From Classical Mechanics to Quantum Field Theory

A Concise Introduction to Quantum Field Theory 217

gives an extra divergent contribution to the vacuum energy

ΔE 0 =

λ

512 π^4

(

Λ^4 +Λ^2 m^2

(

log

m^2

2Λ^2 +2

)

+

m^4

4

(

log

m^2

2Λ^2 +2

) 2 )

+O

(

m^2

Λ^2

)

. (3.47)

That contribution has to be subtracted from the Hamiltonian to renormalize the
vacuum energy to zero.

3.6.1 Renormalization of excited states

The vacuum energy is not the only divergent quantity of the theory. The energy
of one-particle states gets a perturbative correction which is also UV divergent.
The energy of the excited state|fn〉=a†n| 0 〉in the free theory isωn. The first
order correction to this energy is

ΔEn=〈fn|Hˆint|fn〉.

Using Wick’s theorem, a simple calculation shows that

ΔEn=ΔE 0 +

λ

8

ωn (^1) ∑,ωn 2 <Λ
n 1 ,n 2 ∈Z^3
〈fn|φ−n 1 | 0 〉〈 0 |φn 1 |fn〉〈 0 |φ−n 2 φn 2 | 0 〉.
Both terms are divergent. The first term corresponds to the vacuum energy
correction, which is removed by the previous renormalization of vacuum energy
Eq. (3.47). The second term gives a new type of UV quadratic divergence. In the
sharp momentum cutoff it is given by
λ
64 π^2 ω^2 n

(

Λ^2 +

m^2

2

(

log

m^2

2Λ^2

+2

))

.

This renormalization of the divergence can be absorbed by a renormalization of
the mass of the theory. Indeed if we redefine the Hamiltonian of the theory as

Hˆintren=^1

2

Δm^2

∑

n∈Z^3

|φn|^2 +

λ

4!

ωn 1 ,ωn∑ 2 ,ωn 3 <Λ

n 1 ,n 2 ,n 3 ∈Z^3

φn 1 φn 2 φn 3 φ−n 1 −n 2 −n 3 ,

where

Δm^2 =−

λ

32 π^2

(

Λ^2 −m^2

(

log

m^2

2Λ^2

−

1

2

))

,