14
2 Phenomenology of Jet Substructure
to the fermion f , respectively. Due to electroweak unification, the coupling of the
Z boson to fermions is very different from the one of the W boson. Inserting g V, f
and g A, f , one obtains the following branching fractions at LO: B Z →ν ¯
ν = 20.4%,
B Z → + − = 10.2% and B Z →had = 69.2%. Again, the branching fraction to hadrons
is dominant, with more than two thirds of all Z bosons decaying hadronically. Note
that there is a sizeable branching fraction of 15.2% into bb, which often results in
a non-negligible background in analyses targeting the H → bb decay. Owing to
the Z bosons’ importance in electroweak precision tests, higher order corrections to
the partial widths have been calculated in EW two-loop order [108–111], including
mixed EW/QCD one-loop diagrams [112–116], leading three- and four-loop corrections in the large top quark mass limit [117–124], factorisable final state two-loop
QED and four-loop QCD radiation [125–127], and mixed EW/QCD one-loop nonfactorisable vertex corrections [128–133]. Including all corrections, the branching
fractions become B Z →ν ¯
ν = 20.105 ± 0.004%, B Z → + − = 10.092 ± 0.002% and
B Z →had = 69.804 ± 0.059% [111], in excellent agreement with experimental measurements [15].
Similar to the W boson case, the angular distribution in the centre-of-mass frame
of the decay fermions can be calculated [134],
1
σ
dσ
d cos θ ∗ = f −
3
8
1 + cos
2
θ
∗
−
2(c
2
L − c
2
R )
c
2
L + c
2
R
cos θ
∗
+ f +
3
8
1 + cos
2
θ
∗
+
2(c
2
L − c
2
R )
c
2
L + c
2
R
cos θ
∗
+ f 0
3
4
sin
2
θ
∗
,
(2.7)
where the angle θ
∗ is defined equivalent to (2.5). The couplings to left- and righthanded chiral states are given by c L and c R , respectively. Since the Z boson couples
to both, left- and right-handed fermions, in principal the decay angle distributions of
the outgoing quarks differs for left- and right-handed Z polarisation states. However,
due to the inability of distinguishing quarks from anti-quarks, only absolute values
| cos θ
∗
| can be measured and the decay angle distribution becomes identical for leftand right-handed polarisations,
1
σ
dσ
d|cos θ ∗ |
= f ±
3
4
1 + | cos θ
∗
|
2
+ f 0
3
2
| sin θ
∗
|
2
,
(2.8)
where the dependence on c L and c R drops out. It follows that the decay angle distribution is identical to the one for W bosons, (2.5). Thus, similar as for W bosons,
transversely polarised Z bosons show a more pronounced tail towards larger opening
angles α when compared to longitudinally polarised Z bosons. The only difference
between the α distribution from W and Z boson decays arises from the mass difference m W and m Z , resulting in a shift towards somewhat larger values of α in the
case of Z bosons at a given momentum (the shift is between 0.1 and 0.01 radians
2 Phenomenology of Jet Substructure
to the fermion f , respectively. Due to electroweak unification, the coupling of the
Z boson to fermions is very different from the one of the W boson. Inserting g V, f
and g A, f , one obtains the following branching fractions at LO: B Z →ν ¯
ν = 20.4%,
B Z → + − = 10.2% and B Z →had = 69.2%. Again, the branching fraction to hadrons
is dominant, with more than two thirds of all Z bosons decaying hadronically. Note
that there is a sizeable branching fraction of 15.2% into bb, which often results in
a non-negligible background in analyses targeting the H → bb decay. Owing to
the Z bosons’ importance in electroweak precision tests, higher order corrections to
the partial widths have been calculated in EW two-loop order [108–111], including
mixed EW/QCD one-loop diagrams [112–116], leading three- and four-loop corrections in the large top quark mass limit [117–124], factorisable final state two-loop
QED and four-loop QCD radiation [125–127], and mixed EW/QCD one-loop nonfactorisable vertex corrections [128–133]. Including all corrections, the branching
fractions become B Z →ν ¯
ν = 20.105 ± 0.004%, B Z → + − = 10.092 ± 0.002% and
B Z →had = 69.804 ± 0.059% [111], in excellent agreement with experimental measurements [15].
Similar to the W boson case, the angular distribution in the centre-of-mass frame
of the decay fermions can be calculated [134],
1
σ
dσ
d cos θ ∗ = f −
3
8
1 + cos
2
θ
∗
−
2(c
2
L − c
2
R )
c
2
L + c
2
R
cos θ
∗
+ f +
3
8
1 + cos
2
θ
∗
+
2(c
2
L − c
2
R )
c
2
L + c
2
R
cos θ
∗
+ f 0
3
4
sin
2
θ
∗
,
(2.7)
where the angle θ
∗ is defined equivalent to (2.5). The couplings to left- and righthanded chiral states are given by c L and c R , respectively. Since the Z boson couples
to both, left- and right-handed fermions, in principal the decay angle distributions of
the outgoing quarks differs for left- and right-handed Z polarisation states. However,
due to the inability of distinguishing quarks from anti-quarks, only absolute values
| cos θ
∗
| can be measured and the decay angle distribution becomes identical for leftand right-handed polarisations,
1
σ
dσ
d|cos θ ∗ |
= f ±
3
4
1 + | cos θ
∗
|
2
+ f 0
3
2
| sin θ
∗
|
2
,
(2.8)
where the dependence on c L and c R drops out. It follows that the decay angle distribution is identical to the one for W bosons, (2.5). Thus, similar as for W bosons,
transversely polarised Z bosons show a more pronounced tail towards larger opening
angles α when compared to longitudinally polarised Z bosons. The only difference
between the α distribution from W and Z boson decays arises from the mass difference m W and m Z , resulting in a shift towards somewhat larger values of α in the
case of Z bosons at a given momentum (the shift is between 0.1 and 0.01 radians