2282 relevant interactions lets look at φ 3 theory

22.8.2 relevant interactions
Let’s look at φ 3 theory first, then the mass terms. We are interested in radiative corrections, so let’s
couple φ to something else, say a fermion ψ . We will set the mass of φ to zero for simplicity. The
Lagrangian is
L = −
1
2
φ square φ +
g
3!
φ
3
+ λφψ
¯
ψ + ψ
¯
( i∂ − M ) ψ (22.59)
22.8 Super-renormalizable field theories 259

Now consider the renormalization of g . We can compute this by evaluating the 3-point function
(
0 | T { φ ( p ) φ ( q
1
) φ ( q
2
) }| 0
)
= g + (22.60)
There is a radiative correction from the loop of φ . This is UV finite, proportional to g
3
and not particu-
larly interesting. A more important radiative correction comes from the loop of the fermion.
M
2
=
p
q
2
k
q
1
k − q
1
k + q
2
= λ 3
integraldisplay
d
4
k
(2 π ) 4
Tr [
1
k − M
1
k − q
1
− M
1
k + q
2
− M
] (22.61)
=
λ 3
8 π 2
M
integraldisplay
0
1
dz
integraldisplay
0
1 − z
dy
bracketleftBigg
M 2 + p 2 ( y + z − 3 yz −
1
2
)
M 2 − p 2 yz
+ 3 log
Λ 2
M 2 − p 2 yz
bracketrightBigg
(22.62)
Taking p 2 = 0 , this gives
( 0 | T { φ ( p ) φ ( q
1
) φ ( q
2
) }| 0 ) = g +
λ
3
16 π 2
M
parenleftbigg
1 + 3 log
Λ
2
M 2
parenrightbigg
(22.63)
So we find a (divergent) shift in g proportional to the mass of the fermion M . This is fine if M ∼ g , but
parametrically it’s really weird. As M gets larger, the correction grows. That means that the theory is
sensitive to scales much larger than the scale associated with g . So,
• Super-renormalizable theories are sensitive to UV physics
Is this a problem? Not in the technical sense, because the correction can be absorbed in a renormalization
of g . We just add a counterterm by writing
g = g
R
−
λ 3
16 π 2
M
parenleftbigg
1 + 3 log
Λ 2
M 2
parenrightbigg
(22.64)
This will cancel the radiative correction exactly (at p = 0 ). Again, this is totally reasonable if g
R
∼ M , and
theoretically and experimentally consistent for all g , but if g
R
≪ M it looks strange .
Will there be problems due to the quantum corrections? We can look at the Green’s function at a dif-
ferent value of p
2
. This would contribute to something like the scalar Coulomb potential. We find that of,
course, the divergence cancels, and the result at small p is something like
( 0 | T { φ ( p ) φ ( q
1
) φ ( q
2
) }| 0 ) ≈ g
R
+
λ 3
16 π
2
bracketleftbigg
p 2
M
+ 3 M log
parenleftbigg
1 −
p 2
M
2
parenrightbigg
+
bracketrightbigg
(22.65)
≈ g
R
+
λ
3
16 π 2
p
2
M
(22.66)
So for p ≪ M these corrections are tiny and do in fact decouple as M → ∞ .
As for the non-renormalizable case, we enjoy speculating that the ultimate theory of nature is finite.
Then Λ is physical. If we suppose that there is some bare g which is fixed, then we need a very special
value of Λ to get the measured coupling to be g
R
≪ M . Of course, there is some solution for Λ , for any g
0
and g
R
. But if we change Λ by just a little bit, then we get
g
R
→ g
R
+
λ
3
16 π 2
M log
Λ
old
2
Λ
new
2
(22.67)
which is a huge correction if M ≫ g
R
.