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Two different analysis strategies will be presented in the next two sections. The first one has been used in the previous studies [13] and underwent many refinement steps [110]. It is based on the acoplanarity (ϕ∗_{) distribution which matches a cosine whose phase-shift} is determined by the Higgs boson parity. The second strategy is independent of such theoretical prerequisites as it measures directly the difference in the physical observables of both parity states.

Analysis based on a Cosine fitted to the Acoplanarity

It was shown in section 2.2 that the distribution of events in the acoplanarity angle follows a cosine function with some vertical offset if events are selected according to y1y2 > 0 or

7.4 Higgs Boson Parity Measurement at a Future Linear Collider 163 analysis methods presented here is to generate the clearest possible cosine signal in the acoplanarity distribution. Two components have to be taken into account: On the one hand the amplitude of the cosine should be maximised (by rejecting events which only contribute to a flat background). On the other hand as many events as possible should be used to fit the cosine function because the uncertainty of the amplitude is high with too low statistics. A fitting procedure determines the significance of the cosine signal for a given event set by taking both factors into account: The significance is defined as

s = A

σA

, (7.4)

the amplitude of the fitted cosine function divided by its uncertainty which is derived from the fit (compare figure 7.28). For the fit the vertical offset of the cosine function is fixed to the mean value (mean number of events per bin) and no phase shift of the cosine function is allowed. A shift would correspond to a mixing of states but this is a different analysis and will not be discussed here [111].

Different training targets and preprocessing cuts will be compared in the next section. The starting point of the analysis is the acoplanarity histogram with “scalar events” for which y1y2 > 0 or equivalently with “pseudo-scalar events” for which y1y2 < 0. The

remaining half of the events can be used in the same way, but the acoplanarity is shifted by 180 degrees (compare the discussion in section 2.2). Using the τ± _{impact parameters} in the reconstruction process means replacing y1 and y2 by y1i and yi2, respectively, and

improves the significance slightly [13].

Different preselections will be used in different combinations to improve the significance which is obtained from the histogram described above:

(a) A significant improvement can be achieved by selecting events according toy1y1i >0

and y2y2i > 0. This means that those events generate a very clear cosine signal for

which both reconstruction methods return the same sign.

(b) The significance can be improved further if the pion energies from ρ± _{are compared} in a second way. y1y2 >0 meant that for bothρ’s the major energy part (relative to

their energy sum) should be either in the charged or in the neutral pion. We define

z1 =Eπ+ −E_π0 z₂ =E_π−−E_π0 (7.5)

where theπ0’s are from the respectiveρ±decays. A selectionsign(z1z2) = sign(y1y2)

improves the significance by demanding the same relation between the pion energies not relatively but in terms of absolute energies as measured in the laboratory frame. According to the basic selection withsign(y1y2) (or sign(y1iyi2)) the training targets for

statistical learning methods are chosen. The source for the target values is the simulation which knows about the generated energies and momenta, in contrast to the experimen- tal situation where only reconstructed quantities are available. The values x1 and x2 are

derived from generated values and correspond to y1 and y2, respectively. A near optimal

significance should therefore be obtained by selecting according to sign(x1x2). The sta-

tistical learning methods will thus use sign(x1x2) as the basis for a training target. The

164 7. Analysis and Results (A) Preselect according to selection (a) above, then use the selection according tosign(yi

1y2i)

if this selection is validated (output > cut) by the statistical learning method (else discard the event). The training target is thereforesign(yi

1y2i) = sign(x1x2) (defined

as 0 if false and 1 if true).

(B) Preselect according to selection (b) above and then select according to (output >

cut) or (output <1₋cut) since the training target is then (x1x2 >0).

(C) No preselection, select directly according to (output>cut) or (output<1₋cut), the training target is again (x1x2 >0).

The output distributions are usually symmetric so that indeed (output > cut) performs likeyi

1yi2 >0 and (output<1−cut) likey1iyi2 <0 in (B) and (C). To control this behaviour

all event types of the test set are used. This means that we use scalar and pseudo-scalar events to test the classification. To clarify the event selection according to sign(yi

1y2i) (or

the respective cut in the output) we write down case by case which events are entered into the histogram to calculate the significance:

• scalar events with yi

1y2i >0 (or the respective cut in the output),

• scalar events with yi

1yi2 <0 (or the respective cut in the output) with a phase shift

in acoplanarity of 180 degrees,

• pseudo-scalar events withyi

1yi2 <0 (or the respective cut in the output),

• pseudo-scalar events withyi

1yi2 >0 (or the respective cut in the output) with a phase

shift in acoplanarity of 180 degrees.

Effectively a phase-shift needs to be applied for each event depending on its simulated parity and depending onsign(yi

1yi2) or the output of the statistical learning method. Using

all these event types assures that any detection method is sensitive to both parity types. Analysis based on a direct Discrimination of Parity States

In contrast to the analysis strategy described above, the training target can be directly defined by the parity: 0 for scalar events and 1 for pseudo-scalar events. The inputs used for the training are the same as mentioned above. The resulting output distributions like the one in figure 7.26 show an accumulation around the mean value 0.5 with different tails to the left and right side. By construction the left tail is higher for scalar events because they were trained to 0 and the right tail is higher for pseudo-scalar events because they were trained to 1.

The significance of a small (N events) event set is

s= ∆µ σ∆µ = µN −µtot σµN = µNσ−µ0/µ1 tot √ N . (7.6)

Here the difference ∆µ of observed mean µN to the overall mean µtot (from both classes, close to 0.5) is divided by its uncertainty σ∆µ which is calculated by transforming the standard deviation of each class σµ0/1 to the lower statistics N. The values µtot and σµ0/1

7.4 Higgs Boson Parity Measurement at a Future Linear Collider 165 0 0.2 0.4 0.6 0.8 1 10-2 10-1 1 10 102 output events [%] scalar events 0 0.2 0.4 0.6 0.8 1 10-2 10-1 1 10 102 output events [%] pseudoscalar events

Figure 7.26: Direct discrimination of scalar and pseudo-scalar events: Example of output distributions after training (high statistics).

Technically, the overall mean valueµtot and the variances of the output distributions for scalar and pseudo-scalar events σµ0/1 are calculated by using the whole test set consisting

of 160.000 events of each type. As described below several pseudo experiments are done with small parts of the test set. These have their own mean value µN and determine thus the difference to the overall mean ∆µ and its uncertainty σ∆µ given by the number of events N.

Compared to the analysis based on the acoplanarity angle this direct discrimination has the advantage that much less theoretical input is necessary. In addition, no fit is necessary, only means and variances have to be calculated.

Significance obtained from the Pseudo Experiments

The significances are obtained by performing a number of pseudo experiments (300 with scalar and 300 with pseudo-scalar events in the results presented below): A total number of observed events is simulated which has a Gaussian distribution around its mean given by the luminosity, cross section and detection efficiency assumptions (this mean will be 500 events in the results presented below). The events of each of these pseudo experiments are passed through the selection cuts and a significance for the Higgs parity (equation 7.6) is calculated. The events for the pseudo experiments are taken from the large test set which has not been used in the training. Performing many pseudo experiments does not only result in a mean significance but also in a standard deviation of the distribution of significances (compare figure 7.27). With this information a confidence interval can be constructed for the significance to be determined in a future linear collider experiment.