The Electron-Extrapolation Method

In this work, photon identification efficiencies have been measured using theelectron-extrapolation method. It makes use of the similarity of shower shapes of electrons and photons. One can derive transformations that transform the shower shape of electrons such that the resulting objects have distributions of shower-shape variables that resemble closely those of photons.

These shower-shape transformations are based onSmirnov transformations[162]. By applying the shower-shape transformations to electrons in data, which can be selected fromZ →e⁺e⁻ events using a tag-and-probe method as described in Section 4.5.2, one obtains a sample of pseudo-photons, which can be used to determine the photon identification efficiency in data.

It is possible to collect a pure and unbiased sample of photons fromZ→`<sup>+</sup>`⁻γevents, using a method that is briefly described in Section 4.3. Therefore, it is warranted to ask why such a detour over electrons and shower-shape transformations should be made. The answer is that the method of collecting photons fromZ→`<sup>+</sup>`⁻γ events results in a sample of photons with relatively low p_T, as these photons correspond to emissions from one of the leptons from the Z-boson decay. Electrons inZ →e⁺e⁻ events, on the other hand, tend to have comparatively large p_T. Therefore, by transforming these electrons to pseudo-photons, one obtains a collection of photon-like objects with higher pTthan is the case in the method based onZ→`<sup>+</sup>`⁻γevents.

The statistical uncertainty of the measured photon identification efficiency is therefore lower at intermediate to high pT in the electron-extrapolation method than in the method based on Z→`<sup>+</sup>`⁻γevents.

The transformations used to transform electrons to photon-like objects are based on samples of simulatedZ→e⁺e⁻events electrons and on samples of inclusive-photon events, see Section 4.4.2.

For each of the considered shower-shape variables and for converted and unconverted photons

separately, the transformations are derived in 7 |η|bins × 12 p_T bins, see Tables 4.4 and 4.2, respectively. The relatively fine binning in|η|is used because shower-shape variable distributions can vary significantly in different regions of|η|. The principle of the shower-shape transformations

|η|bin number 1 2 3 4 5 6 7

lower boundary 0 0.6 0.80 1.15 1.52 1.81 2.01 upper boundary 0.6 0.8 1.15 1.37 1.81 2.01 2.37

Table 4.4.|The upper and lower boundaries of the seven|η|bins for which the electron transformations are derived.

is illustrated in Figure 4.2. An example based on simulation is shown in Figure 4.3. A description

Shower-shape variable

p.d.f.

γ e

c.d.f.

Shower-shape variable

e

γ

Shower-shape variable(e)

Shower-shapevariable(γ)

(a) (b) (c)

Figure 4.2.|Illustration of the electron-extrapolation method. In (a), the distribution of a shower-shape variable is shown for both photons and electrons. In (b), the corresponding c.d.f.s are shown. The orange arrow illustrates the transformation procedure based on the c.d.f.s. The corresponding mapping from electron values for the shower-shape variable to the corresponding values for photons, (F_γ)⁻¹(F_e(xe)), is shown in (c).

of the principle is given below for a shower-shape variable denoted byxand for a p_Tbiniand a

|η|bin j:

1. Based on simulated samples of electrons and photons, normalized shower-shape variable distributions, i.e. probability density functions (p.d.f.s),

• f_e(x|p_Tbini;|η|bin j)

• f_γ^(un)conv(x|p_Tbini;|η|bin j)

are determined for electrons and converted (unconverted) photons. See Figure 4.2 (a).

0.2 0.4 0.6 0.8 1 1.2 1.4 Fside

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045

p.d.f.

e γ [GeV]<60 pT

|<1.37, 50<

1.15<|η

0.2 0.4 0.6 0.8 1 1.2 1.4 Fside

0 0.2 0.4 0.6 0.8 1

c.d.f. 1.2

e γ

0.2 0.4 0.6 0.8 1 1.2 1.4 Fside

0 0.2 0.4 0.6 0.8 1 1.2

γpseudo-e )(FFsideside1.4

0.2 0.4 0.6 0.8 1 1.2 1.4 Fside

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045

p.d.f. γpseudo-

γpseudo-γ

Figure 4.3. |Example of the derivation and application of a shape transformation. The shower-shape variable in this example is chosen to beF_side. The red and black markers correspond to distributions of photons and electrons in simulation, respectively. In the top-left plot, the p.d.f.s, and in the top-right plot, the corresponding c.d.f.s are shown. The mapping from electron values for the F_side variable to corresponding pseudo-photon values is shown in the bottom-left plot. The result of the application of this transformation on a sample of electrons and a comparison with the original photon p.d.f. is shown in the bottom-right plot.

2. These distributions are the basis for the computation ofcumulative distribution functions (c.d.f.s) of the shower-shape variable distributions [163], see Figure 4.2 (b). A c.d.f. is defined as

Fe(x|pTbini;|η|bin j)= Z x

−∞

fe(x⁰|pTbini;|η|bin j) dx⁰ (4.3) for electrons and similarly for converted (unconverted) photons:

F^(un)conv_γ (x|p_Tbini;|η|bin j)=Z x

−∞

f_γ^(un)conv(x⁰|p_Tbini;|η|bin j) dx⁰. (4.4) Given the electron and photon c.d.f.s for the shower-shape variable, a given electron can now be transformed to a pseudo-photon by applying the following operations, summarized also in Figure 4.2 (b) and (c) in form of an arrow:

1. For a given electron, the valuex_measof the shower-shape variable xis determined.

2. In order to obtain the shower-shape variable value of the to-be-created pseudo-photon, the inverted photon c.d.f. is evaluated at the value of the electron c.d.f. at the measured shower-shape variable value x_meas:

x^(un)conv_γ =(F^(un)conv_γ )⁻¹( F_e(x_meas) ), (4.5) where the dependence of the c.d.f.s on thepT and|η|of the electron is implicit.

While the transformations reproduce the one-dimensional photon shower-shape distributions, the transformations preserve the electron-like correlations between shower-shape variables. A correction for this bias is applied to the measured efficiency.

The transformation procedure is applied to electrons from Z→e⁺e⁻events collected with a tag-and-probe method in data, see Section 4.5.2. Given an electron to be transformed, the transformations for the various shower-shape variables are selected based on the p_Tand|η|of that electrons. The photon identification efficiency measurement is based on the invariant-mass spectrum of the tagged electron and transformed electron-candidate probe. Based on this mass spectrum, a signal-plus-background fit is performed in order to extract signal yields for both probe selections relevant for the efficiency determination, following Eq. (4.1). The efficiency

then is given by the ratio of the signal yield when applying the photon identification selection to the probe pseudo-photons and the signal yield when no identification selection is applied.

Since the efficiency measurement is performed for isolated photons, the isolation criteria are part of both numerator and denominator selections applied to the probe pseudo-photon. Separate invariant-mass distributions are defined forN_bins^pT ×N_bins^|η| = 12×4 kinematic regions of the probe.

The electron-extrapolation method relies on electrons fromZdecays in sufficient numbers;

therefore, it is currently constrained to values of transverse momentum below about 250 GeV.

There is, however, no fundamental reason against expanding this upper limit once more data is taken. A lower p_Tboundary of 25 GeV is used since the level of background events increases towards low pT.

In Appendix C, the distributions of the various shower-shape variables for photons and electrons are shown for a representative selection of kinematic regions. Additionally, the effect of the shower-shape transformation is shown in terms of the difference between the variable value after and before the transformation was applied.