III Advanced Quantum Field Theory - Wilsonian renormalization

4Wilsonian renormalization

III Advanced Quantum Field Theory

4.2 Integrating out modes

Suppose, for some magical reason, we know exactly what the theory at the

energy scale Λ

is, and they are given by the coupling coefficients

(Λ

). What

happens when we integrate out some high energy modes?

We pick some Λ

, and split our field

into “low” and “high” energy

modes as follows:

ϕ(x) =

|p|≤Λ

(2π)

˜ϕ(p)e

ip·x

0≤|p|≤Λ

(2π)

˜ϕ(p)e

ip·x

Λ<|p|≤Λ

(2π)

˜ϕ(p)e

ip·x

We thus define

φ(x) =

0≤|p|≤Λ

(2π)

˜ϕ(p)e

ip·x

χ(x) =

Λ<|p|≤Λ

(2π)

˜ϕ(p)e

ip·x

and so

ϕ(x) = φ(x) + χ(x).

Let’s consider the effective theory we obtain by integrating out

. As before, we

define the scale Λ effective action

[φ] = −~ log

∞

(M)

Λ<|p|<Λ

Dχ e

−S

[ϕ,χ]/~

. (∗)

Of course, this can be done for any Λ

, and so defines a map from [0

] to

the “space of all actions”. More generally, for any

, this procedure allows us to

take a scale Λ action and produce a scale Λ

− ε

effective action from it. This is

somewhat like a group (or monoid) action on the “space of all actions”, and thus

the equation (∗) is known as the Wilsonian renormalization group equation.

Just as we saw in low-dimensional examples, when we do this, the coupling

constants of the interactions will shift. For each Λ

, we can define the

shifted coefficients g

(Λ), as well as Z

and δm

, by the equation

[φ] =

(∂φ)

d−d

(Λ)O

(φ, ∂φ)

where n

is the number of times φ or ∂φ appears in O

Note that we normalized the

(Λ) in terms of the new Λ and

. So even if,

by some miracle, our couplings receive no new corrections, the coefficients still

transform by

(Λ) =





d−d

(Λ

The factor

account from the fact that there could be new contributions to

the kinetic term for

. This is called wavefunction renormalization. The factor

is not to be confused with the partition function, which we denote by a

calligraphic Z instead. We will explore these in more detail later.

We define

Z(Λ, g

(Λ)) =

∞

(M)

≤Λ

Dϕ e

−S

[ϕ]/~

Then by construction, we must have

Z(Λ

, g

(Λ

)) = Z(Λ, g

(Λ))

for all Λ

. This is a completely trivial fact, because we obtained

(Λ

, g

(Λ))

simply by doing part of the integral and leaving the others intact.

We will assume that

varies continuously with Λ (which is actually not the

case when the allowed modes are discrete, but whatever). It is then convenient

to write the above expression infinitesimally, by taking the derivative. Instead

of the usual

dΛ

, it is more convenient to talk about the operator Λ

dΛ

instead,

as this is a dimensionless operator.

Differentiating the above equation, we obtain

dΛ

(Λ, g

(Λ)) = Λ

∂Z

∂Λ



∂Z

∂g



∂g

∂Λ

= 0. (†)

This is the Callan-Symanzik equation for the partition function.

It is convenient to refer to the following object:

Definition (Beta function). The beta function of the coupling g

) = Λ

∂g

∂Λ

As mentioned before, even if our coupling constants magically receive no

corrections, they will still change. Thus, it is convenient to separate out the

boring part, and write

) = (d

− d)g

+ β

quantum

({g

}).

Notice that perturbatively, the

quantum

(

}

) come from loops coming from

integrating out diagrams. So generically, we expect them to depend on all other

coupling constants.

We will later need the following definition, which at this point is rather

unmotivated:

Definition (Anomalous dimension). The anomalous dimension of φ by

= −

∂ log Z

∂Λ

Of course, at any given scale, we can absorb, say,

by defining a new field

ϕ(x) =

so as to give

(

) canonically normalized kinetic terms. Of course, if we do

this at any particular scale, and then try to integrate out more modes, then the

coefficient will re-appear.