††thanks: Corresponding author

# Neutrinoless double beta decay in the minimal type-I seesaw model: How the
enhancement or cancellation happens?

Dong-Liang Fang<EMAIL_ADDRESS>Institute of Modern Physics, Chinese
Academy of sciences, Lanzhou, 730000, China  School of Nuclear Science and
Technology, University of Chinese Academy of Sciences, Beijing 100049, China
Yu-Feng Li<EMAIL_ADDRESS>Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing 100049, China  School of Physical Sciences,
University of Chinese Academy of Sciences, Beijing 100049, China  Yi-Yu Zhang
<EMAIL_ADDRESS>Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, China  School of Physical Sciences, University of
Chinese Academy of Sciences, Beijing 100049, China

( , )

###### Abstract

We discuss the contribution of right-handed neutrinos (RHNs) to the effective
neutrino mass of the neutrinoless double beta decay within the minimal type-I
seesaw model using the intrinsic seesaw relation of neutrino mass and mixing
parameters and the relative mass dependence of the nuclear matrix elements. In
the viable parameter space, we find the possibilities of both the enhancement
and cancellation to the effective neutrino mass from RHNs. The bounds on the
parameter space of the RHNs can be determined with the effective neutrino mass
extracted from neutrinoless double beta decay experiments.

##  I Introduction

The observation of neutrino oscillations from the atmospheric Fukuda  _et al._
(1998), solar Ahmad  _et al._ (2001, 2002a, 2002b), reactor Eguchi  _et al._
(2003); Abe  _et al._ (2012); Kim  _et al._ (2012); An  _et al._ (2012) and
accelerator Ahn  _et al._ (2003); Abe  _et al._ (2011); Adamson  _et al._
(2016) neutrinos is one of the most important discoveries in particle physics,
which indicates that neutrinos are massive, and is currently the only evidence
for physics beyond the Standard Model (SM). But neutrino oscillation
experiments do not allow us to determine the absolute neutrino mass scale, as
well as the origin of neutrino masses. To determine the absolute neutrino mass
scale, three complementary methods can be used. The first one is the
cosmological observation, which probe the direct sum of three neutrino masses
Alam  _et al._ (2017); Aghanim  _et al._ (2020); Palanque-Delabrouille  _et
al._ (2020); Di Valentino  _et al._ (2021). The second one is the
$\beta$-decay experiments, such as the KATRIN experiment, which gives the
limit on the effective neutrino mass from the spectral fine structure near the
$\beta$-decay endpoint Kraus  _et al._ (2005); Aseev  _et al._ (2012); Aker
_et al._ (2019). The third is the neutrinoless double beta decay (i.e.,
$0\nu\beta\beta$) experiments Andreotti  _et al._ (2011); Alfonso  _et al._
(2015); Klapdor-Kleingrothaus  _et al._ (2001); Aalseth  _et al._ (2000);
Agostini  _et al._ (2013); Albert  _et al._ (2014); Gando  _et al._ (2013)
that we will discuss in this work, for recent reviews see Ref. Dolinski  _et
al._ (2019); Bilenky and Giunti (2015); Dell’Oro  _et al._ (2016).

Neutrino masses are several orders of magnitude smaller than the masses of
charged leptons and quarks, and may not be (or not only) of the SM Higgs
origin. Therefore, alternative new mechanisms of the neutrino mass generation
have been proposed, of which the most plausible one is the type-I seesaw
mechanism Minkowski (1977); Yanagida (1979); Gell-Mann  _et al._ (1979);
Mohapatra and Senjanovic (1980). According to this mechanism, the small
neutrino masses are related to the gauge invariant masses of the SM singlets,
i.e., right-handed neutrinos (RHNs), which violates the total lepton number at
a mass scale much heavier than the electroweak interaction and generates the
Majorana masses of massive neutrinos. Within such mechanism, the neutrinos are
Majorana particles, and consequently, this leads to the above mentioned lepton
number violating (LNV) $0\nu\beta\beta$-decay process. The nature of
neutrinos, that is whether the neutrino mass is of the Majorana or Dirac type,
is important for our understanding of the origin for small neutrino masses.
With the neutrino Majorana nature, more phases (namely the Majorana phases)
will be included in the neutrino mixing matrix Xing (2020); Bilenky  _et al._
(1980). Neither results from neutrino oscillation experiments, cosmology
probes, nor that from $\beta$-decay experiments depend on these Majorana
phases. However, from the effective neutrino mass ($m_{\mathrm{eff}}$) of the
$0\nu\beta\beta$ decay, such information could be extracted Xing  _et al._
(2015); Xing and Zhao (2017); Ge and Lindner (2017); Penedo and Petcov (2018);
Cao  _et al._ (2020).

So far, the $0\nu\beta\beta$-decay hasn’t been detected yet and the lower
bounds of the decay half-lives for various isotopes are obtained from
different collaborations: such as CUORE Adams  _et al._ (2020), EXO-200 Anton
_et al._ (2019), GERDA Agostini  _et al._ (2020a) and KamLAND-Zen Gando  _et
al._ (2013); Abe  _et al._ (2022). These results help to extract
$m_{\mathrm{eff}}$ with account of nuclear matrix elements (NMEs) from nuclear
structure calculations. The most stringent bound on $m_{\mathrm{eff}}$ at
present is 36-156 meV obtained from the KamLAND-Zen experiment Abe  _et al._
(2022) with the uncertainties from NMEs included. Future experiments will
further push the limit down to 10 meV right below the inverted hierarchy (IH)
region Chen  _et al._ (2017); Akerib  _et al._ (2020); Agostini  _et al._
(2020b); Abgrall  _et al._ (2021).

The minimal seesaw model is the simplest model of the seesaw mechanism of
neutrino masses, with the inclusion of two additional RHNs King (2000);
Frampton  _et al._ (2002); Guo  _et al._ (2007); Xing and Zhao (2021), which
could be important contribution to the $0\nu\beta\beta$-decay process
Zel’dovich and Khlopov (1981); Ibarra  _et al._ (2010); Abada  _et al._
(2019); Bolton  _et al._ (2020); Ibarra  _et al._ (2011). When the masses of
RHNs are small, because of the intrinsic seesaw model, their contribution will
cancel that of active neutrinos and lead to extremely small $m_{\mathrm{eff}}$
approaching zero, then the $0\nu\beta\beta$-decay is forbidden Halprin  _et
al._ (1983); Leung and Petcov (1984); Blennow  _et al._ (2010). In contrast,
when the masses of RHNs are large enough, they can be regarded as short-range
contribution, thus the effective neutrino mass from each mass eigenstate will
be inversely proportional to the neutrino mass. However, when the masses of
RHNs are between these two extreme cases, the phenomenology of the effective
neutrino mass will be rather rich and complicated, which strongly depends on
the intrinsic seesaw relation of neutrino mass and mixing parameters and the
relative mass dependence of the NMEs. In this sense, considering the
contribution of RHNs, even active neutrino masses are with the inverted
hierarchy (IH), the possibility of non-observation of future experiments still
exists due to this cancellation. On the other hand, if active neutrino masses
are with the normal hierarchy (NH), the future experiments still have the
opportunity to observe this process with the enhancement to $m_{\mathrm{eff}}$
from RHNs. Therefore, thorough investigation is needed to clarify the roles of
the parameters of RHNs in mapping the viable parameter space.

In this paper, we study the contribution of RHNs to the $0\nu\beta\beta$-decay
in the minimal type-I seesaw model. First, we give a brief introduction on the
theoretical framework on the neutrino mass generation and the
$0\nu\beta\beta$-decay in the minimal type-I seesaw model in Sec. II. Then in
Sec. III we show the numerical results taking account of the effects of RHNs
on $m_{\mathrm{eff}}$. Possible parameter regions for the enhancement and
cancellation of $m_{\mathrm{eff}}$ are identified. Finally, we summarize our
main conclusion in Sec.IV.

##  II Theoretical Framework

The minimal type-I seesaw model is the SM extended by two RHNs
$N_{\mathrm{R}I}~{}(\mathrm{for}~{}I=1,2)$, and the Lagrangian is given by
King (2000); Frampton  _et al._ (2002); Guo  _et al._ (2007); Xing and Zhao
(2021)

$\mathcal{L}=\mathcal{L}_{\mathrm{SM}}-\left(Y_{\alpha}\overline{L_{\alpha}}\tilde{\Phi}N_{\mathrm{R}}+\frac{1}{2}\overline{N_{\mathrm{R}}^{c}}M_{R}N_{\mathrm{R}}+\rm{h.c.}\right),$
(1)

where $L_{\alpha}=(\nu_{L\alpha},e_{L\alpha})^{\mathrm{T}}(\alpha=e,\mu,\tau)$
and $\Phi$ are the weak doublets of left-handed lepton and Higgs particles
respectively, $\tilde{\Phi}=i\sigma_{2}\Phi^{*}$ where $\sigma_{2}$ is the
second Pauli matrix, and $Y_{\alpha}$ and $M_{\mathrm{R}}$ are the Yukawa
coupling matrix and the Majorana mass matrix of RHNs respectively. The seesaw
mechanism works if the Dirac masses
$M_{\mathrm{D}}=Y_{\alpha}\langle\Phi\rangle$ are much smaller than the
Majorana masses of RHNs Minkowski (1977); Yanagida (1979); Gell-Mann  _et al._
(1979); Mohapatra and Senjanovic (1980). After spontaneous symmetry breaking,
the resultant neutrino mass term becomes

$\mathcal{L}_{\mathrm{mass}}=-\frac{1}{2}\overline{\left(\nu_{\mathrm{L}},N_{\mathrm{R}}^{c}\right)}\left(\begin{array}[]{cc}0&M_{\mathrm{D}}\\\
M_{\mathrm{D}}^{\mathrm{T}}&M_{\mathrm{R}}\end{array}\right)\left(\begin{array}[]{c}\nu_{\mathrm{L}}^{c}\\\
N_{\mathrm{R}}\end{array}\right)+\rm{h.c.}\,.$ (2)

The overall $5\times 5$ Majorana neutrino mass matrix can be diagonalized by a
unitary mixing matrix,

$\left(\begin{array}[]{cc}0&M_{\mathrm{D}}\\\
M_{\mathrm{D}}^{\mathrm{T}}&M_{\mathrm{R}}\end{array}\right)=\left(\begin{array}[]{cc}U&R\\\
S&V\end{array}\right)\left(\begin{array}[]{cc}\hat{M}_{\nu}&0\\\
0&\hat{M}_{\mathrm{R}}\end{array}\right)\left(\begin{array}[]{cc}U&R\\\
S&V\end{array}\right)^{\mathrm{T}},$ (3)

where $\hat{M}_{\nu}=\mathrm{Diag}\\{m_{1},m_{2},m_{3}\\}$ stands for the
masses of the three active neutrinos and
$\hat{M}_{\mathrm{R}}=\mathrm{Diag}\\{M_{1},M_{2}\\}$ stands for those of
RHNs, while $U$, $R$, $S$, $U$ are the upper-left $3\times 3$, upper-right
$3\times 2$, lower-left $2\times 3$, lower-right $2\times 2$ sub-matrices of
the unitary matrix respectively Xing (2008, 2012). $U$ is the mixing matrix of
active neutrinos in the SM charged current interactions, named as the PMNS
matrix Maki  _et al._ (1962); Pontecorvo (1957), and $R$ is that of RHNs. In
the seesaw model, $U$ is not exactly unitary in the presence of $R$, which
will lead to a well-known intrinsic relation between mixing elements in the
seesaw mechanism:

$\sum_{i}U_{\alpha i}^{2}m_{i}+\sum_{I}R_{\alpha I}^{2}M_{I}=0.$ (4)

The left-handed flavor neutrinos are then written as decompositions of mass
eigenstates:

$\nu_{\mathrm{L}\alpha}=\sum_{i}U_{\alpha i}\nu_{i}+\sum_{I}R_{\alpha
I}N_{I}^{c},$ (5)

where $N_{I}$ is the mass eigenstate of RHNs.

Considering the $0\nu\beta\beta$-decay process in the minimal type-I seesaw
model, the decay half-life is then given by Vergados  _et al._ (2016)

$\tau_{{1}/{2}}=Gg_{A}^{4}\left\lvert
M_{0\nu}(0)\cdot\frac{m_{\mathrm{eff}}}{m_{e}}\right\rvert^{2},$ (6)

where $G$ is a kinematic phase space factor Kotila and Iachello (2012); Stoica
and Mirea (2013), $g_{A}$ is the axial vector coupling constant, $m_{e}$ is
the mass of the electron. $M_{0\nu}(0)$ is the NME when the neutrino mass in
the intermediate propagator is zero, which contributes the largest uncertainty
Barea  _et al._ (2015); Šimkovic  _et al._ (2013); Menendez  _et al._ (2009);
Rath  _et al._ (2013); Rodriguez and Martinez-Pinedo (2010); Fang  _et al._
(2018); Faessler  _et al._ (2014). For more details on the NMEs please see the
recent reviews in Ref.Engel and Menéndez (2017); Yao  _et al._ (2021). The
corresponding effective neutrino mass is:

$m_{\mathrm{eff}}=m_{\mathrm{eff}}^{\nu}+m_{\mathrm{eff}}^{N},$ (7)

where the first term in the right-hand side represents the contribution from
the active neutrinos:

$m_{\mathrm{eff}}^{\nu}=\sum_{i=1}^{3}U_{ei}^{2}m_{i}f_{\beta}(m_{i}),$ (8)

and the contribution of RHNs is:

$m_{\mathrm{eff}}^{N}=\sum_{I=1}^{2}R_{eI}^{2}M_{I}f_{\beta}(M_{I}),$ (9)

where the normalized NME dependence is expressed as

$f_{\beta}(M)=\frac{M_{0\nu}(M)}{M_{0\nu}(0)}\,.$ (10)

In this work, only the normalized NME dependence is relevant in the effective
neutrino mass, which can be empirically expressed as

$f_{\beta}(M)=\frac{\left\langle p^{2}\right\rangle}{\left\langle
p^{2}\right\rangle+M^{2}},$ (11)

where the parameter $\left\langle p^{2}\right\rangle$ stands for the mean
Fermi momentum of nucleons in a nucleus, and the value of $\sqrt{\langle
p^{2}\rangle}$ is around 200 MeV Faessler  _et al._ (2014). Note that recent
studies suggest new short-range contribution to the $0\nu\beta\beta$-decay
process Cirigliano  _et al._ (2018), which may give large contribution to the
NMEs Wirth  _et al._ (2021), and also a new empirical mass-dependent relation
Dekens  _et al._ (2020). However, in the current discussion we restrict
ourselves to the conventional calculations and will leave this new
contribution to our future studies.

Since the absolute masses of three active neutrinos are much smaller than
$\sqrt{\langle p^{2}\rangle}$, therefore $f_{\beta}(m_{i})=1$. The
contribution of the active neutrinos can be simplified as

$m_{\mathrm{eff}}^{\nu}=\sum_{i=1}^{3}U_{ei}^{2}m_{i}.$ (12)

If we assume neutrino mixing between the active and sterile ones is small,
then the contribution of the active neutrinos can be obtained as

$\left|m_{\mathrm{eff}}^{\nu}\right|=\bigg{|}\sum_{i=1}^{3}U_{ei}^{2}m_{i}\bigg{|}\simeq\left|\cos^{2}{\theta_{13}}\cos^{2}{\theta_{12}}m_{1}+\cos^{2}{\theta_{13}}\sin^{2}{\theta_{12}}e^{i\alpha_{2}}m_{2}+\sin^{2}{\theta_{13}}e^{i\alpha_{3}}m_{3}\right|,$
(13)

where $\theta_{ij}$ are the mixing angles of three active neutrinos in the
standard parameterization, which can be obtained from the latest neutrino
oscillation experiments, see the 2020 Review of Particle Physics Zyla  _et
al._ (2020). The mass ordering of active neutrinos is not determined yet
Esteban  _et al._ (2020). In the minimal type-I seesaw model, the lightest
neutrino mass is equal to zero, so there exist two possibilities: the NH
($m_{3}>m_{2}>m_{1}=0$) and the IH ($m_{2}>m_{1}>m_{3}=0$). Finally
$\alpha_{i}$ stand for the relative Majorana phases associated with the
Majorana mass eigenstates, which are completely unknown, in what follows we
will consider $0<(\alpha_{2},\alpha_{3})<2\pi$ for both NH and IH cases.
Following the latest oscillation experiment results Zyla  _et al._ (2020), we
obtain

$1.45~{}\mathrm{meV}\lesssim\left|m_{\mathrm{eff}}^{\nu}\right|\lesssim
3.68~{}\mathrm{meV}\quad({\rm NH})\quad{\rm and}\quad
18.6~{}\mathrm{meV}\lesssim\left|m_{\mathrm{eff}}^{\nu}\right|\lesssim
48.4~{}\mathrm{meV}\quad({\rm IH})\,,$ (14)

for the NH and IH cases, respectively.

Using the intrinsic relation between the mass and mixing elements in the
seesaw mechanism as shown in Eq. (4), we have

$m_{\mathrm{eff}}=m_{\mathrm{eff}}^{\nu}-m_{\mathrm{eff}}^{\nu}f_{\beta}(M_{2})+R_{e1}^{2}M_{1}\left[f_{\beta}(M_{1})-f_{\beta}(M_{2})\right]\,.$
(15)

The measured half-life can only give an absolute value of
$|m_{\mathrm{eff}}|$, however, $m_{\mathrm{eff}}$ actually is a complex number
as the combination of various neutrino masses, mixing angles and phases. In
what follows we present some detailed analyses on $m_{\mathrm{eff}}$ case by
case:

  * •

Degenerate masses of RHNs: $M_{1}=M_{2}=M_{N}$, the effective neutrino mass is
given by

$m_{\mathrm{eff}}=m_{\mathrm{eff}}^{\nu}[1-f_{\beta}(M_{N})],$ (16)

which means that when the masses of RHNs are degenerated, their contribution
to the effective mass is always smaller than that of the active ones
$m_{\mathrm{eff}}^{\nu}$, and thus RHNs reduce $m_{\mathrm{eff}}$. When the
degenerated mass is compatible with the active neutrino masses $m_{i}$, the
effective mass will be strongly suppressed, approaching to zero. When the
degenerated mass is very large $M_{N}\gg\sqrt{\langle p^{2}\rangle}$, the
contribution of RHNs is close to zero and the effective mass will become
$m_{\mathrm{eff}}^{\nu}$.

  * •

The masses of RHNs are both very large but not degenerated:
$M_{1},M_{2}\gg\sqrt{\langle p^{2}\rangle}$, the effective neutrino mass comes
completely from the active neutrino part

$m_{\mathrm{eff}}=m_{\mathrm{eff}}^{\nu}\,,$ (17)

which is the same as the degenerated case with $M_{N}\gg\sqrt{\langle
p^{2}\rangle}$. In this case, the active neutrino contribution can be
expressed as a function of the parameters of RHNs:
$m_{\mathrm{eff}}=-R_{e1}^{2}M_{1}-R_{e2}^{2}M_{2}$, which can help us to
obtain strict limits on the parameters of RHNs Xing (2009).

  * •

Only one of the masses of RHNs is large enough, $M_{2}\gg\sqrt{\langle
p^{2}\rangle}$, the effective neutrino mass is contributed by $N_{1}$ while
that of $N_{2}$ is vanishing:

$m_{\mathrm{eff}}=m_{\mathrm{eff}}^{\nu}+R_{e1}^{2}M_{1}f_{\beta}(M_{1}),$
(18)

where the phenomenon on $m_{\mathrm{eff}}$ is very different from the
degenerated case, but similar as the case with one sterile neutrino Li and Liu
(2012); Girardi  _et al._ (2013); Giunti and Zavanin (2015); Liu and Zhou
(2018); Ge  _et al._ (2017).

  * •

The masses of RHNs are both small enough compared to $\sqrt{\langle
p^{2}\rangle}$. In this case, the effective neutrino mass always equals to 0
by virtue of the intrinsic relation in Eq. (4), in this case the
$0\nu\beta\beta$ decay is forbidden and $0\nu\beta\beta$ experiments can not
help distinguish the mass hierarchy of the active neutrinos even if the
sensitivity is smaller than 10 meV. One should consider other LNV processes,
such as the rare meson decays Zyla  _et al._ (2020) or same-sign dilepton
events at colliders Cai  _et al._ (2018).

Next the most complicated cases are one or two masses of RHNs are close to
$\sqrt{\langle p^{2}\rangle}$, which will be numerically discussed in the next
section.

##  III Numerical Results

Although there are three phases in $m_{\mathrm{eff}}^{\nu}$, $R_{e1}$,
$R_{e2}$, only the relative phase between $R_{e1}^{2}$ and
$m_{\mathrm{eff}}^{\nu}$ is relevant for the absolute value of the effective
neutrino mass $m_{\mathrm{eff}}$, then Eq. (15) will become

$|m_{\mathrm{eff}}|=\bigg{|}|m_{\mathrm{eff}}^{\nu}|-|m_{\mathrm{eff}}^{\nu}|f_{\beta}(M_{2})+|R_{e1}^{2}|e^{2i\delta_{14}}M_{1}\left[f_{\beta}(M_{1})-f_{\beta}(M_{2})\right]\bigg{|},$
(19)

where $2\delta_{14}=\arg(R_{e1}^{2})-\arg(m_{\mathrm{eff}}^{\nu})$ is the
relative phase of $R_{e1}^{2}$ and $m_{\mathrm{eff}}^{\nu}$. To simplify our
following analytical discussion, we just consider two cases with
$\delta_{14}=0$ and $\delta_{14}=\pi/2$ as our representative choices. All the
other cases are between these two extreme values. Therefore, the intrinsic
seesaw relation will become

$|m_{\mathrm{eff}}^{\nu}|\pm|R_{e1}^{2}|M_{1}\pm|R_{e2}^{2}|M_{2}=0.$ (20)

In the case of $\delta_{14}=0$, considering the seesaw relation, we find that
the phase of $R_{e2}$ can only be $\pi/2$, and the intrinsic seesaw relation
will become

$|R_{e2}^{2}|M_{2}-|R_{e1}^{2}|M_{1}=|m_{\mathrm{eff}}^{\nu}|,$ (21)

and thus the contribution of two RHNs is opposite.

In the case of $\delta_{14}=\pi/2$, the phase of $R_{e2}$ can be $0$ or
$\pi/2$. This implies that the value of the phase of $R_{e2}$ cannot be
uniquely determined by the seesaw relation, but also depends on the relative
size of $|R_{e1}^{2}|M_{1}$ and $|m_{\mathrm{eff}}^{\nu}|$. When
$|R_{e1}^{2}|M_{1}>|m_{\mathrm{eff}}^{\nu}|$, the seesaw relation will become

$|R_{e1}^{2}|M_{1}-|R_{e2}^{2}|M_{2}=|m_{\mathrm{eff}}^{\nu}|,$ (22)

where the contribution of $N_{2}$ will cancel a part of the contribution of
$N_{1}$ since the individual effective neutrino mass is proportional to
$|R_{ei}^{2}|M_{i}$. The other case is
$|R_{e1}^{2}|M_{1}<|m_{\mathrm{eff}}^{\nu}|$, which means

$|R_{e2}^{2}|M_{2}+|R_{e1}^{2}|M_{1}=|m_{\mathrm{eff}}^{\nu}|,$ (23)

the contribution of $N_{2}$ will have the same sign as $N_{1}$, leading to the
additive contribution of $N_{1}$ and $N_{2}$.

With the help of these analytical discussions, in the following we shall
present our results of numerical analyses. According to Eq. (19), there are in
total five physical parameters in the effective neutrino mass
$|m_{\mathrm{eff}}|$, i.e., $|m_{\mathrm{eff}}^{\nu}|$, $|R^{2}_{e1}|$,
$\delta_{14}$, $M_{1}$ and $M_{2}$. Since we are mainly interested in those
parameters related to RHNs, $|m_{\mathrm{eff}}^{\nu}|$ will be fixed at its
central values in Eq. (14) for both NH and IH cases.

(a) NH, with $\delta_{14}=\pi/2$

(b) NH, with $\delta_{14}=0$

(c) IH, with $\delta_{14}=\pi/2$

(d) IH, with $\delta_{14}=0$

Figure 1: The 4D plots of effective neutrino mass of the
$0\nu\beta\beta$-decay as a function of $M_{1}$, $M_{2}$ and $|R_{e1}^{2}|$
for both the NH (upper panels) and IH (lower panels) cases and with both
$\delta_{14}=\pi/2$ (left panels) and $\delta_{14}=0$ (right panels). The
value of $m_{\mathrm{eff}}$ is under the logarithmic coordinates.

(a) NH, with $M_{2}=10~{}\mathrm{MeV}$

(b) NH, with $M_{2}=200~{}\mathrm{MeV}$

(c) NH, with $M_{2}=1~{}\mathrm{TeV}$ (d) IH, with $M_{2}=10~{}\mathrm{MeV}$
(e) IH, with $M_{2}=200~{}\mathrm{MeV}$ (f) IH, with $M_{2}=1~{}\mathrm{TeV}$

Figure 2: $m_{\mathrm{eff}}$ as a function of $M_{1}$ for the NH (upper
panels) and IH (lower panels) cases. The left, middle and right panels show
the cases with three different values of $M_{2}=0.01,0.2,1000~{}\mathrm{GeV}$,
respectively. We also take $\delta_{14}$ as a free parameter with
$\delta_{14}=[0,\pi/2]$ and $|R_{e1}^{2}|=10^{-8}$ (black),
$|R_{e1}^{2}|=10^{-10}$ (blue), $|R_{e1}^{2}|=10^{-12}$ (red).

Fig. 1 shows the 4D plots of the effective neutrino mass with respect to the
three free parameters $M_{1}$, $M_{2}$ and $|R_{e1}^{2}|$ in both the NH
(upper panels) and IH (lower panels) cases and with both $\delta_{14}=\pi/2$
(left panels) and $\delta_{14}=0$ (right panels). From these figures, one may
roughly figure out the allowed range of the effective neutrino mass. Firstly,
we find that there is a plane with $M_{1}=M_{2}$ which stands for the mass
degeneracy. The behavior for $M_{1}>M_{2}$ is very different from that of
$M_{1}<M_{2}$ in both the NH and IH cases. In the region of $M_{1}<M_{2}$, the
effective neutrino mass $m_{\mathrm{eff}}$ will increase as $M_{2}$ increases
when $M_{2}<\sqrt{\langle p^{2}\rangle}$, and it barely changes when
$M_{2}>1~{}\mathrm{GeV}$. Also, $m_{\mathrm{eff}}$ will increase as $M_{1}$
increases if $M_{1}<\sqrt{\langle p^{2}\rangle}$, and decreases otherwise.
While in the region of $M_{1}>M_{2}$, $m_{\mathrm{eff}}$ increases as $M_{1}$
increases no matter $M_{1}$ or $M_{2}$ are larger or smaller than
$\sqrt{\langle p^{2}\rangle}$. Secondly, we also find there is a special
region, where the full cancellation happens. By requiring
$|m_{\mathrm{eff}}|=0$, Eq. (19) becomes

$-\frac{|R_{e1}^{2}|e^{2i\delta_{14}}M_{1}\left[f_{\beta}(M_{1})-f_{\beta}(M_{2})\right]}{1-f_{\beta}(M_{2})}=|m_{\mathrm{eff}}^{\nu}|.$
(24)

This suggests that the full cancellation happens only when $\delta_{14}=0$ and
$\delta_{14}=\pi/2$. In addition, $M_{1}>M_{2}$ requires $\delta_{14}=0$ and
when $M_{1}<M_{2}$, one should take $\delta_{14}=\pi/2$. If $M_{1}<M_{2}$
(thus $\delta_{14}=\pi/2$) and $M_{2}>1~{}\mathrm{GeV}$, the value of
$f_{\beta}(M_{2})$ is close to 0, and there are always full cancellation lines
for certain values of $M_{1}$:
$|R_{e1}^{2}|=|m_{\mathrm{eff}}^{\nu}|/[M_{1}f_{\beta}(M_{1})]$. For example,
when $M_{1}=1~{}\mathrm{MeV}$, $|R_{e1}^{2}|$ is around $10^{-9}$ in NH
(upper-left panel) and $10^{-7}$ in IH (lower-left panel). The similar
discussion can also be found in Ref. Asaka  _et al._ (2021).

To uncover more detailed dependence of $m_{\mathrm{eff}}$ on the RHNs
parameters, we want to fix some parameters and analyze the variations of
others. Fig. 2 shows the effective neutrino mass as a function of $M_{1}$ with
fixed $M_{2}$ and $|R_{e1}^{2}|$ for the cases of NH (upper panels) and IH
(lower panels). The left, middle and right panels show the cases with three
different values of $M_{2}=0.01,0.2,1000~{}\mathrm{GeV}$, respectively. The
shadow regions illustrate the effect of the phase variation with
$\delta_{14}=[0,\pi/2]$. In each plot, we also take three absolute values of
the mixing matrix element $|R_{e1}^{2}|=10^{-8},10^{-10},10^{-12}$. Comparing
these plots from the horizontal lines, we observe that there is a clear
boundary at $M_{1}=M_{2}$ for the case of $M_{2}=200~{}\mathrm{MeV}$ (middle
panels), and $m_{\mathrm{eff}}$ evolves differently above and below this
boundary. When $M_{1}<M_{2}$, it is similar to the case of
$M_{2}=10~{}\mathrm{MeV}$, but when $M_{1}>M_{2}$, it is similar to the case
of $M_{2}=1~{}\mathrm{TeV}$. In the case of $M_{2}=10~{}\mathrm{MeV}$ (left
panels), a tiny mass of $M_{1}$ will make $m_{\mathrm{eff}}$ close to zero, as
we have already discussed. Then $m_{\mathrm{eff}}$ grows as $M_{1}$ and
$|R_{e1}^{2}|$ increase. In the case of $M_{2}=1~{}\mathrm{TeV}$ (right
panels), at the two extreme ends of $M_{1}$, $m_{\mathrm{eff}}$ is close to
contribution of the active neutrinos $m_{\mathrm{eff}}^{\nu}$, which means the
disappearance of the contribution from RHNs. When $M_{1}$ is around $\langle
p^{2}\rangle$ with large $|R_{e1}^{2}|$, the contribution of RHNs increases
significantly which indicates that the enhancement happens. We also find the
regions with the mass-degeneracy effect, where $m_{\mathrm{eff}}$ will quickly
converge to the value of $m_{\mathrm{eff}}^{\nu}[1-f_{\beta}(M_{N})]$ in the
vicinity of $M_{1}=M_{2}$. There are also regions of full cancellation, when
the condition of Eq. (24) is fulfilled. In these cases, the phase
$\delta_{14}$ will become crucial on the effective neutrino mass around the
points of $m_{\mathrm{eff}}\rightarrow 0$.

(a) NH, with $|m_{\mathrm{eff}}|<|m_{\mathrm{eff}}^{\nu}|$

(b) NH, with $|m_{\mathrm{eff}}|>200~{}\mathrm{meV}$

(c) IH, with $|m_{\mathrm{eff}}|<|m_{\mathrm{eff}}^{\nu}|$

(d) IH, with $|m_{\mathrm{eff}}|>200~{}\mathrm{meV}$

Figure 3: Contours of the cancellation (left panels) and strong enhancement
(right panels) regions of $m_{\mathrm{eff}}$ for the NH (upper panels) and IH
(lower panels) cases. In each plot, we also take two cases with
$\delta_{14}=\pi/2$ and $\delta_{14}=0$.

Next we are going to take a closer look at the properties of enhancement and
cancellation. To be more specific, the cancellation is defined as the region
with $|m_{\mathrm{eff}}|<|m_{\mathrm{eff}}^{\nu}|$, and the strong enhancement
is defined as the region with $|m_{\mathrm{eff}}|>{\rm 200\;meV}$, which is
roughly the region excluded by the current experiments Dell’Oro  _et al._
(2016). Between these two parts, there is a region of moderate enhancement
with $|m_{\mathrm{eff}}^{\nu}|<|m_{\mathrm{eff}}|<{\rm 200\;meV}$. For this
purpose, in Fig. 3 we present the 3D contour plots of the cancellation (left
panels) and strong enhancement (right panels) regions of $m_{\mathrm{eff}}$
for the NH (upper panels) and IH (lower panels) cases. In each plot, we also
take two special cases with $\delta_{14}=\pi/2$ and $\delta_{14}=0$. From the
figure, we can roughly see the correlations among $M_{1}$, $M_{2}$ and
$|R_{e1}^{2}|$. In general, with decreasing $|R_{e1}^{2}|$, the cancellation
ranges will be expanded. The plots are divided into two different regions by
the boundary $M_{1}=M_{2}$. In the region of $M_{1}<M_{2}$, the behavior of
the parameters is very different for the two cases with $\delta_{14}=\pi/2$
and $\delta_{14}=0$. There is an obvious hollow around $M_{1}=\sqrt{\langle
p^{2}\rangle}$ for $\delta_{14}=\pi/2$, but it disappears with
$\delta_{14}=0$. In the region of $M_{1}>M_{2}$, the behavior is similar
between $\delta_{14}=\pi/2$ and $\delta_{14}=0$. We also observe that the
shape difference of the cancellation and strong enhancement contours is very
small between the NH and IH cases.

(a) NH, with $|R_{e1}^{2}|=10^{-8}$

(b) NH, with $|R_{e1}^{2}|=10^{-10}$

(c) NH, with $|R_{e1}^{2}|=10^{-12}$

(d) IH, with $|R_{e1}^{2}|=10^{-8}$

(e) IH, with $|R_{e1}^{2}|=10^{-10}$ (f) IH, with $|R_{e1}^{2}|=10^{-12}$

Figure 4: Contours of $|m_{\mathrm{eff}}|$ in the $0\nu\beta\beta$-decay as
functions of $M_{1}$ and $M_{2}$ for the NH (top) and IH (bottom) cases with
two specific values of $\delta_{14}$ with $\delta_{14}=\pi/2$ (colored regions
without slash lines) and $\delta_{14}=0$ (regions with slash lines). The black
solid line and dashed line represent the special $m_{\mathrm{eff}}=0$ cases
for $\delta_{14}=\pi/2$ and $\delta_{14}=0$, respectively. The blue regions
(either light or dark) stand for
$|m_{\mathrm{eff}}|<|m_{\mathrm{eff}}^{\nu}|$, the red regions stand for
$|m_{\mathrm{eff}}|>200~{}\mathrm{meV}$, and the white regions correspond to
the moderate enhancement in $|m_{\mathrm{eff}}|$.

(a) NH, $M_{2}=10~{}\mathrm{MeV}$ (b) NH, $M_{2}=200~{}\mathrm{MeV}$ (c) NH,
$M_{2}=1~{}\mathrm{TeV}$ (d) IH, $M_{2}=10~{}\mathrm{MeV}$

(e) IH, $M_{2}=200~{}\mathrm{MeV}$ (f) IH, $M_{2}=1~{}\mathrm{TeV}$

Figure 5:  Contours of $|m_{\mathrm{eff}}|$ in the $0\nu\beta\beta$-decay as
functions of $|R_{e1}^{2}|$ and $M_{1}$ for the NH (upper panels) and IH
(lower panels) cases. We take $\delta_{14}=\pi/2$ (colored regions without
slash lines) and $\delta_{14}=0$ (regions with slash lines). The black solid
line and dashed line stand for $m_{\mathrm{eff}}=0$ in $\delta_{14}=\pi/2$ and
$\delta_{14}=0$, respectively. The blue regions stand for
$|m_{\mathrm{eff}}|<|m_{\mathrm{eff}}^{\nu}|$, the red regions stand for
$|m_{\mathrm{eff}}|>200~{}\mathrm{meV}$, and the white regions correspond to
the moderate enhancement in $|m_{\mathrm{eff}}|$.

In order to reveal the quantitative parameter dependence regarding the
contours of enhancement and cancellation, we further cut the contours of Fig.
3 in the horizontal direction by fixing the parameter of $|R_{e1}^{2}|$ in
Fig. 4 and in the vertical direction by fixing the parameter of $M_{2}$ in
Fig. 5.

In Fig. 4, we illustrate the contours of $|m_{\mathrm{eff}}|$ in the left,
middle and right panels with the fixed values of
$|R_{e1}^{2}|=10^{-8},10^{-10},10^{-12}$. The upper and lower panels show the
results for the NH and IH cases respectively. In each plot, we also take the
phase $\delta_{14}=\pi/2$ (colored region without slash lines) and
$\delta_{14}=0$ (region with slash lines) respectively. The red regions
represent the contours of strong enhancement with
$|m_{\mathrm{eff}}|>200~{}\mathrm{meV}$ and the blue regions show the
cancellation contours with $|m_{\mathrm{eff}}|<|m_{\mathrm{eff}}^{\nu}|$. The
special case with the full cancellation, which means $|m_{\mathrm{eff}}|=0$ is
also shown on the figures with black lines (solid for $\delta=\pi/2$ and
dashed for $\delta=0$). The white regions in between are for the case of
moderate enhancement allowed by current experiments. The strong enhancement
always happens in the case with large $M_{1}$ and small $M_{2}$, and it is
positively correlated to the mixing matrix element $|R_{e1}^{2}|$ as well as
the hierarchy of the two masses of RHNs. In the case of large mixing
$|R_{e1}^{2}|$, the regions of strong enhancement also appear around the
island at $M_{1}=0.1~{}\mathrm{GeV}$ with $M_{2}>0.1~{}\mathrm{GeV}$. The
boundary between the white and blue regions is shown for
$|m_{\mathrm{eff}}|=|m_{\mathrm{eff}}^{\nu}|$, which stands for the case where
the contribution of RHNs disappears. As we discussed above, the cancellation
always happens when both of the RHNs are light, and the cancellation region
expands to a larger region when $|R_{e1}^{2}|$ becomes smaller. When the value
of $M_{1}$ is large, the cancellation also happens around the mass degeneracy
regions. In the mass degeneracy case, $|m_{\mathrm{eff}}|$ is always smaller
than $|m_{\mathrm{eff}}^{\nu}|$ and the cancellation always happens, but if
there is a tiny difference between $M_{1}$ and $M_{2}$, the cancellation is
rapidly changing to enhancement around $M_{1}=\sqrt{\langle p^{2}\rangle}$
with large mixing $|R_{e1}^{2}|$. These special situations can be clearly
found in Fig. 4(a) and 4(d). And of course, it will disappear when
$|R_{e1}^{2}|$ is small enough due to the fact that $|m_{\mathrm{eff}}|$ is
smaller than $|m_{\mathrm{eff}}^{\nu}|$ in almost all regions. Comparing the
case of $\delta_{14}=0$ with that of $\delta_{14}=\pi/2$, the regions of
strong enhancement change slightly, while the regions of cancellation change
drastically, covering quite a different parameter space. The reason is that
when the contribution of RHNs is large enough, it will be dominant in
$|m_{\mathrm{eff}}|$ so that $|m_{\mathrm{eff}}|$ is not sensitive to the
relative phase $\delta_{14}$. On the other hand, it is sensitive when
$|m_{\mathrm{eff}}^{N}|$ is close to $|m_{\mathrm{eff}}^{\nu}|$, and in this
case, the phase $\delta_{14}$ will determine whether there is enhancement or
cancellation.

Here we want to remark that the intrinsic relation in the seesaw mechanism
leads to the removal of the free parameter $R_{e2}^{2}$, which also breaks the
commutative symmetry of $N_{1}$ and $N_{2}$. With a given set of $M_{1}$
,$M_{2}$ and $R_{e1}^{2}$, exchanging the masses of $N_{1}$ and $N_{2}$ can
not ensure the values of mixing matrix elements are also exchanged. Because of
this, the absolute values of $m_{\mathrm{eff}}^{N}$ and regions of
cancellation or enhancement are not symmetric, and strong enhancement close to
$|m_{\mathrm{eff}}^{\nu}|$ can not happen in case of $\delta_{14}=0$ with
small mixing $|R_{e1}^{2}|$.

(a) NH, $M_{1}=10~{}\mathrm{MeV}$ (b) NH, $M_{1}=200~{}\mathrm{MeV}$ (c) IH,
$M_{1}=10~{}\mathrm{MeV}$ (d) IH, $M_{1}=200~{}\mathrm{MeV}$

Figure 6: Contours of $|m_{\mathrm{eff}}|$ in the $0\nu\beta\beta$-decay as
functions of $|R_{e1}^{2}|$ and $\delta_{14}$ for the NH (upper panels) and IH
(lower panels) cases. We take $M_{1}=10,200~{}\mathrm{MeV}$ with
$M_{2}=1~{}\mathrm{GeV}$ (colored regions without slash lines) and
$M_{2}=1~{}\mathrm{TeV}$ (regions with slash lines). The blue regions stand
for $|m_{\mathrm{eff}}|<|m_{\mathrm{eff}}^{\nu}|$, the red regions stand for
$|m_{\mathrm{eff}}|>200~{}\mathrm{meV}$, and the white regions correspond to
the moderate enhancement in $|m_{\mathrm{eff}}|$.

Now in Fig. 5 we discuss the role of the mixing element $|R_{e1}^{2}|$ in the
contours of cancellation and enhancement. If future experiments determine a
precise effective neutrino mass $m_{\mathrm{eff}}$, the masses of RHNs and the
mixing matrix element can be well constrained. The values of $|R_{e1}^{2}|$
can be directly determined by a measurement of $|m_{\mathrm{eff}}|$, a given
$|m_{\mathrm{eff}}^{\nu}|$ and two masses of RHNs, in which
$|m_{\mathrm{eff}}^{\nu}|$ can be properly determined from neutrino
oscillation data. In this sense, we can explore the allowed and excluded
regions of $|R_{e1}^{2}|$ in Fig. 5 as functions of a given mass $M_{1}$ for
both NH and IH cases. We take the mass of $N_{2}$ to be
$M_{2}=0.01,0.2,1000~{}\mathrm{GeV}$, and assume the observed values of the
effective neutrino mass to $|m_{\mathrm{eff}}|>200~{}\mathrm{meV}$ (strong
enhancement), $|m_{\mathrm{eff}}|<|m_{\mathrm{eff}}^{\nu}|$ (cancellation) or
in between. We find that strong enhancement regions are similar in both the
cases of NH and IH. When the strong enhancement happens,
$|m^{N}_{\mathrm{eff}}|$ is dominant, and the effect of
$|m_{\mathrm{eff}}^{\nu}|$ or the phase $\delta_{14}$ can be ignored. In
contrast, when the cancellation actually happens, even though the active
neutrino mixing and masses could be well determined by neutrino oscillation
experiments, the cancellation regions will be drastically changed since the
value of $|R_{e1}^{2}|$ is sensitive to the phase $\delta_{14}$. The mass
degeneracy effect is also clearly shown in the figure. When $M_{1}=M_{2}$, the
value of $|R_{e1}^{2}|$ cannot be constrained by the measurement of
$|m_{\mathrm{eff}}|$ since in this case, $|R_{e1}^{2}|$ becomes singular. The
black lines stand for $|m_{\mathrm{eff}}|=0$ where the $0\nu\beta\beta$-decay
disappears. If the two RHNs are with large enough masses, $|R^{2}_{e1}|$ can
be well constrained with a known $|m_{\rm{eff}}|$ indirectly by the intrinsic
seesaw relation. For $M_{2}$ around $\langle p^{2}\rangle$, we also find that
the phase will change the cancellation regions, smaller $|R^{2}_{e1}|$ are
needed for the cancellation to happen with larger $\delta_{14}$ in the case of
$M_{1}>M_{2}$, contrary to the case of $M_{1}<M_{2}$. In the case of
$M_{2}=1~{}\mathrm{TeV}$, the cancellation disappears with $\delta_{14}=0$ for
$|R^{2}_{e1}|$ as large as $10^{-12}$.

Finally, to get a better understanding on the role of $\delta_{14}$, we show
in Fig. 6 more details of the effect of $\delta_{14}$ on the effective
neutrino mass $|m_{\mathrm{eff}}|$. We discuss the cases of
$M_{1}=10,200~{}\mathrm{MeV}$ by taking the values of $M_{2}$ at
$1~{}\mathrm{GeV}$ and $1~{}\mathrm{TeV}$. The reason for this consideration
is that we could avoid the mass degeneracy by forcing $M_{1}<M_{2}$. We find
that the cases of $\delta_{14}=0$ and $\delta_{14}=\pi/2$ in the previous
figures present two extreme cases, all the other choices of $\delta_{14}$ are
in between these two sets of results. Therefore, choosing these two values
generally gives a full view of the $|m_{\rm{eff}}|$ on the phase. For the
cases with $M_{1}<M_{2}$, we find that the strong enhancement regions are
basically independent of $\delta_{14}$, as already noted in the lower-right
parts of Fig. 3. The cancellation happens for smaller $|R^{2}_{e1}|$ if we
decrease $\delta_{14}$. The mass difference of the two RHNs will affect the
behaviour of $m_{\rm{eff}}$ as stated before. We observe that the cancellation
is very sensitive not only to the absolute masses of the RHNs but also their
mass difference. A larger mass difference leads to weaker cancellation, in
particular for smaller values of $\delta_{14}$.

Before finishing this section, we would like to summarize the experimental
constraints on the parameter space of RHNs from other probes and discuss the
rationale of our choices of the typical parameter range. Firstly, assuming the
electron neutrino mixing dominance, the mixing $|R^{2}_{e1}|$ is constrained
to the level of [$10^{-8}$, $10^{-9}$] by NA62 Cortina Gil  _et al._ (2020)
for the $M_{1}$ range between 150 MeV and 400 MeV, and the level of
[$10^{-6}$, $10^{-7}$] by CHARM Bergsma  _et al._ (1986) for $400\;{\rm
MeV}<M_{1}<2\;{\rm GeV}$. Constraints from Belle Liventsev  _et al._ (2013),
DELPHI Abreu  _et al._ (1997), ATLAS Aad  _et al._ (2019), CMS Sirunyan  _et
al._ (2018) can extend to the region of $M_{1}>2\;{\rm GeV}$, which present a
limit of $|R^{2}_{e1}|$ at the level of $10^{-4}$. Secondly, regarding the
neutrino probe, T2K Abe  _et al._ (2019) is sensitive to $|R^{2}_{e1}|$ for
the mass range of $150\;{\rm MeV}<M_{1}<500\;{\rm MeV}$, in which the limit
varies from $10^{-7}$ to $10^{-9}$. Meanwhile MicroBooNE Abratenko  _et al._
(2020) and ArgoNeuT Acciarri  _et al._ (2021) have also made the searches of
RHNs at the same mass range, but their constraints are for the muon and tau
neutrino channels respectively. Thirdly, the process of charged lepton flavor
violation is sensitive to the combination of $|\Sigma_{i}R_{\alpha
i}R^{*}_{\beta i}|$ Lindner  _et al._ (2018), where $\alpha$ and $\beta$ are
the final and initial flavors of the process, respectively, and $i$ is the
index of the RHNs. The present bound on $|\Sigma_{i}R_{\alpha i}R^{*}_{\beta
i}|$ is at the level of [$10^{-3}$, $10^{-4}$] for the GeV-scale RHNs Alonso
_et al._ (2013). However, the above experimental searches of RHNs are
typically carried out under the assumption of a single RHN mixing with a
single flavor, which cannot be used in the realistic seesaw model. A recent
study has shown that the limit on the mixing element could be several orders
weaker than the experimental reported ones after marginalizing over the
unknown model parameters Tastet  _et al._ (2021). Therefore, our choices of
$|R^{2}_{e1}|$ are motivated by the above experimental limits and the largest
one of $10^{-8}$ can be considered near the edge of the current upper bound,
which is also motivated by the seesaw relation where strong cancellation
between two RHNs is required to produce the correct active neutrino masses.

##  IV Conclusion

In this work, we have studied the contribution to the effective neutrino mass
of the $0\nu\beta\beta$-decay from two RHNs within the minimal type-I seesaw
model using the intrinsic seesaw relation of neutrino mass and mixing
parameters and the relative mass dependence of NMEs. The effective neutrino
mass as a function of the parameters of the RHNs is illustrated. We have
demonstrated that both the enhancement and cancellation to the effective
neutrino mass from the RHNs can happen, with a quantitative exploration on
their dependence on the masses $M_{1}$, $M_{2}$, the mixing element
$|R^{2}_{e1}|$ and the phase $\delta_{14}$. In general, strong enhancement
usually needs the large mass hierarchy of two RHNs, but it also happens for
the masses of the RHNs at around $0.2~{}\mathrm{GeV}$ with large mixing
element $|R^{2}_{e1}|$. The cancellation happens in the case that the masses
of the RHNs and the mixing element $|R^{2}_{e1}|$ are relatively small, but
there are also exception for large masses in the presence of strong
cancellation between the RHNs themselves, in which the phase plays a crucial
role. We believe our studies in this work can be useful to analyze the current
and future experimental results and help to pin down the viable parameter
space of the seesaw model. We defer this to future separated works.

###### Acknowledgements.

The work of Y.F. Li and Y.Y. Zhang is supported by the National Natural
Science Foundation of China under Grant No. 12075255 and No. 11835013, and by
Beijing Natural Science Foundation under Grant No. 1192019. Y.F. Li is also
grateful for the support by the CAS Center for Excellence in Particle Physics
(CCEPP). The work of Y.Y. Zhang is also supported by China Postdoctoral
Science Foundation under Grant No. 2021T140669. DLF is supported by CAS from
the ”Light of West” program and the ”from zero to one” program.

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