Physical Foundations of Cosmology

4.2 Quantum chromodynamics and quark–gluon plasma 143

figure out the structure of perturbative expansion (4.22) and find a way to resum
this series, at least partially. This can be done using simple physical arguments.
Let us note that the value of the coupling constantα(q^2 )should not depend on the
normalization pointμ^2 ,which is arbitrary. Therefore, the derivative of the right
hand side of (4.22) with respect toμ^2 should be equal to zero:

d

dμ^2

(∞

∑

n= 0

αn+^1 (μ^2 )fn

(

q^2

μ^2

))

= 0. (4.23)

Differentiating and rearranging the terms, we obtain the following differential equa-
tion forα(μ^2 ):

dα(μ^2 )

dlnμ^2

=α^2

⎛

⎜⎜

⎝

∑∞

l= 0

xfl′+ 1 (x)αl

∑∞

l= 0

(l+ 1 )fl(x)αl

⎞

⎟⎟

⎠=α

2 (μ 2 )

(∞

∑

l= 0

fl+′ 1 ( 1 )αl(μ^2 )

)

, (4.24)

where a prime denotes the derivative with respect tox≡q^2 /μ^2 .The ratio of sums
in (4.24) should not depend onxbecause the left hand side of the equation is
x-independent. Therefore, to obtain the second equality in (4.24) we setx= 1 .The
requirement that the ratio does not depend onximposes rather strong restrictions on
the admissible functionsfn(x).From the second equality, we derive the following
recurrence relations:

dfn+ 1 (x)

dlnx

=

∑n

k= 0

(k+ 1 )fn′+ 1 −k( 1 )fk(x). (4.25)

Problem 4.6Verify that the general solution of these recurrence relations is given
by

fn(x)=

∑n

l= 0

cl(lnx)l, (4.26)

where the numerical coefficient in front of the leading logarithm is equal to( cn=
f 1 ′( 1 )

)n

.

The running constantα(q^2 ) depends onq^2 in the same way thatα(μ^2 ) depends
onμ^2 .Henceα(q^2 ) satisfies the equation

dα

(

q^2

)

dlnq^2

=α^2 (q^2 )

(∞

∑

l= 0

fl′+ 1 ( 1 )αl(q^2 )

)

, (4.27)

which follows from (4.24) by the substitutionμ^2 →q^2 .Equations (4.24) and
(4.27) are the well known Gell-Mann–Lowrenormalization groupequations and