# The Dirac Equation and Its Application in Quantum Field Theories

Julio Cesar Jaramillo Quiceno

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Julio Cesar Jaramillo Quiceno. The Dirac Equation and Its Application in Quantum Field Theories. 2025. hal-05096158v2

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# The Dirac Equation and Its Application in Quantum Field Theories

Julio Cesar Jaramillo Quiceno*

¹Departamento de Física, Universidad Nacional de Colombia, Edificio Yu Takeuchi, Bogotá

## Abstract

This work presents a comprehensive review of the Dirac equation and its foundational role in the development of quantum field theories. Starting from the incompatibility of the Schrödinger equation with special relativity, we derive the Dirac equation and explore its relativistic and covariant formulations. We analyze the structure and properties of Dirac spinors, gamma matrices, and conserved currents, highlighting their physical significance. The interaction of fermions with external electromagnetic fields is addressed via minimal coupling, leading to the formulation of quantum electrodynamics (QED). We examine charge conjugation and the emergence of antiparticles, emphasizing the theoretical basis for positrons. The discussion extends to gauge invariance, renormalization, and scattering amplitudes through Feynman diagrams. Furthermore, we introduce quantum chromodynamics (QCD) as the gauge theory of the strong interaction, detailing its SU(3) symmetry, color confinement, and non-Abelian structure. Special attention is given to the electroweak sector and neutrino physics, including Dirac, Majorana, and Weyl neutrino formalisms, as well as their electromagnetic form factors. This exposition provides a unified and pedagogical overview of the Dirac formalism's application to modern quantum field theory.

## Keywords:

Dirac equation. QED, QCD, Neutrinos

<footnote>*jcjaramilloq@unal.edu.co</footnote>
# 1 The Dirac Equation

In the framework of non-relativistic quantum mechanics, the equation describing the dynamics of particles in a potential $V(x, t)$ is the Schrödinger equation, which is typically written as:

$$i\hbar \frac{\partial}{\partial t} \psi(x, t) = -\frac{\hbar^2}{2m} \nabla^2 \psi(x, t) + V(x, t) \psi(x, t). \quad (1)$$

It is evident that this equation is not compatible with Lorentz transformations. In the framework of special relativity, we define the space-time differential operators as follows:

$$p_{\mu} = i\hbar\partial_{\mu} := \left( i\hbar\frac{\partial}{\partial t}, -i\hbar\frac{\partial}{\partial x} \right) = (p^0, \mathbf{p}). \quad (2)$$

Recall that in equation (1), the Hamiltonian is:

$$\hat{H} = \frac{\mathbf{p}^2}{2m} + V(x, t). \quad (3)$$

Whereas in the relativistic framework, the Hamiltonian becomes:

$$\hat{H} = \pm \sqrt{(\mathbf{p}c)^2 + (mc^2)^2} + V(x, t), \quad (4)$$

for a free particle, we have:

$$\hat{H}\psi(x, t) = \pm \sqrt{(\mathbf{p}c)^2 + (mc^2)^2} \psi(x, t) = E, \quad \psi(x, t) \quad (5)$$

in terms of differential operators, equation (5) can be written as:

$$-(\hbar c)^2 \left( \frac{1}{c^2} \frac{\partial^2}{\partial t^2} - \nabla^2 \right) \psi(x, t) = (mc^2)^2 \psi(x, t), \quad (6)$$

# 2 The Dirac Equation (Continued)

Considering the following quantity:

$$\hbar c = 197.327 \text{ MeV} \cdot \text{fm}. \quad (7)$$

Equation (6) is known as the Klein-Gordon equation for free particles. The characteristic feature of equation (6) is that it lacks a spin term, indicating that the wave function $\psi(x, t)$ describes scalar particles or particles with zero spin. To describe spin-1/2 particles, we consider the linearization of equation (6) to include spin terms. The equation describing spin-1/2 particles is the Dirac equation, whose functional form is given by:
$$i\hbar \frac{\partial}{\partial t} \psi(\mathbf{x}, t) = -i\hbar c \left( \alpha_i \frac{\partial}{\partial x_i} \right) \psi(\mathbf{x}, t) + \beta mc^2 \psi(\mathbf{x}, t), \quad (8)$$

where $\alpha_i, \beta$ (for $i = 1, 2, 3$) are matrices that must satisfy the following conditions:

* For a correct interpretation of the momentum and energy of a free relativistic particle ($E^2 = (pc)^2 + (mc^2)^2$), the following must hold:

$$\{\alpha_i, \alpha_j\} = 2\delta_{ij}\mathbf{1}, \quad (9)$$

$$\{\alpha_i, \beta\} = 0, \quad (10)$$

$$\alpha_i^2 = \beta^2 = \mathbf{1}. \quad (11)$$

* The requirement $H = H^\dagger$ implies that:

- $\alpha_i, \beta$ are Hermitian matrices

- $\mathrm{Tr}(\alpha_i) = \mathrm{Tr}(\beta) = 0$

- Their eigenvalues are $\pm 1$

- Their dimension must be even and greater than 2

The simplest dimension satisfying these conditions is 4. Therefore, the matrices that satisfy these conditions are related to the Pauli matrices, and we define $\alpha_i$ and $\beta$ as:

$$\alpha_i = \begin{pmatrix} 0 & \sigma_i \\ \sigma_i & 0 \end{pmatrix}, \quad \beta = \begin{pmatrix} \mathbf{1} & 0 \\ 0 & -\mathbf{1} \end{pmatrix}, \quad \text{for } i = 1, 2, 3. \quad (12)$$

The wave function $\psi(\mathbf{x}, t)$ is a four-component column matrix:

The wave function in Dirac theory is represented as a four-component spinor:

$$\psi(\mathbf{x}, t) = \begin{pmatrix} \psi_1(\mathbf{x}, t) \\ \psi_2(\mathbf{x}, t) \\ \psi_3(\mathbf{x}, t) \\ \psi_4(\mathbf{x}, t) \end{pmatrix} = \begin{pmatrix} \psi_+(\mathbf{x}, t) \\ \psi_-(\mathbf{x}, t) \end{pmatrix} \quad (13)$$

This object is called the Dirac spinor. In particle physics, the Dirac equation is essential due to its relativistic nature in describing particle interactions. The Dirac spinor describes two energy states - one with positive energy and one with negative energy - which must satisfy the relativistic relation $E = \pm\sqrt{p^2c^2 + (mc^2)^2}$, corresponding to particle and antiparticle states respectively.

Using relativistic notation, we can write the Dirac spinor simply as:

$$\psi(x) \equiv \psi(\mathbf{x}, t) \quad (14)$$
In momentum space, the Dirac spinors describing a free electron are given by:

$$ \psi(x) = u(p)e^{-ip \cdot x/\hbar} \qquad (15) $$

Upon quantization, these are interpreted as creation and annihilation operators. Notably, squaring the Dirac operator yields the Klein-Gordon equation.

## 2.1 Charge and Current Conservation in the Dirac Equation

The Klein-Gordon equation admits a conserved current, which implies the Dirac equation must also have a conserved current. To derive this:

1. Start with equation (8) using a plane wave solution

2. Take the Hermitian conjugate of equation (8):

$$ -i\hbar \frac{\partial}{\partial t} \psi^{\dagger}(x) = i\hbar c \left( \alpha_i \frac{\partial}{\partial x_i} \right) \psi^{\dagger}(x) + \beta mc^2 \psi^{\dagger}(x) \qquad (16) $$

Multiplying the original equation by $\psi^{\dagger}(z)$ and the conjugated equation by $\psi(z)$, then subtracting, we obtain:

$$ \frac{\partial}{\partial t}(\psi^{\dagger}(x)\psi(x)) = -\nabla \cdot (\psi^{\dagger}(x)\alpha\psi(x)). \qquad (17) $$

This corresponds to the continuity equation:

$$ \partial_{\mu} j^{\mu} = \nabla \cdot \mathbf{j} + \frac{\partial \rho}{\partial t} = 0, \qquad (18) $$

where $\rho = \psi^{\dagger}\psi$ is the probability density and $\mathbf{j} = c\psi^{\dagger}\alpha\psi$ is the probability current.

## 2.2 Current and Charge Density

We define:

$$ \mathbf{j} = c\psi^{\dagger}(x)\alpha\psi(x), \quad \rho = \psi^{\dagger}(x)\psi(x) \qquad (19) $$

The physical interpretation is that the continuity equation implies conservation of probability charge density and current, where $\rho$ represents a positive-definite probability charge density.
## 2.3 The Dirac Gamma Matrices

Starting with equation (8) and multiplying through by $\beta$:

$$i\hbar\beta\frac{\partial}{\partial t}\psi(x) = -i\hbar c\beta\alpha \cdot \nabla\psi(x) + mc^2\psi(x), \qquad (20)$$

from this we define the Dirac gamma matrices:

$$\gamma^0 := \beta = \begin{pmatrix} \mathbf{1} & 0 \\ 0 & -\mathbf{1} \end{pmatrix}, \quad \gamma := \beta\alpha = \begin{pmatrix} 0 & \boldsymbol{\sigma} \\ -\boldsymbol{\sigma} & 0 \end{pmatrix}, \qquad (21)$$

In covariant (4-vector) notation:

$$\gamma^{\mu} := (\gamma^{0}, \gamma) \qquad (22)$$

These are known as the Dirac gamma matrices in the Weyl (or chiral) representation. Mathematically, the gamma matrices satisfy a Clifford algebra through the anticommutation relations:

$$\{\gamma^{\mu}, \gamma^{\nu}\} = \gamma^{\mu}\gamma^{\nu} + \gamma^{\nu}\gamma^{\mu} = 2g^{\mu\nu}, \qquad (23)$$

where $g^{\mu\nu}$ is the Minkowski metric tensor. We also define the spin tensor:

$$\sigma^{\mu\nu} = \frac{i}{2}[\gamma^{\mu}, \gamma^{\nu}] \qquad (24)$$

The Dirac gamma matrices satisfy several important properties:
$$ \gamma^{\mu}\gamma^{\nu} = g^{\mu\nu} - \gamma^{\nu}\gamma^{\mu}, \qquad (25) $$

$$ \gamma^{\mu}\gamma_{\mu} = 4 \cdot 1, \qquad (26) $$

$$ \gamma^{\mu}\gamma^{\nu}\gamma_{\mu} = -2\gamma^{\nu}, \qquad (27) $$

$$ \gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma_{\mu} = 4g^{\nu\rho}, \qquad (28) $$

$$ \gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma_{\mu} = -2\gamma^{\sigma}\gamma^{\rho}\gamma^{\nu}, \qquad (29) $$

$$ \mathrm{Tr}(\gamma^{\mu}) = 0, \qquad (30) $$

$$ \gamma^{\mu\dagger} = \gamma^{0}\gamma^{\mu}\gamma^{0}, \qquad (31) $$

$$ (\gamma^0)^2 = 1, \qquad (32) $$

$$ (\gamma^i)^2 = -1, \qquad (33) $$

$$ \gamma^0 = \gamma^{0\dagger}, \qquad (34) $$

$$ \gamma^i = -\gamma^{i\dagger}, \qquad (35) $$

$$ \mathrm{Tr}(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}) = 4(g^{\mu\nu}g^{\rho\sigma} - g^{\mu\rho}g^{\nu\sigma} + g^{\mu\sigma}g^{\nu\rho}), \qquad (36) $$

$$ \mathrm{Tr}(\gamma^{\mu}\gamma^{\nu}) = 4g^{\mu\nu}, \qquad (37) $$

$$ \gamma^5 := i\gamma^0\gamma^1\gamma^2\gamma^3, \qquad (38) $$

$$ \mathrm{Tr}(\gamma^5) = 0, \qquad (39) $$

$$ [\gamma^5, \sigma^{\mu\nu}] = 0, \qquad (40) $$

$$ \{\gamma^5, \gamma^\mu\} = 0, \qquad (41) $$

$$ (\gamma^5)^2 = 1, \qquad (42) $$

$$ \{\gamma^5, \gamma^0\} = 0. \qquad (43) $$

The matrix $\gamma^5$ is Dirac's gamma-five matrix, which in neutrino physics represents the chirality matrix. We will examine it in more detail in Section 4. The Dirac equation in terms of gamma matrices is given by:

$$ i\hbar\gamma^0 \frac{\partial}{\partial t} \psi(x) = (-i\hbar c \vec{\gamma} \cdot \vec{\nabla} + mc^2) \psi(x) \qquad (44) $$

We will now write the Dirac equation in its covariant formulation.

## 2.4 Covariant Formulation of the Dirac Equation

Given the relativistic nature of the Dirac equation, we can express it using the 4-vector notation. If we use the space-time differential operators defined in (2) and the definition of the gamma matrices (22), then equations like (44) become:

$$ i\hbar c\gamma^{\mu}\partial_{\mu}\psi(x) = mc^{2}\psi(x), \qquad (45) $$
or equivalently:

$$ \left( \gamma^{\mu} p_{\mu} - \frac{mc}{\hbar} \right) \psi(x) = 0. \qquad (46) $$

We can simplify the notation using Feynman's slash notation. For any 4-vector $u_{\mu}$:

$$ \gamma^{\mu} u_{\mu} = g_{\mu\nu} \gamma^{\mu} u^{\nu} = \gamma^{0} u^{0} - \gamma \cdot u \equiv \not{\mu} \qquad (47) $$

Thus, the Dirac equation in Feynman slash notation becomes:

$$ \left( i \not{\partial} - \frac{mc}{\hbar} \right) \psi(x) \equiv \left( \gamma^{\mu} p_{\mu} - \frac{mc}{\hbar} \right) \psi(x) = 0. \qquad (48) $$

This compact notation is widely used in quantum field theories, which we will explore in Section 3.

## 2.5 Covariant Current Conservation

Starting with the Dirac equation (45) in natural units ($\hbar = c = 1$):

$$ i\gamma^{\mu}\partial_{\mu}\psi_{s}(x) - m_{s}\psi_{s}(x) = 0 \qquad (49) $$

$$ (-i\partial_{\mu}\psi_{t}^{\dagger}(x))(\gamma^{\mu})^{\dagger} - m_{t}\psi_{t}^{\dagger}(x) = 0, \qquad (50) $$

using property (31) on equation (50):

$$ i\partial_{\mu}\psi_{t}^{\dagger}(x)\gamma^{0}\gamma^{\mu}\gamma^{0} + m_{t}\psi_{t}^{\dagger}(x) = 0, \qquad (51) $$

multiplying from the right by $\gamma^0$:

$$ i\partial_{\mu}\psi_{t}^{\dagger}(x)\gamma^{0}\gamma^{\mu} + m_{t}\psi_{t}^{\dagger}(x)\gamma^{0} = 0, \qquad (52) $$

introducing the Dirac adjoint $\bar{\psi}(x) \equiv \psi^{\dagger}(x)\gamma^0$:

$$ i\partial_{\mu}\bar{\psi}_{t}(x)\gamma^{\mu} + m_{t}\bar{\psi}_{t}(x) = 0, \qquad (53) $$

multiplying from the right by $\psi_s(x)$:

$$ i\partial_{\mu}\bar{\psi}_{t}(x)\gamma^{\mu}\psi_{s}(x) + m_{t}\bar{\psi}_{t}(x)\psi_{s}(x) = 0, \qquad (54) $$

and multiplying from left to right by $\bar{\psi}_t(x)$ in equation (49):

$$ \bar{\psi}_{t}(x)i\gamma^{\mu}\partial_{\mu}\psi_{s}(x) - m_{s}\bar{\psi}_{t}\psi_{s}(x) = 0, \qquad (55) $$

adding expressions (54) and (55) gives the result:
$$i\partial_{\mu}(\bar{\psi}_{t}(x)\gamma^{\mu}\psi_{s}(x)) = (m_{t} - m_{s})\bar{\psi}_{t}(x)\psi_{s}(x). \qquad (56)$$

For the physical case where $t = s$, we obtain:

$$\partial_{\mu}(\bar{\psi}(x)\gamma^{\mu}\psi(x)) = 0 \qquad (57)$$

This corresponds to the continuity equation $\partial_{\mu}j^{\mu} = 0$, where:

$$j^{\mu}(x) = -e\bar{\psi}(x)\gamma^{\mu}\psi(x) \qquad (58)$$

This represents a conserved current. The term $\bar{\psi}_{k}(x)\psi_{k}(x)$ is a Lorentz scalar, and $j^{\mu}$ transforms as a 4-vector. Equation (58) is known as the Dirac current. Many texts include a factor of $-e$ in the Dirac current, and in momentum space it becomes:

$$j^{\mu} = -e\bar{u}(p)\gamma^{\mu}u(p) \qquad (59)$$

which transforms as a vector under Lorentz transformations and satisfies the continuity equation.

## 2.6 Plane Wave Solutions for Free Particles

Consider an electron in its rest frame, which is an eigenstate of 3-momentum with $p = 0$. The Dirac equation in this case reduces to:

$$i\hbar \frac{\partial}{\partial t} \gamma^0 \psi = mc^2 \psi \qquad (60)$$

We propose a Dirac spinor of the form:

$$\psi = \begin{pmatrix} \chi \\ \varphi \end{pmatrix} \qquad (61)$$

This leads to coupled equations for the components:

## 2.7 Spinor Decomposition and Energy Solutions

We can add and subtract the coupled equations by introducing two new spinor components:

$$\psi_{+} = \frac{1}{\sqrt{2}}(\varphi + \chi), \quad \psi_{-} = \frac{1}{\sqrt{2}}(\varphi - \chi) \qquad (62)$$

The $\sqrt{2}$ factor ensures proper normalization of the new Dirac spinors. With these components, we obtain:
$$i\hbar \frac{\partial \psi_+}{\partial t} = mc^2 \psi_+ \qquad (63)$$

$$i\hbar \frac{\partial \psi_-}{\partial t} = -mc^2 \psi_- \qquad (64)$$

The positive energy solutions ($E = +mc^2$) are:

$$\Psi_+ = \begin{pmatrix} \psi_+ \\ 0 \end{pmatrix} e^{-imc^2t/\hbar} \qquad (65)$$

and the negative energy solutions ($E = -mc^2$):

$$\Psi_- = \begin{pmatrix} 0 \\ \psi_- \end{pmatrix} e^{+imc^2t/\hbar} \qquad (66)$$

The appearance of both energy values is a fundamental feature of relativistic wave equations. We can write these solutions more generally as:

$$\psi = \begin{pmatrix} \Psi_+ \\ \Psi_- \end{pmatrix} = \begin{pmatrix} \psi_+ \\ \psi_- \end{pmatrix} e^{i(\lambda kx + \omega t)} \qquad (67)$$

with $\lambda = \pm 1$. To obtain the complete solution of the Dirac equation for a free electron, we substitute (67) into (44):

$$\left\{ \begin{bmatrix} -\lambda E & 0 \\ 0 & \lambda E \end{bmatrix} + \begin{bmatrix} 0 & \boldsymbol{\sigma} \cdot \mathbf{p} \\ -\boldsymbol{\sigma} \cdot \mathbf{p} & 0 \end{bmatrix} - m \begin{bmatrix} \psi_+ \\ \psi_- \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \right. \qquad (68)$$

This gives a system of coupled equations:

$$(\lambda E - m)\psi_+ - \boldsymbol{\sigma} \cdot \mathbf{p} \psi_- = 0 \qquad (56)$$

$$-(\lambda E + m)\psi_- + \boldsymbol{\sigma} \cdot \mathbf{p} \psi_+ = 0 \qquad (57)$$

Solving for $\psi_-$ (positive energy solutions) and $\psi_+$ (negative energy solutions):

$$\psi_- = \frac{\boldsymbol{\sigma} \cdot \mathbf{p}}{E + m} \psi_+ \qquad (69)$$

$$\psi_+ = \frac{-\boldsymbol{\sigma} \cdot \mathbf{p}}{E + m} \psi_- \qquad (70)$$

For physical momentum $\mathbf{p} \neq 0$, we express (67) as:
$$ \Psi_+ = N \left( \frac{\psi_+}{\frac{\boldsymbol{\sigma} \cdot \boldsymbol{p}}{E+m} \psi_+} \right) e^{i(Et - \boldsymbol{p} \cdot \boldsymbol{x})/\hbar} \quad (58) $$

$$ \Psi_- = N \left( \frac{-\frac{\boldsymbol{\sigma} \cdot \boldsymbol{p}}{E+m} \psi_-}{\psi_-} \right) e^{-i(Et - \boldsymbol{p} \cdot \boldsymbol{x})/\hbar} \quad (59) $$

We thus define the momentum-dependent spinors for positive and negative energy states:

$$ u(\lambda = +1, \boldsymbol{p}, s) = N \left( \frac{\chi_s}{\frac{\boldsymbol{\sigma} \cdot \boldsymbol{p}}{E+m} \chi_s} \right) \quad (60) $$

$$ u(\lambda = -1, \boldsymbol{p}, s) = N \left( \frac{-\frac{\boldsymbol{\sigma} \cdot \boldsymbol{p}}{E+m} \chi_s}{\chi_s} \right) \quad (61) $$

In general form:

$$ u^{(r)}(\boldsymbol{p}) = N \left( \frac{v_{\sigma}}{\frac{\boldsymbol{\sigma} \cdot \boldsymbol{p} v_{\sigma}}{E+m}} \right) \quad (71) $$

$$ v^{(r)}(\boldsymbol{p}) = N \left( \frac{\frac{\boldsymbol{\sigma} \cdot \boldsymbol{p}}{E+m} v_{\sigma}}{v_{\sigma}} \right), \quad (72) $$

For convenience, we write these spinors in the form:

$$ u(\lambda, \boldsymbol{p}, \sigma) = N u(\lambda, \boldsymbol{p}) v_{\sigma} \quad (73) $$

where $u(\lambda, \boldsymbol{p})$ is a 4×2 matrix acting on the two-component spinors $v_\sigma$, yielding the four-component Dirac spinors (71) and (72). Note that $\sigma$ does not represent eigenvalues along a fixed axis (e.g., the z-axis). Unlike the non-relativistic case, Dirac spinors depend on the particle's physical momentum. The factor $N$ is a normalization constant. The general solution of the Dirac equation can then be expressed as:

$$ \psi_{\sigma}(x) = u(\lambda, p, \sigma) \exp \left[ i(-\lambda Et + \boldsymbol{p} \cdot \boldsymbol{x}) \right] \quad (74) $$

Using this formalism, we can compute the conserved current for transition processes using perturbation theory to determine the corresponding probability amplitudes. Following the procedure from Section 2.5 but using the general solution (74) for initial (i) and final (f) states, with their respective energy eigenvalues, physical momenta $\boldsymbol{p}_i, \boldsymbol{p}_f$, and spin indices $\sigma_i, \sigma_f$, the Dirac conserved current becomes:
$$j_{fi}^{\mu}(x) = \bar{u}_f \gamma^{\mu} u_i \exp(iq_{\mu}x^{\mu}) \qquad (75)$$

where $q^{\mu} = p_f^{\mu} - p_i^{\mu}$ is the four-momentum transfer. In momentum space:

$$-e\bar{u}_f\gamma^\mu u_i = -e\bar{u}(\lambda_f, p_f, \sigma_f)\gamma^\mu u(\lambda_i, p_i, \sigma_i) \qquad (76)$$

Applying the four-divergence to (75) yields:

$$\partial_{\mu} j_{fi}^{\mu}(x) = q_{\mu} \bar{u}_{f} \gamma^{\mu} u_{i} \exp(iq_{\mu} x^{\mu}) = 0 \qquad (77)$$

which gives the Ward identity:

$$q_{\mu} \bar{u}_{f} \gamma^{\mu} u_{i} = 0 \qquad (78)$$

This identity will prove essential when examining the electromagnetic form factors of neutrinos in Section 2.3.1 of the reference [1].

## 2.8 Electron in an External Electromagnetic Field

The Dirac equation for an electron in an external electromagnetic field is obtained through the minimal coupling substitution:

$$E \rightarrow E - e\Phi, \qquad (79)$$

$$\mathbf{p} \rightarrow \mathbf{p} - e\mathbf{A}, \qquad (80)$$

In covariant formulation:

$$p_{\mu} \rightarrow p_{\mu} - eA_{\mu}, \qquad (81)$$

where $A_{\mu}$ admits the gauge transformation:

$$A_{\mu} \rightarrow A_{\mu} - \partial_{\mu} \Lambda, \qquad (82)$$

In electrodynamics, $A^{\mu}$ contains the complete information about the electric and magnetic fields. The Dirac equation for an electron in an external electromagnetic field becomes:

$$\gamma^0 \left( i\hbar \frac{\partial}{\partial t} - e\Phi \right) \psi = c\gamma \cdot \left( -i\hbar \nabla - e \frac{\mathbf{A}}{c} \right) \psi + mc^2 \psi, \qquad (83)$$

or in covariant form:

$$[\gamma^{\mu}(i\partial_{\mu} - eA_{\mu}) - m]\psi = 0, \qquad (84)$$
we can rewrite this using the covariant derivative:

$$D_{\mu} = \partial_{\mu} + ieA_{\mu}, \qquad (85)$$

thus, (84) becomes:

$$(i\gamma^{\mu}D_{\mu} - m)\psi = 0. \qquad (86)$$

Physically, the covariant derivative describes the coupling between the electro-magnetic field and the particle (in this case, the electron). The quantum field theory describing this interaction is Quantum Electrodynamics (QED), which we will examine in Section 3.1. An important application of the Dirac equation coupled to electromagnetic fields is the analysis of spin-orbit interaction and the magnetic moments of nucleons in the quark model, where nucleons are treated as Dirac particles [2].

## 2.9 Charge Conjugation

The "hole theory" provides a framework for interpreting negative energy states while respecting the Pauli exclusion principle. This theory describes the vacuum state where all positive energy states are unoccupied, while negative energy states are completely filled - known in the literature as the *Dirac sea*. Phenomenologically:

* A negative-energy electron absorbs radiation and gets excited to a positive energy state

* This leaves both:

- An electron with charge $-|e|$ and energy $+E$

- A hole in the Dirac sea (absence of an electron with charge $-|e|$ and energy $-E$)

* To an observer, this hole appears as a particle with charge $+|e|$ and energy $+E$ - the *positron*

This reveals a fundamental symmetry: every particle has an antiparticle (for electrons, the positron). This symmetry is called *charge conjugation*.

Quantitatively, we examine the charge conjugation of (84):

$$[\gamma^{\mu}(\partial_{\mu} + ieA_{\mu}) - m]\psi^{*}(x) = 0. \qquad (87)$$

We require an invertible matrix $C\gamma^0$ satisfying:
$$ (C\gamma^0)\gamma^{\mu*}(C\gamma^0)^{-1} = -\gamma^\mu \Rightarrow \gamma^{\mu*} = -(C\gamma^0)\gamma^\mu(C\gamma^0)^{-1}, \qquad (88) $$

Substituting (88) into (87):

$$ [-(C\gamma^0)\gamma^\mu(\partial_\mu + ieA_\mu)(C\gamma^0)^{-1} - m] \psi^*(x) = 0, \qquad (89) $$

this yields:

$$ [\gamma^\mu(\partial_\mu + ieA_\mu) - m] (C\gamma^0)^{-1} \psi^*(x) = 0, \qquad (90) $$

we define the charge-conjugated spinor:

$$ \psi^c(x) = (C\gamma^0)^{-1}\psi^*(x), \qquad (91) $$

thus:

$$ (\gamma^\mu\partial_\mu + ie\gamma^\mu A_\mu - m)\psi^c(x) = 0. \qquad (92) $$

Comparing equations (90) and (92), we observe that the fundamental difference lies in the sign of the electric charge. The charge-conjugated spinor $\psi^c(x)$ can be naturally extended to neutrino physics, where it becomes essential for characterizing Majorana neutrinos a subject we will explore in Section 4. From an experimental perspective, the positron was first discovered in 1933 through cosmic ray experiments. This seminal discovery led to Dirac and Schrödinger being jointly awarded the Nobel Prize in Physics in 1936

# 3 Quantum Field Theories

Quantum field theories (QFTs) emerge from the fundamental idea of describing particle interactions through fields. The Dirac equation has an inherent limitation - it cannot by itself describe interactions between particles, making a field theory purely based on electrons impossible. Historically, the photon concept predated the 1950s, with key experiments leading to electromagnetic field quantization including:

* The photoelectric effect

* Compton scattering

* Radiation-matter interactions
Following Dirac’s equation, Feynman, Schwinger, Tomonaga, and Dyson (between 1915-1985) developed perturbative extensions known as *Quantum Electrodynamics* (QED). However, QED cannot describe strong interactions, necessitating the development of *Quantum Chromodynamics* (QCD) - the theory of strong interactions between quarks mediated by gluons (the QCD analogs of photons). All modern field theories are formulated covariantly.

## 3.1 Quantum Electrodynamics (QED)

QED describes the interaction of charged particles with electromagnetic fields and stands as one of physics’ most precise theories, achieving predictions up to 20 significant digits. Its key experimental verifications include:

* The electron and muon magnetic moments

* The Lamb shift in hydrogen energy levels

Theoretically, QED provides a detailed framework for photon-fermion interactions, particularly for electrons. It describes fundamental processes like:

* Rutherford scattering

* Møller scattering ($e^-e^- \to e^-e^-$)

* Bhabha scattering ($e^+e^- \to e^+e^-$)

* Electron-proton scattering ($e^-p \to e^-p$)

* Hadron production in $e^+e^-$ collisions

* Compton scattering

The formal foundation begins with approximate solutions to the Dirac equation in external electromagnetic fields (86), The complete solution of the Dirac equation in an external field can be expressed as:

$$ \psi(x) = \psi_0(x) - i\epsilon \int d^4 y D_F(x-y) A(y) \psi(y) + (-i\epsilon)^2 \int d^4 y D_F(x-y) A(y) \int d^4 y_1 D_F(y-y_1) \psi(y_1) + \dots \quad (93) $$

where:

* $\psi(x)$ is the full interacting solution
* $\psi_0(x)$ is the free Dirac equation solution

* $D_F(x - y)$ is the Feynman propagator

* $A(y)$ represents the external electromagnetic field

Quantum field theory reduces this problem to a perturbative expansion in powers of the coupling constant $\epsilon$, enabling precise calculations of scattering cross-sections. Each term in this expansion admits a graphical representation known as Feynman diagrams. Therefore, the lowest-order transition matrix element in QED is:

$$S_{fi} \approx \delta_{fi} \mp i\epsilon \int d^4 y \bar{\psi}_f(y) A(y) \psi_i(y), \qquad (94)$$

and at second order:

$$S_{fi} \approx \delta_{fi} \mp i\epsilon \int d^4 y \bar{\psi}_f(y) A(y) \psi_i(y) \mp (-i\epsilon)^2 \int d^4 y_1 d^4 y_2 \bar{\psi}_f(y_1) A(y_1) D_F(y_1-y_2) A(y_2) \psi_i(y_2) + \dots \qquad (95)$$

From these, we derive the differential cross-section for electromagnetic processes $ab \to cd$ [2, 3]

$$d\sigma \approx |\mathcal{M}_{fi}|^2 \delta^4(p_c + p_d - p_a - p_b) \frac{d^3 p_c d^3 p_d}{(2\pi)^6 2E_c 2E_d}, \qquad (96)$$

for fermionic particles $abcd$ interacting via the electromagnetic field, equation (93) describes the scattering cross-section, which depends on the Feynman invariant amplitude $|\mathcal{M}|^2$ and the reference frame of the reaction. Considering electron-positron interactions specifically, we establish Feynman rules for amplitude calculations. The matrix element can be derived from Green's functions or propagators $D_F$, yielding:

$$S_{fi} = \delta^4(p_c + p_d - p_a - p_b)\mathcal{M}, \qquad (97)$$

where the invariant Feynman amplitude is:

$$i\mathcal{M} \approx (-ie\gamma^\mu)_{ac} \left( \frac{-ig_{\mu\nu}}{q^2 + i\epsilon} \right) (-ie\gamma^\nu)_{bd}, \qquad (98)$$

with $q = p_d - p_b$ being the momentum transfer. This amplitude is computed using Feynman diagrams.
## 3.2 Decay Rates in QED

For a decay process $a \to \sum_{k=1}^{n} f_k$ (where $f_k$ are final state particles), the decay rate per unit time is:

$$d\Gamma = \frac{1}{2E_a} |\mathcal{M}|^2 \prod_{i=1}^{n} \frac{d^3p_i}{2E_i(2\pi)^3} (2\pi)^4 \delta^4 \left( p_a - \sum_{k=1}^{n} p_k \right), \qquad (99)$$

For the special case of two-body decay in the rest frame:

$$\Gamma(a \to 1+2) \approx \frac{p_f}{m_a^2} \int d\Omega |\mathcal{M}|^2, \qquad (100)$$

the particle's lifetime is then $\tau = \Gamma^{-1}$.

## 3.3 Gauge Invariance in QED

QED is a gauge theory requiring invariance under local U(1) transformations. The free Lagrangian:

$$\mathcal{L} = \bar{\psi}(i\gamma^{\mu}\partial_{\mu} - m)\psi, \qquad (101)$$

transforms under:

$$\psi(x) \to e^{-ie\alpha(x)}\psi(x), \qquad (102)$$

necessitating the introduction of the gauge covariant derivative:

$$\partial_{\mu} \to D_{\mu} = \partial_{\mu} - ieA_{\mu}, \quad A_{\mu} \to A_{\mu} + \frac{1}{e}\partial_{\mu}\alpha, \qquad (103)$$

thus, the QED Lagrangian is given by:

$$\mathcal{L}_{\text{QED}} = \bar{\psi}(i\gamma^{\mu}D_{\mu} - m)\psi - \frac{1}{4}F^{\mu\nu}F_{\mu\nu}, \qquad (104)$$

here, $F^{\mu\nu}$ is the electromagnetic field tensor. It is important to note that quantum electrodynamics does not provide specific predictions about what will happen in a given experiment, but rather probabilities of different outcomes for a particular type of situation. That is why experiments use a relatively large number of particles, which are scattered statistically in accordance with the probabilities predicted by the theory. From the distribution of scattered particles, one can measure the effective cross-section and compare it with the predictions of the theory. The predictions of quantum electrodynamics have been confirmed by experiments to an extraordinary level of precision: typically, one obtains experimental results that agree to 12 decimal places with the theoretical predictions. This makes quantum
electrodynamics one of the most precise theories ever developed by humankind. The dynamics and fundamental properties of a field theory are determined by the form of its Lagrangian. The selection of the Lagrangian depends on the symmetry of the gauge group and the requirement that the theory describes the interaction among charged fermions. In a theory describing fermionic fields interacting through a bosonic gauge field associated with massless particles (photons), whose gauge group is commutative, the Lagrangian in (94) can be taken as the starting point. The physical interpretation is that $\psi$ and its adjoint $\bar{\psi}$ represent the charged fermionic fields, more specifically the electron and the positron through the Dirac spinor. The electromagnetic part of the Lagrangian contains the electromagnetic field tensor that describes the evolution of the electromagnetic field, while the covariant derivative $D_\mu$ in the Dirac equation describes the free evolution of fermionic fields, along with their interaction with the electromagnetic field. The equations of motion are obtained by applying the Euler–Lagrange equations to the covariant formulation:

$$(i\gamma^\mu D_\mu - m)\psi = 0, \qquad (105)$$

$$\partial_\nu F^{\mu\nu} = e\bar{\psi}\gamma^\mu\psi. \qquad (106)$$

A problem that arises in QED is the renormalization in Feynman rules. The Feynman rules relate terms and diagrams with the most divergent amplitudes. The renormalization procedure establishes precise relations between calculated terms and physical observables, providing additional rules to "regularize" calculations and ensure finite numerical results comparable with experimental data. QED is a renormalizable theory, meaning it contains rules to eliminate unobservable divergent terms and yield finite physical predictions.

## 3.4 Quantum Chromodynamics (QCD)

Quantum Chromodynamics (QCD) extends the framework of QED to describe nucleon interactions through quark-gluon dynamics, where gluons serve as the QCD analogs of photons. The strong interaction is characterized by:

* SU(3) color charge symmetry, with quark fields transforming as SU(3) triplets

* Asymptotic freedom - quarks behave as nearly free particles at short distances

* Increasing interaction strength with separation distance (contrasting QED)

* Gluon self-interactions (unlike photons in QED)
• Color charge conservation via Noether's theorem

• Color confinement due to gluon self-coupling

• Short interaction range compared to QED

The QCD Lagrangian is given by:

$$ \mathcal{L}_{\text{QCD}} = \bar{q}(i\gamma^{\mu}D_{\mu} - m)q - \frac{1}{4}G_{\mu\nu}^{a}G_{a}^{\mu\nu} \quad (107) $$

with local gauge transformation:

$$ q(x) \rightarrow e^{-i\alpha_a(x)T^a}q(x) \quad (108) $$

The covariant derivative in QCD takes the form:

$$ D_{\mu} = \partial_{\mu} + igT^{a}G_{\mu}^{a}, \quad G_{\mu}^{a} \rightarrow G_{\mu}^{a} - \frac{1}{g}\partial_{\mu}\alpha^{a} - f^{abc}G_{\mu}^{b}\alpha^{c} \quad (109) $$

where $G_{\mu\nu}^a$ represents the gluonic field strength tensor (QCD analog of the electromagnetic tensor):

$$ G_{\mu\nu}^{a} = \partial_{\mu}G_{\nu}^{a} - \partial_{\nu}G_{\mu}^{a} - gf^{abc}G_{\mu}^{b}G_{\nu}^{c} \quad (110) $$

The $f^{abc}$ represent the structure constants of the SU(3) Lie algebra, where indices a, b, c range from 1 to 8 denoting gluon types. The $G_{\mu}^a$ correspond to the eight vector potential components, with g as the color field coupling constant. The three color charges are conventionally labeled R (red), B (blue), and G (green) - though these designations are purely formal and unrelated to visual color (an electromagnetic phenomenon associated with specific wavelengths). When perturbative methods fail, lattice QCD provides an alternative formulation using discretized spacetime, originally proposed by Kenneth Wilson in 1974 [3]. This non-perturbative approach enables numerical computations through:

• Discretization of QCD on a spacetime lattice

• Monte Carlo integration techniques

• Supercomputing implementations.
## 3.5 Hadronic Electrodynamics

### 3.5.1 Hadronic Currents

The hadronic current emerges as a phenomenological quantity constructed from general kinematic constraints, independent of specific dynamical assumptions. For quark transitions, we replace the current with an appropriate transition matrix element incorporating strong interaction effects:

$$ \langle p(p') | h^\mu(0) | n(p) \rangle, \qquad (111) $$

where $h^\mu(x) = \bar{q}(x)\gamma^\mu q(x)$ represents the quark current. The general hadronic matrix element combines vector and axial-vector components:

$$ h^{\mu}(x) = V^{\mu}(x) - A^{\mu}(x). \qquad (112) $$

### 3.5.2 Neutrino-Proton Scattering

For neutrino-proton interactions (with both particles treated as free fermions), the momentum-space Dirac equations govern their behavior:

$$ (\not p - m)\nu(p) = 0, \qquad (113) $$

$$ (\not p - m)\ell(p) = 0. \qquad (114) $$

Considering the most general bilinear spinor form $\bar{u}(p')\Gamma^\mu u(p)$, the hadronic vector current matrix element takes the form:

$$ \langle p(p') | V^\mu(0) | n(p) \rangle \approx \bar{u}(p') [f_1(q^2)\gamma^\mu + f_2(q^2)\sigma^{\mu\nu}q_\nu + f_3(q^2)q^\mu] u(p), \qquad (115) $$

where $q^\mu = p'^\mu - p^\mu$ is the momentum transfer and $f_i(q^2)$ are form factors characterizing the hadronic structure. The form factors $f_1(q^2), f_2(q^2), f_3(q^2)$ depend on the Lorentz-invariant quantity constructed from $p_n$ and $p_p$:

$$ p_n \cdot p_p = \frac{1}{2}[m_n^2 + m_p^2 - q^2] \quad \text{where} \quad q^\mu := p_p^\mu - p_n^\mu, \qquad (116) $$

here $m_p$ and $m_n$ are the constant proton and neutron masses, making the form factors depend solely on $q^2$. Applying the Dirac equation for spinors yields:

$$ \bar{u}_p(p_p)\gamma^\mu u_n(p_n) = \bar{u}_p(p_p) \left[ \frac{p_p^\mu + p_n^\mu}{2m_N} - \frac{(m_n + m_p)\gamma^\mu}{2m_N} \right] u_n(p_n). \qquad (117) $$

Using the approximation $m_p \approx m_n \approx m_N$ (nucleon mass), we rewrite (115) as:
$$ \langle p(p_p)|V^\mu(0)|n(p_n)\rangle \approx \bar{u}_p(p_p) \left[ \gamma^\mu F_1(Q^2) + i \frac{\sigma^{\mu\nu} q_\nu}{2m_N} F_2(Q^2) + \frac{q^\mu}{m_N} F_3(Q^2) \right] u_n(p_n). \quad (118) $$

Similarly, the axial-vector current becomes:

$$ \langle p(p_p)|A^\mu(0)|n(p_n)\rangle \approx \bar{u}_p(p_p) \left[ \gamma^\mu \gamma^5 G_A(Q^2) + \frac{q^\mu}{m_N} \gamma^5 G_P(Q^2) + \frac{p_p^\mu + p_n^\mu}{m_N} \gamma^5 G_3(Q^2) \right] u_n(p_n). \quad (119) $$

## 3.6 Kinematic Constraints and Symmetries

The form factors are typically expressed as functions of $Q^2 := -q^2$. For neutron decay ($q^2 > 0$), the kinematic range is:

$$ m_e^2 \le q^2 \le (m_n - m_p)^2 \quad (m_e \approx 0.51 \text{ MeV}, m_n - m_p \approx 1.29 \text{ MeV}) \quad (120) $$

Key symmetry implications:

* Time-reversal invariance of strong interactions requires $F_1, F_2, F_3, G_A, G_P, G_3$ to be real

* Isospin symmetry implies $F_3(Q^2) = 0$ and $G_3(Q^2) = 0$ (second-class currents)

Thus, the simplified vector and axial matrix elements become:

$$ \langle p(p')|V^\mu(0)|n(p)\rangle \approx \bar{u}_p(p') \left[ \gamma^\mu F_1(Q^2) + i \frac{\sigma^{\mu\nu} q_\nu}{2m_N} F_2(Q^2) \right] u_n(p), \quad (121) $$

$$ \langle p(p')|A^\mu(0)|n(p)\rangle \approx \bar{u}_p(p') \left[ \gamma^\mu \gamma^5 G_A(Q^2) + \frac{q^\mu}{m_N} \gamma^5 G_P(Q^2) \right] u_n(p). \quad (122) $$

The complete nuclear matrix element for beta decay combines these currents:

$$ \langle p(p')|h^\mu(0)|n(p)\rangle \approx \bar{u}_p(p') \left\{ \begin{array}{l} \gamma^\mu F_1(Q^2) + i \frac{\sigma^{\mu\nu} q_\nu}{2m_N} F_2(Q^2), \\ - \gamma^\mu \gamma^5 G_A(Q^2) - \frac{q^\mu}{m_N} \gamma^5 G_P(Q^2) \end{array} \right\} u_n(p), \quad (123) $$

here $F_1, F_2, G_A$, and $G_P$ are the weak charged current form factors of nucleons.
## 3.7 Electromagnetic Current Analogy

The vector hadronic current relates to the electromagnetic current through:

$$ \langle N(p') | j_{em}^{\mu} | N(p) \rangle \approx \bar{u}_N(p') \left( \gamma^{\mu} F_1^N(Q^2) + i \frac{\sigma^{\mu\nu} q_{\nu}}{2m_N} F_2^N(Q^2) \right) u_N(p), \quad (124) $$

where $N = p, n$ and $j_{em}^{\mu}$ represents the quark electromagnetic current:

$$ j_{em}^{\mu} = \frac{2}{3} \sum_{a=u,c,t} \bar{q}_a \gamma^{\mu} q_a - \frac{1}{3} \sum_{a=d,s,b} \bar{q}_a \gamma^{\mu} q_a, \quad (125) $$

The isovector combinations $F_i^{p-n}(Q^2) \equiv F_i^p(Q^2) - F_i^n(Q^2)$ are known as the Dirac ($F_1$) and Pauli ($F_2$) electromagnetic form factors. The conservation of electromagnetic current ($\partial_{\mu}j_{em}^{\mu} = 0$) eliminates terms proportional to $q^{\mu}\bar{u}_N(p')u_N(p)$.

### 3.7.1 Special Case $q^2 = 0$

Using the spinor normalization convention:

$$ \bar{u}^{(s')}(p) \gamma^0 u^{(s)}(p) = 2E_p \delta^{s's} \quad (126) $$

we derive the form factor constraints at zero momentum transfer:

$$ \langle N(p) | j_{em}^0 | N(p) \rangle = 2E_p F_1^N(0) \quad (127) $$

$$ = \frac{1}{V} \int d^3 x \langle N(p) | e^{-ip \cdot x} j_{em}^0(x) e^{ip \cdot x} | N(p) \rangle \quad (128) $$

$$ = \frac{1}{V} \langle N(p) | Q | N(p) \rangle \quad (129) $$

$$ = 2E_p q_N, \quad (130) $$

Thus, the Dirac form factors yield the nucleon electric charges:

$$ F_1^p(0) = 1, \quad F_1^n(0) = 0, \quad (131) $$

while the Pauli form factors give the anomalous magnetic moments:

$$ F_2^p(0) = \frac{\mu_p}{\mu_N} - 1 \approx 1.792, \quad (132) $$

$$ F_2^n(0) = \frac{\mu_n}{\mu_N} \approx -1.913, \quad (133) $$
where $\mu_N = e\hbar/2m_p$ is the nuclear magneton, and the measured values are:

$$ \mu_p \approx 2.792\mu_N, \qquad (134) $$

$$ \mu_n \approx -1.913\mu_N. \qquad (135) $$

## 3.8 Sachs Form Factors

We define the Sachs electromagnetic form factors:

$$ G_E^N(Q^2) = F_1^N(Q^2) - \frac{Q^2}{4m_N^2} F_2^N(Q^2), \qquad (136) $$

$$ G_M^N(Q^2) = F_1^N(Q^2) + F_2^N(Q^2), \qquad (137) $$

which provide:

* $G_E^N(Q^2)$: Electric charge distribution

* $G_M^N(Q^2)$: Magnetic moment distribution

The Sachs form factors satisfy the following normalization conditions:

$$ G_E^p(0) = 1, \qquad (138) $$

$$ G_E^n(0) = 0, \qquad (139) $$

$$ G_M^p(0) = \frac{\mu_p}{\mu_N}, \qquad (140) $$

$$ G_M^n(0) = \frac{\mu_n}{\mu_N}. \qquad (141) $$

## 3.9 Dipole Form Factor Parameterization

Electron scattering data are typically parameterized using the dipole form factor $G_D(Q^2)$ for nucleon electromagnetic structure:

$$ G_E^p(Q^2) = G_D(Q^2), \qquad (142) $$

$$ G_E^n(Q^2) = 0, \qquad (143) $$

$$ G_M^p(Q^2) = \frac{\mu_p}{\mu_N} G_D(Q^2), \qquad (144) $$

$$ G_M^n(Q^2) = \frac{\mu_n}{\mu_N} G_D(Q^2), \qquad (145) $$
where the dipole form factor is defined as:

$$G_D(Q^2) = \left(1 + \frac{Q^2}{M_V^2}\right)^{-2}, \quad M_V \approx 0.84 \text{ GeV}. \qquad (146)$$

## 3.10 Breit Frame Analysis

The electromagnetic form factors characterize the current matrix elements in the Breit frame (special reference frame where $q^0 = 0$ and $|q| = \sqrt{Q^2}$):

$$\langle N(p_f)|j_{em}^0|N(p_i)\rangle = \bar{u}_N(p_f)\gamma^0 u_N(p_i)G_E^N(Q^2), \qquad (136)$$

$$\langle N(p_f)|\dot{j}_{em}|N(p_i)\rangle = \bar{u}_N(p_f)\gamma u_N(p_i)G_M^N(Q^2). \qquad (147)$$

The form factor $G_E^N(Q^2)$ can be interpreted as the Fourier transform of a spherically symmetric charge distribution $\rho_N(r)$:

$$G_E^N(Q^2) = \int d^3 r e^{i \mathbf{q} \cdot \mathbf{r}} \rho_N(r),$$

The electric form factor can be expanded for small momentum transfers ($\sqrt{Q^2} < 1$ GeV):

$$G_E^N(Q^2) = \int d^3 r \rho_N(r) e^{-i \mathbf{q} \cdot \mathbf{r}} \approx 4\pi \int dr r^2 \rho_N(r) \frac{\sin(|\mathbf{q}|r)}{|\mathbf{q}|r} \approx 1 - \frac{1}{6} Q^2 \langle r_E^2 \rangle \quad (148)$$

where $\langle r_E^2 \rangle$ represents the mean squared charge radius. For the proton's dipole form factor approximation, the electric charge radius is approximately 1 fm [4].

# 4 Neutrino Physics

## 4.1 Neutrino Properties

Neutrinos (a term coined by Enrico Fermi meaning "little neutral ones" in Italian) are spin-1/2 fermions with:

* Zero electric charge

* Measured masses $m_\nu \approx 5.34$ eV (from recent experiments)

* Mass eigenstates determined by the neutrino mass terms in the Lagrangian

The discovery of neutrino mass emerged from the 1957 Wu experiment (confirming parity violation) and subsequent solar neutrino observations.
## 4.2 Dirac Neutrino Dynamics

For Dirac neutrinos, the dynamics are described by:

$$ (i\gamma^{\mu}\partial_{\mu} - m_{\nu})\nu(x) = 0, \qquad (149) $$

where $\nu(x)$ is the Dirac spinor. We can decompose it into Weyl spinors:

$$ \nu(x) = \begin{pmatrix} \nu_L(x) \\ \nu_R(x) \end{pmatrix}. \qquad (150) $$

These chiral components are eigenstates of $\gamma^5$:

$$ -\nu_L(x) = \gamma^5 \nu_L(x), \qquad (151) $$

$$ +\nu_R(x) = \gamma^5 \nu_R(x). \qquad (152) $$

For a generic spinor $\psi = \psi_L + \psi_R$, the projection operators yield:

$$ \nu_R(x) = \frac{1 + \gamma^5}{2} \nu_R(x), \qquad (153) $$

$$ \nu_L(x) = \frac{1 - \gamma^5}{2} \nu_L(x), \qquad (154) $$

where $\frac{1\pm\gamma^5}{2}$ are the projection operators. The Dirac Lagrangian for neutrinos is:

$$ \mathcal{L}_D = \bar{\nu}(x)(i\gamma^\mu\partial_\mu - m)\nu(x), \qquad (155) $$

Substituting (155) into the Euler-Lagrange equations yields:

$$ i\gamma^\mu\partial_\mu\nu_L(x) - m\nu_R(x) = 0, \qquad (156) $$

$$ i\gamma^\mu\partial_\mu\nu_R(x) - m\nu_L(x) = 0. \qquad (157) $$

For massless neutrinos:

$$ i\gamma^\mu\partial_\mu\nu_L(x) = 0, \qquad (158) $$

$$ i\gamma^\mu\partial_\mu\nu_R(x) = 0. \qquad (159) $$

These are known as the Weyl equations. Decomposing (158) and (159):

$$ (\gamma^0 E - \gamma \cdot p)\nu_R(x) = 0, \qquad (160) $$
$$(-\gamma^0 E - \gamma \cdot p)\nu_L(x) = 0, \qquad (161)$$

multiplying by $\gamma^5$ gives:

$$\Sigma \cdot p \nu_R = \gamma^5 E \nu_R, \qquad (162)$$

$$\Sigma \cdot p \nu_L = -\gamma^5 E \nu_L, \qquad (163)$$

Equations (162) and (163) describe massless chiral fields in terms of the helicity operator $\Sigma$. The chiral fields can be momentum-dependent. The interpretation is as follows: If the chiral fields are right-handed, the helicity has a positive value, while left-handed chiral fields have negative helicity. For Majorana neutrinos, we define a complex spinor satisfying the Dirac equation (149):

$$\nu^c(x) = C \bar{\nu}^T(x), \qquad (164)$$

where $C$ is the charge conjugation matrix. The Dirac equation (149) can be written as:

$$(i\gamma^\mu \partial_\mu - m)\nu^c(x) = 0. \qquad (165)$$

The charge conjugation matrix satisfies the following relations with gamma matrices:

$$C^\dagger = C^T = C^{-1} = -C, \qquad (166)$$

$$C\gamma^{\mu}C^{-1} = -\gamma^{\mu}, \qquad (167)$$

$$C\gamma^{5T}C^{-1} = \gamma^{5}, \qquad (168)$$

$$C\gamma^{\mu}\gamma^{5}C^{-1} = -(\gamma^{\mu}\gamma^{5})^{T}. \qquad (169)$$

There are also important properties for the complex spinors:

$$(\nu^c)^c = \nu, \qquad (170)$$

$$\nu^c = \nu^T C, \qquad (171)$$

$$\bar{\nu}_1 \nu_2^c = \bar{\nu}_2^c \nu_1, \qquad (172)$$

$$(\nu_L)^c = (\nu^c)_R, \qquad (173)$$

$$(\nu_R)^c = (\nu^c)_L. \qquad (174)$$

The gamma matrices in the Majorana representation are known as Majorana matrices. Comparing (149) with (165) we find:
$$ \nu(x) = (\nu(x))^c \qquad (175) $$

This is known as the Majorana condition. In contrast to Dirac neutrinos, three important consequences emerge:

i) If the field $\nu_L(x)$ is independent of $\nu_R(x)$, it is called a Dirac Neutrino

ii) If $\nu_L = (\nu_R)^c$ it is called a Majorana Neutrino

iii) If $m = 0$ we recover the Weyl equations

Another important but unaddressed aspect is the mass problem for Majorana vs. Dirac neutrinos. Phenomenologically, the Dirac and Majorana mass matrices are obtained from the Dirac Lagrangian using diagonalizable unitary matrices that can be interpreted as neutrino flavor mixing. Note that the unitary mixing matrices differ between Dirac and Majorana neutrinos.

### 4.2.1 Electromagnetic Form Factors for Neutrinos

**Dirac Neutrino Form Factors** For Dirac neutrinos, we consider the current matrix element $\langle \mathbf{p}', \mathbf{s}' | j_{\mu}(0) | \mathbf{p}, \mathbf{s} \rangle$ at $x = 0$:

$$ \langle \mathbf{p}', \mathbf{s}' | j_{\mu}(0) | \mathbf{p}, \mathbf{s} \rangle = \bar{u}_{s'}(\mathbf{p}') \Gamma_{\mu}(p', p) u_{s}(\mathbf{p}). \qquad (176) $$

The effective Lagrangian's hermiticity requires:

$$ \Gamma_{\mu}^{\dagger} = \gamma_{0} \Gamma_{\mu}(p', p) \gamma_{0}, \qquad (177) $$

Originally this matrix element is given by:

$$ \langle \mathbf{p}', \mathbf{s}' | j_{\mu}(x) | \mathbf{p}, \mathbf{s} \rangle = \bar{u}_{s'}(\mathbf{p}') \Gamma_{\mu}(p', p) u_{s}(\mathbf{p}) e^{iq \cdot x}, \qquad (178) $$

applying the 4-derivative:

$$ \begin{aligned} \langle \mathbf{p}', \mathbf{s}' | \partial^{\mu} j_{\mu}(x) | \mathbf{p}, \mathbf{s} \rangle &= \bar{u}_{s'}(\mathbf{p}') \Gamma_{\mu}(p', p) u_{s}(\mathbf{p}) \partial^{\mu} e^{iq^{\mu}x_{\mu}} \\ &= \bar{u}_{s'}(\mathbf{p}') q^{\mu} \Gamma_{\mu}(p', p) u_{s}(\mathbf{p}) e^{iq^{\mu}x_{\mu}}, \end{aligned} $$

this implies current conservation, where $q^{\mu} = p^{\mu} - p'^{\mu}$. The matrix element for the effective electromagnetic interaction current, expressed via the Gordon identity, is given by:

$$ \Gamma_{\mu}(q, q') \approx (q^2 \gamma_{\mu} - q_{\mu} \not{q}) [F_0(q^2) + F_1(q^2) \gamma_5] + \sigma_{\mu\nu} q^{\nu} [G_0(q^2) + iG_1(q^2) \gamma_5] \qquad (179) $$
Here $F_0, F_1, G_0$, and $G_1$ represent the Lorentz-invariant form factors dependent on $q^2$. Their physical meaning becomes apparent in the non-relativistic limit when using plane-wave solutions and Dirac gamma matrices:

* $G_0(0)\mathbf{B} \cdot \boldsymbol{\sigma}$ reveals $G_0(0) := D_M(q^2) \equiv \mu_m$ as the magnetic dipole moment.

* $G_1(q^2) := D_E(q^2) \equiv \mu_e$ as the electric dipole moment.

* $F_0(q^2) := R(q^2)$ corresponds to the charge radius.

* $F_1(q^2) := r_A(q^2)$ represents the axial charge radius.

### 4.2.2 Form Factors for Majorana Neutrinos

For Majorana neutrinos, the form factors must satisfy:

$$R = D_M = D_E = 0, \qquad (180)$$

leading to:

$$\Gamma_{\nu}(q, q') \approx (q^2\gamma_{\nu} - q_{\nu}q)F_1(q^2)\gamma_5. \qquad (181)$$

The charge radius term $R(q^2)$ originates from interactions involving a fermion bilinear $\bar{\psi}\gamma_5\psi$ coupled to the electromagnetic field (photon). Similarly:

* The axial charge radius emerges from $\bar{\psi}\gamma_\nu\gamma_5\psi$ interactions

* The magnetic dipole moment $D_M$ from $\bar{\psi}\sigma_{\mu\nu}\psi$

* The electric dipole moment from $\bar{\psi}\sigma_{\mu\nu}\gamma_5\psi$

Considering the bilinear $\bar{\psi}\gamma_\nu\gamma_5\psi$, we define a 4 × 4 matrix $F$ satisfying:

$$\bar{\psi}F\psi = \bar{\psi}CF^T C^{-1}\psi. \qquad (182)$$

For the specific case $F = \gamma_\mu\gamma_5$:

$$\gamma_\mu\gamma_5 = C(\gamma_\mu\gamma_5)^T C^{-1}. \qquad (183)$$

Following an analogous procedure to Dirac neutrino form factors, we examine the matrix element:

$$\langle \boldsymbol{p}', \boldsymbol{s}' | \bar{\psi} F \psi | \boldsymbol{p}, \boldsymbol{s} \rangle = \bar{u}_v(\boldsymbol{p}')Fu_v(\boldsymbol{p}) - \bar{v}_v(\boldsymbol{p}')Fv_v(\boldsymbol{p}), \qquad (184)$$

alternatively expressed through matrix $F$:

$$\langle \boldsymbol{p}', \boldsymbol{s}' | \bar{\psi} F \psi | \boldsymbol{p}, \boldsymbol{s} \rangle = \bar{u}_v(\boldsymbol{p}') (F + CF^T C^{-1}) u_v(\boldsymbol{p}). \qquad (185)$$

This shows that Majorana neutrino form factors are associated with dipole moments from the fermion bilinear $\bar{\psi}\sigma_{\mu\nu}\gamma_5\psi$.
## 4.3 Form Factors for Weyl Neutrinos

Electromagnetic form factors for Weyl neutrinos involve the bilinears $\bar{\psi}\sigma_{\mu\nu}\psi$ and $\bar{\psi}\sigma_{\mu\nu}\gamma_5\psi$ for dipole moments. Using chiral projection operators:

$$ \bar{\psi}\sigma_{\mu\nu}\psi = \bar{\psi}_R\sigma_{\mu\nu}\psi_L + \bar{\psi}_L\sigma_{\mu\nu}\psi_R \quad (186) $$

$$ \bar{\psi}\sigma_{\mu\nu}\gamma_5\psi = \bar{\psi}_R\sigma_{\mu\nu}\gamma_5\psi_L + \bar{\psi}_L\sigma_{\mu\nu}\gamma_5\psi_R. \quad (187) $$

Neglecting charge and axial radii, the form factors reduce to:

$$ \bar{\psi}\gamma_{\mu} \left[ (R-r) \left( \frac{1-\gamma_5}{2} \right) + (R+r) \left( \frac{1+\gamma_5}{2} \right) \right] \psi \approx \bar{\psi}_L \gamma_{\mu} [R-r] \psi_L + \bar{\psi}_R \gamma_{\mu} [R+r] \psi_R. \quad (188) $$

## 4.4 Key Physical Aspects

Two crucial features emerge:

* Weyl neutrinos possess non-zero electromagnetic form factors, similar to Majorana neutrinos, combining Lorentz-covariant and gauge-invariant structures

* Neutrino dipole moments must vanish in the massless limit. For massive neutrinos, dipole moments scale proportionally with mass - lighter neutrinos exhibit smaller dipole moments [4, 5].

## References

[1] M. De Sanctis. An Introduction to Relativistic Quantum Mechanics I. From Relativity to Dirac Equation. arXiv:0708.0052v3 [physics.gen-ph] 30 Sep 2011

[2] Bjorken. J. D, Drell. S.D. Relativistic Quantum Mechanics. McGraw Hill (1965)

[3] Francis. H, Martin. A. Quarks and Leptons: An Introductory Course in Modern Particle Physics. John Wiley and Sons. (1984)

[4] Giunti. C, Kim. C. Fundamental of Neutrino Physics on Astrophysics. Oxford University Press

[5] Mohapatra.R, Palash.P. Massive Neutrinos in Physics and Astrophysics Third Edition (2004). World Scientific