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$N_{\textrm{eff}}$ Constraint on Pseudo-Dirac Neutrinos

Chee Sheng Fong, Yago Porto

TL;DR

The paper investigates whether neutrinos can be pseudo-Dirac, forming three nearly degenerate pairs with small mass-squared splittings $\delta m_j^2$. It derives an effective 3+1 Hamiltonian under maximal active-sterile mixing (one nonzero $\delta m_j^2$ at a time) and treats the neutrino ensemble in the early Universe with quantum kinetic equations to compute $N_{\rm eff}$. The main result is a new cosmological bound of $|\delta m_3^2| < 2 \times 10^{-6}\,\mathrm{eV}^2$, with future CMB experiments potentially reaching $\sim 10^{-7}\,\mathrm{eV}^2$. This establishes a powerful link between pseudo-Dirac neutrino scenarios and precise cosmological measurements, offering complementary constraints to oscillation data and motivating combined probes with BBN and lepton asymmetry scenarios.

Abstract

After the electroweak symmetry breaking, we can write down two types of mass for the Standard Model neutrinos, Dirac or Majorana. It is often said that both types of mass cannot be distinguished in neutrino oscillation phenomena. This is in fact not true if neutrinos are pseudo-Dirac (strictly speaking still Majorana) where they mix almost maximally with sterile neutrinos to form pseudo-Dirac pairs. If this is indeed realized in Nature, what we observe experimentally as three mass eigenstates are actually three pairs of mass eigenstates with yet-to-be-measured new mass splitting among each pair. While the new mass squared splitting of the first and second mass eigenstates have stringent constraints from solar neutrino to be $|δm_{1,2}^2| \lesssim10^{-11}\,\textrm{eV}^{2}$, the one regarding the third mass eigenstate has a weaker constraint $|δm_3^2| \lesssim10^{-5}\,\textrm{eV}^{2}$. By keeping only one nonzero pseudo-Dirac mass squared splitting at a time, we derive an effective 3+1 description for the pseudo-Dirac scenario. Then we use the Cosmic Microwave Background (CMB) constraint on neutrino relativistic degrees of freedom $N_{\textrm{eff}}$ to derive a new constraint $|δm_3^2| < 2 \times 10^{-6}\,{\rm eV}^2$ and show that the future CMB-S4 and CMB-HD can improve this bound by an order of magnitude.

$N_{\textrm{eff}}$ Constraint on Pseudo-Dirac Neutrinos

TL;DR

The paper investigates whether neutrinos can be pseudo-Dirac, forming three nearly degenerate pairs with small mass-squared splittings . It derives an effective 3+1 Hamiltonian under maximal active-sterile mixing (one nonzero at a time) and treats the neutrino ensemble in the early Universe with quantum kinetic equations to compute . The main result is a new cosmological bound of , with future CMB experiments potentially reaching . This establishes a powerful link between pseudo-Dirac neutrino scenarios and precise cosmological measurements, offering complementary constraints to oscillation data and motivating combined probes with BBN and lepton asymmetry scenarios.

Abstract

After the electroweak symmetry breaking, we can write down two types of mass for the Standard Model neutrinos, Dirac or Majorana. It is often said that both types of mass cannot be distinguished in neutrino oscillation phenomena. This is in fact not true if neutrinos are pseudo-Dirac (strictly speaking still Majorana) where they mix almost maximally with sterile neutrinos to form pseudo-Dirac pairs. If this is indeed realized in Nature, what we observe experimentally as three mass eigenstates are actually three pairs of mass eigenstates with yet-to-be-measured new mass splitting among each pair. While the new mass squared splitting of the first and second mass eigenstates have stringent constraints from solar neutrino to be , the one regarding the third mass eigenstate has a weaker constraint . By keeping only one nonzero pseudo-Dirac mass squared splitting at a time, we derive an effective 3+1 description for the pseudo-Dirac scenario. Then we use the Cosmic Microwave Background (CMB) constraint on neutrino relativistic degrees of freedom to derive a new constraint and show that the future CMB-S4 and CMB-HD can improve this bound by an order of magnitude.
Paper Structure (7 sections, 27 equations, 1 figure)

This paper contains 7 sections, 27 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Pseudo-Dirac neutrinos
  3. Effective 3+1 pseudo-Dirac model
  4. Neutrino ensemble evolution in the early Universe
  5. $N_{\textrm{eff}}$ determination
  6. Conclusions
  7. Derivation of effective pseudo-Dirac Hamiltonian

Figures (1)

  • Figure 1: $N_{\rm eff} - N_{\rm eff,SM}$ as a function of $x = m_e/T$ for various values of $|\delta m_3^2|$. The red solid line denotes at 2$\sigma$ level, the latest bound from ACT + Planck + BAO Planck:2018vygAtacamaCosmologyTelescope:2025nti, the dashed black line denotes the sensitivity of CMB-S4 CMB-S4:2016ple and the dotted black line denotes the sensitivity of CMB-HD Sehgal:2019ewc.