Estimate for the neutrino magnetic moment from pulsar kick velocities induced at the birth of strange quark matter neutron stars

Compact objects are a class of astrophysical objects that include White Dwarfs (WD), Neutron Stars (NS), and Black Holes (BH). A significant proportion of their physical properties remain unknown since their initial appearance in the astronomical landscape. These types of objects are particularly interesting because many areas of physics have converged into their study.

One of the main aspects that have been studied in recent years is the internal composition of WD and NS, due to the extremely high densities, temperatures, and magnetic fields that are found in these systems, which terrestrial laboratories MPD:2022qhn have little chance of reproducing. One of the aims of studying these kinds of systems is to probe the not yet understood phases of strongly interacting matter Gutierrez:2013sta; Pasztor:2024dpv; Du:2024wjm in the regime of high density, low temperature, and large magnetic fields Radice:2024gic, which determine the proposed scenarios of the exotic phases of matter that compose the internal layers of NS Lugones:2024ryz; Issifu:2024fuw; Schramm:2011aa.

NS are born after a massive star, with a mass larger than 8 solar masses, explodes as a Type II supernova Burrows:2020qrp; Boccioli:2024abp. It has been realised that the properties of NS are related to the continuous neutrino emission during the first 100 years after NS formation, which rapidly cools down the star from dozens of MeV to only dozens of keV. If the emission of neutrinos takes place in an anisotropic way, it can be responsible for the so-called pulsar kicks Hobbs:2005yx.

Several scenarios have been proposed to explain the proper motion of pulsars Fukushima:2024cpg; Berdermann:2006rk; Janka:2016nak, including hydrodynamic perturbations Gessner:2018ekd; Nordhaus:2010ub; Page:2020gsx, asymmetric electromagnetic emissions from an off-centered magnetic dipole 1975Natur.254..676T; 1975ApJ...201..447H; Lai:2000pk; Agalianou:2023lvv, the breaking of a binary system 1970ApJ...160L..91G; Igoshev:2021bxr, and anisotropic neutrino emission Wongwathanarat:2012zp; Sagert:2007as; Fryer:2005sz, among others Wang:2005jg; Khokhlov:1999ka; Colpi:2002cu; Fryer:2003jj; Farzan:2005yp; Lambiase:2023hpq. One of the main scenarios is a natal kick, which takes place during the formation of the proto-NS, in which neutrinos are emitted in a preferred direction and, hence, provide the proto-NS with momentum in its opposite direction. This scenario is favored by Refs. Janka:2017vlw; Sieverding:2019qet, where it has been shown that there is a peak in the electron neutrino luminosity of about 12 s after the core bounce stage of proto-NS formation. It has also been proposed in Refs. Sumiyoshi:2017fpf; Keranen:2004vj that the formation of quark matter during the core collapse could explain the observed NS kicks.

In a minimal extension of the standard model (SM), where neutrinos are massive and can have a non-vanishing magnetic moment Fujikawa:1980yx; Alok:2022pdn; Alok:2023sfr, we recently computed the NS kick velocities by considering the case in which the anisotropic neutrino emission has its origins in the presence of a strong magnetic field in the interior of an NS Ayala:2018kie. If the neutrinos, due to their magnetic moment, interact in equilibrium with the NS medium, they can flip their chirality, becoming right-handed and, hence, suppressing their interactions with matter. As the inverse process happens out of equilibrium and the detailed balance is lost, the right-handed neutrinos cannot flip back onto the left-handed ones, making the no-go theorem non-applicable Kusenko:1998yy. Furthermore, because matter effects dominate in the core of NS, the right-handed neutrinos cannot resonate back into left-handed ones, as this process is suppressed as a consequence of the presence of magnetic fields. A fraction of the emitted left-handed neutrinos can escape from NS as right-handed ones if the typical time that takes them to flip their chirality is smaller than the time needed to travel one mean free path, which is small compared with the NS core radius. This mechanism was implemented in Refs. Ayala:2019sbt; Ayala:2021nhx and was used to establish a lower bound for the neutrino magnetic moment that, together with the most stringent upper bound for this magnetic moment Ayala:1998qz; Ayala:1999xn, allowed for setting the range $4.7\times 10^{-15}\leq\mu_{\nu}/\mu_{B}\leq$ (0.1–0.4) $\times$ $10^{-11}$. In this work, we complement that study by further elaborating on the idea that a neutrino chirality flip, produced by the existence of a neutrino magnetic moment, can explain the observed kick velocities. As SN explosion is mainly driven by neutrino emission, not all of the neutrinos should flip their chirality;  otherwise, the explosion itself would not take place. We show that even if a small percentage of neutrinos become right-handed from the original left-handed state, the observed kick velocities can be reproduced even for a neutrino magnetic moment of the order of SM bound for a neutrino mass of a few eV Fujikawa:1980yx.

The chirality flip within the core of NS can be considered in the same way as it was discussed in Refs. Ayala:1998qz; Ayala:1999xn, ion order to place an upper bound on the neutrino magnetic moment from the chirality flip in supernovae. To this end, we consider a thermally equilibrated plasma at a temperature $T$ and electron chemical potential $\mu_{e}\gg m_{e}$ in which neutrinos are being produced, with $m_{e}$ being the electron mass. For the sake of simplicity, throughout this work, we consider the neutrinos as massless. The production rate of a right-handed neutrino, from a left-handed one, whose energy is $p_{0}$ and its momentum is $\vec{p}$, is given by

$$\Gamma(p_{0})=\frac{\tilde{f}(p_{0})}{2p_{0}}\text{Tr}\left[\not{P}R\text{Im}\Sigma\right],$$

(1)

where $\tilde{f}(p_{0})$ is the Fermi–Dirac distribution for right-handed neutrinos, such that $P_{\mu}=(p_{0},\vec{p})$, $|\vec{p}|=p$ , where we use the Feynman slash notation $\not{a}=\gamma^{\mu}a_{\mu}$, with $\gamma^{\mu}$ as the Dirac matrices. In the previous equation, the operators $R,\>L=\frac{1}{2}(1\pm\gamma_{5})$ projects onto right (left)-handed fermion components, where the chiral matrix $\gamma^{5}$ is defined as $\gamma^{5}\equiv i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$. The neutrino self-energy $\Sigma$ is given by

$$\Sigma(P)=T\sum_{n}\int\frac{d^{3}k}{(2\pi)^{3}}V^{\rho}(K)\,S_{F}(\not{P}-\not{K})\,L\,V^{\lambda}(K)\,D_{\rho\lambda}(K),$$

(2)

where $K_{\alpha}=(k_{0},\vec{k})$, $|\vec{k}|=k$, $V^{\mu}$ is the neutrino–photon vertex function, $S_{F}$ is the neutrino propagator and $D_{\rho\lambda}$ is the photon propagator. We describe the neutrino–photon interaction by means of a magnetic dipole interaction $V_{\mu}(K)=\mu_{\nu}\,\sigma_{\alpha\mu}K^{\alpha}$, where $\mu_{\nu}$ is the neutrino magnetic moment and $\sigma_{\alpha\mu}=\frac{i}{2}\left[\gamma_{\alpha},\gamma_{\mu}\right]$. As usual, we split the photon propagator into its longitudinal and transverse components

$$D_{\rho\lambda}(K)=\Delta_{L}(K)P^{L}_{\rho\lambda}+\Delta_{T}(K)P^{T}_{\rho\lambda}.$$

(3)

The sums over Matsubara frequencies that need to be evaluated are

$$M_{L,T}=T\sum_{n}\Delta_{L,T}(i\omega_{n})\tilde{\Delta}_{F}(i(\omega-\omega_{n})).$$

(4)

By writing the neutrino and photon propagators in their spectral representation, the imaginary part of $M_{L,T}$ can be computed in a straightforward manner, as follows: {IEEEeqnarray}rCl Im