# A Systematic Study on the Resonance-like Structure in Collisional Neutrino
Flavor Instability

Jiabao Liu  Department of Physics and Applied Physics, School of Advanced
Science & Engineering, Waseda University, Tokyo 169-8555, Japan  Masamichi
Zaizen  Faculty of Science and Engineering, Waseda University, Tokyo 169-8555,
Japan  Shoichi Yamada  Department of Physics, School of Advanced Science &
Engineering, Waseda University, Tokyo 169-8555, Japan  Research Institute for
Science and Engineering, Waseda University, Tokyo 169-8555, Japan

###### Abstract

Investigations on the resonance-like phenomenon in the collisional flavor
instability (CFI) of neutrinos, which were observed in the linear phase
recently, are reported. We show that it occurs not only for the isotropy-
preserving modes as pointed out in the previous work but also for the
isotropy-breaking modes and that it enhances the linear growth rate of CFI.
Employing the linear analysis and nonlinear numerical simulations in the two-
flavor scheme and under the relaxation approximation for the collision term,
we discuss the criterion for the resonance-like phenomenon observed in the
linear phase, its effect on the subsequent nonlinear evolution as well as the
influences of homogeneity-breaking ($k\neq 0$) perturbations as well as of
anisotropy in the background on the resonance-like structure. We will also
touch the cohabitation of the resonance-like structure with the fast flavor
conversion (FFC).

##  I Introduction

Neutrino flavor oscillations, particularly the fast flavor conversion (FFC),
have been studied extensively these days due to their more rapid growth
compared to other conversion modes [1, 2, 3, 4, 5, 6]. The neutrino-flavor-
lepton-number (NFLN) crossing, that is, the situation, in which one flavor is
dominant over the other in certain momentum directions while the opposite is
true in other directions at least for a pair of neutrino flavors, is known to
be the criterion for FFC [7]. In its application to core-collapse supernovae
(CCSNe), the electron-lepton-number (ELN) crossing has been conveniently
searched for in many papers [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], since
other flavor-lepton-numbers are normally much smaller. The ordinary non-
forward scatterings of neutrinos were once thought to destroy the coherence
among neutrinos, thus, working against the neutrino flavor oscillations. The
interplay between FFC and the ordinary collisions has been investigated from
various directions [19, 20, 21, 22, 23, 24, 25, 26, 27]. Interestingly, some
numerical simulations, in which the collision rate was artificially modified
by orders of magnitudes, found that FFC can be enhanced by the collisions
[22]. It was also pointed out that the collisions may modify the neutrino
spectra so that FFC could be driven [28]. The so-called collisional dilemma,
i.e., enhancement or damping of FFC by the ordinary collision, has not been
fully resolved so far.

Recently the existence of a new type of flavor conversion referred to as the
collisional flavor instability (CFI), which is driven by the collisions
themselves and can occur without the NFLN crossing and hence FFC, was pointed
out [29]. Properties of CFI were investigated in the same frame work in
another paper [27]. It was also shown that the onset of CFI could be hastened
by FFC [30]. For homogeneous, isotropic, and monochromatic neutrino
distributions the condition for CFI is thought to be that the collision rate
for neutrino should be different from that for antineutrino [29]. The
existence of CFI was later confirmed numerically for homogeneous, isotropic
but non-monochromatic neutrino distributions [31]. The authors added to the
criterion for CFI a condition that there exists at least one ELN crossing in
neutrino $\it{energy}$. The first global simulation of CFI was performed for
static backgrounds taken from different stages in a core-collapse supernova
simulation [32], demonstrating that the flavor oscillation due to CFI could
occur faster than advection. On the other hand, the enhancement of CFI by an
asymmetry in the collision rates between neutrino and anti-neutrino was first
noticed in [27]. The resonance-like behavior was pointed out in [33].

In the following sections, we derive the exact linear growth rate of the CFI
mode with $k=0$ in the resonance-like structure based on the dispersion
relation for the homogeneous, isotropic and monochromatic neutrino
distributions in the background and give the criterion for the resonance-like
phenomenon as well. We consider not only the isotropy-preserving mode but also
the isotropy-breaking mode, which has been somehow overlooked in the
literature so far, and demonstrate that they also give rise to the resonance-
like phenomenon. For the isotropy-preserving mode, on the other hand, we solve
the QKE numerically in the resonance-like region and study the nonlinear
evolution of the system there. We show that the saturation is reached more
rapidly in that case but that the saturation level is hardly affected.

We then procced to the non-monochromatic case. We adopt the Fermi-Dirac
distribution for the neutrino energy. Numerically evaluating the dispersion
relation, we find that the growth rate in and the criterion for the resonance-
like phenomenon obtained in the monochromatic case remain good approximations
if the monochromatic collision rates are simply replaced by the mean collision
rates. Finally, the effect of non-vanishing wave numbers ($k\neq 0$) in the
perturbation or of the anisotropy in the background angular distributions in
momentum space is studied again based on the dispersion relation. It is found
that CFI gets weaker in both cases. We also touch the case, in which the ELN
crossing is present initially and the FFC coexists with the resonance-like
structure of CFI.

##  II Dispersion Relation

The neutrino flavor content in the two flavor approximation is described by
the neutrino flavor density matrix

$\rho(x,P)=\left(\begin{array}[]{cc}f_{\nu_{e}}&S\\\
S^{*}&f_{\nu_{x}}\end{array}\right),$ (1)

where the star means the complex conjugate; the diagonal elements are neutrino
occupation numbers in the individual flavor eigenstates whereas the off-
diagonal elements represent the coherence between the two flavors;
$x=(x^{\mu}$) is the spacetime position and $P=(E,\bm{v})$ is the 4-momentum
vector of neutrinos, in which neutrinos are assumed to be ultra-relativistic
particles traveling at the speed of light $|\bm{v}|=1$ in natural units, which
we will employ hereafter through the paper. We use the signature convention of
$\eta_{\mu\nu}=\text{diag}(+1,-1,-1,-1)$ for the Minkowski metric. In the
flavor isospin convention, the negative energy $E<0$ stand for antineutrinos
as $\rho(E)=-\bar{\rho}(-E)$. Note that quantities associated with
antineutrinos are indicated by the bar.

The evolution of the flavor density matrix is described by the quantum kinetic
equation

$\mathrm{i}v\cdot\partial\rho=[H,\rho]+\mathrm{i}C,$ (2)

where $H$ is the Hamiltonian and $C$ is the collision term. The Hamiltonian
has the vacuum, matter and neutrino contributions given as

$\begin{split}&H=H_{\text{vac}}+H_{\text{mat}}+H_{\nu},\\\
&H_{\text{vac}}(x,P)=\frac{M^{2}}{2E},\\\
&H_{\text{mat}}(x,P)=\sqrt{2}G_{\text{F}}v\cdot\text{diag}(j_{e}(x),j_{x}(x)),\\\
&H_{\nu}(x,P)=\sqrt{2}G_{\text{F}}v\cdot\int\mathrm{d}P^{\prime}\rho(x,P^{\prime})v^{\prime},\\\
\end{split}$ (3)

where $M^{2}$ is the neutrino mass-squared matrix; $j_{\alpha}(x)$ is the
lepton number 4-current of the charged lepton species $\alpha$; the integral
over 4-momentum is abbreviated as

$\int
dP=\int_{-\infty}^{\infty}\frac{E^{2}dE}{2\pi^{2}}\int\frac{d\bm{v}}{4\pi}.$
(4)

The collision term $C$ can be written for neutrinos in the relaxation
approximation as

$C(x,P)=\frac{1}{2}\\{\text{diag}(\Gamma_{e}(x,P),\Gamma_{x}(x,P)),\rho_{\text{eq}}-\rho\\},$
(5)

where the curly bracket denotes anti-commutator; $\Gamma_{\alpha}(x,P)$ is the
collision rate for the charged lepton $\alpha$; $\rho_{\text{eq}}$ is the
density matrix for the equilibrium state that is approached through the
collision. The collision term for antineutrinos is written in the same manner.

The quantum kinetic equation 2 may be linearized with respect to $S$ if
$|S|\ll f_{i}$ as

$v\cdot(\partial-\Lambda_{0e}+\Lambda_{0x})S_{ex}+(f_{\nu_{e}}-f_{\nu_{x}})\sqrt{2}G_{\text{F}}\int\mathrm{d}P^{\prime}v\cdot
v^{\prime}S_{ex}(P^{\prime})+\frac{1}{2E}\sum_{z=e,x}(M^{2}_{ez}S_{zx}-S_{ez}M^{2}_{zx})+\mathrm{i}\Gamma_{ex}S_{ex}=0,$
(6)

where
$\Lambda_{0z}=\sqrt{2}G_{\text{F}}[j_{z}(x)+\int\mathrm{d}Pf_{\nu_{z}}(x,P)v]$
and $\Gamma_{ex}(E)=\left[\Gamma_{e}(E)+\Gamma_{x}(E)\right]/2$. Assuming the
plane wave solution as usual as

$S(x,P)=S(k,P)e^{\mathrm{i}k\cdot x},$ (7)

where $k=(\omega,\bm{k})$ is the 4-wavevector and ignoring the vacuum term,
which is important for the slow instabilities [34] but plays a minor role to
give initial perturbations for the fast instability, we obtain

$\\{v\cdot(k-\Lambda_{0e}+\Lambda_{0x})+\mathrm{i}\Gamma_{ex}\\}S_{ex}+(f_{\nu_{e}}-f_{\nu_{x}})v\cdot
a=0.$ (8)

where the vector $a$ is defined as

$a=\sqrt{2}G_{\text{F}}\int\mathrm{d}PS_{ex}(P)v.$ (9)

Then $S$ can be solved as

$S_{ex}(k,P)=-\frac{(f_{\nu_{e}}-f_{\nu_{x}})v\cdot
a}{v\cdot(k-\Lambda_{0e}+\Lambda_{0x})+\mathrm{i}\Gamma_{ex}}.$ (10)

Substitution of this expression back into the definition of $a$ in Eq. 9 leads
to the following homogeneous equation:

$\Pi_{ex}(k)a_{ex}(k)=0,$ (11)

where the matrix $\Pi_{ex}$is defined as

$\Pi_{ex}(k)=\eta+\sqrt{2}G_{\text{F}}\int\frac{\mathrm{d}P(f_{\nu_{e}}-f_{\nu_{x}})v\bigotimes
v^{\intercal}}{v\cdot(k-\Lambda_{0e}+\Lambda_{0x})+\mathrm{i}\Gamma_{ex}}.$
(12)

$\Lambda$’s in the denominator may be absorbed into $k$, shifting the real
part of $k$ alone and unaffecting the instability [35]. Note that the so-
called zero mode with $k=0$ does not mean a mode homogeneous in space after
this shift. In the isotropic case the neutrino contribution to the shift
vanishes.

Nontrivial solutions of $a$ exist if and only if

$\det\Pi_{ex}(k)=0,$ (13)

which gives us the dispersion relation $\omega=\omega(\bm{k})$. The positive
imaginary part of $\omega$, $\mathrm{Im}\,\omega>0$, implies that the flavor
eigenstate is unstable and the perturbation in $S$ grows exponentially in
time, an indication of CFI. Introducing the spherical coordinates for the
neutrino velocity $\bm{v}$, we can write down the matrix $\Pi_{ex}$ as

$\begin{split}\Pi_{ex}(k)=\eta+\sqrt{2}G_{\text{F}}\int_{-\infty}^{\infty}\frac{E^{2}\mathrm{d}E}{2\pi^{2}}\int_{-1}^{1}\frac{\mathrm{d}c_{\theta}}{2}\int_{0}^{2\pi}\frac{\mathrm{d}\phi}{2\pi}\frac{f_{\nu_{e}}(E,\bm{v})-f_{\nu_{x}}(E,\bm{v})}{\omega-\bm{v}\cdot\bm{k}+\mathrm{i}\Gamma_{ex}(E)}\\\
\times\left(\begin{array}[]{cccc}1&\sqrt{1-c_{\theta}^{2}}c_{\phi}&\sqrt{1-c_{\theta}^{2}}s_{\phi}&c_{\theta}\\\
\sqrt{1-c_{\theta}^{2}}c_{\phi}&\left(1-c_{\theta}^{2}\right)c_{\phi}^{2}&\left(1-c_{\theta}^{2}\right)s_{\phi}c_{\phi}&\sqrt{1-c_{\theta}^{2}}c_{\theta}c_{\phi}\\\
\sqrt{1-c_{\theta}^{2}}s_{\phi}&\left(1-c_{\theta}^{2}\right)s_{\phi}c_{\phi}&\left(1-c_{\theta}^{2}\right)s_{\phi}^{2}&\sqrt{1-c_{\theta}^{2}}c_{\theta}s_{\phi}\\\
c_{\theta}&\sqrt{1-c_{\theta}^{2}}c_{\theta}c_{\phi}&\sqrt{1-c_{\theta}^{2}}c_{\theta}s_{\phi}&c_{\theta}^{2}\end{array}\right),\end{split}$
(14)

where the following abbreviations are used:
$c_{\theta}=\mathrm{cos}(\theta),s_{\theta}=\mathrm{sin}(\theta),c_{\phi}=\mathrm{cos}(\phi),s_{\phi}=\mathrm{sin}(\phi)$.

The expression is significantly simplified if the background neutrino is
isotropic in momentum space. In fact, for $k=0$, $\Pi_{ex}$ becomes even
simpler, being diagonal. The equation to be solved is reduced in this case to

$I=\sqrt{2}G_{\text{F}}\int_{-\infty}^{\infty}\frac{E^{2}\mathrm{d}E}{2\pi^{2}}\frac{f_{\nu_{e}}(E)-f_{\nu_{x}}(E)}{\omega+\mathrm{i}\Gamma_{ex}(E)}=-1,3.$
(15)

Note that the solutions for $I=3$ are degenerate with the multiplicity of 3.
The solutions of Eq. 15 are collectively referred to as the homogeneity-
preserving modes although the wave vector is shifted. It should be noted that
only the solution branch for $I=-1$ has been studied in previous papers [31,
33]. This is because the authors of these papers assumed tacitly that the
perturbation is isotropic. In fact, the spatial components of $a$ vanish in
that case so that the three spatial components of Eq. 11 become trivial. This
is equivalent to ignoring the modes for I = 3. If one allows anisotropic
perturbations instead, the spatial components of $a$ become nonzero and the
solution branch for $I=3$ is recovered. For this reason, we call the solution
branch for $I=-1$ the isotropy-preserving branch and refer to the solution for
$I=3$ as the isotropy-breaking branch.

The analysis of the real and imaginary parts of Eq. 15 leads to the necessary
condition for the occurrence of CFI, i.e, $\mathrm{Im}[\omega]>0$, that the
following function, $F(E)$, of $E(>0)$,

$\begin{split}F(E)=&\bigl{[}f_{\nu_{e}}(E)-f_{\nu_{x}}(E)\bigr{]}\bigl{[}\mathrm{Im}\,\omega+\Gamma_{ex}(E)\bigr{]}\bigl{[}\bigl{(}\mathrm{Re}\,\omega\bigr{)}^{2}+\bigl{(}\mathrm{Im}\,\omega+\bar{\Gamma}_{ex}(E)\bigr{)}^{2}\bigr{]}\\\
&-\bigl{[}f_{\bar{\nu}_{e}}(E)-f_{\bar{\nu}_{x}}(E)\bigr{]}\bigl{[}\mathrm{Im}\,\omega+\bar{\Gamma}_{ex}(E)\bigr{]}\bigl{[}\bigl{(}\mathrm{Re}\,\omega\bigr{)}^{2}+\bigl{(}\mathrm{Im}\,\omega+\Gamma_{ex}(E)\bigr{)}^{2}\bigr{]}\end{split}$
(16)

should have at least one zero point, or crossing in energy. When the energy-
dependent collision rates, $\Gamma_{ex}(E)$ and $\bar{\Gamma}_{ex}(E)$, are
identical for neutrinos and antineutrinos, this condition is reduced to the
criterion derived in Ref. [31]

##  III Monochromatic Neutrinos

###  III.1 Exact Solutions in the Linear Analysis

In this section, we consider monochromatic neutrinos and solve Eq. 15
analytically. We assume the following background distributions:

$f_{\nu_{e}}(E)-f_{\nu_{x}}(E)=\frac{2\pi^{2}}{\sqrt{2}G_{\text{F}}E^{2}}[{\color[rgb]{0,0,0}\mathfrak{g}}\delta(E-\epsilon)-\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}\delta(E+\bar{\epsilon})],$
(17)

where ${\color[rgb]{0,0,0}\mathfrak{g}},\
\bar{{\color[rgb]{0,0,0}\mathfrak{g}}},\ \epsilon,\ \bar{\epsilon}$ are model
parameters. Then Eq. 15 becomes

$\frac{{\color[rgb]{0,0,0}\mathfrak{g}}}{\omega+\mathrm{i}\Gamma}-\frac{\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}}{\omega+\mathrm{i}\bar{\Gamma}}=-1,3,$
(18)

where
$\Gamma=\Gamma_{ex}(E=\epsilon),\bar{\Gamma}=\bar{\Gamma}_{ex}(E=-\bar{\epsilon})$.
Note that in the typical situation in the supernova core of our concern
following inequalities hold by several orders of magnitude:
${\color[rgb]{0,0,0}\mathfrak{g}},\
\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}\gg\Gamma,\ \bar{\Gamma}$, the fact used
in the following. Eq. 18 can be solved easily to produce

$\omega_{\pm}=-A-\mathrm{i}\gamma\pm\sqrt{A^{2}-\alpha^{2}+\mathrm{i}2G\alpha},$
(19)

for the isotropy-preserving modes and

$\omega_{\pm}=\frac{A}{3}-\mathrm{i}\gamma\pm\sqrt{\left(\frac{A}{3}\right)^{2}-\alpha^{2}-\mathrm{i}\frac{2}{3}G\alpha},$
(20)

for the isotropy-breaking modes. In the above equations, the following
notations are introduced:

$G=\frac{{\color[rgb]{0,0,0}\mathfrak{g}}+\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}}{2},\
A=\frac{{\color[rgb]{0,0,0}\mathfrak{g}}-\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}}{2},\
\gamma=\frac{\Gamma+\bar{\Gamma}}{2},\ \alpha=\frac{\Gamma-\bar{\Gamma}}{2}.$
(21)

Note that when the collision rates are identical between neutrino and
antineutrino, or $\alpha=0$, the imaginary part of $\omega_{\pm}$ is
$-\gamma<0$, implying that they are all stable. It follows hence that the
inequality of the collision rates is needed for CFI, which was the criterion
postulated in Ref. [29].

The complex frequencies given in Eqs. 19 and 20 are approximated as

$\omega_{\pm}=\begin{cases}-A-\mathrm{i}\gamma\pm\left(|A|+\mathrm{i}\frac{G|\alpha|}{|A|}\right),&\text{if
}A^{2}\gg G|\alpha|,\\\
-A-\mathrm{i}\gamma\pm\sqrt{\mathrm{i}2G\alpha},&\text{if }A^{2}\ll
G|\alpha|,\end{cases}$ (22)

for the isotropy-preserving branch and

$\omega_{\pm}=\begin{cases}\frac{A}{{\color[rgb]{0,0,0}3}}-\mathrm{i}\gamma\pm\left(\frac{|A|}{3}-\mathrm{i}\frac{G|\alpha|}{|A|}\right),&\text{if
}A^{2}\gg G|\alpha|,\\\
\frac{A}{3}-\mathrm{i}\gamma\pm\sqrt{-\mathrm{i}\frac{2}{3}G\alpha},&\text{if
}A^{2}\ll G|\alpha|,\end{cases}$ (23)

for the isotropy-breaking branch. The results for the isotropy-preserving
modes were derived previously outside the resonance-like region in [29, 31]
and inside it in [33], whereas those for the isotropy-breaking modes have not
been presented so far because the authors assumed isotropy not only for the
background but also for perturbations as we mentioned earlier.

We plot the real and imaginary parts of $\omega$ for the two branches in Figs.
1 and 2, respectively, for the following parameter values:
${\color[rgb]{0,0,0}\mathfrak{g}}=1\,\mathrm{cm}^{-1},\ \gamma=2.05\times
10^{-7}\,\mathrm{cm}^{-1},\ \alpha=-4.5\times 10^{-8}\,\mathrm{cm}^{-1}$. In
these plots we take $\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}$ as a free
parameter. It is apparent from the plots of imaginary part that there is a
rather narrow region of $\bar{g}$, in which the growth rate is enhanced
roughly by two orders of magnitude. This is the resonance-like structure that
Ref. [33] first pointed out and we will focus on in this paper. In fact, the
resonance-like peak occurs near the point, at which the real parts of
$\omega_{+}$ and $\omega_{-}$ come close to each other. Note that although the
resonance-like structure was reported only for the isotropy-preserving modes
in Ref. [33], it occurs also for the isotropy-breaking modes as shown in Fig.
2.

Figure 1: Real and imaginary parts of the complex frequency $\omega$ for the
isotropy-preserving branch as a functio of
$\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}$. For (a) and (b) we take
${\color[rgb]{0,0,0}\mathfrak{g}}=1\ \mathrm{cm}^{-1},\gamma=2.05\times
10^{-7}\ \mathrm{cm}^{-1},\alpha=-4.5\times 10^{-8}\ \mathrm{cm}^{-1}$,
whereas for (c) we artificially magnify the collision rate of antineutrinos to
$\bar{\Gamma}=2.5\times 10^{-2}\ \mathrm{cm}^{-1}$, so that $|\alpha|\sim
2.5\times 10^{-2}\ \mathrm{cm}^{-1}$ which implies
$\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}\approx 1.025\ \mathrm{cm}^{-1}$ at the
resonance-like peak. See the text for the notational details.

Figure 2: Same as Fig. 1 but for the isotropy-breaking branch.

To locate where the enhancement occurs exactly, we differentiate
$\omega_{+}({\color[rgb]{0,0,0}\mathfrak{g}},\
\bar{{\color[rgb]{0,0,0}\mathfrak{g}}},\ \Gamma,\ \bar{\Gamma})$ of the
isotropy-preserving branch in Eq. 19 with respect to
$\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}$ as

$\begin{split}\frac{\partial\omega}{\partial\bar{g}}=&\frac{1}{2}+\frac{1}{2}(A^{2}-\alpha^{2}+\mathrm{i}2G\alpha)^{-\frac{1}{2}}(-A+\mathrm{i}\alpha)\\\
=&\frac{1}{2}+\frac{1}{2}|C|^{-\frac{1}{2}}\left[\left(-A\mathrm{cos}\frac{\theta[C]}{2}+\alpha\mathrm{sin}\frac{\theta[C]}{2}\right)\right.+\\\
&\left.\mathrm{i}\left(\alpha\mathrm{cos}\frac{\theta[C]}{2}-A\mathrm{sin}\frac{\theta[C]}{2}\right)\right],\end{split}$
(24)

where we define $C$ as

$C=A^{2}-\alpha^{2}+\mathrm{i}2G\alpha=|C|e^{\mathrm{i}\theta[C]}.$ (25)

The imaginary part of the derivative vanishes at the unique value of
$\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}$ that corresponds to
$A^{2}=\alpha^{2}$. Thus the resonance-like peak occurs when
$|A|\approx|\alpha|$, i.e., the number densities of neutrino and antineutrino
come close to each other. Note that the difference between $\Gamma$ and
$\bar{\Gamma}$ is usually orders of magnitude smaller than the difference
between $g$ and $\bar{g}$, i.e., $|\alpha|\ll|A|$.

In the isotropy-preserving branch, the imaginary part of $\omega_{+}$ is given
approximately as

$\mathrm{Im}\,\omega_{+}\approx\begin{cases}-\gamma+\frac{G|\alpha|}{|A|},&\text{if
}A^{2}\gg G|\alpha|,\\\ -\gamma+\sqrt{G|\alpha|},&\text{if }A^{2}\ll
G|\alpha|,\end{cases}$ (26)

In the situation of our concern, where
${\color[rgb]{0,0,0}\mathfrak{g}},\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}\gg\Gamma,\bar{\Gamma}$,
the first case is non-resonant, since $G\gg|\alpha|$. On the other hand, the
second case may fall in the resonance-like region. Note that the imaginary
part is positive in the second regime unless $\alpha$ is much smaller than
$\gamma$. In the first regime, on the other hand, the signature of the
imaginary part is determined by the competition of difference between $g$ and
$\bar{g}$ and that between $\Gamma$ and $\bar{\Gamma}$.

Although we have so far regarded $\omega$ as a function of
$\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}$, one can also take other parameters.
If one varies ${\color[rgb]{0,0,0}\mathfrak{g}}$ instead of
$\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}$, the same criterion
$|A|\approx|\alpha|$ is obtained. This resonance-like peak, if regarded as a
function of one of the collision rates, it reaches the maximum height at a
certain value of this rate. Note also that the resonance-like peak occurs at
${\color[rgb]{0,0,0}\mathfrak{g}}\sim\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}$
but not exactly at
${\color[rgb]{0,0,0}\mathfrak{g}}=\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}$. To
see this more clearly, we choose a different parameter set, in which we
artificially raise the collision rate of antineutrinos to an unrealistically
high value $\bar{\Gamma}=0.025\ \mathrm{cm}^{-1}$ so that $|\alpha|\approx
0.025\ \mathrm{cm}^{-1}$. We then obtain
$\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}\approx 1.025\ \mathrm{cm}^{-1}$ at the
peak. This deviation from ${\color[rgb]{0,0,0}\mathfrak{g}}=1\
\mathrm{cm}^{-1}$ is confirmed in Fig. 1. One should also notice that the peak
amplitude in the resonance-like structure is much larger, and the peak width
is also much broader in this extreme case. A similar analysis shows that the
resonance-like structure occurs also for the isotropy-breaking branch. The
peak location is a bit different from that for the isotropy-preserving mode
and satisfies $A^{2}=(3\alpha)^{2}$, as demonstrated in Fig. 2 again for the
exaggerated value of $\bar{\Gamma}=0.025\ \mathrm{cm}^{-1}$. It is also
observed that the peak width in the resonance-like structure for the isotropy-
breaking mode is not appreciably different from that for the isotropy-
preserving mode whereas the peak amplitude is smaller by a factor $\sim 2$ for
the isotropy-breaking mode than that for the isotropy-preserving mode.

So far we have employed the dispersion relation. The behavior discussed above
can be also derived directly from the original quantum kinetic equation. Since
it provides a different insight into the resonance-like structure, we will
look at it below. We assume that the background is homogeneous, isotropic, and
monochromatic. The vacuum and matter terms are ignored to focus on CFI. Then
the Hamiltonian and the collision term are given as

$\begin{split}&H=H_{\nu}=H_{\text{d}}+H_{\text{o}}=\\\
&\left(\begin{array}[]{cc}{\color[rgb]{0,0,0}\mathfrak{g}}_{\nu_{e}}-{\color[rgb]{0,0,0}\mathfrak{g}}_{\bar{\nu}_{e}}&0\\\
0&{\color[rgb]{0,0,0}\mathfrak{g}}_{\nu_{x}}-{\color[rgb]{0,0,0}\mathfrak{g}}_{\bar{\nu}_{x}}\end{array}\right)+\left(\begin{array}[]{cc}0&S-\bar{S}\\\
S^{*}-\bar{S}^{*}&0\end{array}\right),\end{split}$ (27)
$\begin{split}C=\left(\begin{array}[]{cc}\Gamma_{e}(\displaystyle{\color[rgb]{0,0,0}\mathfrak{g}}_{\nu_{e},eq}-{\color[rgb]{0,0,0}\mathfrak{g}}_{\nu_{e}})&\displaystyle\frac{\Gamma_{e}+\Gamma_{x}}{2}(S_{eq}-S)\\\
\displaystyle\frac{\Gamma_{e}+\Gamma_{x}}{2}(S^{*}_{eq}-S^{*})&\displaystyle\Gamma_{x}({\color[rgb]{0,0,0}\mathfrak{g}}_{\nu_{x},eq}-{\color[rgb]{0,0,0}\mathfrak{g}}_{\nu_{x}})\end{array}\right),\\\
\bar{C}=\left(\begin{array}[]{cc}\displaystyle\bar{\Gamma}_{e}({\color[rgb]{0,0,0}\mathfrak{g}}_{\bar{\nu}_{e},eq}-{\color[rgb]{0,0,0}\mathfrak{g}}_{\bar{\nu}_{e}})&\displaystyle\frac{\bar{\Gamma}_{e}+\bar{\Gamma}_{x}}{2}(\bar{S}_{eq}-\bar{S})\\\
\displaystyle\frac{\bar{\Gamma}_{e}+\bar{\Gamma}_{x}}{2}(\bar{S}^{*}_{eq}-\bar{S}^{*})&\displaystyle\bar{\Gamma}_{x}({\color[rgb]{0,0,0}\mathfrak{g}}_{\bar{\nu}_{x},eq}-{\color[rgb]{0,0,0}\mathfrak{g}}_{\bar{\nu}_{x}})\end{array}\right).\end{split}$
(28)

Note that the Hamiltonian is divided into the diagonal part $H_{\text{d}}$ and
the off-diagonal part $H_{\text{o}}$. The equilibrium distributions are set to
the unperturbed states:

$\begin{split}&{\color[rgb]{0,0,0}\mathfrak{g}}_{\nu_{i},\text{eq}}={\color[rgb]{0,0,0}\mathfrak{g}}_{\nu_{i}},\\\
&{\color[rgb]{0,0,0}\mathfrak{g}}_{\bar{\nu}_{i},\text{eq}}={\color[rgb]{0,0,0}\mathfrak{g}}_{\bar{\nu}_{i}},\\\
&\bar{S}^{(*)}_{\text{eq}}={S}^{(*)}_{\text{eq}}=0.\end{split}$ (29)

The linearized equations for $S$ and $\bar{S}$ are given as

$\mathrm{i}\partial_{t}\bm{V}=\bm{WV},$ (30)

where $\bm{V}$ is vector defined as

$\bm{V}=\left(\begin{array}[]{cc}S\\\ \bar{S}\end{array}\right),$ (31)

and $\bm{W}$ is the matrix expressed as

$\left(\begin{array}[]{cc}-({\color[rgb]{0,0,0}\mathfrak{g}}_{\bar{\nu}_{e}}-{\color[rgb]{0,0,0}\mathfrak{g}}_{\bar{\nu}_{x}})-\mathrm{i}\frac{1}{2}(\Gamma_{e}+\Gamma_{x})&({\color[rgb]{0,0,0}\mathfrak{g}}_{\nu_{e}}-{\color[rgb]{0,0,0}\mathfrak{g}}_{\nu_{x}})\\\
-({\color[rgb]{0,0,0}\mathfrak{g}}_{\bar{\nu}_{e}}-{\color[rgb]{0,0,0}\mathfrak{g}}_{\bar{\nu}_{x}})&({\color[rgb]{0,0,0}\mathfrak{g}}_{\nu_{e}}-{\color[rgb]{0,0,0}\mathfrak{g}}_{\nu_{x}})-\mathrm{i}\frac{1}{2}(\bar{\Gamma}_{e}+\bar{\Gamma}_{x})\end{array}\right).$
(32)

Note that we assume that the perturbation is also isotropic in deriving these
equations here.

The general solution is given as the superposition of two independent
solutions:

$\bm{V}(t)=\bm{V}_{+}e^{-\mathrm{i}\omega_{+}t}+\bm{V}_{-}e^{-\mathrm{i}\omega_{-}t},$
(33)

where $\bm{V}_{\pm},\ \omega_{\pm}$ are the eigenvectors and eigenvalues to
the matrix $W$. The eigenvalues are obtained as [31]

$\omega_{\pm}=-A-\mathrm{i}\gamma\pm\sqrt{A^{2}-\alpha^{2}+2\mathrm{i}G\alpha},$
(34)

where the notations are identical to those for Eq. 19.

It is interesting to point out that the elimination of the diagonal part of
the Hamiltonian does not change the two eigenvalues, which suggests that CFI
is driven by an interplay of the off-diagonal part of the Hamiltonian and the
collision terms. That may be illuminated more clearly by treating the
collision term as a perturbation. Les us first consider the case with
$H=H_{\text{o}}$ and no collision term, $C=0$. The eigenvalues and
eigenvectors in this case are given as

$\begin{split}&\omega_{1}=0,\ \bm{V}_{1}=\left(\begin{array}[]{cc}1\\\
1\end{array}\right),\\\
&\omega_{2}=-{\color[rgb]{0,0,0}\mathfrak{g}}-\bar{{\color[rgb]{0,0,0}\mathfrak{g}}},\
\bm{V}_{2}=\left(\begin{array}[]{cc}{\color[rgb]{0,0,0}\mathfrak{g}}\\\
\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}\end{array}\right).\end{split}$ (35)

where and below we introduce the following notations:

$\begin{split}{\color[rgb]{0,0,0}\mathfrak{g}}&={\color[rgb]{0,0,0}\mathfrak{g}}_{\nu_{e}}-{\color[rgb]{0,0,0}\mathfrak{g}}_{\nu_{x}},\
\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}={\color[rgb]{0,0,0}\mathfrak{g}}_{\bar{\nu}_{e}}-{\color[rgb]{0,0,0}\mathfrak{g}}_{\bar{\nu}_{x}},\\\
\Gamma&=\frac{1}{2}(\Gamma_{e}+\Gamma_{x}),\
\bar{\Gamma}=\frac{1}{2}(\bar{\Gamma}_{e}+\bar{\Gamma}_{x}),\end{split}$ (36)

This is a stable flavor evolution with no growth of amplitudes.

Now we reinstate the collision terms but as a perturbation. The characteristic
equation is then written as

$\bigl{[}-{\color[rgb]{0,0,0}\mathfrak{g}}-(\omega_{0}+\Delta\omega)-\mathrm{i}\Gamma\bigr{]}\left[\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}-(\omega_{0}+\Delta\omega)-\mathrm{i}\bar{\Gamma}\right]+{\color[rgb]{0,0,0}\mathfrak{g}}\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}=0,$
(37)

where the eigenvalue is expressed as the sum of the unperturbed value given in
Eq. 35 and a (small) shift. To the linear order, the shifts are obtained for
the two modes as

$\begin{split}&\Delta\omega_{1}\approx\frac{\mathrm{i}\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}\Gamma-\mathrm{i}{\color[rgb]{0,0,0}\mathfrak{g}}\bar{\Gamma}}{{\color[rgb]{0,0,0}\mathfrak{g}}-\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}},\\\
&\Delta\omega_{2}\approx\frac{\mathrm{i}{\color[rgb]{0,0,0}\mathfrak{g}}\Gamma-\mathrm{i}\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}\bar{\Gamma}}{\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}-{\color[rgb]{0,0,0}\mathfrak{g}}}.\end{split}$
(38)

They are divergent at
${\color[rgb]{0,0,0}\mathfrak{g}}=\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}$
unless $\Gamma=\bar{\Gamma}$, an indication of the resonance-like structure in
the current setting, i.e., the collision term is assumed to be small and so is
the shift, the latter of which is no longer correct at $g\approx\bar{g}$,
however. Note that the two shifts have opposite signatures, in qualitative
agreement with the exact solution.

###  III.2 Numerical Simulations

Now we go beyond the linear analysis. We solve the quantum kinetic equation

$i\partial_{t}\rho=[H_{\nu},\rho]+iC$ (39)

for the homogeneous, isotropic, and monochromatic neutrino distributions. The
vacuum and matter terms are omitted again. The following simulations are meant
to study the nonlinear evolution of the isotropy-preserving mode both outside
and in the resonance-like regions. For this purpose we vary rather arbitrarily
$\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}$, the distribution function of
antineutrino. For comparison, we also run linear simulations, in which the
Hamiltonian is fixed to the initial value.

Since the vacuum term, which would produce perturbations to the flavor
eigenstate automatically, is neglected here, we give an initial perturbation
by hand as follows:

$\begin{split}&{\color[rgb]{0,0,0}\mathfrak{g}}={\color[rgb]{0,0,0}\mathfrak{g}}_{\nu_{e}}=1\
\mathrm{cm}^{-1},\\\
&\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}={\color[rgb]{0,0,0}\mathfrak{g}}_{\bar{\nu}_{e}}\text{
is a free variable},\\\ &S=\bar{S}=10^{-8}+0\mathrm{i}\
\mathrm{cm}^{-1}.\end{split}$ (40)

For simplicity, we assume that all neutrinos are initially in the electron
flavor and the initial perturbation is isotropic so that the isotropy-breaking
mode does not appear in this simulation. As discussed in the previous section,
the occurence of CFI is dictated by the four quantities:
${\color[rgb]{0,0,0}\mathfrak{g}},\ \bar{{\color[rgb]{0,0,0}\mathfrak{g}}},\
\Gamma$ and $\bar{\Gamma}$ (see Eq. 36). Note that in the linear simulations,
${\color[rgb]{0,0,0}\mathfrak{g}}$ and
$\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}$ are conserved quantities and
unchanged in time.

The collision rates are chosen as

$\begin{split}&\Gamma_{e}=\Gamma_{x}=\frac{\Gamma}{2}=1.6\times 10^{-7}\
\mathrm{cm}^{-1},\\\
&\bar{\Gamma}_{e}=\bar{\Gamma}_{x}=\frac{\bar{\Gamma}}{2}=2.5\times 10^{-7}\
\mathrm{cm}^{-1},\end{split}$ (41)

which correspond to

$\begin{split}&\gamma=2.05\times 10^{-7}\ \mathrm{cm}^{-1},\\\
&\alpha=-4.5\times 10^{-8}\ \mathrm{cm}^{-1}.\end{split}$ (42)

Since the $\Gamma$ and $\bar{\Gamma}$ are much smaller than
${\color[rgb]{0,0,0}\mathfrak{g}}$ and
$\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}$, the resonance-like peak occurs
essentially at
${\color[rgb]{0,0,0}\mathfrak{g}}=\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}$.

The results of the linear simulations are shown first in Fig. 3 for three
different values of $\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}$:
$1.000\,\mathrm{cm}^{-1},1.009\,\mathrm{cm}^{-1}$ and $1.100\
\mathrm{cm}^{-1}$. The first one almost corresponds to the resonance-like peak
whereas the second and third values give the edge of and a point outside the
resonance-like region, respectively. The blue curves are the modulus of flavor
coherence, $|S|$, plotted against time while the yellow lines indicate for
comparison the exponential growths with the values of $\omega_{+}$ given in
Eq. 19 for these settings. As should be clear, $|S|$ shows the exponential
growth as expected in all cases. In particular, the growth is much faster
indeed in the resonance-like region. Note also that the initial conditions are
not exactly the eigenmodes corresponding to $\omega_{+}$ and some deviations
from the exact exponential growth are seen.

Figure 3: The time evolutions of $|S|$, the modulus of the flavor coherence,
for three values of $\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}$ in the linear
simulations. We take $\alpha=-4.5\times 10^{-8}\ \mathrm{cm}^{-1}$ and
${\color[rgb]{0,0,0}\mathfrak{g}}=1\ \mathrm{cm}^{-1}$. In the plots, the
yellow straight lines are the exponential evolution at the growth rate, and
the blue curves are numerical results. (a)
$\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}=1.000\ \mathrm{cm}^{-1}$
approoximately corresponds to the resonance-like peak, (b)
$\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}=1.009\ \mathrm{cm}^{-1}$ gives a an
edge of the resonance-like region, and (c)
$\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}=1.100\ \mathrm{cm}^{-1}$ is outside
the resonance-like region.

Now we procced to the results of the fully nonlinear simulations run for the
first two initial conditions employed in the linear simulations presented just
above. They correspond to the peak and edge of the resonance-like structure,
respectively. In Fig. 4 we plot not only $|S|$, the modulus of the flavor
coherence and the off-diagonal component of the density matrix, but also the
distribution functions of all neutrinos, which are also the diagonal
components of the density matrix, as functions of time. For reference, the
exponential growths with the values of $\omega_{+}$ in Eq. 19 for the current
settings are again exhibited.

One recognizes clearly that there are two distinct phases, the linear phase,
in which the flavor coherence grows exponentially at the rate given in the
linear analysis, and the nonlinear saturation phase, where the $|S|$ peaks out
and levels off thereafter and the distribution functions of neutrinos and
antineutrinos are also settled gradually to states that are steady in the
statistical sense and are different from the initial ones (see the insets in
each panel). The transition from the linear phase to the nonlinear phase may
be characterized by $|S|\sim\mathfrak{g}$. It is found from the result in the
resonance-like structure that the saturation level of the flavor coherence is
not particularly large (actually smaller) compared with the case for the edge
of the resonance-like region and that the role of resonance-like phenomenon is
just to shorten the time it takes to reach the saturation.

This may be supported by another simulation, a variant of the second case, in
which the $\bar{\Gamma}$ is changed to $3.6\times 10^{-7}\ \mathrm{cm}^{-1}$
with other parameters, particularly ${\color[rgb]{0,0,0}\mathfrak{g}}$ and
$\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}$, fixed so that it should give the
point closer to the resonance-like peak. The result is presented as Fig. 4.
Note that the linear growth rate is twice as large as that in the second case
and is close to that in the first case. The saturation occurs earlier
consistently with the enhanced linear growth rate. On the other hand, the
asymptotic state is almost the same as that in the second case. It is mostly
determined by the initial neutrino distributions and little affected by the
resonance-like phenomenon.

It shold be noted that the asymptotic state may be oscillating in time and is
steady not in the literal sense but in the statistical sense. The
substantially smaller populations of $\nu_{x}$ and $\bar{\nu}_{x}$ in the
asymptotic states in these models are the consequence of our choice of the
equilibrium state in the relaxation approximation adopted in the simulation
and are rather artificial. It should be also mentioned that in reality, where
the background matter and neutrinos themselves may change significantly due to
advection over the time scale of neutrino oscillations, the asymptotic state
may never be reached. It was found that CFI could win the compitition against
advection (see [32]). What consequences the accelerated saturation may have,
if any, for the core-collapse supernova remain to be studied.

It is interesting to point out that in the first case corresponding
approximately to the resonance-like peak, a very fast flavor bouncing occurs
in the beginning of the nonlinear saturation phase right after the peak-out.
It is due to the diagonal part of the collision term, in which the equilibrium
distributions is imposed. In fact, if we reset the equilibrium values to the
actual asymptotic values when the nonlinear phase is reached, this feature
disappears.

Figure 4: (a) and (b): the nonlinear time evolutions for the same initial
conditions as in Figs. 3 and 3, respectively. Not only the flavor coherence
$|S|$ (blue lines) but also the distribution functions of neutrinos and
antineutrinos (colors indicated in the legends) are presented. (c): a variant
of (b), in which $\bar{\Gamma}$ is changed to $3.6\times 10^{-7}\
\mathrm{cm}^{-1}$ so that the initial condition should be closer to the
resonance-like peak. The longer evolutions leading up to the asymptotic states
are shown in the insets.

##  IV Non-monochromatic neutrinos

We extend the analyses in the previous section to non-monochromatic neutrinos.
We assume that neutrinos and antineutrinos have energy spectra given by the
Fermi-Dirac distributions. We begin with the linear analysis and discuss the
criterion for the resonance-like phenomenon as well as the growth rate at the
resonance-like peak in this case. We then study the nonlinear evolutions of
the system in the resonance-like region for the isotropy-preserving mode with
$k=0$ by numerical simulations. Finally, going back to the linear analysis, we
investigate the effects of nonvanishing wave numbers in the perturbation as
well as of anisotropies in the background on CFI. As an extreme case of the
latter, we discuss a possible interplay of the resonance-like structure in CFI
with FFC on the same footing.

###  IV.1 Linear Analysis

We assume that neutrinos and antineutrinos have continuous energy spectra
given by the Fermi-Dirac distributions:

$f_{i}(E,\,g_{i},\,T_{i},\,\mu_{i})=\frac{g_{i}}{\mathrm{exp}[E/T_{i}-\mu_{i}]+1},$
(43)

where $g_{i}$, $T_{i}$ and $\mu_{i}$ are the model parameters for the neutrino
species $i$, and $E$ is the neutrino energy. Assuming for simplicity that only
electron flavor is present initially, we choose

$\begin{split}&g_{\nu_{e}}=1,\\\ &g_{\bar{\nu}_{e}}\text{ is a free
variable},\\\ &g_{\nu_{x}}=g_{\bar{\nu}_{x}}=0,\end{split}$ (44)

and the shorthand notation

$\begin{split}&g=g_{\nu_{e}},\\\ &\bar{g}=g_{\bar{\nu}_{e}}\\\ \end{split}$
(45)

will be used in the following; the temperatures are set to

$\begin{split}&T_{\nu_{e}}=4\ \mathrm{MeV},\\\ &T_{\bar{\nu}_{e}}=5\
\mathrm{MeV};\end{split}$ (46)

we further assume that $\mu=0$. Although it is not presented here, we
confirmed that a non-zero $\mu$ does not change qualitatively the result in
the following. The collision term is now assumed to depend on the energy
quadratically as

$\Gamma(E)=\Gamma_{0}\left(\frac{E}{10\ \mathrm{MeV}}\right)^{2},$ (47)

with

$\Gamma_{0}=10^{-5}\ \mathrm{cm}^{-1}$ (48)

common to all flavors.

For later convenience we introduce the number density multiplied by
$\sqrt{2}G_{\mathrm{F}}$, $n_{i}$, the mean energy, $\langle E\rangle_{i}$,
and the mean collision rate, $\langle\Gamma\rangle_{i}$ for the neutrino
species $i$ as

$\begin{split}n_{i}&={\color[rgb]{0,0,0}\sqrt{2}G_{\mathrm{F}}}\int\frac{E^{2}\mathrm{d}E}{2\pi^{2}}f(E,\,g_{i},\,T_{i},\,\mu_{i}),\\\
\langle
E\rangle_{i}&=\frac{{\color[rgb]{0,0,0}\sqrt{2}G_{\mathrm{F}}}}{n_{i}}\int\frac{E^{3}\mathrm{d}E}{2\pi^{2}}f(E,\,g_{i},\,T_{i},\,\mu_{i}),\\\
\langle\Gamma\rangle_{i}&=\frac{{\color[rgb]{0,0,0}\sqrt{2}G_{\mathrm{F}}}}{n_{i}}\int\frac{E^{2}\mathrm{d}E}{2\pi^{2}}\Gamma(E)f(E,\,g_{i},\,T_{i},\,\mu_{i}).\end{split}$
(49)

These three quantites defined above have same unit. Note that a change in
$g_{i}$ influences only $n_{i}$ while a variation in $T_{i}$ or $\mu_{i}$
influences all of $n_{i},\ \langle E\rangle_{i},\ \langle\Gamma\rangle_{i}$.
We also introduce the following quantities:

$G=\frac{n+\bar{n}}{2},\ A=\frac{n-\bar{n}}{2},\
\gamma=\frac{\langle\Gamma\rangle+\bar{\langle\Gamma\rangle}}{2},\
\alpha=\frac{\langle\Gamma\rangle-\bar{\langle\Gamma\rangle}}{2}.$ (50)

We solve Eq. 15 numerically to obtain the dispersion relation. In doing so,
the energy range from $0\ \mathrm{MeV}$ to $80\ \mathrm{MeV}$ is divided into
nonuniform 128 bins concentrated more at low energies.

We plot in Fig. 5 the contour lines for $\mathrm{Re}\ I=-1$ or $3$ (blue) and
$\mathrm{Im}\ I=0$ (orange) in the complex $\omega$ plane for some
representative values of $\bar{g}$. The intersection of the two contour lines
gives the dispersion relation $\omega(k)$ at $k=0$. One of the two solutions
(except for the origin) with a positive imaginary part is the unstable mode.
The first three plots in the figure are for the isotropy-preserving branch at
$\bar{g}=0.51,\ 0.511,\ 0.512$, respectively, and the last plot is for the
isotropy-breaking branch at $\bar{g}=0.512$. For the current choice of
parameters, which correspond to

$\begin{split}&\langle\Gamma\rangle=2.07031\times 10^{-6}\
\mathrm{cm}^{-1},\\\ &\langle\bar{\Gamma}\rangle=3.23531\times 10^{-6}\
\mathrm{cm}^{-1},\\\
&\alpha=\frac{\langle\Gamma\rangle-\langle\bar{\Gamma}\rangle}{2}\approx-5.825\times
10^{-7}\ \mathrm{cm}^{-1},\end{split}$ (51)

the resonance-like peak is expected to occur approximately at $\bar{g}=0.512$
(panels (c) and (d)) giving $n\approx\bar{n}\approx 4.88691\
\mathrm{cm}^{-1}$.

Figure 5: The contours lines for $\mathrm{Re}\ I=-1$ or $3$ (blue) and
$\mathrm{Im}\ I=0$ (orange) in Eq. 15. Their intersections give the dispersion
relation $\omega(k=0)$. Panels (a)-(c) are for the isotropy-preserving branch
whereas panel (d) displays the result for the isotropy-breaking branch.

This is confirmed in Fig. 6, where we plot the real and imaginary parts of
$\omega$ for the unstable isotropy-preserving branch as a function of
$\bar{g}$. One can see a familiar enhancement of the imaginary part, or the
growth rate of the instability.

Figure 6: The real (a) and imaginary (b) parts of the complex frequency
$\omega$ for the unstable isotropy-preserving branch as a function of
$\bar{g}$. For the imaginary part, the numerical solution $\mathrm{Im}\
\omega_{\text{num}}(\bar{g})$ shown in blue is compared with the approximate
one $\mathrm{Im}\ \omega_{\text{appr}}(\bar{g})$ exhibited in orange. In panel
(c), the collision rate $\bar{\Gamma}$ of antineutrinos is artificially
magnified by $10^{5}$.

We also investigate an extreme case, in which we artificially magnify the
collision rate $\bar{\Gamma}$ of antineutrino by five orders of magnitude,
$\Gamma_{0}=1\ \mathrm{cm}^{-1}$ in Eq. 48, while other parameters are
unchanged. This gives $\alpha=-1.6174\ \mathrm{cm}^{-1}$. The result is shown
in Fig. 6. The resonance-like peak occurs at $\bar{g}=0.74$, which corresponds
to $A=-1.0081\ \mathrm{cm}^{-1}$. The number density of antineutrino is
noticably different from that of neutrino in this case just as inferred from
the result of the monochromatic case. Again the approximate formula (orange
line) is compared with the numerical result shown in blue. This time the
approximation is not so successful as before although the peak shift as well
as the broadening of resonance-like region are captured qualitatively. It is
pointed out that thanks to this broadening the case with $n=\bar{n}$ is still
in the resonance-like region, although not at the peak, and hence the growth
rate is enhanced. We explored other neutrino spectra, such as the Fermi-Dirac
distributions with non-vanishing chemical potentials or the Gaussian and
polynomial distributions that are not too irregular. The results are
qualitatively the same as those given above and hence will not be presented
here.

Finally, it is worth pointing out that the growth rate of the resonance-like
structure is well approximated by the formula for the monochromatic case, Eq.
19 for $\omega_{+}$, with the following substitutions:

$\begin{split}&\overset{\scriptscriptstyle(-)}{{\color[rgb]{0,0,0}\mathfrak{g}}}\rightarrow\overset{\scriptscriptstyle(-)}{n}\\\
&\overset{\scriptscriptstyle(-)}{\Gamma}\rightarrow\langle\overset{\scriptscriptstyle(-)}{\Gamma}\rangle.\end{split}$
(52)

Note that this approximate formula agrees with the one derived previously in
[33] under the assumption of $|\Gamma|,\ |\bar{\Gamma}|\ll|\omega|$ if
$\gamma$ and $\alpha^{2}$ are neglected accordingly. This is demonstrated in
Fig. 6, where the numerical solution displayed in blue is compared with the
approximate one presented in orange. Their agreement matches what was shown in
[33]. Although the criterion for resonance-like phenomenon posited in that
paper is appropriate only when the collision rates are small, the condition
$|A|\sim|\alpha|$ is always satisfied at the resonance-like peak regardless of
the parameter values.

###  IV.2 Numerical Simulations

We again solve Eq. 39 numerically for homogeneous and isotropic neutrinos with
the Fermi-Dirac distributions as the energy spectra. The background
distributions are actually the same as those employed for the linear analysis
in the previous section, Eqs. 44-48. Since the vacuum term is omitted in the
kinetic equation, we need to set an initial perturbation by hand, which is
also assumed to be isotropic and homogeneous. Hence only the isotropy-
preserving branch is considered. We take $\bar{g}$ as a free parameter and
vary it so that the unperturbed state should be either near the resonance-like
peak or near the edge of the resonance-like region.

In the simulation, the energy range from $1\ \mathrm{MeV}$ to $100\
\mathrm{MeV}$ is divided uniformly into 100 bins this time. We adopt the
following initial perturbations only to the off-diagonal components of the
density matrix

$\begin{split}&S(E)=(1+0.8\mathrm{i})10^{-5}f(E),\\\
&\bar{S}(E)=(-0.5+\mathrm{i})10^{-5}\bar{f}(E).\end{split}$ (53)

The numbers in the functions are chosen rather arbitrarily.

The results of the linear simulations, in which the Hamiltonian is fixed to
the initial value, is shown in Fig. 7 for two choices of $\bar{g}$, one
corresponding approximately to the resonance-like peak and the other giving
the edge of the resonance-like region. The value of $A/\alpha$ measures how
close the initial configuration is to the resonance-like peak: the closer to
unity it is, the nearer to the resonance-like peak the initial state is. It is
evident that the flavor coherence grows at a common rate for all energies
[31], just as expected from the energy-independent nature of $\omega(k)$. In
the resonance-like region, it grows much faster indeed.

The corresponding results for the nonlinear simulations are presented in Figs.
8 and 9. Again one recognizes that the linear phase is followed by the
nonlinear saturation phase also in this case with the non-monochromatic energy
spectra. In the linear phase the flavor coherence grows at a common rate for
all energies just as in the linear simulations given above (see Fig. 8). In
the case for the resonance-like peak (panel (a)), the nonlinear phase is
reached much faster than in the case for the edge of the resonance-like region
(panel (b)).

It is also found that the beginning of the nonlinear phase is marked for each
energy when $|S(E)|=f(E)$ is satisfied individually, which is a direct
extension of the monochromatic case. The saturation level is not much
different between the two cases (see Fig. 9), indicating again that the main
effect of the resonance-like phenomenon is to accelerate the linear growth and
the asymptotic state is hardly affected. The bouncing in the nonlinear phase
is observed only for the resonance-like peak. The bouncing amplitude seems to
be related with the difference between the equilibrium distribution and the
asymptotic distribution although the exact mechanism of bouncing is not clear
at the moment. This is induced by the diagonal part of the collision term. In
fact, if we discard the diagonal part effectively by resetting the equilibrium
distributions to the asymptotic ones, the bouncing feature is gone just as in
the monochromatic case.

Figure 7: The time evolutions of $|S|$, the modulus of the flavor coherence,
at different neutrino energies in the linear simulations for the continuous
energy spectrum. Two values of $\bar{g}$ are adopted: $\bar{g}=0.512$ and
$0.519$; $\alpha=-1.165\times 10^{-6}\ \mathrm{cm}^{-1}$, and $g=1$ giving
$n_{\nu_{e}}=4,887\ \mathrm{cm}^{-1}$. The violet lines indicate for reference
the exponential evolution at the growth rate given by the linear analysis.

Figure 8: The nonlinear time evolutions for the same initial conditions as in
Fig. 7. Not only the flavor coherence $|S|$ but also the distribution
functions of neutrinos and antineutrinos are presented for different energies
(line types given in the legend). The purple solid line indicates the
exponential growth at the rate given by the linear analysis.

Figure 9: The longer-term evolutions of the same simulations as in Fig. 8.

###  IV.3 $k\neq 0$ Perturbation/Anisotropy in Background

In this last section, we investigate the resonance-like structure either for
$k\neq 0$ perturbations in the homogeneous and isotropic background or for
$k=0$ modes in the homogeneous but anisotropic background. We work on the
dispersion relations in the linear regime. The neutrino energy spectra are
again assumed to be the Fermi-Dirac distributions given in Eq. 43, with the
same parameters as in Eqs. 44-46 and, in addition, with
$\bar{g}=g_{\bar{\nu}_{e}}=0.512$, corresponding to the resonance-like peak at
$k=0$. The collision rates are given by Eqs. 47-48.

We begin with the $k\neq 0$ perturbation in the homogeneous and isotropic
background. In the following the wave vector $\bm{k}$ is assumed to be
parallel to the z-axis. We solve Eq. 14 numerically to obtain the dispersion
relation. For $k\neq 0$, there appear two nonvanishing off-diagonal components
in $\Pi_{ex}$ in addition to the diagonal ones, which survive in the
$k\rightarrow 0$ limit. In order to see how the isotropy-preserving and
isotropy-breaking branches at $k=0$ are mixed with each other to new modes in
the $k\neq 0$ case, we decompose $\Pi_{ex}$ as

$\Pi^{\mu\nu}_{ex}=\eta^{\mu\nu}+A^{\mu\nu}.$ (54)

Then its determinant can be written explicitly in a simple form as

$\mathrm{det}\,\Pi_{ex}=(A^{11}-1)^{2}\left[(A^{00}+1)(A^{33}-1)-(A^{03})^{2}\right],$
(55)

where the following relations

$\begin{split}&A^{11}=A^{22},\\\ &A^{30}=A^{03},\end{split}$ (56)

are used. Note that in the limit of $k=0$, the following relations hold
further:

$\begin{split}&A^{30}=A^{03}=0,\\\ &A^{33}=A^{11}=A^{22},\end{split}$ (57)

and the dispersion relation is obtained from

$\mathrm{det}\,\Pi_{ex}=(A^{00}+1)(A^{11}-1)^{3}=0.$ (58)

In fact, the isotropy-preserving branch is derived from the first factor and
the isotropy-breaking branch is originated from the second factor. In the case
of $k\neq 0$, while $A^{11}-1=0$ is unchanged,
$(A^{00}+1)(A^{11}-1)-(A^{03})^{2}=0$ now mixes the isotropy-preserving and
isotropy-breaking branches to produce four branches in general. In the
following we look into these modes in detail.

From the first factor in Eq. 55, a pair of solutions are obtained, which take
the following form $a=(0,a_{x},a_{y},0)$ and is perpendicular to $\bm{k}$, and
they hence break the isotropy in the $x-y$ plane. On the other hand, the
second factor can vanish in three different manners: $(1)A^{00}+1=A^{03}=0,\
(2)A^{33}-1=A^{03}=0$, or $(3)(A^{00}+1)(A^{33}-1)=(A^{03})^{2}\neq 0$. The
first case produces a solution of the form: $a=(a_{t},0,0,0)$, which is
isotropy-preserving. The second case leads to $a=(0,0,0,a_{z})$, which is
hence isotropy-breaking in the z direction, that is, the direction of
$\bm{k}$. The last one yields a solution with the form of
$a=(a_{t},0,0,a_{z})$ in general, which is also isotropy-breaking. If the
first and second factors vanish simultaneously, the solution takes a combined
form: for example, the combination of the first factor with case (1) for the
second factor gives a solution of the following form:
$a=(a_{t},a_{x},a_{y},0)$. These cases are exceptional, though, and occur only
for special spectra/collision rates. In fact, we find that case (3) is always
satisfied in the second factor of Eq. 55 for the ranges of $k$ and $\bar{g}$
considered in this paper.

In Fig. 10, we display plots of the contours indicated in each panel for
$k=0.001\,\mathrm{cm}^{-1}$. The left panel gives the solutions for the first
factor in Eq. 55 whereas the right panel shows the solutions for the second
factor. The latter corresponds to case (3) as mentioned above. At this small
$k$, all the modes are not much different from the counterparts at $k=0$. For
later convenience, we refer to the solutions for unstable modes in these plots
as Q, W, and Y. Mode Q originated from the first factor in Eq. 55 merges at
$k=0$ with Y from the second factor to give the isotropy-breaking modes. On
the other hand, mode W is reduced to the isotropy-preserving mode at $k=0$.
These modes are all isotropy-breaking at $k\neq 0$. In Fig. 11 we plot the
linear growth rates, $\mathrm{Im}\,\omega$, for these three modes as a
function of $\bar{g}$ at the same value of $k=0.001\,\mathrm{cm}^{-1}$. The
resonance-like structure is evident in all cases around $\bar{g}=0.512$.

In Fig. 12, we show the $k$-dependence of these growth rates at
$\bar{g}=0.512$, i.e., around the resonance-like peak. It is apparent that
they decrease monotonically with $k$, indicating that the resonance-like peak
gets weaker at non-vanishing $k$ for all modes. This is actually true outside
the resonance-like region as well. The non-zero $k$ tends to reduce the CFI
itself. It is noted that the presence of $\bm{k}$ in the denominator of Eq. 14
poses a challenge in the numerical integration by discretization in the region
$-k<\mathrm{Re}\,\omega<k,\ 0<\mathrm{Im}\,\omega<\Gamma(E_{max})$. We do not
think that the numerical solutions in these ranges are reliable and hence do
not consider those solutions outside the resonance-like region or inside it
but at large values of $k$ that approach this problematic region, which is
indicated as the gray bands around the origin in Fig.10.

Figure 10: The plots of the contour lines denoted in each panel in the complex
$\omega$ plane at $k=0.001\ \mathrm{cm}^{-1}$. The gray band around the origin
is the numerical unreliable region that should be discarded.

Figure 11: The resonance-like structures for the unstable branches Q in (a), W
in (b), and Y in (c) at $k=0.001\,\mathrm{cm}^{-1}$.

Figure 12: The linear growth rates for the unstable branches Q in (a), W in
(b), and Y in (c) as functions of the wave number $k$.

Figure 13: The resonance-like structures for the anisotropic neutrino
backgrounds. The degree of anisotropy is identical for $\nu_{e}$ and
$\bar{\nu}_{e}$: $\delta=\bar{\delta}=0.4$. (a)-(c) correspond to the unstable
banches Q, W, and Y respectively. Note that Q and Y are no longer degenerate.

Figure 14: The linear growth rates for mode W studied as functions of the
degree of anisotropy $\delta$. (a) at the resonance-like peak
($\bar{g}=0.512$) and (b) at roughly half the peak amplitude ($\bar{g}=0.51$).

Next we study the effect of the anisotropy in the homogeneous neutrino
background on the resonance-like structure. We introduce the following
angular-dependence to the neutrino distributions in momentum space:

$f_{i}(E,\,g_{i},\,T_{i},\,\mu_{i},\,\theta,\,\delta_{i})=(1+\delta_{i}\mathrm{cos}\theta)f_{i}(E,\,g_{i},\,T_{i},\,\mu_{i}),$
(59)

where $f_{i}(E,\,g_{i},\,T_{i},\,\mu_{i})$ on the right hand side is the
Fermi-Dirac distribution for neutrino species $i$; $0<\theta<\pi$ is the angle
that the neutrino velocity makes with the radially outward direction; the
factor $\delta_{i}$ controls the degree of anisotropy. Since we are interested
in the resonance-like structure in CFI in this paper, we first consider a case
with no angular crossing, and hence no FFC. We then look at two cases with
different angular crossings to see the interplay between the resonance-like
structure in CFI and FFC. We solve Eq. 14 numerically for $k=0$ to obtain the
dispersion relation $\omega(k=0)$ for the anisotropic (but homogeneous)
background just described. Note that the integration over the solid angle can
be done analytically for $k=0$.

We first present the results for the first case, in which only electron-type
neutrinos and antineutrinos are present initially with the same anisotropy:
$\delta=\bar{\delta}$. We choose the model parameters as in Eqs. 44-48
together with $\bar{g}=0.512$ and $k=0$ so that the unperturbed state
corresponds to the resonance-like peak at $\delta=0$. We vary the value of
$\delta$ to see how the anisotropy affects CFI in the resonance-like region.
The choice of $k=0$ simplifies the analysis a lot. In fact, only
$\Pi_{ex}^{03}$, which is linear in $\delta$, is non-vanishing as the off-
diagonal components of $\Pi_{ex}$ just as in the previous case with $k\neq 0$,
and the diagonal components are unchanged from those for the isotropic case.
Since the dispersion relation is obtained from
$\mathrm{det}\,\Pi_{ex}{\color[rgb]{0,0,0}=0}$ given as Eq. 55 again, we refer
to the corresponding modes as Q, W, and Y. Under the current setting, all
modes are isotropy-breaking for $\delta\neq 0$. Note that mode Q is
independent of $\delta$ and $\bar{\delta}$ in this setting, and is always
identical to the isotropy-breaking mode for isotropic neutrinos.

In Fig. 13 we plot the linear growth rates for the three modes Q, W and Y as
functions of $\bar{g}$ at $\delta=0.4$ to show that the resonance-like
structure occurs indeed also in this case. Note that Q and Y are strictly
distinct from each other at $\delta\neq 0$. In Fig.14 we demonstrate how the
background anisotropy affects CFI. It is observed that it tends to reduce CFI.
This time the effect is pretty minor, though.

Figure 15: The linear growth rates as functions of $\bar{g}$ for anisotropic
backgrounds with an ELN crossing. The collision term is either turned on (red)
or off (blue). (a): $\delta={\color[rgb]{0,0,0}0.}1$ and $\bar{\delta}=0.08$
and (b): $\delta={\color[rgb]{0,0,0}0.}1$ and $\bar{\delta}=0.095$.

Now we go on to the study of a possible interplay of the resonance-like
phenomenon in CFI with FFC. For this purpose we choose

$\begin{split}&\delta=0.1,\\\ &\overline{\delta}=0.08.\end{split}$ (60)

The values of other model parameters are unnchanged from the previous case for
$\delta=\bar{\delta}$. We adjust $\bar{g}$ so that there should be an ELN
crossing in the angular distributions of $\nu_{e}$ and $\bar{\nu}_{e}$. There
is indeed an interval of $\bar{g}$, in which the angular ELN crossing and thus
FFC occur. In this model, $\nu_{e}$ is dominant in the radially outward
direction while the opposite is true in the inward direction. Note that the
mean energies and collision rates for $\bar{\nu}$ are unchanged by the
variation of $\bar{g}$. We study at $k=0$ the behavior of the fast-unstable
branch in the presence of the collision term.

We show in Fig. 15 the growth rates of this mode as functions of $\bar{g}$
both with (red) and without (blue) the collision term. As is evident from the
latter, FFC occurs in this mode for $0.505\lesssim\bar{g}\lesssim 0.517$,
where $\mathrm{Im}\,\omega$ is positive. Interestingly, the resonance-like
structure in CFI takes place almost in the same region as could be inferred
from the red line, in which the foot of the resonance-like peak can be
recognized near the both ends of the FFC range. It is also apparent that the
resonance-like phenomenon has a very small impact on the growth rate when the
FFC is in operation: it only modifies the amplitude very slightly and shifts
the peak position only a bit.

This is more evident when we take a different parameter set: $\delta=0.1$ and
$\bar{\delta}=0.095$. The result is shown in Fig. 15. In this case, the
degrees of anisotropy are not much different between $\nu_{e}$ and
$\bar{\nu}_{e}$ and, as a consequence, the range for the ELN crossing is much
narrower in $\bar{g}$ (see the blue line). Now the shift in the peak position
by the resonance-like phenomenon in CFI is apparent. Note that without FFC the
height and width of the resonance-like structure are essentially unchanged
with this variation of $\bar{\delta}$. Because of this shift, the amplitude of
FFC, on the other hand, is enhanced or reduced, depending on where we look at.
Such a shift becomes also noticeable for larger deviations of $\bar{\delta}$
from $\delta$ if the difference in the collision rates is large enough. As
mentioned above, it seems that when FFC and the resonance-like phenomenon in
CFI are simultaneously in operation, the growth rate is set by the former
(recall that the growth rate of CFI at the resonance-like peak for the current
setting at $\delta=\bar{\delta}$ is $\sim 0.005\,\mathrm{cm}^{-1}$, i.e., 10
times smaller than the growth rate of FFI alone). More systematic
investigations in a broader parameter range are certainly needed to see how
generic this is, but they will be deferred to future studies.

##  V Conclusion

We have presented in this paper the results of a systematic study on the
resonance-like structure in CFI. Employing the two-flavor approximation for
simplicity, we have done both linear analysis and nonlinear numerical
simulations. The collision is taken into account in the relaxation
approximation, which is actually exact for the emission/absorption as well as
iso-energetic scatterings of neutrinos. We have always assumed that the
neutrino distributions are homogeneous initially but have considered both
isotropic and anisotropic distributions in momentum space. We have also taken
into account the continuous energy distributions of neutrinos. By changing
rather arbitrarily the number densities or the collision rates for the
antineutrinos, we have produced both configurations in and out of the
resonance-like region freely.

Starting with the linear analysis of the simplest case, i.e., the
monochromatic, homogeneous and isotropic background with $k=0$ perturbations,
we have analytically obtained the dispersion relations both for the isotropy-
preserving and isotropy-breaking modes. Note that the latter has been
overlooked in the literature so far. We have confirmed that the resonance-like
structure shows up at $A\sim\alpha$ (see Eq. 21 for the notations) for the
isotropy-preserving branch, where the two modes, $\omega_{+}$ and $\omega_{-}$
(Eq. 19), come close to each other. We have demonstrated that a similar
feature occurs at $A\sim 3\alpha$ also for the isotropy-breaking branch. In
both cases, the resonance-like peak obtains at
${\color[rgb]{0,0,0}\mathfrak{g}}\sim\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}$
(Eq. 17) but not exactly at
${\color[rgb]{0,0,0}\mathfrak{g}}=\bar{{\color[rgb]{0,0,0}\mathfrak{g}}}$. In
fact, the deviation becomes larger for a greater difference between the
collision rates, $\Gamma$ for neutrino and $\bar{\Gamma}$ for antineutrinos.
The resonance-like region is also broadened in that case.

We have then conducted numerical simulations for the same background setting
to investigate the nonlinear evolutions of the perturbations, which were again
assumed to have $k=0$ and added only to the off-diagonal components of the
density matrix. We have confirmed the exponential growth at the rate given by
the linear analysis in the linear phase, which we have found is followed by
the nonlinear saturation phase, where the flavor coherence levels off and the
distribution functions are settled to new steady states asymptotically. We
have observed a bouncing with large amplitudes only around the resonance-like
peak. Its mechanism is unclear for the moment but we have demonstrated that it
is induced mainly by the diagonal part of the collision term.

We have then proceeded to the non-monochromatic case, in which we assumed that
neutrinos have Fermi-Dirac distributions as their energy spectra. In the
linear analysis for the homogeneous and isotropic background with the $k=0$
perturbation, we have shown that there are again isotropy-preserving and
isotropy-breaking branches and that both of them give resonance-like
structure. We have also demonstrated that the growth rate in the resonance-
like structures is well approximated by the exact formula for the
monochromatic case with an appropriate substitution of variables as long as
the collision rates are not much different between neutrino and antineutrino.
This was pointed out in the previous work [33] for small collision rates. Our
results have extended its validity.

The nonlinear simulations have been done for the same background as for the
linear analysis above. The perturbation was assumed to have $k=0$ and to be
also isotropic. Only the isotropy-preserving mode has been hence calculated.
As expected from the dispersion relation, we have observed that the
perturbation grows exponentially at the same rate for all energies of
neutrinos. We have found that for different energies of neutrinos the
nonlinear saturation phase begins when their flavor coherence becomes of the
similar amplitude to the distribution functions at their energies, $|S(E)|\sim
f(E)$ (Eq. 1), a direct extension of the monochromatic case. We have found
that the saturation level is not much affected by the resonance-like
phenomenon and its main role is to shorten the time it takes to get to the
saturation. We have also seen the bouncing of the flavor coherence after its
peak-out near the resonance-like peak alone again.

We finally conducted the linear analysis either for the $k\neq 0$ perturbation
to the isotropic background or for the $k=0$ perturbation to anisotropic
background configurations. In the former we have shown how the isotropy-
preserving and isotropy-breaking branches at $k=0$ are mixed for $k\neq 0$ to
produce new branches and demonstrated that resonance-like structure occurs
just in the same way for all these modes. We have also found that the CFI is
weaker for modes with $k\neq 0$ than $k=0$.

For the anisotropic background with no electron-lepton-number, or ELN,
crossings, we have shown that resonance-like structure shows up again in the
same way and that the CFI tends to be weaker, albeit slightly, for the
anisotropic background. With an ELN crossing, on the other hand, the growth
rate seems to be set by the fast flavor conversion, or FFC, and the resonance-
like phenomenon in CFI, the region of which tends to overlap with the range of
FFC, shifts the peak position and broadens the range of instability. The
growth rate may be enhanced or reduced, depending on the position in the
unstable range.

There remain many issues to be addressed further. Not to mention, we need to
extend the analysis to three flavors. The relaxation approximation should be
removed to incorporate non-isoenergetic collisions. The parameter regions
considered in this paper are rather limited. For instance, we have invetigated
rather small $k$ in the $k\neq 0$ perturbation although new modes with
different properties may emerge for larger $k$. Interplays of the resonance-
like phenomenon in CFI with FFC in the nonlinear phase should be studied,
probably numerically. Indeed the growth of the isotropy-breaking modes may
induce FFC in that phase. The mechanism of the large-amplitude bouncing in the
resonance-like peak case needs to be understood. The asymptotic state in the
nonlinear saturation phase should be characterized. Eventually we are
interested in what consequences, if any, the resonance-like structure in CFI
may have for core-collapse supernovae and compact object mergers. In fact, the
advection, which is ignored in this paper, occurs in these realistic
situations and the build-up time of flavor instabilities becomes crucially
important [32]. They all will be future works.

##  VI Acknowledgement

J.L. thanks Taiki Morinaga for introducing this field of research to him and
providing well constructed basic codes for dispersion relation analysis. This
work is partially supported by Grants-in-Aid for Scientific Research
(21H01083) and the Grant-in-Aid for Scientific Research on Innovative areas
“Unraveling the History of the Universe and Matter Evolution with Underground
Physics” (19H05811) from the Ministry of Education, Culture, Sports, Science
and Technology (MEXT), Japan. M.Z. is supported by the Japan Society for
Promotion of Science (JSPS) Grant-in-Aid for JSPS Fellows (Grants No.
22J00440) from the Ministry of Education, Culture, Sports, Science, and
Technology (MEXT) in Japan. S.Y. is supported by Institute for Advanced
Theoretical and Experimental Physics, Waseda University, and the Waseda
University Grant for Special Research Projects (project No. 2022C-140).

## References

  * Pantaleone [1992] J. Pantaleone, Dirac neutrinos in dense matter, Phys. Rev. D 46, 510 (1992).
  * Sigl and Raffelt [1993] G. Sigl and G. Raffelt, General kinetic description of relativistic mixed neutrinos, Nuclear Physics 406, 423 (1993).
  * Duan  _et al._ [2010] H. Duan, G. M. Fuller, and Y.-Z. Qian, Collective neutrino oscillations, Annual Review of Nuclear and Particle Science 60, 569 (2010), https://doi.org/10.1146/annurev.nucl.012809.104524 .
  * Chakraborty  _et al._ [2016] S. Chakraborty, R. Hansen, I. Izaguirre, and G. Raffelt, Collective neutrino flavor conversion: Recent developments, Nuclear Physics B 908, 366 (2016), neutrino Oscillations: Celebrating the Nobel Prize in Physics 2015\.
  * Tamborra and Shalgar [2021] I. Tamborra and S. Shalgar, New developments in flavor evolution of a dense neutrino gas, Annual Review of Nuclear and Particle Science 71, 165 (2021), https://doi.org/10.1146/annurev-nucl-102920-050505 .
  * Richers and Sen [2022] S. Richers and M. Sen, Fast flavor transformations (2022).
  * Morinaga [2022] T. Morinaga, Fast neutrino flavor instability and neutrino flavor lepton number crossings, Phys. Rev. D 105, L101301 (2022).
  * Abbar  _et al._ [2019] S. Abbar, H. Duan, K. Sumiyoshi, T. Takiwaki, and M. C. Volpe, On the occurrence of fast neutrino flavor conversions in multidimensional supernova models, Phys. Rev. D 100, 043004 (2019).
  * Nagakura  _et al._ [2019] H. Nagakura, T. Morinaga, C. Kato, and S. Yamada, Fast-pairwise collective neutrino oscillations associated with asymmetric neutrino emissions in core-collapse supernovae, The Astrophysical Journal 886, 139 (2019).
  * Delfan Azari  _et al._ [2020] M. Delfan Azari, S. Yamada, T. Morinaga, H. Nagakura, S. Furusawa, A. Harada, H. Okawa, W. Iwakami, and K. Sumiyoshi, Fast collective neutrino oscillations inside the neutrino sphere in core-collapse supernovae, Phys. Rev. D 101, 023018 (2020).
  * Abbar  _et al._ [2020] S. Abbar, H. Duan, K. Sumiyoshi, T. Takiwaki, and M. C. Volpe, Fast neutrino flavor conversion modes in multidimensional core-collapse supernova models: The role of the asymmetric neutrino distributions, Phys. Rev. D 101, 043016 (2020).
  * Morinaga  _et al._ [2020] T. Morinaga, H. Nagakura, C. Kato, and S. Yamada, Fast neutrino-flavor conversion in the preshock region of core-collapse supernovae, Phys. Rev. Res. 2, 012046 (2020).
  * Glas  _et al._ [2020] R. Glas, H.-T. Janka, F. Capozzi, M. Sen, B. Dasgupta, A. Mirizzi, and G. Sigl, Fast neutrino flavor instability in the neutron-star convection layer of three-dimensional supernova models, Phys. Rev. D 101, 063001 (2020).
  * Abbar [2020] S. Abbar, Searching for fast neutrino flavor conversion modes in core-collapse supernova simulations, Journal of Cosmology and Astroparticle Physics 2020 (05), 027.
  * Capozzi  _et al._ [2021] F. Capozzi, S. Abbar, R. Bollig, and H.-T. Janka, Fast neutrino flavor conversions in one-dimensional core-collapse supernova models with and without muon creation, Phys. Rev. D 103, 063013 (2021).
  * Nagakura  _et al._ [2021] H. Nagakura, A. Burrows, L. Johns, and G. M. Fuller, Where, when, and why: Occurrence of fast-pairwise collective neutrino oscillation in three-dimensional core-collapse supernova models, Phys. Rev. D 104, 083025 (2021).
  * Harada and Nagakura [2022] A. Harada and H. Nagakura, Prospects of fast flavor neutrino conversion in rotating core-collapse supernovae, The Astrophysical Journal 924, 109 (2022).
  * Akaho  _et al._ [2022] R. Akaho, A. Harada, H. Nagakura, W. Iwakami, H. Okawa, S. Furusawa, H. Matsufuru, K. Sumiyoshi, and S. Yamada, Protoneutron star convection simulated with a new general relativistic boltzmann neutrino radiation-hydrodynamics code (2022).
  * Martin  _et al._ [2021] J. D. Martin, J. Carlson, V. Cirigliano, and H. Duan, Fast flavor oscillations in dense neutrino media with collisions, Phys. Rev. D 103, 063001 (2021).
  * Sigl [2022] G. Sigl, Simulations of fast neutrino flavor conversions with interactions in inhomogeneous media, Phys. Rev. D 105, 043005 (2022).
  * Johns and Nagakura [2022] L. Johns and H. Nagakura, Self-consistency in models of neutrino scattering and fast flavor conversion, Phys. Rev. D 106, 043031 (2022).
  * Shalgar and Tamborra [2021] S. Shalgar and I. Tamborra, Change of direction in pairwise neutrino conversion physics: The effect of collisions, Phys. Rev. D 103, 063002 (2021).
  * Kato  _et al._ [2021] C. Kato, H. Nagakura, and T. Morinaga, Neutrino transport with the monte carlo method. ii. quantum kinetic equations, The Astrophysical Journal Supplement Series 257, 55 (2021).
  * Sasaki and Takiwaki [2022] H. Sasaki and T. Takiwaki, A detailed analysis of the dynamics of fast neutrino flavor conversions with scattering effects, Progress of Theoretical and Experimental Physics 2022, 10.1093/ptep/ptac082 (2022), 073E01, https://academic.oup.com/ptep/article-pdf/2022/7/073E01/44400755/ptac082.pdf .
  * Hansen  _et al._ [2022] R. S. L. Hansen, S. Shalgar, and I. Tamborra, Enhancement or damping of fast neutrino flavor conversions due to collisions, Phys. Rev. D 105, 123003 (2022).
  * Kato and Nagakura [2022] C. Kato and H. Nagakura, Effects of energy-dependent scatterings on fast neutrino flavor conversions, Physical Review D 106, 10.1103/physrevd.106.123013 (2022).
  * Padilla-Gay  _et al._ [2022] I. Padilla-Gay, I. Tamborra, and G. G. Raffelt, Neutrino fast flavor pendulum. ii. collisional damping, Phys. Rev. D 106, 103031 (2022).
  * Capozzi  _et al._ [2019] F. Capozzi, B. Dasgupta, A. Mirizzi, M. Sen, and G. Sigl, Collisional triggering of fast flavor conversions of supernova neutrinos, Phys. Rev. Lett. 122, 091101 (2019).
  * Johns [2021] L. Johns, Collisional flavor instabilities of supernova neutrinos (2021).
  * Johns and Xiong [2022] L. Johns and Z. Xiong, Collisional instabilities of neutrinos and their interplay with fast flavor conversion in compact objects, Phys. Rev. D 106, 103029 (2022).
  * Lin and Duan [2022] Y.-C. Lin and H. Duan, Collision-induced flavor instability in dense neutrino gases with energy-dependent scattering (2022).
  * Xiong  _et al._ [2023] Z. Xiong, M.-R. Wu, G. Martínez-Pinedo, T. Fischer, M. George, C.-Y. Lin, and L. Johns, Evolution of collisional neutrino flavor instabilities in spherically symmetric supernova models, Phys. Rev. D 107, 083016 (2023).
  * Xiong  _et al._ [2022] Z. Xiong, L. Johns, M.-R. Wu, and H. Duan, Collisional flavor instability in dense neutrino gases (2022).
  * Airen  _et al._ [2018] S. Airen, F. Capozzi, S. Chakraborty, B. Dasgupta, G. Raffelt, and T. Stirner, Normal-mode analysis for collective neutrino oscillations, Journal of Cosmology and Astroparticle Physics 2018 (12), 019.
  * Izaguirre  _et al._ [2017] I. Izaguirre, G. Raffelt, and I. Tamborra, Fast pairwise conversion of supernova neutrinos: A dispersion relation approach, Phys. Rev. Lett. 118, 021101 (2017).
  * Zaizen and Nagakura [2022] M. Zaizen and H. Nagakura, Simple method for determining asymptotic states of fast neutrino-flavor conversion, arXiv e-prints , arXiv:2211.09343 (2022), arXiv:2211.09343 [astro-ph.HE] .
  * Shalgar and Tamborra [2019] S. Shalgar and I. Tamborra, On the occurrence of crossings between the angular distributions of electron neutrinos and antineutrinos in the supernova core, The Astrophysical Journal 883, 80 (2019).
  * Shalgar and Tamborra [2022] S. Shalgar and I. Tamborra, Neutrino flavor conversion, advection, and collisions: The full solution (2022).