Alberto Saborido Patiño and Eduard Costa Reina

Measure of matter antimatter differences with the LHCb

experiment
The aim of this experiment is to see the differences in the interactions of matter and antimatter.

PART 1. First steps

In the first place, we work with simulated data. This is done to develop and polish the code that is
going to be used in the analysis of real data. The main difference between simulated and real data is
that the latter contains not only signal events but background too, apart from the limits imposed from
the detector resolution, making its analysis more complicated. Furthermore, this procedure helps
avoiding fine-tuning the analysis code with the real data to obtain specific or desired results.

The first part of the lab consists of the familiarisation with the data, understanding the format and
trying out basic calculations with it. It is paramount to understand that the data is given in a table
where the rows contain information of single events, name we give to the collision of two proton
bunches, where multiple collisions could take place and several particles can be created. In the
columns there is information about different quantities that will be of interest for our analysis.

The given information contains candidates for B meson (B + or B-) decays into 3 kaons (combination of
K+ and K-), in particular about the reconstructions of the tracks that might be from kaons.

DOUBT 1. Add information about the different quantities in the table?

For the general event, we have the travelled distance of the B meson candidate and the quality of the
vertex fit for the 3 kaon candidates. Then, we have kinematic information about each of the three
particles involved, such as the momentum components, its probabilities of being a kaon, muon and pi
meson, its charge and finally the impact parameter $\Chi^2$.

PART 2. Reconstruction of the invariant mass

DOUBT 2. Why do we have this shape for the momentum in the direction of the beam pipe? Why
negative momentum if we only have detector in the forward direction? It might be because there is
part of the detector before the collision point, and some particles could go backwards, being captured
by this part of the detector. Regarding the histogram shapes of the components of the momentum, we
have to remember that the collision products will go mainly in the direction of the beam. For this
reason, in the beam axis (z-axis), almost all the particles have a positive component, and they reach
higher values than in the transverse plane. For the latter, the distribution approximates a gaussian
centred at 0, which means that the most part of the particles are not heavily dispersed, having lower
components in both the x and y directions.

PLOT: Histogram of the different components of the momentum for the first particle. Explanation of
the histograms above.

Energy and mass

From the momentum components of the kaon candidates, we can trivially compute the magnitude of
the momentum. Afterwards, their energy can be computed using the energy-momentum relation ($E 2 =
p2 + m2$). In this step we are assuming they are actually kaons and therefore extracting their known
mass from the PDG: $m_K = 493.677 \pm 0.016 MeV$).

PLOT: Distribution of the energy of the kaons (assuming their nature). One can see that the most
encountered energy is around 5000 MeV. From there, the range of energy spans quite far, decaying
significantly as the energy increases.
Alberto Saborido Patiño and Eduard Costa Reina

Adding features of the B meson

In order to know if the presumed mother particle of the kaons could be a B meson, one can make use
of the conservation of the energy and momentum (as a vectorial quantity). Hence, adding the energies
of the 3 daughter particles and the different components of the momentum, the total energy and
momentum of the presumed mother particle can be computed. Again, using the energy-momentum
relation, its invariant mass can be obtained.

PLOT: Histogram of the invariant mass of the 3-particle system, which corresponds to the candidate
mother particle. A quite sharp peak appears at the mass of the B meson, which is what we expect if the
mother particle is actually a B meson. EXTRA: Fit a gaussian in the peak to for the central value and
the error associated with the value.

Because the data used until now is simulated, there are no background events in the plots. This makes
the plots be cleaner. If real data was used, a contribution of the background is to be expected. This
contribution would have a descending shape in figure [], where we should still be able to identify the
gaussian corresponding to the resonance of the B meson.

PART 3. Working with real data

The data used in this part is from the first major data taking period in the LHC, in 2011. The used data
from LHCb has already been filtered to the events that are likely to have come from a B meson. The
aim is now to keep only the events where the decay results are 3 charged kaons. This is the
information that will be required to measure the CP violation.

In order to filter out the events that don’t contain 3 charged kaons in the final state, cuts to the data
need to be applied. This procedure is not as straightforward as it might seem because neither the data
is pure signal from the event we are interested in, nor the detector is perfect. This means that a
compromise is to be reached in order to get rid of enough background while still keeping enough
statistics to get significant results.

Initially, one can make use of the variables that describe the nature of the particles. The first cut
applied has been that the probability of being a $\pi$ meson is below $42%$ and the one to be a kaon
surpasses $50%$. From the binary variable that tells us if the particle is a muon, all the positives have
been discarded.

Possible cuts: Why not set the kaon probability to at least 90%? Most of the data is on the right of that
threshold. Also, the pion probability could be set to a maximum value of 20% for the cut (not a lot of
signal nor background will be removed, but the ratio might be better).

PLOT: Invariant mass of the B meson with all data, the pregiven cuts and a cut of my own.

EXTRA: Fit a gaussian in the peak to for the central value and the error associated with the value.

After experimenting with different cuts, the final preselection has been: ?. The B meson peak for this
selection is showed, with more detail, in figure [].

PLOT: Invariant mass of the B meson with the final cut and zoomed in.

PART 4. Searching for global matter anti-matter differences

CP violation is the name we give to the differences observed between matter and antimatter (since CP
it is the symmetry that relates them). The term global refers to the kinematics, to the fact that the
search is carried out across all ranges of energy and momentum.
Alberto Saborido Patiño and Eduard Costa Reina

In this analysis, the asymmetry of the process is the quantity used to measure the CP violation. It
consists of the difference in the number of events for $B^+$ and $B^-$ decays with respect to the total
amount of events, given by []. These events have been identified from the values of the charge of the
daughter particles.

$A = \frac{N^+ - N^-}{N^+ + N^-}$

The statistical uncertainty of the asymmetry is given by equation [].

$\sigma_A = \sqrt{\frac{1-A^2}{N^+ + N^-}}$

Finally, one can extract, from these parameters, the significance of the result, computed with [].

$s = \frac{A}{\sigma_A}$

If we take this result directly, the value obtained is $s = 5.48$, which is customary considered a
discovery. Unfortunately, this does not reflect the whole truth. Other sources of error exist in this data,
which fall in the category of systematic uncertainties. They take into account the possible biases in the
measurements. Among the possible sources, a new estimation has been made considering the
production asymmetry. It is due to the fact that the initial state of the collision is not matter-antimatter
symmetric, given that the collision is not proton-antiproton but proton-proton. As discussed in
[REFERENCE FOR THE LHCb PAPER], it is estimated to be of 1%. We add then this uncertainty
with [] because it is not correlated to the statistical error, providing a new result for the asymmetry
(only the error varies) and the significance [].

$BOXED RESULT?$

As expected, the significance is lower, but still falls in the range to consider the result as evidence of
CP violation.

PART 5. Dalitz plots and two body resonances

An interesting technique for the analysis of 3-body decays is the Dalitz plot, because it allows to
identify resonances. These correspond to the formation of an intermediate particle in the decay before
reaching the 3-body final state [].

$ B^+ → K^+K^+K^− $
$ B^+ \to K^+R^0 where R^0 \to K^+K^-$

This is possible due to the fact that the kinematics in a 3-body decay only has two independent
variables. For instance, the energy and momentum of two of the daughter particles together with the
ones of the mother particle determine unambiguously the energy and momentum of the third body. By
plotting the squared invariant masses of two different pairs, resonances should appear as bands in the
plot.

(In our data, the kaon candidates are numbered.) There are three possible pairs to make from the 3
kaons. The doubly charged resonance is not possible from the decay of a meson, though, because no
baryons can be obtained in the process (quarks are produced or destroyed in pairs) and those are the
only ones that can be doubly charged. Therefore, only the two combinations leading to a neutral
resonance will be able to be observed.

DOUBT 3. Why does it say that we will not encounter any double charged resonance? Why does the
intermediate particle be a meson? I think that the answer to the second is that quarks must create or
Alberto Saborido Patiño and Eduard Costa Reina

annihilate in pairs, so we will not have an odd number of quarks and therefore no baryons can be
created (they are the only ones that can have a charge of 2 units!).

Computing the invariant mass of the two neutral combinations, for the simulated data, we obtained the
Dalitz plot in figure []. As the possible resonances were not included in the simulation, no bands can
be identified in the plot.

PLOT: Dalitz plot with simulated data.

Adding plot for real data

Following the same procedure but using the real data, the plot in figure [] has been obtained. We can
see 3 main differences with respect to the one with simulation data. Firstly, we can see clearly defined
bands that correspond to resonances. Also, the distribution of data points is no longer homogeneous
(apart from the bands). Finally, dispersed points at higher squared invariant masses can be
encountered.

PLOT: Dalitz plot with real data.

We can go one step further regarding the visibility in the Dalitz plots. As both resonances plotted
(R_{12} and R_{13}) contain both one K^+ and one K^-, they are symmetric, so we can arrange them
so that we plot the higher invariant mass always in one of the axis. This will graphically fold the plot,
emphasising the difference in density, making resonances more visible. The folded Dalitz plot with
ordered resonances is shown in the figure [].

PLOT: Ordered Dalitz plot.

Another possible visualisation of this plots is by binning the plot and converting it into a two-
dimensional histogram (see figure []).

PLOT: Binned Dalitz plot.

Two-body resonances

The clearest two-body resonance can be found between $(0.335-0.360)10^7 MeV^2$, corresponding
to $1830-1897 MeV$. In this range, one particle that decays to $K^+K^-$ can be found: the neutral
meson $D^0$, with $m_D = 1864.84\pm0.05 MeV$.

Another resonance that has been spotted is between $(1.15-1.175)10^7 MeV^2$, corresponding to
$3391-3427 MeV$. The neutral meson $\chi_{c0}(1P)$, with mass $m_\chi = 3414.75\pm0.31 MeV$
would tally with that description, given that it can also decay into $K^+K^-$.

Finally, a weaker resonance could be identified between $(0.95-1.00)10^7 MeV^2$ ($3082-3162

MeV$). Fulfilling the decay requisites and with a mass of $m = 3096.916\pm0.011$, the famous
meson $J/\Psi (1S)$ could be creating that resonance.

PART 6. Searching for local matter anti-matter differences

Considering that CP violation appears as a result of different decay modes with a common final state,
the computed asymmetry can depend on the kinematic region. In this sense, one can check the CP
violation locally to see if more significant results can be obtained, even if they are not so general.

Removing charm resonances

Alberto Saborido Patiño and Eduard Costa Reina

In our study, we are only interested in charmless $B$ meson decays, reason why they have been
removed from the data. Taking into account the 3 found resonances, the Dalitz plot obtained is shown
in figure [].

PLOT: Dalitz plot without resonances

Comparing Dalitz plots

The last step to find the local CP violation is to compare the Dalitz plots for the decays of $B^+$ and
the ones of $D^-$. Both Dalitz plots can be superposed to let us have a first picture of this asymmetry
in the different kinematic regions, but as seen in [], it is not a good visualisation of the data. For that
reason, a binned Dalitz plot has been created where the asymmetry and the significance have been
computed independently from bin to bin. It is worth noting that, in the case where very few data points
are found inside a single bin, the statistical error might get an unrealistic value. To mitigate this effect,
a thicker binning has been used (losing kinematic space resolution in the way).

PLOT: Superposition of both Dalitz plots

In the figure [], plots for the asymmetry, statistical error and significance of the asymmetry are shown.
It is paramount to remark that a large value of asymmetry is not necessarily related to a significant
value owing to the large uncertainty it could have associated. Hence, the local CP violation must be
checked in [c].

PLOT: Asymmetry plots

Observing CP violation

From the results obtained in figure [], we can look further into the most promising regions. Hence,
targeting the data in the bin with more significance ($0.805 MeV^2 \lt R_H \lt 1.307 MeV^2$ and
$0.098 MeV^2 \lt R_H \lt 0.376 MeV^2$), the value obtained for the asymmetry are, following the
same procedure as before: $A = 0.093\pm0.018$ with a significance $s=5.03\sigma$.

PART 7. Further analysis

Adding extra sophistication

Further analysis

DOUBTS
- Resonances?
- Significance that goes nuts
- Cuts?
Alberto Saborido Patiño and Eduard Costa Reina

CORRECTION STUFF

1.1.A. The longer lifetime is only for the B-meson decay we are interested with, then?
Correct.
1.3.A. Has the interaction with the detector been included in the end? (Page 5 of the
instructions.) Nope.
2.2.A. Then, in order to explain it, we are thinking that it could be that there are no other
detectors in the backwards direction, so the momentum is not determined and therefore not
plotted. If that was the case, why would we get some negative values? Not sure.
2.3.A. Is it an estimation of the probability instead of a probability? Likelihood. Indicator.
2.4.A. We have checked and we have no minus sign. Both the determination of the charge and
the formula for the asymmetry should be correct. Should we obtain a negative value? No.
2.6.Observing CP violation. A. Understandeable have a good day.

Don’t put information for the familiarising with the data.

Alberto Saborido Patiño and Eduard Costa Reina

\section{Introduction}

The matter-antimatter asymmetry in the Universe is still an open question in physics. Such
phenomenon is yet still to be explained but has been measured in several processes and
experiments. The focus of this project is on the measurement of CP violation in 3-body decays
of B mesons of the shape $B^{\pm} \to h^{\pm}h^+h^-$, where the hadrons $h$ can either
correspond to pions $\pi^{\pm}$ or kaons $K^{\pm}$. The data employed comes from the
LHC$b$ experiment and it has been used to measure global and local matter-antimatter
asymmetries.

\subsection{The LHCb detector}

\label{subsec:LHCb}

The LHC$b$ experiment is one of the four main experiments of the LHC, at CERN. It is
a single-arm forward spectrometer originally designed for precision measurements of CP
violation and rare decays of beauty and charm hadrons, produced with strong boosts in the
forward direction. In Figure \ref{im:LHCb} the different detector elements can be seen. For
the purposes of this project, the determination of the momentum, impact parameter, type of
charged hadron and velocity are paramount.

\begin{figure}[H]
\centering
\includegraphics[width=0.8\linewidth]{chapters/im/LHCb.png}
\caption{\begin{small} View of the LHCb detector \cite{LHCb:2014set}. In the figure
the Vertex Locator (VELO), the Tracking System (composed by the TT and T1-T3 stations),
the Ring Imaging Cherenkov (RICH) detectors, the calorimeters (ECAL and HCAL), the
muon system (composed by stations M1-M5) and the magnet are shown. The collisions take
place inside the VELO. \end{small}}
\label{im:LHCb}
\end{figure}

Regarding the momentum, it is determined from the curvature of the tracks of charged
particles in the magnetic field generated by the dipole magnet. The track is reconstructed
based on several individual detector signals. The vertex locator (VELO) allows for the
determination of the primary vertex as well as the decay location of the $b$ hadron (which is
a secondary vertex). Together with the signals in the different tracking detectors (TT before
the magnet and T1, T2 and T3 after it), the trajectory reconstruction and impact parameter
determination can be done, important to distinguish the searched final state tracks of B
mesons we are interested in, which will have larger impact parameter than background tracks.

The distinction of charged hadrons, in particular between pions and kaons, is achieved
using the Ring-Imaging Cherenkov (RICH) detectors. The characteristic light cone produced
with the Cherenkov effect gives an indirect measurement of the velocity of the particle,
strongly related to the angle of emission of this cone. Combining this information with the
momentum of the particles, the mass can be inferred and the particle can therefore be
identified.

For the identification of photons, electrons and hadrons, the calorimeter system is used.
Particles interact with a heavy material (iron and lead), creating a particle shower and being
absorbed by the latter, emitting photons in the process. The amount of scintillation is closely
Alberto Saborido Patiño and Eduard Costa Reina

related to the energy of the particles, thus the latter can be estimated. The signature of photons
and electrons is their full absorption in the Electromagnetic Calorimeter (ECAL) while
hadrons do so in the Hadronic Calormiter (HCAL). Finally, muons are the only particles that
can reach the muon chambers and also leave a trace in the muon chambers (unlike neutrinos).

\subsection{CP violation}

As stated, the CP violation is an interesting feature contained in the SM worth of

attention, given that it cannot explain the matter-antimatter imbalance in the universe, and
might even be the answer to other fields.

CP is the resulting symmetry of combining C (charge conjugation, that interchanges

particles for antiparticles) and P (spatial parity). Therefore, its violation can be assessed by
comparing the rate of decays of particles and their antiparticle version. For this experiment,
the processes $B^+ → K^+K^+K^−$ and $B^- → K^+K^-K^−$ have been used.

\subsection{Data}
\label{subsec:data}

The data used in this part is from the first major data taking period of the LHC, in 2011.
A key aspect of the data used in this project is that it has already undergone a selection so that
only events of interest for the analysis are kept. The importance of this point relies on the
large amount of data generated in each event. A trigger, which is the system used to decide
which events are stored, helps with keeping the most interesting events by focusing on
interesting signatures. Among these, the characteristic secondary vertex of b-hadrons and
events containing three charged hadrons have been used to filter the data.

With respect to the simulated data, only simulation for the physical process has been
carried out, the interaction with the detector has not been included.

% \subsection{Outline}

% The analysis in this study is focused on the B meson decay into 3 kaons: $B^{\pm} \to
K^{\pm}K^+K^-$. This process has been selected over others containing pions because more
Feynman diagrams are possible and therefore the background level is higher, complicating the
analysis.

% LITTLE OUTLINE

\section{Analysis}
\label{sec:analysis}

\subsection{First steps}

In the first place, simulated data is treated to develop and polish the code that is going to
be used in the analysis of real data. The main difference between simulated and real data is
that the latter contains not only signal events but background too, making its analysis more
complicated. Furthermore, this procedure helps avoiding fine-tuning the analysis code with
the real data to obtain specific or desired results.
Alberto Saborido Patiño and Eduard Costa Reina

The given dataset contains candidates for B meson (B$^+$ or B$^-$) decays into 3
kaons (combination of $K^+$ and $K^-$), in particular about the reconstructions of the tracks
that might be from kaons. Regarding the general event, there is the flight distance of the B
meson candidate and the quality of the vertex fit for the 3 kaon candidates. Then, kinematic
information about each of the three particles involved is provided, such as the momentum
components, its probabilities of being a kaon, muon, and pi meson, its charge, and finally the
impact parameter's $\chi^2$.

\subsection{Reconstruction of the invariant mass}

Before delving into the determination of the invariant mass, it is interesting to see the
shape of the momentum components for one of the candidates (see Figure \
ref{fig:momentumcomponents}). As expected, the distribution for the components transverse
to the beam pipe are gaussians centered at 0, which only means that the most part of the
particles are not heavily dispersed, having lower components in both the x and y directions. It
is interesting to note that there is measured negative momentum in the z-axis, i.e. the beam
pipe. Even if the collision is symmetric, most of the detectors are only in the forward direction
(see Section \ref{subsec:LHCb}). Only the VELO is symmetric, and the contribution for low
momentum in the experimental data could come from the reconstruction of low momentum
particles that are deviated by residual magnetic field inside this detector. Apart from that, most
of the particles go in the beam direction, making the distribution lean largely to its positive
(and large) values.

\begin{figure}[H]
\centering
\includegraphics[width=0.8\linewidth]{chapters/im/momentumcomponents.png}
\caption{Histogram of the different components of the momentum for $H_1$.}
\label{fig:momentumcomponents}
\end{figure}

\textbf{Energy and mass}

From the momentum components of the kaon candidates, the magnitude of the
momentum can be computed. Afterwards, their energy can be calculated using the energy-
momentum relation ($E^2 = p^2 + m^2$), obtaining Figure \ref{fig:kaonenergy}. Because
only kaons are simulated, their known mass can be used to deduce its energy. From the PDG \
cite{PDG}, the mass of the kaon can be retrieved: $m_K = 493.677 \pm 0.016\ \unit{\mega\
electronvolt}$).

\begin{figure}[H]
\centering
\includegraphics[width=0.6\linewidth]{chapters/im/kaonenergy.png}
\caption{Distribution of the energy of the kaons (assuming their nature). Even
though it is not visible in the plot, the most encountered energy is around $\SI{5000}{\mega\
electronvolt}$. From there, the range of energy spans quite far, decaying significantly as the
energy increases.}
\label{fig:kaonenergy}
\end{figure}
Alberto Saborido Patiño and Eduard Costa Reina

\textbf{Adding features of the B meson}

In order to know if the presumed mother particle of the kaons could be a $B$ meson,
one can make use of the conservation of the energy and momentum (the latter as a vectorial
quantity). Hence, adding the energies of the 3 daughter particles and the different components
of the momentum, the total energy and momentum of the presumed mother particle can be
computed. Again, using the energy-momentum relation, its invariant mass can be obtained.
The resonance encountered at approximately $m = \SI{5279.25}{\mega\electronvolt}$ in
Figure \ref{fig:binvmass} is compatible with the nominal value of the B meson mass given in
the PDG of $m_B = 5279.34\pm0.12 \unit{\mega\electronvolt}$.

\begin{figure}[H]
\centering
\includegraphics[width=0.6\linewidth]{chapters/im/binvmass.png}
\caption{Histogram of the invariant mass of the 3-particle system, which
corresponds to the candidate mother particle. A quite sharp peak appears near the mass of the
B meson, which is expected if the mother particle is actually a B meson.}
\label{fig:binvmass}
\end{figure}

Because the data used until now has been simulated, there are no background events in
the plots. This makes the plots cleaner. If real data was used, a contribution of the background
is to be expected. This contribution would have a descending shape in figure \
ref{fig:binvmass}, where composed of the gaussian corresponding to the resonance of the B
meson covered by the background contribution. The resonance might only be visible after a
selection.

\subsection{Working with real data}

The real data used for the analysis has to be filtered before using it for the desired
calculations, since the interest is on the decay of B mesons into kaons, particles for which the
simulation was carried out. The first preselection of the data (briefly discussed in Subsection \
ref{subsec:data}) was aimed at keeping events containing particles with great momentum and
transverse momentum (characteristic of B decays), having total invariant mass similar to the
one of the B meson and accounting for significant values of the impact parameter. The aim is
now to keep only the events where the decay results are 3 charged kaons, given the similarity
in the kinematics for kaons and pions did not allow for a distinction from the previous
variables.

In order to filter out the events that don’t contain 3 charged kaons in the final state, cuts
to the data need to be applied. This procedure is not as straightforward as it might seem
because neither the data is pure signal from the event we are interested in, nor the detector is
perfect. This means that a compromise is to be reached to get rid of enough background while
still keeping enough statistics to get significant results.

Initially, one can make use of the variables that describe the nature of the particles. The
first cut applied has been that the likelihood of being a $\pi$ meson is below $42\%$ and the
one to be a kaon surpasses $50\%$. From the binary variable that tells us if the particle is a
muon or not, all the positives have been discarded. The final histogram obtained can be seen
in Fig \ref{fig:binvmassreal}. One can see that the background at higher but specially at lower
Alberto Saborido Patiño and Eduard Costa Reina

energies is visible. This is, as expected, completely different from the non-existing
background in Fig \ref{fig:binvmass}, given that there was no background simulated.

\begin{figure}[H]
\centering
\includegraphics[width=0.8\linewidth]{chapters/im/binvmassreal.png}
\caption{Histogram with the B meson invariant mass obtained using the real data and
cuts in particle identification.}
\label{fig:binvmassreal}
\end{figure}

\subsection{Searching for global matter anti-matter differences}

In this section, the determination of global CP violation is computed. The term global
refers to the kinematic region studied, initially carried out across all ranges of energy and
momentum. The asymmetry of the process is the quantity used to measure the CP violation. It
just consists of the difference in the number of events for $B^+$ and $B^-$ decays with
respect to the total number of events, given by

\begin{equation*}
A = \frac{N^+ - N^-}{N^+ + N^-},
\end{equation*}

where $N^+$ and $N^-$ represent the number of $B^+$ and $B^-$ decays, respectively.
A selection has been done checking the values of the total charge of the daughter particles,
which is conserved and must therefore coincide with the one of the B meson. The statistical
uncertainty of the asymmetry is given

\begin{equation*}
\sigma_A = \sqrt{\frac{1-A^2}{N^+ + N^-}},
\end{equation*}

Finally, one can extract, from these parameters, the significance of the result, computed
with the following simple expression:

\begin{equation*}
s = \frac{A}{\sigma_A}.
\end{equation*}

The value obtained following this process is a obtained is $A_{global} = 0.037 \pm
0.007$ with a significance $s = 5.48$, which would be considered a discovery. Unfortunately,
this does not reflect the whole truth. Other sources of error exist in this data, which fall in the
category of systematic uncertainties. These take into account the possible biases in the
measurements. Among the possible sources, a new estimation has been made considering the
production asymmetry. The latter is due to the fact that the initial state of the collision is not
matter-antimatter symmetric, given that the collision is not proton-antiproton but proton-
proton. As discussed in \cite{LHCbpaper}, it is estimated to be of 1\%. Adding this
uncertainty (and treating it as uncorrelated to the statistical one), the new result for the
asymmetry is $A_{global} = 0.037 \pm 0.012$ with a significance $s = 3.06$. As expected,
Alberto Saborido Patiño and Eduard Costa Reina

the significance is lower, but still falls in the range to consider the result as evidence of CP
violation.

\subsection{Dalitz plots and two body resonances}

\label{subsec:dalitz}

An interesting technique for the analysis of 3-body decays is the Dalitz plot, because it
allows to identify resonances. These correspond to the formation of an intermediate particle in
the decay before reaching the 3-body final state. Both possibilities are represented in
equation \ref{eq:posdecays}, where the first line corresponds to the direct three-body decay
and the second one undergoes an intermediate state, a resonance.

\begin{align}
B^+ &\to K^{+}K^{+}K^{-}\\
B^+ &\to K^+R^0\ \text{where}\ R^0 \to K^+K^-
\label{eq:posdecays}
\end{align}

The Dalitz plot takes advantage of the fact that the kinematics in a 3-body decay only has
two independent variables to identify the resonances. For instance, the energy and momentum
of two of the daughter particles together with the ones of the mother particle determine
unambiguously the energy and momentum of the third body. By plotting the squared invariant
masses of two different pairs, resonances should appear as bands in the plot.

A priori, there are three possible pairs of kaons. The doubly charged resonance, though, is
not possible from the decay of a meson, because no baryons can be obtained in the process
(quarks are produced or destroyed in pairs) and those are the only ones that can be doubly
charged. Therefore, only the two combinations leading to a neutral resonance can be
observed. Hence, computing the invariant mass of the two neutral combinations of kaons for
the simulated data, we obtained the Dalitz plot in Figure \ref{fig:dalitzsim}. As the possible
resonances were not included in the simulation, no bands can be identified in the plot. No
background can be seen either for the same reason.

\begin{figure}[H]
\centering
\includegraphics[width=0.5\linewidth]{chapters/im/dalitzsim.png}
\caption{Dalitz plot obtained for the simulated data. No background or resonances are
encountered and therefore the decays are evenly distributed and the shape is clearly defined
by the mass limits. The $R_{12}$ and $R_{13}$ denote that the particles used for the
calculation of the invariant mass are labeled as 1 and 2 or 1 and 3 respectively in the dataset.}
\label{fig:dalitzsim}
\end{figure}

\textbf{Adding plot for real data}

Following the same procedure but using the real data, the plot in Figure \
ref{fig:dalitzreal} has been obtained. 3 main differences can be spotted with respect to the one
with simulation data. Firstly, clearly defined bands that correspond to resonances appear in
different regions. Also, the distribution of data points is no longer homogeneous. Finally,
dispersed points outside of the four-momentum constrained region can be encountered.
Alberto Saborido Patiño and Eduard Costa Reina

\begin{figure}[H]
\centering
\includegraphics[width=0.5\linewidth]{chapters/im/dalitzreal.png}
\caption{Dalitz plot obtained for the real data. Several resonances as well as a non-
negligible background can be observed.}
\label{fig:dalitzreal}
\end{figure}

One can go one step further regarding the visibility in the Dalitz plots. As both
resonances plotted ($R_{12}$ and $R_{13}$) contain both one $K^+$ and one $K^-$, they
are symmetric, so we can arrange them so that we plot the higher invariant mass always in
one of the axis. This will graphically fold the plot, emphasising the difference in density,
making resonances more visible. The folded Dalitz plot with ordered resonances is shown in
Figure \ref{fig:dalitzfold}.

\begin{figure}[H]
\centering
\includegraphics[width=0.5\linewidth]{chapters/im/dalitzfold.png}
\caption{Dalitz plot for the ordered Dalitz variables obtained for the real data. $R_H$
and $R_L$ represent the higher and lower invariant mass of the two for each event.}
\label{fig:dalitzfold}
\end{figure}

Another possible visualisation of this plots is by binning the plot and converting it into a
two-dimensional histogram (see figure \ref{fig:dalitzbinned}).

\begin{figure}[H]
\centering
\includegraphics[width=0.5\linewidth]{chapters/im/dalitzbinned.png}
\caption{Binned Dalitz plot for the ordered Dalitz variables obtained for the real data.
The resolution chosen is solely to emphasise the binning nature of the plot.}
\label{fig:dalitzbinned}
\end{figure}

\textbf{Two-body resonances}

The clearest two-body resonance can be found between $\SI{0.335e7}{\squared\mega\

electronvolt}/c^2$ and $\SI{0.360e7}{\squared\mega\electronvolt}/c^2$, corresponding to an
energy range of $\SI{1830}{\mega\electronvolt}$ to $\SI{1897}{\mega\electronvolt}$. In
this range, one particle that decays to $K^+K^-$ can be found: the neutral meson $D^0$, with
$m_D = (1864.84\pm0.05) MeV$.

Another resonance that has been spotted is between $\SI{1.150e7}{\squared\mega\

electronvolt}/c^2$ and $\SI{1.175e7}{\squared\mega\electronvolt}/c^2$, corresponding to $\
SI{3391}{\mega\electronvolt}$ to $\SI{3427}{\mega\electronvolt}$. The neutral meson $\
chi_{c0}(1P)$, with mass $m_\chi = (3414.75\pm0.31) MeV$ would tally with that
description, given that it can also decay into $K^+K^-$.
Alberto Saborido Patiño and Eduard Costa Reina

Finally, a weaker resonance could be identified between $\SI{0.95e7}{\squared\mega\

electronvolt}/c^2$ and $\SI{1.00e7}{\squared\mega\electronvolt}/c^2$ ($\SI{3082}{\mega\
electronvolt}$ to $\SI{3162}{\mega\electronvolt}$). Fulfilling the decay requisites and with a
mass of $m_{J/\Psi} = (3096.916\pm0.011)$ MeV, the famous meson $J/\Psi (1S)$ could be
creating that resonance.

\subsection{Searching for local matter anti-matter differences}

Considering that CP violation appears as a result of different decay modes with a

common final state, the computed asymmetry can depend on the kinematic region. In this
sense, one can check the CP violation locally to see if more significant results can be
obtained, even if they are not so general.

\textbf{Removing charm resonances}

The charm resonances search performed in Subsection \ref{subsec:dalitz} was not in

vain. The removal of these resonances from the data has been considered adequate because
the CP violation in the charmed sector is much smaller than in the beauty sector. Thus,
refraining from using them, should lead to clearer results. The resulting Dalitz plot without
the charm resonances can be seen in Figure \ref{fig:dalitznoc}.

\begin{figure}[H]
\centering
\includegraphics[width=0.5\linewidth]{chapters/im/dalitznoc.png}
\caption{Dalitz plot for the ordered Dalitz variables obtained for the real data where
the charm resonances have been removed.}
\label{fig:dalitznoc}
\end{figure}

\textbf{Comparing Dalitz plots}

The last step to find the local CP violation is to compare the Dalitz plots for the decays of
$B^+$ and the ones of $D^-$. Both Dalitz plots can be superposed to let us have a first
picture of this asymmetry in the different kinematic regions, but as seen in Figure \
ref{fig:dalitznocpm}, it is not a good visualisation of the data.

\begin{figure}[H]
\centering
\includegraphics[width=0.4\linewidth]{chapters/im/dalitznocpm.png}
\caption{Dalitz plot for the ordered Dalitz variables obtained for the real data where
the charm resonances have been removed. The decays coming from $B^+$ and $B^-$ are
distinguished.}
\label{fig:dalitznocpm}
\end{figure}

For that reason, a binned Dalitz plot has been created where the asymmetry and the
significance have been computed independently from bin to bin. It is worth noting that, in the
case where very few data points are found inside a single bin, the statistical error might get an
unrealistic value. To mitigate this effect, a wider binning has been used. Furthermore, it is
worth noting that, for this final part of the analysis, only the D meson resonance has been
Alberto Saborido Patiño and Eduard Costa Reina

removed from the dataset, given that the results have a greater significance with it. This can
be done because the omission or not of the charm resonances is a choice; taking charmed
resonances for the CP violation calculation does not represent a methodological error, they
take part in the physical process and in this case contribute positively to the asymmetry. In
addition, the recognition of the omitted charm resonances was done by hand, and this could
have discarded valuable information.

\begin{figure}[H]
\centering
\subfigure[Local asymmetry values.]{\includegraphics[width=0.49\linewidth]
{chapters/im/asym.png}}
\subfigure[Local asymmetry uncertainty values.]{\includegraphics[width=0.49\
linewidth]{chapters/im/unce.png}}
\subfigure[Local asymmetry significance values.]{\includegraphics[width=0.49\
linewidth]{chapters/im/sign.png}}
\caption{Binned plots for the local asymmetry in the phase space.}
\label{fig:asymplots}
\end{figure}

In Figure \ref{fig:asymplots}, plots for the asymmetry, statistical error and significance
of the asymmetry are shown. It is paramount to remark that a large value of asymmetry is not
necessarily related to a significant value owing to the large uncertainty it could have
associated. Hence, the local CP violation must be checked in the significance plot.

\textbf{Observing CP violation}

From the results obtained in figure \ref{fig:asymplots}, we can look further into the most
promising regions. Hence, targeting the data in the bin with more significance ($\SI{0.805}{\
square\mega\electronvolt} < R_H < \SI{1.307}{\square\mega\electronvolt}$ and $\SI{0.098}
{\square\mega\electronvolt} < R_H < \SI{0.376}{\square\mega\electronvolt}$), the value
obtained for the asymmetry are, following the same procedure as before: $A = 0.093\
pm0.018$ with a significance $s=5.03$. The events in this regions can be clearly seen in
Figure \ref{fig:local}.

\begin{figure}[H]
\centering
\includegraphics[width=0.6\linewidth]{chapters/im/local.png}
\caption{Invariant mass distributions for the $B^+$ and $B^-$ events in the region
with most significance obtained in Figure \ref{fig:asymplots}.}
\label{fig:local}
\end{figure}

A crucial comment is to be made regarding this last figure: the background that can be
seen for energies lower than $\SI{5200}{\mega\electronvolt}$ has actually been used for all
the calculations. There are two key reasons for that. Firstly, the physics studied in this project
does not change. In second place, the plots for the asymmetries are much clearer when this
(not too large) background is used because too little signal remains and even a very low
background can easily get labeled as a large asymmetry with no uncertainty (and therefore
high significance) when it is really not the case. All in all, even though the specific results for
the asymmetry and significance values could be higher, specially in the local asymmetry
Alberto Saborido Patiño and Eduard Costa Reina

study, they can also get easily diluted by this random peaks of significance caused by a very
small amount of data points in certain regions. There are several ways that can help mitigating
this problem, but a rather precise equilibrium in all the aspects needs to be reached so that this
problem disappears. Just to set some example, a complex combination of cuts both for the
events and energies could be found. These would need to keep enough points (or none) in
regions with very small signal in order not to generate false high significance values for the
local asymmetries. Other ways that could help are creating a tailor-made binning that allows
for thinner binning in the regions of the phase space where the focus is and larger bins cover
the outskirts. Even so, the results obtained are still significant enough to call this result an
observation or discovery of CP violation in a local region, which was the main goal of this
study.

Last but not least, there are other aspects of this study that can be improved. For
example, a more thorough treatment of the systematic uncertainties could be carried out,
taking into account production detection asymmetry or polishing the analysis technique. Also,
it has been seen in \ref{fig:binvmassreal} that some background was present at low energies.
Instead of cutting out the regions with more background, a real estimation of the amount of
background below the actual signal could help improve the results. Finally, fitting the
distributions is always helpful to get more precise results and a better estimation of the error
associated to the values.