Cabibbo Theory, SU(3) Symmetry, and Weak N–Y Transition FormFactors

For the ∆S = 0 processes,

and for the |∆S| = 1 processes,

the matrix elements of the vector (Vμ) andthe axial vector (Aμ) currents between anucleon or a hyperon and a nucleon N = n,p are written as:

and

where and are the masses of the nucleon and hyperon, respectively. and arethe vector, weak magnetic and induced scalar N − Ytransition form factors and and are the axial vector, induced tensor (orweak electric), and induced pseudoscalar form factors, respectively.

In the Cabibbo theory, the weak vector(Vμ) and the axial vector(Aμ) currents corresponding tothe ∆S = 0 and ∆S = 1hadronic currents whose matrix elements are defined between the states areassumed to belong to the octet representation of SU(3).

Accordingly, they are defined as:

Where are the generators of flavor SU(3) and is are the well-knownGell–Mann matrices written as

The generators obey the following algebra of SU(3) generators

are the structure constants, and are antisymmetric and symmetric,

respectively, under the interchange of any two indices. These are obtainedusing the λi given in Eq. (B.9) and have beentabulated in Table B.1.

From the property of the SU(3) group, it follows that there are threecorresponding SU(2) subgroups of SU(3) which must be invariant under theinterchange of quark pairs ud, ds, andus respectively, if the group is invariant under theinterchange of u, d, ands quarks. Each of these SU(2) subgroups has raising andlowering operators. One of them is SU(2)I , generated bythe generators (λ1,λ2,λ3) to be identified with the isospinoperators (I1, I2,I3) in the isospin space. For example,I± of isospin space is givenby

The other two are defined as SU(2)U and SU(2)V generated by the generators ,respectively, in the U-spin and V-spin space with (d s) and(u s) forming the basic doublet representation ofSU(2)U and SU(2)V .

Introduction

In 1951, Lyman, Hanson, and Scott [541] were the first to observe elasticelectron scattering from different nuclei using a 15.7 MeV beam obtainedwith the help of the betatron accelerator facility at Illinois. Furtherextensive studies using electron beams started with the development of theMark III linear accelerator (LINAC) in 1953 at the High Energy PhysicsLaboratory (HEPL), Stanford. Hofstadter, Fechter, and McIntyre studied theeffect of electron scattering (Ee ~ 125− 150 MeV) on various nuclear targets andconcluded that these nuclei have finite charge distribution. With increasedenergy (550 MeV) available for scattering, the first evidence of elasticscattering from the proton was observed by Chambers and Hofstadter at HEPLin 1956 [542], using a polyethylene target. Assuming that proton had anexponential density distribution, they found the r.m.s. (root mean squared)radius of the proton to be about 0.8 fm. Later, Yearian and Hofstadterperformed electron scattering experiments on deuteron targets and determinedthe magnetic moment of the neutron [543]. At that time, the investigation ofthe structure of the proton and neutron was a major objective of HEPL, whichwas upgraded to achieve electron beams of energies up to 20 GeV; today, HEPLis known as the Stanford Linear Accelerator Center (SLAC) [544]. By the1960s, it became possible to perform both elastic and inelastic scatteringexperiments at high energies and for a wide range of four-momentum transfersquared (Q2). Thus, by the end of the 1960s,nuclear physics entered the ‘deep inelastic scattering’ (DIS)era, when experiments with 20–40 times higher energies were beingperformed at SLAC; it became possible to probe the hadron. DIS is thescattering of charged leptons/neutrinos from hadrons in the kinematic regionof very high Q2 and energy transferν. During the late 1960s, experiments byMIT-SLAC collaboration, led by Taylor, Kendall, and Friedman [544] confirmedthe scaling phenomenon in the deep inelastic region, which was theoreticallypredicted by Bjorken [545, 546]. They received the 1990 Nobel Prize inPhysics for the experiments. These experimental results confirmed that, inthe scaling region, the constituents of protons behave like free particlescalled partons. Charged partons are identified as quarks and neutral partonsare identified as gluons. Later, many experiments using electron and muonbeams were performed at CERN, DESY, Fermilab, etc.

Cross Section

Consider a two body scattering process with four momenta. There areN particles in the final state with four momenta.

The general expression for the cross section is given by

where

In the laboratory frame, where, say, particle“b” is at rest (i.e.vb = 0, Eb =mb)

va = c = 1, if the incidentparticle is relativistic, that is, Ea ≫ma.

In the center of mass frame, where particles“a” and “b”approach each other from exactly opposite direction, that is,θab = 180o, withthe same magnitude of three momenta, that is, such that

where is the center of mass energy.

More conveniently, this is also written as

Two body scattering

For a reaction, where “a” and“b” are particles in the initial stateand “1” and “2” are particles in the finalstate, that is,

the general expression for the differential scattering cross section is givenby

In any experiment, one observes either particle “1” or particle“2”; therefore, the kinematical quantities of the particlewhich is not to be observed are fixed by doing phase space integration. Forexample, if particle “2” is not to be observed, then

which gives the constraint on where is the three momentum transfer and, whichresults in

Integrating over the energy of particle “1”, using the deltafunction integration property, we Get

Thus,

Which result in,

If the scattering takes place in the lab frame, and, it result in

If the scattering takes place in the center of mass frame, where, is thecenter of mass energy, we write

Energy distribution of the outgoing particle“1”

Here, we evaluate energy distribution in the lab frame.

Historical Introduction to Neutrinos

Neutrino hypothesis

Beginning with the neutrino hypothesis proposed by Pauli in 1930, the storyof the neutrino has been an amazing one [1]. It all started with a letterwritten by Pauli to the participants of a nuclear physics conference inTubingen, Germany, on December 4, 1930 [1], in which he proposed theexistence of a new neutral weakly interacting particle of spin and called it“neutron” as a “verzweifelten Ausweg” (desperateremedy), to explain the two outstanding problems in contemporary nuclearphysics which posed major difficulties with respect to thescientists’ theoretical interpretations. These two problems wererelated with the puzzle of energy conservation inβ-decays of nuclei [2, 3], discovered by Chadwick in1914 [4], and anomalies in understanding the spin–statistics relationin the case of 14N and 6Li nuclei within the contextof the nuclear structure model that was prevalent in the early decades ofthe twentieth century [5, 6] in which electrons and protons were consideredto be nuclear constituents.

This proposed ‘verzweifelten Ausweg’ was considered sotentative by Pauli himself that he postponed its scientific publication byalmost three years. Today, neutrinos, starting from being a mere theoreticalidea of an undetectable particle, are known to be the most abundantparticles in the universe after photons, being present almost everywherewith a number density of approximately 330/cm3 pan universe. Thehistory of the progress of our understanding of the physics of neutrinos isfull of surprises; neutrinos continue to challenge our expectationsregarding the validity of certain symmetry principles and conservation lawsin particle physics. The study of neutrinos and their interaction withmatter has made many important contributions to our present knowledge ofphysics, which are highlighted by the fact that ten Nobel Prizes have beenawarded for physics discoveries in topics either directly in the field ofneutrino physics or in the topics in which the role of neutrino physics hasbeen very crucial.

In this chapter, we will provide a historical introduction to the developmentof our understanding of neutrinos and their properties as they have emergedfrom the theoretical and experimental studies made over the last 90years.

Introduction

There are many sources of neutrinos in the universe. For example, neutrinosare produced inside the sun, the earth, and the entire atmosphere. Neutrinosare also produced during the birth, collision, and death of stars.Particularly huge flux of neutrinos is emitted during supernovae explosions.Most of the neutrinos that pass through the earth come from the sun and areproduced in the nuclear fusion reactions going on inside the sun'score. In addition, (anti)neutrinos are produced in nuclear power plants. Thebeams of high energy protons striking a target material produce intense fluxof pions and kaons, which then decay to produce neutrinos. There areneutrinos around us which were born almost 13.8 billion years ago,

soon after the birth of the universe; they constitute cosmic neutrinos (relicof the Big Bang). Neutrinos produced inside the core of the earth are calledgeoneutrinos.

Figure 17.1 represents the different sources of neutrinos. Besides thesesources, neutrinos are also produced by our body; they are also present invegetables and fruits, etc. In fact, we are living in a sea of neutrinoshaving a wide range of energies. Simply speaking, through an area of 1cm2 of our body, almost a billion neutrinos pass everysecond. The theoretically obtained energy ranges of these neutrinos arequite broad, ranging from micro electron volts for the neutrinos left overfrom the Big Bang, right up to peta electron volts for the neutrinosproduced in the violent gamma-ray bursts in the universe. Figure 17.2 showsthe predicted neutrino flux as a function of neutrino energy from a varietyof neutrino sources. Some of these neutrinos have been studied by variousneutrino experiments. In this chapter, we will discuss some of the importantneutrino sources and their detection techniques in various energyregions.

Solar Neutrinos

Production

The sun is the main source of energy for the earth. It was first suggested byEddington [884] that nuclear fusion is the source of solar energies as wellas the source of energies of other stars. However, it was Bethe [885] in1939, who explained the mechanism of thermonuclear fusion reactions, statingthat hydrogen burning takes place inside the core of the sun, in whichneutrinos are being produced, along with helium and positrons.

Introduction

Neutrino oscillation is purely a quantum mechanical phenomenon, in which aneutrino of one flavor, say να develops acomponent of a neutrino of another flavor, sayνβ, whereα, β =e, μ,τ, and α ≠ β.This phenomenon of neutrino oscillation implies that neutrinos have definitemass and their flavor eigenstates are different from their masseigenstates.

The idea of neutrino masses, mixing, and oscillations was first proposed in1957–58 [179] just after the two-component neutrino theory wasconfirmed by the Goldhaber et al. [153] experiment. Pontecorvo [179]conjectured the idea of neutral particle (not necessarily being elementaryparticle) oscillation based on the analogy with kaon regeneration, andargued that : ‘If the two- component neutrino theory should turn outto be incorrect (which at present seems to be rather improbable) and if theconservation law of neutrino charge would not apply, then in principleneutrino ↔ antineutrino transitions could take placein vacu’. At that time only one type of neutrino existed, therefore,for the two-component theory, there were only left-handed neutrinos andright-handed antineutrinos, implying oscillations between. These transitionsare however not possible because of the conservation of total angularmomentum.

Around the same time, Davis [968] conducted an experiment using reactorantineutrino beams and looked for the electron in the reaction

which showed negative results. Contrary to the negative observations, a rumorreached Pontecorvo that Davis has observed the aforementioned event beforethe actual publication of the work. Based on the assumed positive results,in 1958, Pontecorvo, in an another paper [969], concluded that: “Theresult of the Davis experiment was nonzero probability for the process understudy, if confirmed, definitely shows that the (strict) law of neutrinocharge conservation is not valid.” He suggested that the events couldbe due to the transitions. However, this explanation contradicted thetwo-component neutrino theory, where only the left-handed neutrinos or theright-handed antineutrinos participate in the weak interaction. Therefore,the suggestion of νR and similarly, byPontecorvo implies the existence of a class of neutrinos which arenoninteracting; they are called sterile neutrinos.

The unified theory of weak and electromagnetic interactions was formulatedindependently by Weinberg [157] and Salam [37] for leptons. It was laterextended to the quark sector using the GIM mechanism of quark mixingproposed by Glashow, Iliopoulos, and Maiani (GIM) [64]. The theory ispopularly known as the “standard model of electroweakinteractions”. (ii) L. Glashow, A. Salam, and S. Weinberg wereawarded the Nobel Prize in Physics in 1979 for the formulation of thistheory. In this chapter, we describe the formulation of the standard modelfor leptons and quarks and its applications to other interactions. Thepredictions of the model and their experimental confirmations are alsopresented. The limitations of the model and some processes suggesting theneed for the physics beyond the standard model are discussed later inChapter 20.

In this section, the essential features of the electromagnetic interactionsand the phenomenological V – Atheory of electroweak interactions, well-known from the experimental andtheoretical studies of the various weak processes are summarized. They areused as inputs in formulating the standard model. They are as follows:

• 1. Both electromagnetic and weak interactions involve, in general,all the elementary particles, that is, leptons and quarks, unlikethe strong interaction, which affect only the quarks and the hadronsbuilt from these quarks.

• 2. Both electromagnetic and weak interactions are mediated by vectorfields. Electromagnetic interaction is known to be mediated byphotons described by the electromagnetic fieldAμ, which is massless.The observed weak interactions are presumed to be mediated bycharged intermediate vector bosons (IVB), Which are massive withmass MW.

• 3. Weak interactions involve a pair of leptons, like in which thecharged vector bosons, interact with the charged lepton currents,where only the left-handed leptons participate, that is,right-handed leptons, that is, are not involved in the interaction.The interaction Lagrangian is written as:

• 4. Electromagnetic interactions involve the electromagnetic fieldAm, which interacts with the electromagneticcurrent, implying that both the left- and the righthanded componentsof the electron participate, because The interaction Lagrangian iswritten as:

• 5. Both electromagnetic and weak interactions are universalinteractions, that is, they have the same coupling strength forleptons and quarks. However, the coupling strengths of theelectromagnetic and weak interactions to the quarks (leptons) aredifferent from each other.

Introduction

Our present understanding of the physical phenomena in nature and the lawsgoverning them is based on the assumption that quarks and leptons are thebasic constituents of matter which interact with each other through strong(quarks only), weak, electromagnetic, and gravitational interactions. One ofthe major aims of scientists in the physics community has been to formulatea unified theory of all these fundamental interactions to describe thenatural phenomenon. The first and earliest step in this direction was tounify electromagnetic and gravitational interactions, both of them beinglong range interactions, that is, proportional to; many attempts were madeto unify them in the early twentieth century. It was then believed thatthese were the only two fundamental interactions and the interactions couldbe described by field theories based on the principle of invariance undercertain transformations called local gauge transformations because of theirexplicit dependence on space–time coordinates. In this type of fieldtheories, the electromagnetic interaction between two charged particles isdescribed by the exchange of a massless vector fieldAμ (x) asproposed by Weyl [40], while the gravitational interaction between the twoobjects is described by the exchange of a tensor fieldgμν (x)as proposed by Weyl [322] and Einstein [323]. Later, after the discovery ofthe atomic nucleus and the experimental studies of the structure of nucleiand the phenomenon of nuclear radioactivity, two more fundamentalinteractions, viz., strong and weak interactions were revealed. Theexistence of strong interaction is responsible for binding neutrons andprotons together and the weak interaction enables them to decay inside thenucleus. Both the interactions were found to be of short range. The need wasfelt to formulate a unified theory of all the four fundamental interactions,viz., the electromagnetic, strong, weak, and gravitational interactions.

Introduction

The study of the various inelastic processes induced by photons, electrons,pions, and (anti) neutrinos from nucleons is very important as it providesinformation about the excitation mechanism of nucleons. This enables us toinvestigate the structure of nucleons as a composite of quarks and the roleof gluons in the quark–quark forces using quantum chromodynamics(QCD). The inelastic processes induced by (anti)neutrinos are a uniquesource to determine the axial vector aspects of the nucleon structure andrelate it to the pion physics. Moreover, in recent years, inelasticreactions induced by (anti)neutrinos leading to the production of mesonslike pions, kaons, and ηs have become more relevantin the search for proton decay and for neutrino oscillations.

Within the standard model, proton stability is associated with baryon numberconservation, as the proton being the lightest baryon, cannot decay into anyother baryon when it is in the free state. It is believed that the baryonnumber conservation law is not a fundamental law; and there are models ofgrand unified theory (GUT) [503], which predict the lifetime of the protonto be of the order of 1031 years. To estimate a proton'slifetime, some calculations have also been performed using supersymmetry(SUSY) GUTs like SUSY SU(5) and SUSY SO(10). Generally, SUSY GUT modelsfavor kaons (K0 orK+) in the final state if a protondecays. However, some of these models also predict decay modes where an etameson is produced in the final state [504]. In the minimal SU(5) GUT, thepredicted proton lifetime and decay to e+π0 is 1031±1 years,which has been ruled out by IMB [505], Kamiokande [506, 507], andSuper-Kamiokande [508]. Although these experiments have not observed anyevent for the decay p → e+π0, they have provided the limit for theproton's lifetime to be 1033±1 years. The bestlimit on the proton's lifetime comes from Super-Kamiokande [509] forthe channel p → νK+> 5.9 × 1033years. Soudan-2 [510] looked for eta in the final state in the proton years.Soudan-2 [510] looked for eta in the final state in the proton decaysearches and found the best limit for the channels p →μ+η and p →e+η to be 8.9 ×1031 years and 8.1 × 1031years, respectively.

Introduction

Neutrinos play a very important role in astrophysics. Due to their weaklyinteracting nature, they give information about the interior of stars,supernova explosions, the distant galaxies, the possible origin of thecosmic rays, etc. In Chapter 17, we have observed that astrophysical sitesare the major contributors to low energy neutrinos. In this chapter, wediscuss the importance of neutrinos in the creation of the chemical elements(see Figure 19.1) in the universe. We know that the universe started around13.8 billion years ago with a singularity, that is, Big Bang and since then,it has been expanding and becoming cooler. The very early universe consistedonly of radiation, which during the expansion and cooling phase gave rise toquark–antiquark and lepton–antilepton pairs. With further fallin temperature, the quarks combined to form

nucleons, which in turn fused to form the lighter elements like hydrogen,helium, and lithium, the first ever nuclei created in the universe. Thus,the early universe consisted of about 75% hydrogen nuclei, 25% heliumnuclei, and traces of lithium nuclei. It should be noted that hydrogen isthe only element that was solely created during the Big Bangnucleosynthesis; all the other elements including helium and lithium aresynthesized by several processes as shown in Figure 19.1. Therefore, all thehydrogen in water molecules were produced during the first few minutes ofthe Big Bang.

The process of creation of new nuclei from pre-existing nucleons is known asnucleosynthesis. The nucleosynthesis of the lighter elements does notrequire the emission or absorption of neutrinos while all the elementsheavier than lithium require neutrinos directly or indirectly in theirsynthesis. The nucleosynthesis of intermediate and heavy elements require avery high temperature and pressure environment. Elements up to iron were/aresynthesized in the core of stars through the nuclear fusion reaction and itis believed that the heavier elements were synthesized outside the newlyformed neutron star in a core collapse supernova. Without neutrinos, wecannot think of energy from the stars. Moreover, neutrino properties figureprominently in many astrophysical environment. Neutrinos are involved indifferent types of nucleosynthesis processes like theν-process, νp-process,etc., in the creation of proton-rich nuclei as well as in the synthesis ofneutron-rich nuclei through the r-process and s-process.

Introduction

In the previous chapter, we have described free particles and fields in termsof Lagrangian, equations of motion, their solutions and quantization.However, in the physical world, particles and fields are visualized by theirinteractions with other particles and fields or among themselves. Forexample, the simple processes of Compton scattering, photoelectric effect,and Coulomb scattering involving photons and electrons, are well known. Allthe known elementary particles interact with each other through the fourfundamental interactions, that is, electromagnetic, weak, strong, andgravitational, through the exchange of gauge fields. Since the particlesthemselves can be described in terms of fields, all the physical processesgoverned by the four fundamental interactions are examples of various typesof fields in interaction with each other including self interaction, allowedby the general principles of physics. These interactions are described by aninteraction Lagrangian Lint(x),to be included along with the free LagrangianLfree(x), described inChapter 2 for a quantum description of the evolution of physical systems.The interaction Lagrangians can be obtained by using the symmetry propertiesof the physical system defined by certain transformations called local gaugetransformations and imposing the invariance of the free Lagrangian underthese transformations. These will be discussed in some detail in Chapter 8,in the case of electromagnetic, weak, and strong interactions of scalar,vector, and spin particles.

In this chapter, we give some simple examples of interaction Lagrangiansinvolving spin 0, spin and spin 1 particles and illustrate the generalprinciples to write them. We use the example of electromagnetic interactionto demonstrate the general method of the relativistic perturbation theory tofind out the solution of the equations of motion of fields in the presenceof the interaction LagrangianLint(x). It is assumed thatthe strength of the interaction Lagrangian can be quantified by a parameterwhich is small, so that perturbation theory can be applied. This is normallythe situation in the case of electromagnetic and weak interactions; in alimited range of kinetic variables, it is also true in the case of stronginteractions. The relativistic perturbation theory has been very useful indescribing physical processes.

Introduction

We have seen in Chapter 8, that the standard model for the leptons developedby Salam [37] and Weinberg [157], which was later extended to the quarksector by Glashow [64], unifies weak and electromagnetic interactions. Itpredicts, in a unique way, the interaction Lagrangian for charge changing(CC) weak interactions of leptons and neutrinos of all flavors with chargedgauge vector bosons, W± and theelectromagnetic interactions of charged leptons with photons. It alsopredicts the existence of neutral current(NC) weak interactions of chargedleptons and neutrinos of all flavors with the neutral gauge boson,Z0. The strength of the interaction of thecharged, neutral, and electromagnetic currents with theW±,Z0, and A gauge bosons aredescribed in terms of the weak coupling constants g,electromagnetic coupling constant e, and a free parameterθW called the weak mixing angle.Specifically, the interaction Lagrangian discussed in Chapter 8 is writtenhere again as:

where Zμ, andAμ are the charged, neutral andelectromagnetic gauge fields and

with is the fine structure constant. In the following sections, we use theinteraction Lagrangian in Eq. (9.1) to calculate the cross sections for someweak and electromagnetic processes using point particles, that is, chargedleptons and neutrinos.

Scattering

This scattering process can take place through an electromagnetic processmediated by a virtual photon as well as by the weak neutral current mediatedby a Z0 boson in the standard model [400].

When an electron interacts with a photon field (Figure 9.1(a)), theinteraction Lagrangian is given by:

and when it interacts with the Z0 boson field (Figure 9.2(a)), theLagrangian is given by:

Using the aforementioned Lagrangians corresponding to Figure and followingthe Feynman rules the transition amplitude for the process

mediated through virtual photon exchange of momentum in the lowest order, maybe written as:

and for the process mediated through the virtual Z0 exchange,shown by the Feynman diagram in Figure 9.2(b) as:

where. The process proceeding through Z0 is highly suppressed ascompared to the photon exchange therefore, in the present case, we presentthe cross section for the process given in Eq. (9.7) mediating through-exchange.

Lorentz Transformations

Lorentz transformation (L.T.) is the rotation in Minkowski space. Minkowskispace is the mathematical representation of the space time inEinstein's theory of relativity. Unlike the Euclidean space,Minkowski space treats space and time on a different footing and the spacetime interval between the two events would be the same in all frames ofreference.

If an event is observed in an inertial frame S at thecoordinates (x, y, z,t) (Fig.A.1) and in another inertial frame at thecoordinates and the frame is moving with a constant velocityv with respect to S in the directionof x, then the measurements in S andS’ are related by

where

The L.H.S. of these equations describe the transformation of the coordinatesfrom S to S’ and theR.H.S. of the equations describe the transfer of coordinates fromS’ to S.Equation (A.1) may also be written as

The invariant length element squared ds2 is defined as

where is a contravariant vector and is a covariant vector. The metric tensoris defined as

One may rewrite Eq. (A.2) as

Where is the Lorentz transformation matrix. Any set of quantities which havefour components and transform like Eq. (A.5) under L.T. form a four vector,that is,

with the properties

For an infinitesimal may also be written as The tensor generates theinfinitesimal L.T. and it is considered to be a quantity that is smallerthan unity. The major contributor is (unit matrix) and generatesinfinitesimal boost or infinitesimal rotation or both. This can beunderstood as demonstrated in the next section.

Rotation of coordinates around the zaxis in 3D

If we consider the rotation around the z-axis by an angleθ, then

In the matrix form, Eq. (A.7) can be written as

For an infinitesimal rotation, that is,

where represents an infinitesimal rotation if. Equation (A.7) can berewritten as

The matrix representing the infinitesimal change is antisymmetric.Reconsidering Eq. (A.2),

When and Now and is given by

For an infinitesimal boost or rotation, that is,

Therefore, Eq. (A.11) implies that

Now,

The second term on the R.H.S. of Eq. (A.13) can be neglected as it is theproduct of two infinitesimal numbers, that is,

Since should be an invariant quantity,

that is, Eq. (A.14) shows antisymmetry. Therefore, the generator of an L.T.must be an antisymmetric matrix.

Introduction

The phenomenon of neutrino oscillation discussed in Chapter 18 implies thatneutrinos have finite mass. The phenomenology of neutrino oscillationdetermines only the mass square differences but not the absolute masses ofthe neutrinos. The scalar and fermionic structure of the standard model doesnot allow neutrinos to have non-zero mass. There is also the possibilitythat neutrinos may have magnetic moments (intrinsic or transition)considerably larger than the prediction of the standard model. Moreover,there are anomalous results in the measurements of neutrino and antineutrinooscillation parameters in various regions of energies which suggest that thestandard model description in terms of three weak flavor doublets of leptonsand quarks is not adequate to describe weak interactions and there may existadditional flavor of neutrinos which are non-interacting i.e. sterile. Allthese processes suggest that although the standard model has been aspectacular success in describing most electroweak processes, there is needfor physics beyond the standard model.

Moreover, there are many other rare physical processes driven by weakinteractions which have been studied for a long time theoretically as wellas experimentally and are not explained by the standard model. Theexperimental observations of these processes would establish the physicsbeyond the standard model. The subject of physics beyond the standard modelis too vast to be described in space of a chapter but we discuss here, someof the neutrino processes to introduce the subject.

• (i) Neutrinoless double beta decay (NDBD) and Majorana neutrinos

• (ii) Lepton flavor violating (LFV) decays of elementary particles

• (iii) Flavor changing neutral current (FCNC)

• (iv) Existence of non-standard interaction in high precision weakprocesses.

Netrinoless Double-beta Decay

General considerations

The problem of double-beta decays (DBD) involving two-neutrino double-betadecay (2νββ) and neutrinolessdouble-beta decay (0νββ) has been withus for more than 80 years after it was first discussed by Goeppert-Mayer inthe case of (2νββ) in 1935 [1062] soonafter the Fermi theory of β-decay was formulated[23] and the process of 0νββ wasdiscussed by Furry in 1939[1063] after a new theory of neutrino was given byMajorana [121].

Trace Properties of Pauli and Dirac Matrices

Here, we have used the properties of cyclic permutation of gammamatrices.

Spin Density Matrix

Consider that a beam of spin particles is produced from the interaction withanother system. When the interaction is over, the wavefunction of the wholesystem can be written as a sum of the products of wavefunctions of the othersystems in free state:

where

and represents the wavefunction of the states with momentum p; helicityrepresents the wavefunction of the other system and are constants. Thewavefunctions and are normalized by the conditions:

Operator acts on the spin variable. The mean value of the operator isobtained as

Using Eq. (C.2), can be obtained as

Using Eq. (C.2) and (C.6) and the normalization condition for from Eq. (C.4),the denominator of Eq. (C.5) can be written as:

with

Similarly, the numerator of Eq. (C.5) can be obtained as

Using Eqs. (C.9) and (C.7) in Eq. (C.5), we have

With

This matrix ρ(p) is called the spindensity matrix.

Some properties of the spin density matrix

1. Hermiticity

2. Normalization

Now, in order to obtain the expression for the spin density matrix, we expandρ(p) in terms of bilinearcovariants as

Using the properties of the spin density matrix, we can evaluate theconstants a to e in Eq. (C.19).

To evaluate constant a, we take the trace of Eq. (C.19)

Using the trace properties of gamma matrices as given in Section C.1, weget

Using the normalization condition forρ(p) matrices as given in Eq.(C.15), the constant a can be evaluated as

To calculate, we multiply Eq. (C.19) by from the right side and then thetrace of the resulting equation can be calculated as:

Using Eq. (C.18) in the aforementioned expression, we get

where

and the anti- commutation relation of matrices have been used.