3 The Standard Model of Electroweak Interactions
61
Even leaving aside QCD corrections, a set of important quantitative contributions
to the radiative corrections arise from large logarithms [e.g. terms of the form
(α/π ln (m Z /m f l l )) n where f ll is a light fermion]. The sequences of leading and
close-to-leading logarithms are fixed by well-known and consolidated techniques (β
functions, anomalous dimensions, penguin-like diagrams, etc.). For example, large
logarithms from pure QED effects dominate the running of α from m e , the electron
mass, up to m Z . Similarly large logarithms of the form [α/π ln (m Z /μ)] n also
enter, for example, in the relation between sin
2 θ W at the scales m Z (LEP, SLC)
and μ (e.g. the scale of low-energy neutral-current experiments). Also, large logs
from initial state radiation dramatically distort the line shape of the Z resonance as
observed at LEP1 and SLC and this effect was accurately taken into account for
the measurement of the Z mass and total width. The experimental accuracy on m Z
obtained at LEP1 is δm Z = ±2.1 MeV (see Chap. 6). Similarly, a measurement of
the total width to an accuracy δδ = ±2.3 MeV has been achieved. The prediction of
the Z line-shape in the SM to such an accuracy has posed a formidable challenge to
theory, which has been successfully met. For the inclusive process e + e − → f ¯
f X,
with f = e (for a concise discussion, we leave Bhabha scattering aside) and X
including γ ’s and gluons, the physical cross-section can be written in the form of a
convolution [42]:
σ (s) =
1
z 0
dz ˆ
σ (zs)G(z, s) ,
(3.95)
where ˆ
σ is the reduced cross-section, and G(z, s) is the radiator function that
describes the effect of initial-state radiation; ˆ
σ includes the purely weak corrections,
the effect of final-state radiation (of both γ ’s and gluons), and also non-factorizable
terms (initial- and final-state radiation interferences, boxes, etc.) which, being small,
can be treated in lowest order and effectively absorbed in a modified ˆ
σ . The radiator
G(z, s) has an expansion of the form
G(z, s) = δ(1 − z) + α/π(a 11 L + a 10 ) + (α/π)
2 (a 22 L
2
+ a 11 L + a 20 ) + . . . +
+ (α/π)
n
n
i=0
a ni L
i ,
(3.96)
where L = ln s/m 2
e 24.2 for
√
s m Z . All first- and second-order terms
are known exactly. The sequence of leading and next-to-leading logs can be
exponentiated (closely following the formalism of structure functions in QCD). For
m Z ≈ 91 GeV, the convolution displaces the peak by +110 MeV, and reduces it
by a factor of about 0.74. The exponentiation is important in that it amounts to an
additional shift of about 14 MeV in the peak position with respect to the one loop
radiative correction.
Among the one loop EW radiative corrections, a very remarkable class of
contributions are those terms that increase quadratically with the top mass. The
sensitivity of radiative corrections to m t arises from the existence of these terms. The
61
Even leaving aside QCD corrections, a set of important quantitative contributions
to the radiative corrections arise from large logarithms [e.g. terms of the form
(α/π ln (m Z /m f l l )) n where f ll is a light fermion]. The sequences of leading and
close-to-leading logarithms are fixed by well-known and consolidated techniques (β
functions, anomalous dimensions, penguin-like diagrams, etc.). For example, large
logarithms from pure QED effects dominate the running of α from m e , the electron
mass, up to m Z . Similarly large logarithms of the form [α/π ln (m Z /μ)] n also
enter, for example, in the relation between sin
2 θ W at the scales m Z (LEP, SLC)
and μ (e.g. the scale of low-energy neutral-current experiments). Also, large logs
from initial state radiation dramatically distort the line shape of the Z resonance as
observed at LEP1 and SLC and this effect was accurately taken into account for
the measurement of the Z mass and total width. The experimental accuracy on m Z
obtained at LEP1 is δm Z = ±2.1 MeV (see Chap. 6). Similarly, a measurement of
the total width to an accuracy δδ = ±2.3 MeV has been achieved. The prediction of
the Z line-shape in the SM to such an accuracy has posed a formidable challenge to
theory, which has been successfully met. For the inclusive process e + e − → f ¯
f X,
with f = e (for a concise discussion, we leave Bhabha scattering aside) and X
including γ ’s and gluons, the physical cross-section can be written in the form of a
convolution [42]:
σ (s) =
1
z 0
dz ˆ
σ (zs)G(z, s) ,
(3.95)
where ˆ
σ is the reduced cross-section, and G(z, s) is the radiator function that
describes the effect of initial-state radiation; ˆ
σ includes the purely weak corrections,
the effect of final-state radiation (of both γ ’s and gluons), and also non-factorizable
terms (initial- and final-state radiation interferences, boxes, etc.) which, being small,
can be treated in lowest order and effectively absorbed in a modified ˆ
σ . The radiator
G(z, s) has an expansion of the form
G(z, s) = δ(1 − z) + α/π(a 11 L + a 10 ) + (α/π)
2 (a 22 L
2
+ a 11 L + a 20 ) + . . . +
+ (α/π)
n
n
i=0
a ni L
i ,
(3.96)
where L = ln s/m 2
e 24.2 for
√
s m Z . All first- and second-order terms
are known exactly. The sequence of leading and next-to-leading logs can be
exponentiated (closely following the formalism of structure functions in QCD). For
m Z ≈ 91 GeV, the convolution displaces the peak by +110 MeV, and reduces it
by a factor of about 0.74. The exponentiation is important in that it amounts to an
additional shift of about 14 MeV in the peak position with respect to the one loop
radiative correction.
Among the one loop EW radiative corrections, a very remarkable class of
contributions are those terms that increase quadratically with the top mass. The
sensitivity of radiative corrections to m t arises from the existence of these terms. The