5Perturbative renormalization

III Advanced Quantum Field Theory


5.3 Renormalization of the φ
4
coupling
We now try to renormalize the
φ
4
coupling. At 1-loop, in momentum space, we
receive contributions from
x
1
k
1
x
2
k
2
x
4
x
3
p
p+k
1
+k
2
and also a counter-term:
×
We first do the first loop integral. It is given by
g
2
µ
4−d
2
Z
d
4
p
(2π)
4
1
p
2
+ m
2
1
(p + k
1
+ k
2
)
2
+ m
2
.
This is a complicated beast. Unlike the loop integral we did for the propagator,
this loop integral knows about the external momenta
k
1
,
k
2
. We can imagine
ourselves expanding the integrand in
k
1
and
k
2
, and then the result involves
some factors of
k
1
and
k
2
. If we invert the Fourier transform to get back to
position space, then multiplication by
k
i
becomes differentiation. So these gives
contributions to terms of the form, say, (
∂φ
)
2
φ
2
, in addition to the
φ
4
, which is
what we really care about.
One can check that only the
φ
4
contribution is divergent in
d
= 4. This is
reflecting the fact that all these higher operators are all irrelevant.
So we focus on the contribution to
φ
4
. This is
k
i
-independent, and is given
by the leading part of the integral:
g
2
µ
4−d
2(2π)
d
Z
d
4
p
(p
2
+ m
2
)
2
=
1
2
g
2
(4π)
d/2
µ
m
4−d
Γ
2 −
d
2
.
How about the other two loop integrals? They give different integrals, but they
differ only in where the
k
i
appear in the denominator. So up to leading order,
they are the same thing. So we just multiply our result by 3, and find that the
loop contributions are
− δλ +
3g
2
2(4π)
d/2
µ
m
4−d
Γ
2 −
d
2
∼ −δλ +
3g
2
32π
2
2
ε
− γ + log
4πµ
2
m
2

+ O(ε).
Therefore, in the MS scheme, we choose
δλ =
3g
2
32π
2
2
ε
− γ + log 4π
,
and so up to O(~), the loop contribution to the φ
4
coupling is
3g
2
32π
2
log
µ
2
m
2
.
So in
λφ
4
theory, to subleading order, with an (arbitrary) dimensional regular-
ization scale µ, we have
∼ + + two more +
×
+ ···
−
g
~
+
3g
2
32π
2
log
µ
2
m
2
+ O(~)
Now note that nothing physical (such as this 4-point function) can depend on
our arbitrary scale µ. Consequently, the coupling g(µ) must run so that
µ
∂
∂µ
−
g
~
+
3g
2
32π
2
log
µ
2
m
2
+ O(~)
= 0.
This tells us that we must have
β(g) =
3g
2
~
32π
2
.
Note that this is the same
β
-function as we had when we did local potential
approximation!
We can solve this equation for the coupling
g
(
µ
), and find that the couplings
at scales µ and µ
0
are related by
1
g(µ)
=
1
g(µ
0
)
+
3
16π
2
log
µ
0
µ
.
Thus, if we find that at energy scale
µ
, the coupling takes value
g
0
, then at the
scale
µ
0
= µe
16π
2
/(3g
0
)
,
the coupling g(µ
0
) diverges. Our theory breaks down in the UV!
This is to be taken with a pinch of salt, because we are just doing perturbation
theory, with a semi-vague interpretation of what the running of
g
signifies.
So claiming that
g
(
µ
0
) diverges only says something about our perturbative
approximation, and not the theory itself. Unfortunately, more sophisticated
non-perturbative analysis of the theory also suggests the theory doesn’t exist.