#  Neutrino effective potential and damping in a fermion and scalar background
in the resonance region

José F<EMAIL_ADDRESS>
Laboratory of Theoretical Physics, Department of Physics
University of Puerto Rico, Río Piedras, Puerto Rico 00936
Sarira<EMAIL_ADDRESS>
Instituto de Ciencias Nucleares
Universidad Nacional Autónoma de Mexico
Circuito Exterior, C. U.
A. Postal 70-543, 04510 Mexico DF, Mexico

(January 2022)

###### Abstract

We consider the propagation of a neutrino or an antineutrino in a medium
composed of fermions ($f$) and scalars ($\phi$) interacting via a Yukawa-type
coupling of the form $\bar{f}\nu\phi$, for neutrino energies at which the
processes like $\nu+\phi\leftrightarrow f$ or
$\nu+\bar{f}\leftrightarrow\bar{\phi}$, and the corresponding ones for the
antineutrino, are kinematically accessible. The relevant energy values are
around $|m^{2}_{\phi}-m^{2}_{f}|/2m_{\phi}$ or
$|m^{2}_{\phi}-m^{2}_{f}|/2m_{f}$, where $m_{\phi}$ and $m_{f}$ are the masses
of $\phi$ and $f$, respectively. We refer to either one of these regions as a
_resonance energy range_. Near these points, the one-loop formula for the
neutrino self-energy has a singularity. From a technical point of view, that
feature is indicative that the self-energy acquires an imaginary part, which
is associated with damping effects and cannot be neglected, while the integral
formula for the real part must be evaluated using the principal value of the
integral. We carry out the calculations explicitly for some cases that allow
us to give analytic results. Writing the dispersion relation in the form
$\omega=\kappa+V_{\text{eff}}-i\gamma/2$, we give the explicit formulas for
$V_{\text{eff}}$ and $\gamma$ for the cases considered. When the neutrino
energy is either much larger or much smaller than the resonance energy,
$V_{\text{eff}}$ reduces to the effective potential that has been already
determined in the literature in the high or low momentum regime, respectively.
The virtue of the formula we give for $V_{\text{eff}}$ is that it is valid
also in the _resonance energy range_ , which is outside the two limits
mentioned. As a guide to possible applications we give the relevant formulas
for $V_{\text{eff}}$ and $\gamma$, and consider the solution to the
oscillation equations including the damping term, in a simple two-generation
case.

##  1 Introduction and Summary

As is well known, the properties of neutrinos that propagate through a medium
differ from those in the vacuum. In particular, the energy-momentum for
massless neutrinos $\omega=\kappa$, where $\omega$ is the energy and $\kappa$
the magnitude of the momentum vector, is not valid in the medium[1, 2]. The
modifications of the neutrino dispersion relation can be represented in terms
of an index of refraction, or more suitable for our purposes, in terms of an
effective potential ($V_{\text{eff}}$) and a damping ($\gamma$), by writing it
in the generic form

$\omega=\kappa+V_{\text{eff}}-i\gamma/2\,.$ (1.1)

It is now well established that an efficient method to determine the
dispersion relation, is to compute $V_{\text{eff}}$ and $\gamma$ from the
calculation of the neutrino thermal self-energy[3, 4, 5, 6] in the framework
of thermal field theory[7].

In several models and extensions of the standard electro-weak theory the
neutrinos interact with scalar particles ($\phi$) and fermions ($f$) via a
coupling of the form $\bar{f}_{R}\nu_{L}\phi$ or just with neutrinos
$\bar{\nu}^{c}_{R}\nu_{L}\phi$. Couplings of the latter form have been
explored recently in various contexts[8, 9, 10, 11, 12, 13, 14, 15, 16]. Such
couplings produce additional contributions beyond the standard ones to the
neutrino effective potential when the neutrino propagates in a neutrino
background, as it occurs in the environment of a supernova, where the effect
leads to the collective neutrino oscillations and related phenomena(see for
example Refs. [17] and [18] and the works cited therein), or in the hot plasma
of the Early-Universe before the neutrinos decouple[19, 20]. Couplings of the
form $\bar{f}_{R}\nu_{L}\phi$ produce additional contributions to the neutrino
effective potential when the neutrino propagates in a background of $\phi$ and
$f$ particles and their possible effects have been considered in the context
of Dark Matter-neutrino interactions[21, 22, 23, 24, 25, 26, 27, 28, 29]. More
recently it has been pointed out that observable effects of such scalar
interactions, although precluded in terrestrial experiments, are still
possible in future solar and supernovae neutrino data, and in cosmological
observations such as cosmic microwave background and big bang nucleosynthesis
data[30].

Motivated by these developments, we carried out in previous work a systematic
calculation of the neutrino effective potential in such models[31]. We
considered various cases, depending on the magnitude of $\kappa$ relative to
other parameters such as the masses of the particles and the temperature of
the background. In the limit of small $\kappa$, the effective potential
becomes independent of $\kappa$ and has a form that is reminiscent of the
Wolfenstein potential[1]. In the opposite limit (relatively large $\kappa$),
the effective potential has a term proportional to $1/\kappa$ that mimics a
contribution to the neutrino mass[32].

The main point that is relevant to the present work, is that in the
intermediate $\kappa$ region, to be defined precisely below, neither limiting
case is a valid approximation to the effective potential. From a physical
point of view, there is a region of neutrino energies at which the processes
like $\nu+\phi\leftrightarrow f$ or $\nu+\bar{f}\leftrightarrow\bar{\phi}$,
and the corresponding ones for the antineutrino, become kinematically
accessible. The relevant energy values are around
$|m^{2}_{\phi}-m^{2}_{f}|/2m_{\phi}$ or $|m^{2}_{\phi}-m^{2}_{f}|/2m_{f}$, to
which we refer as a _resonance energy range_. In those ranges, the one-loop
integral formula for the neutrino self-energy has a singularity, as has been
emphasized recently in Ref. [33].

From a technical point of view, the singularity is indicative of two things.
Firstly, at those points the self-energy acquires an imaginary part that
cannot be neglected. The imaginary part of the self-energy is associated with
damping effects, and determines the damping term $\gamma$ in the dispersion
relation. A systematic calculation of the damping terms was carried out in
Ref. [34].

Secondly, and what is our main observation here, the effective potential,
which is determined from the real (dispersive) part of the self-energy, must
be evaluated using the principal value of the integral formula for the self-
energy. The principal value prescription allows us to give a well-defined
meaning to (the real part of) the integral for values of $\kappa$ around the
singularities. Our purpose in this work is to carry out the calculation of the
effective potential in the resonance regions using the strategy just
explained. Writing the dispersion relation in the form given in Eq. (1.1), we
give the explicit formulas for $V_{\text{eff}}$ and $\gamma$ for some cases
that allow us to give analytic results, and indications for carrying out
extensions and generalizations to other cases of interest. When the neutrino
energy is either much larger or much smaller than the resonance energy,
$V_{\text{eff}}$ reduces to the effective potential that has been already
determined in the literature in the high or low momentum regime, respectively.
The virtue of the formula we give for $V_{\text{eff}}$ is that it is valid
also in the _resonance energy range_ , which is outside the two limits
mentioned. As a guide to possible applications to neutrino oscillations in the
case that the neutrino energy is in the resonance region, we give the relevant
formulas for the $V_{\text{eff}}$ and $\gamma$ terms that enter in the
oscillation equations in a simple two-generation case, and consider their
solution including the damping term.

In Section 2 we summarize our notation and conventions, and the context in
which we carry out the calculations. In Section 3 we calculate the effective
potential, paying special attention to the contribution from the resonance
terms, which are expressed as an integral over the background particle
momentum distribution functions. We consider in detail the evaluation of the
relevant integral for the case that the resonance term is the fermion
background contribution, and for concreteness we give the explicit formulas
for the case of a non-relativistic and degenerate Fermi gas. Such formulas can
be particularly useful for considering the implications and/or setting limits
on the neutrino interactions with light particle dark matter backgrounds from
their effect on the phenomenology in reactor, solar, atmospheric, and
accelerator experiments. In Section 4 we consider the damping term. We discuss
some generalizations and possible extensions of the results in Section 5.1,
and the two-generation example case mentioned above in Section 5.2.

##  2 Preliminaries

In this section we review the context of the present work and state the
problem on which we focus.

###  2.1 Context

For definiteness we consider only one neutrino flavor coupling to the fermion
and scalar, which we denote simply by $\nu$, and write

$L^{(f\nu\phi)}=\lambda\bar{f}_{R}\nu_{L}\phi+h.c\,.$ (2.1)

We denote by $k^{\mu}$ the momentum four-vector of the propagating neutrino
and by $u^{\mu}$ the velocity four-vector of the background medium. In the
background medium’s own rest frame, $u^{\mu}$ takes the form

$u^{\mu}=(1,\vec{0})\,,$ (2.2)

and in this frame we write

$k^{\mu}=(\omega,\vec{\kappa})\,.$ (2.3)

Since we are considering only one background medium, we can take it to be at
rest and therefore we adopt Eqs. (2.2) and (2.3) throughout. For completeness,
we briefly review and borrow from Refs. [31] and [34] the formulas that we
will use to determine the dispersion relations from the self-energy
calculation. We remind that the calculations are based on the application of
the real-time Thermal Field Theory methods.

The neutrino dispersion relation is determined by the solution of the equation

$(k\hbox to0.0pt{\hss/}-\Sigma_{\text{eff}})\psi_{L}(k)=0\,,$ (2.4)

where $\Sigma_{\text{eff}}$ is the neutrino thermal self-energy.
$\Sigma_{\text{eff}}$ can be decomposed into its dispersive ($\Sigma_{r}$) and
absorptive ($\Sigma_{i}$) parts,

$\Sigma_{\text{eff}}=\Sigma_{r}+i\Sigma_{i}\,.$ (2.5)

$\Sigma_{r}$ is given by the real (dispersive) part of the 11 element of the
neutrino thermal self-energy matrix, while $\Sigma_{i}$ is determined from the
12 element.

The chirality of the neutrino interactions implies that $\Sigma_{\text{eff}}$
has the form

$\Sigma_{\text{eff}}=V\hbox to0.0pt{\hss/}L\,,$ (2.6)

where $L=\frac{1}{2}(1-\gamma_{5})$. Corresponding to the decomposition of
$\Sigma_{\text{eff}}$ in Eq. (2.5),

$V^{\mu}=V^{\mu}_{r}+iV^{\mu}_{i}\,,$ (2.7)

where $V^{\mu}_{r}$ and $V^{\mu}_{i}$ are real. $V_{r}$ and $V_{i}$ are
functions of $\omega$ and $\vec{\kappa}$, but to simplify the notation we omit
writing the arguments unless it is necessary to indicate them.

Equation (2.4) has two solutions. Denoting them by $\omega^{(\lambda)}$
($\lambda=\pm$), they are determined by the equation

$\omega^{(\lambda)}=V^{0}(\omega^{(\lambda)},\vec{\kappa})+\lambda[(\vec{\kappa}-\vec{V}(\omega^{(\lambda)},\vec{\kappa}))\cdot(\vec{\kappa}-\vec{V}(\omega^{(\lambda)},\vec{\kappa}))]^{1/2}\,,$
(2.8)

or to lowest order,

$\omega^{(\lambda)}=V^{0}(\omega^{(\lambda)},\vec{\kappa})+\lambda(\kappa-\hat{\kappa}\cdot\vec{V}(\omega^{(\lambda)},\vec{\kappa}))\,.$
(2.9)

The neutrino ($\omega^{(\nu)}$) and antineutrino
($\omega^{(\bar{\nu})}$)dispersion relations are identified as

$\displaystyle\omega^{(\nu)}(\vec{\kappa})$ $\displaystyle=$
$\displaystyle\omega^{(+)}(\vec{\kappa})\,,$
$\displaystyle\omega^{(\bar{\nu})}(\vec{\kappa})$ $\displaystyle=$
$\displaystyle\left(-\omega^{(-)}(-\vec{\kappa})\right)^{\ast}\,.$ (2.10)

Decomposing $\omega^{(\nu)}$ and $\omega^{(\bar{\nu})}$ in terms of their real
and imaginary parts in the form ($x=\nu,\bar{\nu}$),

$\omega^{(x)}=\omega^{(x)}_{r}-\frac{i}{2}\gamma^{(x)}\,,$ (2.11)

and assuming that it is a valid approximation to set

$\displaystyle\gamma^{(x)}_{i}$ $\displaystyle\ll$
$\displaystyle|\omega^{(x)}_{r}|\,,$ $\displaystyle\omega^{(x)}_{r}$
$\displaystyle\simeq$ $\displaystyle\kappa\,,$ (2.12)

Eq. (2.9) gives, for the real part

$\displaystyle\omega^{(\nu)}_{r}$ $\displaystyle=$
$\displaystyle\kappa+V^{(\nu)}_{\text{eff}}(\vec{\kappa})\,,$
$\displaystyle\omega^{(\bar{\nu})}_{r}$ $\displaystyle=$
$\displaystyle\kappa+V^{(\bar{\nu})}_{\text{eff}}(\vec{\kappa})\,,$ (2.13)

while for the imaginary part

$\displaystyle-\frac{1}{2}\gamma^{(\nu)}$ $\displaystyle=$
$\displaystyle\frac{n\cdot
V_{i}(\kappa,\vec{\kappa})}{1-n\cdot\left.\frac{\partial
V_{r}(\omega,\vec{\kappa})}{\partial\omega}\right|_{\omega=\kappa}}\,,$
$\displaystyle-\frac{1}{2}\gamma^{(\bar{\nu})}$ $\displaystyle=$
$\displaystyle\frac{n\cdot
V_{i}(-\kappa,-\vec{\kappa})}{1-n\cdot\left.\frac{\partial
V_{r}(\omega,-\vec{\kappa})}{\partial\omega}\right|_{\omega=-\kappa}}\,,$
(2.14)

where

$\displaystyle V^{(\nu)}_{\text{eff}}(\vec{\kappa})$ $\displaystyle=$
$\displaystyle n\cdot V_{r}(\kappa,\vec{\kappa})\,,$ $\displaystyle
V^{(\bar{\nu})}_{\text{eff}}(\vec{\kappa})$ $\displaystyle=$
$\displaystyle-n\cdot V_{r}(-\kappa,-\vec{\kappa})\,,$ (2.15)

with

$n^{\mu}=(1,\hat{\kappa})\,.$ (2.16)

In those cases in which the correction due to the $n\cdot\partial
V_{r}(\omega,\vec{\kappa})/\partial\omega$ in the denominator can be
neglected, the formulas in Eq. (2.1) simplify to

$\displaystyle-\frac{\gamma^{(\nu)}(\vec{\kappa})}{2}$ $\displaystyle=$
$\displaystyle n\cdot V_{i}(\kappa,\vec{\kappa})\,,$
$\displaystyle-\frac{\gamma^{(\bar{\nu})}(\vec{\kappa})}{2}$ $\displaystyle=$
$\displaystyle n\cdot V_{i}(-\kappa,-\vec{\kappa})\,,$ (2.17)

which are the ones we will use here, borrowing from the work in Ref. [34].

###  2.2 Statement of the problem

To state the problem in concrete terms and set the stage for the work that
follows, we recall (see, e.g, Ref. [31]) the following expression for the
background-dependent part of the 11 element of the thermal self-energy matrix
in the $f$ and $\phi$ background,

$\Sigma_{11}(k)=\Sigma^{(f)}_{11}(k)+\Sigma^{(\phi)}_{11}(k)\,,$ (2.18)

where333We take the opportunity to mention that by an abuse in notation the
symbols $\eta_{F}(p,\alpha_{f})$ and $\eta_{B}(p,\alpha_{\phi})$ used in Eqs
(20) and (21) in Ref. [31] are the same as the $\eta_{f}(p)$ and
$\eta_{\phi}(p)$ defined in Eq (17) of that reference, and reproduced below in
Eq. (2.21).

$\displaystyle\Sigma^{(f)}_{11}$ $\displaystyle=$
$\displaystyle-|\lambda|^{2}\int\frac{d^{4}p}{(2\pi)^{3}}\frac{p\hbox
to0.0pt{\hss/}L}{(p-k)^{2}-m^{2}_{\phi}+i\epsilon}\delta(p^{2}-m^{2}_{f})\eta_{f}(p)\,,$
(2.19) $\displaystyle\Sigma^{(\phi)}_{11}$ $\displaystyle=$
$\displaystyle|\lambda|^{2}\int\frac{d^{4}p}{(2\pi)^{3}}\frac{(p\hbox
to0.0pt{\hss/}+k\hbox
to0.0pt{\hss/})L}{(p+k)^{2}-m^{2}_{f}+i\epsilon}\delta(p^{2}-m^{2}_{\phi})\eta_{\phi}(p)\,.$
(2.20)

Using the label $x$ to stand for either $f$ or $\phi$, the functions
$\eta_{x}(p)$ are given by

$\eta_{x}(p)=\theta(p^{0})f_{x}(p^{0})+\theta(-p^{0})f_{\bar{x}}(-p^{0})\,,$
(2.21)

where $f_{x,\bar{x}}(p^{0})$ are the equilibrium momentum distribution
functions of the background particles and antiparticles,

$\displaystyle f_{f,\bar{f}}(p^{0})$ $\displaystyle=$
$\displaystyle\frac{1}{e^{\beta p^{0}\mp\alpha_{f}}+1}\,,$ $\displaystyle
f_{\phi,\bar{\phi}}(p^{0})$ $\displaystyle=$ $\displaystyle\frac{1}{e^{\beta
p^{0}\mp\alpha_{\phi}}-1}\,,$ (2.22)

where $\beta=1/T$ and $\alpha_{x}=\beta\mu_{x}$, with $T$ being the
temperature and $\mu_{x}$ the chemical potentials.

To be precise, we mention that in Eq. (2.18) we are neglecting the term that
involves the product of the two thermal parts of the propagators, which does
not contribute to the real part of $\Sigma_{11}$. Thus, going back to Eq.
(2.5), the dispersive part $\Sigma_{r}$ is given by

$\Sigma_{r}=\Sigma^{(f)}_{r}+\Sigma^{(\phi)}_{r}\,,$ (2.23)

where

$\displaystyle\Sigma^{(f)}_{r}$ $\displaystyle=$
$\displaystyle-|\lambda|^{2}\int\frac{d^{4}p}{(2\pi)^{3}}\frac{p\hbox
to0.0pt{\hss/}L}{(p-k)^{2}-m^{2}_{\phi}}\delta(p^{2}-m^{2}_{f})\eta_{f}(p)\,,$
(2.24) $\displaystyle\Sigma^{(\phi)}_{r}$ $\displaystyle=$
$\displaystyle|\lambda|^{2}\int\frac{d^{4}p}{(2\pi)^{3}}\frac{(p\hbox
to0.0pt{\hss/}+k\hbox
to0.0pt{\hss/})L}{(p+k)^{2}-m^{2}_{f}}\delta(p^{2}-m^{2}_{\phi})\eta_{\phi}(p)\,.$
(2.25)

In Eqs. (2.24) and (2.25), and the integrals that follow, the integrations are
to be interpreted in the sense of their principal value.

Carrying out the integral over $p^{0}$, we obtain

$\displaystyle n\cdot V_{r}(\omega,\vec{\kappa})$ $\displaystyle=$
$\displaystyle
v_{f}(\omega,\vec{\kappa})+v_{\bar{f}}(\omega,\vec{\kappa})+v_{\phi}(\omega,\vec{\kappa})+v_{\bar{\phi}}(\omega,\vec{\kappa})\,,$
(2.26)

where

$\displaystyle v_{f}(\omega,\vec{\kappa})$
$\displaystyle=-|\lambda|^{2}\int\frac{d^{3}p}{(2\pi)^{3}2E_{f}}(n\cdot
p)\frac{f_{f}(\vec{p})}{D_{f}(k,p)}\,,$ (2.27a) $\displaystyle
v_{\bar{f}}(\omega,\vec{\kappa})$
$\displaystyle=|\lambda|^{2}\int\frac{d^{3}p}{(2\pi)^{3}2E_{f}}(n\cdot
p)\frac{f_{\bar{f}}(\vec{p})}{D_{f}(k,-p)}\,,$ (2.27b) $\displaystyle
v_{\phi}(\omega,\vec{\kappa})$
$\displaystyle=|\lambda|^{2}\int\frac{d^{3}p}{(2\pi)^{3}2E_{\phi}}\frac{(n\cdot
p+(\omega-\kappa))f_{\phi}(\vec{p})}{D_{\phi}(k,p)}\,,$ (2.27c) $\displaystyle
v_{\bar{\phi}}(\omega,\vec{\kappa})$
$\displaystyle=|\lambda|^{2}\int\frac{d^{3}p}{(2\pi)^{3}2E_{\phi}}\frac{(-n\cdot
p+(\omega-\kappa))f_{\bar{\phi}}(\vec{p})}{D_{\phi}(k,-p)}\,,$ (2.27d)

with

$E_{x}=\sqrt{{\vec{p}\,}^{2}+m^{2}_{x}}\,,\qquad(x=f,\phi)\,,$ (2.28)
$\displaystyle D_{f}(k,p)$ $\displaystyle=$
$\displaystyle(p-k)^{2}-m^{2}_{\phi}=k^{2}-2p\cdot
k+m^{2}_{f}-m^{2}_{\phi}\,,$ $\displaystyle D_{\phi}(k,p)$ $\displaystyle=$
$\displaystyle(p+k)^{2}-m^{2}_{f}=k^{2}+2p\cdot k+m^{2}_{\phi}-m^{2}_{f}\,,$
(2.29)

and

$n\cdot p=E_{x}(1-\hat{\kappa}\cdot\vec{v}_{x})\,.$ (2.30)

In Eq. (2.30) $\vec{v}_{x}$ stands for the velocity of the background
particle.

To bring out the issue that we want to address, consider for example the
contribution from the $\bar{f}$ background, and suppose that the conditions
are such that it can be treated in the non-relativistic limit. Then
approximating

$p^{\mu}\rightarrow(m_{f},\vec{0})\,,$ (2.31)

in the integrand, $v_{\bar{f}}(\kappa,\vec{\kappa})$ is inversely proportional
to

$\left.D_{f}(k,-p)\right|_{\omega=\kappa}\simeq
2m_{f}\omega-(m^{2}_{\phi}-m^{2}_{f})\,.$ (2.32)

Identifying the effective potential by Eq. (2.1), in the low momentum (_heavy
background_) limit this gives a momentum-independent contribution to the
effective potential reminiscent of the standard Wolfenstein term. In the
opposite limit, the high momentum (or _light background_) limit this gives a
term proportional to $1/\kappa$ that mimics a contribution to the neutrino
mass[32]. But in the intermediate region the expression is not valid and
actually undefined at

$\omega\sim\frac{m^{2}_{\phi}-m^{2}_{f}}{2m_{f}}\,.$ (2.33)

If $m_{\phi}>m_{f}$, physically this feature reflects the fact that in that
regime the process $\nu+\bar{f}\rightarrow\bar{\phi}$ is kinematically
accessible. In some cases the singularity appears for $\omega$ negative, which
corresponds to the antineutrino dispersion relation. This is the case, in the
example above, if $m_{f}>m_{\phi}$, and in that case the singularity
corresponds to the process $\bar{f}\rightarrow\bar{\nu}+\bar{\phi}$. Similar
considerations apply to the $v_{f}$ and the $\phi$ background terms. An
exhaustive list of the various possibilities is summarized in Table 1.

| $v_{f}$ | $v_{\bar{f}}$ | $v_{\phi}$ | $v_{\bar{\phi}}$
---|---|---|---|---
$m_{\phi}>m_{f}$ | $-\Omega_{f}$ | $\Omega_{f}$ | $-\Omega_{\phi}$ | $\Omega_{\phi}$
| $\bar{\nu}+f\rightarrow\phi$ | $\nu+\bar{f}\rightarrow\bar{\phi}$ | $\phi\rightarrow\bar{\nu}+f$ | $\bar{\phi}\rightarrow\nu+\bar{f}$
$m_{f}>m_{\phi}$ | $\Omega_{f}$ | $-\Omega_{f}$ | $\Omega_{\phi}$ | $-\Omega_{\phi}$
| $f\rightarrow\nu+\phi$ | $\bar{f}\rightarrow\bar{\nu}+\bar{\phi}$ | $\nu+\phi\rightarrow f$ | $\bar{\nu}+\bar{\phi}\rightarrow\bar{f}$
Table 1:  List of possible resonance conditions in each background
contribution and the associated physical processes. The parameters
$\Omega_{x}$ are defined as $\Omega_{x}=|m^{2}_{\phi}-m^{2}_{f}|/2m_{x}$ (for
$x=f,\phi)$, and they give the value of $\omega$ for which the indicated $v$
term, given in Eq. (2.27), is undefined, in the sense discussed in the text.

The bottom line is that near the resonance ranges indicated in the Table 1,
the integrals in Eq. (2.27) must be handled following the principal value
prescription, and approximations such as those we have indicated in Eqs.
(2.31) and (2.32), which are commonly employed, are not valid in the cases we
are considering.

Moreover, in those energy ranges, the corresponding damping term is not
negligible. This can be seen from the calculation of the imaginary part of the
self-energy, or equivalently $V^{\mu}_{i}$, in Ref. [34]. We will borrow the
results of those calculations here without further ado. But regarding $V_{r}$,
our proposal is to go back to Eq. (2.27) and evaluate those terms in a
systematic way that is valid through the entire neutrino energy range.

##  3 Neutrino effective potential in the $\Omega_{f}$ region

For definiteness, we consider first the neutrino case in detail. To be clear
and precise, in what follows we assume

$m_{\phi}>m_{f}\,,$ (3.1)

throughout. The opposite case can be treated in a similar way by making
appropriate changes.

According to Table 1, the $v_{\bar{f}}$ and $v_{\bar{\phi}}$ terms have a
resonance for $\omega=\Omega_{f}$ and $\omega=\Omega_{\phi}$, respectively,
which we write in the form

$\displaystyle\Omega_{f}$ $\displaystyle=$
$\displaystyle\frac{\Delta^{2}_{\phi f}}{2m_{f}}\,,$
$\displaystyle\Omega_{\phi}$ $\displaystyle=$
$\displaystyle\frac{\Delta^{2}_{\phi f}}{2m_{\phi}}\,,$ (3.2)

where

$\Delta^{2}_{\phi f}=m^{2}_{\phi}-m^{2}_{f}\,.$ (3.3)

We assume that $m_{f}$ and $m_{\phi}$ are significantly different, such that
$\Omega_{f,\phi}$ are sufficiently far apart and the two resonance regions do
not overlap. For the purpose of evaluating the integrals this assumption is
not strictly necessary, but the physical picture is conceptually clearer if we
adopt it. Under this assumption we can consider each region separately. Thus
we consider first the region

$\omega\sim\kappa\sim\Omega_{f}\,,$ (3.4)

which is the resonance region of $v_{\bar{f}}(\omega,\vec{\kappa})$.

From Eqs. (2.1) and (2.26) we then have

$V^{(\nu)}_{\text{eff}}(\vec{\kappa})=U+r\,,$ (3.5)

where

$U\equiv v_{\bar{f}}(\kappa,\vec{\kappa})\,,$ (3.6)

and

$r=v_{f}(\kappa,\vec{\kappa})+v_{\phi}(\kappa,\vec{\kappa})+v_{\bar{\phi}}(\kappa,\vec{\kappa})\,.$
(3.7)

As already mentioned, the evaluation of $r$ is straightforward. For example,
let us consider the case that the $f$ background can be treated in the non-
relativistic limit. In this case $v_{f}(\omega,\vec{\kappa})$ reduces to

$v_{f}(\omega,\vec{\kappa})=\frac{\frac{1}{8}|\lambda|^{2}n_{f}}{m_{f}(\omega+\Omega_{f})}\,.$
(3.8)

Since we are considering the case $m_{\phi}>m_{f}$, the $\phi$ and
$\bar{\phi}$ backgrounds must be considered in the non-relativistic limit as
well. Thus in this case,

$\displaystyle v_{\phi}(\omega,\vec{\kappa})$
$\displaystyle=\frac{\frac{1}{4}|\lambda|^{2}n_{\phi}}{m_{\phi}(\omega+\Omega_{\phi})}\,,$
(3.9a) $\displaystyle v_{\bar{\phi}}(\omega,\vec{\kappa})$
$\displaystyle=\frac{\frac{1}{4}|\lambda|^{2}n_{\bar{\phi}}}{m_{\phi}(\omega-\Omega_{\phi})}\,.$
(3.9b)

Since we are considering $m_{\phi}>m_{f}$ and assuming that these masses are
such that the resonance regions $\omega\sim\Omega_{f}$ and
$\omega\sim\Omega_{\phi}$ are well separated, for the case of neutrino
propagation near $\omega\sim\Omega_{f}$ we can put $\omega\gg\Omega_{\phi}$ in
Eq. (3.9). Thus,

$\displaystyle v_{\phi}(\omega,\vec{\kappa})$ $\displaystyle\simeq$
$\displaystyle\frac{\frac{1}{4}|\lambda|^{2}n_{\phi}}{m_{\phi}\omega}\,,$
$\displaystyle v_{\bar{\phi}}(\omega,\vec{\kappa})$ $\displaystyle\simeq$
$\displaystyle\frac{\frac{1}{4}|\lambda|^{2}n_{\bar{\phi}}}{m_{\phi}\omega}\,,$
(3.10)

and therefore

$r=|\lambda|^{2}\left[\frac{n_{f}}{8m_{f}(\kappa+\Omega_{f})}+\frac{(n_{\phi}+n_{\bar{\phi}})}{4m_{\phi}\kappa}\right]\,.$
(3.11)

For later reference, it is useful to record that for $\omega$ and $\kappa$
away from the resonance region, $v_{\bar{f}}$ is given by a formula analogous
to those quoted above for the other potential terms,

$v_{\bar{f}}(\omega,\vec{\kappa})=\frac{\frac{1}{8}|\lambda|^{2}n_{\bar{f}}}{m_{f}(\omega-\Omega_{f})}=\frac{\frac{1}{8}|\lambda|^{2}n_{\bar{f}}}{m_{f}}\left\\{\begin{array}[]{cc}\frac{1}{\omega}&(\omega\gg\Omega_{f})\\\
-\frac{1}{\Omega_{f}}&(\omega\ll\Omega_{f})\,.\end{array}\right.$ (3.12)

Regarding the damping terms defined in Eqs. (2.11) and (2.1) we can borrow
literally the results given in Ref. [34].

###  3.1 Solution

Away from the kinematic points where $D_{f}(k,-p)$ does not vanish, the
procedure of taking the principal value is not necessary. But if the
kinematics is such that the integration covers the point at which
$D_{f}(k,-p)=0$, the principal value operation defines the integral around
that point.

We assume the $\bar{f}$ gas can be treated in the non-relativistic (NR) limit.
Therefore we take

$p^{\mu}=(m_{f},\vec{p})\,.$ (3.13)

Then doing the angular integral, remembering the principal value prescription,
we get

$v_{\bar{f}}(\omega,\vec{\kappa})=\frac{|\lambda|^{2}}{8\pi^{2}}\int
dp\;p^{2}\frac{f_{\bar{f}}(p)}{2p\kappa}\log\left|\frac{k^{2}+2m_{f}\omega+2p\kappa-\Delta^{2}_{\phi
f}}{k^{2}+2m_{f}\omega-2p\kappa-\Delta^{2}_{\phi f}}\right|\,,$ (3.14)

Further, we put

$k^{2}=(\omega+\kappa)(\omega-\kappa)\rightarrow 2\kappa(\omega-\kappa)\,,$
(3.15)

so that

$k^{2}+2m_{f}\omega\pm 2p\kappa-\Delta^{2}_{\phi
f}=2\kappa\left[\left(1+\frac{m_{f}}{\kappa}\right)(\omega-\kappa)\pm
p+\frac{m_{f}}{\kappa}(\kappa-\Omega_{f})\right]\,,$ (3.16)

and therefore

$v_{\bar{f}}(\omega,\vec{\kappa})=\frac{|\lambda|^{2}}{16\pi^{2}\kappa}\int
dp\;pf_{\bar{f}}(p)\log\left|\frac{\omega-\kappa+b+ap}{\omega-\kappa+b-ap}\right|\,,$
(3.17)

with

$\displaystyle a$ $\displaystyle=$
$\displaystyle\frac{1}{1+\frac{m_{f}}{\kappa}}\,,$ $\displaystyle b$
$\displaystyle=$
$\displaystyle\frac{\kappa-\Omega_{f}}{1+\frac{\kappa}{m_{f}}}\,.$ (3.18)

Our next job is to evaluate the integral

$I(\omega,\kappa)=\int
dp\;pf_{\bar{f}}(p)\log\left|\frac{\omega-\kappa+b+ap}{\omega-\kappa+b-ap}\right|\,,$
(3.19)

which is of the form encountered in the original calculation by Weldon[3] and
similar calculations of the fermion self-energy in various physical
contexts[35].

###  3.2 Evaluation of $v_{\bar{f}}$ for a Fermi gas

For definiteness, we consider the case in which the $\bar{f}$ background can
be treated in the completely degenerate limit. We then write

$v_{\bar{f}}(\omega,\vec{\kappa})=\frac{|\lambda|^{2}}{16\pi^{2}\kappa}I_{F}(\omega,\kappa)\,,$
(3.20)

where

$I_{F}(\omega,\kappa)=\int^{p_{F}}_{0}dp\;p\log\left|\frac{p+A}{p-A}\right|\,,$
(3.21)

with

$A=\frac{\omega-\kappa+b}{a}=\omega-\kappa+\frac{m_{f}}{\kappa}(\omega-\Omega_{f})\,.$
(3.22)

A straightforward evaluation of the integral in Eq. (3.21) leads to

$I_{F}(\omega,\kappa)=\frac{1}{2}(p^{2}_{F}-A^{2})\log\left|\frac{p_{F}+A}{p_{F}-A}\right|+Ap_{F}\,.$
(3.23)

We can now use this to find the expression for the effective potential (in a
NR Fermi gas) which, we repeat, is valid for the entire range of the neutrino
momentum. $v_{\bar{f}}(\omega,\vec{\kappa})$ is given by Eq. (3.20), with $A$
defined in Eq. (3.22).

###  3.3 Formula for $v_{\bar{f}}(\kappa,\vec{\kappa})$

First of all, as a check, let us consider the limit of small $p_{F}$.
Expanding the $\log$ function up to terms of order $p^{3}_{F}$, it follows
that

$I_{F}=\frac{2p^{3}_{F}}{3A}\,.$ (3.24)

From Eq. (3.20), this gives

$v_{\bar{f}}(\omega,\vec{\kappa})\simeq\frac{\frac{1}{8}|\lambda|^{2}n_{\bar{f}}}{m_{f}(\omega-\Omega_{f})}\,,$
(3.25)

where we have used $p^{3}_{F}=3\pi^{2}n_{\bar{f}}$, and from Eq. (3.22),
$A\sim\frac{m_{f}}{\kappa}(\omega-\Omega_{f})$ for $\omega\sim\kappa$. It is
reassuring to see that the formula for $v_{\bar{f}}(\omega,\vec{\kappa})$ in
Eq. (3.25) coincides with Eq. (3.12), which is obtained by taking
$p^{\mu}\rightarrow(m_{f},\vec{0})$ from the beginning in the integrand.
However, as we have emphasized, this limiting form is not valid for values of
$\kappa$ near the resonance point.

In the general case, going back to Eq. (3.23),

$\frac{I_{F}(\kappa,\kappa)}{p^{2}_{F}}=\frac{1}{2}(1-\eta^{2})\log\left|\frac{1+\eta}{1-\eta}\right|+\eta\,,$
(3.26)

where

$\eta=\frac{m_{f}}{p_{F}}\frac{\xi-1}{\xi}\,,$ (3.27)

with

$\xi=\frac{\kappa}{\Omega_{f}}\,.$ (3.28)

Using Eq. (3.26) in Eq. (3.20),

$U(\kappa)=v_{\bar{f}}(\kappa,\vec{\kappa})=U_{0}\frac{1}{\xi}\left[\frac{1}{2}(1-\eta^{2})\log\left|\frac{1+\eta}{1-\eta}\right|+\eta\right]\,,$
(3.29)

where

$U_{0}=\frac{|\lambda|^{2}p^{2}_{F}}{16\pi^{2}\Omega_{f}}\,.$ (3.30)

The formula in Eq. (3.29) is our main result. For reference, a plot of
$U(\kappa)$ is shown in Fig. 1.

Figure 1:  Plot of $U(\kappa)$, for $\kappa\sim\Omega_{f}$, with some example
values of $p_{F}/m_{f}$.

We note the following. Away from the resonance region, $\eta$ has the limiting
values

$\eta=\frac{m_{f}}{p_{F}}\left\\{\begin{array}[]{ll}-\frac{1}{\xi}&\mbox{(low
$\kappa$)}\\\ 1&\mbox{(high $\kappa$)}\,.\end{array}\right.$ (3.31)

Therefore, either in the high ($\kappa\gg\Omega_{f}$) or low
($\kappa\ll\Omega_{f}$) momentum limit, $\eta$ is large and we can approximate
$I_{F}(\kappa,\vec{\kappa})$ by its limiting value for large $\eta$, which
gives

$\frac{I_{F}(\kappa,\vec{\kappa})}{p^{2}_{F}}\simeq\frac{2}{3\eta}=\frac{2p_{F}}{3m_{f}}\left(\frac{\xi}{\xi-1}\right)\,.$
(3.32)

Then from Eq. (3.20), this reproduces, again, Eq. (3.25). Thus, in these
asymptotic limits, namely high or low momentum as specified above, the
expression for $v_{\bar{f}}(\kappa,\vec{\kappa})$ given in Eq. (3.29)
coincides with the results that are obtained by approximating from the
beginning the integrals for $v_{\bar{f}}$ under the same conditions, namely,
away from the resonance and for the non-relativistic limit. However, as we can
see from Eq. (3.27), $|\eta|<1$ for values of $\kappa$ such that
$|\xi-1|<p_{F}/m_{f}$, so that in this range the asymptotic form of $I_{F}$
given in Eq. (3.32) is not valid.

The virtue of Eq. (3.29) is that it is valid also in the resonance region,
interpolating between the asymptotic expressions corresponding to high or low
$\kappa$ mentioned above, and they allow us to consider the propagation in the
resonance region, including the range $|\kappa/\Omega_{f}-1|<p_{F}/m_{f}$
mentioned. On the other hand, the imaginary part of the dispersion relation is
important in that region, as we have remarked, and it must be included in the
treatment of the propagation equation. As a guide to possible applications we
consider that next.

##  4 Damping term

The damping term $\gamma^{(\nu)}$ in Eq. (2.11) is not negligible in the
resonance region. As indicated in Eq. (2.1), it is determined from the
absorptive part $V^{\mu}_{i}$ of the effective potential which in turn is
determined from the $\Sigma_{12}$ element of the thermal self-energy matrix.
That calculation was carried out in detail in Ref. [34]. As shown in that
reference, the resulting formula for $\gamma^{(\nu)}$ is related to the
transition probabilities for the various processes in which the neutrino may
be annihilated or created, such as $\nu+\bar{f}\leftrightarrow\bar{\phi}$ and
$\nu+\phi\leftrightarrow f$, in the forward and reverse directions. The
formulas involve the phase space integrals weighted by the appropriate
momentum distribution functions. Here we just need to borrow the results of
those calculations. Quoting the results that are summarized in Eq. (3.38) of
Ref. [34],

$\frac{\gamma^{(\nu)}(\kappa)}{2}=\frac{|\lambda|^{2}}{32\pi\kappa^{2}\beta}\left\\{\begin{array}[]{ll}\Delta^{2}_{f\phi}\left[\log\left(1+e^{-\beta
E^{(I)}_{min}+\alpha_{f}}\right)-\log\left(1-e^{-\beta\Omega^{(I)}_{min}+\alpha_{\phi}}\right)\right]&(m_{f}>m_{\phi})\\\\[12.0pt]
\Delta^{2}_{\phi f}\left[\log\left(1+e^{-\beta
E^{(II)}_{min}-\alpha_{f}}\right)-\log\left(1-e^{-\beta\Omega^{(II)}_{min}-\alpha_{\phi}}\right)\right]&(m_{\phi}>m_{f})\,.\end{array}\right.$
(4.1)

The corresponding formulas for $\frac{\gamma^{(\bar{\nu})}(\kappa)}{2}$ are
obtained by making the substitutions

$\alpha_{f,\phi}\rightarrow-\alpha_{f,\phi}\,.$ (4.2)

To be consistent with Eq. (3.1), we focus on the second formula in Eq. (4.1).
In that formula,

$\displaystyle\Omega^{(II)}_{min}$ $\displaystyle=$
$\displaystyle\frac{\Delta^{2}_{\phi f}}{4\kappa}+\frac{\kappa
m^{2}_{\phi}}{\Delta^{2}_{\phi f}}\,,$ $\displaystyle E^{(II)}_{min}$
$\displaystyle=$ $\displaystyle\frac{\Delta^{2}_{\phi
f}}{4\kappa}+\frac{\kappa m^{2}_{f}}{\Delta^{2}_{\phi f}}\,,$ (4.3)

with $\Delta^{2}_{\phi f}$ defined in Eq. (3.3). The term involving
$\alpha_{f}$ corresponds to the contribution from the $\bar{f}$ gas in the
background, while the term with $\alpha_{\phi}$ corresponds to the
$\bar{\phi}$ gas, which are associated with the processes
$\nu+\bar{f}\leftrightarrow\bar{\phi}$ and $\nu+\phi\leftrightarrow f$,
respectively.

To complement our calculation of the effective potential in the previous
section, here we want to calculate the $\bar{f}$ background contribution to
the damping in the case that it can be considered as a completely degenerate
fermion gas. In order to bring out the physical picture in a clearer way, let
us consider first the case that both the $\bar{\phi}$ and $\bar{f}$ gases can
be treated in the classical and non-relativistic limit.

###  4.1 Damping in the classical and non-relativistic (NR) limit

In that case

$\displaystyle\frac{\gamma^{(\nu)}(\kappa)}{2}$ $\displaystyle=$
$\displaystyle\frac{|\lambda|^{2}}{32\pi\kappa^{2}\beta}\Delta^{2}_{\phi
f}\left[e^{-\beta
E^{(II)}_{min}-\alpha_{f}}+e^{-\beta\Omega^{(II)}_{min}-\alpha_{\phi}}\right]\,,$
(4.4)

where in the non-relativistic limit (as we have assumed in Section 3.1), the
chemical potentials are444These are obtained by requiring
$n_{\bar{x}}=g_{x}\int\frac{d^{3}p}{(2\pi)^{3}}e^{-\beta
E_{x}-\alpha_{x}}\qquad(x=f,\phi)\,,$ with $g_{f}=2g_{\phi}=2$.

$\displaystyle e^{-\alpha_{f}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}n_{\bar{f}}\left(\frac{2\pi\beta}{m_{f}}\right)^{\frac{3}{2}}e^{\beta
m_{f}}\,,$ $\displaystyle e^{-\alpha_{\phi}}$ $\displaystyle=$ $\displaystyle
n_{\bar{\phi}}\left(\frac{2\pi\beta}{m_{\phi}}\right)^{\frac{3}{2}}e^{\beta
m_{\phi}}\,.$ (4.5)

Therefore,

$\frac{\gamma^{(\nu)}(\kappa)}{2}=\frac{\Omega^{2}_{f}}{\kappa^{2}}\frac{\gamma^{(0)}_{\bar{f}}}{2}e^{-\Lambda_{f}}+\frac{\Omega^{2}_{\phi}}{\kappa^{2}}\frac{\gamma^{(0)}_{\bar{\phi}}}{2}e^{-\Lambda_{\phi}}\,,$
(4.6)

where

$\displaystyle\frac{\gamma^{(0)}_{\bar{f}}}{2}$ $\displaystyle=$
$\displaystyle\frac{|\lambda|^{2}}{16}\sqrt{\frac{2\pi
T}{m_{f}}}\frac{n_{\bar{f}}}{T\Omega_{f}}\,,$
$\displaystyle\frac{\gamma^{(0)}_{\bar{\phi}}}{2}$ $\displaystyle=$
$\displaystyle\frac{|\lambda|^{2}}{8}\sqrt{\frac{2\pi
T}{m_{\phi}}}\frac{n_{\bar{\phi}}}{T\Omega_{\phi}}\,,$
$\displaystyle\Lambda_{f}$ $\displaystyle=$
$\displaystyle\frac{m_{f}}{2T}\frac{\left(\frac{\kappa}{\Omega_{f}}-1\right)^{2}}{\left(\frac{\kappa}{\Omega_{f}}\right)}\,,$
$\displaystyle\Lambda_{\phi}$ $\displaystyle=$
$\displaystyle\frac{m_{\phi}}{2T}\frac{\left(\frac{\kappa}{\Omega_{\phi}}-1\right)^{2}}{\left(\frac{\kappa}{\Omega_{\phi}}\right)}\,.$
(4.7)

with $\Omega_{f,\phi}$ defined in Eq. (3).

The picture that emerges is consistent with our previous discussions regarding
the resonances and complements it in a practical way. Outside of either
resonance range the damping is exponentially small and can be neglected in the
formula for the dispersion relation. Therefore, if we are considering a
neutrino propagating in the $\Omega_{f}$ resonance region, we can discard the
$\bar{\phi}$ contribution to the damping term, assuming, as we do, that
$m_{f}$ and $m_{\phi}$ are sufficiently different that the resonances at
$\Omega_{f}$ and $\Omega_{\phi}$ do not overlap.

For illustrative purposes we show a plot of $\gamma^{(\nu)}(\kappa)$ in Fig. 2
in the $\kappa\sim\Omega_{f}$ resonance region, obtained as follows. As
already stated, we assume that $m_{f}$ and $m_{\phi}$ are sufficiently
different that the resonances at $\Omega_{f}$ and $\Omega_{\phi}$ do not
overlap; e.g., the $\kappa=\Omega_{\phi}$ point falls outside the range shown
in the plot. Under this condition, in the $\Omega_{f}$ resonance region we can
neglect the $\bar{\phi}$ contribution in Eq. (4.6) and consider simply

$\frac{\gamma^{(\nu)}}{\gamma^{(0)}_{\bar{f}}}=\frac{1}{\xi^{2}}e^{-\frac{m_{f}}{2T}\frac{(\xi-1)^{2}}{\xi}}\,,$
(4.8)

with $\xi$ defined in Eq. (3.28). Equation (4.8) is plotted in Fig. 2.

Figure 2:  Plot of the damping term for $\kappa\sim\Omega_{f}$, given by Eq.
(4.8), for some example values of $T/m_{f}$.

###  4.2 Damping in the degenerate limit

Armed with the results of the previous section, we thus ignore the
$\bar{\phi}$ contribution to the damping, and as a complement to the formula
for the effective potential in Eq. (3.29) here we calculate the $\bar{f}$
background contribution to $\gamma^{(\nu)}$, that is

$\frac{\gamma^{(\nu)}(\kappa)}{2}=\frac{|\lambda|^{2}\Delta^{2}_{\phi
f}}{32\pi\kappa^{2}\beta}\log\left(1+e^{-\beta
E^{(II)}_{min}-\alpha_{f}}\right)\,,$ (4.9)

in the limit of a completely degenerate fermion gas. This formula, and the
results discussed below, hold in all the kinematic regime of the fermion gas,
so they can be used in the NR case, or in any other case as well.

Setting

$\alpha_{f}=-\beta E_{F}\,,$ (4.10)

where $E_{F}$ is the Fermi energy, and taking the degenerate limit
($\beta\rightarrow\infty$),

$\frac{\gamma^{(\nu)}(\kappa)}{2}=\frac{|\lambda|^{2}\Delta^{2}_{\phi
f}}{32\pi\kappa^{2}}\left(E_{F}-E^{(II)}_{min}\right)\theta\left(E_{F}-E^{(II)}_{min}\right)\,.$
(4.11)

The step function in Eq. (4.11) implies that $\gamma^{(\nu)}(\kappa)$ is non-
zero if $\kappa$ lies in the range such that

$\frac{1}{2}(E_{F}-p_{F})\leq\frac{\kappa m^{2}_{f}}{\Delta^{2}_{\phi
f}}\leq\frac{1}{2}(E_{F}+p_{F})\,,$ (4.12)

or it is zero otherwise555To prove Eq. (4.12) we rewrite the condition
$E_{F}\geq E^{(II)}_{min}$ in the form
$\frac{1}{4x}+x\leq\frac{E_{F}}{m_{f}}\,,$ (4.13) where  $x\equiv\frac{\kappa
m_{f}}{\Delta^{2}_{\phi f}}\,.$ (4.14) The left-hand-side of Eq. (4.13) can be
written in the form  $\frac{1}{4x}+x=\frac{1}{x}(x-x_{+}(t))(x-x_{-}(t))+t$
(4.15) where $x_{\pm}(t)$, given by
$x_{\pm}(t)\equiv\frac{t}{2}\pm\frac{1}{2}\sqrt{t^{2}-1}\,,$ (4.16) satisfy
$\frac{1}{4x_{\pm}}+x_{\pm}=t\,,$ (4.17) for any $t\geq 1$. The functions
$x_{\pm}(t)$ are positive, have the same value at $t=1$, and as $t$ increases
$x_{+}$ increases while $x_{-}$ decreases. Therefore for any value of $x$ such
that  $x_{-}(t)\leq x\leq x_{+}(t)\,,$ (4.18)
$\frac{1}{x}(x-x_{+}(t))(x-x_{-}(t))\leq 0$, and therefore from Eq. (4.15)
$\frac{1}{4x}+x<t\,,$ (4.19) for any such value of $x$. It then follows that
all the values of $x$ that lie between $x_{-}(E_{F}/m_{f})$ and
$x_{+}(E_{F}/m_{f})$ satisfy Eq. (4.13), while the values outside that range
will violate it. Using the fact that
$x_{\pm}(E_{F}/m_{f})=\frac{1}{2}\left(\frac{E_{F}}{m_{f}}\right)\pm\frac{1}{2}\sqrt{\left(\frac{E_{F}}{m_{f}}\right)^{2}-1}=\frac{1}{2}\left(\frac{E_{F}}{m_{f}}\right)\pm\frac{1}{2}\frac{p_{F}}{m_{f}}\,,$
(4.20) proves Eq. (4.12).. Equation (4.11) can be written in the form

$\gamma^{(\nu)}=\gamma^{(0)}_{f}\frac{1}{\xi^{2}}\left[\frac{E_{F}}{m_{f}}-\frac{1}{2}\left(\xi+\frac{1}{\xi}\right)\right]\,,$
(4.21)

for $\xi$, defined in Eq. (3.28), in the range

$\frac{1}{m_{f}}(E_{f}-p_{F})\leq\xi\leq\frac{1}{m_{f}}(E_{F}+p_{F})\,,$
(4.22)

and

$\frac{\gamma^{(0)}_{f}}{2}=\frac{|\lambda|^{2}m^{2}_{f}}{16\pi\Omega_{f}}\,.$
(4.23)

As already stated, the result given in Eq. (4.21) holds in all the kinematic
regime of the fermion gas.

Eq. (4.21) is ploted in Fig. 3. The damping becomes smaller as $p_{F}$
decreases. From Fig. 1 we see that, at the same time, the width of the
resonance peaks in the effective potential become narrower. These features
indicate that the resonance effects are more important for relatively large
values of $p_{F}$ (high density) and less important as $p_{F}$ decreases
(lower density). A similar effect occurs with the damping in the classical
case, which reduces as the temperature decreases, as shown in Fig. 2.

Figure 3:  Plot of the damping term in the case of a degenerate $\bar{f}$
background, given by Eq. (4.21), for some example values of $p_{F}/m_{f}$.

##  5 Discussion

Here we discuss some generalizations and extensions of our work. On one hand,
the case of antineutrinos, as well as the other resonance region, around
$\omega\sim\Omega_{\phi}$ in the notation of Table 1, can be treated in
analogous form. In order to point out possible differences in details and/or
implementation we consider them briefly here.

On the other hand, since we have restricted ourselves to the case of one
neutrino flavor propagating and interacting in the $f$ and $\phi$ background,
as a guide and example to possible applications and generalizations, here we
will consider the application to the oscillation equations including the
damping term, in a simple two-generation case.

###  5.1 Anti-neutrino propagation near $\Omega_{f}$

Again we assume that $m_{\phi}>m_{f}$, and we consider the propagation at
energies $\omega^{(\bar{\nu})}\sim\Omega_{f}$. This is the resonance region of
the term $v_{f}$, which is the term that must be singled out for special
consideration. From Eqs. (2.1) and (2.26) we have in the present case

$V^{(\bar{\nu})}_{\text{eff}}(\vec{\kappa})=\bar{U}+\bar{r}\,,$ (5.1)

where

$\bar{U}=-v_{f}(-\kappa,-\vec{\kappa})\,,$ (5.2)

and

$\bar{r}=-v_{\bar{f}}(-\kappa,-\vec{\kappa})-v_{\phi}(-\kappa,-\vec{\kappa})-v_{\bar{\phi}}(-\kappa,-\vec{\kappa})\,.$
(5.3)

For $v_{\phi}$ and $v_{\bar{\phi}}$, we can simply borrow the formulas given
in Eqs. (3.9a) and (3), while for $v_{\bar{f}}(\omega,\vec{\kappa})$ the
corresponding formula in this case is Eq. (3.12). Thus, mimicking the steps
leading to Eq. (3.11), we obtain

$\bar{r}=|\lambda|^{2}\left[\frac{n_{\bar{f}}}{8m_{f}(\kappa+\Omega_{f})}+\frac{(n_{\phi}+n_{\bar{\phi}})}{4m_{\phi}\kappa}\right]\,.$
(5.4)

Regarding $v_{f}$, we go back to Eq. (2.27a). Carrying out the angular
integral, remembering the principal value prescription,

$v_{f}(\omega,\vec{\kappa})=-\frac{|\lambda|^{2}}{8\pi^{2}}\int
dp\;p^{2}\frac{f_{f}(p)}{2p\kappa}\log\left|\frac{k^{2}-2m_{f}\omega+2p\kappa-\Delta^{2}_{\phi
f}}{k^{2}-2m_{f}\omega-2p\kappa-\Delta^{2}_{\phi f}}\right|\,.$ (5.5)

As we can see, $-v_{f}(-\omega,-\vec{\kappa})$ is given by the same expression
given in Eq. (3.14) for $v_{\bar{f}}(\omega,\vec{\kappa})$, with the
substitution $f_{\bar{f}}\rightarrow f_{f}$ in the integrand. Thus, for
example, in the case that $f$ gas can be treated in the NR and degenerate
limit, the net result of this is that the final formula for $\bar{U}$ is the
same as the formula for $U$ given in Eq. (3.29), but of course with $p_{F}$
given in terms of the $f$ number density, $p^{3}_{F}=3\pi^{2}n_{f}$.

The damping can be treated similarly to the case of neutrinos in Section 4,
but in the present case the relevant number densities are $n_{f}$ and
$n_{\phi}$. Explicitly, remembering Eq. (4.2), in the classical and NR limit
the damping is

$\frac{\gamma^{(\bar{\nu})}(\kappa)}{2}=\frac{\Omega^{2}_{f}}{\kappa^{2}}\frac{\gamma^{(0)}_{f}}{2}e^{-\Lambda_{f}}+\frac{\Omega^{2}_{\phi}}{\kappa^{2}}\frac{\gamma^{(0)}_{\phi}}{2}e^{-\Lambda_{\phi}}\,,$
(5.6)

where

$\displaystyle\frac{\gamma^{(0)}_{f}}{2}$ $\displaystyle=$
$\displaystyle\frac{|\lambda|^{2}}{16}\sqrt{\frac{2\pi
T}{m_{f}}}\frac{n_{f}}{T\Omega_{f}}\,,$
$\displaystyle\frac{\gamma^{(0)}_{\phi}}{2}$ $\displaystyle=$
$\displaystyle\frac{|\lambda|^{2}}{8}\sqrt{\frac{2\pi
T}{m_{\phi}}}\frac{n_{\phi}}{T\Omega_{\phi}}\,,$ (5.7)

with $\Lambda_{f,\phi}$ given by Eq. (4.1). For a degenerate $f$ gas, the
formula for $\gamma^{(\bar{\nu})}$ is the same as Eq. (4.21), but with the
reinterpretation of $p_{F}$ as the Fermi momentum associated with the $f$ gas
number density $n_{f}$, as we have stated above. The sketches of the damping
and the effective potential in this case are therefore similar to those shown
in Figs. 1, 2 and 3 with the corresponding identification of the parameters
involved.

In the case of a neutrino or antineutrino propagating near the $\Omega_{\phi}$
energy region, the same method can be applied to calculate the effective
potential. In this case the term that must be singled out for special
treatment is $v_{\bar{\phi}}$ (or $v_{\phi}$ for antineutrinos). The relevant
integrals are of the same form given in Eq. (3.19), but they involve the
$\bar{\phi}$ or $\phi$ distribution functions. In the classical limit of the
distribution functions, the task involves the computation of the generic
integrals

$\int dx\,e^{-x^{n}}x\log\left|\frac{x+A}{x-A}\right|\,,$ (5.8)

with $n=1,2$ in the ultra-relativistic and non-relativistic limits,
respectively. Integrals of this form appear in similar calculations in other
contexts as already mentioned[35]. We do not pursue this case any further
here.

###  5.2 Two-generation example

We consider a two-generation case, assuming that only the first generation
(e.g., electron neutrino) couples to $f$ and $\phi$. Working at the level of
the evolution of the flavor spinor, the equation, including the damping term,
is

$i\partial_{t}\phi=\left(H_{r}-\frac{i}{2}\Gamma\right)\phi\,.$ (5.9)

Up to a term proportional to identity matrix, the Hamiltonian is

$H_{r}=\frac{1}{2}\frac{\Delta
m^{2}_{21}}{2\kappa}\left(\begin{array}[]{cc}-\cos 2\theta&\sin 2\theta\\\
\sin 2\theta&\cos
2\theta\end{array}\right)+\frac{1}{2}\left(\begin{array}[]{cc}V^{(\nu)}_{\text{eff}}&0\\\
0&-V^{(\nu)}_{\text{eff}}\end{array}\right)\,,$ (5.10)

where $\Delta m^{2}_{21}=m^{2}_{2}-m^{2}_{1}$, while

$\Gamma=\left(\begin{array}[]{cc}\gamma^{(\nu)}&0\\\
0&0\end{array}\right)\equiv\gamma^{(\nu)}I_{e}\,,$ (5.11)

with

$I_{e}\equiv\left(\begin{array}[]{cc}1&0\\\ 0&0\end{array}\right)\,.$ (5.12)

$V^{(\nu)}_{\text{eff}}$ and $\gamma^{(\nu)}$ are understood to be the
effective potential and damping term that we have obtained.

Following the usual steps, $H_{r}$ can be written in the standard form

$H_{r}=\frac{h}{2}\left(\begin{array}[]{cc}-\cos 2\theta_{m}&\sin
2\theta_{m}\\\ \sin 2\theta_{m}&\cos 2\theta_{m}\end{array}\right)\,,$ (5.13)

where

$h=\frac{\Delta^{2}_{m}}{2\kappa}\,,$ (5.14)

and

$\displaystyle\cos 2\theta_{m}$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta^{2}_{m}}\left(\Delta m^{2}_{21}\cos
2\theta-2\kappa V^{(\nu)}_{\text{eff}}\right)\,,$ $\displaystyle\sin
2\theta_{m}$ $\displaystyle=$ $\displaystyle\frac{\Delta m^{2}_{21}\sin
2\theta}{\Delta^{2}_{m}}\,,$ (5.15)

with

$\Delta^{2}_{m}\equiv\sqrt{(\Delta m^{2}_{21}\sin 2\theta)^{2}+\left(\Delta
m^{2}_{21}\cos 2\theta-2\kappa V^{(\nu)}_{\text{eff}}\right)^{2}}\,.$ (5.16)

Writing the solution in the form

$\phi(t)=G\phi(0)\,,$ (5.17)

in the absence of the damping term

$G=\sum_{s=\pm}u_{s}u^{\dagger}_{s}e^{-i\lambda_{s}t}\,,$ (5.18)

where $\lambda_{s}=sh$ are the eigenvalues of $H_{r}$, and $u_{s}$ the
corresponding eigenspinors

$\displaystyle u_{+}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\sin\theta_{m}\\\
\cos\theta_{m}\end{array}\right)\,,$ (5.21) $\displaystyle u_{-}$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{c}\cos\theta_{m}\\\
-\sin\theta_{m}\end{array}\right)\,.$ (5.24)

In the spirit of a perturbative treatment of the damping term, we construct
the solution by taking the eigenvectors of $H$ to be the same as the
eigenvectors of $H_{r}$, but with the eigenvalues modified by the first order
corrections; i.e.,

$\lambda_{s}=sh-\frac{i}{2}u^{\dagger}_{s}\Gamma u_{s}\,.$ (5.25)

By explicitly calculation,

$\lambda_{s}=\left\\{\begin{array}[]{cc}h-\frac{i}{2}\gamma_{s}&(s=+)\\\
-h-\frac{i}{2}\gamma_{c}&(s=-)\,,\end{array}\right.$ (5.26)

where we have defined

$\displaystyle\gamma_{s}$ $\displaystyle=$
$\displaystyle\gamma^{(\nu)}\sin^{2}\theta_{m}\,,$ $\displaystyle\gamma_{c}$
$\displaystyle=$ $\displaystyle\gamma^{(\nu)}\cos^{2}\theta_{m}\,.$ (5.27)

We then obtain the following explicit expression for the evolution matrix

$G=u_{+}u^{\dagger}_{+}e^{-iht}e^{-\frac{1}{2}\gamma_{s}t}+u_{-}u^{\dagger}_{-}e^{iht}e^{-\frac{1}{2}\gamma_{c}t}\,.$
(5.28)

So, for example, for

$\phi(0)=\left(\begin{array}[]{c}1\\\ 0\end{array}\right)\,,$ (5.29)

we have the persistence and transition amplitudes,

$\displaystyle\phi_{1}(t)$ $\displaystyle=$
$\displaystyle\sin^{2}\theta_{m}e^{-iht}e^{-\frac{1}{2}\gamma_{s}t}+\cos^{2}\theta_{m}e^{iht}e^{-\frac{1}{2}\gamma_{c}t}\,,$
$\displaystyle\phi_{2}(t)$ $\displaystyle=$
$\displaystyle\sin\theta_{m}\cos\theta_{m}\left(e^{-iht}e^{-\frac{1}{2}\gamma_{s}t}-e^{iht}e^{-\frac{1}{2}\gamma_{c}t}\right)\,,$
(5.30)

and the corresponding oscillation probabilities can be written in the
form666We have used  $\displaystyle\sin^{4}\theta_{m}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(1-\cos 2\theta_{m})^{2}=\frac{1}{4}(2-2\cos
2\theta_{m}-\sin^{2}2\theta_{m})$ $\displaystyle\cos^{4}\theta_{m}$
$\displaystyle=$ $\displaystyle\frac{1}{4}(1+\cos
2\theta_{m})^{2}=\frac{1}{4}(2+2\cos 2\theta_{m}-\sin^{2}2\theta_{m})$

$\displaystyle P_{1}\equiv|\phi_{1}|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(e^{-\gamma_{c}t}+e^{-\gamma_{s}t}\right)+\frac{1}{2}\cos
2\theta_{m}\left(e^{-\gamma_{c}t}-e^{-\gamma_{s}t}\right)$
$\displaystyle\mbox{}-\frac{1}{2}\sin^{2}2\theta_{m}\left[\frac{1}{2}\left(e^{-\gamma_{c}t}+e^{-\gamma_{s}t}\right)-e^{-\frac{1}{2}\gamma^{(\nu)}t}\cos
2ht\right]\,,$ $\displaystyle P_{2}\equiv|\phi_{2}|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sin^{2}2\theta_{m}\left[\frac{1}{2}\left(e^{-\gamma_{c}t}+e^{-\gamma_{s}t}\right)-e^{-\frac{1}{2}\gamma^{(\nu)}t}\cos
2ht\right]\,.$ (5.31)

The importance of the damping terms depends on the interplay between $h$,
defined in Eq. (5.14), and $\gamma^{(\nu)}$. To be more specific, we can
consider, for example, the propagation of neutrinos near the $\Omega_{f}$
region, with $V^{(\nu)}_{\text{eff}}$ given by Eq. (3.5) (with $r$ and $U$
given by Eqs. (3.11) and (3.29), respectively) and $\gamma^{(\nu)}$ by Eq.
(4.8). For small values of $\gamma^{(\nu)}$, many oscillation length cycles
are required for the damping effects to be observable. A distinctive feature
of Eq. (5.2) is the energy dependence of the damping and the oscillation
terms, based on the formulas for $\gamma^{(\nu)}$ and $V^{(\nu)}_{\text{eff}}$
that we have given. For example, if we consider a pure $\bar{f}$ background
(no $f$ and no $\phi,\bar{\phi}$ backgrounds) then $V^{(\nu)}_{\text{eff}}$ is
given only by $U$ [Eqs. (3.5) and (3.29)], which becomes zero for neutrino
energies very close to the resonance point $\Omega_{f}$. In that regime $h$
reduces to its vacuum value $\Delta m^{2}_{21}/2\kappa$ while the damping term
reaches its largest value.

##  6 Conclusions and outlook

In this work we have been concerned with the propagation of a neutrino in a
background of fermions ($f$) and scalars ($\phi$), interacting via a Yukawa-
type interaction. Our particular goal was to obtain the dispersion relation in
the case that the neutrino energy lies in the range in which the absorption
and production processes become kinematically accessible, such as
$\nu+\bar{f}\rightarrow\bar{\phi}$ or the crossed counterparts, and the
corresponding ones for the antineutrino. The relevant energy values are around
$|m^{2}_{\phi}-m^{2}_{f}|/2m_{f}$ or $|m^{2}_{\phi}-m^{2}_{f}|/2m_{\phi}$, to
which we refer as a resonance energy range. The distinguishing aspect of these
energy ranges is that the one-loop formula for the neutrino self-energy has a
singularity, which is the indication that the self-energy acquires an
imaginary part that cannot be ignored. Technically, the imaginary part is
associated with the damping effects, while the integral formula for the real
part must be evaluated using the principal value of the integral.

Writing the dispersion relation in the form
$\omega=\kappa+V_{\text{eff}}-i\gamma/2$, we gave the explicit formulas for
the effective potential ($V_{\text{eff}}$) and damping ($\gamma$) for some
cases that allowed us to give analytic results. In particular we considered in
detail the evaluation of those quantities for a neutrino propagating with the
energy near ($m^{2}_{\phi}-m^{2}_{f})/2m_{f}$ (corresponding, for
$m_{\phi}>m_{f}$, to the processes $\nu+\bar{f}\rightarrow\bar{\phi}$ becoming
kinematically accessible), in the case that the $\bar{f}$ background can be
treated as a non-relativistic degenerate Fermi gas. We also considered the
analogous case for an antineutrino.

The formulas obtained have the property that, when the neutrino energy is
either much larger or much smaller than the resonance energy,
$V^{(\nu)}_{\text{eff}}$ reduces to the effective potential that has been
already determined in the literature in the high or low momentum regime,
respectively. The virtue of the formula we give for $V^{(\nu)}_{\text{eff}}$
is that it is valid also in the _resonance energy range_ , which is outside
the two limits mentioned. We outlined how the same strategy can be applied to
consider the case of a neutrino propagating with an energy in the other
resonance region $|m^{2}_{\phi}-m^{2}_{f}|/2m_{\phi}$, in which case the terms
in the effective potential that require special consideration are those
corresponding to the contribution from the scalar backgrounds. For example,
for $m_{f}>m_{\phi}$, the resonance shows up in the $v_{\phi}$ contribution,
which corresponds to the processes $\nu+\phi\rightarrow f$ becoming
kinematically accessible.

For definiteness we restricted ourselves to the calculation of the dispersion
relation in the case that only one neutrino flavor interacts with the $f$ and
$\phi$ background particles. As a guide and example to possible applications
and generalizations, we gave the relevant formulas for the $V_{\text{eff}}$
and $\gamma$ matrices, and considered the solution to the oscillation
equations including the damping term, in a simple two-generation case.

The same strategy we have used to determine the effective potential for a
neutrino propagating in an $f$ background in the energy range to produce a
$\phi$ particle, can be applied to the case of a neutrino propagating in an
electron background with an energy in the Glashow resonance region[36].
Several technical aspects of the calculations are of course different, but the
idea of determining the effective potential for such energy range by the
method we have followed can be applied to that case as well.

The work of S. S. is partially supported by DGAPA-UNAM (Mexico) PAPIIT project
No. IN103522.

## References

  * [1] L. Wolfenstein, _Neutrino oscillations in matter_ , Phys.Rev. D17, 2369 (1978).
  * [2] P. Langacker, J. P. Leveille, and J. Sheiman, _On the detection of cosmological neutrinos by coherent scattering_ , Phys. Rev. D 27, 1228 (1983).
  * [3] H. Arthur Weldon, _Effective fermion masses of order $gT$ in high-temperature gauge theories with exact chiral invariance_, Phys. Rev. D 26, 2789 (1982)
  * [4] D. Notzold and G. Raffelt, _Neutrino dispersion at finite temperature and density_ , Nucl. Phys. B307, 924 (1988).
  * [5] P. B. Pal and T. N. Pham, _Field-theoretic derivation of Wolfenstein’s matter oscillation formula_ , Phys. Rev. D 40, 259 (1989).
  * [6] José F. Nieves, _Neutrinos in a medium_ , Phys. Rev. D40, 866 (1989)
  * [7] See for example, N. P. Landsman and C. G. van Weert, _Real and Imaginary Time Field Theory at Finite Temperature and Density_ , Phys. Rept. 145, 141 (1987); J. I. Kapusta, _Finite Temperature Field Theory_ , (Cambridge University Press,Cambridge, 1989); A. K. Das, _Finite Temperature Field Theory_ , (World Scientific Singapore, 1997); M. L. Bellac, _Thermal Field Theory_ , (Cambridge University Press, Cambridge, 2011).
  * [8] Jeffrey M. Berryman, André de Gouvea, Kevin J. Kelly and Yue Zhang, _Lepton-Number-Charged Scalars and Neutrino Beamstrahlung_ , Phys. Rev. D 97, 075030 (2018), [arXiv:1802.00009]
  * [9] Y. Farzan, M. Lindner, W. Rodejohann and X. J. Xu, _Probing neutrino coupling to a light scalar with coherent neutrino scattering_ , JHEP 1805, 066 (2018) [arXiv:1802.05171]
  * [10] G. J. Stephenson, Jr. and J. T. Goldman, _Observable consequences of a scalar boson coupled only to neutrinos_ , report No: LA-UR-93-3348 [arXiv:hep-ph/9309308]
  * [11] C. Boehm, A. Olivares-Del Campo, S. Palomares-Ruiz and S. Pascoli, _Phenomenology of a Neutrino-DM Coupling: The Scalar Case_ , NuPhys2016 [arXiv:1705.03692]
  * [12] L. Heurtier and Y. Zhang, _Supernova Constraints on Massive (Pseudo)Scalar Coupling to Neutrinos_ , JCAP 1702, no. 02, 042 (2017) [arXiv:1609.05882]
  * [13] R. F. Sawyer, _Bulk viscosity of a gas of neutrinos and coupled scalar particles, in the era of recombination_ , Phys. Rev. D 74, 043527 (2006) [arXiv:astro-ph/0601525].
  * [14] P. S. Pasquini and O. L. G. Peres, _Bounds on Neutrino-Scalar Yukawa Coupling_ , Phys. Rev. D 93, no. 5, 053007 (2016); Erratum: [Phys. Rev. D 93, no. 7, 079902 (2016)] [arXiv:1511.01811]
  * [15] Shao-Feng Ge, Manfred Lindner and Werner Rodejohann, _Atmospheric Trident Production for Probing New Physics_ , Phys. Lett. B 772, 164 (2017); [arXiv:1702.02617]
  * [16] Vedran Brdar, Joachim Kopp, Jia Liu, Pascal Prass and Xiao-Ping Wang, _Fuzzy dark matter and nonstandard neutrino interactions_ , Phys. Rev. D 97, 043001 (2018) [arxiv: 1705.09455].
  * [17] See for example, H. Duan, G. M. Fuller and Y. Z. Qian, _Collective Neutrino Oscillations_ , Ann. Rev. Nucl. Part. Sci. 60, 569 (2010) [arXiv:1001.2799 [hep-ph]], and references therein
  * [18] S. Chakraborty, R. Hansen, I. Izaguirre and G. Raffelt, _Collective neutrino flavor conversion: Recent developments_ , Nucl. Phys. B 908, 366 (2016) [arXiv:1602.02766].
  * [19] Y. Y. Y. Wong, _Analytical treatment of neutrino asymmetry equilibration from flavor oscillations in the early universe_ , Phys. Rev. D 66, 025015 (2002) [arXiv:hep-ph/0203180].
  * [20] G. Mangano, G. Miele, S. Pastor, T. Pinto, O. Pisanti and P. D. Serpico, _Effects of non-standard neutrino-electron interactions on relic neutrino decoupling_ , Nucl. Phys. B 756, 100 (2006) [arXiv:hep-ph/0607267].
  * [21] G. Mangano, A. Melchiorri, P. Serra, A. Cooray and M. Kamionkowski, _Cosmological bounds on dark matter-neutrino interactions_ , Phys. Rev. D 74, 043517 (2006) [arXiv:astro-ph/0606190].
  * [22] T. Binder, L. Covi, A. Kamada, H. Murayama, T. Takahashi and N. Yoshida, _Matter Power Spectrum in Hidden Neutrino Interacting Dark Matter Models: A Closer Look at the Collision Term_ , JCAP 1611, 043 (2016) [arXiv:1602.07624].
  * [23] R. Primulando and P. Uttayarat, _Dark Matter-Neutrino Interaction in Light of Collider and Neutrino Telescope Data_ , JHEP 1806, 026 (2018) [arXiv:1710.08567].
  * [24] A. Olivares-Del Campo, C. Bœhm, S. Palomares-Ruiz and S. Pascoli, _Dark matter-neutrino interactions through the lens of their cosmological implications_ , Phys. Rev. D 97, 075039 (2018) [arXiv:1711.05283].
  * [25] T. Brune and H. Päs, _Massive Majorons and constraints on the Majoron-neutrino coupling_ , Phys. Rev. D 99, 096005 (2019) [arXiv:1808.08158].
  * [26] T. Franarin, M. Fairbairn and J. H. Davis, _JUNO Sensitivity to Resonant Absorption of Galactic Supernova Neutrinos by Dark Matter_ , [arXiv:1806.05015].
  * [27] P. S. Bhupal Dev et al., _Neutrino Non-Standard Interactions: A Status Report_ , SciPost Phys. Proc. 2, 001 (2019) [arXiv:1907.00991]
  * [28] S. Pandey, S. Karmakar and S. Rakshit, _Interactions of Astrophysical Neutrinos with Dark Matter: A model building perspective_ , JHEP 1901, 095 (2019) [Erratum: JHEP 11, 215 (2021)] [arXiv:1810.04203]
  * [29] S. Karmakar, S. Pandey and S. Rakshit, _Are We Looking at Neutrino Absorption Spectra at IceCube?_ , [arXiv:1810.04192]
  * [30] K. S. Babu, Garv Chauhan, P. S. Bhupal Dev, _Neutrino Non-Standard Interactions via Light Scalars in the Earth, Sun, Supernovae and the Early Universe_ , Phys. Rev. D 101, 095029 (2020) [arXiv:1912.13488].
  * [31] J. F. Nieves and S. Sahu, _Neutrino effective potential in a fermion and scalar background_ , Phys. Rev. D 98, 063003 (2018) [arXiv:1808.01629].
  * [32] Shao-Feng Ge and Stephen J. Parke, _Scalar Non-Standard Interactions in Neutrino Oscillation_ , Phys. Rev. Lett. 122, 211801 (2019) [arXiv: 1812.08376]
  * [33] A. Y. Smirnov and V. B. Valera _Resonance refraction and neutrino oscillations_ , JHEP 09 (2021) 177 [arXiv:2106.13829]
  * [34] J. F. Nieves and S. Sahu, _Neutrino damping in a fermion and scalar background_ , Phys. Rev. D 99, 095013 (2019) [arXiv:1812.05672]
  * [35] See for example, C. Quimbay and S. Vargas-Castrillon, _Fermionic Dispersion Relations in the Standard Model at Finite Temperature_ , Nucl. Phys. B451, 265 (1995) [arXiv:hep-ph/9504410]; M. Laine, _Thermal right-handed neutrino production rate in the relativistic regime_ , JHEP 1308, 138 (2013) [arXiv:1307.4909]; J. Ghiglieri and M. Laine, _Neutrino dynamics below the electroweak crossover_ , JCAP 1607, 015 (2016) [arXiv:1605.07720].
  * [36] IceCube collaboration, _Detection of a particle shower at the Glashow resonance with IceCube_ , Nature 591, 220 (2021).