140
15. QCD II: Asymptotic Freedom, the Renormalization Group
are parallel). This is a divergence of the ‘collinear’ type, in the terminology of
section 14.4.2 – or, as there, a ‘mass singularity’, occurring in the zero quark
mass limit. If we simply replace the propagator factor t ˆ −1 = [(q − p
′ )
2 ]
−1 by
[(q − p
′ )
2
− m
2 ]
−1 , where m is a quark mass, then (15.92) becomes
∫ 1
dc
∼
(15.93)
(1 + 2m 2 z/Q 2 ) − c
which will produce a factor of the form ln(Q
2 /m
2 ) as m
2
→ 0. Thus m regulates the divergence. We have here an uncancelled mass singularity, and it
violates scaling. This crucial physical result is present in the lowest-order QCD
correction to the parton model, in this case. As we are learning, such logarithmic violations of scaling are a characteristic feature of all QCD corrections
to the free (scaling) parton model.
We may calculate the coefficient of the ln Q
2 term by retaining in (15.87)
only the terms proportional to t ˆ −1 :
∫ 1 dc
1 ≈
i
(
)
2
α s (μ
2 ) 4 1 + z
· ·
2π
3 1 − z
2F ˆ
2
i
(15.94)
e
1 − c
−1
and so, for just one quark species, this QCD correction contributes (from
(15.90)) a term
1
2
i
e α s (μ
2 )
to 2F 1 , where
(
)
2
4 1 + z
P ˆ qq (z) =
,
(15.96)
3 1 − z
and C(x/y) has no mass singularity.
Our result so far is therefore that the ‘free’ quark distribution function
q(x), which depended only on the scaling variable x, becomes modified to
∫ 1
α s (μ
2 )
dy
∫
{
}
dy
ˆ
q(y) P qq (x/y) ln(Q
2 /m
2 ) + C(x/y)
y
(15.95)
2π
x
{
(
)
}
P ˆ qq (x/y) ln Q
2 /m
2 + C(x/y)
q(x) +
q(y)
(15.97)
2π
x y
∫
∫
1
1
α s (μ
2 )
dy
dzδ(zy − x)q(y){P ˆ qq (z) ln(Q
2 /m
2 )
0
= q(x) + 2π
0
+ C(z)}
(15.98)
due to lowest-order gluon radiation. Clearly, this corrected distribution function violates scaling because of the ln Q
2 term. But the result as it stands
cannot represent a well-controlled approximation, since it contains divergences
as z → 1 and as m
2
→ 0.
We postpone discussion of the mass divergence until the next section. The
divergence as z → 1 is a standard infrared divergence (the quark momentum
15. QCD II: Asymptotic Freedom, the Renormalization Group
are parallel). This is a divergence of the ‘collinear’ type, in the terminology of
section 14.4.2 – or, as there, a ‘mass singularity’, occurring in the zero quark
mass limit. If we simply replace the propagator factor t ˆ −1 = [(q − p
′ )
2 ]
−1 by
[(q − p
′ )
2
− m
2 ]
−1 , where m is a quark mass, then (15.92) becomes
∫ 1
dc
∼
(15.93)
(1 + 2m 2 z/Q 2 ) − c
which will produce a factor of the form ln(Q
2 /m
2 ) as m
2
→ 0. Thus m regulates the divergence. We have here an uncancelled mass singularity, and it
violates scaling. This crucial physical result is present in the lowest-order QCD
correction to the parton model, in this case. As we are learning, such logarithmic violations of scaling are a characteristic feature of all QCD corrections
to the free (scaling) parton model.
We may calculate the coefficient of the ln Q
2 term by retaining in (15.87)
only the terms proportional to t ˆ −1 :
∫ 1 dc
1 ≈
i
(
)
2
α s (μ
2 ) 4 1 + z
· ·
2π
3 1 − z
2F ˆ
2
i
(15.94)
e
1 − c
−1
and so, for just one quark species, this QCD correction contributes (from
(15.90)) a term
1
2
i
e α s (μ
2 )
to 2F 1 , where
(
)
2
4 1 + z
P ˆ qq (z) =
,
(15.96)
3 1 − z
and C(x/y) has no mass singularity.
Our result so far is therefore that the ‘free’ quark distribution function
q(x), which depended only on the scaling variable x, becomes modified to
∫ 1
α s (μ
2 )
dy
∫
{
}
dy
ˆ
q(y) P qq (x/y) ln(Q
2 /m
2 ) + C(x/y)
y
(15.95)
2π
x
{
(
)
}
P ˆ qq (x/y) ln Q
2 /m
2 + C(x/y)
q(x) +
q(y)
(15.97)
2π
x y
∫
∫
1
1
α s (μ
2 )
dy
dzδ(zy − x)q(y){P ˆ qq (z) ln(Q
2 /m
2 )
0
= q(x) + 2π
0
+ C(z)}
(15.98)
due to lowest-order gluon radiation. Clearly, this corrected distribution function violates scaling because of the ln Q
2 term. But the result as it stands
cannot represent a well-controlled approximation, since it contains divergences
as z → 1 and as m
2
→ 0.
We postpone discussion of the mass divergence until the next section. The
divergence as z → 1 is a standard infrared divergence (the quark momentum