Observation of neutrino oscillations implies a finite mass...

Observation of neutrino oscillations implies a finite mass for neutrinos and thus physics beyond the Standard Model. One idea for extending the Standard Model is to add the missing right-handed component of the neutrino, which is assumed to be very massive since it is not seen. By some elementary arguments, it is speculated that the 2-flavor neutrino mass matrix M (see Section 12.2) for such a minimal extension of the Standard Model to include a heavy righthanded neutrino may be written $$ \mathscr{M} \simeq\left(\begin{array}{cc} 0 & m \\ m & M \end{array}\right), $$ where it is expected that $m$ is of order $100 \mathrm{GeV}$ and $M$ is of order $10^{15} \mathrm{GeV}$. (a) Find the eigenvalues of this matrix and show that this gives one very light and one very massive eigenvalue for the neutrino masses. Hint: See Problem 11.5. (b) Find the corresponding eigenvectors and argue that they can be interpreted approximately as a left-handed Standard Model neutrino but with a small but non-zero mass, and a right-handed neutrino with mass much too large to be detected in present experiments.

Observation of neutrino oscillations implies a finite mass for neutrinos and thus physics beyond the Standard Model. One idea for extending the Standard Model is to add the missing right-handed component of the neutrino, which is assumed to be very massive since it is not seen. By some elementary arguments, it is speculated that the 2-flavor neutrino mass matrix M (see Section 12.2) for such a minimal extension of the Standard Model to include a heavy righthanded neutrino may be written
$$
\mathscr{M} \simeq\left(\begin{array}{cc}
0 & m \\
m & M
\end{array}\right),
$$
where it is expected that $m$ is of order $100 \mathrm{GeV}$ and $M$ is of order $10^{15} \mathrm{GeV}$. (a) Find the eigenvalues of this matrix and show that this gives one very light and one very massive eigenvalue for the neutrino masses. Hint: See Problem 11.5. (b) Find the corresponding eigenvectors and argue that they can be interpreted approximately as a left-handed Standard Model neutrino but with a small but non-zero mass, and a right-handed neutrino with mass much too large to be detected in present experiments.

Key Concepts

Eigenvector Interpretation

Eigenvector interpretation involves analyzing the composition of the mass eigenstates obtained from the diagonalization of the mass matrix. In the seesaw framework, the eigenvectors show that the light neutrino state is predominantly the left-handed neutrino with a small admixture of the heavy state, while the heavy eigenstate is primarily the right-handed neutrino. This interpretation helps in understanding the physical implications of the neutrino mass generation mechanism and its detectability in experiments.

Mass Matrix Diagonalization

Mass matrix diagonalization is a mathematical procedure used to determine the physical mass eigenstates of a system from its Lagrangian. In the context of neutrino mass generation, constructing and diagonalizing a 2x2 neutrino mass matrix reveals the emergence of one very light and one very heavy state. This process is essential for linking theoretical models with the observed behavior of neutrino masses and mixings.

Seesaw Mechanism

The seesaw mechanism is a theoretical model used to explain the small masses of the observed neutrinos. It postulates a mass matrix with a large Majorana mass for the right-handed neutrino and a relatively small Dirac mass coupling. Diagonalizing this matrix yields one very light neutrino, corresponding largely to the left-handed state, and one very massive neutrino, predominantly the right-handed state, thereby ‘seesawing’ the masses between the two scales.

Standard Model Extension

Extending the Standard Model involves incorporating new particles or interactions to explain phenomena that the Standard Model cannot, such as neutrino masses. One common approach is to add right?handed neutrino components. This extension is crucial in explaining why neutrinos have a small but nonzero mass and in exploring additional symmetries or mechanisms beyond those in the Standard Model.

Neutrino Oscillations

Neutrino oscillations refer to the phenomenon where neutrinos change their flavor as they propagate. This observation implies that neutrinos have mass, which is not accounted for in the original formulation of the Standard Model. The study of neutrino oscillations motivates extensions of the Standard Model and provides key experimental evidence that guides theories addressing the neutrino mass problem.

Right-Handed Neutrino

Right-handed neutrinos are hypothetical particles that are not included in the Standard Model. They are introduced in many extensions as sterile neutrinos that do not participate in weak interactions but can be very massive. Their inclusion provides a natural explanation for the smallness of the observed neutrino masses through mechanisms that couple heavy and light neutrino states.

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