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Neutrino Physics

Field: Particle Physics

The study of neutrinos, subatomic particles with no electric charge and very small mass.

Sequence of Expressions

Define the Lagrangian density L\mathcal{L} for the neutrino sector, including the mass term mνm_{\nu}. For a massive neutrino field ν\nu, the mass term contributes to the Lagrangian as: \nLmass=−12νˉ(Mν)ijν+h.c.\mathcal{L}_{mass} = -\frac{1}{2} \bar{\nu} (M_{\nu})_{ij} \nu + \text{h.c.} \nwhere ν\nu is the neutrino field, and (Mν)ij(M_{\nu})_{ij} is the 3×33 \times 3 neutrino mass matrix in the flavor basis, whose eigenvalues represent the physical masses mim_i of the mass eigenstates.
Consider the evolution of the neutrino state ∣ν(L)⟩|\nu(L)\rangle over a distance LL. The flavor state ∣να⟩|\nu_{\alpha}\rangle evolves according to the Hamiltonian HH: \niddL∣ν(L)⟩=H∣ν(L)⟩i \frac{d}{dL} |\nu(L)\rangle = H |\nu(L)\rangle \nwhere H=Udiag(E1,E2,E3)U†H = U \text{diag}(E_1, E_2, E_3) U^{\dagger}, with UU being the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix, and EiE_i are the energies of the mass eigenstates, related to the squared mass differences Δmij2\Delta m_{ij}^2 by Ei≈p+mi22pE_i \approx p + \frac{m_i^2}{2p}. The probability of transition Pα→βP_{\alpha \to \beta} is given by the survival probability formula derived from this evolution.
For neutrino oscillations observed over a short baseline LL, the probability Pα→βP_{\alpha \to \beta} is approximated by considering the small parameter Δm2L/E\Delta m^2 L / E. The oscillation probability is generally given by: \nPα→β≈sin⁡2θeffsin⁡2(Δmeff2L4E)P_{\alpha \to \beta} \approx \sin^2 \theta_{\text{eff}} \sin^2 \left(\frac{\Delta m^2_{\text{eff}} L}{4E}\right) \nwhere θeff\theta_{\text{eff}} and Δmeff2\Delta m^2_{\text{eff}} are effective mixing angles and mass-squared differences, respectively, constrained by the short distance LL and the neutrino energy EE. This regime often requires extending the standard three-flavor model to include sterile neutrinos.
The Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix UU is a 3×33 \times 3 unitary matrix that relates the flavor eigenstates νf=(νe,νμ,ντ)T\nu_{f} = (\nu_e, \nu_{\mu}, \nu_{\tau})^T to the mass eigenstates νm=(ν1,ν2,ν3)T\nu_{m} = (\nu_1, \nu_2, \nu_3)^T: \n\n$$ \nu_{f} = U \nu_{m} \tag{7} \text{where } U_{fi} = \langle \nu_{f} | \nu_{i} \rangle \text{ and } U^{\dagger} U = I. \tag{8} \text{The matrix elements contain the mixing angles } \theta_{ij} \text{ and the CP-violating phase } \delta_{CP}. \tag{9}
For a Dirac neutrino mass term MDM_D, the mass matrix MM is defined by the coupling between the left-handed and right-handed components of the neutrino field ν\nu: \n\n$$ \mathcal{L}_{mass} = - \frac{1}{2} \bar{\nu} M \nu^c + \text{h.c.} \tag{10} \text{If the mass is parameterized by a mixing angle } \beta \text{ and mass } m \text{ such that } M \propto \frac{\beta^2}{2} m^2 \text{ (as specified), the effective mass term is:} \tag{11} \mathcal{L}_{eff} = - \frac{\beta^2}{2} m^2 \bar{\nu} \nu^c + \text{h.c.} \tag{12} \text{This structure relates the mass scale } m \text{ to the mixing parameter } \beta. \tag{13}
Let ψ\psi be the four-component Dirac spinor representing the neutrino field, and let mm be its mass. The Dirac equation in natural units (ℏ=c=1\hbar = c = 1) is given by: \n(iγμ∂μ−m)ψ=0\left(i \gamma^{\mu} \partial_{\mu} - m\right) \psi = 0 \nwhere γμ\gamma^{\mu} are the Dirac gamma matrices satisfying the anti-commutation relation {γμ,γν}=2ημνI\left\{ \gamma^{\mu}, \gamma^{\nu} \right\} = 2 \eta^{\mu \nu} \mathbb{I}, and ημν=diag(+1,−1,−1,−1)\eta^{\mu \nu} = \text{diag}(+1, -1, -1, -1) is the Minkowski metric.
Principle

Vacuum Energy

The vacuum energy density ρvac\rho_{vac} is related to the expectation value of the Lagrangian L\mathcal{L} in the vacuum state ∣0⟩|0\rangle. If neutrino masses arise from spontaneous symmetry breaking (SSB) via a mechanism like the seesaw, the effective mass matrix MνM_{\nu} is determined by the vacuum expectation values (VEVs) of the relevant scalar fields ⟨ϕ⟩\langle \phi \rangle. Specifically, the effective potential VeffV_{eff} must yield a non-zero minimum ⟨ϕ⟩≠0\langle \phi \rangle \neq 0, leading to the mass term: \nLmass=−12νˉMνν\mathcal{L}_{mass} = -\frac{1}{2} \bar{\nu} M_{\nu} \nu \nwhere Mν∝1⟨ϕ⟩⟨Higgs⟩M_{\nu} \propto \frac{1}{\langle \phi \rangle} \langle \text{Higgs} \rangle (schematically).
Let νf\nu_{f} be the flavor eigenstate vector and νm\nu_{m} be the mass eigenstate vector. The mixing is described by the unitary PMNS matrix UU: νf=Uνm\nu_{f} = U \nu_{m}. The effective four-fermion interaction Lagrangian Leff\mathcal{L}_{eff} responsible for neutrino oscillations is given by:\n\n$$ \mathcal{L}_{eff} = \frac{G_{F}}{\sqrt{2}} \sum_{f, f'} ( \bar{\nu}_{f} \gamma^{\mu} \nu_{f} ) ( \bar{f} \gamma_{\mu} (1 - \sin^2\theta_W) \nu_{f'} ) + \text{h.c.} \quad \text{where } \nu_{f} = \sum_{i=1}^{3} U_{fi} \nu_{i} \text{ and } \nu_{i} \text{ are mass eigenstates.} \tag{1}
A Majorana neutrino field ν\nu is defined by the condition that it is equal to its charge conjugate νc\nu^c: ν=νc\nu = \nu^c. Consequently, the Lagrangian density LM\mathcal{L}_{M} must be invariant under charge conjugation and contain a symmetric mass term MM: \n\n$$ \mathcal{L}_{M} = \frac{1}{2} \bar{\nu} (i \gamma^{\mu} \partial_{\mu}) \nu - \frac{1}{2} \bar{\nu} M \nu^c + \text{h.c.} \tag{2} \text{where } M \text{ is a real, symmetric mass matrix.} \tag{3}
Define the neutrino field νs\nu_{s} as a hypothetical singlet under the Standard Model gauge group SU(3)C×SU(2)L×U(1)YSU(3)_C \times SU(2)_L \times U(1)_Y. Its Lagrangian interaction term Lint\mathcal{L}_{int} with Standard Model fermions is suppressed or zero, implying no coupling to WW or ZZ bosons:\n\n$$ \mathcal{L}_{int} = \bar{\nu}_{s} (i \gamma^{\mu} \partial_{\mu}) \nu_{s} + \text{mixing terms} \tag{4} \text{where the mixing term } \mathcal{L}_{mix} \text{ is typically proportional to } \epsilon \bar{\nu}_{s} \nu_{L} + \text{h.c.} \text{ and } \epsilon \ll 1. \tag{5} \text{The absence of } SU(2)_L \text{ coupling is the defining characteristic.} \tag{6}