Master Thesis
Electron identi cation e ciency in B (s) 0 → e ± µ ∓
August 2016
Author:
Rosa Kappert S2048051 Supervisor:
dr. ir. C.J.G. (Gerco) Onderwater Second corrector:
dr. J.G. (Johan) Messchendorp
Abstract
The Standard Model describes all fundamental particles we know of, together with three of the four fundamental forces. Despite large successes the model has shortcomings. There are many New Physics models, like Seesaw and supersymmetry, trying to explain these. The experimental search for New Physics is an active eld as well, among these the search for charged Lepton Flavor Violating decays is a viable approach. The decay B(s)0 → e±µ∓ is such a decay and is studied by the LHCb collaboration. One of the e ciencies that needs to be known in the analysis is the e ciency of the identi cation of electrons (PIDe). A new method is developed to extract the e ciency without knowing the shape of the signal, resulting in an e ciency of 0.9229 ± 0.0018 for the LHCb data of 2011 and 2012. Further the traditional method, that makes use of Crystal Ball functions, is evaluated and compared to the new method.
Contents
1 Introduction 3
2 Particle physics and the SM 5
2.1 The SM . . . 5
2.1.1 Quark mixing . . . 6
2.1.2 The Standard Model and symmetry groups . . . 7
2.2 Shortcomings of the SM . . . 8
2.2.1 Experimental issues . . . 8
2.2.2 Theoretical issues . . . 8
3 The physics behind B(s)0 → e±µ∓ 10 3.1 Neutrino oscillations and Lepton Flavor (Violation) . . . 10
3.1.1 Conservation Laws . . . 10
3.1.2 Neutrino oscillations . . . 11
3.1.3 cLFV revisited . . . 13
3.2 New Physics in theory . . . 14
3.2.1 Seesaw models . . . 14
3.2.2 Supersymmetry . . . 18
3.2.3 Leptoquarks . . . 23
3.2.4 Leptogenesis . . . 24
3.3 Experimental status of cLFV . . . 25
3.3.1 Muon decays . . . 25
3.3.2 Tauon decays . . . 27
3.3.3 Meson decays . . . 27
3.3.4 Other cLFV decay modes . . . 28
4 The LHCb detector 29 4.1 The Large Hadron Collider . . . 29
4.2 The LHCb detector . . . 30
4.2.1 Tracking system . . . 31
4.2.2 Cherenkov detectors (RICH1 and RICH2) . . . 32
4.2.3 The calorimeter system . . . 35
4.2.4 The Muon system (M1-M5) . . . 36
4.2.5 The LHCb trigger and o ine reconstruction . . . 36
4.3 Particle Identi cation . . . 38
5 Electron Identi cation E ciency for B(s)0 → e±µ∓ 39 5.1 Conceptual approach of B(s)0 → e±µ∓ . . . 39
5.2 Practical approach of B(s)0 → e±µ∓ . . . 40
5.3 Signal and background . . . 40
5.4 Data preparation . . . 41
5.4.4 Why a new method? . . . 45
5.4.5 B+− J/ψmass correction . . . 47
5.4.6 Further cuts . . . 49
5.4.7 Bremsstrahlung and the BremAdder . . . 50
5.5 E ciency calculation without use of signal shape . . . 51
5.5.1 The new method . . . 51
5.5.2 Analysis and results . . . 53
5.6 E ciency calculation with the traditional method . . . 58
5.6.1 The method . . . 58
5.6.2 Analysis and results . . . 59
5.7 Comparison of methods and results . . . 67
6 Summary and outlook 68
A Toy Monte Carlo's new method 69
B Tables with e ciencies extracted with new method 73
C Plots of all bins with new method 75
Chapter 1
Introduction
Neutrinos they are very small.
They have no charge and have no mass And do not interact at all.
The earth is just a silly ball
To them, through which they simply pass, Like dustmaids down a drafty hall Or photons through a sheet of glass.
John Updike, 1960 [1]
The existence of the electron neutrino was rst postulated by Pauli in 1930 to understand β decays.
The rst experiment that observed neutrinos was an experiment by Reines and Cowan in 1956 [2]. This experiment and other early neutrino experiments never gave a hint that neutrinos might have a mass.
When Updike wrote the poem quoted at the beginning of this chapter, scientists assumed that neutrinos had no mass. In the last 20 years experimenters have found evidence that neutrinos do have mass, in contradiction with the postulates of the Standard Model (SM).
The SM is a widely accepted theory that describes the properties of the fundamental particles, quarks and leptons like electrons and neutrinos, as well as describing how they interact according to three of the four fundamental forces, namely the electromagnetic force and the strong and weak forces. The SM has been very successful in particle physics in the 20th century. A well known prediction of the last years is the Higgs mechanism, which describes how particles acquire mass. The existence of the Higgs boson was con rmed at the European Organization for Nuclear Research (CERN) in 2012.
But for all its power, the SM has some shortcomings as well. One of the unanswered questions is thus the masses of neutrinos, and besides that, the measured neutrino oscillations. The biggest limitation though is that gravity, the fourth fundamental force, is not included in its formulation. Attempts to include gravity have so far been unsuccessful. Other unsolved problems by the SM are, among other things: Dark Matter, Dark Energy, Baryogenesis, Baryon Asysmetry of the Universe (BAU) and lepton universality. While these frictions hint at a hidden better theory, we are yet to nd evidence of it.
To get more insight there is Beyond Standard Model (BSM) research necessary. A viable approach is to look at charged Lepton Flavor Violation (cLFV). The search for cLFV has continued from the early 1940's, when the muon was identi ed as a separate particle, until today. Even though neutrino oscillations did proof that lepton avor is not conserved for neutrinos, a avor conservation law seems to hold for the charged leptons. Typical SM branching fractions for cLFV processes are of the order O(10−54)or smaller, making their contribution unmeasurably small. A measured cLFV decay will thus unambiguously claim New Physics (NP). The decay channel B(s)0 → e±µ∓ is such a cLFV decay. The LHCb detector at CERN has provided data to study this decay. In the searches for cLFV, the aim is to set a smaller upper limit onto this ratio than the previous one, or to ultimately obtain a positive result.
To be able to extract the upper limit of the branching fraction a lot of cuts need to be performed on the data. One of these cuts is a cut on the Particle Identi cation of the electron (P IDe).
To achieve this, rst a brief introduction to the SM is given in Chapter 2. In Chapter 3 we will study the physics behind B(s)0 → e±µ∓ and have a look at the theoretical and experimental status of NP.
In chapter 4, an overview is given about the LHCb detector. Chapter 5 contains the main topic, here two dierent methods will be explained and used to analyze the data. One of these methods is a new developed method to extract the e ciency without knowing the shape of the signal. Finally, a summary and outlook is given in chapter 6.
Chapter 2
Particle physics and the SM
2.1 The SM
Particle physics is the study of the fundamental constituents of matter and the forces between them.
For more than 40 years these have been described by the SM. The SM is a quantum eld theory which contains two types of particles, namely fermions with half integer spin and bosons with integer spin. The SM contains three ` avors' of fermions. Each avor group consists of two leptons, two quarks and for each particle an antiparticle. All matter we know is composed of (anti)quarks and (anti)leptons. Apart from the fermions, the SM contains gauge bosons which are force carriers of the strong, electromagnetic and weak forces and the Higgs boson, responsible for the masses of SM particles. The three fundamental forces known in the SM are, in order of descending strength, the strong force, the electromagnetic force and weak force. Processes due to the strong force occur within 10−22 seconds and processes due to the electromagnetic force take place in 10−14 to 10−20 seconds. Processes with the weak interaction are relatively `slow' and happen typically within 10−8 and to 10−13 seconds, with extremely faster and slower exceptions [3].
The force carrier of the strong force is the gluon. The gluons comes in 8 types, called colors, and are all massless. The force associated with each of these particles has in nite range. The carrier of the electromagnetic force is the photon, the carriers of the weak force are the W+, W−, and Z bosons. The weak force has a very short range. These bosons should be massless to conserve the gauge symmetry, however, the symmetry breaking induced by the Higgs eld changes the W+, W−, and Z into particles with mass and leaves the photon massless. All four of them have spin S = 1. For an explanation of the Higgs mechanism we refer to [4]. In 2012 the Higgs boson was detected and therefore added to the SM. One of the largest shortcomings1 of the SM is the absence of gravitation. The graviton has been postulated as the carrier of the gravitational force, which would include gravity.
The lepton family includes the well known electron e−. Associated with this electron is an elusory particle named the electron neutrino νe. The other members of the lepton family are the the tauon τ− with its neutrino ντ and the muon µ with its neutrino νµ. All together those leptons give three doublets
νe e−
, νµ
µ−
, ντ
τ−
. (2.1)
The electron, muon and tauon have an electric charge of −1e and interact with other charged particles via the electromagnetic force and the weak force, whereas the neutral charged neutrinos interact only via the weak force. These properties are replicated for each doublet, called generation. The only dis- tinctive feature between the generations is the increasing masses of the particles. The electron is stable, the muon has a lifetime of 2.197 × 10−6 seconds and the tau of 290.3 × 10−15 seconds. All six lep- tons have antiparticles with opposite charge. The antiparticle of the electron (e+) is called the positron.
The antiparticles of the muon (µ+), tauon (τ+) and neutrinos (¯νe, ¯νµand ¯ντ) do not have speci c names.
Figure 2.1: Overview of the dierent elementary particles and force carriers [5].
Just like the leptons, quarks come in six types and are divided into three generations. The six types are the up u, down d, strange s, charmed c, bottom b, and top t quarks, each with their own antiquark, ¯u, ¯d,
¯
s, ¯c, ¯b and ¯t, respectively. The b and t quarks are sometimes called beauty and truth, too. These quarks can be represented by three doublets as well:
u d
, s
c
, b
t
. (2.2)
Each doublet contains one up-type and one down-type quark with electric charges equal to +2/3e and
−1/3e respectively. Figure 2.1 shows an overview of all elementary particles. Quarks are sensitive to the strong force, but are sensitive to the electromagnetic and weak force as well. Since the strong force has in nite range with strength that does not decrease with increasing distance, an isolated quark would radiate in nite energy. So quarks are, with exception of the top quark which decays too fast, never found alone [6]. The particles composed of quarks and interacting by the strong interaction are known collectively as hadrons. Among the hadrons, the proton and neutron are well-known members of the group of particles called baryons, which are made up of three quarks. Another family of strongly interacting particles are the mesons, which are made up of a quark-antiquark pair. In the last few years tetraquarks and pentaquarks particles, consisting of 4 and 5 quarks respectively, have been observed as well. Most recently, the LHCb experiment has detected four pentaquarks which are between four and
ve times more massive than a proton [7, 8].
2.1.1 Quark mixing
To understand a bit more how the SM provides the tools for particle decays of dierent particles we introduce quark mixing. In gure 2.2 the Feynman diagram is shown for the decay of a neutron in a proton, an electron and an electron antineutrino, so we have the reaction n → pe+ν¯e. The contribution of each diagram to a given physical process can be calculated precisely in quantum eld theory with use of the Feynman rules. In this diagram the gW and the gud represent the coupling constants. To explain experimentally observed decays the conclusion was that the mass eigenstates (u0, d0, t0) of the quarks are
linear combinations of the avor eigenstates (u, d, t):
d0 s0 b0
=
Vud Vus Vub
Vcd Vcs Vcb
Vtd Vts Vtb
d s b
. (2.3)
This phenomena is called quark mixing and the matrix given by Vαβ (α = u, c, t; β = d, s, b) is known as the CKM matrix, named after the founders Cabibbo, Kobayashi and Maskawa [9]. This should be a unitary matrix to make sure that d0, s0 and b0 are orthonormal single quark states as well. The values of the coupling constants of the quarks are now obtained by [2]
gαβ= gWVαβ. (2.4)
So, the CKM matrix describes the probability amplitude of a transition from one up-type quark α to a down-type quark β . These transition rates are proportional to |Vαβ|2. The most recent experiments and t sets the magnitudes of the nine CKM matrix elements [9]
VCKM =
0.94727 ± 0.00014 0.22536 ± 0.00061 0.00355 ± 0.00015 0.22522 ± 0.00061 0.97343 ± 0.00015 0.0414 ± 0.0012
0.00886+0.00033−0.00032 0.0405+0.0011−0.0012 0.99914 ± 0.00005
. (2.5)
For neutrinos there exists a similar mixing matrix, called the PMNS matrix, which will be treated in section 3.1.2. If a complex phase δCP appears in the CKM or PMNS matrix Charge Parity (CP) violation2 is allowed in the SM. A complex phase appears only if there are at least three generations.
Since there are three generations in the SM CP violation is possible. The phase δCP has been measured in the quark sector, values can be found in [10].
Figure 2.2: Feynman diagram of the dominant neutron decay [2].
2.1.2 The Standard Model and symmetry groups
The standard model needs to be invariant under speci c local phase transformations and can therefore be represented as the symmetry group3
U (1) × SU (2) × SU (3) (2.6)
Each symmetry is associated with a conserved quantity, such as energy, momentum, or charge. Each interaction has its own symmetry group. U(1) is the symmetry group of the electromagnetic interac- tion, SU(2) the symmetry group of the weak interaction and SU(3) the symmetry group of the strong interaction. The conserved quantity of a gauge symmetry is the charge (electric charge, isospin, color)
[13]. Uni cation attempts to treat all interactions as one, with the same coupling constant and the same symmetry group. A grand uni ed theory (GUT) uni es all three interactions of the SM at high energies, where the coupling constants approach each other. The most popular symmetry group for uni cation is SU (5)[14]. It is up to experiments to nd out what the correct representation is.
2.2 Shortcomings of the SM
No amount of experimentation can ever prove me right; a single experiment can prove me wrong.
Albert Einstein The SM has been very successful. One of these large successes is the detection of the predicted Higgs boson a few years ago. Even though the SM is currently the best description of the subatomic world, it does not explain the complete picture. There are several experimental observations that can not be explained by the SM and many theoretical aspects remain unsatisfactory. A few of these issues will be discussed brie y.
2.2.1 Experimental issues
• One of the most straigtforward examples of unexplained measurements is the evidence of dark matter and dark energy. Many cosmological observations cannot be explained, so dark matter is necessary to explain the expansion of our Universe. Already back in 1933 it was measured that there should be more luminous matter than we know [15] and in the 1970s there was found evidence of non-luminous matter [16, 17]. For a complete overview of dark matter we refer to [18], but since the matter we know only makes up less than 5% of the Universal content it is certain that there is still a lot to be discovered [19].
• Measurements of neutrino avor mixing, explained in section 3.1, suggest that neutrinos have mass, whereas the SM postulates massless neutrinos. However, this can be solved by a minimal extension of the SM. In fact, the detection that neutrinos do have mass is an example of detected Beyond the Standard Model (BSM) physics.
• In our universe there is a large unexplained and unexpected asymmetry between matter and anti- matter, which is known as the Baryogenesis problem. In 1967 three conditions on the baryon- generating interaction were proposed such that this asymmetry can be explaind [20]. One of these conditions is a CP-violation with a magnitude larger than the SM allows.
• With the current Standard Model the Baryon Asymmetry of the Universe (BAU), which gives rise to a baryon/photon ratio of ηB ≈ 6.2 × 10−10 [21], can not be explained qualitatively.
• Several hints for lepton non-universality are measured. The LHCb has for example measured the ra- tios R(D∗) ≡ B(B0→ D∗+τ−ντ)/B(B0→ D∗+µ−νµ)and R(K+) ≡ B(B+→ K+µ+µ−)/B(B+→ K+e+e−)and found results of R(D∗) = 0.336 ± 0.027 (stat) ± 0.030 (syst)and
R(K+) = 0.745+0.090−0.074(stat) ± 0.036 (syst) respectively, hinting towards lepton non-universality within 2 − 3σ [22, 23].
2.2.2 Theoretical issues
• The most obvious shortcoming of the SM is the absence of gravity. The SM explains only three out of the four fundamental forces, and can not explain gravity due to the lack of a quantum description [24]. The SM is regarded as incompatible with general relativity, which is the best theory of gravity we have today. Note that, although the name suggests otherwise, even the weak force is a lot stronger than gravity, so gravity is eectively negligible on the scale of particle physics.
• The experimental fact that the proton charge is equal but opposite in sign to the electron charge
• The gauge couplings of the electromagnetic, weak and strong force do not match at high energies, which would seem natural. This is shown in gure 2.3. GUTs and supersymmetry are examples of NP models where couplings at high energies do match.
• Three of the most fundamental constants, namely the speed of light c, Planck constant h, and Newton gravitational constant GN, can be combined to give a quantity with the dimension of mass, the Planck mass: Mpl =p
~c/GN = 1.2 × 1019 GeV/c2. Hence it is expected that dimensionful parameters are zero if forced by a symmetry and else of the order of Mpl. In the SM, the electroweak symmetry is broken and the Higgs boson mass is non-zero, but mh= 126GeV/c2. The SM provides no answer for the question why mh Mpl [6, 26]. This is called the gauge hierarchy problem.
• The large range of masses in the fermion mass spectrum, ranging from ∼ 170 GeV/c2 for the top-quark to ∼ 10−3 GeV/c2for the electron [25].
• An unsatisfactory large number of free parameters in the SM, mainly coming from the Yukawa couplings4 and avor mixing parameters.
Figure 2.3: Relative strengths of the electromagnetic force (α1), the weak force (α2) and the strong force (α3) coupling constants [29].
We just named a few of the shortcomings of the SM, so although the SM accurately describes the phenomena within its domain, it is still believed to be an incomplete theory. Perhaps the SM is a low-energy version of a more complete theory that still needs to be constructed. There are a lot of BSM models that try to explain the issues. Several of these theories will be discussed in section 3.2.
The question remains which theory is the right one, or is at least a step closer towards a Theory of Everything. This question can only be answered via experiments, where there is searched for decays or observables that are forbidden by the SM to nd more information on NP. Next we will discuss the physics behind the decay B0(s)→ e±µ∓, which is such a forbidden decay.
Chapter 3
The physics behind B (s) 0 → e ± µ ∓
3.1 Neutrino oscillations and Lepton Flavor (Violation)
3.1.1 Conservation Laws
This section relies strongly on reference [2].
Particles accelerated to high-energy in modern accelerators collide to produce an astounding variety of new particles. Based on years of experiments conservation laws are de ned and describe which reac- tions can and cannot occur. Energy, momentum and charge are for example three conserved quantities, which are also conserved in other elds of science. Another quantity that is conserved in particle re- actions is the baryon number, which can be expressed in terms of the number of quarks N(q) and the number of antiquarks N(¯q) by the following formula:
B = 1
3[N (q) − N (¯q)] (3.1)
Since baryons consist of three quarks they have baryon number +1. Antibaryons consist of three anti- quarks and thus have baryon number −1. For the mesons made of a quark-antiquark pair we get baryon number zero.
Baryon number is a conserved quantity in strong, weak and electromagnetic interactions because in these interactions quarks and antiquarks are created or destroyed only in particle/antiparticle pairs. In a similar manner we have Lepton number conservation, with N(l) the number of leptons and N(¯l) the number of antileptons:
L = N (l) − N (¯l). (3.2)
An example of a forbidden reaction is the decay (3.3). Although the baryon number is conserved (B = +1 at both sides), the lepton number is not conserved, since we have L = −1 at the left-hand side and L = +1 at the right-hand side.
¯
νen → e−p (3.3)
This reaction is thus inconsistent with lepton conservation and indeed has not been found to occur, whereas νen → e−p conserves lepton number and has indeed been detected. However, some reactions are consistent with all the conservation laws discussed so far and still do not occur. An example of such a reaction is
µ− → e−γ. (3.4)
This and other unobserved reactions led to an extension of the conservation law on lepton number. Not only the total lepton number needs to be conserved, but the lepton number of each avor needs to be conserved as well. So we introduce muon number Lµ, electron number Leand tauon number Lτ, de ned in the same way as L, which all need to be conserved. Now it can easily be seen that the decay (3.4) is forbidden. At the left-handside we have L = 1, L = 0, whereas the righthandside has L = 0, L = 1.
3.1.2 Neutrino oscillations
The standard solar model is a model of our sun that predicts the sun's observable features, like solar neutrino uxes on earth. Solar neutrinos were measured for the rst time in the Homestake Chlorine experiment in 1968 [30, 31]. Since then in dierent experiments solar neutrinos were measured, like the gallium experiments [32, 33], the (Super)Kamiokande [2, 34] and the Sudbury Neutrino Observatory (SNO) experiment [33, 35].
Figure 3.1: Solar neutrino energy spectrum for the solar model BS05(OP) [36].
Figure 3.1 shows the solar neutrino energy spectrum that is calculated using the BS05(OP) solar model, which may be taken as the currently preferred solar model [37]. A discussion about the dierences be- tween dierent solar models can be found in [36]. The BS05(OP) solar model gives an accurate prediction of the solar neutrino uxes.
Figure 3.2: Predicted versus experimental neutrino uxes [38].
The results of abovementioned experiments are compared to those predicted by the preferred standard
is the oscillation of neutrinos. The oscillation of neutrinos can occur because the mass eigenstates of the neutrinos are linear combinations of the avor eigenstates, just like the mixing of quarks. For the mixing of neutrinos we have a matrix similar to the CKM matrix for quark mixing. This is the PontecorvoMakiNakagawaSakata (PMNS) matrix [39]:
νe
νµ
νtau
=
Ue1 Ue2 Ue3
Uµ1 Uµ2 Uµ3
Uτ 1 Uτ 2 Uτ 3
ν1
ν2
ν3
. (3.5)
The current best- t values for the magnitudes of the PMNS components, using direct and indirect measurements are [40, 41]
UP M N S=
0.82 ± 0.01 0.54 ± 0.02 0.15 ± 0.03 0.35 ± 0.06 0.70 ± 0.06 0.62 ± 0.06 0.44 ± 0.06 0.45 ± 0.06 0.77 ± 0.06
. (3.6)
The phase δCP is not yet determined in the neutrino case. Neutrino mixing and neutrino oscillations require that neutrinos have nonzero masses, in contrast with the rst postulates of the SM. Instead of considering mixing between all three avor states it is simpler, and often a good approximation, to consider the mixing between just two of them. The following derivation relies strongly on [2] and is done in natural units1. If we denote the two states ναand νβ, we can write
|ναi = |νii cos θij+ |νji sin θij|νβi = − |νii sin θij+ |νji cos θij (3.7) where νiand νj are the two mass eigenstates involved and θijis a mixing angle that must be determined from experiment. To illustrate neutrino oscillations we will consider a να produced with momentum ¯p at time t = 0, we then have initial state
|να, ¯pi = |νi, ¯pi cos θij+ |νj, ¯pi sin θij. (3.8) After time t this will become
ai(t) |ναi |νi, ¯pi θij+ aj(t) |νj, ¯pi sin θij, (3.9) where ai(t) = e−Eitand aj(t) = e−Ejtare the usual oscillating time factors associated with any quantum mechanical stationary state2. For t 6= 0 the linear combination in equation (3.9) does not correspond to a pure ναstate, but can be written as the linear combination
A(t) |να, ¯pi + B(t) |νβ, ¯pi (3.10) of να and νβ states, where |νβ, ¯pi is given by
|νβ, ¯pi = − |νi, ¯pi sin θij+ |νj, ¯pi cos θij. (3.11) Using formulas (3.8)−(3.11) we nd:
A(t) = ai(t) cos2θij+ aj(t) sin2θijB(t) = sin θijcos θij[aj(t) − ai(t)] (3.12) The probability of nding a νβ state at time t is therefore
P (να→ νβ) = |B(t)|2= sin2(2θij) sin2[1
2(Ej− Ei)t]. (3.13)
This probability thus oscillates with time, while the probability of nding a να particle is reduced by a corresponding oscillating factor. The lengths of the oscillations are typically of order 100km or more, so that oscillations can be safely neglected under normal laboratory conditions. Nevertheless, they become important if the travel distance is as large as the distance to the sun. With use of the neutrino oscillations
the predictions of the standard solar model are experimentally con rmed.
Figure 3.3: Pattern of neutrino masses for the normal and inverted hierarchies is shown as mass squared. Flavor composition of the mass eigenstates as the function of the unknown CP phase δCP is indicated. ∆m2atm ∼ |∆m231| ∼ |∆m232| and
∆m2sol ∼ |∆m21|2 stands for the atmospheric and the solar mass-squared splitting, respectively [44].
The rst experiment that observed neutrinos was an experiment by Reines and Cowan in 1956. In fact, they used antineutrinos emitted from a nuclear reactor at Savannah River, North Carolina [2]. This experiment and other early neutrino experiments never gave a hint that neutrinos might be massive, so the SM rst postulated a nonzero mass for neutrinos. The existence of the avor changing neutrino oscillations, and by implication nonzero neutrino masses, is now generally accepted. The neutrino oscil- lations are used to measure the mass dierence between the dierent neutrino mass states as well. With use of the special relativity equation E2= m2+ p2the term (Ej− Ei)becomes
Ej− Ei= (m2j+ p2)1/2− (m2i + p2)1/2≈ ∆m2ji
2p , (3.14)
where ∆m2ji≡ (m2j− m2i). Up to now it cannot be determined whether the ν3 neutrino mass eigenstate is heavier or lighter than the ν1and ν2neutrino mass eigenstates. The scenario in which the ν3is heavier is called the normal mass hierarchy (NH). The other scenario, in which the ν3 is lighter, is called the inverted mass hierarchy (IH). Both scenarios are shown in gure 3.3, calculated numbers can be found in [44]. It is known from cosmological observations that a neutrino must be at least a million times lighter than an electron, but the exact mass of each type of neutrino remains unknown [45].
It should be emphasized that the oscillations of neutrinos directly mean that lepton avor conservation is violated for neutrinos, and thus for the neutral leptons. The question now comes up if we have Lepton Flavor Violation for charged leptons (cLFV) as well.
3.1.3 cLFV revisited
The discovery of neutrino mass and neutrino oscillations guarantees that in the SM cLFV must occur through neutrino oscillations in loops. Figure 3.4a and 3.4b show examples of Feynman diagrams of the cLFV decays µ− → e−γ and Bs0 → e+µ− in the SM. Such transitions are suppressed by sums over (∆mij/mW)4, which de nes the branching fraction3 of the cLFV decay µ− → e−γ of the order of
Figure 3.4c shows for example the Feynman diagram of the cLFV decay ¯Bs0 → e+µ− with use of a leptoquark. In the next section several NP models that predict measurable cLFV will be discussed.
(a) The decay µ− → e−γ in the
SM. (b) The decay B0s → e+µ− in the
SM. (c) The decay ¯Bs0 → e+µ− with
use of a leptoquark.
Figure 3.4: Feynman diagrams for cLFV decays.
3.2 New Physics in theory
The Standard Model describes everything we see in the laboratory. Aside from leaving gravity out, it's a complete theory of what we see in nature. But it's not an entirely satisfactory theory, because it has a number of arbitrary elements. For example, there are a lot of numbers in this SM that appear in the equations, and they just have to be put in to make the theory t the observation. For example, the mass of the electron, the masses of the dierent quarks, the charge of the electron. If you ask, "Why are those numbers what they are? Why, for example, is the top quark, which is the heaviest known elementary particle, something like 300,000 times heavier than the electron?" The answer is, "We don't know. That's what ts experiment." That's not a very satisfactory picture.
Steven Weinberg This section relies strongly on thesis [47]. A study was done on which new physics theories were most discussed. Here we give a summary of the given overview.
A general rule that will come back in the dierent models is that the more vertexes, with each their own coupling constant, are present in a Feynman diagram the more the reaction is suppressed. Tree diagrams will in general thus have a larger amplitude than loop or penguin diagrams.
3.2.1 Seesaw models
One of the problems the SM cannot account for, is the small but non-zero mass of neutrinos. In the framework of the SM neutrinos would have zero mass. To explain the small but nonzero mass of the neutrinos, the SM is often extended with the Seesaw mechanism. All Seesaw models contribute to branching fractions of lepton avor violating processes. Due to the large number of unknown parameters, calculating branching fractions is very hard. Since these branching fractions often depend on the same parameters, calculating ratios can be done without making major assumptions on the size of these parameters.
Type I Seesaw
The Type I Seesaw model expands the SM with three fermions which are singlet under the SU(2) gauge group. These singlets are better known as the right-handed neutrinos (νR). Adding these particles results in a neutrino mass matrix
Mν =
0 mV
mTD mM
(3.15) Where mD and mM are 3 × 3 matrices relating to the Dirac and Majorana masses4. Just like in the
to the larger mass of νR, branching fractions of cLFV processes are increased. Calculated branching fractions are shown in gure 3.5.
(a) Branching fraction ratios for B(l → l0γ)/B(l → 3l0), where l stands for an arbitrary lepton and l0 also an arbitrary, but dierent, lepton.
(b) Branching fraction ratios for B(lN → l0N )/B(l → 3l0), where N stands for an arbitrary nucleus.
Figure 3.5: Branching fractions in Type I Seesaw [48].
Type II Seesaw
The Type II Seesaw mechanism adds a SO(2) scalar triplet ∆ with hypercharge5 Y = 2 to the SM.
This triplet contributes to the neutrino mass by generating a Majorana mass as a consequence of adding Yukawa term LSSII to the Lagrangian of the SM. This mechanism has four lepton interactions (l → 3l0) at the tree level due to the exchange of the scalar triplet ∆ and other cLFV interactions at the loop level. This results in rather large branching fraction for l0 → lllcompared to the other Seesaw models.
Calculated branching fractions are shown in gure 3.6.
Figure 3.6: Branching fraction ratios for B(µN → eN)/B(µ → eγ) against m∆ in Type II Seesaw.
Type III Seesaw
Type III Seesaw model introduces fermionic SU(2) triplets Σi with hypercharge Y = 0:
Σi=
Σ0k
√i
2 Σ+k
i
Σ−k
i −Σ0k
√i
2
(3.16)
At least two of those triplets have to be introduced to generate non-zero neutrino mass. Couplings generated in a Type III Seesaw framework can generate cLFV directly at the tree level. The only cLFV process that cannot happen at tree level is l → l0γ. This process still happens only on the loop level as a result of the avor diagonal coupling. With the Type III Seesaw model the following ratios can be calculated:
BR(µ → eγ) = 1.3 × 10−3× BR(µ → eee) BR(τ → µγ) = 1.3 × 10−3× BR(τ → µµµ)
= 2.1 × 10−3× BR(τ−→ e−e+µ−) BR(τ → eγ) = 1.3 × 10−3× BR(τ → eee)
= 2.1 × 10−3× BR(τ−→ µ−µ+e−) BR(µ → eee) = 2.4 × 10−1× BR(µN → eN )
BR(µ → eγ) = 2.4 × 10−4× BR(µN → eN )
(3.17)
Inverse Seesaw
The Inverse Seesaw model (ISS) introduces two SU(3) × SU(2) × U(1) singlets νC and S. This model has a neutrino mass matrix:
Mν=
0 mD 0
mTD 0 mM
0 mTM µ
(3.18)
with mD, µ and mM being 3 × 3 Dirac, lepton conserving, and Majorana mass matrices, respectively.
This Mν can be diagonalized to obtain the light neutrino masses. If an U(1) global lepton number symmetry is assumed, µ will be equal to 0, which would result in massless light neutrinos. However if this symmetry is broken, µ 6= 0, light neutrinos become massive particles. The mass matrix Mν can be used to calculate branching fractions for LFV processes. In this model the largest contributions to LFV processes come from the heavy singlet neutrinos νC and S in one-loop diagrams. The dierence between the branching fraction within the SM and the Inverse Seesaw, is the sum over nine neutrinos, three light (ν) and six heavy (νC and S) in ISS, instead of the sum over just the three light neutrinos in the SM.
Calculated branching fractions for µ → eγ are shown in gure 3.7.
Figure 3.7: BR(µ → eγ) against µ in the Inverse Seesaw and against νL in the Linear Seesaw. The continuous line represents mM = 100 GeV, the dashed line mM = 200GeV and the dot-dashed line mM = 1000GeV [49].
Linear Seesaw
The Linear Seesaw mechanism is similar to the Inverse Seesaw mechanism, since it adds a νc and a S neutrino, too. But its neutrino eective mass matrix is dierent,
Mν=
0 mD νLT
mTD 0 mM
νLT mTM µ
(3.19)
Just like in the Inverse Seesaw, light neutrinos become massive particles when the symmetry is broken and νL6= 0. In this model the ratio between the branching fractions of the decays τ → eγ and τ → µγ can be calculated to give
BR(τ → µγ)
BR(τ → eγ) = 4. (3.20)
Calculated branching fractions for µ → eγ are shown in gure 3.7.
LRSM model
The Left-Right Symmetric Model (LRSM) makes use of the Type I and II Seesaw models. This particular model shows how Seesaw mechanics, multiple coupling constants and masses can arise from the breaking of symmetries in a GUT theory. Additionally it has in uences on cLFV processes and neutrinoless double beta decay. The LRSM is based on the gauge group SU(2)L× SU (2)R× U (1)B−L which is embedded in a SO(10) GUT with Pati-Salam symmetry. In a Pati-Salam model the number of quark colors is extended from three to four colors. Besides the usual red, green and blue color the fourth color violet or lilac is added. This fourth color is better known as the leptons within our Standard Model. The scale of the breaking of the SU(2)L× SU (2)R× U (1)B−L symmetries govern the masses of neutrinos and scalar triplets in a LRSM. If the symmetries break at the TeV scale, right-handed neutrinos, scalar triplets, and the right-handed gauge bosons WR and ZR acquire TeV scale masses and Type I and Type II Seesaw mechanisms contribute to the neutrino mass. Calculated branching fractions for µ → eγ are shown in
gure 3.8.
Figure 3.8: The ratio BR(µN → eN)/BR(µ → eγ) for dierent sets of parameters [50].
While most LRSM models assume an explicit left-right symmetry (D-parity) at the TeV scales, i.e.
gL = gR, some of the LRSMs assume a spontaneous breaking of this D-parity at a higher scale than the SU(2)R breaking. Due to the breaking of D-parity the branching fractions will gain an extra factor
g
R
gL
4 .
3.2.2 Supersymmetry
In the model of supersymmetry (SUSY) every fundamental particle gets a supersymmetric partner. This partner can be found with a supersymmetric transformation. This transformation changes a bosonic state into a fermionic and vice versa, resulting in a fermionic partner for every bosonic fundamental particle and a bosonic partner for every fermionic fundamental particle. Quarks (S = 12) will partner up with squarks (S = 0), a gluon (S = 1) with a gluino (S = 12) et cetera. The Higgs scalar boson (S = 0) is the only particle with two supersymmetric partners instead of one, namely the two Higgs supermultiplets with S = 12 and S = −12. The model with the particles described above added is called the Minimal Supersymmetry Model (MSSM), shown in gure 3.9. To have supersymmetric particles, which are not degenerate with their SM partner, the supersymmetry must be broken. Gauge hierarchy can be maintained if the breaking is soft and the breaking mass parameters are no larger than a few TeV. One of the problems with SUSY is the `supersymmetric CP-problem'. The Electric Dipole Moment (EDM) for neutrons, electrons, and other fundamental systems predicted by SUSY models are typically larger than the current experimental bounds on these EDMs. Adding more particles to the Standard Model obviously adds more possible processes, including cLFV proceses. The main reason to introduce a supersymmetry is to solve to the hierarchy problem.
Figure 3.9: The particles contained in the MSSM [51].
νRMSSM
In νRMSSM the MSSM is extended by a Type I Seesaw mechanism, introducing right-handed neutrinos and right-handed sneutrinos to explain the small neutrino masses. In this model cLFV is induced by three mechanisms, namely the heavy neutrinos, heavy sneutrinos and soft SUSY breaking terms. The neutrino mass matrix is the same as in Seesaw Type I,
Mν =
0 mV
mTD mM.
(3.21) The sneutrino mass Lagrangian is used to calculate a sneutrino mass matrix in terms of the neutrino mass matrix Mν,
Mν˜=MνMν† 06×6
06×6 MνMν†.
(3.22) With this model the ratios of the branching fractions become
B → l0llC B → l0γ = α
3π
logm2l
m2l0
− 3
, (3.23)
B → l0l0l0C B → l0γ = α
3π
log m2l
m2l0
−11 4
, (3.24)
where α is the ne structure constant, mland ml0 are the masses of the leptons l and l0 respectively. lC is the conjugate of l, such that l and lC are a particle-antiparticle couple.
MSSM without R-parity
R-parity is de ned as R = (−1)3B−L+2S, where B, L and S are baryon number, lepton number and spin, respectively. In many MSSM models this R-parity is assumed to be conserved to prevent proton decay and prevent the decay of the supersymmetric particle with lowest mass. If this particle is stable, it becomes a potential candidate for Dark Matter. However, R-parity does not have to be conserved in a generic supersymmetric model to meet the constraints set by current experiments. The decay h0→ µ±τ∓ becomes possible at tree level in an R-parity violating MSSM via R-parity violating (RPV) neutral-scalar charged-lepton couplings. Branching fractions for this model have been calculated for multiple Feynman Diagrams. The most dominant decays have branching fractions in the range of 10−5− 10−3.
Compacti ed M-theory
In string theory the material making up all energy and matter is thought to consist of tiny vibrating strings that exist in a multi-dimensional (10 or 26 dimensions) hyperspace. The extra dimensions are thought to be compacti ed, or curled up, inside observable space. The vibrations of the strings within this multidimensional hyperspace are thought to correspond to particles that form the basis of all matter and energy. M-theory (the `M' stands for the mother of all theories, magic, mystery, or matrix, depending on the source) is an adaptation of superstring theory developed by Ed Witten of Princeton and Paul Townsend of Cambridge and is a single framework that subsumes all consistent versions of string theory.
LFV proccesses can occur in compacti ed M-Theory due to the avor structure of the Kähler potential, explained in [52]. l → l0γ decays can occur via loop diagrams. Numerical results can be found in gure 3.11, here the three shaded regions are restricted for a gravitino mass of 25 TeV (I), 35 TeV (II) and 50 TeV (III).
ISS-MSSM
The ISS-MSSM is a Minimal Supersymmetric Model extended with an Inverse Seesaw mechanism. As a result the νC and S are promoted to super elds ˜νC and ˜S. The mass matrix Mν is obtained in the same way as in the normal ISS model. Branching ratios for this model can be found in gure 3.12.
Non-Abelian RVV
The non-abelian Ross-Velasco Sevilla-Vives (RVV) model provides all the information about the quark- Yukawa matrices by assuming a non-abelian SU(3) avor symmetry that is embedded in an SO(10) GUT supersymmetric model. It solves the CP and avor problems that occur in SUSY models. The RVV describes the full mass and mixing matrices for quarks and leptons, and predicts large right-handed currents. Calculated branching fractions in this model are shown in gure 3.10.
Figure 3.10: RVV predictions for BR(µ → eγ) with respect to electric dipole mo- ment and to BR(τ → µγ) [53]. (More yellow means increased intensity).
AKM
The Antusch-King-Malinsky (AKM) model makes use of a gauged SU(3) family symmetry to solve problems concerning CP and avor within the supersymmetric model. Just like in the RVV model, a nonabelian SU(3) symmetry gives rise to the mass and mixing angle matrices for quarks and leptons.
The dierence between RVV and AKM is the freedom to suppress soft avor-changing terms. This model
(a) BR(µ → eγ)
(b) BR(τ → eγ) (which equals BR(τ → µγ)).
(a) BR(µ → eγ)
(b) BR(µ → 3e).
(c) BR(µ → e) in Aluminium.
Figure 3.13: AKM predictions for BR(µ → eγ) with respect to electric dipole moment and to BR(τ → µγ) [53]. (More yellow means increased intensity).
3.2.3 Leptoquarks
Leptoquarks are particles with both baryon number B and lepton number L. They can possibly couple to multiple avor generations and thus induce cLFV processes. Leptoquarks are currently not present within the standard Model, but are predicted by various extensions of the SM. Some of these extensions are the Pati-Salam model, GUTs based on SU(5) or SO(10) with Pati-Salam color SU(4), and leptoquark-type Yukawa-coupling in R-parity violating SUSY models. Some of these predicted leptoquarks can couple to right- and left-handed fermions, causing increased LFV branching fractions.
A leptoquark model with leptoquark scalars χ13 and ω23 was introduced to explain the small neutrino masses. The light neutrino masses are generated by two-loop radiative corrections due to the leptoquarks.
The scalar leptoquarks have SU(3) × SU(2) × U(1) representation:
ω23 ω−13
∼ (3, 1, −1
6), χ13 ∼ (3, 1, −1
3) (3.25)
Minimum branching fractions can be calculated for µ → eee using bounds on µN → eN and neutrino mass ttings, shown in gure 3.14.
Figure 3.14: Branching fraction for BR(µ → eee) using neutrino mass tting and constraints from BR(µ → eee) [56].
More recently lepton avor violating B meson decay via scalar leptoquark were studied, predicting upper bounds on the branching fraction in the range 10−10− 10−6. The details and results of this study can be found in [57].
3.2.4 Leptogenesis
Leptogenesis is similar to Baryogenesis with an asymmetry between leptons and anti-leptons instead of an asymmetry in baryons. A leptogenesis model that introduces heavy Majorana neutrinos could explain the baryon asymmetry of the universe, since these neutrinos can decay rapidly in a fast expanding universe6. The Resonant Leptogenesis (RL) model is a low-scale thermal leptogenesis model, where the low-scale is produced by a dynamical mechanism in which heavy-neutrino self energy eects on the leptonic asym- metry become dominant and get resonantly enhanced by a pair of heavy Majorana neutrinos that have a mass dierence equal to the decay width. A modi cation of this model, where a single avor symmetry is resonantly produced, could lead to an explanation of the Baryoleptogenesis by means of sphalarons.
Sphalerons preserve the quantum number13B −Li. As a result the excess of leptons can be converted into the observed BAU. The existence of three dierent lepton avors results in three possible models, RµL, ReL, and RτL, which generate the baryon asymmetry from an excess of µ, e and τ leptons, respectively.
Breaking of the SO(3) symmetry on the singlet Majorana basis gives rise to the subgroup of lepton symmetries: U(1)Li+Lj × U (1)Lk. With these symmetries one can obtain a neutrino Yukawa-coupling matrix for each of the models. Branching fractions for the three possible models have been calculated with a normal and an inverted hierarchy of the light neutrino masses. Parameters are chosen to maxi- mize the overlap between the required generated baryon asymmetry and high LFV rates. The calculated branching fractions of µ → e conversion in4822Ti are shown in table 3.1.
RµL ReL Rτ L
normal hierarchy ≈ 2 × 10−17 ≈ 3 × 10−17 ≤ ×10−16 inverted hierarchy ≈ 5 × 10−16 ≈ 7 × 10−16 ≤ ×10−16 Table 3.1: Branching fractions for the µ → e conversion in4822Ti
6Neutrinos are Majorana if they are their own anti-particle.
The three branching fractions l → l0γ are calculated to be:
ReL : BR(τ → µγ) ≈ 10−14 RµL : BR(τ → eγ) ≈ 10−12 Rτ L : BR(µ → eγ) ≤ 10−10
(3.26)
It needs to be stressed that we only studied the most discussed theories in this section. Although these theories get a lot of attention, these theories are still speculative. There are many more interesting theories, for example renormalizable models [58], that are less discussed, but could however be closer to the truth. In the end experiments will decide which theories remain feasible.
3.3 Experimental status of cLFV
This section relies strongly on thesis [59]. Here we give a summary of the given overview.
In this section we will give an overview of the experimental status and prospects on the search for charged Lepton Flavor Violation (cLFV). Well-researched cLFV channels are, among others, rare muon decays, tau decays, rare kaon decays, lepton conversions and processes involving hadronic resonances7or heavy quarks. No cLFV has been observed up to now, but the limit on the branching fractions is getting more stringent after each new result. In the searches for cLFV, the aim is to set a smaller upper limit onto this ratio than the previous one, or to ultimately obtain a positive result. The rst experiment ever intended to detect cLFV was done in the late 1940s by Hincks and Pontecorvo. Nowadays it is a very active eld of research. We will discuss the most recent experiments and future projects concerning muon decays, tauon decays, meson decays and other relevant cLFV experiments.
3.3.1 Muon decays
For muon decays just a few channels are possible due to energy-momentum conservation. The most dominant decays are µ → eγ, µ → eee and muon conversion. An overview of the history of the upper limits found for these decays is shown in gure 3.15. A muon has a lifetime about 2.2µs, which makes it relatively easy to detect. The most recent result on µ → eγ was found at the MEG (Muon to Electron and Gamma) experiment, located at the Paul Scherrer Institute (PSI) in Switzerland and published in 2013. The current best upper limit on the branching ratio is 5.7 × 10−13 (90% con dence level), which is four times more stringent than the previous best limit set by MEG. An upgrade proposal for the MEG experiment, which will be called MEG II, has been approved in 2013. The MEG II experiment is expected to run for three years starting from 2016 and tends to get a branching fraction of 6 × 10−14, so one order of magnitude smaller than the current upper limit.
The current best upper limit of 1.0×10−12 on the µ → eee branching fraction was set back in 1986 at the former Swiss Institute for Nuclear Research (SIN, nowadays a part of PSI) with the SINDRUM magnetic spectrometer, which was operating since 1983. At PSI a new project that will look for the µ → eee decay is currently under development. The experiment, called Mu3e, will be performed in two phases, namely an exploratory rst phase that started this year and a second phase with a new high-intensity beamline starting after 2017. Besides commissioning and validating the experimental techniques, the aim of the rst phase is to push the existing upper limit by three orders of magnitude, meaning a limit on the branching fraction of 10−15. During this stage only some parts of the nal setup are used.
The nal aim is to exclude a branching fraction larger than 10−16at 90% con dence level during phase II.
7Hadronic resonances are hadrons in an excited state.
Figure 3.15: Overview of the history of the branching fractions in cLFV searches in muons [46].
Another possible muonic cLFV channel is the process of a negative muon being trapped by an atomic nucleus after which the muon converts into an electron, known as muon conversion. Additionally, the muon can undergo two other possible non avor violating processes, namely muon decay in or- bit (µ−(A, Z) → e−νuν¯e(A, Z)) and muon capture (µ−(A, Z) → e−νµ(A, Z − 1)). The most recent project to search for muon to electron conversion has been executed with the SINDRUM II spectrometer at PSI. The search resulted in upper limits on the branching fractions of µ−e conversion in muonic gold, titanium and lead, leading to the branching fractions
BR(µ−P b → e−P b) < 4.6 × 10−11, BR(µ−T i → e−T i) < 6.1 × 10−13, BR(µ−Au → e−Au) < 7.0 × 10−13,
(3.27)
extracted in 1996, 1999 and 2006, respectively. A design for another facility to search for muon conversion has been proposed by the Mu2e Collaboration at the Fermi National Accelerator Laboratory (Fermilab) in the United States which aims for an estimated upper limit of 6×10−17. This is four orders of magnitude smaller than the SINDRUM II upper limit. The Mu2e commissioning phase is scheduled in 2019 and data taking should start in 2020 with an expected three year operating period.
Another future muon conversion experiment with a similar setup to Mu2e and almost the same sensitivity aim of 10−16is COMET (COherent Muon to Electron Transition), currently under development at Japan Proton Accelerator Research Complex (JPARC). This experiment will be staged in two phases so that COMET Phase I can provide useful information settling the uncertainties related to new and therefore unknown techniques for the full-sized COMET experiment (COMET Phase II). Apart from collecting experimental data to prepare for the nal phase experiment, a search for muon conversion during COMET Phase I with an expected sensitivity of 3 × 10−15 will take place starting in 2018. In the Phase I stage there will be a parallel search for the µ−A → e+Adecay and the, never before measured, µ−µ− → e−e− decay besides the muon conversion. A longterm upgrade for the COMET experiment targetting for a sensitivity of order 10−18 and beyond is the PRISM (Phase Rotated Intense Slow Muon source) project.
In 2010 an experimental search for muon conversion at the JPARC has been proposed aspiring for a sensitivity of 10−14. This would be an improvement of nearly two orders of magnitude with respect to the current experimental upper limit. Due to less complex design, shorter time scale and lower costs this experiment, called DeeMe (Direct Emission of Electrons by Muon-Electron conversion), should be able to provide an intermediate result earlier than the Mu2e and COMET experiments aiming for an another two orders of magnitude smaller upper limit.
3.3.2 Tauon decays
Since the tauon mass is about 17 times the muon mass, it has many more decay modes than the muon.
In particular hadrons are now possible decay products as well. Therefore a wide variety of tauonic cLFV modes are being and have been studied. An overview is shown in gure 3.16.
Figure 3.16: 90% C.L. upper limits of cLFV tauon decay branching ratios as sum- marized in summer 2014 by HFAG [60].
The upper limits of the tauon decays, reaching down to 10−8, are signi cantly larger than recent upper limits on muon branching fractions. Due to its short lifetime (2.99 × 10−13s) a tauon beam cannot be realized which results in higher upper limits. Tauons are, for example, produced as tau-antitau pairs at e+e− colliders such as Belle (Japan) and BaBar (United States). These e+e− colliders are actually B meson factories, but they serve as τ factories as well, since almost as many τ+τ− pairs as b¯b pairs are produced. The LHCb experiment at CERN produces tauons from proton collisions, mainly via leptonic Dsdecays [61]. Furthermore, the larger amount of possible decay modes of the tauon leads to a relatively smaller branching fraction for each mode, which reduces the respective branching fraction. Besides this a tauon lacks a conversion process due to the small lifetime, which also leads to larger upper limits than searches for muon decays. Future perspectives in Japan, Italy and at LHCb are expected to lower the upper limits to an order of 10−9 to 10−10.
3.3.3 Meson decays
Whereas the previously discussed muon cLFV processes are only lepton avor violating, meson decays can be lepton number violating as well, which is inherently lepton avor violating. Meson cLFV searches are performed quite often. A complete overview of the current most stringent upper limits can be found in [25] or at the website [62]. An indicative selection of this overview is summarized in table 3.2, ordered by their sensitivities. Most of the present upper limits are of comparable order of magnitude as the tauon limits and thus far above the sensitivities reached in muon experiments. Note that the three most sensitive decays owe their sensitiviy to the ability to produce beams of the initial mesons.
Decay Upper limit (90% C.L) Experiment Year Ref.
DS+→ K+e−µ+ 9.7 × 10−6 BaBar 2011 [63]
Υ(3S) → µ±τ∓ 3.1 × 10−6 BaBar 2010 [64]
φ(1020) → e±µ∓ 2 × 10−6 SND 2010 [65]
D+→ K+e+µ− 1.2 × 10−6 BaBar 2011 [63]
J/ψ → e±µ∓ 1.6 × 10−7 BESIII 2013 [66]
B+/0→ K+/0e±µ∓ 3.8 × 10−8 BaBar 2006 [67]
D0→ e±µ∓ 1.6 × 10−8 LHCb 2016 [68]
Bs0→ e±µ∓ 1.1 × 10−8 LHCb 2013 [69]
B0→ e±µ∓ 2.8 × 10−9 LHCb 2013 [69]
π0→ e+µ−+ e−µ+ 3.6 × 10−10 KTeV 2008 [70]
K+→ π+µ+e− 1.3 × 10−11 BNL E865 2005 [71]
KL0 → e±µ∓ 4.7 × 10−12 BNL E871 1998 [72]
Table 3.2: Selection of current lowest upper limits on cLFV decays.
Mesons are heavy particles and thus have many dierent decay modes. Both tauons and heavy mesons cannot be detected directly due to their short lifetimes and thus they can only be detected through their decay products. Hence meson experiments require a higher level of particle identi cation (PID) compared to muon decays, leading to lower upper limits. Decay modes containing a tau lepton tend to be more di cult to access experimentally because of the multiple decay modes of the tauon and missing energy caused by neutrinos. This is why decay modes with a tauon are less studied than decays with electrons and/or muons in their nal state, as can be seen in table 3.2. Consequently searches for meson decays involving tauons require special experimental techniques which leads to lower upper limits due to limited signal e ciencies.
Future prospects for meson decay are at Belle II, the upgrade of the Belle experiment in Japan, the SuperB project in Italy, an upgraded accelerator complex at Fermilab and further searches at LHCb.
3.3.4 Other cLFV decay modes
In 1998 there has been searched for the spontaneous muonium to antimuonium conversion µ+e−→ µ+e at PSI. An upper limit of 8.2 × 10−11(90% C.L.) on the branching fraction has been established. Future experiments could be planned at facilities with high-intensity pulsed muon beams.
Another cLFV channel which has been comprehensively searched for is the A → A∗+ 2e− decay, called neutrinoless double beta decay. This decay is obviously LFV since it is lepton number violated by
∆L = 2. Results for this decay can be found in [73].
A recent search for Z0→ eµhas been conducted at ATLAS, one of the detectors at the Large Hadron Collider (LHC) at CERN. The resulting upper limit on the branching fraction of 7.5×10−7(95% C.L.) has been published in 2014. Only one other experiment searched for Z0cLFV decays. This was in 1996 at the DELPHI experiment, once located at CERN too. Future searches for cLFV Z0 decays can be expected from the particle detectors at CERN, as well as from Belle II and other prospective meson factories as a by-product of other Z0decays. The same statement holds for the Higgs decay h0→ τ µ. CERN plans to perform precision measurements on the Higgs boson with ATLAS and CMS and to increase the luminosity of the LHC by a factor 10. This would increase the feasibility to search for a cLFV decay as a by-product.
It is clear that the four main detectors at CERN are in uential players in the search for a cLFV decay.
Next we will discuss the setup of one of these four experiments, namely the LHCb detector. Because of its excellent vertex and momentum resolution coupled with very good particle identi cation LHCb can play an important role in the search for cLFV.
