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253a/Schwartz
Due October 5, 2010
Problem Set 4
1. Consider the Lagrangian for
φ
3
theory
L
=
−
1
2
φ
(
s
+
m
2
)
φ
+
g
3!
φ
3
(1)
a) Draw a tree level feynman diagram for the decay
φ
→
φφ
. Write down the corre
sponding amplitude using the Feynman rules.
b) Now consider the one loop correction, given by
Write down the corresponding amplitude using the Feynman rules.
c) Now start over and write down the diagram from part
b
in position space, in terms
of integrals over the intermediate points and Wick contractions, represented with
D
F
′
s
.
d) Show that after you apply LSZ, what you got in
c
)
reduces to what you got in
b
)
,
by integrating the phases into
δ
functions, and integrating over those
δ
functions.
2. Nonrelatavistic Moller scattering:
e
−
e
−
→
e
−
e
−
. If the electron and photon were spinless,
we can write the Lagrangian as
L
=
−
1
2
φ
e
(
s
+
m
e
2
)
φ
e
−
1
2
A
s
A
+
e
2
m
e
Aφ
e
φ
e
(2)
the factor of
m
e
comes from the nonrelativistic limit (or by dimensional analysis!).
a) Draw the three treelevel
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This note was uploaded on 02/04/2012 for the course PHYS 253A at Harvard.
 '10
 SCHWARTZ

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