102
G. Altarelli and S. Forte
For some Green function G, normalised to one in lowest order, (like V/e with V the
3-g vertex function at the symmetric point p 2 = q 2 = r 2 , considered in the previous
section) we typically find at 1-loop:
G bare = 1 + α 0 (
−μ 2
p 2 )
[B(
1
+ log 4π − γ E ) + A + o(()]
(4.54)
In MS one rewrites this at 1-loop accuracy (diagram by diagram: this is a virtue of
the method):
G bare = ZG ren
Z = 1 + α [B(
1
+ log 4π − γ E )]
G ren = 1 + α [B log
−μ 2
p 2 + A]
(4.55)
Here Z stands for the relevant product of renormalisation factors. In the original MS
prescription only 1// was subtracted (that clearly plays the role of a cutoff) and not
also log 4π and γ E . Later, since these constants always appear from the expansion
of functions it was decided to modify MS into MS. Note that the MS definition
of α is different than that in the momentum subtraction scheme because the finite
terms (those beyond logs) are different. In particular here δG ren does not vanish at
p 2 = −μ 2 .
The third [12] and fourth [13] coefficients of the QCD β function are also
known in the MS prescription (recall that only the first two coefficients are scheme
independent). The calculation of the last term involved the evaluation of some
50,000 4-loop diagrams. Translated in numbers, for n f = 5 one obtains :
β(α) = − 0.610α
2
[1 + 1.261 . . .
α
π
+ 1.475 . . . (
α
π
)
2
+ 9.836 . . . (
α
π
)
3 . . .]
(4.56)
It is interesting to remark that the expansion coefficients are all of order 1 or (10 for
the last one), so that the MS expansion looks reasonably well behaved.
It is important to keep in mind that the QED and QCD perturbative series,
after renormalisation, have all their coefficients finite, but the expansion does not
converge. Actually the perturbative series are not even Borel summable. After Borel
resummation for a given process one is left with a result which is ambiguous
by terms typically down by exp −n/(bα), with n an integer and b the first β
function coefficient. In QED these corrective terms are extremely small and not very
important in practice. On the contrary in QCD α = α s (Q 2 ) ∼ 1/(b log Q 2 // 2 )
and the ambiguous terms are of order (1/Q 2 ) n , that is are power suppressed. It is
interesting that, through this mechanism, the perturbative version of the theory is
able to somehow take into account the power suppressed corrections. A sequence
G. Altarelli and S. Forte
For some Green function G, normalised to one in lowest order, (like V/e with V the
3-g vertex function at the symmetric point p 2 = q 2 = r 2 , considered in the previous
section) we typically find at 1-loop:
G bare = 1 + α 0 (
−μ 2
p 2 )
[B(
1
+ log 4π − γ E ) + A + o(()]
(4.54)
In MS one rewrites this at 1-loop accuracy (diagram by diagram: this is a virtue of
the method):
G bare = ZG ren
Z = 1 + α [B(
1
+ log 4π − γ E )]
G ren = 1 + α [B log
−μ 2
p 2 + A]
(4.55)
Here Z stands for the relevant product of renormalisation factors. In the original MS
prescription only 1// was subtracted (that clearly plays the role of a cutoff) and not
also log 4π and γ E . Later, since these constants always appear from the expansion
of functions it was decided to modify MS into MS. Note that the MS definition
of α is different than that in the momentum subtraction scheme because the finite
terms (those beyond logs) are different. In particular here δG ren does not vanish at
p 2 = −μ 2 .
The third [12] and fourth [13] coefficients of the QCD β function are also
known in the MS prescription (recall that only the first two coefficients are scheme
independent). The calculation of the last term involved the evaluation of some
50,000 4-loop diagrams. Translated in numbers, for n f = 5 one obtains :
β(α) = − 0.610α
2
[1 + 1.261 . . .
α
π
+ 1.475 . . . (
α
π
)
2
+ 9.836 . . . (
α
π
)
3 . . .]
(4.56)
It is interesting to remark that the expansion coefficients are all of order 1 or (10 for
the last one), so that the MS expansion looks reasonably well behaved.
It is important to keep in mind that the QED and QCD perturbative series,
after renormalisation, have all their coefficients finite, but the expansion does not
converge. Actually the perturbative series are not even Borel summable. After Borel
resummation for a given process one is left with a result which is ambiguous
by terms typically down by exp −n/(bα), with n an integer and b the first β
function coefficient. In QED these corrective terms are extremely small and not very
important in practice. On the contrary in QCD α = α s (Q 2 ) ∼ 1/(b log Q 2 // 2 )
and the ambiguous terms are of order (1/Q 2 ) n , that is are power suppressed. It is
interesting that, through this mechanism, the perturbative version of the theory is
able to somehow take into account the power suppressed corrections. A sequence