
L = − Fµν F µν − (∂µ Aµ )2 + (Dµ φ)∗ (Dµ φ) − m2 φ∗ φ − (φ∗ φ)2 ,
4
2ξ
4
where φ is a complex scalar, Aµ is an Abelian gauge field, Dµ = ∂µ + ieAµ is a covariant
derivative, and Fµν = ∂µ Aν − ∂ν Aµ is the field strength tensor.
(i) Show that the Feynman rules for the interaction vertices are
↔ −iλ,
(1)
↔ −ie(kµ + kµ0 ),
(2)
↔ 2ie2 gµν .
(3)
µ
q
k0
k
µ
ν
Indicate also the momenta of the legs and show how momentum is conserved at each
vertex.
[6 marks]
(ii) Draw the 1PI diagrams that contribute to the scalar and photon two-point functions ,
and
at one-loop level.
,
[3 marks]
(iii) Use the Feynman gauge propagators
µ
ν
k
k
−ig µν
,
k2
i
↔ 2
,
k − m2
↔
to write down the integrals corresponding to the 1PI diagrams in (ii) in dimensional
regularisation.
[7 marks]
(iv) Using the expressions given in Question 1.(i), calculate the photon two-point function at
k = 0. Explain the significance of your result.
[4 marks]
[Total 20 marks]
4
Please go to the next page
3. Consider again scalar electrodynamics (see Question 2)
λ
1
L = − Fµν F µν + (Dµ φ)∗ (Dµ φ) − m2 φ∗ φ − (φ∗ φ)2 ,
4
4
this time in d spacetime dimensions.
(i) Find the dimensionalities of the fields Aµ and φ and parameters e, λ and m.
[3 marks]
(ii) Show that the dimensionality of the momentum-space 1PI correlator Γ̃EA ,Eφ with EA
external photon legs and Eφ external scalar legs is
[Γ̃EA ,Eφ ] = d − (EA + Eφ )
d−2
.
2
[4 marks]
(iii) Using momentum conservation or other arguments, show that the number of loops in a
diagram with EA external photon legs and Eφ external scalar legs, and V1 , V2 and V3
vertices of the types (1), (2) and (3), respectively (see Question 2) is
1
1
L = V1 + V2 + V3 − (EA + Eφ ) + 1.
2
2
[4 marks]
(iv) Using dimensional analysis, show that the the superficial degree of divergence of the
diagram is
D = 4 − (EA + Eφ ) + (d − 4)L.
[4 marks]
(v) Use the previous result to explain how the renormalisability of the theory depends on
d?
[3 marks]
(vi) What counterterms do you need to introduce to renormalise the theory in d = 3 dimensions? Explain.
[2 marks]
[Total 20 marks]
5
Please go to the next page
4. Consider the Georgi-Glashow model, with an SU(2) gauge field Aµ and a scalar field Φ in the
adjoint representation. The Lagrangian is
2
1
L = − Tr Fµν F µν + Tr (Dµ Φ)(Dµ Φ) − λ TrΦ2 − v 2 ,
2
with field strength tensor Fµν = ∂µ Aν − ∂ν Aµ + ig[Aµ , Aν ] and covariant derivate Dµ Φ =
∂µ Φ + ig[Aµ , Φ]. This Lagrangian is invariant under the SU(2) gauge symmetry
i
Aµ → AUµ = U Aµ U † − U ∂µ U † ,
g
Φ → ΦU = U ΦU † .
(i) Express the kinetic terms of the gauge field Aµ in terms of its components Aaµ defined as
Aµ = Aaµ ta , where the 2 × 2 matrices T a satisfy [ta , tb ] = (i/2)abc tc and Tr ta tb = δ ab /2.
Show that without gauge fixing the tree-level two-point function hAaµ (k)Abν (q)i is not well
defined, and explain why.
[6 marks]
(ii) Explain why any gauge-invariant expectation value can be calculated using the modified
Lagrangian
1
Lξ = L − (∂ µ Aaµ )2 + ∂ µ ca∗ ∂µ ca − gabc ∂ µ ca∗ Abµ cc .
2ξ
What are the properties of the field c, and what type of particle does it describe? How
is the value of the parameter ξ determined?
[11 marks]
(iii) Show that with Lagrangian Lξ , the tree-level gauge-field two-point function is
−iδ ab
kµ kν
a
b
4
hAµ (k)Aν (q)i = (2π) δ(k + q) 2
gµν − (1 − ξ) 2 .
k
k
[3 marks]
[Total 20 marks]
6
End of examination paper