3Interacting fields
III Quantum Field Theory
3.1 Interaction Lagrangians
How do we introduce field interactions? The short answer is to throw more
things into the Lagrangian to act as the “potential”.
We start with the case where there is only one real scalar field, and the field
interacts with itself. Then the general form of the Lagrangian can be given by
L =
1
2
∂
µ
φ∂
µ
φ −
1
2
m
2
φ
2
−
∞
X
n=3
λ
n
n!
φ
n
.
These λ
n
are known as the coupling constants.
It is almost impossible to work with such a Lagrangian directly, so we want to
apply perturbation theory to it. To do so, we need to make sure our interactions
are “small”. The obvious thing to require would be that
λ
n
1. However, this
makes sense only if the
λ
n
are dimensionless. So let’s try to figure out what the
dimensions are.
Recall that we have
[S] = 0.
Since we know S =
R
d
4
xL, and [d
4
x] = −4, we have
[L] = 4.
We also know that
[∂
µ
] = 1.
So the ∂
µ
φ∂
µ
φ term tells us that
[φ] = 1.
From these, we deduce that we must have
[λ
n
] = 4 − n.
So
λ
n
isn’t always dimensionless. What we can do is to compare it with some
energy scale
E
. For example, if we are doing collisions in the LHC, then the
energy of the particles would be our energy scale. If we have picked such an
energy scale, then looking at
λ
n
E
n−4
would give us a dimensionless parameter
since [E] = 1.
We can separate this into three cases:
(i) n
= 3: here
E
n−4
=
E
−1
decreases with
E
. So
λ
3
E
−1
would be small at
high energies, and large at low energies.
Such perturbations are called relevant perturbations at low energies. In
a relativistic theory, we always have
E > m
. So we can always make the
perturbation small by picking
λ
3
m
(at least when we are studying
general theory; we cannot go and change the coupling constant in nature
so that our maths work out better).
(ii) n
= 4: Here the dimensionless parameter is just
λ
4
itself. So this is small
whenever λ
4
1. This is known as marginal perturbation.
(iii) n >
4: This has dimensionless parameter
λ
n
E
n−4
, which is an increasing
function of E. So this is small at low energies, and high at large energies.
These operators are called irrelevant perturbations.
While we call them “irrelevant”, they are indeed relevant, as it is typically difficult
to avoid high energies in quantum field theory. Indeed, we have seen that we
can have arbitrarily large vacuum fluctuations. In the Advanced Quantum Field
Theory course, we will see that these theories are “non-renormalizable”.
In this course, we will consider only weakly coupled field theories — one that
can truly be considered as small perturbations of the free theory at all energies.
We will quickly describe some interaction Lagrangians that we will study in
more detail later on.
Example (φ
4
theory). Consider the φ
4
theory
L =
1
2
∂
µ
φ∂
µ
φ −
1
2
m
2
φ
2
−
λ
4!
φ
4
,
where
λ
1. We can already guess the effects of the final term by noting that
here we have
[H, N] 6= 0.
So particle number is not conserved. Expanding the last term in the Lagrangian
in terms of
a
p
, a
†
p
, we get terms involving things like
a
p
a
p
a
p
a
p
or
a
†
p
a
†
p
a
†
p
a
†
p
or
a
†
p
a
†
p
a
†
p
a
p
, and all other combinations you can think of. These will create or
destroy particles.
We also have a Lagrangian that involves two fields — a real scalar field and
a complex scalar field.
Example
(Scalar Yukawa theory)
.
In the early days, we found things called
pions that seemed to mediate nuclear reactions. At that time, people did not
know that things are made up of quarks. So they modelled these pions in terms
of a scalar field, and we have
L = ∂
µ
ψ
∗
∂
µ
ψ +
1
2
∂
µ
φ∂
µ
φ − M
2
ψ
∗
ψ −
1
2
m
2
φ
2
− gψ
∗
ψφ.
Here we have
g m, M
. Here we still have [
Q, H
] = 0, as all terms are at
most quadratic in the
ψ
’s. So the number of
ψ
-particles minus the number of
ψ
-antiparticles is still conserved. However, there is no particle conservation for
φ.
A wary — the potential
V = M
2
ψ
∗
ψ +
1
2
m
2
φ
2
+ gψ
∗
ψφ
has a stable local minimum when the fields are zero, but it is unbounded below
for large −gφ. So we can’t push the theory too far.