The method of the precise neutrino reconstruction has not yet been established in theH→ τ τ analysis, and therefore several alternative methods are proposed to reconstruct the acoplanarity angle according to the di-τ lepton decay mode [135–143]. This analysis uses two typical methods for theπ±π∓andρ±ρ∓ decay modes, referred to as the impact parameter and theρ→π±π0decay plane method, respectively.
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Impact Parameter Method
The main target of the impact parameter method [135–138] is theτ±τ∓→π±π∓decay mode, which has 2.8%branching fraction in theH →τhadτhadchannel. The method constructs three types of CP sensitive angles using pseudo planes of theτ±→ π±ντ decay, where neutrino directions are approximated from the primary vertex and impact parameter information. The reconstruction procedure of three angles is as follows:
The first angleϕlabIP is defined as:
ϕlab.IP = arccos(n+lab.·n−lab.), (5.5) wheren±lab. are impact parameter vectors. The impact parameter vector is defined as the perpendicular vector to theπ±vector from the primary vertex position, illustrated in Fig. 5.6.
z
~ π
±Vertex
~
n ~ τ
~ ν
τFig. 5.6: Schematic diagram of the definition of the impact parameter vector n±lab., where the z axis is defined as the proton beam direction The impact parameter vector⃗nis defined as the perpendicular vector from the primary vertex position to theπ±vector.
The second angle isϕIP, which is defined as:
ϕIP= arccos( n+⊥·n−⊥
|n+⊥||n−⊥|
), (5.6)
where n±⊥ are perpendicular components of impact parameter vectors in the π+π− rest frame. The impact parameter vectors defined in laboratory frame are boosted into the rest frame using measuredπ± four-momenta (n±lab. → n±). Then, then± is decomposed into parallel (n±∥) and perpendicular (n±⊥)
components to eachp±, expressed by:
n± =r⊥·n±⊥+r∥·n±∥, (5.7)
wherer∥ andr⊥are fractions of parallel and perpendicular components, respectively. Then±∥ andn±⊥ are unit vectors.
The third angle isψIP, which is defined as:
ψIP= arccos(p−·(n+⊥×n−⊥)), (5.8) wherep−represents the unit vector of the π−direction. TheψIPvariable is especially sensitive to the CP mixing state of the Higgs boson because it includes CP-odd and T-odd triple correlations [135].
Figure 5.7 shows theϕlabIP andϕIP distributions for theτ±τ∓ → π±π∓ and theτ±τ∓ → ℓ±π∓decay from the ggFH → τ τ process. The impact parameter method focuses on the τ±τ∓ → π±π∓decay mode, while these angles can be also reconstructed in theτ±τ∓ → ℓ±π∓decay mode by replacing the π± vector with theℓ± vector. TheϕlabIP distributions of theZ boson in Fig. 5.7 (a) and (c) have the tendency to be close to the distribution of the CP-even Higgs boson. This is due to the spin correlation ofZ →τ τprocess as the following: The spin directions of di-τleptons are the same due to theZ boson spin of 1. In the τ±τ∓ → π±π∓ decay, pions are emitted in theτ spin direction because of the left-handed nature of the neutrino. Hence, the impact parameter vectors tend to have the opposite directions i.e. ϕlabIP tends to be a large angle. On the other hand, in theℓ±π∓decay, the tendency is opposite due to the presence of two neutrinos in the leptonic decay. TheϕIPgives more sensitivity than theϕlabIP and the flat distribution of theZ boson thanks to moving from the laboratory frame to theπ+π−rest frame.
In order to discriminate the CP-even and CP-odd Higgs boson, the ϕIP variable is mainly used in this analysis. The ψIP is dedicated to distinguish the CP mixing Higgs boson as shown in Fig. 5.11 (see Section 5.3.1), so that this variable is useful to a CP mixing measurement in future analysis.
τ±τ∓ →ρ±ρ∓ decay plane Method
The main target of this method [139–142] is theτ±τ∓ →ρ±ρ∓decay mode, which has16.0%branching fraction in theH → τhadτhad channel. In this decay mode, the neutrino has relatively small momentum compared with other decay modes due to theρ meson mass of∼ 770MeV. Therefore, theρ+ρ−rest frame is useful to approximate theτ+τ− rest frame. As discussed in Section 1.2.4, aρ meson decays into detectable particles: one charged pion and one neutral pion. The approximated acoplanarity angle ϕρ[139] is defined as an angle between theρ+-π+plane and theρ−-π−plane, expressed by:
ϕρ = arccos( n+·n−
|n+||n−| )
, (5.9)
n± = pπ±×pπ0, (5.10)
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(IP)
lab.
φCP
Acoplanarity
0 0.5 1 1.5 2 2.5 3
A.U.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
ντ
π -τ + ν π+
→ τ
-τ+ CP even (H0)
0) CP odd (A
τ τ
→ Z
(a)
(IP) φCP
Acoplanarity
0 0.5 1 1.5 2 2.5 3
A.U.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
ντ
π -τ + ν π+
→ τ
-τ+ CP even (H0)
0) CP odd (A
τ τ
→ Z
(b)
(IP)
lab.
φCP
Acoplanarity
0 0.5 1 1.5 2 2.5 3
A.U.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
ντ
π± τ +
lν
±ν
→ l τ
-τ+ CP even (H0)
0) CP odd (A
τ τ
→ Z
(c)
(IP) φCP
Acoplanarity
0 0.5 1 1.5 2 2.5 3
A.U.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
ντ
π± τ +
lν
±ν
→ l τ
-τ+ CP even (H0)
0) CP odd (A
τ τ
→ Z
(d)
Fig. 5.7: TheϕlabIP andϕIPangle distributions reconstructed by the impact parameter method, for the (a,b) τ±τ∓→π±π∓and the (c,d)τ±τ∓→ℓ±π∓decay modes from the CP-even (blue), CP-odd (red) Higgs boson and theZ boson (black), where the Higgs boson is produced from the ggF process with a mass of 125GeV.
y′
x′
z′
ρ−
ρ+ π−
π+ φρ
Fig. 5.8: Definition of the acoplanarity angleϕρ. Thez′ axis is defined as theρmeson direction in the ρ+-ρ−rest frame, Thex′, y′ axes are defined as horizontal and vertical axes in the transverse plane with respect to thez′axis, respectively. Theϕρis defined as the angle between theρ→π±π0decay planes.
wherepπ±andpπ0 are momentum vectors ofπ±andπ0, respectively. The definition ofϕρis illustrated in Fig. 5.8, and it takes the range of [0,2π]. Theπ± ρ∓ is also folded into2π around π by the same procedure with the equation (5.3).
If a spin correlation in theτ± →ρ±(
→π±π0)
ντ decay is not taken into account, this variable has no sensitivity to the CP state of the Higgs boson. In order to cope with this, a charged energy asymmetry (y±) variable, which reflects to the spin correlation, is defined as the following:
y± = Eπ±−Eπ0
Eπ±+Eπ0, (5.11)
whereEπ±andEπ0 are energies of theπ±and theπ0, respectively. According to a product of charged energy asymmetries (y1×y2), theϕρdistributions withy1×y2 >0andy1×y2 <0are shown as Figure 5.9 for theτ±τ∓→ρ±ρ∓decay mode from the CP-even and CP-odd Higgs boson and theZ boson. In order to deal with the effect, theϕρangle is re-defined as the following:
ϕρ =
ϕρ (y+×y−>0)
π−ϕρ−π (y+×y−<0). (5.12)
(5.13)
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ρ)
CP ( φ Acoplanarity
0 0.5 1 1.5 2 2.5 3
A.U.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
ντ
ρ± τ + ν ρ± -→ τ τ+
ντ
π0
π± τ + ν π0
π±
→
0) CP even (H
0) CP odd (A
τ τ
→ Z
(a)ϕρangle (y1×y2>0)
ρ)
CP ( φ Acoplanarity
0 0.5 1 1.5 2 2.5 3
A.U.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
ντ
ρ± τ + ν ρ± -→ τ τ+
ντ
π0
π± τ + ν π0
π±
→
0) CP even (H
0) CP odd (A
τ τ
→ Z
(b)ϕρangle (y1×y2<0)
Fig. 5.9: Theϕρangle distributions withy1×y2 >0(a) and withy1×y2 <0(b) for theτ±τ∓ →ρ±ρ∓ decay mode from the CP-even (blue) and CP-odd (red) Higgs boson and theZ boson (black), where the Higgs boson is produced from the ggF process with a mass of125GeV.
Mixture Method
In order to cover one of the remaining decay mode ofτ±τ∓→π±ρ∓, which has non-negligible branch-ing fraction of13.6%in theH → τhadτhadchannel, an another acoplanarity angleϕIP-ρis reconstructed by mixing the impact parameter method and the ρ → π±π0 decay plane method. The definition is expressed by:
ϕρ= arccos( nIP·nρ
|nIP||nρ|
), (5.14)
wherenIPis the cross product vector of the impact parameter vector and theπ±momentum vector, and nρis the cross product vector ofπ±andπ0momentum vectors. TheϕIP-ρis also folded into2πaround πby the same procedure with the equation (5.3).
This method also needs to take into account the effect of the spin correlation of theτ±→ρ±(
→π±π0) ντ
decay, and the acoplanarity angle is classified corresponding to the sign of the charge asymmetryy as the following:
ϕIP-ρ =
ϕIP-ρ (y <0) π−ϕIP-ρ (y >0)
y = Eπ±−Eπ0
Eπ±−Eπ0
. (5.15)
Figure 5.10 shows theϕIP-ρdistributions withy >0andy <0for theτ±τ∓ →π±ρ∓decay mode from
the CP-even, CP-odd Higgs boson and theZboson.
The approximated acoplanarity angles are used as discriminant variables in this analysis, according to the di-τ lepton decay mode. The separation ability of the impact parameter method is equivalent to Fig. 5.5 because the method indirectly uses the same neutrino information using the impact parameter vector. The separation ability of the ρ → π±π0 decay plane and the mixture methods are inferior to the impact parameter method because these methods ignore neutrinos in the reconstruction of the τ±→ρ±(
→π±π0)
ντ decay plane. The reconstruction method for each decay mode with its branching ratio is summarized in Table 5.2.
ρ) (IP + φCP
Acoplanarity
0 0.5 1 1.5 2 2.5 3
A.U.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
ντ
ρ± τ + ν π±
→ τ
-τ+
ντ
π0
π± τ + ν π±
→
0) CP even (H
0) CP odd (A
τ τ
→ Z
(a)ϕIP-ρangle (y <0)
ρ) (IP + φCP
Acoplanarity
0 0.5 1 1.5 2 2.5 3
A.U.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
ντ
ρ± τ + ν π±
→ τ
-τ+
ντ
π0
π± τ + ν π±
→
0) CP even (H
0) CP odd (A
τ τ
→ Z
(b)ϕIP-ρangle (y >0)
Fig. 5.10: TheϕIP-ρangle distributions withy <0(a) and withy >0(b) or theτ±τ∓ →π±ρ∓decay mode from the CP-even (blue) and CP-odd (red) Higgs boson and theZ boson (black), where the Higgs boson is produced from the ggF process with a mass of125GeV.
Decay Method B.R.
H→τhadτhad 42.0%
τ±τ∓→π±π∓ Impact Parameter 1.2%
τ±τ∓→ρ±ρ∓ ρ→π±π0decay plane 6.5%
τ±τ∓→π±ρ∓ Mixture 5.5%
H→τℓτhad 45.6%
τ±τ∓→ℓ±π∓ Impact Parameter 7.6%
Table 5.2: Summary of the reconstruction method for each decay mode with its branching ratio.
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