Early Universe production of $W$ bosons in neutrino decays

1. Problem Statement

The paper addresses the phenomenon of particle production in the early Universe, specifically focusing on the generation of massive $W^{\pm}$ bosons via perturbative electroweak processes in a de Sitter spacetime (representing the rapidly expanding early Universe).

• The Core Issue: In standard Minkowski (flat) spacetime, the decay of a neutrino into a $W$ boson and a charged lepton ($\nu \to W + e$) is kinematically forbidden because the neutrino is effectively massless while the $W$ boson is massive ($m_\nu \ll M_W$).
• The Hypothesis: The authors investigate whether the expansion of the Universe (modeled by the de Sitter metric) breaks time-translation invariance sufficiently to allow these "forbidden" transitions to occur. They aim to calculate the transition rates and the resulting number density of $W$ bosons produced during the early stages of the Universe.

2. Methodology

The study employs perturbative Quantum Field Theory (QFT) adapted to curved spacetime, specifically the Poincaré patch of de Sitter space.

• Metric and Background: The analysis uses the de Sitter metric $ds^2 = dt^2 - e^{2\omega t} d\vec{x}^2$, where $\omega$ is the Hubble parameter (expansion factor). The conformal time $t_c$ is used for calculations.
• Field Solutions: The authors utilize exact solutions for the Dirac field (for neutrinos and electrons) and the Proca field (for $W$ bosons) in de Sitter space. These solutions are expressed in terms of Hankel functions and helicity bispinors.
• Transition Amplitude:
  • They compute the first-order transition amplitude for the process $\nu \to W^+ + e^-$.
  • The calculation is split into transversal modes ($\lambda = \pm 1$) and longitudinal modes ($\lambda = 0$). The paper focuses primarily on transversal modes for the main rate calculation, noting that longitudinal modes introduce complications in the large expansion limit.
  • The amplitude involves integrals over time and space. The spatial integral yields a Dirac delta function (momentum conservation), while the temporal integral involves products of Hankel functions.
• Regularization and Renormalization:
  • The total transition rate involves divergent integrals. The authors employ Dimensional Regularization (extending space to $D$ dimensions) and the Minimal Subtraction (MS) scheme to extract finite physical results.
  • They introduce a renormalization mass scale $\mu$.
• Limiting Cases:
  • Minkowski Limit ($\omega \to 0$): To verify consistency with flat spacetime theory.
  • Large Expansion Limit ($\omega \gg M_W, m_e$): This corresponds to the very early Universe where the expansion rate dominates particle masses. In this limit, the indices of the Bessel/Hankel functions simplify, allowing for analytical solutions.

3. Key Contributions

1. First Calculation of Neutrino-Induced $W$ Production: This is the first study to explicitly calculate the perturbative production of $W$ bosons specifically from neutrino decays in de Sitter spacetime. Previous works focused on vacuum production or emission by charged fermions.
2. Analytical Transition Rates: The authors derive explicit analytical expressions for the transition rates in the large expansion limit, simplifying complex hypergeometric functions (Appell $F_4$) into manageable forms involving logarithms and constants.
3. Density Number Definition: They propose a method to define the number density of produced $W$ bosons not via thermal equilibrium (which is ill-defined in rapid expansion), but as the ratio of the production rate to the decay rate ($R_{prod} / R_{decay}$).
4. Longitudinal Mode Analysis: While the main results focus on transversal modes, the paper provides the full amplitude derivation for longitudinal modes in Appendix C, explaining why they were excluded from the final density calculation (due to singular behavior as $\omega \to \infty$).

4. Key Results

A. Transition Amplitude and Rates

• Forbidden Transitions Allowed: The transition amplitude is non-zero in de Sitter space. The loss of time-translation invariance allows energy conservation to be satisfied by the background expansion, enabling $\nu \to W + e$.
• Minkowski Limit: As the Hubble parameter $\omega \to 0$, the transition rate vanishes. This confirms that the process is strictly forbidden in flat spacetime.
• Large Expansion Limit: In the limit $\omega \gg M_W$, the transition rate $R_{\nu \to W + e}$ is finite and depends on the particle momenta ($p, p'$) and the renormalization scale $\mu$.
  • The rate is proportional to $G_F M_W^2 p'$ (where $G_F$ is the Fermi constant).
  • The rate decreases as the ratio of particle mass to Hubble parameter ($m/\omega$) increases.

B. Density Number of $W$ Bosons

The authors calculate the net density of $W$ bosons by balancing production (from vacuum, electron emission, and neutrino decay) against decay (into leptons and quarks).

• Formula: The density number $\Delta N_W / (\Delta t \Delta V)$ is derived as a function of momenta and $\mu$.
• Dependence on Parameters:
  • Momentum: The production of $W$ bosons is favored for particles with small momenta (non-relativistic speeds). As particle speeds approach the speed of light, the ratio of produced-to-decaying bosons drops significantly.
  • Renormalization Scale ($\mu$): The density number is highly sensitive to $\mu$. Graphical analysis shows peaks in the production-to-decay ratio at specific values of $\mu$ relative to particle masses.
  • Expansion: As the Universe expands and $\omega$ decreases (eventually becoming smaller than $M_W$), the production rate drops, and the density of $W$ bosons decreases.

C. Graphical Analysis

• Figures 1 & 2: Show that the transition rate peaks when the mass-to-Hubble ratio is small and vanishes as the ratio increases.
• Figures 3–6: Illustrate how transition rates increase with the renormalization mass $\mu$ and depend on particle momenta.
• Figures 7–12: Demonstrate that the ratio of produced to decaying $W$ bosons is significantly higher for non-relativistic particles (low speeds) compared to relativistic ones.

5. Significance and Conclusions

• Validation of Curved Spacetime QFT: The study successfully demonstrates that perturbative methods used in flat spacetime can be adapted to de Sitter space to predict physical phenomena (particle production) that are impossible in flat space.
• Early Universe Cosmology: The results suggest that in the very early Universe (high $\omega$), massive gauge bosons like the $W$ could be generated in significant numbers via neutrino decays, even though they are kinematically forbidden today.
• Stability of Heavy Particles: The analysis supports the hypothesis that heavy particles could exist as "stable" (or long-lived) entities in the early Universe when the background energy density exceeded their rest mass, delaying their decay until the expansion rate slowed down.
• Limitations: The authors note that the current analysis is restricted to transversal modes and perturbative first-order processes. A complete picture would require including longitudinal modes (which are difficult to regularize in the large expansion limit) and non-perturbative effects.

In summary, the paper provides a rigorous theoretical framework showing that the expansion of the early Universe acts as a catalyst for the production of massive $W$ bosons from neutrino decays, with the production efficiency heavily dependent on the expansion rate, particle momenta, and renormalization scales.