Let us consider a neutrino within the SM. To make a neutrino massive we choose the most simple way, that is, add a right-handed neutrino singlet νlR to the lepton sector. Then, the Yukawa Lagrangian takes the form

LY = − ∑ a

haalaLϕlaR − ∑ a,b

h′abνaLϕ cνbR + h.c.. (15.1)

The nondiagonal YCCs h′ab (a 6= b) in (15.1) call forth not only mixing of neutrino generations, but lead to the existence of neutrino processes with the lepton flavor violation at the tree level as well. To date the processes that violate both the total and partial lepton flavors are nor observed. However, we do not have any fundamental principle that demands the existence of similar phenomena. This circumstance prompts experimentalists to look for such processes. Because the accuracy of any experiment is limited, then the absence of positive results in searches of processes with the lepton flavor violation should be interpreted no more than the establishment of upper boundaries on the cross sections of these processes, which, in turn, results in constraints on values of nondiagonal YCCs. We consider the one-dimensional case and assume that at time t = 0 we get a νl neutrino

with momentum p perfectly defined. Then, in the absence of external fields, the set of the de Broglie’s waves with the constant amplitude νj(0) is associated with this neutrino

νl(z, t = 0) = 3∑

Uljνj(0) exp (ipz). (15.2)

After a time t we shall have instead of (15.2)

νl(z, t) =

Uljνj(0) exp [i(pz − Ejt)]. (15.3)

In most cases we shall deal with, the neutrino energy belongs to the ultrarelativistic region, that is, Ej ≫ mj . At that, it is difficult to resist the temptation and not expand all available expressions as a power series in the infinitesimally small parameter m2j/p

2 (p =| p |). Then, the neutrino energy and velocity are given by the expressions:

Ej = p

√ 1 +

≃ p+ m 2 j

2p , (15.4)

Particle and

vj ≃ 1− m2j 2p2

≃ 1. (15.5) All this allows us to rewrite the neutrino wave function in the following form: (t ≃ z)

νl(z) = 3∑

Ulj exp (−im2jz/2p)U∗l′jνl′(0), (15.6)

where νl(z, z) ≡ νl(z) and we have returned to the flavor basis (recall that UU † = 1). So, after traveling a distance z, the wave function of the neutrino, born with the flavor l, represents a superposition of states with all the neutrino flavors. It means that there is a nonvanishing probability of finding the νl′ 6=l components in the neutrino beams, at a distance z from its source, if originally this beam consisted of νl particles. To put it differently, neutrinos can exhibit oscillation transitions. The neutrino oscillation phenomenon resembles microparticles diffraction on several slits (in our case the number of slits is equal to that of the neutrino generations), which, in accordance with the Huygens-Fresnel principle, is explained by the appearance of secondary de Broglie waves with amplitudes UljU

and their subsequent interference. Let us examine whether the expression (15.6) is changed when the neutrino was born

with the definite energy, that is, when all the states in the physical basis have the same energy. Then, in the νj state the momentum is defined by the expression

pj = √ E2 −m2j ≈ E −

m2j 2E


and, again, we have the same phase shift for different mass eigenstates as in Eq. (15.6). So, in the ultrarelativistic case, there is no distinction between the situations when the neutrino is born in the state with the definite momentum value or when the one is born in the state with the definite energy value. Using the expression for the wave function (15.6), we can find the probabilities of tran-

sitions between states with different flavor values, that is, the quantities measured directly in neutrino oscillation experiments. There are two kinds of such experiments, namely, appearance experiments and disappearance experiments. In both cases, the total number and spectrum of charged leptons produced by a neutrino beam at different distances from a source are measured. In appearance experiments, they are charged leptons that do not correspond to an initial neutrino type. In the latter, they are charged leptons belonging to the same generation l as an initial neutrino νl, that is, the weakening of a beam is looked for. The probability of finding the neutrino to have flavor l′ at a distance z from its source, if

originally it had flavor l will look like:

Pl→l′ (z) =| ∑ j

| Ulj |2| Ul′j |2 +

+ ∑ j 6=j′

[ Re(UljU

( ∆m2jj′

2E z

) + Im(UljU

( ∆m2jj′

2E z

)] , (15.8)

where we have set E ≈ p and introduced the designation ∆m2jj′ = m2j − m2j′ (hereafter Greek letters denote physical states). If one assumes that the CP parity is the conserved quantum number, then the matrix U could be chosen as real. Then, the expression (15.8) takes the form:

Pl→l′ (z) = ∑ j

( 2pi

ljj′ z

) , (15.9)

where the quantities

|ljj′ | = 4piE ∆m2jj′


are called the oscillation lengths between the νj and νj′ states; those give a distance scale over which the oscillation effects can be appreciable. In disappearance experiments the probability of surviving a neutrino emitted by a source

is measured. Assume that we are dealing with an electron neutrino. If we fulfill the observation of a neutrino flux at a distance L from its source, then the survival probability of νe will be

Pνeνe = 1− ∑ j,j′

( piL

) . (15.11)

When L ≪ ljj′ , then the second term in (15.11) may be neglected and Pνeνe = 1, that is, oscillations will be unobservable. This puts some idea into us how to define the sensitivity to measuring ∆m2jj′ for one or another neutrino oscillation experiments. In the sine argument of the expression (15.11) we pass on from a natural system of units to ordinary units


( ∆m2jj′L


) = sin2

( 1.27∆m2jj′(eV



) . (15.12)

Consequently, to improve the experiment accuracy (measuring small values of ∆m2jj′ ) the ratio L/E should be increased. The most optimal solution is to use low-energy neutrinos and place the detector at large distances from a source. Thus, for example, in the experiment with a neutrino beam having the energy E ≈ 1 GeV at the distance L = 1 km, one can measure ∆m2jj′ with an accuracy of 1 eV

2. Under the fulfillment

L≫ ljj′ , (15.13) or

∆m2jj′ ≫ 4piE

L , for all (j, j′) pairs (15.14)

the sines in (15.12) quickly oscillate allowing us to set

sin2 ( piL

) ≈ 1

2 (15.15)

and time-averaged (distance-averaged) oscillations are obtained. Making use of the explicit form of the matrix U , we find

Pνeνe ≥ 1

3 . (15.16)

The analysis shows, in the case of N neutrino generations, the expression for Pνeνe takes the form

Pνeνe ≥ 1

N . (15.17)

From the obtained expressions it follows that the appearance of transitions between different kinds of neutrinos, as they travel in a vacuum, is possible only under the realization of the following conditions: (i) neutrinos must have masses and a mass-degeneration multiplicity does not equal to the number of neutrino generations; (ii) some or all mixing angles must be nonvanishing; and (iii) for oscillations to appear a neutrino beam must pass a distance comparable with an oscillation length.