Mass Reconstruction

The invariant mass of di-τ (mτ τ) is one of the most important variables in the H → τ τ search, but it cannot be directly reconstructed due to the presence of neutrinos fromτ decays.

The most simplest reconstruction method is the invariant mass of detectable (visible) decay products, referred to as the visible mass (mvis). However, this method completely ignores the effect of neutrinos, so that a peak of them_visdistribution significantly differs from the original boson mass.

The collinear mass approximation [112] is one of the frequently used methods instead of them_vis, based on two important assumptions. One is that neutrinos from τ decays are collinear with the visible τ decay products. The other is that the missing transverse energy in the event is only produced from neutrino energies. Two neutrinos from a leptonically decaying τ are treated as one neutrino system in this method. Thus, neutrino four-momenta are estimated by following equations:

Region VBF Boosted

Signal Region (SR) Opposite Sign Opposite Sign

≥2jets (p_T>50/30GeV) Not VBF

∆η(jet₁,jet₂)>3.0 p^H_T >100GeV m_T<70GeV m_T<70GeV

b-jet veto b-jet veto

m_vis >40GeV m_vis >40GeV

Z →τ τ SR cut and SR cut and

Control Region mT<40GeV mT<40GeV

(CR) m_{τ τ} <110GeV m_{τ τ} <110GeV

W+jets CR As SR, but As SR, but

mT>70GeV mT>70GeV

multi-jet CR As SR, but As SR, but

leptonI(pT,0.4)>0.06 leptonI(pT,0.4)>0.06 remove leptonI(E_T,0.2)cut remove leptonI(E_T,0.2)cut looseID only forH →τeτhad looseID only forH →τeτhad

Top CR As SR, but As SR, but

≥1b-tagged jet and ≥1b-tagged jet and mT>70GeV mT>70GeV

Top(jet→τhad)CR As SR, but As SR, but

≥1b-tagged jet and ≥1b-tagged jet and mT<70GeV mT<70GeV

Z/γ^∗ →ℓℓ As SR, but As SR, but

(jet→τ_had)CR 2 OS same-flavor leptons 2 OS same-flavor leptons 61< m_ℓℓ <121GeV 61< m_ℓℓ <121GeV

FakeτhadCR As SR, but As SR, but

Same sign Same sign

Table 4.4: Summary of signal and control regions for VBF and Boosted categories.

64

E_T^miss_x = pmis1sinθvis1cosϕvis1+pmis2sinθvis2cosϕvis2, E_T^miss_y = pmis1sinθvis1sinϕvis1+pmis2sinθvis2sinϕvis2,

ϕmis1 ≈ ϕvis1, θmis1≈θvis1,

ϕ_mis2 ≈ ϕ_vis2, θ_mis2≈θ_vis2, (4.2)

where E_T^miss

x(y) is the x(y) component of the missing transverse energy. ϕ and θ represents polar and azimuthal angles. Subscripts of mis1,2 and vis1,2 denote neutrino systems and visible products fromτ decays, respectively. The di-τ invariant mass is then calculated as:

m_{τ τ} = mvis

√x₁x₂, x_1,2 = p_vis1,2

p_vis1.2+p_mis1,2, (4.3)

wherex1,2 are momentum fractions of the visibleτ decay products. Although the collinear method is possible to fully reconstruct them_{τ τ} including neutrino four-momenta, a reasonable mass resolution is given only in a small phase space: the di-τ system is highly boosted because of associated highp_T jets.

Indeed, the equation (4.3) cannot be solved (x1.2 < 0 or x1.2 > 1)) in case that di-τ are emitted as back-to-back. Moreover, the method is sensitive to theE_T^miss resolution, and it makes long tails of the reconstructed mass distribution due to theE_T^missworse resolution.

The missing mass calculator (MMC) [113] is an improved program to overcome weak points of the collinear method, such as no solution and long tails. The MMC is possible to reconstruct the di-τ invariant mass for any event topology. The MMC imposes an assumption that visibleτ decay products are not affected from detector resolutions and an origin of neutrinos is only fromτ decays. Under the assumptions, the reconstruction of neutrino four-momenta requires to solve 6 to 8 unknown parameters.

The parameters are constrained by following equations:

E_T^miss_x = p_mis1sinθ_mis1cosϕ_mis1+p_mis2sinθ_mis2cosϕ_mis2, E_T^miss_y = p_mis1sinθ_mis1sinϕ_mis1+p_mis2sinθ_mis2sinϕ_mis2,

m²_τ1 = m²_mis1+m²_vis1+ 2

√

p²_vis1+m²_vis1

√

p²_mis1+m²_mis1

−2p_vis1p_mis1cos ∆θ_vis1,mis1, m²_τ2 = m²_mis2+m²_vis2+ 2

√

p²_vis2+m²_vis2

√

p²_mis2+m²_mis2

−2pvis2pmis2cos ∆θ_vis2,mis2, (4.4) wherem_τ1,2 are the invariant mass of theτ (m_τ = 1.77GeV),m_mis1,2are the invariant mass of neutrino

systems (mmis1,2= 0in case thatτ decays hadronically).∆θvis,misis an polar angle difference between the visible product and the neutrino system. Although the equation (4.4) reduces unknown parameters, there are still 2 to 4 unknown parameters: ϕmis1,2 (mmis1,2). An additional knowledge ofτ decay kine-matics can be used to distinguish more likely solutions from less likely ones. These parameters are determined by global event likelihood using∆θ_3Das probability density functions (PDFs), where∆θ_3D is a three-dimensional angle between the visible product and the neutrino system. The PDFs are obtained from simulatedZ →τ τ events with respect to the momentum of the originalτ lepton (p(τ)). Figure 4.6 shows PDFs,P(∆θ_3D, p(τ)), for the leptonic, 1-prong and 3-prongτ decays with45< p(τ)≤50GeV.

[rad]

θ3D

∆

0 0.02 0.04 0.06 0.08

Arbitrary units

0 0.01 0.02 0.03

ATLAS Simulation Simulation

τ τ

→ Z

Probability function decay τ Leptonic

50 [GeV]

τ≤ 45<p

(a)

[rad]

θ3D

∆

0 0.05 0.1 0.15 0.2

Arbitrary units

0 0.005 0.01

0.015 ATLAS Simulation

Simulation τ τ

→ Z

Probability function decay τ 1-prong

50 [GeV]

τ≤ 45<p

(b)

[rad]

θ3D

∆

0 0.05 0.1 0.15 0.2 0.25

Arbitrary units

0 0.005 0.01 0.015 0.02

ATLAS Simulation Simulation τ τ

→ Z

Probability function decay τ 3-prong

50 [GeV]

τ≤ 45<p

(c)

Fig. 4.6: Probability density functionsP(∆θ_3D, p_T)for (a) the leptonic, (b) 1-prong and (c) 3-prongτ decays with45 < p(τ) ≤50GeV. Red dot shows the PDFs obtained from simulated Z → τ τ events, while black line shows a fit result with a Gaussian plus landau function.

The MMC includes an effect of finite resolution of the E_T^miss measurement in order to improve the m_{τ τ} resolution. The effect is taken into account by introducing two probability functions of the E_T^miss resolution to the likelihood:

P(∆E_T^miss_x,y) =exp (

− (

∆E_T^miss_x,y)2

2σ² )

, (4.5)

where∆E_T^miss_x,y are variations between a measured and trueE_T^miss for x and y components, andσ is the E_T^missresolution measured from calibration data samples. Thus, the likelihood is defined as:

L=P(∆θ_3D,1, p_T,1)×P(∆θ_3D,2, p_T,2)×P(∆E^miss_Tx )×P(∆E_T^miss_y ). (4.6) The MMC scans about10⁵ phase space of(ϕmis1,2, E_T^miss_x , E_T^miss_y (, mmis1,2)), and then a number ofm_{τ τ} andL are produced at all scan points. The peak value of the m_{τ τ} ×Lhistogram is used as the final estimatedmτ τ. The MMC is able to estimate physicalmτ τ value with high reconstruction efficiency around97%∼99%.

66

[GeV]

τ

mτ

0 50 100 150 200 250 300

Fraction of Events / 5 GeV

0 0.1 0.2

1jet

≥ Preselection + τhad

µ

had + τ = 8TeV , e s

= 36.7%) = 90.4GeV , FWHM/mpeak MMC (mpeak

= 48.2%) = 91.2GeV , FWHM/mpeak Collinear Mass (mpeak

= 45.3%) = 61.5GeV , FWHM/mpeak Visible Mass (mpeak

(a)

[GeV]

τ

mτ

0 50 100 150 200 250 300

Fraction of Events / 5 GeV

0 0.05 0.1 0.15

1jet

≥ Preselection + τhad

µ

had + τ = 8TeV , e s

= 37.9%) = 124.8GeV , FWHM/mpeak MMC (mpeak

= 41.3%) = 118.4GeV , FWHM/mpeak Collinear Mass (mpeak

= 56.4%) = 78GeV , FWHM/mpeak Visible Mass (mpeak

(b)

Fig. 4.7:mτ τ distributions using the visible (green), the collinear (blue) and the MMC (red) methods for (a) theZ → τ τ → ℓτ_had and (b) theH → τ_ℓτ_had with the Higgs boson mass of125GeV. Events are required to pass the preselection and to have at least one jet withp_T>30GeV.

Figure 4.7 showsm_{τ τ}distributions of different three methods, separately forZ →τ τ →ℓτ_hadandH→ τ_ℓτhadwith the Higgs boson mass of125GeV processes. Events are required to pass the preselection and to have at least one jet with pT > 30GeV. This jet requirement increases physical solutions of the collinear method from 25.2±0.3% to42.3±0.3%. On the other hand, the efficiency of the MMC is∼99%in spite of the jet requirement. Figure 4.8 shows the linearity of them_{τ τ} peak and the Root-Mean-Square (RMS) of them_{τ τ}histogram, as a function of the input Higgs boson mass from100GeV to 150GeV in a5GeV step. The peaks can be almost correctly reconstructed by the MMC, while the peaks of collinear method are shifted under∼5GeV, depending on the Higgs boson mass values. The RMSs of the MMC and the collinear method are18% ∼ 27%and41% ∼ 52%, respectively. Although the RMSs of the visible mass are of the order of14%∼15%, the peaks of the visible mass are significantly smaller from the Higgs boson mass of the order of40GeV ∼ 50GeV due to not considering neutrino four-momenta. This analysis uses the MMC method to reconstruct them_{τ τ} because of better mass peak reconstruction with the smaller RMS than the collinear method.