Numerical modeling of non-equilibrium neutrino radiation field by solving quantum kinetic equation

Numerical modeling of neutrino quantum kinetics is a new frontier in the study of core-collapse supernova (CCSN) and binary neutron star merger (BNSM). The noticeable feature in the quantum kinetics is neutrino flavor conversion driven by neutrino self-interactions. Although there remain many unresolved issues in their non-linear properties of flavor conversions, rapid progress has been made in the last few years. In this article, we introduce the current status for the study of collective neutrino oscillations, which are representative characteristics in self-induced flavor conversions, and then we show our recent results: global quantum kinetic simulations in CCSN and BNSM environments.


Introduction
Neutrinos are the most mysterious elementary particles among those known to exist in our universe.There are comelling experimental evidences that the intriguing particle can undergo flavor conversions or oscillations, representating one of the quantum kinetic features beyond the standard model of particle physics.The flavor conversion implies that the eigenstates between flavors and masses are different from each other.Determining neutrino masses, mixing angles, and nature (Dirac or Majorana) are important but remaining fundamental issues in not only neutrino physics but also cosmology and even astrophysics.In fact, the neutrino oscillation was a key process to solve the solar neutrino problem, and nowadays it is highly likely that they also play pivotal roles in high-energy astrophysical phenomena such as core-collapse supernova (CCSN) and binary neutron star merger (BNSM).Neutrino quantum kinetics in these explosive phenomena are the subject in this article.
During formations of CCSN and BNSM, matter density can exceed nuclear saturation density (ρ ∼ 10 14 g/cm 3 , where ρ denotes the mass density of baryons) and its temperature can also reach > 10 11 K.In such extremely hot and dense environments, neutrinos can be produced abundantly through various weak interactions.Since neutrinos are the least interacting particles, they are the dominant lepton-and energy carriers in the vicinity of proto-neutron star (NS) or black hole (BH).This exhibits that different fluid elements can exchange their energy and leptons via neutrinos, which has an influence on fluid dynamics and nucleosynthesis.It should also be stressed that neutrinos are observable particles, in which key information on microphysical properties in high density environments (equation-of-state, neutrino-matter interactions, and nuclear reactions) are imprinted.Developing accurate models of neutrino kinetics is, hence, mandatory for the study of CCSN and BNSM.During the last decades, neutrino-radiation hydrodynamic simulations have been a frontier in computational astrophysics (see [1] as a recent review), which is due to the complex, multiphysics, and multi-dimensional natures of CCSN and BNSM physics.It has been expected that neutrino kinetics in these phenomena can be fully determined by solving Boltzmann equation.In this theory, the distribution function of neutrinos provides all fundamental quantities to describe neutrino kinetics including neutrino-matter interactions.Although their direct numerical simulations are computationally expensive, the numerical method and implementations on massive parallel supercomputer facilities have been already established (see, e.g., [2,3]).It is worthy to note that the first axisymmetric CCSN simulations with full Boltzmann neutrino transport in our group was carried out in 2018 [4].Nowadays, we performed the Boltzmann CCSN simulations in 3D [5] and with general relativistic treatment [6].With improving other input physics (equation-of-state and neutrino-matter interactions; see, e.g., our recent review of [10]), the community, no doubt, keeps progressing towards first-principles CCSN models.
One thing we do notice here is, however, that neutrino flavor conversions induced by neutrino self-interactions have been neglected in CCSN and BNSM models.It is important to note that linear stability analyses of flavor conversions based on recent CCSN models have commonly suggested that neutrino-flavor conversions occur in post-shock regions, driven by fast mode [8,7,9] and collisional instability [11].In BNSM environments, both instabilities would also be ubiquiouts [12,13].These facts suggest that neutrino kinetics should be modeled with taking into account flavor conversions, which can not be handled by Boltzmann equation.The most straightforward and simplest extension to include effects of flavor conversions is the mean-field quantum kinetic equation, or QKE (see, e.g., [14,15]).The QKE has abilities to capture essential features of various flavor instabilities, including fast mode and collisional instability.This has motived us to develop QKE solvers for modeling of neutrino radiation fields in CCSNe and BNSMs.
Although there are many previous works solving QKE, they mostly focused on local properties of flavor conversions.This is due to the fact that there is a large disparity of scales between neutrino flavor conversions and astrophysical phenomena, if neutrino flavor conversions are driven by neutrino self-interactions.This is the subject of this article.Although these studies focusing on local properties are very important, radiation fields in non-linear phases would be qualitatively different from those in CCSN and BNSM.This is because the global advection of neutrinos characterizes the neutrino angular distributions and energy spectram.This motivated us to develop a new code: GRQKNT (General-Relativistic Quantum-Kinetics Neutrino Transport) [16], in which we solve general relativistic QKE with multi-energy, multiangle, three-flavor frameworks, and essential neutrino-matter interactions (or collision terms).We also applied the code to understand global properties of neutrino radiation fields accompanied by fast neutrino flavor conversion (FFC).Below, let us at first describe the essence of GRQKNT code, and then we will present some essential findings in our recent studies.

GRQKNT
A new general-relativistic quantum-kinetics neutrino transport code, GRQKNT, was developed, aiming at studying global neutrino radiation field with neutrino oscillations in CCSN and BNSM environments.The details of input physics, methodologies, and basic tests in GRQKNT are presented in [16].Here, we describe the design briefly.
We solve QKE (see Fig. 1) with a conservative form.The formalism is essentially the same as that used in our Boltzmann code.Although many modules in GRQKNT are transplanted from our Boltzmann code, there are mainly four different features.First, we solve density matrices of neutrinos and antineutrinos with a flavor-basis, in which the off-diagonal components represent the coherency of different flavor states.This indicates that the number of dimension in flavor space needs to be increased from 6 (for Boltzmann) to 18 (for QKE).Next, GRQKNT includes neutrino oscillation Hamiltonian, which has vacuum, matter, and neutrino self-interaction potentials.Third, the emission, absorption, and scattering processes are incorporated in GRQKNT code.Although they are a minimum set of neutrino-matter interactions for the study of CCSN and BNSM, its formulations and implementations are consistent with quantum Figure 3. Net neutrino heating-cooling rate as functions of radius [20].Black line represents the case without FFC, while others denote the result with FFC.Our result suggests that neutrino cooling is enhanced in optically thick region by FFC, whereas the heating is hampered in the gain region.See the text for more details.kinetics of neutrinos; in fact, we need to extend the collision term from classical to quantum one (see, e.g., [14,15]).Fourth, we employ an explicit time-integration scheme (WENO) to solve QKE in GRQKNT, whereas implicit treatments are mandatory for Boltzmann neutrino transport.This is attributed to the fact that the QKE is not a stiff equation, if flavor instabilities are triggered by neutrino self-interactions.In these cases, the neutrino advection needs to be solved with the same time scale as neutrino-oscillation one.
One thing we do notice here is that flavor conversions, in particular for fast mode (or FFC), occur in much shorter time-and length-scales than those in CCSNe and BNSMs.The disparity between the two scales reaches several orders of magnitude, indicating that global simulations are intractable in the current computational powers.In [18,19], we proposed a novel approach to overcome this issue.We introduced an attenuation parameter, which reduces a coupling constant of neutrino oscillation Hamiltonian.Although the reduced coupling results in artificial outcomes, the realistic features can be extracted by convergence studies with respect to the attenuation parameter.In [18,19], we showed that the small-scale structures of flavor conversions are sensitive to the attenuation parameter, whereas their time-averaged and global features are not affected (see Fig. 2).In the next section, we introduce some noticeable results in our recent study of FFCs in CCSN and BNSM environments, while we used the attenuation prescription in all simulations.We refer readers to [18,19] for the detailed properties in their convergence studies.

Fast neutrino-flavor conversion in CCSN and BNSM
Figure 3 summarizes the most important result for the study of FFC in our recent paper [20].We find that FFC can change neutrino radiation fields radically from those obtained from classical (Boltzmann) neutrino transport, which gives impacts on both neutrino cooling and heating in post-shock regions.In optically thick region, the colored lines (corresponding the cases with FFC; see [20] for the detail of color code) are always lower than the black one (the case with no FFC), indicating that the neutrino cooling is enhanced by FFC.At large radii, on the other hand, the net neutrino heating is suppressed by FFC.One thing we should point out here is that these two effects would compete each other for CCSN explosions.It has been suggested that the large neutrino cooling in the gain region accelerates PNS cooling, resulting in fast PNS contraction.This also increases the average energy of neutrinos, which facilitates neutrino absorptions, i.e., heating in the gain region (it should be noted, however, that this effect is not taken into account in our simulations, since we need to include feedback effects from neutrino radiation field to matter, but we fix the matter background in time).However, the neutrino fluxes of electron-and antielectron type neutrinos are reduced by FFC, which reduces the neutrino heating in the gain region, as shown in Fig. 3.At the moment, we do not give a conlusive answer at the moment whether FFC gives a positive or negative effect on CCSN explosions.However, our result clearly shows that the fluid dynamics including shock revival must be impacted by FFC.
In Fig. 4, we show the main results for the study of FFC in BNSM environments [21].It should be emphasized that this is the first axisymmetric global FFC simulation.We find that the dynamical features of FFCs are very different from those expected from local simulations.There are mainly two noticeable features.First, an ELN-XLN Zero Surface (EXZS), corresponding to a surface where ELN and heavy-leptonic one (XLN) number densities become the same, emerges, and then the neutrinos undergo flavor swaps when they pass through it.The flavor swap is strong evidence that FFC can give a large impact on neutrino radiation field, since ELN and XLN can be completely exchanged at this surface.Next, FFC occurs only in narrow regions, while linear stability analysis (based on the radiation field from Boltzmann neutrino transport) suggests that FFC occurs in almost everywhere above the accretion disk.This represents that non-local neutrino advection changes the overall property of FFC in BNSM environments.This finding presents an importance of including effects of global advection of neutrinos when we study actual impacts of FFC on BNSM.

Summary
There is a long history of neutrino transport simulations for the modeling of central engines of CCSN and BNSM phenomena.These numerical modelings rely on Boltzmann-base kinetic theory, in which the neutrinos are treated as classical particles and their flavor conversions have been neglected.This is due to the fact that matter potential is much stronger than the vacuum contribution, which can effectively reduce the difference between mass-and flavor eigenstates.On the other hand, effects of neutrino-self interactions can induce rapid flavor conversions even in dense matter environments.In fact, linear stability analysis of flavor instabilities suggest that these types of flavor conversions can occur in both CCSN and BNSM environments ubiquitously.Motivated by these facts, we developed a new code, GRQKNT, which has the ability to solve mean-field QKE for neutrino transport.
By using the code, we demonstrated QKE simulations for FFC in both CCSN and BNSM environments.Our results clearly show that FFC changes substantially the neutrino radiation field.We also find that global advections should be taken into account appropriately when we discuss impact of FFC on these phenomena.Last but not least, we would like to mention that the community of neutrino flavor conversion is rapidly evolving.Although there still remain many open issues in non-linear features of neutrino flavor instabilities, we will address all these issues one by one in collaborations with CCSN and BNSM modellers.

Figure 2 .
Figure 2. Spherically symmetric FFC simulations.The top and bottom panels show the results of the same simulations but with different attenuation parameters.The bottom one represents a more realistic result than that in the top panels.See the text for more details.

15thFigure 4 .
Figure 4. Global features of FFCs in BNSM environments[21].In the top panels, the color contour displays the difference of electron-neutrino lepton number (ELN) number density from those in the initial (which corresponds to a steady state obtained from Boltzmann simulation) profile.In the bottom panel, we show the off-diagonal component of density matrix of neutrinos, which represents the coherency of flavor states; see the text for more details.