a) Draw the three tree-level
e
−
e
−
→
e
−
e
−
diagrams following from this Lagrangian.
b) Which one of the diagrams would be forbidden in real QED?
c) Evaluate the other two diagrams, and express the answers in terms of
s, t, u
. Give
the diagrams an extra relative minus sign, because electrons are fermions.
d) Now let’s put back the spin. In the non-relativistic limit, the electron spin is con-
served. This should be true at each vertex, even if the photon is virtual. For each
of the 16 possible sets of spins for the 4 electrons (for example
|↑↓) → |↑↑)
), which
of the diagrams can contribute for each set?
e) Add up all the contributions to
dσ
d
Ω
for each spin channel. Express the answer in
terms of
E
CM
and the scattering angle
θ
. Sketch the angular distribution.
3. Use the Lagrangian
L
=
−
1
2
φ
1
square
φ
1
−
1
2
φ
2
square
φ
2
+
λ
2
φ
1
(
∂
μ
φ
2
)(
∂
μ
φ
2
) +
g
2
φ
1
2
φ
2
(3)
to calculate the differential cross section
dσ
d
Ω
(
φ
1
φ
2
→
φ
1
φ
2
)
(4)
at tree level.
4. * Consider a Feynman diagram that looks like a regular tetrahedron, with the external
lines coming out of the 4 corners. This can contribute to
2
→
2
scatttering in a scalar
field theory with interaction
λ
4!
φ
4
. Write down the corresponding amplitude including the
appropriate symmetry factor. You can take
φ
to be massless. What would the symmetry
factor be for the same diagram in
φ
3
theory without the external lines?
1

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- SCHWARTZ
- Quantum Field Theory, Fundamental physics concepts, Feynman, corresponding amplitude