INSTRUMENTOS DE EVALUACIÓN

Reconstruction ofτlepton energy Theτlepton is identified by disentangling its production and decay vertices. Due to its short lifetime, its energy must be reconstructed starting from its daughter.

For the lepton number conservation, the decay of a_τresults in the production of at least one (invisible) neutrino. However, the kinematics of the decay can still be closed if there is only one neutral undetected particle.

A pure one prong decay, i.e. with no emitted neutral particles at the decay vertex, represents 11.5% of the totalτdecay probabilities, while the pure three prong decay represents 9.8% of the total branching fractions. In all the other cases, at least one neutral particle is emitted.

When theτlepton decays into a muon or into an electron, two neutrinos are emitted: these leptonic decay modes have thus been excluded from the following analysis. They account for ∼ 35% of the total tau decays.

(a) (b)

Figure 4.14: Energy calibration of the visible reconstructed energy as a function of the true MC energy (a) and ∆E_E distribution (b) for electron neutrinos.

When theτlepton decays into one or more hadrons it can originate alsoπ0s and K0s. If I assume to be able to reconstruct the_π0 starting from the_γs produced in its decay, I can close the kinematics and thus correctly evaluate theτenergy. The only hadronic decay modes where I have an unidentified neutral particle are those in which a K0is emitted. Since they constitute less than 2% of the total tau lepton branching ratios, I can identify them as pure hadronic events with no other neutral particle emitted than theν_τmaking an error that can be at the moment easily neglected for the following analysis.

Tau decays with only one neutral particle For the previously pointed out reasons, for this analysis I consider only those events where the_τlepton decays into hadrons.

The subscript d refers to variables associated to theτlepton daughters, while the subscript m (as for "missing") refers to variable associated to the undetected neutrino. Therefore,~pd(eq.

4.4) and E_d(eq. 4.5) are respectively the momentum and the energy of the system formed by the

τdaughters: (4.4) ~p_d=X i ~pi (4.5) E_d=X i Ei

(4.6) md= q E2_{d− (}p~d)2 = v u u t à X i Ei !2 − à X i ~pi !2

The conservation of the four-momentum at theτdecay vertex gives:

(4.7)    ~pτ=~pm+~pd E_{τ= Ed+ Em}

Recalling that~p_τand E_τmust satisfy the on shell condition, i.e. m2_{τ= E}2_τ−~p2_τ. Using eq. 4.7 and the on shell condition, we get that p_τis:

(4.8) p_τ=−(m 2 τ+ m2d)pdcosθk± q E2_d[(m2_{τ− m}2_d)2_{− 4m}2 τp2dsin 2_θ k] 2(p2_dcos2_θ k− E2_d)

Solving equation 4.8, leads to two possible solutions and, except for some peculiar cases, it is not possible to discard one of them a priori.

Let p_τ,1be the solution of eq. 4.8 with the "+" sign, while p_τ,2is the solution with the "-" sign. I can discard and take p_τ,2as the correct momentum of theτlepton every time p_τ,1> 400 GeV/c, and vice-versa I can consider p_τ,1as the true solution when p_{τ,2< 0. In all the other cases, I have} to take both:

(4.9) p_τ=w1· pτ,1+ w2· pτ,2

w_{1+ w2}

where w₁ is the fraction of cases in which p_τ,1has a lower discrepancy from the true MC value of p_τand w2 is the fraction of cases in which it is instead pτ,2that has a lower discrepancy from the true value. The discrepancy from the MC p_τvalue is estimated as:

p_τ− p_τMC p_τMC

In the decay channel of aτin one single hadron w1h₁ and w1h₂ are 52% and 48% respectively. In the decay channel of a_τin three hadrons w3h₁ and w3h₂ are 48% and 52% respectively. However, when the two solutions differ significantly (e.g: |pτ,1− pτ,2| >30 GeV/c), in order to better estimate the momentum of theτ, I used as discriminator the measuredτflight length (∆l):

∆lτ= pτ mc∆tτ being∆t_τthe mean life time of theτlepton.

Comparing the one computed using both p_τ,1and p_τ,2to the measured one, I took as better p_τestimation the solution which gave me the smaller discrepancy between the estimated flight length and the measured one.

The_τlepton reconstructed energy can be obtained by its momentum.

Once again I perform a calibration of the true MC and the visibleτenergy, as shown in fig. 4.15 with on the left side the calibration for theτ→ 1h decay channel and on the right side the one for the_{τ→ 3h decay channel.}

(a) (b)

Figure 4.15: Energy calibration of the visibleτlepton visible energy as a function of EMC_τ for the

τ→ 1h decay channel (a) and for theτ→ 3h one (b).

In both cases, the visible energy of theτis parametrized as:

Evis_{τ = p}0+ p1EMC_τ

With this parametrisation the resolution∆E/E (def in eq. 4.3) on theτlepton reconstructed energy, which is defined as:

Erec_{τ =}E

vis

τ − a

b

In theτ→ h channel the resolutionσEτ= (35 ± 3)% (see fig. 4.16(a)), while in theτ→ 3h (see

fig. 4.16(b)):_{σEτ= (22 ± 2)%. In theτ→ h decay channel, the discrepancy between the solutions} p_τ,1and p_τ,2is indeed larger than in theτ_{→ 3h case, as it can be seen in fig. 4.4.3.}

Reconstruction of neutrino energy Considering only the tau neutrino events with theτ lepton decaying in the hadronic channels and using the previously found resolutions on the

(a) (b)

Figure 4.16:∆E/E distribution for theτ→ 1h decay channel (a) and for theτ→ 3h one (b).

Figure 4.17: Absolute difference between the two solutions of equation 4.8 for the_{τ→ h channel} in black and for theτ→ 3h channel in red. On average, the discrepancy between pτ,1and pτ,2is larger in the 1-prong channel.

reconstruct the_ντenergy starting from visible tracks at the neutrino interaction vertex. Results of the calibration between reconstructed and true MC energy are shown in fig. 4.18(a). The resolution on the reconstructed E_ντ is (see fig. 4.18(b)):

σE_ντ= (22 ± 2)%

As for the_νealso in this case the resolution on the_τlepton energy is the major component in the totalν_τenergy resolution.

(a) (b)

Figure 4.18: Energy calibration of the visible reconstructed energy as a function of the true MC energy (a) and ∆E_E distribution (b) for tau neutrinos.

ντ ν¯τ decay channel Nex p Nbk g R Nex p Nbk g R τ→µ 1031 41 25 756 122 6 τ→ h 1807 108 17 1325 325 4 τ→ 3h 385 43 9 282 124 2 total 3224 192 17 2364 570 4

Table 4.5: Number of expected reconstructedν_τ and ¯ν_τsignal (Nex p) events together with the expected number of charm background events (Nbk g) for the different_τdecay channels, except theτ→ e, where the lepton number cannot be determined. The signal to background ratio R is also reported.