APC8 Quantum Field Theory - Take Home Final (Problem 1) Consider the following classical Lagrangian for a massive vector particle
1
1
L = − Fαβ F αβ + M 2 Aα Aα − Aα J α ,
4
2
where Fαβ = ∂α Aβ − ∂β Aα , Aα is the vector potential and J α is an
external source.
(a) (10 points) Show that Aν satisfies the so-called Proca equation
n   o g µν  + M 2 − ∂ µ ∂ ν Aν = J µ . Assuming ∂µ J µ = 0, find the constraint equation imposed on Aν .
(b) (10 points) For a static point source Jµ at the origin, show that the
solution for the potential Aµ is
−ie Z ∞ kdk
eikr ,
A0 (r) = 2
4π r −∞ k 2 + M 2 Ai = 0 . (b) (10 points) Evaluate the above integral using contour integration
to obtain an explicit form of A0 (r). Show that one can reproduce the
Coulomb potential by taking M → 0.
(d) (10 points) The propagator Πµν = (g µν ( + M 2 ) − ∂ µ ∂ ν )−1 is defined as
    g µα x + M 2 − ∂xµ ∂xα Παν (x, y) = gνµ δ 4 (x − y) . Show that the solution is
!
Z
kα kν ik·(x−y)
−1
d4 k
gαν −
e
.
Παν (x, y) =
(2π)4 k 2 − M 2
M2
Can you guess what the Feynman propagator is?
(Problem 2) (20 points) Consider the following interaction Lagrangian for a real
quantum scalar field φ
1
Lint = − λ φ4 .
4!
Use the Lagrangian approach to compute the following 2-point function
hΩ|T {φ(x1 )φ(x2 )}|Ωi
to first order in the coupling λ.
1 φ(p3 ) e− (p1, s1) µ
γ ν φ∗ (p4) e+ (p2, s2) (Problem 3) Consider the following process
e− (p1 , s1 ) + e+ (p2 , s2 ) → φ(p3 ) + φ∗ (p4 )
in quantum electrodynamics to 2nd order in perturbation theory, as
shown above. Here e− is the electron and φ is a hypothetical scalar
particle with the same charge as the electron.
(a) (10 points) Write down the Lorentz invariant matrix element iM
for this process using the Feynman rules listed in the text.
(b) (10 points) Compute the unpolarized matrix element squared
1 X
|M|2 .
4 s1 ,s2
Express your answer in terms of the Mandelstam variables
s = (p1 + p2 )2 , t = (p1 − p3 )2 and u = (p1 − p4 )2 .
(c) (10 points) Calculate the unpolarized differential cross section dσ/dΩ
for the process in the center-of-mass frame.
(d) (10 points) Set the masses me = 0 and mφ = 0 and obtain the total
cross section in the center-of-mass frame. Due 9:00 AM, May 3, 2017
Be Honest & Good Luck!
2

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