CHAPTER 2
Asymptotic Freedom and the
Renormalization Group
2.1 Principal Chiral Fieldst
In this section we shall study the large p limit for the ^ = 2 principal
chiral field described by the Lagrangian:
I /Trid^g ^ c^g)
(l/e^ =
I and g e G)
( 2.1)
As we discussed in Chapter 1, the infrared interaction of the massless
particles described by (2.1) is logarithmically strong. It is our aim now
to reveal the structure of this logarithmic interaction.
Let us study the effective Lagrangian which arises from (2.1) in the
loop approximation. In order to find it we write the quantum field g{x)
in the form:
g(x) = h(x)-g,^(x)
(2.2)
where ^ci(-^) is some classical solution for the Lagrangian (2.1), namely
— yj
(2.3)
=0
f^^ = c^g’ g
Our programme is to integrate over the field h{x) so as to obtain an
effective action depending on g^x{x). This approach is more or less
standard in field theory but it requires some clarification. At the first
sight, due to the invariance properties of the integration measure,
^h{x) = ^(h(x)g^x(x)X the result of such integration would seem not to
be dependent on
^Iso it is not quite clear, what kind of
t A principal chiral field is one which defines the principal bundle over the base space.
19
DOI: 10.1201/9780203755082-2
Asymptotic Freedom and the
Renormalization Group
2.1 Principal Chiral Fieldst
In this section we shall study the large p limit for the ^ = 2 principal
chiral field described by the Lagrangian:
I /Trid^g ^ c^g)
(l/e^ =
I and g e G)
( 2.1)
As we discussed in Chapter 1, the infrared interaction of the massless
particles described by (2.1) is logarithmically strong. It is our aim now
to reveal the structure of this logarithmic interaction.
Let us study the effective Lagrangian which arises from (2.1) in the
loop approximation. In order to find it we write the quantum field g{x)
in the form:
g(x) = h(x)-g,^(x)
(2.2)
where ^ci(-^) is some classical solution for the Lagrangian (2.1), namely
— yj
(2.3)
=0
f^^ = c^g’ g
Our programme is to integrate over the field h{x) so as to obtain an
effective action depending on g^x{x). This approach is more or less
standard in field theory but it requires some clarification. At the first
sight, due to the invariance properties of the integration measure,
^h{x) = ^(h(x)g^x(x)X the result of such integration would seem not to
be dependent on
^Iso it is not quite clear, what kind of
t A principal chiral field is one which defines the principal bundle over the base space.
19
DOI: 10.1201/9780203755082-2