124
15. QCD II: Asymptotic Freedom, the Renormalization Group
15.3 Back to QCD: asymptotic freedom
15.3.1 One loop calculation
The reader will of course have realized, some time back, that the quantity β 0
introduced in (15.3) must be precisely the coefficient of α
2 in the one-loop
s
contribution to the β-function of QCD defined by
∂α s
β s = μ
2
;
(15.47)
∂μ 2
|
|
|
|
fixed bare αs
that is to say,
β s (one loop) = −β 0 α
2
(15.48)
s
with
33 − 2N f
β 0 =
.
(15.49)
12π
For N f ≤ 16 the quantity β 0 is positive, so that the sign of (15.48)) is opposite
to that of the QED analogue, equation (15.36). Correspondingly, (15.44) is
replaced by
α s (μ
2 )
α s (|q
2
|) =
,
(15.50)
[1 + α s (μ 2 )β 0 ln(Q 2 /μ 2 )]
2
where Q
2 = |q
2
|. Then replacing α s in (15.1) by (15.50) leads to (15.7).
Thus in QCD the strong coupling runs in the opposite way to QED, becoming smaller at large values of Q
2 (or small distances) – the property of
asymptotic freedom. The justly famous result (15.49) was first obtained by
Politzer (1973), Gross and Wilczek (1973), and ’t Hooft. ’t Hooft’s result,
announced at a conference in Marseilles in 1972, was not published. The
published calculation of Politzer and of Gross and Wilczek quickly attracted
enormous interest, because it immediately offered a way to understand how
the successful parton model could be reconciled with the undoubtedly very
strong binding forces between quarks. The resolution, we now understand,
lies in quite subtle properties of renormalized quantum field theory, involving
first the exposure of ‘large logarithms’, then their re-summation in terms of
the running coupling, and of course the crucial sign of the β-function. Not
only did the result (15.49) explain the success of the parton model: it also,
we repeat, opened the prospect of performing reliable perturbative calculations in a strongly interacting theory, at least at high Q
2 . For example, at
sufficiently high Q
2 , we can reliably compute the β function in perturbation
theory. The result of Politzer and of Gross and Wilczek, when combined with
2 Except that in (15.50) αs is evaluated at large spacelike values of q 2 , whereas in (15.7)
it is wanted at large timelike values. Readers troubled by this may consult Peskin and
Schroeder (1995) section 18.5. The difficulty is evaded in the approach of section 15.6
below.
15. QCD II: Asymptotic Freedom, the Renormalization Group
15.3 Back to QCD: asymptotic freedom
15.3.1 One loop calculation
The reader will of course have realized, some time back, that the quantity β 0
introduced in (15.3) must be precisely the coefficient of α
2 in the one-loop
s
contribution to the β-function of QCD defined by
∂α s
β s = μ
2
;
(15.47)
∂μ 2
|
|
|
|
fixed bare αs
that is to say,
β s (one loop) = −β 0 α
2
(15.48)
s
with
33 − 2N f
β 0 =
.
(15.49)
12π
For N f ≤ 16 the quantity β 0 is positive, so that the sign of (15.48)) is opposite
to that of the QED analogue, equation (15.36). Correspondingly, (15.44) is
replaced by
α s (μ
2 )
α s (|q
2
|) =
,
(15.50)
[1 + α s (μ 2 )β 0 ln(Q 2 /μ 2 )]
2
where Q
2 = |q
2
|. Then replacing α s in (15.1) by (15.50) leads to (15.7).
Thus in QCD the strong coupling runs in the opposite way to QED, becoming smaller at large values of Q
2 (or small distances) – the property of
asymptotic freedom. The justly famous result (15.49) was first obtained by
Politzer (1973), Gross and Wilczek (1973), and ’t Hooft. ’t Hooft’s result,
announced at a conference in Marseilles in 1972, was not published. The
published calculation of Politzer and of Gross and Wilczek quickly attracted
enormous interest, because it immediately offered a way to understand how
the successful parton model could be reconciled with the undoubtedly very
strong binding forces between quarks. The resolution, we now understand,
lies in quite subtle properties of renormalized quantum field theory, involving
first the exposure of ‘large logarithms’, then their re-summation in terms of
the running coupling, and of course the crucial sign of the β-function. Not
only did the result (15.49) explain the success of the parton model: it also,
we repeat, opened the prospect of performing reliable perturbative calculations in a strongly interacting theory, at least at high Q
2 . For example, at
sufficiently high Q
2 , we can reliably compute the β function in perturbation
theory. The result of Politzer and of Gross and Wilczek, when combined with
2 Except that in (15.50) αs is evaluated at large spacelike values of q 2 , whereas in (15.7)
it is wanted at large timelike values. Readers troubled by this may consult Peskin and
Schroeder (1995) section 18.5. The difficulty is evaded in the approach of section 15.6
below.