# Neutrino meets ultralight dark matter: $\boldsymbol{0\nu\beta\beta}$ decay
and cosmology

Guo-yuan Huang<EMAIL_ADDRESS>Max-Planck-Institut für
Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany  Newton Nath
<EMAIL_ADDRESS>Istituto Nazionale di Fisica Nucleare, Via Orabona 4,
70126 Bari, Italy

###### Abstract

We explore the neutrinoless double beta ($0\nu\beta\beta$) decay induced by an
ultralight dark matter field coupled to neutrinos. The effect on
$0\nu\beta\beta$ decay is significant if the coupling violates the lepton
number, for which the $\Delta L=2$ transition is directly driven by the dark
matter field without further suppression of small neutrino masses. As the
ultralight dark matter can be well described by a classical field, the effect
features a periodic modulation pattern in decay events. However, we find that
in the early Universe such coupling will be very likely to alter the standard
cosmological results. In particular, the requirement of neutrino free-
streaming before the matter-radiation equality severely constrains the
parameter space, such that the future $0\nu\beta\beta$ decay experiments can
hardly see any signal even with a meV sensitivity to the effective neutrino
mass.

##  I  Introduction

In spite of its remarkable success, the Standard Model (SM) of particle
physics fails to address several fundamental issues, e.g., the nature of dark
matter (DM). The mass of viable DM candidates can cover a huge landscape
spanning from the primordial black holes Bird:2016dcv ; Carr:2016drx ;
Chen:2016pud ; Georg:2017mqk  to the ultralight regime Hu:2000ke . At present,
the DM direct detection experiments such as Xenon1T XENON:2020gfr , PandaX-II
PandaX-II:2017hlx  come out with null signal of weakly interacting massive
particle (WIMP), one of the most promising DM candidates, hence setting
stringent limits on the available WIMP parameter space. Another particular DM
class, the ultralight DM Ferreira:2020fam ; Urena-Lopez:2019kud  with a mass
ranging from $\mathcal{O}(10^{-22})~{}{\rm eV}$ 111This is often called fuzzy
dark matter Hu:2000ke ; Hui:2016ltb . The term “fuzzy” corresponds to a huge
Compton wavelength, $\lambda=2\pi/m_{\phi}\simeq 0.4\,{\rm
pc}\times(10^{-22}\,\text{eV}/m_{\phi})$ with $m_{\phi}$ being the DM mass.
to $\mathcal{O}(1)~{}{\rm eV}$, for example the well-motivated quantum
chromodynamics (QCD) axion Kim:2008hd ; Graham:2015ouw ; Irastorza:2018dyq ;
Marsh:2015xka , is receiving renewing attention. Different from WIMP, the
ultralight DM is produced non-thermally in the early Universe and can be well
described by a classical-number field.

Beside DM, the nature of massive neutrinos, i.e., whether they are Majorana or
Dirac fermions, is yet to be answered. At present, the only feasible process
that can uncover the nature of neutrinos is the neutrinoless double beta
($0\nu\beta\beta$) decay, which violates the lepton number by two units
DellOro:2016tmg ; Vergados:2016hso .

There have been many attempts to connect the mentioned two mysterious sectors,
i.e., neutrinos and ultralight DM Berlin:2016woy ; Brdar:2017kbt ;
Krnjaic:2017zlz ; Liao:2018byh ; Capozzi:2018bps ; Reynoso:2016hjr ;
Huang:2018cwo ; Pandey:2018wvh ; Farzan:2018pnk ; Choi:2019ixb ; Baek:2019wdn
; Choi:2019zxy ; Choi:2020ydp ; Dev:2020kgz ; Baek:2020ovw ; Losada:2021bxx ;
Smirnov:2021zgn ; Alonso-Alvarez:2021pgy ; Huang:2021zzz . In this work, we
investigate $0\nu\beta\beta$ decays in the presence of a coupling between
neutrinos and the ultralight DM with a general mass spanning from
$10^{-22}~{}{\rm eV}$ to sub-eV. Among general interaction forms, the scalar
and tensor couplings are able to violate the lepton number (or equivalently,
the DM particle carries two units of lepton number), and hence important for
$0\nu\beta\beta$-decay experiments. The Lagrangian reads

$\displaystyle-\mathcal{L}^{\rm M}_{\rm
int}=\begin{dcases}\frac{g_{\phi}^{\alpha\beta}}{2}\,\phi\,\overline{\nu_{\rm
L\alpha}}\nu^{\rm c}_{\rm L\beta}+{\rm h.c.}\;,\\\\[2.15277pt]
\frac{g_{T}^{\alpha\beta}}{2}T^{\mu\nu}\overline{\nu_{\rm
L\alpha}}\sigma_{\mu\nu}\nu^{\rm c}_{\rm L\beta}+{\rm h.c.}\;,\end{dcases}$
(1)

where $\alpha,\beta=e,\mu,\tau$ are the flavor indices, and $\phi$ and
$T^{\mu\nu}$ represent real scalar and tensor fields with neutrino coupling
constants $g_{\phi}^{\alpha\beta}$ and $g_{T}^{\alpha\beta}$, respectively. We
have assumed that neutrinos are Majorana particles, such that only the field
$\nu_{\rm L}$ is involved at low energies.

In analogy with the electromagnetic force, the tensor interaction can be
generated by a dark photon field $A^{\prime}$ coupled to neutrinos with a dark
magnetic dipole moment Capozzi:2018bps . However, for Majorana neutrinos, only
transition magnetic moment exists and the diagonal part of the tensor coupling
constant $g_{T}^{\alpha\beta}$ is vanishing, thus not contributing to the
$0\nu\beta\beta$ process which exchanges two electron neutrinos. Hence, we
will restrict our following discussions to only the ultralight scalar DM. It
is worth mentioning that the transition magnetic moment can be relevant for
the lepton-number-violating meson decays by exchanging different flavors of
neutrinos.

For Majorana neutrinos, $\nu\equiv\nu_{\rm L}+{\nu^{\rm c}_{\rm L}}$, in the
background of scalar DM, the Hamiltonian density of the system is

$\displaystyle\mathcal{H}=\frac{1}{2}(\partial\phi)^{2}+\frac{1}{2}m^{2}_{\phi}\phi^{2}+\frac{1}{2}\overline{\nu}(\mathrm{i}\not{\partial}+\widehat{M}_{\nu}+g\phi)\nu\;,$
(2)

where $m_{\phi}$ is the DM mass, $\widehat{M}_{\nu}$ is the neutrino mass
matrix in vacuum, and the coupling constant $g$ is a real symmetric matrix in
general. Note that the pseudoscalar coupling has been ignored in the
expression, which will induce a similar effect. The corresponding equation of
motion is given by

$\displaystyle(\partial^{2}+m^{2}_{\phi})\phi=-\frac{1}{2}g\overline{\nu}\nu\;,$
(3) $\displaystyle i\not{\partial}\nu=(\widehat{M}_{\nu}+g\phi)\nu\;.$ (4)

As long as the neutrino number density is not too large in Eq. (3), the
ultralight DM evolves as free classical field in the Universe as
$\phi=\phi_{0}\cos{m_{\phi}t}$ after the scalar mass “freeze-in”, i.e.,
$H<m_{\phi}$, with $H$ being the Hubble expansion rate. The field strength is
given by $\phi_{0}=\sqrt{2\rho}/m_{\phi}$, where $\rho\approx
m_{\phi}n_{\phi}\approx 0.3~{}{\rm GeV\cdot cm^{-3}}$ is the DM energy density
in our local galaxy.

The large occupation number of local ultralight DM, namely
$N=n_{\phi}\lambda^{3}\approx 6\times 10^{36}\,(10^{-10}~{}{\rm
eV}/m_{\phi})^{4}$ within the cube of Compton wavelength, guarantees the
classical-field description as an excellent approximation to the more
fundamental quantum field. The formation of the classical field in our local
galaxy will generate a time-varying effective Majorana mass $m_{\nu,{\rm
eff}}=g\phi$ to neutrinos, which is very similar to the Higgs mechanism. At
the particle level, the $0\nu\beta\beta$ decays induced by ultralight DM are
shown as diagrams in Fig. 1. The illustrated leading-order contribution is
accurate enough since the coupling of our concern is set to be extremely
small. The emission process has been discussed before in the context of
Majoron model Blum:2018ljv ; Brune:2018sab ; Cepedello:2018zvr . However,
different from the “spontaneous” Majoron emission process, in our case both
the absorption and emission of scalars benefit from the large occupation
number of ultralight DM, which gives rise to a dramatic stimulation
enhancement.

Figure 1: Feynman diagrams of $0\nu\beta\beta$ decays induced by the
absorption and emission of ultralight scalar DM. Because these transitions
take place in a background of identical scalar bosons, there will be an
enhancement of stimulation.

In the rest of the work, we first describe how the coupled system of
ultralight DM and neutrinos evolves in the early Universe in Sec. II. Based on
the robust cosmic microwave background (CMB) observations, we confine the
allowed parameter space of the mass and coupling of ultralight DM. In Sec.
III, we explore the possible impacts on $0\nu\beta\beta$ decays. Conclusions
are presented in Sec. IV.

##  II Cosmological evolution

A popular way to generate the coherent scalar field in the early Universe is
via the misalignment mechanism, which is well known for the QCD axion
Kim:2008hd ; Graham:2015ouw ; Irastorza:2018dyq ; Marsh:2015xka . The
production is typically associated with a global symmetry, which is broken
spontaneously and results in a Goldstone mode. The later dynamics will break
the shift-symmetry of the Goldstone by selecting a preferred phase. A mass
term will then be generated to the field, which populates the DM energy
density after $H\lesssim m_{\phi}$. The initial condition of a misaligned
scalar in our scenario reads Marsh:2015xka

$\displaystyle\phi(t_{\rm i},x)=f_{\phi}\theta_{\phi}\;,~{}\dot{\phi}(t_{\rm
i},x)=0\;,$ (5)

where $t_{\rm i}$ is the time set by the spontaneous symmetry breaking of the
global symmetry, $f_{\phi}$ represents the energy scale of the symmetry
breaking, $\theta_{\phi}$ is the initial phase, and
$\dot{\phi}\equiv\mathrm{d}\phi/\mathrm{d}t$.

At the quantum field level, the ultralight DM produced by the misalignment
mechanism (from the dynamical symmetry breaking) should be descried by a
coherent state of the form Marsh:2015xka ; Ferreira:2020fam ; Miransky1994 ;
Ferreira:lecture ; Davidson:2014hfa ; Veltmaat2020GalaxyFW ; Guth:2014hsa ;
Zhang:1999is ; Barnett:1997

$\displaystyle\left|\phi\right\rangle=\mathrm{e}^{-\frac{N}{2}}\mathrm{e}^{\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\tilde{\phi}(k)\hat{a}^{\dagger}_{k}}\left|0\right\rangle\;,$
(6)

where $\mathrm{e}^{-\frac{N}{2}}$ is a normalization factor, $\tilde{\phi}(k)$
is a general complex field with the wave-vector $k$, and
$\hat{a}^{\dagger}_{k}$ is the corresponding particle creation operator. The
coherent state $|\phi\rangle$ has been thought of as the most field-like
solution of quantum states. The property of this state can be largely captured
by the expectation value of the field operator $\hat{\phi}$, namely

$\displaystyle\phi(t,x)\equiv\left\langle\phi\right|\hat{\phi}\left|\phi\right\rangle$
$\displaystyle=$
$\displaystyle\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\frac{1}{\sqrt{2E_{k}}}\left[\tilde{\phi}(k)\mathrm{e}^{-i(\omega
t-k\cdot x)}+\right.$ (7)
$\displaystyle\left.+\tilde{\phi}^{*}(k)\mathrm{e}^{i(\omega t-k\cdot
x)}\right]\;,$

which is a classical-number field. This non-vanishing expectation value can be
interpreted in a way similar to the Higgs mechanism. The evolution of the
classical field $\phi(t,x)$ right follows the equation of motion in Eq. (3).

For the isotropic DM field without kinematic energy, i.e., $k=0$, we have
$\left|\phi\right\rangle=\mathrm{e}^{-\frac{N}{2}}\mathrm{e}^{\sqrt{N}\mathrm{e}^{\mathrm{i}\theta}\hat{a}^{\dagger}_{0}}\left|0\right\rangle$.
It should be emphasized that the coherent state $|\phi\rangle$ is an
eigenstate of $\hat{a}_{0}$, i.e.,
$\hat{a}_{0}|\phi\rangle=\sqrt{N}\mathrm{e}^{\mathrm{i}\theta}|\phi\rangle$,
but not that of the particle number $\hat{N}=\hat{a}^{\dagger}_{0}\hat{a}_{0}$
in the Fock space. However, for a large enough ensemble of DM, there is a
well-defined mean particle number
$\langle\hat{a}^{\dagger}_{0}\hat{a}_{0}\rangle=N$ with a quantum fluctuation
proportional to $1/\sqrt{N}$. In fact, for the state with a definitive
particle number, e.g., $\hat{a}^{\dagger}_{0}\hat{a}_{0}|N\rangle=N|N\rangle$,
the field expectation value $\left\langle
N\right|\hat{\phi}\left|N\right\rangle$ is vanishing. The difference between
$|\phi\rangle$ and $|{N}\rangle$ is critical and should be highlighted, as
they will induce very distinct signals experimentally. For instance, for
$|{N}\rangle$ the absolute quantum phase
$\mathrm{exp}(-\mathrm{i}\,m_{\phi}t)$ is meaningless in practice, but the
classical phase of $|\phi\rangle$ is physical and will lead to time-varying
signals DirectMeasurementofLightWaves ; Barnett1996 .

In the early Universe, the problem is ascribed to solving the evolution of
classical field. Because the ultralight DM should be homogeneously produced,
the evolution equation of Eq. (3) can be recast to Esteban:2021ozz

$\displaystyle\ddot{\phi}+3H\dot{\phi}+m^{2}_{\phi}\phi=-g\int\frac{\mathrm{d}^{3}\bm{p}}{(2\pi)^{3}}\frac{g\phi}{\sqrt{\bm{p}^{2}+(g\phi)^{2}}}f_{\nu}(a|\bm{p}|),\hskip
14.22636pt$ (8)

where $a$ is the scale factor, connected to the redshift $z$ via $a\equiv
a_{0}/(z+1)$ with $a_{0}$ being the scale factor at present. Here, the
neutrino distribution function simply follows
$f_{\nu}(a|\bm{p}|)=D_{\nu}/(\mathrm{e}^{|\bm{p}|/T_{\nu}(a)}+1)$ with the
temperature $T_{\nu}(a)\propto 1/a$ and the factor $D_{\nu}=6$ accommodating
three generations of neutrinos and antineutrinos. The source term on the
right-hand side of Eq. (8) modifies the evolution of a free scalar. Note that
we have assumed the vacuum neutrino mass to be vanishing in the above
equation, and a larger effect of neutrino source will be expected if the
vacuum neutrino mass is taken into account.

###  II.1 Observable impacts

If $T_{\nu}\gg g\phi$ is satisfied (i.e., neutrinos in the plasma are
relativistic), the source term is equivalent to a screening mass of the
scalar, with Davoudiasl:2018hjw ; Esteban:2021ozz

$\displaystyle m^{2}_{\phi,\rm
s}(a)=\int\frac{\mathrm{d}^{3}\bm{p}}{(2\pi)^{3}}\frac{g^{2}}{|\bm{p}|}f_{\nu}(a|\bm{p}|)\sim
g^{2}T^{2}_{\nu}.\hskip 14.22636pt$ (9)

The screening mass describes how the neutrino plasma should response to the
change of the scalar potential, and its main effect will be altering the
oscillating frequency of scalar field. In this case, we have

$\displaystyle\ddot{\phi}+3H\dot{\phi}+\left[m^{2}_{\phi}+m^{2}_{\phi,\rm
s}(a)\right]\phi=0\;,\hskip 14.22636pt$ (10)

with an effective mass varying as the Universe expands, $m_{\phi,{\rm
eff}}=[m^{2}_{\phi}+m^{2}_{\phi,\rm s}(a)]^{1/2}$. For the numerical
convenience, it is useful to replace the physical time $t$ with the scale
factor $a=m\,x$, where $m$ is an arbitrary scale and $x$ is dimensionless. The
equation of motion can be recast into

$\displaystyle
x^{2}\phi^{\prime\prime}+\left(4x+x^{2}\frac{H^{\prime}}{H}\right)\phi^{\prime}+\frac{m^{2}_{\phi,{\rm
eff}}}{H^{2}}\phi=0\;,$ (11)

where ${\phi}^{\prime}\equiv\mathrm{d}\phi/\mathrm{d}x$.

The scalar field $\phi$ starts to oscillate when $H\lesssim m_{\phi,{\rm
eff}}$. If $\phi$ is not coupled to neutrinos, its amplitude will simply scale
as $\phi_{0}\propto a^{-3/2}$ Marsh:2015xka , and the energy density
$\rho_{\phi}\approx m^{2}_{\phi}\phi^{2}_{0}/2$ will be proportional to
$a^{-3}$, which justifies its capability of being the cold DM. However, if the
scalar mass is significantly screened, this benefit will be lost due to the
behavior $m_{\phi,{\rm s}}\propto a^{-1}$. In the completely screened case,
i.e. $m_{\phi,{\rm s}}\gg m_{\phi}$, we find 222In the fast-oscillating
regime, to obtain the evolution of the amplitude we decompose the scalar field
as $\phi(t)=\phi_{0}(t)\cos{(m_{\phi,{\rm eff}}\cdot t)}$ Marsh:2015xka .
Keeping the leading order in the assumption of $H\sim\dot{\phi}_{0}\ll
m_{\phi,{\rm eff}}$, we obtain
$2\dot{\phi_{0}}=-3H\phi_{0}-\phi_{0}\dot{m}_{\phi,{\rm eff}}/m_{\phi,{\rm
eff}}$. For $m_{\phi,{\rm eff}}\propto a^{-1}$, we have
$\dot{\phi_{0}}=-H\phi_{0}$, and hence $\phi_{0}\propto a^{-1}$.

$\displaystyle\phi_{0}\propto
a^{-1}\;,~{}\rho_{\phi}\sim\frac{m^{2}_{\phi,{\rm s}}\phi^{2}_{0}}{2}\propto
a^{-4}\;,$ (12)

which indicates that $\phi$ evolves more like a radiation instead of cold DM
with the screened mass. The screening effect is thus to be constrained by the
cosmological observations if we require the scalar to be DM.

An example of $\phi^{2}$ ($\geq 0$) evolution with respect to the scale factor
$a$ is given in Fig. 2 by numerically solving Eq. (11). The lighter fast-
oscillating curves stand for the magnitude of $\phi^{2}$ with (red) or without
(blue) the screening effect, and the darker ones are produced by averaging
over fast-oscillating modes to see the overall amplitude. As the screening
mass $m_{\phi,{\rm s}}$ is dominant for $a<500$, the scalar field scales as
$a^{-1}$ instead of $a^{-3/2}$ and contributes to the radiation component.
After $a=500$ the vacuum mass starts to dominate over the screening one, and
the scalar field plays the role of DM as normal.

Figure 2:  A demonstration of evolution of ultralight DM as a function of the
scale factor. For demonstration we give $\phi^{2}$ in the logarithmic scale.
The case with the screening effect of neutrinos is shown as the red curve,
while that without the screening effect is given as the blue one. The lighter
curves are the actual solutions, and the darker ones are averaged over certain
scale factor to see the magnitude of the field.

Another effect to avoid is the impact on neutrino velocity $v_{\nu}$, which is
related to the neutrino free-streaming length before the decoupling of CMB. As
has been mentioned, in the background of ultralight DM, neutrinos acquire an
effective mass $m_{\nu,{\rm eff}}=g\phi$. The neutrino velocity in the early
Universe will be given by

$\displaystyle\frac{v_{\nu}}{c}=\frac{|\bm{p}|}{\sqrt{\bm{p}^{2}+m^{2}_{\nu,{\rm
eff}}}}\;,$ (13)

where the average momentum of neutrino plasma reads
$\langle|\bm{p}|\rangle\sim 3T_{\nu}$. Notice that the effective neutrino mass
$m_{\nu,{\rm eff}}\propto a^{-3/2}$ increases faster than $T_{\nu}\propto
a^{-1}$ as we go to higher redshift $z$. This means that neutrinos will become
more and more non-relativistic and hence less and less free-streaming at
higher $z$. In comparison, the CMB data are consistent with neutrino free-
streaming with a speed of light up to a redshift as high as $z=10^{5}$
Hannestad:2004qu ; Hannestad:2005ex ; Bell:2005dr ; Basboll:2008fx ;
Archidiacono:2013dua ; Forastieri:2015paa ; Cyr-Racine:2013jua ;
Oldengott:2014qra ; Forastieri:2017oma ; Oldengott:2017fhy ; Choudhury:2020tka
; Brinckmann:2020bcn ; Das:2020xke ; Mazumdar:2020ibx . For simplicity, in the
following we will consider neutrinos as free-streaming with the speed of light
if $T_{\nu}>g\phi$, and vise versa.

It is worth noting that the CMB constraint on the sum of neutrino masses is
not as competitive as the above two effects. The combined analysis of
$Planck~{}{\rm TT,TE,EE+lowE+lensing+BAO}$ data leads to $\Sigma
m_{\nu}<0.12~{}{\rm eV}$ Planck:2018vyg , which actually takes account of data
at very low redshift long after the recombination $z_{\rm rc}=1100$. In fact,
around the recombination era neutrinos with a mass $\mathcal{O}(0.1)~{}{\rm
eV}$ are still relativistic. Since $m_{\nu,{\rm eff}}\propto a^{-3/2}$, the
impact of effective neutrino masses turns out to be less and less important at
lower $z$. Hence, we do not attempt to set the bound according to $\Sigma
m_{\nu,\rm eff}(z)<0.12~{}{\rm eV}$ at recombination $z_{\rm rc}=1100$, which
has been considered previously.

Figure 3:  The evolution of average neutrino velocity in terms of the redshift
in the early Universe. The colorful curves stand for the cases with a coupling
between neutrinos and the ultralight DM, with the vacuum neutrino mass
vanishing and DM-induced neutrino mass $g\phi$ at present indicated close to
the curve. The black curve represents the neutrino velocity of the standard
case with a mass $m_{\nu}=0.04~{}{\rm eV}$.

###  II.2 Updated CMB constraints

Depending on the parameter choice of $g$ and $m_{\phi}$, there can be four
different scenarios in the early Universe for the evolution equation of Eq.
(8):

  * •

Case I: $g\phi<T_{\nu}$ and $g^{2}T^{2}_{\nu}<m^{2}_{\phi}$. The effective
neutrino mass is negligible compared to the neutrino temperature and the
screening effect is also not important. In this case, the scalar field evolves
like matter with $\phi_{0}\propto a^{-3/2}$ while $T_{\nu}\propto a^{-1}$, and
these two conditions will maintain themselves as the Universe expands.

  * •

Case II: $g\phi<T_{\nu}$ and $g^{2}T^{2}_{\nu}>m^{2}_{\phi}$. Neutrinos are
relativistic but the screening scalar mass $m_{\phi,\rm s}\sim gT_{\nu}$
dominates the oscillation of $\phi$. In such scenario, the scalar field
evolves like radiation with $\phi_{0}\propto a^{-1}$. This state is not stable
and will collapse into Case I eventually.

  * •

Case III: $g\phi>T_{\nu}$ and $gT^{3}_{\nu}<m^{2}_{\phi}\phi$. The screening
effect is not important compared to the vacuum mass of $\phi$, but neutrinos
are non-relativistic (thus not free-streaming with $c$) due to the large
effective mass. This case will also end up into Case I as the Universe
expands.

  * •

Case IV: $g\phi>T_{\nu}$ and $gT^{3}_{\nu}>m^{2}_{\phi}\phi$. Both the
screening effect and the effective neutrino mass are dominant factors, which
is to be avoided over the observable history of CMB.

For Cases I and III, in terms of redshift, the neutrino temperature and the
scalar field strength read as $T_{\nu}\approx 1.68\times 10^{-4}~{}{\rm
eV}(z+1)$ and $\phi\approx 2.15\times 10^{-3}~{}{\rm
eV}^{2}/m_{\phi}\left[(z+1)/45\right]^{3/2}$, respectively, assuming the
ultralight scalar accommodates all the DM abundance.

The evolution of the Universe can be in a sequence of Case III$\to$I or
II$\to$I. In Fig. 3, we illustrate the neutrino velocity evolution $\langle
v_{\nu}\rangle/c$, i.e., Eq. (13), in terms of the redshift $z$ from Case III
to I, where the screening effect is not important but the neutrino effective
mass is severely affected. This can be achieved by properly choosing the DM
parameter. It can be noticed that the neutrino velocity drops rapidly as we go
back to higher $z$, which should be avoided to be in accordance with CMB free-
streaming requirement. In comparison, we also give the neutrino velocity in
the standard case with a constant mass $m_{\nu}=0.04~{}{\rm eV}$ corresponding
to the Planck limit $\Sigma m_{\nu}<0.12~{}{\rm eV}$. In this case, the
neutrino velocity evolves in an opposite manner compared to that with a
coupling to the ultralight DM. As has been mentioned, around the recombination
era $z_{\rm rc}\approx 1100$, neutrinos of the Planck limit are still ultra-
relativistic, and it is the data at lower $z$ that contribute to this limit.

In order not to significantly alter the CMB observations, two criteria should
be satisfied: (1) the ultralight scalar must evolve as matter before the
matter-radiation equality around $z_{\rm eq}=3000$; (2) neutrinos should be
free-streaming up to the redshift $z_{\rm eq}=3000$ (conservative) or $z_{\rm
fs}=10^{5}$ (aggressive). The aggressive redshift for neutrino free-streaming
is inspired by the investigation of neutrino self-interactions
Hannestad:2004qu ; Hannestad:2005ex ; Bell:2005dr ; Basboll:2008fx ;
Archidiacono:2013dua ; Forastieri:2015paa ; Cyr-Racine:2013jua ;
Oldengott:2014qra ; Forastieri:2017oma ; Oldengott:2017fhy ; Choudhury:2020tka
; Brinckmann:2020bcn ; Das:2020xke ; Mazumdar:2020ibx , which similarly
shrinks the neutrino free-streaming length. One may argue that the scalar
becomes a dynamical field (acquiring the shift-symmetry breaking terms) only
after a certain redshift, e.g., $z=10^{4}$, such that the aggressive free-
streaming limit does not apply in general. But there is no way to avoid the
robust limit set at $z_{\rm eq}=3000$, because the CMB data require the
dominance of matter component after that redshift.

If these two criteria are satisfied, during the observable history of CMB the
ultralight scalar will play the role of cold DM and neutrinos will stream as
free particles. We can evolve the current Universe back to higher redshift for
the CMB constraints. To be more specific, we require $g\phi<T_{\nu}$ and
$g^{2}T^{2}_{\nu}<m^{2}_{\phi}$ before $z=3000$ for the conservative limit,
and $g\phi<T_{\nu}$ before $z=10^{5}$ for the aggressive one. We obtain our
CMB limits as

$\displaystyle g$ $\displaystyle<$ $\displaystyle 0.4\cdot\frac{m_{\phi}}{\rm
eV}\hskip 14.22636pt({\rm conservative}),$ (14) $\displaystyle g$
$\displaystyle<$ $\displaystyle 0.09\cdot\frac{m_{\phi}}{\rm eV}\hskip
9.67383pt({\rm aggressive}),$ (15)

which correspond to the limits on the effective neutrino mass $g\phi$ in our
local galaxy

$\displaystyle m_{\nu,{\rm eff}}$ $\displaystyle<$ $\displaystyle 9\times
10^{-4}~{}{\rm eV}\hskip 11.38092pt({\rm conservative}),$ (16) $\displaystyle
m_{\nu,{\rm eff}}$ $\displaystyle<$ $\displaystyle 2\times 10^{-4}~{}{\rm
eV}\hskip 11.38092pt({\rm aggressive}).$ (17)

These limits are dominated by the neutrino free-streaming requirement. We find
that, for the aggressive limit, as we go to higher redshift the state will
transit from Case I to II around $z=6\times 10^{4}$, after which the free-
streaming condition $g\phi<T_{\nu}$ becomes stabilized as $\phi_{0}\propto
a^{-1}$ for Case II. For $g$ larger than the aggressive limit Case I will
transit to III instead before $z_{\rm fs}=10^{5}$, and the free-streaming
requirement is spoiled. We need to point out that a thorough analysis of the
CMB spectrum by simulating the evolution of perturbations in this context will
make the results more robust, and we leave it as a possible future work.

##  III Implication for $\boldsymbol{0\nu\beta\beta}$ decays

There are many dedicated experiments that are looking for the signatures of
$0\nu\beta\beta$ decays, e.g., GERDA Phase-II Agostini:2018tnm , CUORE
Alduino:2017ehq , SuperNEMO Barabash:2011aa , KamLAND-Zen KamLAND-Zen:2016pfg
and EXO Agostini:2017jim . The discovery of lepton-number-violating processes
in the future can reveal the Majorana nature of massive neutrinos
Dolinski:2019nrj .

As has been shown in Fig. 1, even without the vacuum Majorana neutrino mass
the $0\nu\beta\beta$ transition can still take place in the background of
ultralight DM. The induced transition is very similar to the Majoron emission
process in vacuum. However, because the vacuum in our case is occupied by
dense coherent scalars, the dominant process will be the “stimulated” emission
and absorption of the scalar particle, in analogy with the atomic transition
in a laser field.

Because the occupation number of scalars $N=n_{\phi}\lambda^{3}$ is very
large, such a transition will leave the DM field almost unaffected. This is
equivalent to simply taking the expectation value for all relevant operators
in the Lagrangian or scattering matrix. In the presence of ultralight DM, the
effective neutrino mass ($m_{ee}$) will hence receive an additional time-
varying contribution

$\displaystyle{m}^{\prime}_{ee}(t)=\langle\phi|m_{ee}+g\hat{\phi}(t,x)|\phi\rangle=m_{ee}+g{\phi}(t,x)\;,$
(18)

The $0\nu\beta\beta$-decay process typically takes place with a very small
uncertainty of time. It is therefore a good approximation to take the factor
$\cos{m_{\phi}t}$ as a constant during the transition. But the long-term
events will feature a time-varying pattern if enough data have been collected.

The above result should be able to be reproduced from the more fundamental
quantum field theory. However, it is not apparent to obtain the time-varying
effect from a view of the Feynman diagrams as illustrated in Fig. 1, where the
emission and absorption processes are themselves time-independent. To
understand this, we note that the mass of the ultralight DM ($<{\rm eV}$),
namely the change in the energy of final electrons, is much smaller than the
energy resolution of $0\nu\beta\beta$-decay experiments ($\sim{\rm keV}$).
Hence, it is not possible at the quantum level to distinguish between the
emission and absorption diagrams, which will lead to an interference. One can
find that it is this interference that gives rise to the time variation. To be
more precise, after the $0\nu\beta\beta$ transition via the vertex
$(m_{ee}+g\hat{\phi})\overline{\nu^{\rm c}_{\rm L}}\nu_{\rm L}/2$ the scalar
state will be taken to

$\displaystyle\left[m_{ee}+g\hat{\phi}(t,x)\right]\left|\phi\right\rangle$
$\displaystyle=\left[m_{ee}+g\sqrt{\frac{2}{m_{\phi}V}}\left(\hat{a}\mathrm{e}^{-\mathrm{i}m_{\phi}t}+\hat{a}^{\dagger}\mathrm{e}^{\mathrm{i}m_{\phi}t}\right)\right]|\phi\rangle\;.$
(19)

Note that $|\phi\rangle$, $\hat{a}|\phi\rangle$ and
$\hat{a}^{\dagger}|\phi\rangle$ almost completely overlap with each other,
e.g.,
$\langle\phi|\hat{a}\,\hat{a}|\phi\rangle=N\mathrm{e}^{2\mathrm{i}\theta}$.
Together with other terms, this feature results in a probability proportional
to $\Sigma_{N}|\left\langle N\right|m_{ee}+g\hat{\phi}(t,x)|\phi\rangle|^{2}$,
namely $\left|m_{ee}+g{\phi}(t,x)\right|^{2}$. The above observations are also
applicable to the impact on neutrino oscillation experiments.

As for our $0\nu\beta\beta$ process, the half-life of a given isotope can be
expressed as Rodejohann:2011mu

$\dfrac{1}{T^{0\nu}_{1/2}}=G_{0\nu}\cdot\left|{M}_{0\nu}\right|^{2}\cdot\frac{\left|m^{\prime}_{ee}(t)\right|^{2}}{M^{2}_{e}}\;,$
(20)

where $G_{0\nu}$ represents the phase-space factor, $M_{0\nu}$ is the nuclear
matrix element (NME), and $M_{e}\approx 0.51~{}{\rm MeV}$ is the electron
mass. The present lower limit on the half-life of $0\nu\beta\beta$ decays from
KamLAND-Zen collaboration KamLAND-Zen:2016pfg  is $T^{0\nu}_{1/2}>1.07\times
10^{26}$ yr at 90% C. L. Using the phase-space factor $G_{0\nu}=3.793\times
10^{-14}~{}{\rm yr^{-1}}$ Kotila:2012zza  with $g=1.27$ and NME
$M_{0\nu}\in(1.5-4.2)$ Guzowski:2015saa  for ${}^{136}{\rm Xe}$, the upper
bound on the effective neutrino mass in standard three-flavor neutrino
scenario is obtained as $|m_{ee}|<(61-169)$ meV. Here, $m_{ee}$ is given by

$\displaystyle{m}_{ee}$ $\displaystyle\equiv$ $\displaystyle
m_{1}\cos^{2}\theta_{13}\cos^{2}\theta_{12}e^{{\rm
i}\rho}+m_{2}\cos^{2}\theta_{13}\sin^{2}\theta_{12}$ (21)
$\displaystyle+m_{3}\sin^{2}\theta_{13}e^{{\rm i}\sigma}\;,$

where $m_{i}$ (for $i=1,2,3$) stand for the absolute masses of three
neutrinos, $\\{\theta_{12},\theta_{13}\\}$ are the leptonic mixing angles, and
$\\{\rho,\sigma\\}$ are the Majorana CP phases.

Figure 4:  Exclusion and sensitivities of various experiments on the parameter
space of the ultralight DM mass $m_{\phi}$ and the coupling to neutrinos $g$.
The exclusion using the result of KamLAND-Zen are given by cyan lines, while
the CMB bounds in present work are shown as the dashed brown and solid black
lines. See the text for details of other limits.

To find out how the ultralight scalar exactly modifies the rate, we expand the
vacuum and DM contributions

$\displaystyle|{m}^{\prime}_{ee}|^{2}$ $\displaystyle=$
$\displaystyle\left|m_{ee}\right|^{2}+g^{2}\phi^{2}_{0}\cos^{2}{m_{\phi}t}$
(22) $\displaystyle+2|m_{ee}|g\phi_{0}\cos\eta\cos{m_{\phi}t}\;,$

where $\eta$ is the relative phase between $m_{ee}$ and $g\phi$, the second
term is induced solely by the DM field, and the third term is the interference
with the vacuum neutrino mass. Such time-varying effect is in principle
observable, if enough $0\nu\beta\beta$-decay events with proper time
resolution have been obtained. When the ultralight DM oscillates fast enough,
the long-term decay rate depends on the time average of
$|{m}^{\prime}_{ee}|^{2}$, i.e.,

$\displaystyle\overline{|{m}^{\prime}_{ee}|^{2}}=\left|m_{ee}\right|^{2}+\frac{1}{2}g^{2}\phi^{2}_{0}\;.$
(23)

Note that the interference term has been averaged out in such case. The
coupling constant $g$ is in general not related to $m_{ee}$. Hence, when the
cancellation of effective neutrino mass occurs, i.e., $m_{ee}=0$ for normal
ordering, the DM-induced term can dominate the transition.

We present our noteworthy results as Fig. 4 in the $m_{\phi}$-$g$ parameter
plane. The exclusion regions of $0\nu\beta\beta$-decay experiments and
cosmology are shown along with bounds and sensitivities arising from different
experimental searches. The vertical blue band shows the constraint based on
the results of Lyman-$\alpha$ forest. It has been pointed out (see for
detailed analysis in Ref. Hui:2016ltb ) that fuzzy DM lighter than $\sim
10^{-21}$ eV is in tension with observations of the Lyman-$\alpha$ forest. The
gray shaded region gives the limit from Super-K and Sudbury Neutrino
Observatory (SNO) neutrino oscillation experiments Berlin:2016woy ; Super-
Kamiokande:2003snd ; SNO:2005ftm . The sensitivity from forthcoming neutrino
oscillation experiments Krnjaic:2017zlz ; Dev:2020kgz  like DUNE DUNE:2015lol
and JUNO JUNO:2015zny  have also been presented using the magenta shaded
region and orange dot-dashed line, respectively.

The constraint from $0\nu\beta\beta$-decay searches (KamLAND-Zen KamLAND-
Zen:2016pfg ) is shown as the solid cyan line. The projection with an ultimate
sensitivity goal of $0\nu\beta\beta$-decay experiments
$|m^{\prime}_{ee}|=1~{}{\rm meV}$ Ge:2016tfx ; Penedo:2018kpc ; Cao:2019hli ;
Huang:2020mkz  is shown as the dotted cyan line. In order to translate results
into the $m_{\phi}$-$g$ plane, we have used Eq. (23). We present our CMB
bounds as the dashed brown and solid black lines, respectively. Two different
limits corresponding to conservative and aggressive considerations are shown,
as given by Eqs. (14) and (15), respectively. It can be noticed that JUNO can
cover the limit of $0\nu\beta\beta$-decay experiment for most of the parameter
spcae. However, for the region where the ultralight DM mass is greater than
$10^{-11}~{}{\rm eV}$, $0\nu\beta\beta$-decay experiments set tighter bound
than neutrino oscillation experiments. On the other hand, Super-K and SNO put
the best constraint for ultralight DM below $10^{-17}~{}{\rm eV}$, whereas the
CMB analysis provides the most stringent limit for ultralight DM above
$10^{-17}$ eV.

Figure 5:  The effective Majorana neutrino mass as a function of the lightest
active neutrino mass $m_{1}$. The orange region stands for the standard case
with normal mass ordering for neutrinos, while the blue region is for the case
with $g\phi_{0}=0.01~{}{\rm eV}$.

Finally, as a visual guide for the ultralight DM searches at
$0\nu\beta\beta$-decay experiments, we present the effective Majorana neutrino
mass, in presence of ultralight DM, in Fig. 5 Graf:2020cbf . The horizontal
green band indicates the current experimental limit from KamLAND-Zen,
$(61-165)~{}{\rm meV}$ KamLAND-Zen:2016pfg . The dashed lines correspond to
the future sensitivities projected by SNO+ with $(19-46)~{}{\rm meV}$
Andringa:2015tza , LEGEND with $(10.7-22.8)~{}{\rm meV}$ Abgrall:2017syy , and
nEXO with $(5.7-17.7)~{}{\rm meV}$ Albert:2017hjq , respectively. The vertical
gray band represents the current neutrino mass limit of cosmological data from
the Planck collaboration Aghanim:2018eyx . The orange region depicts the
standard three-flavor neutrino scenario. The impact of ultralight DM has been
demonstrated using the blue region for a benchmark choice of
$g\phi_{0}=0.01~{}{\rm eV}$. It turns out that even the averaged effect of
ultralight DM can be tested by the next generation of $0\nu\beta\beta$-decay
experiments.

##  IV Conclusion

In this work, we investigate the impact of ultralight DM on
$0\nu\beta\beta$-decay experiments. We find that the scalar DM can directly
induce the $0\nu\beta\beta$-decay transition, even without the vacuum neutrino
mass term. The effect can be significant for very small couplings between the
ultralight DM and neutrinos, due to the stimulated absorption and emission of
scalars in the phase space filled with identical particles. As the
consequence, the vacuum Majorana neutrino mass will be modified with a time-
varying term.

We also explore the consequence of such a scalar field on cosmology. There are
mainly two effects. First, the neutrino plasma will induce a screening mass to
the ultralight DM, such that the DM field evolves as radiation instead of
matter component. Second, the dense ultralight DM in the early Universe will
shrink the neutrino free-streaming length significantly and is in tension with
the CMB observations.

The constraints and sensitivities of different experiments are given in Fig.
4. We conclude that $0\nu\beta\beta$-decay experiments can give stringent
constraint compared to the forthcoming neutrino oscillation experiments like
DUNE and JUNO for the DM mass above $\thicksim 10^{-11}$ eV. In addition, the
CMB observations have the best constraining capability for the mass range
above $\thicksim 10^{-17}$ eV. Furthermore, we observe from Fig. 5 that our
framework of ultralight DM can be tested even without analyzing the time-
varying effect by the next generation of $0\nu\beta\beta$-decay experiments.

###### Acknowledgements.

Authors are thankful to Werner Rodejohann for his valuable comments and
careful reading of the manuscript. Authors would also like to thank Eligio
Lisi, Pablo Martínez-Miravé, Manibrata Sen, and Shun Zhou for fruitful
discussions. GYH is supported by the Alexander von Humboldt Foundation. NN is
supported by the Istituto Nazionale di Fisica Nucleare (INFN) through the
“Theoretical Astroparticle Physics” (TAsP) project.

## References

  * (1) S. Bird, I. Cholis, J. B. Muñoz, Y. Ali-Haïmoud, M. Kamionkowski, E. D. Kovetz, A. Raccanelli, and A. G. Riess, “Did LIGO detect dark matter?,” Phys. Rev. Lett. 116 (2016) no. 20, 201301, arXiv:1603.00464.
  * (2) B. Carr, F. Kuhnel, and M. Sandstad, “Primordial Black Holes as Dark Matter,” Phys. Rev. D 94 (2016) no. 8, 083504, arXiv:1607.06077.
  * (3) L. Chen, Q.-G. Huang, and K. Wang, “Constraint on the abundance of primordial black holes in dark matter from Planck data,” JCAP 12 (2016) 044, arXiv:1608.02174.
  * (4) J. Georg and S. Watson, “A Preferred Mass Range for Primordial Black Hole Formation and Black Holes as Dark Matter Revisited,” JHEP 09 (2017) 138, arXiv:1703.04825.
  * (5) W. Hu, R. Barkana, and A. Gruzinov, “Cold and fuzzy dark matter,” Phys. Rev. Lett. 85 (2000) 1158–1161, arXiv:astro-ph/0003365.
  * (6) XENON, E. Aprile et al., “Search for Coherent Elastic Scattering of Solar 8B Neutrinos in the XENON1T Dark Matter Experiment,” Phys. Rev. Lett. 126 (2021) 091301, arXiv:2012.02846.
  * (7) PandaX-II, X. Cui et al., “Dark Matter Results From 54-Ton-Day Exposure of PandaX-II Experiment,” Phys. Rev. Lett. 119 (2017) no. 18, 181302, arXiv:1708.06917.
  * (8) E. G. M. Ferreira, “Ultra-light dark matter,” Astron. Astrophys. Rev. 29 (2021) no. 1, 7, arXiv:2005.03254.
  * (9) L. A. Ureña López, “Brief Review on Scalar Field Dark Matter Models,” Front. Astron. Space Sci. 6 (2019) 47.
  * (10) L. Hui, J. P. Ostriker, S. Tremaine, and E. Witten, “Ultralight scalars as cosmological dark matter,” Phys. Rev. D 95 (2017) no. 4, 043541, arXiv:1610.08297.
  * (11) J. E. Kim and G. Carosi, “Axions and the Strong CP Problem,” Rev. Mod. Phys. 82 (2010) 557–602, arXiv:0807.3125. [Erratum: Rev.Mod.Phys. 91, 049902 (2019)].
  * (12) P. W. Graham, I. G. Irastorza, S. K. Lamoreaux, A. Lindner, and K. A. van Bibber, “Experimental Searches for the Axion and Axion-Like Particles,” Ann. Rev. Nucl. Part. Sci. 65 (2015) 485–514, arXiv:1602.00039.
  * (13) I. G. Irastorza and J. Redondo, “New experimental approaches in the search for axion-like particles,” Prog. Part. Nucl. Phys. 102 (2018) 89–159, arXiv:1801.08127.
  * (14) D. J. E. Marsh, “Axion Cosmology,” Phys. Rept. 643 (2016) 1–79, arXiv:1510.07633.
  * (15) S. Dell’Oro, S. Marcocci, M. Viel, and F. Vissani, “Neutrinoless double beta decay: 2015 review,” Adv. High Energy Phys. 2016 (2016) 2162659, arXiv:1601.07512.
  * (16) J. D. Vergados, H. Ejiri, and F. Šimkovic, “Neutrinoless double beta decay and neutrino mass,” Int. J. Mod. Phys. E 25 (2016) no. 11, 1630007, arXiv:1612.02924.
  * (17) A. Berlin, “Neutrino Oscillations as a Probe of Light Scalar Dark Matter,” Phys. Rev. Lett. 117 (2016) no. 23, 231801, arXiv:1608.01307.
  * (18) V. Brdar, J. Kopp, J. Liu, P. Prass, and X.-P. Wang, “Fuzzy dark matter and nonstandard neutrino interactions,” Phys. Rev. D 97 (2018) no. 4, 043001, arXiv:1705.09455.
  * (19) G. Krnjaic, P. A. N. Machado, and L. Necib, “Distorted neutrino oscillations from time varying cosmic fields,” Phys. Rev. D 97 (2018) no. 7, 075017, arXiv:1705.06740.
  * (20) J. Liao, D. Marfatia, and K. Whisnant, “Light scalar dark matter at neutrino oscillation experiments,” JHEP 04 (2018) 136, arXiv:1803.01773.
  * (21) F. Capozzi, I. M. Shoemaker, and L. Vecchi, “Neutrino Oscillations in Dark Backgrounds,” JCAP 07 (2018) 004, arXiv:1804.05117.
  * (22) M. M. Reynoso and O. A. Sampayo, “Propagation of high-energy neutrinos in a background of ultralight scalar dark matter,” Astropart. Phys. 82 (2016) 10–20, arXiv:1605.09671.
  * (23) G.-Y. Huang and N. Nath, “Neutrinophilic Axion-Like Dark Matter,” Eur. Phys. J. C 78 (2018) no. 11, 922, arXiv:1809.01111.
  * (24) S. Pandey, S. Karmakar, and S. Rakshit, “Interactions of Astrophysical Neutrinos with Dark Matter: A model building perspective,” JHEP 01 (2019) 095, arXiv:1810.04203.
  * (25) Y. Farzan and S. Palomares-Ruiz, “Flavor of cosmic neutrinos preserved by ultralight dark matter,” Phys. Rev. D 99 (2019) no. 5, 051702, arXiv:1810.00892.
  * (26) K.-Y. Choi, J. Kim, and C. Rott, “Constraining dark matter-neutrino interactions with IceCube-170922A,” Phys. Rev. D 99 (2019) no. 8, 083018, arXiv:1903.03302.
  * (27) S. Baek, “Dirac neutrino from the breaking of Peccei-Quinn symmetry,” Phys. Lett. B 805 (2020) 135415, arXiv:1911.04210.
  * (28) K.-Y. Choi, E. J. Chun, and J. Kim, “Neutrino Oscillations in Dark Matter,” Phys. Dark Univ. 30 (2020) 100606, arXiv:1909.10478.
  * (29) K.-Y. Choi, E. J. Chun, and J. Kim, “Dispersion of neutrinos in a medium,” arXiv:2012.09474.
  * (30) A. Dev, P. A. N. Machado, and P. Martínez-Miravé, “Signatures of ultralight dark matter in neutrino oscillation experiments,” JHEP 01 (2021) 094, arXiv:2007.03590.
  * (31) S. Baek, “A connection between flavour anomaly, neutrino mass, and axion,” JHEP 10 (2020) 111, arXiv:2006.02050.
  * (32) M. Losada, Y. Nir, G. Perez, and Y. Shpilman, “Probing scalar dark matter oscillations with neutrino oscillations,” arXiv:2107.10865.
  * (33) A. Y. Smirnov and V. B. Valera, “Resonance refraction and neutrino oscillations,” arXiv:2106.13829.
  * (34) G. Alonso-Álvarez and J. M. Cline, “Sterile neutrino dark matter catalyzed by a very light dark photon,” arXiv:2107.07524.
  * (35) G.-y. Huang and W. Rodejohann, “Tritium beta decay with modified neutrino dispersion relations: KATRIN in the dark sea,” arXiv:2110.03718.
  * (36) K. Blum, Y. Nir, and M. Shavit, “Neutrinoless double-beta decay with massive scalar emission,” Phys. Lett. B 785 (2018) 354–361, arXiv:1802.08019.
  * (37) T. Brune and H. Päs, “Massive Majorons and constraints on the Majoron-neutrino coupling,” Phys. Rev. D 99 (2019) no. 9, 096005, arXiv:1808.08158.
  * (38) R. Cepedello, F. F. Deppisch, L. González, C. Hati, and M. Hirsch, “Neutrinoless Double-$\beta$ Decay with Nonstandard Majoron Emission,” Phys. Rev. Lett. 122 (2019) no. 18, 181801, arXiv:1811.00031.
  * (39) V. A. Miransky, Dynamical Symmetry Breaking In Quantum Field Theories.  World Scientific, 1994.
  * (40) “Lecture on ultralight dark matter,”. https://wwwmpa.mpa-garching.mpg.de/~komatsu/lecturenotes/Elisa_Ferreira_on_SuperfluidDarkMatter.pdf.
  * (41) S. Davidson, “Axions: Bose Einstein Condensate or Classical Field?,” Astropart. Phys. 65 (2015) 101–107, arXiv:1405.1139.
  * (42) J. Veltmaat, “Galaxy Formation With Ultralight Bosonic Dark Matter,”  2020\.  https://d-nb.info/121070269X/34.
  * (43) A. H. Guth, M. P. Hertzberg, and C. Prescod-Weinstein, “Do Dark Matter Axions Form a Condensate with Long-Range Correlation?,” Phys. Rev. D 92 (2015) no. 10, 103513, arXiv:1412.5930.
  * (44) W.-M. Zhang, “Coherent states in field theory,” arXiv:hep-th/9908117.
  * (45) S. Barnett, K. Burnett, and J. Vaccaro, “Why a Condensate Can Be Thought of as Having a Definite Phase,” Journal of Research of the National Institute of Standards and Technology 101 .
  * (46) E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct Measurement of Light Waves,” Science 305 (2004) no. 5688, 1267–1269.
  * (47) S. Barnett, K. Burnett, and A. Vacarro, “Why a Condensate can be Thought of Having a Definite Phase,,” J. Res. Natl. Inst. Stand. Technol. 101 (1996) 593.
  * (48) I. Esteban and J. Salvado, “Long Range Interactions in Cosmology: Implications for Neutrinos,” JCAP 05 (2021) 036, arXiv:2101.05804.
  * (49) H. Davoudiasl, G. Mohlabeng, and M. Sullivan, “Galactic Dark Matter Population as the Source of Neutrino Masses,” Phys. Rev. D 98 (2018) no. 2, 021301, arXiv:1803.00012.
  * (50) S. Hannestad, “Structure formation with strongly interacting neutrinos - Implications for the cosmological neutrino mass bound,” JCAP 02 (2005) 011, arXiv:astro-ph/0411475.
  * (51) S. Hannestad and G. Raffelt, “Constraining invisible neutrino decays with the cosmic microwave background,” Phys. Rev. D 72 (2005) 103514, arXiv:hep-ph/0509278.
  * (52) N. F. Bell, E. Pierpaoli, and K. Sigurdson, “Cosmological signatures of interacting neutrinos,” Phys. Rev. D 73 (2006) 063523, arXiv:astro-ph/0511410.
  * (53) A. Basboll, O. E. Bjaelde, S. Hannestad, and G. G. Raffelt, “Are cosmological neutrinos free-streaming?,” Phys. Rev. D 79 (2009) 043512, arXiv:0806.1735.
  * (54) M. Archidiacono and S. Hannestad, “Updated constraints on non-standard neutrino interactions from Planck,” JCAP 07 (2014) 046, arXiv:1311.3873.
  * (55) F. Forastieri, M. Lattanzi, and P. Natoli, “Constraints on secret neutrino interactions after Planck,” JCAP 07 (2015) 014, arXiv:1504.04999.
  * (56) F.-Y. Cyr-Racine and K. Sigurdson, “Limits on Neutrino-Neutrino Scattering in the Early Universe,” Phys. Rev. D 90 (2014) no. 12, 123533, arXiv:1306.1536.
  * (57) I. M. Oldengott, C. Rampf, and Y. Y. Y. Wong, “Boltzmann hierarchy for interacting neutrinos I: formalism,” JCAP 04 (2015) 016, arXiv:1409.1577.
  * (58) F. Forastieri, M. Lattanzi, G. Mangano, A. Mirizzi, P. Natoli, and N. Saviano, “Cosmic microwave background constraints on secret interactions among sterile neutrinos,” JCAP 07 (2017) 038, arXiv:1704.00626.
  * (59) I. M. Oldengott, T. Tram, C. Rampf, and Y. Y. Y. Wong, “Interacting neutrinos in cosmology: exact description and constraints,” JCAP 11 (2017) 027, arXiv:1706.02123.
  * (60) S. Roy Choudhury, S. Hannestad, and T. Tram, “Updated constraints on massive neutrino self-interactions from cosmology in light of the $H_{0}$ tension,” arXiv:2012.07519.
  * (61) T. Brinckmann, J. H. Chang, and M. LoVerde, “Self-interacting neutrinos, the Hubble parameter tension, and the Cosmic Microwave Background,” arXiv:2012.11830.
  * (62) A. Das and S. Ghosh, “Flavor-specific Interaction Favours Strong Neutrino Self-coupling,” arXiv:2011.12315.
  * (63) A. Mazumdar, S. Mohanty, and P. Parashari, “Flavour specific neutrino self-interaction: $H_{0}$ tension and IceCube,” arXiv:2011.13685.
  * (64) Planck, N. Aghanim et al., “Planck 2018 results. VI. Cosmological parameters,” Astron. Astrophys. 641 (2020) A6, arXiv:1807.06209. [Erratum: Astron.Astrophys. 652, C4 (2021)].
  * (65) GERDA, M. Agostini et al., “Improved Limit on Neutrinoless Double-$\beta$ Decay of 76Ge from GERDA Phase II,” Phys. Rev. Lett. 120 (2018) no. 13, 132503, arXiv:1803.11100.
  * (66) CUORE, C. Alduino et al., “First Results from CUORE: A Search for Lepton Number Violation via $0\nu\beta\beta$ Decay of 130Te,” Phys. Rev. Lett. 120 (2018) no. 13, 132501, arXiv:1710.07988.
  * (67) A. S. Barabash, “SeperNEMO double beta decay experiment,” J. Phys. Conf. Ser. 375 (2012) 042012, arXiv:1112.1784.
  * (68) KamLAND-Zen, A. Gando et al., “Search for Majorana Neutrinos near the Inverted Mass Hierarchy Region with KamLAND-Zen,” Phys. Rev. Lett. 117 (2016) no. 8, 082503, arXiv:1605.02889. [Addendum: Phys.Rev.Lett. 117, 109903 (2016)].
  * (69) M. Agostini, G. Benato, and J. Detwiler, “Discovery probability of next-generation neutrinoless double- $\beta$ decay experiments,” Phys. Rev. D 96 (2017) no. 5, 053001, arXiv:1705.02996.
  * (70) M. J. Dolinski, A. W. P. Poon, and W. Rodejohann, “Neutrinoless Double-Beta Decay: Status and Prospects,” Ann. Rev. Nucl. Part. Sci. 69 (2019) 219–251, arXiv:1902.04097.
  * (71) W. Rodejohann, “Neutrino-less Double Beta Decay and Particle Physics,” Int. J. Mod. Phys. E 20 (2011) 1833–1930, arXiv:1106.1334.
  * (72) J. Kotila and F. Iachello, “Phase space factors for double-$\beta$ decay,” Phys. Rev. C 85 (2012) 034316, arXiv:1209.5722.
  * (73) P. Guzowski, L. Barnes, J. Evans, G. Karagiorgi, N. McCabe, and S. Soldner-Rembold, “Combined limit on the neutrino mass from neutrinoless double-$\beta$ decay and constraints on sterile Majorana neutrinos,” Phys. Rev. D 92 (2015) no. 1, 012002, arXiv:1504.03600.
  * (74) Super-Kamiokande, J. Yoo et al., “A Search for periodic modulations of the solar neutrino flux in Super-Kamiokande I,” Phys. Rev. D 68 (2003) 092002, arXiv:hep-ex/0307070.
  * (75) SNO, B. Aharmim et al., “A Search for periodicities in the B-8 solar neutrino flux measured by the Sudbury neutrino observatory,” Phys. Rev. D 72 (2005) 052010, arXiv:hep-ex/0507079.
  * (76) DUNE, R. Acciarri et al., “Long-Baseline Neutrino Facility (LBNF) and Deep Underground Neutrino Experiment (DUNE): Conceptual Design Report, Volume 2: The Physics Program for DUNE at LBNF,” arXiv:1512.06148.
  * (77) JUNO, F. An et al., “Neutrino Physics with JUNO,” J. Phys. G 43 (2016) no. 3, 030401, arXiv:1507.05613.
  * (78) S.-F. Ge and M. Lindner, “Extracting Majorana properties from strong bounds on neutrinoless double beta decay,” Phys. Rev. D 95 (2017) no. 3, 033003, arXiv:1608.01618.
  * (79) J. T. Penedo and S. T. Petcov, “The $10^{-3}$ eV frontier in neutrinoless double beta decay,” Phys. Lett. B 786 (2018) 410–417, arXiv:1806.03203.
  * (80) J. Cao, G.-Y. Huang, Y.-F. Li, Y. Wang, L.-J. Wen, Z.-Z. Xing, Z.-H. Zhao, and S. Zhou, “Towards the meV limit of the effective neutrino mass in neutrinoless double-beta decays,” Chin. Phys. C 44 (2020) no. 3, 031001, arXiv:1908.08355.
  * (81) G.-y. Huang and S. Zhou, “Tentative sensitivity of future $0\nu\beta\beta$-decay experiments to neutrino masses and Majorana CP phases,” JHEP 03 (2021) 084, arXiv:2010.16281.
  * (82) L. Graf, S. Jana, M. Lindner, W. Rodejohann, and X.-J. Xu, “Flavored neutrinoless double beta decay,” Phys. Rev. D 103 (2021) no. 5, 055007, arXiv:2010.15109.
  * (83) SNO+, S. Andringa et al., “Current Status and Future Prospects of the SNO+ Experiment,” Adv.High Energy Phys. 2016 (2016) 6194250, arXiv:1508.05759.
  * (84) LEGEND, N. Abgrall et al., “The Large Enriched Germanium Experiment for Neutrinoless Double Beta Decay (LEGEND),” vol. 1894, p. 020027.  2017\.  arXiv:1709.01980.
  * (85) nEXO, J. Albert et al., “Sensitivity and Discovery Potential of nEXO to Neutrinoless Double Beta Decay,” Phys.Rev. C97 (2018) 065503, arXiv:1710.05075.
  * (86) Planck, N. Aghanim et al., “Planck 2018 results. VI. Cosmological parameters,” arXiv:1807.06209.