Chiral Majorana Neutrino Currents

• Chiral Majorana neutrino currents are axial currents arising from the self-conjugate property of Majorana particles, leading to distinctive transport phenomena.
• Their field-theoretic and kinetic analyses reveal key differences from Dirac currents, including interference effects and Berry-curvature-induced contributions.
• Experimental and astrophysical studies, via polarized scattering and supernova observations, provide practical insights on Majorana mass effects and chiral oscillations.

Chiral Majorana neutrino currents refer to the axial (purely chiral) currents formed by Majorana neutrinos, manifesting both in their fundamental interactions and in kinetic and hydrodynamic transport theory. Unlike Dirac neutrino currents, Majorana chiral currents are strictly axial at the electroweak scale and below, are sensitive to neutrino mass and polarization, and are associated with operator structures and observables that distinguish them from vector–axial (V–A) Dirac currents. Their paper encompasses field-theoretic structure, kinetic evolution, observable consequences in scattering, and emergent collective effects in astrophysical and condensed matter environments.

1. Field-Theoretic Structure of Chiral Majorana Currents

At energies well below the weak scale, the effective neutral-current Lagrangian describing neutrino–electron scattering highlights the distinction between Dirac and Majorana neutrino currents. The current–current form is

$$\mathcal{L}_{\nu e} = \frac{G_F}{\sqrt{2}} \bigl[\bar{\nu} \gamma^\mu X \nu \bigr] \left[\bar{e} \gamma_\mu(g_V^\ell - g_A^\ell \gamma^5) e\right]$$

with $X=(1-\gamma^5)$ for Dirac and $X=\gamma^5$ for Majorana neutrinos. The purely axial Majorana current is explicitly

$J_M^\mu = \bar{\nu} \gamma^\mu \gamma^5 \nu$

as opposed to the Dirac structure

$J_D^\mu = \bar{\nu} \gamma^\mu (1-\gamma^5) \nu$

The axiality of Majorana currents results from the self-conjugacy condition $\psi = \psi^c = C\bar\psi^T$, which causes destructive interference between neutrino and antineutrino exchange diagrams and cancels the vector component (Barranco et al., 2014).

Kinetic-theoretical and hydrodynamic treatments (see (Yamamoto, 2015, Yamamoto et al., 2023)) further validate this structure, where the vector and axial components of the phase-space distribution coincide ($f_V = f_A = f_\nu$) for Majorana neutrinos.

2. Chiral Currents in Scattering and Polarization-Dependent Observables

Chiral Majorana currents lead to distinctive signatures in scattering processes. In elastic neutrino–electron scattering,

$$\mathcal{M}^M = +\frac{i2 G_F}{\sqrt{2}} \left[\bar u_e^f \gamma^\mu (g_V^\ell - g_A^\ell \gamma^5) u_e^i\right] \left[ \bar u_\nu^f \gamma_\mu \gamma^5 u_\nu^i \right]$$

Spin-summed differential cross sections exhibit an exact match with the Dirac case for ultrarelativistic, fully left-handed neutrinos ($m_\nu = 0$, $s_{‖}=-1$), but for $m_\nu \neq 0$ or partial depolarization ($s_{‖}>-1$), terms $\propto m_\nu$ and those depending on the polarization components $s_{‖}$, $s_\perp$ survive only in the Majorana amplitude (Barranco et al., 2014).

The fractional difference in cross sections,

$$D(E_\nu, s_{‖}) = \frac{|\sigma^D - \sigma^M|}{\sigma^D}$$

is negligible ($\mathcal{O}(10^{-8})$) for typical terrestrial beams but can reach observable levels ($10-20\%$) for partially depolarized supernova neutrinos or in laboratory experiments involving polarized targets. Scattering of depolarized Majorana neutrinos on polarized electrons amplifies these axial–vector distinctions, with $D_{e_{\rm pol}} \sim 20\%$ at MeV energies when $|s_e|\gtrsim 0.3$ (Barranco et al., 2014).

3. Chiral Kinetics, Berry Curvature, and Anomaly-Induced Effects

Chiral Majorana currents are central to the relativistic kinetic theory of neutrinos, manifesting in Wigner-function (CKT) and hydrodynamic frameworks. The quantum chiral kinetic equations, valid to $\mathcal{O}(G_F)$, decompose the currents into classical (self-energy-induced) and quantum (Berry-curvature) pieces: $J_M^\mu = J_B^\mu + J_{\rm SHE}^\mu$ where

$$J_B^\mu = \int \frac{d^4q}{(2\pi)^3}\Theta(q_0)\delta(q_0-|\mathbf{q}|)\{\Delta\varepsilon_{qB}(q^\mu\partial_{q_0} - \hat q^\mu_\perp) - \bar{\Sigma}_B^\mu \} f_M(q_0)$$

$$J_{\rm SHE}^\mu = \hbar \epsilon^{\mu\nu\rho\sigma} \ell_\nu n_\sigma \partial_\rho V$$

The first term describes the chiral magnetic effect (CME)-type current, and the second, a “spin Hall effect” (SHE) current sourced by Berry curvature and density gradients (Yamamoto et al., 2023). For Majorana neutrinos, the absence of a net lepton-number current and chemical potential is enforced, and the physical current is identified with the axial component.

In hydrodynamic regimes, chiral transport manifests as the chiral vortical effect (CVE): $$J_{\nu}^\mu|_{\rm CVE} = \xi_\nu \omega^\mu, \quad \xi_\nu = \frac{\mu_\nu^2}{8\pi^2} + \frac{T^2}{24}$$ Here, the fluid helicity $h_f = u\cdot\omega$ generated via the CVE acts (in multi-fluid settings) as an effective chiral chemical potential for electrons, enabling helical plasma instabilities and generation of large-scale helical magnetic fields (Yamamoto, 2015). This mechanism has been proposed as a route for magnetar field origin.

4. Chiral Symmetry, Majorana Mass, and Axial Charge

A massless Majorana field exhibits an exact global chiral $U(1)$ symmetry: $$\psi_M \to e^{i\alpha \gamma_5} \psi_M, \qquad \mathcal{L}_0 = \frac{1}{2} \bar \psi_M i\!\not\!\partial \psi_M$$ with associated Noether current

$J_5^\mu = \bar{\psi}_M \gamma^\mu \gamma_5 \psi_M$

Introduction of a Majorana mass explicitly breaks this symmetry, yielding

$$\partial_\mu J_5^\mu = 2i m \bar{\psi}_M \gamma_5 \psi_M$$

No additional gauge anomaly is present beyond the standard ABJ anomaly, and the divergence arises solely from the mass term (Fujikawa et al., 2017). In the Bogoliubov quasiparticle formalism, fermion-number-violating condensates are rotated into an effective Dirac mass for the Majorana quasifermion, preserving the consistency of the chiral current structure.

Distinct Majorana species constructed from different chiralities or charge assignments carry separate $U(1)_A$ charges, resulting in multiple independent axial currents; mass terms break these symmetries in different linear combinations.

5. Chiral Gauge Models and New Currents

Models with additional chiral gauge invariance, such as abelian $U(1)_R$ extensions, allow right-handed Majorana neutrinos to couple to new axial gauge bosons: $$J^\mu_{\rm chiral} = \overline{N_M}\gamma^\mu \gamma^5 N_M = J^\mu_R - J^\mu_L$$ These couplings generate tree-level axial currents leading to modifications in elastic $\nu e$ and coherent $\nu A$ scattering. Experimental bounds on the associated coupling constants—for instance, from TEXONO/CHARM and COHERENT—are stringent, limiting effective couplings to $10^{-6}$–$10^{-4}$ for MeV–GeV scale mediators (Alikhanov et al., 2019).

6. Chiral Oscillation, Matter Effects, and Majorana Phases

Finite neutrino masses induce chiral oscillations between left- and right-chiral states, with probability

$$P(\nu_L^h \rightarrow \nu_R^h; t) = \frac{m^2}{E^2}\sin^2(Et)$$

for vacuum, and with matter-modified dispersion,

$$P(\nu_L^h \rightarrow (\nu_L^h)^c) = \frac{m^2}{E_h^2}\sin^2(E_h t)$$

Chiral oscillation effects are highly suppressed for relativistic neutrinos but can become significant for non-relativistic cosmic neutrinos ($p\sim m$), affecting e.g. PTOLEMY capture rates (Li et al., 2023).

In matter, eigen-energy splitting depends on helicity, producing a resonance condition for the chiral flip distinct from the standard MSW resonance. Majorana phases enter explicitly into the oscillation probabilities in mixed systems, surviving time averaging and potentially measurable in dense astrophysical environments or cosmic background neutrino detection.

7. Experimental and Astrophysical Implications

Observable consequences of chiral Majorana currents include:

• Enhanced cross-section differences in $\nu$–$e$ scattering for partially depolarized neutrinos or polarized targets, particularly in supernova neutrino detection scenarios (Barranco et al., 2014).
• Linear interference terms in cross sections for Majorana neutrinos, strengthening laboratory sensitivity to exotic couplings by up to $30\%$ relative to Dirac cases in Borexino-like setups (Sobków et al., 2016).
• Berry-curvature–induced spin Hall and CME-like effects in supernovae and proto-neutron stars, with implications for collective neutrino transport, fluid helicity, and generation of helical magnetic fields (Yamamoto et al., 2023, Yamamoto, 2015).
• The absence of a true lepton-number chemical potential and net vector current for Majorana neutrinos, requiring axial or energy-momentum current-based observables for transport and hydrodynamic modeling (Yamamoto et al., 2023).
• Sensitivity to absolute Majorana masses and phases in non-relativistic or dense matter environments (Li et al., 2023).

These theoretical predictions motivate future experiments with polarized targets, ultra-low threshold detectors, and astrophysical observations focusing on magnetar birth and core-collapse supernova dynamics.