Φ NE? Can e e → ηπ π bedetectedatDA + − + − UppsalaUniversity

Uppsala University

Department of Physics and Astronomy

Bachelor of Science Degree in Physics

Can e

+

e

−

→ ηπ

+

π

−

be detected

at DAΦNE?

Author:

Viktor Thorén

Supervisor:

Andrzej Kupsc, Division of Nuclear Physics

Subject Reader:

Tord Johansson, Division of Nuclear Physics

Abstract

Studying the annihilation of e

+

_e

−

_{into hadrons is of great interest in}

the search for physics beyond the standard model. The cross sections of

specific hadronic channels can be used in precise test of the Standard Model,

or to estimate the anomalous muon magnetic moment, for which there are

discrepancies between theory and experiment.

This thesis focuses on one particular hadronic channel, e

+

_e

−

_{→ ηπ}

+

_π

−

with the final state π

+

_π

−

_{γγ. As background, the reaction e}

+

_e

−

_{→ π}

+

_π

−

_π

0

is chosen. By simulations using the Monte Carlo event generator PHOKHARA

and the data analysis framework ROOT, the feasibility of detecting the

afore-mentioned reaction with the detector KLOE at the electron-positron collider

DAΦNE is studied. The detector acceptance and expected number of events

is evaluated both for the signal and the background. An analysis program

that can be used for feasibility studies of e

+

e

−

_{→ ηπ}

+

_π

−

_{was written.}

The simulations indicate that the signal significance from an experiment

at DAΦNE would be 2.4σ and thus lower than the confidence level required

for detection. A definitive statement can however only be made after further

investigations into minimizing the background and studies of additional final

states with other decays of the η-meson and their corresponding background

reactions.

Sammanfattning

I sökandet efter fysik bortom standardmodellen är studier av

elektron-positron-annihilation till hadroner av stort intresse. Tvärsnitten för specifika

hadroniska kanaler kan användas för precisa tester av standardmodellen,

el-ler för att uppskatta myonens anomala magnetiska moment, vars teoretiska

värde avviker från det experimentella.

Detta examensarbete fokuserar på en specifik hadronisk kanal, e

+

_e

−

_→

ηπ

+

_π

−

_{med sluttillståndet π}

+

_π

−

_{γγ. Som bakgrund väljs reaktionen e}

+

_e

−

_→

π

+

_π

−

_π

0

_{. Med hjälp av simulationer byggda på den Monte Carlo-baserade}

händelsegeneratorn PHOKHARA, och ROOT, ett ramverk för dataanalys,

studeras om det är genomförbart att detektera den ovan nämnda reaktionen

med detektorn KLOE vid acceleratorn DAΦNE. Detektoracceptansen och

det förväntade antalet händelser utvärderas både för signalen och

bakgrun-den. Ett analysprogram som skulle kunna användas för andra

genomförbar-hetsstudier för e

+

_e

−

_{→ ηπ}

+

_π

−

_{har skrivits.}

Contents

1

Introduction

1

2

Background

2

2.1

Beyond the Standard Model . . . .

2

2.2

DAΦNE Collider . . . .

3

2.3

KLOE Detector . . . .

3

2.3.1

Minimum Transverse Momentum . . . .

5

2.4

The Reaction e

+

_e

−

_{→ ηπ}

+

_π

−

_{. . . .}

₆

2.5

Luminosity, Cross section, and Acceptance . . . .

7

2.6

Signal Significance

. . . .

7

3

Method

7

3.1

PHOKHARA . . . .

8

3.2

ROOT . . . .

8

3.3

Test of the Simulations . . . .

8

3.4

Execution . . . .

9

3.5

Cuts on the Background . . . .

10

3.6

Design of the Analysis Program . . . .

11

4

Result

13

4.1

Particle Distributions for the Signal . . . .

13

4.2

Particle Distributions for the Background . . . .

21

4.3

Cross Sections . . . .

26

4.4

Acceptances . . . .

26

4.5

Number of Events . . . .

27

5

Discussion

28

6

Recommendations

29

7

Conclusion

29

References

31

A Code

33

A.1 Main Class EtaPiPi.C

. . . .

33

A.2 Auxiliary class Detector

. . . .

52

1

Introduction

To this day, no physical theory can match the success of the Standard Model

(hereafter abbreviated to SM) in describing the constituents of matter and

their interactions. It explains a wealth of experimental results, and has been

used to predict the existence of particles that have later been detected; e.g.

the top quark, the tau neutrino [1], and recently the Higgs Boson. [2]

However, the SM is not a complete theory. For instance, it describes

three of the fundamental interactions: the electromagnetic, weak, and strong

interactions, but does not include the fourth: gravity. There are also a few

important issues left unexplained, such as the composition of dark matter, or

why quarks and leptons come in three generations. Furthermore, there are

several examples of experimental results that deviate from SM predictions,

meaning that there is an interest in investigating the possibility of new, exotic

interactions beyond the SM. [1]

Studying e

+

_e

−

_{annihilation into hadrons provides several avenues for}

searching for such new physics signals. The corresponding cross sections

are used for estimating the anomalous muon magnetic moment, for which

discrepancies between experiments and the SM have been detected [3]. It

also serves as a viable option to studying the rare decays of pseudoscalar

mesons, such as η, η

0

, η

c

, or π

0

, into a lepton-antilepton pair.

Discrepan-cies between the SM and experiment have already been detected for the

decay π

0

_{→ e}

+

_e

−

_{[4] [5], and it is consequently desirable to study the}

decays of similar particles. Both η

0

and η

c

could be studied through the

process e

+

_e

−

_{→ η}

0

_/η

c

→ ηπ

+

π

−

. Conducting experiments on the reaction

e

+

_e

−

_{→ ηπ}

+

_π

−

_{is therefore of great interest.}

This thesis focuses on studying e

+

e

−

→ ηπ

+

_π

−

_{with the KLOE detector}

at the DAΦNE collider in Frascati, Italy [6]. Using this particular facility is

desirable as its center of mass energy (1.02 GeV), is close to threshold energy

of the reaction. Furthermore, the reaction has not been detected at such low

energies as of yet.

Before any actual experiment is undertaken, it is necessary to examine the

feasibility of detecting the reaction. Running the collider and taking part in

experiments is both expensive and time consuming, and therefore one needs

to be sure of receiving usable results. This particularly important when it

comes to studying e

+

e

−

→ ηπ

+

_π

−

_{at DAΦNE as there are no previous results}

to indicate that such an experiment is realisable.

reaction e

+

_e

−

_{→ ηπ}

+

_π

−

_{at energy 1 GeV with the integrated luminosities}

available at the DAΦNE using the KLOE detector. The expected number of

events and the detection acceptance will be determined, and the background

evaluated. This information will be used to decide if such an experiment

could be considered meaningful.

The secondary goal of the project is the preparation of an analysis

pro-gram for the e

+

_e

−

_{→ ηπ}

+

_π

−

_{reaction which could also be used for other}

feasibility studies at the KLOE or BESIII [7] experiments.

2

Background

2.1

Beyond the Standard Model

One area of discrepancy between the SM and experiments is the anomalous

muon magnetic moment. It has been very precisely measured at Brookhaven

National Laboratory [8], and the contribution from quantum

electrodynam-ics is known with high accuracy. In the SM, one also expects a significant

contribution from hadronic states to the anomalous muon magnetic moment.

This contribution is difficult to calculate, but can be reliably estimated using

hadronic cross sections in e

+

e

−

collisions. [3] Knowledge of these collision

processes, e.g. e

+

_e

−

_{→ ηπ}

+

_π

−

_{, is therefore important in discerning possible}

physics beyond the SM.

Precision studies of the annihilation of e

+

e

−

into hadrons at low energies,

and in particular the cross sections of specific hadronic channels can provide

insight into the interactions of light quarks and the structure of light hadrons.

This knowledge is in turn necessary for precise tests of the SM [9].

Another possibility for finding exotic contributions beyond the SM is

studying decays of neutral pseudoscalar mesons into a lepton-antilepton pair

such as η, η

0

, η

c

or π

0

into e

+

e

−

. The rare π

0

→ e

+

e

−

decay is of particular

interest from the perspective of searching for physics beyond the SM, as its

experimentally measured branching ratio [4] deviates from the value expected

from the SM [5].

However, studying these decays directly is difficult due to their low

branch-ing ratios. In the case of η

0

and η

c

, examining the formation process via

e

+

e

−

→ η

0

_/η

c

→ ηπ

+

π

−

might therefore competitive alternative to studying

studied at the CMD-3 detector using this particular final state [11].

2.2

DAΦNE Collider

DAΦNE, Double Annular Φ Factory for Nice Experiments, is an electron

positron collider located at the Laboratori Nazionali di Frascati near Rome,

Italy. As the name implies, it was originally built for the study of the φ-meson

as well as the kaons into which it decays. It therefore operates at a center

of mass energy of 1.02 GeV, matching the mass of the φ-meson. DAΦNE

can also be run at 1.00 GeV, but this project focuses on 1.02 GeV, since

that energy is used most frequently. Bunches of electrons and positrons are

accelerated to their final energy; 510 MeV in a linear accelerator whereafter

they are injected into two storage rings. In total, 120 bunches of electrons

and positrons are stored in the rings simultaneously. Once per turn around

the rings, each bunch of electrons collides with one bunch of positrons and

vice versa. There are two interaction points available, one is used for the

DEAR and FINUDA experiments, the other for the KLOE experiment [6].

This project is concerned with the latter.

For the type of studies described in this project, an integrated luminosity

of 250 pb

−1

is obtainable [12].

2.3

KLOE Detector

One of the detectors at DAΦNE is KLOE, with high precision studies of

the extraordinarily long-lived K

L

-meson as its special mission. This purpose

Figure 1: Schematic cross section of the KLOE detector. [13]

By necessity, the design of the detector puts in place certain limits on

what particles can be detected. Firstly, the beam pipes leading the colliding

particles into the detector limits the detection of particles to the angular

region 15

◦

< θ < 165

◦

[6]. However, in order to minimize contributions

from machine background in experiments, a slightly smaller region such as

22

◦

< θ < 158

◦

, used by [14], needs to be chosen.

of particle trajectories should be at least 25 cm. The radius depends on the

velocity of the particle in the transverse plane, and consequently, one can

require a minimum transverse momentum for detection of charged particles.

The minimum transverse momentum required at KLOE is derived below.

2.3.1

Minimum Transverse Momentum

A charged particle moving in the magnetic field of the detector with velocity

~

v experiences a Lorentz force:

~

F = q~

v × ~

B

(1)

The force will cause the particle to move along a circular path with a certain

radius r. The two quantities can be related as:







q~

v × ~

B







=

mv

2 T

r

(2)

The velocity can be seen as having one component in the axial direction, and

one in the transverse direction, v

T

. This allows the simplification of the cross

product to a product of scalars:

qv

T

B =

mv

_T2

r

(3)

In order to be registered in the drift chamber, the radius of the curved track

needs to be larger than the inner radius of the detector, r

i

. From the above

expression, one can derive an expression for the minimum transverse

momen-tum required for the track to have radius larger than r

i

:

qB =

mv

T

r

i

qB =

p

T

r

i

p

T

= qBr

i

(4)

Inserting the inner radius (25 cm) and magnetic field (0.52 T) of the KLOE

detector, one arrives at a minimum transverse momentum:

p

T

= 39

MeV

2.4

The Reaction e

+

e

−

→ ηπ

+

_π

−

This project focuses on the reaction e

+

e

−

→ ηπ

+

_π

−

and the possibility

of studying it with KLOE at DAΦNE based on the background presented

above. Hitherto, the reaction has not been studied at energies as low as the

CMS energy of DAΦNE, but it is of interest to do so as the CMS energy

(1.02 GeV) is close to the threshold energy of the reaction (0.827 GeV). As

mentioned in section 2.1, studies of e

+

e

−

→ hadrons at low energies can yield

knowledge necessary for precise tests of the SM. Since the threshold energy

is the lowest energy at which the reaction can be studied, studies close to it

are desireable. Previous studies at higher energies have been performed at

the CMD-2 detector [9], and at the BABAR detector [15] as an intermediate

state of e

+

_e

−

_{→ π}

+

_π

−

_π

+

_π

−

_π

0

_{, among others.}

When identifying the reaction with the detector, one option is to detect

all products. Another method is to detect all but one of the products and

use their four-momenta to infer the mass of the remaining particle. While π

+

and π

−

can be detected as they are, η has a very short lifetime and decays to

other particles. Therefore, η itself can not be detected, but its decay products

can. The most common decays of η are presented in table 1.

Table 1: The most prominent decay modes of the η-meson and their

respec-tive branching ratios according to Particle Data Group [10].

Decay mode

Branching ratio

η → γγ

39.41 ± 0.20 %

η → π

0

_π

0

_π

0

_{32.68 ± 0.23 %}

η → π

+

π

−

π

0

22.92 ± 0.28 %

η → π

+

_π

−

_γ

4.22 ± 0.08 %

This project focuses on the most prominent decay branch, with η → γγ.

As background, the reaction e

+

_e

−

_{→ π}

+

_π

−

_π

0

_{is chosen. Like η, π}

0

_decays

2.5

Luminosity, Cross section, and Acceptance

A particle accelerator can be characterized by its luminosity, the ratio of the

number of events per unit time to the cross section.

L =

1

σ

dN

dt

(6)

The integrated luminosity gives the ratio of the total number of events over

a given period of time to the cross section:

L

int

=

Z

Ldt

(7)

If the cross section for a reaction is known, the expected number events in

the detector can be calculated as:

N

r

= L

int

σ

r

(8)

where  is the acceptance of the detector, i.e. the percentage of all events

that can be detected after geometric and energetic constraints are imposed.

2.6

Signal Significance

When determining if an observation in particle physics can be considered

significant, one compares the number of events in the signal to the standard

deviation, σ, in the number of background events. If the signal corresponds

to 5σ or more, a significant observation is said to be made [16]. Given a

number of signal events S and a number of background events B, the signal

significance α can be computed as:

α =

√

S

S + B

(9)

3

Method

The study of the reaction e

+

e

−

→ ηπ

+

_π

−

_{conducted in this project is based}

3.1

PHOKHARA

PHOKHARA is a Monte Carlo event generator designed to simulate electron

positron annihilation into hadronic final states at next-to-leading order

ac-curacy. Since its version 8.0 the final state ηπ

+

π

−

is available for study. [17]

A run of the program returns the four-momenta of the final state particles in

an output file, as well as the cross section for the reaction, which is printed in

the terminal window by default. If so desired, it is possible to include initial

state radiation.

3.2

ROOT

ROOT is a data analysis framework developed at CERN since 1994 for

effi-ciently handling large amounts of data from high energy physics. The

frame-work is based on C++ and is therefore object-oriented. This facilitates the

writing of code that is general and reusable. [18]

The features of ROOT that are used in this project are, most importantly,

the class TLorentzVector for handling and manipulating four-vectors, the

class TGenPhaseSpace for generating particle decays from phase space, as

well as its functions for creating and plotting histograms.

3.3

Test of the Simulations

The e

+

e

−

→ ηπ

+

_π

−

_{cross section calculated by PHOKHARA has been tested}

(GeV) c.m. E 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 )(nb) η -_π +_π → -e + (e σ 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 PHOKHARA BaBar

Figure 2: Comparison of experimental and simulated e

+

_e

−

_{→ ηπ}

+

_π

−

_cross

sections.

3.4

Execution

Initially, the reactions e

+

_e

−

_{→ ηπ}

+

_π

−

_{and e}

+

_e

−

_{→ π}

+

_π

−

_π

0

_{were simulated}

with PHOKHARA at the center of mass energy of DAΦNE. In total, three

runs with increasing event counts, 200 000, 2 million, and 20 million, were

made. In all three cases, the calculated cross section was recorded and the

output file containing the four-momenta of the particles saved. Subsequently,

PHOKHARA was run with 2 million events for energies from 1.0250 GeV to

2.9750 GeV in order the compare the calculated cross sections with

experi-mental results from the BaBar detector (see section 3.3).

Thereafter, the analysis program based on ROOT was written. The

func-tions of the program were implemented one at a time, and tested before

pro-ceeding. For the tests, the data files with lowest event counts were used in

order to minimize the runtime.

Firstly, the data files from PHOKHARA are read and the four-momenta

of the particles are stored as TLorentzVectors. Thereafter, a check is

per-formed to see if the particles could be detected given the geometrical and

energitcal constraints at KLOE. For η and π

0

_{, separate methods handle the}

in table 1 are included. This is done in order to facilitate future studies of

all final states of e

+

e

−

→ ηπ

+

_π

−

_{and their corresponding backgrounds.}

Information about the particles is stored in histograms. All histograms

are normalized to the cross section of the reaction that they correspond to

and multiplied with the luminosity to give the number of events. This is

done to ensure that the number of events from the signal and background

are comparable. Following this normalization, the histograms are plotted.

Finally, the detector acceptances for the various branches are calculated. For

testing purposes, the possibility of printing individual variables is included.

With a program that can read and manipulate the desired data in place,

cuts were imposed on the background in order to discard those e

+

_e

−

_→

π

+

π

−

π

0

events that could not be mistaken for e

+

e

−

→ ηπ

+

_π

−

_{events (see}

section 3.5).

3.5

Cuts on the Background

Several methods for minimizing the number of background events are

evalu-ated in this thesis. The first is to impose an invariant mass window on the

π

+

_π

−

_{-pairs from the π}

+

_π

−

_π

0

_{final state. Only pairs with an invariant mass}

in the same range as those from the final state ηπ

+

_π

−

_{are selected.}

Further-more, only those events for which the energy of the photons from the decay

of the π

0

_{lie in the same range as that of the photons from the η are kept.}

]

2

) [GeV/c

-π

+

π

W(

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of Events

0

10

20

30

40

50

60

70

-π + π

Another method is to view all events as a reaction e

+

_e

−

_{→ π}

+

_π

−

_{X, where}

the particle X may be either an η or a π

0

. The magnitude squared of the

four-momentum, p

X

, of the particle X can be computed as:

|p

X

|

2

= |p

0

− p

π+

− p

_π−

|

2

(10)

where, p

0

is the momentum of the electron-positron pair. The

four-momentum squared is by definition equal to the mass of the particle squared.

Thus, for all detectable π

+

π

−

-pairs, one may compute the corresponding

mass of the third particle and compare it to the mass of the η-meson. Only

events that lie within 2 MeV of the actual η mass are kept as candidates.

For these studies, a simulation of the resolution of the KLOE detector is

introduced. As stated in section 2.3, the resolution for measurements of

charged particle momenta at KLOE is 0.4 %. To reflect this, the momenta

of detectable π

+

and π

−

from the simulation are smeared according to the

following formula:

p

smeared

= p

generated

∗ (1 + N (0, 0.004))

(11)

where N(0, 0.004) is a random number from a normal distribution with mean

0 and σ = 0.004.

3.6

Design of the Analysis Program

This section describes the design of the analysis program and explain the

choices made about its structure. For the code itself, see appendix A.

The philosophy behind the design of the code used in this project has

been to first implement all necessary functionality, one function at a time,

so as to arrive at a working code at the earliest possible stage. When such a

stage was reached, the process of making the code more general and reusable

was started.

created, by reading an input file containing filenames for the files where

PHOKHARA data is stored, the calculated cross sections corresponding to

the reactions studied, the constraints of the detector that is to be studied,

as well as the desired luminosity. See appendix B for an example illustrating

the structure of this file.

The functionality described above is implemented as instance methods.

This is done to facilitate the storing of data, as it allows all interesting

infor-mation that is collected throughout the simulation to be stored in instance

variables.

4

Result

4.1

Particle Distributions for the Signal

Figures 4 through 16 show distributions for particles from the signal.

E [GeV]

0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Number of Events

0

5

10

15

20

25

30

35

40

45

+ π

Figure 4: Difference between total and recorded number of π

+

_{from e}

+

_e

−

_→

)

θ

cos(

1

−

−

0.8

−

0.6

−

0.4

−

0.2

0

0.2

0.4

0.6

0.8

1

Number of Events

0

2

4

6

8

10

+

π

Angular Distribution of

Figure 5: Angular distribution of recorded (blue) and total number (red) of

π

−

from e

+

_e

−

_{→ ηπ}

+

_π

−

_.

E [GeV]

0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Number of Events

0

2

4

6

8

10

12

14

16

18

+ π

Figure 6: Difference between total and recorded number of π

+

_{from e}

+

_e

−

_→

ηπ

+

π

−

→ π

+

_π

−

_{γγ. The full line represents the total number of π}

+

_{, and the}

E [GeV]

0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Number of Events

0

5

10

15

20

25

30

35

40

45

-π

Figure 7: Difference between total and recorded number of π

−

from e

+

_e

−

_→

ηπ

+

_π

−

_{. The full line represents the total number of π}

−

_{, and the dashed line}

the recorded π

−

.

)

θ

cos(

1

−

−

0.8

−

0.6

−

0.4

−

0.2

0

0.2

0.4

0.6

0.8

1

Number of Events

0

2

4

6

8

10

-π

Angular Distribution of

E [GeV]

0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Number of Events

0

1

2

3

4

5

6

7

8

γ γ

Figure 9: Difference between total and recorded number of γ

1

(blue) and γ

2

(red) from e

+

_e

−

_{→ ηπ}

+

_π

−

_{→ π}

+

_π

−

_{γγ. The full lines represent the total}

number of events, and the dashed lines the recorded events.

)

θ

cos(

1

−

−

0.8

−

0.6

−

0.4

−

0.2

0

0.2

0.4

0.6

0.8

1

Number of Events

0

0.5

1

1.5

2

2.5

3

γ

Angular Distribution of

]

2

) [GeV/c

-π

+

π

W(

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of Events

0

10

20

30

40

50

60

70

-π + π

Invariant mass distribution of

Figure 11: Invariant mass distribution for π

+

_π

−

_{from e}

+

_e

−

_{→ ηπ}

+

_π

−

_.

) [GeV]

+

π

E(

0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

) [GeV]

-π

E(

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 10 20 30 40 50 60 70 80 0 100 200 300 400 500 600 700 800 900 -π vs + π

Figure 12: π

+

_{energies versus π}

−

_{energies. The group to the left corresponds}

) [GeV]

+

π

E(

0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

) [GeV]

-π

E(

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 10 20 30 40 50 60 0 100 200 300 400 500 600 -π vs + π

Figure 13: π

+

_{energies versus π}

−

_{energies for detected π}

+

_π

−

_{. The group to}

the left corresponds to the signal and the right group to the background.

) [GeV]

+

π

E(

0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Number of events

0

5

10

15

20

25

30

35

) [GeV]

-π

E(

0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Number of events

0

5

10

15

20

25

30

35

Figure 15: Distribution of smeared π

−

from the signal.

]

2

/c

2

[GeV

2

|

X

|p

0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Number of Events

1 −

10

1

10

2

10

3

10

4

10

5

10

6

10

7

10

8

10

Figure 16: Calculated |p

X

|

2

with smearing from the signal from the (blue) and

] 2 /c 2 [GeV 2 | X |p 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Number of Events 1 − 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10

Figure 17: Calculated |p

X

|

2

without smearing from the signal (blue) and from

4.2

Particle Distributions for the Background

Figures 18 through 27 show distributions for particles from the background.

E [GeV]

0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Number of Events

0

500

1000

1500

2000

2500

3000

3

10

×

+ π

Figure 18: Difference between total and recorded number of π

+

from e

+

e

−

→

π

+

_π

−

_π

0

_{. The full line represents the total number of π}

+

_{, and the dashed line}

)

θ

cos(

1

−

−

0.8

−

0.6

−

0.4

−

0.2

0

0.2

0.4

0.6

0.8

1

Number of Events

0

200

400

600

800

1000

1200

1400

1600

3

10

×

+

π

Angular distribution for background

Figure 19: Angular distribution of recorded (blue) and total number (red) of

π

+

_{from e}

+

_e

−

_{→ π}

+

_π

−

_π

0

_.

E [GeV]

0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Number of Events

0

500

1000

1500

2000

2500

3000

3

10

×

-π

Figure 20: Difference between total and recorded number of π

−

from e

+

e

−

→

π

+

_π

−

_π

0

_{. The full line represents the total number of π}

−

_{, and the dashed line}

)

θ

cos(

1

−

−

0.8

−

0.6

−

0.4

−

0.2

0

0.2

0.4

0.6

0.8

1

Number of Particles

0

200

400

600

800

1000

1200

1400

1600

3

10

×

-π

Angular distribution for background

Figure 21: Angular distribution of recorded (blue) and total number (red) of

π

−

from e

+

_e

−

_{→ π}

+

_π

−

_π

0

_.

E [GeV] 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Number of Events 0 500 1000 1500 2000 2500 3000 3 10 × 0 π

Figure 22: Difference between total and recorded number of π

0

from e

+

e

−

→

π

+

_π

−

_π

0

_{. The full line represents the total number of π}

0

_{, and the dashed line}

E [GeV] 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Number of Events 0 200 400 600 800 1000 1200 1400 1600 1800 2000 3 10 ×

γ

γ

Background

Figure 23: Difference between total and recorded number of γ

1

(blue) and

γ

2

(red) from π

0

→ γγ. The full lines represents the total number, and the

dashed lines the recorded number.

] 2 ) [GeV/c -π + π W( 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Number of Events 0 500 1000 1500 2000 2500 3000 3500 3 10 ×

-π

+

π

Invariant mass distribution for background

) [GeV]

+

π

E(

0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Number of events

0

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

2400

3

10

×

Figure 25: Distribution of smeared π

+

_{from the background.}

) [GeV]

-π

E(

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Number of events

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 3 10 ×

] 2 /c 2 [GeV 2 | X |p 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Number of Events 0 10 20 30 40 50 60 6 10 ×

Figure 27: Calculated |p

X

|

2

from the background with smearing.

4.3

Cross Sections

The cross sections computed by PHOKHARA are given in table 2.

Table 2: Cross sections for the signal and background reactions computed

by PHOKHARA, and the corresponding expected number of events with

integrated luminosity 250 pb

−

1.

Even though they are not used in this

project, cross sections for 1.00 GeV are included for comparison.

Reaction

Cross section [nb]

Expected number of events

e

+

e

−

→ ηπ

+

_π

−

_{at 1.02 GeV}

_{2.9701 ·10}

−3

_{± 1.4 ·10}

−6

_{742 ± 0.35}

e

+

_e

−

_{→ ηπ}

+

_π

−

_{at 1.00 GeV}

_{2.0299 ·10}

−3

_{± 3.0 ·10}

−6

_{507 ± 0.75}

e

+

_e

−

_{→ π}

+

_π

−

_π

0

_{at 1.02 GeV}

_{421.78 ± 0.38}

_{105445000 ± 96000}

e

+

e

−

→ π

+

_π

−

_π

0

_{at 1.00 GeV}

_{27.256 ± 0.064}

_{6814000 ± 16000}

4.4

Acceptances

In table 3, the detector acceptance, calculated as

Ndetected

Ntotal

, is given for all

Table 3: Calculated detector acceptances for e

+

_e

−

_{→ ηπ}

+

_π

−

_{with different}

decays of η as well as for e

+

e

−

→ π

+

_π

−

_π

0

_.

Branch

Acceptance

e

+

_e

−

_{→ ηπ}

+

_π

−

_{59.63 %}

e

+

e

−

→ ηπ

+

_π

−

_{→ π}

+

_π

−

_γγ

_{67.96 %}

e

+

_e

−

_{→ ηπ}

+

_π

−

_{→ π}

+

_π

−

_π

0

_π

0

_π

0

_{61.60 %}

e

+

_e

−

_{→ ηπ}

+

_π

−

_{→ π}

+

_π

−

_π

+

_π

−

_π

0

_{55.50 %}

e

+

e

−

→ ηπ

+

_π

−

_{→ π}

+

_π

−

_π

+

_π

−

_γ

_{0.00 %}

e

+

_e

−

_{→ π}

+

_π

−

_π

0

_{70.40 %}

4.5

Number of Events

In table 4 below, the number of events, both total and detected are

pre-sented for all branches at. Table 5 gives the total and detected number of

background events after the invariant mass cut discussed in section 3.5 has

been imposed. Table 6 gives the total and detected number of events with the

cut on π

+

_π

−

_{-pair momenta imposed as described in section 3.5. All numbers}

given are for center of mass energy 1.02 GeV.

Table 4: Number of total and detected events for e

+

e

−

→ ηπ

+

_π

−

_with

different decays of η as well as for e

+

_e

−

_{→ π}

+

_π

−

_π

0

_{with integrated luminosity}

250 pb

−1

.

Branch

Total events

Detected events

e

+

_e

−

_{→ ηπ}

+

_π

−

₇₄₄

₄₄₃

e

+

_e

−

_{→ ηπ}

+

_π

−

_{→ π}

+

_π

−

_γγ

₂₉₄

₁₉₉

e

+

e

−

→ ηπ

+

_π

−

_{→ π}

+

_π

−

_π

0

_π

0

_π

0

₂₄₂

₁₄₉

e

+

_e

−

_{→ ηπ}

+

_π

−

_{→ π}

+

_π

−

_π

+

_π

−

_π

0

₁₇₁

₉₅

e

+

_e

−

_{→ ηπ}

+

_π

−

_{→ π}

+

_π

−

_π

+

_π

−

_γ

₃₇

₀

e

+

_e

−

_{→ π}

+

_π

−

_π

0

_{105.5 ·10}

6

_{74.29 ·10}

6

Table 5: Number of total detected background events after a cut on π

+

π

−

-pair invariant mass has been imposed.

Branch

Total events

Detected events

Table 6: Number of Events after the cut on the momenta of π

+

_π

−

_{-pairs has}

been imposed, with and without smearing. For each number, the statistical

error is given.

Branch

Events with smearing

Events without smearing

e

+

_e

−

_{→ ηπ}

+

_π

−

_{379 ± 19}

_{457 ± 21}

e

+

e

−

→ ηπ

+

_π

−

_{→ π}

+

_π

−

_γγ

_{150 ± 12}

_{180 ± 13}

e

+

_e

−

_{→ π}

+

_π

−

_π

0

_{3969 ± 63}

_{2977 ± 54}

Using the number of events from table 6, the signal significance of the e

+

e

−

→

ηπ

+

_π

−

_{→ π}

+

_π

−

_{γγ channel is computed to be α =}

_√S S+B

=

150 √ 150+3969

= 2.4σ.

5

Discussion

The results of the simulation, table 4 shows that an experiment with KLOE

at DAΦNE with 250 pb

−1

will yield a fairly high number of e

+

_e

−

_{→ ηπ}

+

_π

−

events. There is however a large background that needs to be reduced.

Ta-ble 5 shows that invariant mass cut, while it has a noticaTa-ble effect, does not

allow one to discard enough of the background.

Constraining the |p

X

|

2

= |p

0

− p

π+

− p

_π−

|

2

on the other hand allows for

much better discrimination of the background, as indicated by table 6. Still

however, the signal significance of falls below the minimum of 5σ and is

therefore not enough to claim detection.

It can be noted that the number of background events does not go to

zero without smearing, as one might expect, implying that the only a part

of the background stems from events in the π

0

_{mass range being}

mismea-sured. Rather, as can be seen in figures 16 and 17, both the signal and the

background exhibit significant "tails" with values of |p

2

X

| higher than m

2η

and

m

2

π0

respectively. These are caused by radiative processes, where an initial

state radiation photon is emitted by either the electron or positron before

the collision. Through further analysis, it might be possible to achieve

bet-ter background discrimination. Firstly, one needs to debet-termine the energy

distribution of these additional photons, most importantly those that are

de-tectable. If the energies of the photons corresponding to π

+

_π

−

_{-pairs in the}

m

2

total energy of 1.02 GeV. This way, one could possibly reduce the number of

background events that could be mistaken for e

+

e

−

→ ηπ

+

_π

−

_events.

The error resulting from the analysis stems only from the error in the

cross section calculated by PHOKHARA (see table 2) and is therefore small.

Figure 2 also shows that there is a fairly good correspondence between the

experimentally determined and calculated cross section for e

+

e

−

→ ηπ

+

_π

−

.

For a more complete picture of the accuracy of the simulations, a similar

comparison could be made for the cross section of e

+

_e

−

_{→ π}

+

_π

−

_π

0

_.

6

Recommendations

The first step in further investigations into the possibility of detecting the

reaction e

+

e

−

→ ηπ

+

_π

−

_{at DAΦNE must be to examine possibility of}

dis-carding background events from radiative processes. Furthermore, the

back-ground for the remaining decays of the η-meson should be studied. Of the

decays shown in table 1, the background was only examined for η → γγ. At

least two additional decays, η → π

0

π

0

π

0

and η → π

+

π

−

π

0

, have significant

branching ratios. For a complete picture of the detectability of the reaction,

studies of the background for these final states are necessary.

The study conducted in this project was carried out for a center of mass

energy of 1.02 GeV. However, DAΦNE could also be run at 1.00 GeV. As can

be seen in table 2 the cross section for the background reaction is considerably

lower at 1.00 GeV than at 1.02 GeV, while the cross section for the signal

remains approximately the same. Therefore, investigating the detectability

of the reaction at 1.00 GeV is interesting prospect.

7

Conclusion

In this thesis, the feasibility of studying the reaction e

+

_e

−

_{→ ηπ}

+

_π

−

_with

The results of the simulations show a signal significance of 2.4σ for e

+

_e

−

_→

ηπ

+

π

−

→ π

+

_π

−

_{γγ with e}

+

_e

−

_{→ π}

+

_π

−

_π

0

_{as background. Therefore, a certain}

References

[1] B.R. Martin and G. Shaw. Particle Physics. Wiley, The Atrium,

South-ern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom, 3

edi-tion, 2008.

[2] Serguei Chatrchyan et al. Search for the standard model Higgs boson

produced in association with a W or a Z boson and decaying to bottom

quarks. Phys.Rev., D89(1):012003, 2014.

[3] A. Hoecker and Marciano W.J. The Muon Anomalous Magnetic

Mo-ment. 2013.

[4] E. et al. Abouzaid. Measurement of the rare decay π

0

→ e

+

_e

−

_{. Phys.}

Rev. D, 75:012004, Jan 2007.

[5] Alexander E. Dorokhov and Mikhail A. Ivanov. Rare decay π

0

_{→ e}

+

_e

−

_:

Theory confronts ktev data. Phys. Rev. D, 75:114007, Jun 2007.

[6] F. Bossi, E. De Lucia, J. Lee-Franzini, S. Miscetti, and M. Palutan.

Pre-cision Kaon and Hadron Physics with KLOE. Riv.Nuovo Cim., 31:531–

623, 2008.

[7] D.M. Asner, T. Barnes, J.M. Bian, I.I. Bigi, N. Brambilla, et al. Physics

at BES-III. Int.J.Mod.Phys., A24:S1–794, 2009.

[8] G.W. Bennett et al. Final Report of the Muon E821 Anomalous

Mag-netic Moment Measurement at BNL. Phys.Rev., D73:072003, 2006.

[9] R.R. Akhmetshin et al. Study of the process e+ e- —> pi+ pi+

pi-pi0 with CMD-2 detector. Phys.Lett., B489:125–130, 2000.

[10] K.A. Olive et al. Review of Particle Physics. Chin.Phys., C38:090001,

2014.

[11] R.R. Akhmetshin et al. Search for the process e

+

e

−

→ η

0

_{(958) with the}

CMD-3 detector. Phys.Lett., B740:273–277, 2015.

[13] KLOE in Pictures. http://www.lnf.infn.it/kloe/. [Accessed:

2015-05-27].

[14] F. Ambrosino et al. Study of the process e

+

e

−

→ ωπ

0

_{in the φ-meson}

mass region with the KLOE detector. Phys.Lett., B669:223–228, 2008.

[15] Bernard Aubert et al. The e+ e- —> 2(pi+ pi-) pi0, 2(pi+ pi-) eta, K+

K- pi+ pi- pi0 and K+ K- pi+ pi- eta Cross Sections Measured with

Initial-State Radiation. Phys.Rev., D76:092005, 2007.

[16] Pekka K. Sinervo. Signal significance in particle physics. pages 64–76,

2002.

[17] H. Czyż, M. Gunia, and J.H. Kühn. Simulation of electron-positron

annihilation into hadrons with the event generator PHOKHARA. JHEP,

1308:110, 2013.

A

Code

Below, the code used in the simulations is given.

A.1

Main Class EtaPiPi.C

f g e t s ( line , 50 , f ) ; s s c a n f ( line , " % lf " , & m a x A n g l e ) ; f g e t s ( header , 60 , f ) ; f g e t s ( line , 50 , f ) ; s s c a n f ( line , " % lf " , & m i n T r a n M o m ) ; f g e t s ( header , 50 , f ) ; f g e t s ( line , 50 , f ) ; s s c a n f ( line , " % lf " , & l u m i n o s i t y ) ; f c l o s e ( f ) ; // I n i t i a l i z e C o u n t e r s n E v e n t s E t a P i P i =0 , n D e t E t a P i P i =0 , n E v e n t s P i P i P i =0 , n D e t P i P i P i =0; n E v e n t s 2 p h = n E v e n t s 3 p i 0 = n E v e n t s P p P m P 0 = n E v e n t s P p P m P h = n D e t 2 p h = n D e t 3 p i 0 = 0; n D e t P p P m P 0 = n D e t P p P m P h = n P i P i P a i r D e t B g = n P i P i P a i r D e t M a i n = n E t a P i P i D e t P o s t C u t = n P i P i P i D e t P o s t C u t = 0; n E t a P i P i 2 p h D e t P o s t C u t = n E t a P i P i N o S m e a r = n P i P i P i N o S m e a r = n E t a P i P i N o S m e a r D e t = n P i P i P i N o S m e a r D e t = n E t a 2 p h N o S m e a r D e t = n P i P i P a i r D e t 2 p h = n P i P i P a i r D e t 2 p h N o S m e a r = 0; n E v e n t s I n v M a s s = n D e t I n v M a s s = 0; // I n i t i a l i z e h i s t o g r a m s gStyle - > S e t O p t S t a t (0) ; h _ m a i n _ p p = new T H 1 D ( " # pi ^ { + } " , " " , 100 , 0 , 0 . 5 ) ; h _ m a i n _ p p _ d e t = new T H 1 D ( " # pi ^ { + } _ { det } " , " " , 100 , 0 , 0 . 5 ) ; h _ m a i n _ p p _ e t a 2 p h = new T H 1 D ( " " , " " , 100 , 0 , 0 . 5 ) ; h _ m a i n _ p p _ e t a 2 p h _ d e t = new T H 1 D ( " " , " " , 100 , 0 , 0 . 5 ) ; h _ m a i n _ p p _ t h e t a = new T H 1 D ( " cos (# t h e t a _ {# pi ^ { + } } ) " , " " , 100 , -1 , 1) ; h _ m a i n _ p p _ t h e t a _ d e t = new T H 1 D ( " cos (# t h e t a _ {# pi ^ { + } _ { det }}) " , " " , 100 ,

-1 , 1) ;

h _ m a i n _ p m = new T H 1 D ( " # pi ^{ -} " , " " , 100 , 0 , 0 . 5 ) ;

h _ m a i n _ p m _ d e t = new T H 1 D ( " # pi { -} _ { det } " , " " , 100 , 0 , 0 . 5 ) ;

h _ m a i n _ p m _ t h e t a = new T H 1 D ( " cos (# t h e t a _ {# pi ^{ -}}) " , " " , 100 , -1 , 1) ; h _ m a i n _ p m _ t h e t a _ d e t = new T H 1 D ( " cos (# t h e t a _ {# pi ^{ -} _ { det }}) " , " " , 100 ,

-1 , 1) ;

h _ m a i n _ g a m m a = new T H 1 D ( " # g a m m a _ 1 " , " " , 100 , 0 , 0 . 5 ) ;

h _ m a i n _ g a m m a _ d e t = new T H 1 D ( " # g a m m a _ {1 , det } " , " " , 100 , 0 , 0 . 5 ) ;

h _ m a i n _ g a m m a _ t h e t a = new T H 1 D ( " cos (# t h e t a _ {# g a m m a }) " , " " , 100 , -1 , 1) ; h _ m a i n _ g a m m a _ t h e t a _ d e t = new T H 1 D ( " cos (# t h e t a _ {# g a m m a _ { det }}) " , " " , 100 ,

D o u b l e _ t w e i g h t = e t a P p P m P h . G e n e r a t e () ; T L o r e n t z V e c t o r * Pp = e t a P p P m P h . G e t D e c a y (0) ; T L o r e n t z V e c t o r * Pm = e t a P p P m P h . G e t D e c a y (1) ; T L o r e n t z V e c t o r * Ph = e t a P p P m P h . G e t D e c a y (2) ; if ( D . c h e c k D e t C h a r g e d (* Pp ) && D . c h e c k D e t C h a r g e d (* Pm ) && D . c h e c k D e t (* Ph ) ) { r e t u r n = t r u e ; } } r e t u r n f a l s e ; } // D e c a y of the pi0 - m e s o n B o o l _ t E t a P i P i S i m :: p i 0 D e c a y ( T L o r e n t z V e c t o r pi0 ) { D o u b l e _ t r a n d = gRandom - > U n i f o r m (0.0 , 1 . 0 ) ; // pi0 - > g a m m a g a m m a if ( r a n d <= 0 . 9 8 8 2 3 ) { p i 0 2 p h . S e t D e c a y ( pi0 , 2 , d m 2 p h ) ; D o u b l e _ t w e i g h t = p i 0 2 p h . G e n e r a t e () ; T L o r e n t z V e c t o r * ph1 = p i 0 2 p h . G e t D e c a y (0) ; T L o r e n t z V e c t o r * ph2 = p i 0 2 p h . G e t D e c a y (1) ; h _ b g 1 _ g a m m a _ 1 . F i l l ( ph1 - > E () ) ; h _ b g 1 _ g a m m a _ 2 . F i l l ( ph2 - > E () ) ;

if ( ph1 - > E () >= 0 . 1 5 && ph1 - > E () <=0.45 && ph2 - > E () >=0.15 && ph2 - > E () <= 0 . 4 5 ) { p h I n R a n g e = t r u e ; } if ( D . c h e c k D e t (* ph1 ) ) { h _ b g 1 _ g a m m a _ 1 _ d e t . F i l l ( ph1 - > E () ) ; } if ( D . c h e c k D e t (* ph2 ) ) { h _ b g 1 _ g a m m a _ 2 _ d e t . F i l l ( ph2 - > E () ) ; } if ( D . c h e c k D e t (* ph1 ) && D . c h e c k D e t (* ph2 ) ) { r e t u r n t r u e ; } } r e t u r n f a l s e ; } // M e t h o d for r e a d i n g the e t a p i p i f i l e . v o i d E t a P i P i S i m :: r e a d E t a P i P i () { c h a r h e a d [11] , lin [ 1 1 4 ] ;

T L o r e n t z V e c t o r pp , pm , eta , pXSmear , pX , v , ppSmear , p m S m e a r ; B o o l _ t ppDet , pmDet , e t a D e t ;

p I n i t . S e t P x P y P z E ( 0 . , 0 . , 0 . , 1 . 02 ) ; T R a n d o m r a n d G e n ;

{ p r i n t f ( " E r r o r : ␣ No ␣ s u c h ␣ f i l e ␣ in ␣ d i r e c t o r y " ) ; e x i t ( -1) ; } r = f s c a n f ( f , " % s " , lin ) ; w h i l e ( r != EOF ) { p p D e t = p m D e t = e t a D e t = f a l s e ; D o u b l e _ t E , px , py , pz ; // P h o t o n 1:

f s c a n f ( f , " %10 s % lf % lf % lf % lf " , head , & E , & px , & py , & pz ) ; // P h o t o n 2:

f s c a n f ( f , " %10 s % lf % lf % lf % lf " , head , & E , & px , & py , & pz ) ; // Eta :

f s c a n f ( f , " %10 s % lf % lf % lf % lf " , head , & E , & px , & py , & pz ) ; eta . S e t P x P y P z E ( px , py , pz , E ) ;

// Pi +

f s c a n f ( f , " %10 s % lf % lf % lf % lf " , head , & E , & px , & py , & pz ) ; pp . S e t P x P y P z E ( px , py , pz , E ) ;

// Pi

e l s e if ( e t a D e c P p P m P h ) { n D e t P p P m p h ++; } } v = pp + pm ; h _ m a i n _ i n v m a s s . F i l l ( v . M () ) ; h _ m a i n _ p p . F i l l ( pp . E () ) ; h _ m a i n _ p p _ t h e t a . F i l l ( T M a t h :: Cos ( pp . T h e t a () ) ) ; h _ m a i n _ p m . F i l l ( pm . E () ) ; h _ m a i n _ p m _ t h e t a . F i l l ( T M a t h :: Cos ( pm . T h e t a () ) ) ; r = f s c a n f ( f , " % s " , lin ) ; } f c l o s e ( f ) ; } // M e t h o d for r e a d i n g the p i p i p i f i l e . v o i d E t a P i P i S i m :: r e a d P i P i P i () { c h a r h e a d [11] , lin [ 1 1 4 ] ; B o o l _ t p0Det , ppDet , p m D e t ; T L o r e n t z V e c t o r p0 , pp , pm , pX , pXSmear , v , ppSmear , p m S m e a r ; p I n i t . S e t P x P y P z E ( 0 . , 0 . , 0 . , 1 . 02 ) ; T R a n d o m r a n d G e n ; I n t _ t r ; F I L E * f = f o p e n ( f i l e n a m e 2 , " r " ) ; if ( f == N U L L ) { p r i n t f ( " E r r o r : ␣ No ␣ s u c h ␣ f i l e ␣ in ␣ d i r e c t o r y " ) ; e x i t ( -1) ; } r = f s c a n f ( f , " % s " , lin ) ; w h i l e ( r != EOF ) { p p D e t = p m D e t = p 0 D e t = p h I n R a n g e = f a l s e ; D o u b l e _ t E , px , py , pz ; // P h o t o n 1:

f s c a n f ( f , " %10 s % lf % lf % lf % lf " , head , & E , & px , & py , & pz ) ; // P h o t o n 2:

f s c a n f ( f , " %10 s % lf % lf % lf % lf " , head , & E , & px , & py , & pz ) ; // Pi +:

f s c a n f ( f , " %10 s % lf % lf % lf % lf " , head , & E , & px , & py , & pz ) ; pp . S e t P x P y P z E ( px , py , pz , E ) ;

// Pi

-f s c a n -f ( -f , " %10 s % l-f % l-f % l-f % l-f " , head , & E , & px , & py , & pz ) ; pm . S e t P x P y P z E ( px , py , pz , E ) ;

// Pi0

r e t u r n D o u b l e _ t ( n D e t E t a P i P i ) / D o u b l e _ t ( n E v e n t s E t a P i P i ) * 1 0 0 ; } D o u b l e _ t E t a P i P i S i m :: g e t A c c P i P i P i () { r e t u r n D o u b l e _ t ( n D e t P i P i P i ) / D o u b l e _ t ( n E v e n t s P i P i P i ) * 1 0 0 ; } D o u b l e _ t E t a P i P i S i m :: g e t A c c 2 p h () { r e t u r n D o u b l e _ t ( n D e t 2 p h ) / D o u b l e _ t ( n E v e n t s 2 p h ) * 1 0 0 ; } D o u b l e _ t E t a P i P i S i m :: g e t A c c 3 p i 0 () { r e t u r n D o u b l e _ t ( n D e t 3 p i 0 ) / D o u b l e _ t ( n E v e n t s 3 p i 0 ) * 1 0 0 ; } D o u b l e _ t E t a P i P i S i m :: g e t A c c P p P m P 0 () { r e t u r n D o u b l e _ t ( n D e t P p P m P 0 ) / D o u b l e _ t ( n E v e n t s P p P m P 0 ) * 1 0 0 ; } D o u b l e _ t E t a P i P i S i m :: g e t A c c P p P m P h () { r e t u r n D o u b l e _ t ( n D e t P p P m P h ) / D o u b l e _ t ( n E v e n t s P p P m P h ) * 1 0 0 ; } // M e t h o d s for g e t t i n g the v a l u e s of q u a n t i t i e s r e a d f r o m i n p u t f i l e . D o u b l e _ t E t a P i P i S i m :: g e t C s E t a P i P i () { r e t u r n c s E t a P i P i ; } D o u b l e _ t E t a P i P i S i m :: g e t C s P i P i P i () { r e t u r n c s P i P i P i ; } D o u b l e _ t E t a P i P i S i m :: g e t L u m i n o s i t y () { r e t u r n l u m i n o s i t y ; } D o u b l e _ t E t a P i P i S i m :: g e t M i n T r a n M o m () { r e t u r n m i n T r a n M o m ; }

D o u b l e _ t E t a P i P i S i m :: g e t P i P i 2 p h D e t () { r e t u r n n D e t 2 p h * c s E t a P i P i / n E v e n t s E t a P i P i * l u m i n o s i t y ; } D o u b l e _ t E t a P i P i S i m :: g e t P i P i 2 p h D e t E r r o r () { r e t u r n n D e t 2 p h * c s E t a P i P i E r r o r / n E v e n t s E t a P i P i * l u m i n o s i t y ; } // M e t h o d u s e d for r u n n i n g the e n t i r e s i m u l a t i o n v o i d E t a P i P i S i m :: r u n S i m () { p r i n t f ( " R e a d i n g ␣ eta ␣ pi + ␣ pi - ␣ f i l e . ␣ \ n " ) ; this - > r e a d E t a P i P i () ; p r i n t f ( " eta ␣ pi + ␣ pi - ␣ f i l e ␣ r e a d ␣ s u c c e s f u l l y . ␣ \ n " ) ; p r i n t f ( " R e a d i n g ␣ pi0 ␣ pi + ␣ pi - ␣ f i l e . ␣ \ n " ) ; this - > r e a d P i P i P i () ; p r i n t f ( " pi0 ␣ pi + ␣ pi - ␣ f i l e ␣ r e a d ␣ s u c c e s s f u l l y . ␣ \ n " ) ; this - > n o r m a l i z e () ; this - > p l o t () ;

p r i n t f ( " T o t a l ␣ d e t e c t o r ␣ a c c e p t a n c e ␣ for ␣ eta ␣ pi + ␣ pi -: ␣ % f ␣ % ␣ \ n " , this - > g e t A c c E t a P i P i () ) ;

p r i n t f ( " T o t a l ␣ d e t e c t o r ␣ a c c e p t a n c e ␣ for ␣ pi0 ␣ pi + ␣ pi -: ␣ % f ␣ % ␣ \ n " , this - > g e t A c c P i P i P i () ) ;

p r i n t f ( " D e t e c t o r ␣ a c c e p t a n c e ␣ for ␣ eta ␣ pi + ␣ pi - ␣ - - > ␣ 2 g a m m a ␣ pi + ␣ pi -: ␣ % f ␣ % ␣ \ n " , this - > g e t A c c 2 p h () ) ;

p r i n t f ( " D e t e c t o r ␣ a c c e p t a n c e ␣ for ␣ eta ␣ pi + ␣ pi - ␣ - - > ␣ 3 pi0 ␣ pi + ␣ pi -: ␣ % f ␣ % ␣ \ n " , this - > g e t A c c 3 p i 0 () ) ;

p r i n t f ( " N u m b e r ␣ of ␣ e + ␣ e - ␣ - > ␣ pi + ␣ pi - ␣ pi0 ␣ e v e n t s : ␣ % f ␣ \ n " , h _ b g 1 _ p p . I n t e g r a l () ) ; p r i n t f ( " N u m b e r ␣ of ␣ d e t e c t e d ␣ e + ␣ e - ␣ - > ␣ pi + ␣ pi - ␣ pi0 ␣ e v e n t s : ␣ % f ␣ \ n " , n D e t P i P i P i * c s P i P i P i / n E v e n t s P i P i P i * l u m i n o s i t y ) ; p r i n t f ( " N u m b e r ␣ of ␣ e + ␣ e - ␣ - > ␣ pi + ␣ pi - ␣ pi0 ␣ e v e n t s ␣ in ␣ i n t e r e s t i n g ␣ i n v a r i a n t ␣ m a s s ␣ r a n g e : ␣ % f ␣ + - ␣ % f ␣ \ n " , h _ b g 1 _ i n v m a s s _ c u t . I n t e g r a l () , n E v e n t s I n v M a s s * c s P i P i P i E r r o r / n E v e n t s P i P i P i * l u m i n o s i t y ) ; p r i n t f ( " N u m b e r ␣ of ␣ e + ␣ e - ␣ - > ␣ pi + ␣ pi - ␣ pi0 ␣ e v e n t s ␣ d e t e c t e d ␣ ␣ in ␣ i n t e r e s t i n g ␣ i n v a r i a n t ␣ m a s s ␣ r a n g e : ␣ % f ␣ + - ␣ % f ␣ \ n " , h _ b g 1 _ i n v m a s s _ c u t _ d e t . I n t e g r a l () , n D e t I n v M a s s * c s P i P i P i E r r o r / n E v e n t s P i P i P i * l u m i n o s i t y ) ; p r i n t f ( " N u m b e r ␣ of ␣ c a n d i d a t e ␣ pi + pi - ␣ p a i r s ␣ f r o m ␣ s i g n a l : ␣ % f ␣ + - ␣ % f ␣ \ n " , n P i P i P a i r D e t M a i n * c s E t a P i P i / n E v e n t s E t a P i P i * l u m i n o s i t y , n P i P i P a i r D e t M a i n * c s E t a P i P i E r r o r / n E v e n t s E t a P i P i * l u m i n o s i t y ) ; p r i n t f ( " N u m b e r ␣ of ␣ c a n d i d a t e ␣ pi + pi - ␣ p a i r s ␣ f r o m ␣ b a c k g r o u n d : ␣ % f ␣ + - ␣ % f ␣ \ n " , n P i P i P a i r D e t B g * c s P i P i P i / n E v e n t s P i P i P i * l u m i n o s i t y , n P i P i P a i r D e t B g * c s E t a P i P i E r r o r / n E v e n t s E t a P i P i * l u m i n o s i t y ) ; p r i n t f ( " N u m b e r ␣ of ␣ d e t e c t a b l e ␣ pi + pi - ␣ c a n d i d a t e ␣ e v e n t s ␣ f r o m ␣ s i g n a l : ␣ % f ␣ + - ␣ % f ␣ \ n " , n E t a P i P i D e t P o s t C u t * c s E t a P i P i / n E v e n t s E t a P i P i * l u m i n o s i t y , n E t a P i P i D e t P o s t C u t * c s E t a P i P i E r r o r / n E v e n t s E t a P i P i * l u m i n o s i t y ) ; p r i n t f ( " N u m b e r ␣ of ␣ c a n d i d a t e ␣ pi + pi - ␣ p a i r s ␣ f r o m ␣ eta - > g a m m a ␣ g a m m a ␣ c h a n n e l : ␣ % f ␣ + - ␣ % f ␣ \ n " , n P i P i P a i r D e t 2 p h * c s E t a P i P i / n E v e n t s E t a P i P i * l u m i n o s i t y , n P i P i P a i r D e t 2 p h * c s E t a P i P i E r r o r / n E v e n t s E t a P i P i * l u m i n o s i t y ) ; p r i n t f ( " N u m b e r ␣ of ␣ d e t e c t a b l e ␣ pi + pi - ␣ c a n d i d a t e ␣ e v e n t s ␣ f r o m ␣ g a m m a ␣ g a m m a ␣ c h a n n e l ␣ of ␣ s i g n a l : ␣ % f ␣ + - ␣ % f ␣ ␣ \ n " , n E t a P i P i 2 p h D e t P o s t C u t * c s E t a P i P i / n E v e n t s E t a P i P i * l u m i n o s i t y , n E t a P i P i 2 p h D e t P o s t C u t * c s E t a P i P i E r r o r / n E v e n t s E t a P i P i * l u m i n o s i t y ) ; p r i n t f ( " N u m b e r ␣ of ␣ d e t e c t a b l e ␣ pi + pi - ␣ c a n d i d a t e ␣ e v e n t s ␣ f r o m ␣ b a c k g r o u n d : ␣ % f ␣ + - ␣ % f ␣ \ n " , n P i P i P i D e t P o s t C u t * c s P i P i P i / n E v e n t s P i P i P i * l u m i n o s i t y , n P i P i P i D e t P o s t C u t * c s P i P i P i E r r o r / n E v e n t s P i P i P i * l u m i n o s i t y ) ; p r i n t f ( " N u m b e r ␣ of ␣ c a n d i d a t e ␣ pi + pi - ␣ p a i r s ␣ f r o m ␣ s i g n a l ␣ w i t h o u t ␣ s m e a r i n g : ␣ % f ␣ + - ␣ % f ␣ \ n " , n E t a P i P i N o S m e a r * c s E t a P i P i / n E v e n t s E t a P i P i * l u m i n o s i t y , n E t a P i P i N o S m e a r * c s E t a P i P i E r r o r / n E v e n t s E t a P i P i * l u m i n o s i t y ) ; p r i n t f ( " N u m b e r ␣ of ␣ c a n d i d a t e ␣ e v e n t s ␣ f r o m ␣ g a m m a ␣ c h a n n e l ␣ w i t h o u t ␣ s m e a r i n g : ␣ % f ␣ + - ␣ % f ␣ \ n " , n P i P i P a i r D e t 2 p h N o S m e a r * c s E t a P i P i / n E v e n t s E t a P i P i * l u m i n o s i t y , n P i P i P a i r D e t 2 p h N o S m e a r * c s E t a P i P i E r r o r / n E v e n t s E t a P i P i * l u m i n o s i t y ) ; p r i n t f ( " N u m b e r ␣ of ␣ d e t e c t a b l e ␣ pi + pi - ␣ c a n d i d a t e ␣ e v e n t s ␣ f r o m ␣ s i g n a l ␣ w i t h o u t ␣ s m e a r i n g : ␣ % f ␣ + - ␣ % f ␣ \ n " , n E t a P i P i N o S m e a r D e t * c s E t a P i P i / n E v e n t s E t a P i P i * l u m i n o s i t y , n E t a P i P i N o S m e a r D e t * c s E t a P i P i E r r o r / n E v e n t s E t a P i P i * l u m i n o s i t y ) ; p r i n t f ( " N u m b e r ␣ of ␣ d e t e c t a b l e ␣ e v e n t s ␣ f r o m ␣ g a m m a ␣ g a m m a ␣ c h a n n e l ␣ w i t h o u t ␣ s m e a r i n g : ␣ % f ␣ + - ␣ \ n " , n E t a 2 p h N o S m e a r D e t * c s E t a P i P i / n E v e n t s E t a P i P i * l u m i n o s i t y , n E t a 2 p h N o S m e a r D e t * c s E t a P i P i E r r o r / n E v e n t s E t a P i P i * l u m i n o s i t y ) ; p r i n t f ( " N u m b e r ␣ of ␣ c a n d i d a t e ␣ pi + pi - ␣ p a i r s ␣ f r o m ␣ b a c k g r o u n d ␣ w i t h o u t ␣ s m e a r i n g : ␣ % f ␣ + - ␣ % f ␣ \ n " , n P i P i P i N o S m e a r * c s P i P i P i / n E v e n t s P i P i P i * l u m i n o s i t y , n P i P i P i N o S m e a r * c s P i P i P i E r r o r / n E v e n t s P i P i P i * l u m i n o s i t y ) ; p r i n t f ( " N u m b e r ␣ of ␣ d e t e c t a b l e ␣ pi + pi - ␣ c a n d i d a t e ␣ e v e n t s ␣ f r o m ␣ b a c k g r o u n d ␣ w i t h o u t ␣ s m e a r i n g : ␣ % f ␣ + - ␣ % f ␣ \ n " , n P i P i P i N o S m e a r D e t * c s P i P i P i / n E v e n t s P i P i P i * l u m i n o s i t y , n P i P i P i N o S m e a r D e t * c s P i P i P i E r r o r / n E v e n t s P i P i P i * l u m i n o s i t y ) ; }