The decays of a neutral particle with zero spin
and arbitrary CP parity into ZZ or W − W +
Taras Zagoskin
Kharkov Institute of Physics and Technology,
Kharkov, Ukraine
June 27, 2015
1
Introduction
• In 2012 the ATLAS and CMS collaborations observed a boson
h with the mass around 126 GeV. We call this particle the
Higgs boson. However, clarification of properties of the
observed boson h requires more data.
qh = 0
Sh = 0 or Sh = 2 (very unlikely )
CPh =?
• In the SM for the Higgs boson
q = 0, S = 0, C = P = 1,
but some supersymmetric extensions of the SM assume
existence of Higgs bosons with negative or indefinite CP
parity.
2
Plan of the investigation
In order to clarify the CP properties of h the following way has
been chosen.
• We consider the decay X → Z1∗ Z2∗ → f1 f¯1 f2 f¯2 , where X is a neutral
particle with zero spin and arbitrary CP parity, f1 6= f2 .
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Plan of the investigation
AX →Z1∗ Z2∗ ∼a(e1∗ · e2∗ ) +
c
b
(e1∗ · p2 )(e2∗ · p1 ) + i 2 εµνρσ (p1µ + p2µ )(p1ν − p2ν )e1∗ρ e2∗σ
mX2
mX
e1 and e2 are the polarization 4-vectors of Z1∗ and Z2∗ respectively.
a, b, c are complex-valued functions of the masses of Z1∗ and Z2∗ . These
functions characterize the CP properties of the boson X . At tree level
CPX
1
1 (SM)
-1
indefinite
a
any
1
0
6= 0
any
b
any
0
0
any
6= 0
c
0
0
6 0
=
6= 0
6= 0
• We derive the full distribution of the decay X → Z1∗ Z2∗ → f1 f¯1 f2 f¯2 .
• Experimentalists measure an experimental full distribution of this decay
for X = h.
• Comparing the theoretical and experimental distributions, one can get
constraints on the values of a, b, c at various masses of Z1∗ and Z2∗ .
4
Definitions of θ1 , θ2 , ϕ
θ1 is the angle between the momentum of Z1∗ in a rest frame of X and the
momentum of f1 in a rest frame of Z1∗ ,
θ2 is the angle between the momentum of Z2∗ in a rest frame of X and the
momentum of f2 in a rest frame of Z2∗ ,
ϕ is the azimuthal angle between the planes of the decays Z1∗ → f1 f¯1 and
Z2∗ → f2 f¯2 .
5
Definitions of A0 , Ak , A⊥
Moreover, it is convenient to write down the fully differential width
by means of the following amplitudes:
2
λ(mX2 , a1 , a2 )
m X − a1 − a2
+b
,
A0 ≡ − a
√
√
2 a1 a2
4mX2 a1 a2
√
Ak ≡ 2a,
A⊥ ≡
√
1
2c
λ 2 (mX2 , a1 , a2 )
.
mX2
aj is the mass squared of Zj∗ , i.e. the invariant mass of the pair fj f¯j ,
λ(x, y , z) ≡ x 2 + y 2 + z 2 − 2xy − 2xz − 2yz.
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The differential width with respect to a1 , a2 , θ1 , θ2 , ϕ
Using approximations mf1 = mf2 = 0, we have derived that
d 5Γ
=|A0 |2 f1 + (|Ak |2 + |A⊥ |2 )f2 + (|Ak |2 − |A⊥ |2 )f3
da1 da2 dθ1 dθ2 dϕ
+ Re(A∗0 Ak )f4 + Re(A∗0 A⊥ )f5 + Re(A∗k A⊥ )f6
+ Im(A∗0 Ak )f7 + Im(A∗0 A⊥ )f8 + Im(A∗k A⊥ )f9 .
f1 , f2 , ..., f9 depend on a1 , a2 , θ1 , θ2 , ϕ, but they are independent of a, b and c.
The dependence of the fully differential width on the couplings a, b and c
is concentrated in nine quadratic combinations of the amplitudes A0 , Ak , A⊥ .
How many decays should be measured for obtaining a precise enough
experimental full distribution of the decay?
d n Γ ↔ 10n+1 decays
d 5 Γ ↔ 106 decays
How many decays have been observed?
h → Z1∗ Z2∗ → e − e + µ− µ+
26 decays (ATLAS and CMS together after about 1.5 years of measurements)
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Distributions of four and less variables should be considered
We will probably have a precise enough experimental full distribution
in 60000 years (roughly).
That is why we should try to get constraints on a, b, c by means of measuring
distributions of as little a number of variables as possible.
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a1 a2 -differential width
2
Γ
Figure: dad1 da
of the decay X → Z1∗ Z2∗ → l1− l1+ l2− l2+ as a function of
2
√
√
a1 , a2 , if X is the SM Higgs boson and mX = 125.7 GeV.
l1 , l2 = e, µ, τ, l1 6= l2 .
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a2 -differential width
Integrating
d2Γ
da1 da2
approximately, we derive that
√ 3 9
1
2 , m2 , a )a
2GF mZ
λ 2 (mX
dΓ
2 2
2
2
2
2
Z
≈
(a
+
v
)(a
+
v
)
f1
f1
f2
f2
3
2
4
da2 288π mX ΓZ
(a2 − mZ )2 + (mZ ΓZ )2
q
√
∀a2 | 2mf2 < a2 ≤ mX − mZ2 + 3mZ ΓZ .
X
λ=0,k,⊥
af and vf are constants depending on a fermion f , A0λ ≡ Aλ |a1 =m2 .
Z
In several articles the formula for
approximation when
dΓ
da2
√
and their approach is inaccurate.
q
mh − mZ2 + 3mZ ΓZ ≈ 30.8 GeV
mh − mZ ≈ 34.5 GeV
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has been used in the narrow-Z -width
a2 ≤ mX − mZ ,
|A0λ |2
Relations between the helicity coefficients and observables
We call the ratios of the nine quadratic combinations of the fully differential
P
d5Γ
,
width to λ=0,k,⊥ |Aλ |2 ‘the helicity coefficients’. Integrating da1 da2 dθ
1 dθ2 dϕ
we can relate all the helicity coefficients to observables. For example,

 π
−1 Z2
Zπ
0
2
2
Re(A0∗
dΓ
d Γ 
d Γ
k A⊥ )

(1,2)
O1 (a2 ) ≡
− dθ1,2
 dθ1,2
∼ P
0 2
da2
da2 dθ1,2
da2 dθ1,2
λ |Aλ |
0
=
π
2
0
Re(Fk0∗ F⊥
).
A0
Fλ0 ≡ pP λ
, λ = 0, k, ⊥ .
2
λ |Aλ |
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Relations between the helicity coefficients and observables
(1,2)
(a2 ) ∼
(1,2)
(a2 ) ∼
O1
O2
O3 (a2 ) ∼
O4 (a2 ) ∼
O5 (a2 ) ∼
O6 (a2 ) ∼
(1,2)
O7
(a2 ) ∼
(1,2)
O8 (a2 )
∼
O9 (a2 ) ∼
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
Re(Fk0∗ F⊥0 )




0 2

|F0 | 


0 2
0 2

|Fk | + |F⊥ | 



0 2
0 2

|Fk | − |F⊥ | 


0∗ 0
Im(Fk F⊥ ) → Constraints on a, b, c



Re(F00∗ Fk0 )




0∗ 0 

Im(F0 Fk )




0∗ 0 
Re(F0 F⊥ )



0∗ 0 
Im(F0 F⊥ )
Conclusions
• In order to clarify the CP properties of the Higgs boson we have
considered the fully mass and angular differential width of the decay
X → Z1∗ Z2∗ → f1 f¯1 f2 f¯2 , where X is a neutral particle with zero spin
and arbitrary CP parity, f1 6= f2 .
• Limits of applicability of approximations used when deriving various
differential widths are established.
• All the helicity coefficients are related to observables. We have also
plotted the observables and determined what constraints on a, b, c
can be put by them.
• We should wait for experimentalists measuring the observables
(1,2)
O1 , O2 , ..., O9 and then get constraints on a, b, c using the
(1,2)
shown relations between F00 , Fk0 , F⊥0 and O1 , ..., O9 .
• An analogous analysis has been carried out for the decay
X → W −∗ W +∗ → f1− f¯2− f¯1+ f2+ .
The presentation is based on the paper Zagoskin and Korchin,
arXiv:1504.07187.
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