Signal Modeling and Efficiency Mass resolutionMass resolution

2 As discussed in the introduction, sufficiently strong

6.6 Signal Modeling and Efficiency Mass resolutionMass resolution

−30 −20 −10 0 10 20 30 40

Score

10⁰ 10¹ 10² 10³ 10⁴ 10⁵

En tries

^{Data, cτ=0.1 mm}_{Data, cτ=1 mm}

Data, cτ=10 mm

Signal, cτ=0.1 mm Signal, cτ=1 mm Signal, cτ=10 mm

Figure 6.12: The distribution of the classifier scores for data (markers) and signal MC simulations (solid lines) for dark photon lifetimes corresponding to (top)cτ_A⁰ = 0.1mm, (middle) cτ_A⁰ = 1mm, and (bottom) cτ_A⁰ = 10mm. The MC simulations are arbitrarily normalized.

Event selection results

For signal with prompt decays, a total of 69 events pass all the selection criteria. The corresponding(mΥ_D,m_A⁰)distribution is shown in Fig. 6.14. Most events correspond toe⁺e⁻ → qq¯process. The events nearmΥD ∼0.1 GeV andm_A⁰ ∼0.05 GeV arise frome⁺e⁻ →γγγevents in which all three photons convert toe⁺e⁻ pairs.

For signal with displaced decays, a total of 56, 33, and 31 events are selected for the cτ_A⁰ =0.1, 1, and 10 mm data sample, respectively. The resulting mass distributions are shown in Fig. 6.15.

6.6 Signal Modeling and Efficiency

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10 102

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Figure 6.13: Fits to the classification score distributions of data with a function of the form p(x) = e^a·x2+b·x+c, forC₀(upper),C₁(middle), andC₂(bottom) category of events.

0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5

m

_ΥD

(GeV)

0.0 0.5 1.0 1.5 2.0

m

A0

(GeV)

C0

C1

C2

Figure 6.14: The(mΥD,m_A⁰)distribution for events passing all selection criteria for prompt dark photon decays.

0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5

m

_ΥD

(GeV)

0.00 0.05 0.10 0.15 0.20

m

A0

(GeV)

cτ = 0.1 mm cτ = 1 mm cτ = 10 mm

Figure 6.15: The(mΥ_D,m_A⁰)mass distribution of event candidates passing all selec- tion criteria for the datasets optimized for each dark photon lifetime.

than a simple Gaussian fit. While the fits are not perfect, they are sufficient for the purpose of estimating the mass resolution.

To evaluate the mass resolution as a function of(mΥD,m_A⁰), we interpolate the results at known points using a 2-dimensional smooth spline. Smooth spline function aims to fit a polynomial function on data and guarantees the smoothness of the function at the same time. To minimize the fitting uncertainties, we fit each category of events individually. The results are shown in Fig. 6.17. TheΥ_Dmass resolutions are at the level of 20 MeV. The dark photon mass resolutions are at the level of 3 MeV.

Signal efficiency

The signal efficiency is evaluated by counting the fraction of simulated events passing the selection criteria in the signal window at a given point, normalized to the total number of generated events. The size of the signal window is given by the mass resolutions at that point, namely 4∆_mΥ and 4∆_mA0.

To evaluate the signal efficiency as a function of (mΥ_D,m_A⁰), we also use a 2- dimensional smooth spline technique for interpolation, fitting each category sepa- rately. The degrees of the polynomial functions used for interpolation are increased until the model is stable. The results of the signal acceptance, selection efficiency, and signal efficiency are shown in Fig. 6.18.

The signal acceptance is low when mΥD and m_A⁰ are low, which stops us from improving the signal efficiency in the low mass region. The MVA prefers to select events in the range of 6 GeV ≤ mΥ_D ≤ 9 GeV, thus the selection efficiencies are higher in this region.

For each signal window, the signal efficiencies fromC₀,C₁, andC₂are weighted by branching fractions to obtain the total signal efficiency of the signal window. The total signal efficiency is then used for signal extraction. The result is shown in Fig.

6.19. The two horizontal bands aroundm_A⁰ = 0.78 GeV and 1.0 GeV correspond to the ω and φ resonances. The drop in the branching fraction A⁰ → X⁺X⁻,X = e, µ, πatm_A⁰ =0.78 GeV and 1.0 GeV is due to theA⁰decaying predominantly into π⁺π⁻π⁰andK⁺K⁻, respectively.

Lifetime dependency

The signal efficiencies depend on the dark photon lifetime. When the dark photons have non-zero lifetime, the χ²fit of assuming prompt decay becomes larger and the mass resolution worse. The larger the dark photon lifetime, the smaller the signal

6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 Υ

^D

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^D

mass C ystal Ball Fit Pe fo mance

C ystal Ball Fit M.C. Mass Histog am

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Photon Mass / GeV

0 20 40 60 80 100 120

De nsi ty

Dark Photon mass Crystal Ball Fit Performance

Crystal Ball Fit M.C. Mass Histogram

Figure 6.16: Example of CBF fits to theΥ_D andA⁰mass spectrum (mΥ_D =8.0 GeV andm_A⁰ = 0.5 GeV).

0 2 Υ^D4 mass / GeV6 8 10 0.0

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C2 Dark Upsilon Mass Resolution (MeV)

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2 4ΥD ma / GeV6 8 10 0.25

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Dark photon ma / GeV

C2 Dark Photon Ma Re olution (MeV)

1.0 1.52.0 2.5 3.03.5 4.04.5

Figure 6.17: (Left) TheΥ_Dand (right) the dark photon mass resolution as a function ofmΥD andm_A⁰ for each category of events.

efficiency. We generate and reconstruct the signal MC with non-zero lifetime to estimate and interpolate the impact of lifetime on the signal efficiencies. We use the functional form log() = a ·log(l)+ bto fit the relation between the average signal efficiencies () and the dark photon lifetime (l). We checked that the results obtained for each mass hypothesis are compatible with those obtained by considering the average efficiency, but the later are more robust and were chosen to describe the global dependence on the lifetime. For a given mass hypothesis(mΥ_D,m_A⁰)with prompt signal efficiency ₀, the relation between the signal efficiency and the dark photon lifetime under the mass hypothesis is reasonably described by:

log()= a· (log(l) −log(l₀))+log(₀), (6.10)

Figure 6.18: The acceptance, selection efficiency, and signal efficiency as a function ofmΥD andm_A⁰ for each category of events.

withl₀= 0.1 mm, and₀the efficiency for prompt decays. For a dark photon flight lengths above 100 µm, this interpolated relation will be used to correct the signal efficiency and establish the kinetic mixing strength upper limit. When the dark photon flight length is below 100 µm, we use the efficiency determined for prompt decays, as the displaced decay vertices of the dark photon have negligible effect on the signal efficiencies.