Download Exercises for Quantum Field Theory (TVI/TMP)

Arnold Sommerfeld Center
Ludwig–Maximilians–Universität München
Prof. Dr. Livia Ferro
Summer term 2016
Exercises for Quantum Field Theory (TVI/TMP)
Problem set 7, due June 1, 2016
1 Interacting real scalar field theory: φ4
Consider the theory describing an interacting real scalar field with Lagrangian density
L[φ] =
1
1
g
∂µ φ ∂ µ φ − m2 φ2 − φ4 .
2
2
4!
(1)
During the lectures you have seen how the generating functional can be used to derive n-point functions of
scalar fields in this theory. Consider a perturbative expansion with parameter g, and focus on the second
order contributions (g 2 ).
(i) How many vacuum diagrams can you build using two 4-point vertices? Draw them.
(ii) How do these diagrams contribute to the 2- and 4-point correlation functions?
(iii) Consider the 2-point correlation function. How many second order diagrams can you draw with two
external legs? You should find three of them. Compute their symmetry factor.
(iv) Consider the 4-point correlation function. How many fully connected second order diagrams can you
draw with four external legs? You should find seven of them. Compute their symmetry factor.
2 Interacting complex scalar field theory: (φ∗ φ)2
Consider the theory describing an interacting complex scalar field with Lagrangian density
g
L[φ, φ∗ ] = ∂µ φ∗ ∂ µ φ − m2 φ∗ φ − (φ∗ φ)2 .
4
(2)
This theory has two kinds of sources, J and J ∗ , and so we need a way to tell which is which when we draw
the diagrams. Rather than labeling the source blobs with a J or J ∗ , we will indicate which is which by
putting an arrow on the attached propagator that points towards the source if it is a J ∗ , and away from the
source if it is a J.
(i) How do you draw the two point function?
(ii) What kind of vertex appears in the diagrams for this theory? Hint: your answer should involve those
arrows.
(iii) Compute (up to order g) the normalized generating functional
R
δ
exp iSint [−i δJ
, −i δJδ ∗ ] exp[− d4 x d4 yJ ∗ (x)∆F (x − y)J(y))]
∗
R
Znorm [J, J ] = .
δ
exp iSint [−i δJ
, −i δJδ ∗ ] exp[− d4 x d4 yJ ∗ (x)∆F (x − y)J(y))] |J,J ∗ =0
(3)
Draw all the involved diagrams, and verify that there is no vacuum contribution.
(iv) Evaluate, up to order g, the 2-point correlation function
δ
δ
hφ∗ (x1 )φ(x2 )i = −i
−i ∗
Znorm [J, J ∗ ]
.
δJ(x1 )
δJ (x2 )
J,J ∗ =0
(4)
3 Interacting real scalar field theory: φ3
Consider the following Lagrangian density for an interacting real scalar field:
L=
1
g
1
∂µ φ ∂ µ φ − m2 φ2 − φ3 .
2
2
3!
(5)
(i) What is the mass dimension of g in four space-time dimensions?
Consider the normalized generating functional:
R
δ
exp iSint [−i δJ
] exp[− 12 d4 x d4 yJ(x)∆F (x − y)J(y))]
R
Znorm [J] =
.
δ
{exp iSint [−i δJ
] exp[− 21 d4 x d4 yJ(x)∆F (x − y)J(y))]}|J=0
(6)
(ii) Evaluate explicitly the numerator up to and including the order g and then give the diagrammatic
expression of its various terms.
(iii) The 1-point correlator is non-zero. Evaluate it at order g.
(iv) Evaluate the 2-point correlator up to and including the order g.