Neutrino Emissivity in $\Delta$-direct-Urca Processes Inside Neutron Stars

Technical Summary: Neutrino Emissivity in $\Delta$-direct-Urca Processes Inside Neutron Stars

Problem Statement
Recent astrophysical observations indicate that neutron stars (NS) can possess masses exceeding $2 M_\odot$, implying core densities several times the nuclear saturation density ($n_0$). At these densities, the equation of state (EOS) likely includes exotic degrees of freedom, such as $\Delta$ isobars. While the presence of negatively charged $\Delta^-$ isobars is known to soften the EOS at intermediate densities and potentially lower the threshold for the nucleonic direct-Urca (NdUrca) process, the specific contribution of $\Delta$-direct-Urca ($\Delta$-dUrca) processes to neutrino emissivity has not been explicitly quantified. Existing literature has established that semileptonic decays of $\Delta$ isobars (e.g., $\Delta^- \to n + e^- + \bar{\nu}_e$) become kinematically allowed under certain conditions, yet a rigorous derivation of their cooling efficiency relative to standard nucleonic processes remains absent.

Methodology
The authors derive the neutrino emissivity for $\Delta$-dUrca processes within the framework of relativistic mean-field (RMF) theory, specifically utilizing the DD-ME2 parameterization. The calculation assumes beta equilibrium and charge neutrality in dense matter.

1. Process Identification: The study focuses on three primary $\Delta$-dUrca channels and their inverse processes:

   • $\Delta^- \to n + e^- + \bar{\nu}_e$ (and $n + e^- \to \Delta^- + \nu_e$)
   • $\Delta^- \to \Lambda + e^- + \bar{\nu}_e$ (and $\Lambda + e^- \to \Delta^- + \nu_e$)
   • $\Delta^0 \to p + e^- + \bar{\nu}_e$ (and $p + e^- \to \Delta^0 + \nu_e$)

2. Theoretical Derivation:

   • Unlike standard Urca processes involving spin-1/2 baryons, these transitions involve a spin-3/2 $\Delta$ resonance decaying into a spin-1/2 baryon.
   • The authors construct the weak interaction Hamiltonian using a Rarita–Schwinger spinor ($u_{\Delta, \sigma}$) for the initial state and standard V–A structure for the leptonic current.
   • The hadronic transition vertex $\Gamma_{\mu\sigma}$ is defined to encode the spin-3/2 to spin-1/2 structure, incorporating vector ($f_1$) and axial-vector ($g_1$) weak form factors.
   • The squared matrix element $|\mathcal{M}|^2$ is calculated by summing over final spins and averaging over initial spins. This requires the use of the spin-3/2 completeness relation (Rarita–Schwinger projector) rather than the standard Dirac projector. The authors employed the FeynCalc package in Mathematica to handle the complex Dirac algebra and tensor contractions.

3. Phase Space Integration:

   • The emissivity $Q$ is derived under the assumption of a strongly degenerate, low-temperature limit ($k_B T \ll E_{F,i}$).
   • Phase-space integrals are simplified using decomposition techniques near the Fermi surfaces, converting momentum integrals into density-of-states integrals.
   • The final analytical expression for emissivity (Eq. 17) depends on the Fermi constant ($G_F$), Cabibbo factor ($C$), particle masses, effective masses ($m^*$), temperature ($T^6$ dependence), and the kinematic threshold function $\Theta_{\Delta Be}$.

4. Numerical Implementation:

   • Calculations were performed for a $1.7 M_\odot$ neutron star.
   • Coupling constants were varied to address uncertainties: $f_1 \in [-2, 2]$ and $g_1 \in [-1.5, 1.5]$. For the strangeness-changing process ($\Delta^- \to \Lambda$), specific hyperon weak coupling values were adopted.
   • The results were compared against the standard NdUrca emissivity.

Key Contributions

• First Derivation: This work provides the first explicit derivation of the neutrino emissivity associated with $\Delta$-dUrca processes, incorporating the necessary spin-3/2 Rarita–Schwinger formalism.
• Analytical Expression: The authors present a closed-form expression (Eq. 17) for the emissivity that accounts for the unique kinematic and spin structure of $\Delta$ decays.
• Quantitative Comparison: The study numerically evaluates these emissivities and directly compares them with the standard NdUrca process across relevant density ranges.

Results

• Kinematic Thresholds: Using the DD-ME2 parameterization, the $\Delta^-$ isobars appear at densities $<\sim 2n_0$, triggering the NdUrca process earlier than in nucleonic-only models. The $\Delta$-dUrca processes become operative shortly after the onset of these isobars.
• Emissivity Magnitude: The calculated emissivities for $\Delta$-dUrca processes ($\Delta^- \to n + e^- + \bar{\nu}_e$ and $\Delta^0 \to p + e^- + \bar{\nu}_e$) are found to be slightly higher than, but comparable to, the NdUrca emissivity over the entire density range considered.
• Cooling Efficiency: The results indicate that once kinematically allowed, $\Delta$-dUrca processes act as fast cooling mechanisms with efficiency nearly equivalent to the standard NdUrca process. The emissivity generally decreases with increasing baryon density due to the density dependence of effective masses and chemical potentials.

Significance and Claims
The paper claims that the inclusion of $\Delta$ isobars in the neutron star EOS has a dual impact on thermal evolution:

1. It lowers the threshold density for the standard NdUrca process.
2. It introduces additional, equally powerful rapid cooling channels via $\Delta$-dUrca processes.

The authors conclude that because these processes are as efficient as NdUrca, they cannot be neglected in realistic neutron star cooling simulations. The presence of $\Delta$ isobars significantly alters the predicted thermal evolution of massive neutron stars, necessitating their inclusion to accurately constrain the dense-matter EOS and interpret surface temperature observations. The findings are presented as largely independent of the specific matter parameterization chosen, provided $\Delta$ isobars are present.