Weak interactions and the gravitational collapse

TL;DR

The work investigates whether gravity-induced collapse could be counteracted by a repulsive pressure from the weak interaction, mediated by $Z$ exchange, leading to a new class of ultra-dense, weakly charged compact objects with radii of a few meters and masses around $10^{-3}\,M_\odot$. By deriving the weak-interaction potential, computing the self-energy, and formulating an equation of state where $p\approx\varepsilon$ at ultrahigh densities, the authors solve the Tolman-Oppenheimer-Volkoff equations in both Newtonian and relativistic regimes. They find a universal radius scale of a few meters and a maximum mass near Jupiter scale, with regular, non-singular interiors, suggesting a possible stable end state distinct from black holes under extreme collapse. The results hinge on coherent weak-charge transport and offer a potential link to dark matter phenomenology, while highlighting the need for more realistic density profiles and EOS refinements.

Abstract

The chart of nuclei could be enlarged with a branch describing neutron stars that are huge nuclei of a few solar masses held together by gravity force and sustained by the pressure due to the degenerate Fermi sea. We contend in this manuscript that yet another branch could be added: objects with a large weak charge, with masses around $10^{-3}$ solar masses and having radii of a few meters, very compact, only slightly larger than their Schwarzchild radius, and sustained by the pressure generated by the weak force due to $Z$ exchange. This interaction, insignificant in normal neutron stars, could become dominant when ultrahigh densities are reached due to the action of gravity and lead to stable configurations if the appropriate conditions are met. They would constitute a physical realization of the equation of state proposed by Zeldovich some decades ago.

Figures (3)

• Figure 1: On the left panel we plot the tree-level potential $V(r)$ due to $Z$ exchange (light/orange line) and compare it to the one-loop potential due to neutrino exchange (dark/blue line) as a function of $x=rM$ (Eq. \ref{['oneloop']}). As we can see neutrino exchange indeed dominates at very long distances because it falls as $1/r^5$ (at least up to micron distances where the neutrino mass cuts-off the interaction -this is well outside the range of values of $x$ plotted). However, at $x\simeq 15$ the Yukawa potential takes over and amply dominates the potential down to $x=0$. Actually, if the $1/r^5$ behaviour would stay valid down to $x=0$ it would eventually dominate again the potential because it is extremely singular. However, we see on the right panel that around $x\simeq 6$ the shape of the potential due to neutrinos is no longer of the $1/r^5$ type. It has a dip and it shoots up again, but it is always negligible compared to the leading Yukawa term. Physical units are obtained after multiplying the vertical scale by $M$.
• Figure 2: Comparison between the energy density profiles of the relativistic and newtonian solutions using the same boundary condition at the center. In the relativistic case (left) the energy density and the rest mass density are plotted, while in the newtonian case (right) only the mass density is shown. To restore physical units, the results have to be divided by $D= 2.94 \times 10^{-5}$ GeV$^{-4}$. The results are plotted as a function of the dimensionless variable $y$ (see text). Not only is the maximum value for $y$ larger in newtonian physics but the profile is also fatter. Recall that the value $y_{max}$ is universal in both cases, and that the initial densities have been chosen identical. As a result the relativistic solution is able to sustain substantially less mass/energy.
• Figure 3: The figure shows the dependence of the mass of the star (in units of the solar mass) as a funtion of the central energy density. The physical density is given by $\varepsilon(0)= f(0)/D$ with $D=2.94 \times 10^{-5}$ GeV$^{-4}$ for neutron constituents. This gives central densities of the order of $10^7$ GeV$^4$, many orders of magnitude larger than a typical neutron star, where models predict densities in the $10^{-4}$ to $10^{-3}$ GeV$^4$ range.