Quantum field theory applications, Fall 2014,
Excercise 10.
Consider the Higgs boson decays into or through pairs of W W and ZZ gauge bosons
as shown in Fig. 1. (The appropriate Feynman rules (for the HV V - and Hf f -vertices
and the gauge boson propagators) you can find from any QFT book such as Peskin and
Schröder.)
a) Two body decays: Above kinematical thresholds Higgs dominantly decays into real
gauge boson pairs. Show that the partial width for H → V V with V = W or Z are given
by
M2
GF M 3 √
Γ(H → V V ) = √ H δV 1 − 4x 1 − 4x + 12x2 , x = V2
MH
16 2π
with δW = 2 and δz = 1.
b) Three body decays: Below the pair production threshold one may have decays to one
real and one virtual gauge boson. Show that the partial width for the decay H → V V ∗ →
V f f¯, the charges of the vector bosons V summed over and assuming massless fermions,
is given by
3G2F MV4
MH δV0 RT (x)
Γ(H → V V ∗ ) =
16π 3
7
0
with δW
= 1, δZ0 = 12
− 10
sin2 ΘW + 40
sin4 ΘW and
9
27
3x − 1
3
3(1 − 8x + 20x2 )
1−x
√
arccos
(2−13x+47x2 )− (1−6x+4x2 ) log(x)
RT (x) =
−
3/2
2x
2x
2
4x − 1
c) Four body decays: well below the real VV thresholds both gauge bosons may be
virtual. In this case argue that (for massless fermions), the total decay width proceeding
through gauge boson pairs can be cast into the compact form
Γ(H → V ∗ V ∗ ) =
Z (MH −q1 )2
dq12 MV ΓV
1 Z MH2
dq22 MV ΓV
Γ0 ,
π2 0
(q12 − MV2 )2 + MV2 Γ2V 0
(q22 − MV2 )2 + MV2 Γ2V
where q12 and q22 are the squared invariant masses of the virtual gauge bosons, MV and ΓV
their physical masses and total decay widths, and the effective decay into virtual gauge
bosons Γ0 is
q2q2
GF MH3 q 2 2
√ δV λ(q1 , q2 ; MH2 ) λ(q12 , q22 ; MH2 ) + 12 1 42 ,
MH
16 2π
Γ0 =
where the kinematical function λ(x, y; z) ≡ (1 − x/z − y/z)2 − 4xy/z 2 and δV = 2(1) for
V = W (Z).
d) Put all these different approximations into computer and plot the corresponding decay
widthts. Comment your findings.