b) Which diagrams must vanish or cancel by Furry’s theorem? Show this explicitly
through direct calculation.
c) Show that the QED counterterms cancel all the UV divergences. [Hint: you do not
need to evaluate this integrals for this, just look at their behavior at large loop
momenta]
d) * If the diagrams in part
b
did not identically vanish, would this mean that QED
is sick? If not, what do you do with the UV divergences in these loops?
3. Recall that in the standard model, the Higgs couples to all the fermions through Yukawa
couplings of the form
L
higgs
=
−
1
2
φ
(
square
+
m
h
2
)
φ
+
λφψ
¯
ψ
+
ψ
¯
(
i∂
+
eA
−
m
ψ
)
ψ
(1)
In this problem we are interested in the running of the Yukawa coupling
λ
. This running
is extremely important for grand unified theories and the origin of the peculiar distribu-
tion of particle masses in the standard model. Analog theories with the same Lagrangian
also appear in many condensed matter contexts.
a) Come up with renormalization conditions to define a renormalized Yukawa cou-
pling
λ
R
at a scale
p
0
in terms of the measured value of a 3-point Green’s function.
Which diagrams are relevant for the 1-loop radiative corrections to this Green’s
function?

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