Neutrino Masses and See-Saw Mechanism

A neutrino is a subatomic particle that is very similar to an electron, but has no electrical charge and a very small mass. Neutrinos are one of the most ab-undant particles in the universe. Because they have very little interaction with matter, however, they are incredibly difficult to detect. We present a study of the physics of neutrinos using the Dirac lagrangian. Based on Lorentz invariance we introduce the notion of Majorana spinor. Then we derive the mass terms for both Dirac and Majorana neutrinos. We further discuss the general framework of the See-Saw mechanism considering a simplification of the problem.


Introduction
The neutrino is an elementary particle of the standard model of particle physics.
It's a fermion of spin 1/2. The existence of the neutrino was postulated by Pauli in 1930, in order to explain the non-conservation of energy in beta decay, but its experimental discovery did not occur until 1956, when Reins and Cowan revealed the interactions of neutrinos from the Savannah River nuclear reactor [1] [2].
The study of the properties of neutrinos still presents many unsolved problems, and in particular the fundamental question on the mass of the neutrinos.
In this context, the study of neutrino oscillations plays a privileged role, because their observation implies a non-zero mass. Oscillations are the key to solving different experimental observations, such as the problem of solar neutrinos. In 1962, the muonic neutrino µ ν is discovered at Brookhaven and the tau neutri- With this in mind, in the current context, starting from the above introduction, this paper is organized as follows: In Section 2 we introduce the Dirac lagrangian and the corresponding equations of motion. Then in Section 3 we explain the Majorana spinor in addition to the difference with Dirac spinor. In Section 4, we shall explore and discuss the neutrino mass term with a detailed calculation. In Section 5, we will show and derive the equations of the See-Saw mechanism, and Section 6 is devoted to our conclusions.

The Dirac Equation
Dirac's equation is a relativistic quantum mechanics equation that describes half-integer spin particles [5]. The Dirac lagrangian is given by: From now on, the 4-dimensional unit matrix and the vector (x) of Minkowski space-time will not be marked. One can divide this lagrangian into a kinetic term and another massive term: For ψ : ; we then obtain the conjugate Dirac equation: For ψ : we obtain the Dirac equation:

Majorana Spinors
It is a fermionic type particle which is its own antiparticle. The neutrino could be either a Majorana or Dirac particle. This implies that the spinor ψ is related to its conjugate * ψ . In fact, the equation is not Lorentz invariant, then the solution is to use the charge conjugate spinor [3] [4] [6]. This gives a definition of Majorana Fermions compatible with the Lorentz invariance: then, using the Majorana representation [5] and the charge conjugate matrix 1 , C, we obtain: Hence the four components of the Majorana spinor are real, and it has half the degrees of freedom of a Dirac spinor.

Mass Terms
Charged leptons and quarks, i.e. all fermions of the standard model of particle physics apart from the neutrinos, are Dirac particles. It is tempting to think that neutrinos are too. A 4-component Dirac spinor represents the particle and the antiparticle, each in the states of left and right chirality respectively. When a neutrino is considered massless, the standard model doesn't contain the right chirality field R ν , but only the left chirality field L ν . So, in order to introduce the mass of the neutrino as the mass of the introduced quarks, we add R ν to the model [ Using the proprieties of the projection operator and the γ matrices, we ob- That's give the Dirac term where D m is a constant. Consequently, one can construct the Dirac mass term: 1 Charge conjugate matrix: This matrix is used to define the charge conjugate spinor C is defined within a phase factor, where D m is a complex ( ) where Ĉ is charge conjugate operator (particle-antiparticle).
Using the proprieties of commutation of Dirac γ matrices, we show that the operator Ĉ applied to a chiral field toggles its chirality: Once R ν is added to the model description, we introduce the Majorana mass term: . ., where R m is another constant. This term mix ν and ν and doesn't conserve the leptonic number L.
Note that there is a very important difference between the Dirac and Majorana terms. While the first preserves the electric charges, the baryonic and leptonic numbers; the second (i.e. Majorana term) violates the conservation of all additive quantum numbers of two units.
Since the electric charge is exactly conserved, this means that no charged particle can be of Majorana type. So among all fermions, only neutrinos can be described by Majorana fields, and in this case the leptonic number will be violated.

Partners of Neutrino Masses
The Dirac term, already introduced, conserves the fermionic numbers is the only possible mass term. Then, to conserve ( ) 1 U symmetry, we always need a particle-antiparticle interaction [8].
Since neutrinos do not have electromagnetic charges, it is possible to introduce another mass term that contains two fields of neutrinos (or antineutrinos).
The general mass term contains then the Dirac mass term in addition to the Majorana term which has a left and right contributions [9]. It can be written in the following form: The reason that C  appears in Equation (17) is to connect the conjugate charge field c ψ to ψ and † ψ .
We have:

General Framework
The Equation (17) Using these equations, the Equation (17) can be written with the following compact form: Journal of High Energy Physics, Gravitation and Cosmology ( ) For the 3 generations of neutrinos, the 6 eigenstates of mass i m are eigenvalues of the ( ) M is not necessarily Hermitian, so its diagonalization requires a bi-unitary where L U and R U are ( ) 6 6 × matrices. This diagonalization is accomplished with a change of base: to obtain a new set of fields L η and R η defined by: , .

Problem Simplification
We can simplify our problem by working on a single neutrino family [11]. Suppose that 0 A. Tarhini, M. Mrad so the two eigenstates of mass of the two neutrinos are: We note that the two eigenstates of mass 1 η and 2 η are Majorana states: The state L ν which undergoes essentially the weak interactions is 1L η ; it is the state associated to the eigenstate of the light neutrino This example can be generalized by neglecting the matrix T m (its eigenvalues are negligible), then the neutrino mass matrix takes the form: So the See-Saw mechanism explains, not only, the small mass of the left neutrinos, but it also introduces massive Majorana-type neutrinos.

Conclusion
In the present work we derived the Dirac equation for neutrinos, which represents a relativistic quantum mechanics equation that describes half-integer spin particles. Then we introduced the concept of Majorana neutrinos using the charge conjugate operator. After that, we calculated in details the general mass terms for both types of neutrinos. Finally, we presented the general idea of the See-Saw mechanism and we calculated the mass eigenstates of the Dirac and Majorana neutrinos. We also explained how a spectrum separates into a light and heavy neutrinos.