56
G. Altarelli and S. Forte
A formulation of the standard EW theory with good apparent ultraviolet behaviour
can be obtained by introducing the renormalizable or R ξ gauges, in analogy with
the abelian case discussed in detail in Chap. 2. One parametrizes the Higgs doublet
as:
φ =
φ +
φ 0
=
φ 1 + iφ 2
φ 3 + iφ 4
=
−iw +
v +
H +iz
√
2
,
(3.85)
and similarly for φ † , where w − appears. The scalar fields w ± and z are the pseudo
Goldstone bosons associated with the longitudinal modes of the physical vector
bosons W ± and Z. The R ξ gauge fixing lagrangian has the form:
GF = −
1
ξ
|∂
μ W μ − ξm W w|
2
−
1
2η
(∂
μ Z μ − ηm Z z)
2
−
1
2α
(∂
μ A μ )
2 . (3.86)
The W ± and Z propagators, as well as those of the scalars w ± and z, have exactly
the same general forms as for the abelian case in Eqs. (67)–(69) of Chap. 2, with
parameters ξ and η, respectively (and the pseudo Goldstone bosons w ± and z have
masses ξm W and ηm Z ). In general, a set of associated ghost fields must be added,
again in direct analogy with the treatment of R ξ gauges in the abelian case of
Chap. 2. The complete Feynman rules for the standard EW theory can be found
in a number of textbooks (see, for example, [25]).
The pseudo Goldstone bosons w ± and z are directly related to the longitudinal
helicity states of the corresponding massive vector bosons W ± and Z. This
correspondence materializes in a very interesting “equivalence theorem”: at high
energies of order E the amplitude for the emission of one or more longitudinal gauge
bosons V L (with V = W, Z) becomes equal (apart from terms down by powers of
m V /E) to the amplitude where each longitudinal gauge boson is replaced by the
corresponding Goldstone field w ± or z [26]. For example, consider top decay with
a longitudinal W in the final state: t → bW
+
L . The equivalence theorem asserts that
we can compute the dominant contribution to this rate from the simpler t → bw +
matrix element:
(t → bW
+
L ) = (t → bw
+ )[1 + o(m
2
W /m
2
t )] .
(3.87)
In fact one finds:
(t → bw
+ ) =
h 2
t
32π
m t =
G F m 3
t
8π
√
2
,
(3.88)
where h t = m t /v is the Yukawa coupling of the top quark (numerically very close
to 1), and we used 1/v 2 = 2
√
2G F (see Eq. (3.54)). If we compare with Eq. (3.34),
we see that this expression coincides with the total top width (i.e. including all
polarizations for the W in the final state), computed at tree level, apart from terms
G. Altarelli and S. Forte
A formulation of the standard EW theory with good apparent ultraviolet behaviour
can be obtained by introducing the renormalizable or R ξ gauges, in analogy with
the abelian case discussed in detail in Chap. 2. One parametrizes the Higgs doublet
as:
φ =
φ +
φ 0
=
φ 1 + iφ 2
φ 3 + iφ 4
=
−iw +
v +
H +iz
√
2
,
(3.85)
and similarly for φ † , where w − appears. The scalar fields w ± and z are the pseudo
Goldstone bosons associated with the longitudinal modes of the physical vector
bosons W ± and Z. The R ξ gauge fixing lagrangian has the form:
GF = −
1
ξ
|∂
μ W μ − ξm W w|
2
−
1
2η
(∂
μ Z μ − ηm Z z)
2
−
1
2α
(∂
μ A μ )
2 . (3.86)
The W ± and Z propagators, as well as those of the scalars w ± and z, have exactly
the same general forms as for the abelian case in Eqs. (67)–(69) of Chap. 2, with
parameters ξ and η, respectively (and the pseudo Goldstone bosons w ± and z have
masses ξm W and ηm Z ). In general, a set of associated ghost fields must be added,
again in direct analogy with the treatment of R ξ gauges in the abelian case of
Chap. 2. The complete Feynman rules for the standard EW theory can be found
in a number of textbooks (see, for example, [25]).
The pseudo Goldstone bosons w ± and z are directly related to the longitudinal
helicity states of the corresponding massive vector bosons W ± and Z. This
correspondence materializes in a very interesting “equivalence theorem”: at high
energies of order E the amplitude for the emission of one or more longitudinal gauge
bosons V L (with V = W, Z) becomes equal (apart from terms down by powers of
m V /E) to the amplitude where each longitudinal gauge boson is replaced by the
corresponding Goldstone field w ± or z [26]. For example, consider top decay with
a longitudinal W in the final state: t → bW
+
L . The equivalence theorem asserts that
we can compute the dominant contribution to this rate from the simpler t → bw +
matrix element:
(t → bW
+
L ) = (t → bw
+ )[1 + o(m
2
W /m
2
t )] .
(3.87)
In fact one finds:
(t → bw
+ ) =
h 2
t
32π
m t =
G F m 3
t
8π
√
2
,
(3.88)
where h t = m t /v is the Yukawa coupling of the top quark (numerically very close
to 1), and we used 1/v 2 = 2
√
2G F (see Eq. (3.54)). If we compare with Eq. (3.34),
we see that this expression coincides with the total top width (i.e. including all
polarizations for the W in the final state), computed at tree level, apart from terms