64
G. Altarelli and S. Forte
Beyond tree level, these relations are modified by radiative corrections:
(1 −
m 2
W
m 2
Z
)
m 2
W
m 2
Z
=
πα(m Z )
√
2G F m 2
Z
1
1 − W
m 2
W
m 2
Z cos 2 θ W
= 1 + m
(3.101)
The Z and W masses are to be precisely defined in terms of the pole position in
the respective propagators. Then, in the first relation the replacement of α with the
running coupling at the Z mass α(m Z ) makes W completely determined at 1-loop
by purely weak corrections (G F is protected from logarithmic running as an indirect
consequence of (V-A) current conservation in the massless theory). This relation
defines W unambigously, once the meaning of α(m Z ) is specified (for example,
¯
MS). On the contrary, in the second relation m depends on the definition of
sin
2 θ W beyond the tree level. For LEP physics sin
2 θ W is usually defined from the
Z → μ + μ − effective vertex. At the tree level the vector and axial-vector couplings
g
μ
V and g
μ
A are given in Eqs. (3.29). Beyond the tree level a corrected vertex can be
written down in terms of modified effective couplings. Then sin
2 θ W ≡ sin
2 θ eff is
in general defined through the muon vertex:
g
μ
V /g
μ
A = 1 − 4 sin
2 θ eff
sin
2 θ eff = (1 +
2
0 ,
s
2
0 c
2
0 =
πα(m Z )
√
2G F m 2
Z
g
μ2
A =
1
4
(1 + ρ)
(3.102)
We see that s 2
0 and c 2
0 are “improved” Born approximations (by including the
running of α) for sin 2 θ eff and cos 2 θ eff . Actually, since in the SM lepton
universality is only broken by masses and is in agreement with experiment within
the present accuracy, in practice the muon channel can be replaced with the average
over charged leptons.
We can write a symbolic equation that summarizes the status of what has been
computed up to now for the radiative corrections (we list some recent work on each
item from where older references can be retrieved) W [46], [47] and [48]:
W , ,ρ, ,k = g
2 m 2
t
m 2
W
(1 + α s + α
2
s ) + g
2 (1 + α s + ∼ α
2
s ) + g
4 m 4
t
m 4
W
+ g
4 m 2
t
m 2
W
+ . . .
(3.103)
The meaning of this relation is that the one loop terms of order g 2 are completely
known, together with their first order QCD corrections (the second order QCD
G. Altarelli and S. Forte
Beyond tree level, these relations are modified by radiative corrections:
(1 −
m 2
W
m 2
Z
)
m 2
W
m 2
Z
=
πα(m Z )
√
2G F m 2
Z
1
1 − W
m 2
W
m 2
Z cos 2 θ W
= 1 + m
(3.101)
The Z and W masses are to be precisely defined in terms of the pole position in
the respective propagators. Then, in the first relation the replacement of α with the
running coupling at the Z mass α(m Z ) makes W completely determined at 1-loop
by purely weak corrections (G F is protected from logarithmic running as an indirect
consequence of (V-A) current conservation in the massless theory). This relation
defines W unambigously, once the meaning of α(m Z ) is specified (for example,
¯
MS). On the contrary, in the second relation m depends on the definition of
sin
2 θ W beyond the tree level. For LEP physics sin
2 θ W is usually defined from the
Z → μ + μ − effective vertex. At the tree level the vector and axial-vector couplings
g
μ
V and g
μ
A are given in Eqs. (3.29). Beyond the tree level a corrected vertex can be
written down in terms of modified effective couplings. Then sin
2 θ W ≡ sin
2 θ eff is
in general defined through the muon vertex:
g
μ
V /g
μ
A = 1 − 4 sin
2 θ eff
sin
2 θ eff = (1 +
2
0 ,
s
2
0 c
2
0 =
πα(m Z )
√
2G F m 2
Z
g
μ2
A =
1
4
(1 + ρ)
(3.102)
We see that s 2
0 and c 2
0 are “improved” Born approximations (by including the
running of α) for sin 2 θ eff and cos 2 θ eff . Actually, since in the SM lepton
universality is only broken by masses and is in agreement with experiment within
the present accuracy, in practice the muon channel can be replaced with the average
over charged leptons.
We can write a symbolic equation that summarizes the status of what has been
computed up to now for the radiative corrections (we list some recent work on each
item from where older references can be retrieved) W [46], [47] and [48]:
W , ,ρ, ,k = g
2 m 2
t
m 2
W
(1 + α s + α
2
s ) + g
2 (1 + α s + ∼ α
2
s ) + g
4 m 4
t
m 4
W
+ g
4 m 2
t
m 2
W
+ . . .
(3.103)
The meaning of this relation is that the one loop terms of order g 2 are completely
known, together with their first order QCD corrections (the second order QCD