††thanks: ORCID: 0000-0003-2653-5962.††thanks: ORCID: 0000-0003-4471-2336.

# Dark matter and radiative neutrino masses in conversion-driven scotogenesis

Julian Heeck<EMAIL_ADDRESS>Department of Physics, University of
Virginia, Charlottesville, Virginia 22904-4714, USA  Jan Heisig
<EMAIL_ADDRESS>Department of Physics, University of Virginia,
Charlottesville, Virginia 22904-4714, USA  Anil Thapa<EMAIL_ADDRESS>Department of Physics, University of Virginia, Charlottesville, Virginia
22904-4714, USA

###### Abstract

The scotogenic model generates Majorana neutrino masses radiatively, with dark
matter particles running in the loop. We explore the parameter space in which
the relic density of fermionic dark matter is generated via a conversion-
driven freeze-out mechanism. The necessity for small Yukawa couplings to
initiate chemical decoupling for conversion processes naturally reproduces
small neutrino masses as long as the active neutrinos are hierarchical. The
model can also resolve the recently reported deviation in the $W$-boson mass
while satisfying constraints from direct detection, charged lepton flavor
violation as well as collider bounds. Parts of the parameter space lead to
long-lived particle signatures to be probed at the LHC.

##  I Introduction

The scotogenic model Ma:2006km  is a simple explanation of neutrino masses and
dark matter (DM), connecting these two strongest proofs for physics beyond the
Standard Model (SM). As a particularly welcome feature, neutrino masses
generated in the scotogenic model are naturally small due to loop suppression
factors and the heavy DM mass. DM consists either of the lightest new fermion
or the lightest neutral scalar; the latter is highly reminiscent of the inert-
doublet model Deshpande:1977rw ; Barbieri:2006dq ; LopezHonorez:2006gr .
Constraints from direct and indirect detection as well as charged lepton
flavor violation restrict the parameter space and offer possibilities of
verifying this model (see Refs. Borah:2020wut ; deBoer:2021pon ; Liu:2022byu
for recent work in this direction).

The loop diagrams leading to neutrino masses contain DM and thus link neutrino
parameters to DM couplings. These couplings are severely constrained in the
region of parameter space where DM is produced via thermal freeze-out.
However, our incomplete knowledge of the active-neutrino parameters, notably
the mass of the lightest neutrino, makes it possible to suppress some of the
DM Yukawa couplings to an arbitrary degree. This might be considered fine-
tuning, but opens up the phenomenologically interesting region of _freeze-in_
DM McDonald:2001vt ; Asaka:2005cn ; Hall:2009bx  explored in Refs.
Molinaro:2014lfa ; Hessler:2016kwm ; Baumholzer:2019twf  within the scotogenic
model.

In this article, we point out that there is a region of parameter space in
which the Yukawa couplings are between the freeze-in and freeze-out values,
enabling a _conversion-driven_ Garny:2017rxs  (or coscattering DAgnolo:2017dbv
) freeze-out scenario. No particular fine-tuning is required, all entries of
the Yukawa matrix can even be of similar order of magnitude. The moderately
small Yukawas necessary for this new viable region have implications for the
lightest active neutrino mass and naturally suppress any direct and indirect
detection signatures, as well as charged lepton flavor violation.

Motivated by the recent precision measurement of the $W$-boson mass by CDF
CDF:2022hxs , which exceeds the SM prediction by $7\sigma$, we focus our
analysis on the region of parameter space that can alleviate this tension,
although this is not strictly necessary for successful conversion-driven
freeze-out. The key ingredient for a larger $W$-boson mass is the mass
splitting within the new scalar $SU(2)_{L}$ doublet, which propagates to the
$W$-boson mass at the one-loop level. This solution is of course identical to
many other two-Higgs-doublet solutions of the CDF anomaly, but is connected to
both neutrino mass and DM phenomenology in the scotogenic model and hence less
ad-hoc. Related studies of the impact of the CDF measurement on the scotogenic
and inert-doublet models have already been presented in Ref. Batra:2022pej
and Refs. Fan:2022dck ; Zhu:2022tpr , respectively, but with focus on
different regions of parameter space.

The rest of this article is organized as follows: we introduce the scotogenic
model and our notation in Sec. II. In Sec. III, we discuss how this model can
explain the CDF anomaly. Conversion-driven freeze-out is introduced in Sec. IV
and applied to our model. We discuss our results in Sec. V and conclude in
Sec. VI.

##  II Model

The scotogenic model Ma:2006km  extends the SM by an $SU(2)_{L}\times
U(1)_{Y}$ doublet $\eta\sim({\boldsymbol{2}},1/2)$ – the same quantum numbers
as the Higgs doublet $\Phi$ – and (in the considered case) three right-handed
singlet fermions $N_{1,2,3}\sim({\boldsymbol{1}},0)$. These new particles are
odd under a new $\mathbb{Z}_{2}$ symmetry, while the SM particles are even;
this guarantees the stability of the lightest of the new particles and thus
provides a DM candidate. We will focus on the scenario where DM is made up of
the $N_{k}$ rather than scalars.

The interactions and mass terms of the right-handed fermions $N_{k}$ take the
form

$-\mathcal{L}_{N}=y_{\alpha
j}\bar{L}_{\alpha}\tilde{\eta}N_{j}+\frac{1}{2}\sum_{i}m_{N_{i}}N_{i}N_{i}+\text{h.c.}\,,$
(1)

where we have already diagonalized the Majorana mass matrix of the $N$ without
loss of generality. The most general renormalizable $\mathbb{Z}_{2}$-symmetric
scalar potential $V$ involving the SM Higgs doublet $\Phi$ and the new $\eta$
reads

$\displaystyle V$
$\displaystyle=\mu_{h}^{2}\Phi^{\dagger}\Phi+\mu_{\eta}^{2}\eta^{\dagger}\eta+\frac{\lambda_{1}}{2}(\Phi^{\dagger}\Phi)^{2}+\frac{\lambda_{2}}{2}(\eta^{\dagger}\eta)^{2}$
(2)
$\displaystyle+\lambda_{3}(\Phi^{\dagger}\Phi)(\eta^{\dagger}\eta)+\lambda_{4}(\Phi^{\dagger}\eta)(\eta^{\dagger}\Phi)-\left(\frac{\lambda_{5}}{2}(\Phi^{\dagger}\eta)^{2}+\text{h.c.}\right).$

$\mu_{h}^{2}$ is negative to trigger electroweak symmetry breaking, but
$\mu_{\eta}^{2}$ is positive and $\eta$ does not acquire a vacuum expectation
value. Any phase of $\lambda_{5}$ can be absorbed into $\eta$ and eventually
the lepton fields, allowing us to restrict $\lambda_{5}$ to non-negative
values without loss of generality, similar to the inert-doublet model
Ilnicka:2015jba . The unbroken $\mathbb{Z}_{2}$ symmetry ensures that there is
no mixing between the SM Higgs $h$ and the new neutral scalars in $\eta$,
parameterized via

$\displaystyle\eta=\begin{pmatrix}H^{+}\\\ \frac{1}{\sqrt{2}}\left(H+{\rm
i}A\right)\end{pmatrix}.$ (3)

We note that despite the formal similarity with non-$\mathbb{Z}_{2}$-symmetric
two-Higgs-doublet models Branco:2011iw , $H$ and $A$ cannot be assigned
definite CP properties Belyaev:2016lok  because they do not have diagonal
couplings to fermions. The masses of the scalar fields after electroweak
symmetry breaking are given by Deshpande:1977rw

$\displaystyle m^{2}_{h}=\lambda_{1}v^{2},\,\hskip
11.38109ptm_{H^{+}}^{2}=\mu_{\eta}^{2}+\frac{\lambda_{3}}{2}v^{2},$ (4)
$\displaystyle
m^{2}_{H}=\mu_{\eta}^{2}+\frac{v^{2}}{2}\left(\lambda_{3}+\lambda_{4}-\lambda_{5}\right),$
(5) $\displaystyle
m^{2}_{A}=\mu_{\eta}^{2}+\frac{v^{2}}{2}\left(\lambda_{3}+\lambda_{4}+\lambda_{5}\right).$
(6)

Here, $v=\sqrt{-2\mu_{h}^{2}/\lambda_{1}}\simeq 246\,\mathrm{GeV}$ is the
vacuum expectation value of the SM Higgs doublet, with Higgs mass $m_{h}\simeq
125\,\mathrm{GeV}$ Workman:2022ynf . Our choice $\lambda_{5}\geq 0$ defines
$A$ to be the heavier of the two new neutral scalars without loss of
generality, with mass splitting given by $\lambda_{5}$:
$m_{A}^{2}-m_{H}^{2}=\lambda_{5}v^{2}$. For later convenience we define the
linear combination of couplings

$\displaystyle\lambda_{\text{L}}\equiv\lambda_{3}+\lambda_{4}-\lambda_{5}$ (7)

and note that some of our $\lambda$ couplings are defined differently from
other articles. The scalar couplings of Eq. (2) have to comply with the vacuum
stability conditions Deshpande:1977rw ; Kannike:2012pe

$\displaystyle\lambda_{1}\geq 0\,,\quad\lambda_{2}\geq
0\,,\quad\lambda_{3}+\sqrt{\lambda_{1}\lambda_{2}}\geq 0\,,$ (8)
$\displaystyle\lambda_{3}+\lambda_{4}-\lambda_{5}+\sqrt{\lambda_{1}\lambda_{2}}\geq
0\,,$ (9)

and the perturbativity condition Barbieri:2006dq

$\displaystyle|2\lambda_{3}(\lambda_{3}+\lambda_{4})+\lambda_{4}^{2}+\lambda_{5}^{2}|\lesssim
50\,.$ (10)

Due to the unbroken $\mathbb{Z}_{2}$ symmetry of the Lagrangian, the Majorana
fermions $N_{k}$ do not mix with the left-handed neutrinos, so the latter
remain massless at tree level. However, the simultaneous presence of $y$,
$\lambda_{5}$, and $m_{N}$ explicitly breaks lepton number by two units,
generating a one-loop Majorana neutrino mass matrix $\mathcal{M}_{\nu}$
through the diagrams shown in figure 1 as

$\displaystyle\left(\mathcal{M}_{\nu}\right)_{\alpha\beta}=\sum_{k}y_{\alpha
k}^{*}\Lambda_{k}y^{*}_{\beta k}\,,\text{ with }$ (11)
$\displaystyle\Lambda_{k}=\frac{m_{N_{k}}}{32\pi^{2}}\Bigg{[}\frac{m_{H}^{2}\log\left(\frac{m_{H}^{2}}{m_{N_{k}}^{2}}\right)}{m_{H}^{2}-m_{N_{k}}^{2}}-(m_{H}\leftrightarrow
m_{A})\Bigg{]}.$ (12)

(Our expression is smaller than Ma’s Ma:2006km  by a factor $1/2$, in
agreement with Ref. Merle:2015ica .) In addition to the loop suppression
factor, neutrino masses in the scotogenic model can be suppressed in several
ways, all directly linked to the restoration of lepton number and thus
technically natural in the sense of ’t Hooft tHooft:1979rat : 1) small $y$, 2)
small $\lambda_{5}$, 3) small $m_{N}$ or very large $m_{N}$.

In the present work we are interested in the region of parameter space where
the Yukawas $y_{ik}$ are small, as this enables conversion-driven freeze out;
for this we note that $|\Lambda_{k}|$ is maximized when the mass splitting
between the scalars $H$ and $A$ is as large as possible and $m_{N_{k}}$ is
equal to the _heavier_ one, $A$. This gives
$|\Lambda_{k}|\simeq\sqrt{\lambda_{5}}\frac{v}{32\pi^{2}}\simeq 1\ {\rm
GeV}\times\sqrt{\lambda_{5}}$ and thus a naive lower bound on the Yukawa of
$|y|\gtrsim\sqrt{|\mathcal{M}_{\nu}/\Lambda|}\sim 5\times 10^{-6}$, assuming
$\lambda_{5}$ is close to its perturbativity bound $\lambda_{5}\simeq 4$
Barbieri:2006dq , which also requires large $\lambda_{3}$ and small
$\lambda_{4}$.

Figure 1: Radiative neutrino mass generation at one loop.

A proper evaluation of the Yukawas including their flavor structure can be
obtained via the Casas–Ibarra parametrization Casas:2001sr  adapted to the
scotogenic model Merle:2015ica :

$\displaystyle
y=U_{\text{PMNS}}\sqrt{\mathcal{M}_{\nu}^{\text{diag}}}R\sqrt{\Lambda^{-1}}\,,$
(13)

where $U_{\text{PMNS}}$ is the unitary Pontecorvo–Maki–Nakagawa–Sakata matrix
and $\mathcal{M}_{\nu}^{\text{diag}}$ the diagonal matrix containing the
active-neutrino masses $m_{1,2,3}$, defined through diagonalization of
$\mathcal{M}_{\nu}$. Furthermore, $\Lambda$ is the diagonal matrix containing
the $\Lambda_{k}$ from Eq. (12) and $R$ is some complex orthogonal matrix.
With these definitions, the above $y$ indeed solves Eq. (11) and allows us to
use neutrino masses and mixing angles as input parameters.

Neutrino oscillations have provided experimental access to the three mixing
angles and the Dirac CP phase within $U_{\text{PMNS}}$, as well as the
neutrino mass splittings Workman:2022ynf . The overall mass scale and the mass
ordering are unknown as of yet, although there are preferences for normal
ordering and a hierarchical spectrum, $m_{1}\ll m_{2,3}$, from global fits to
oscillation data Esteban:2020cvm  and cosmology Planck:2018vyg ;
DiValentino:2021hoh , respectively. Assuming normal hierarchy, a vanishing
lightest neutrino mass $m_{1}=0$, and otherwise the best-fit values for the
neutrino parameters from Ref. Esteban:2020cvm , we find

$\displaystyle|y|\simeq 10^{-6}\begin{pmatrix}0&2.1&1.4\\\ 0&2.4&7.0\\\
0&2.3&6.1\end{pmatrix},$ (14)

using a Casas–Ibarra matrix $R=\mathds{1}$, $m_{H}=100$ GeV and
$m_{N_{k}}=m_{A}=266$ GeV (i.e. $\lambda_{5}=1$). The order of magnitude
matches the previous estimate but here we see that one of the $N_{k}$ can be
arbitrarily decoupled in the limit of a vanishing lightest neutrino mass. For
$m_{1}=1\,\mathrm{meV}$, the first column entries are around or below
$10^{-6}$; while this is the same order of magnitude as the other columns, it
is just small enough to make a considerable difference for DM phenomenology.
The situation is qualitatively identical in the inverted-hierarchy case, where
it is column 3 of $y$ that goes to zero with the lightest neutrino mass.

Since the Yukawas are the only portal between the SM and the $N_{k}$, they are
crucial input parameters to study the $N$ abundance in the early universe. For
$|y_{\alpha k}|\ll 10^{-10}$, the $N_{k}$ would be effectively decoupled from
the SM bath and thus non-existent – assuming a vanishing initial abundance
after the Big Bang. The above discussion illustrates that the two measured
neutrino-mass splittings enforce that at least two of the $N_{k}$ have
couplings exceeding $10^{-6}$, which are large enough to put them in thermal
equilibrium with the SM at temperatures around their masses. One of the new
fermions, on the other hand, could be arbitrarily decoupled if the lightest
active-neutrino mass is tiny (as hinted at by cosmology Planck:2018vyg ;
DiValentino:2021hoh ). This was already observed and employed for freeze-in
fermionic DM in Refs. Molinaro:2014lfa ; Hessler:2016kwm ; Baumholzer:2019twf
, for which couplings $|y|\sim 10^{-10}$ are required, considerably smaller
than the Yukawas of the other two $N$. As we will show below, keeping the
couplings of the third $N$ slightly below those of the other two – but of
similar order of magnitude – opens up a qualitatively different DM production
mechanism: conversion-driven freeze out.

##  III Correction to $W$ boson mass

The CDF collaboration has recently released their legacy measurement of the
$W$ mass, $80.4335\,\mathrm{GeV}\pm 9.4\,\mathrm{MeV}$ CDF:2022hxs . This is
not only more precise than any previous measurement, but also deviates from
the SM prediction $M_{W}^{\text{SM}}=80.360\,\mathrm{GeV}\pm 6\,\mathrm{MeV}$
Workman:2022ynf  by an astonishing $7\sigma$. The deviation can be interpreted
as a sign for new physics, which can impact the $W$-mass prediction through
self-energy corrections encoded in the oblique parameters $S$, $T$, and $U$
Peskin:1990zt ; Peskin:1991sw  via Maksymyk:1993zm

$\displaystyle M_{W}\simeq M_{W}^{\rm SM}\left[1-\frac{\alpha_{\rm
EM}(S-2c^{2}_{W}T)}{4(c_{W}^{2}-s^{2}_{W})}+\frac{\alpha_{\rm
EM}}{8s^{2}_{W}}U\right],$ (15)

where $s_{W}\equiv\sin\theta_{W}$, $c_{W}\equiv\cos\theta_{W}$ with the weak-
mixing angle $\theta_{W}\simeq 29^{\circ}$. Since $U$ is suppressed compared
to $S$ and $T$, we will neglect it in the following. The CDF measurement then
prefers $S-2c^{2}_{W}T\simeq-0.25$. Additional constraints on $S$ and $T$ come
from electroweak precision observables such as $\theta_{W}$. We will use the
results of a recent global fit including the CDF result Asadi:2022xiy , which
finds the best-fit value $(S,T)=(0.17,0.27)$ as well as preferred $1$ and
$2\sigma$ regions that we use below. Similar results have been obtained in
other fits Lu:2022bgw .

The oblique parameters of the scotogenic model at one loop only depend on the
scalar doublet $\eta$ and are thus the same as those of the inert-doublet
model Deshpande:1977rw ; Barbieri:2006dq ; LopezHonorez:2006gr ; Grimus:2008nb
. We shall not display them here but note that while it is not difficult to
generate a large $T$ parameter through the doublet, a large positive $S$ is
challenging and would require a very light $H^{+}$. The best-fit value for
$(S,T)$ of Ref. Asadi:2022xiy  can thus not be accommodated in the scotogenic
model and even the $1\sigma$ region is just barely in reach. Given the
$7\sigma$ discrepancy this is hardly of practical concern and still makes the
scalar doublet a highly preferred extension of the SM in light of the CDF
result.

Figure 2: $1\sigma$ (green) and $2\sigma$ (orange) regions preferred by the
CDF measurement of $M_{W}$, assuming $\lambda_{5}=1$. The dashed blue line
shows $m_{A}$. The gray region is excluded by perturbativity Barbieri:2006dq
with $|\lambda_{4}|\lesssim 10$, whereas the black dotted line represents the
bound when $\lambda_{4}\leq 4$.

Focusing on the small $y$ (and thus large $\lambda_{5}$) region of interest in
this article, we show the CDF preferred parameter space in figure 2. The
$1\sigma$ region requires a rather light $H^{+}$ with hierarchy
$m_{H^{+}}<m_{H}<m_{A}$, forcing $H^{+}$ to decay into
$N_{k}\ell_{\alpha}^{+}$, reminiscent of supersymmetric slepton decays into
neutralinos and thus subject to stringent limits from LEP OPAL:2003wxm ;
OPAL:2003nhx , ATLAS ATLAS:2014hep ; ATLAS:2019lff ; ATLAS:2019gti  and CMS
CMS:2018eqb ; CMS:2018yan ; CMS:2020bfa . At $2\sigma$ in the global fit
Asadi:2022xiy , we have ample parameter space that survives collider
constraints, including both hierarchies $m_{A}>m_{H}>m_{H^{+}}$ and
$m_{H^{+}}>m_{A}>m_{H}$. Even at $2\sigma$ this is a substantial improvement
over the SM-only fit given the $7\sigma$ deviation of the CDF result. Note
that perturbativity and unitarity together with the CDF result ultimately put
an upper bound on the masses that depends on the other $\lambda$ couplings, as
can be seen in figure 2.

##  IV Conversion driven freeze-out

Conversion-driven freeze-out Garny:2017rxs  (or co-scattering DAgnolo:2017dbv
) can be realized in regions of parameter space where the preservation of
chemical equilibrium through efficient conversions in the $\mathbb{Z}_{2}$-odd
sector leads to an underabundance of DM. In this region, sufficiently small
conversion rates between DM (here taken to be $N_{1}$) and its coannihilating
partners can initiate the chemical decoupling of DM and hinder its efficient
dilution due to coannihilation effects. As the chemical decoupling of and
within the $\mathbb{Z}_{2}$-odd sector may overlap in time, in general,
abundances of all $\mathbb{Z}_{2}$-odd particles need to be tracked
individually, _i.e._ , a coupled set of Boltzmann equations need to be solved:

$\displaystyle\frac{{\rm d}Y_{i}}{{\rm d}x}$ $\displaystyle=$
$\displaystyle\frac{1}{3\mathcal{H}}\frac{{\rm d}s}{{\rm
d}x}\left[\,\sum_{j}\big{\langle}\sigma_{ij\to\mathrm{SM}}v\big{\rangle}\left(Y_{i}Y_{j}-Y_{i}^{\mathrm{eq}}Y_{j}^{\mathrm{eq}}\right)\right.$
(16) $\displaystyle\qquad\qquad\left.+\sum_{j\neq
i}\frac{\Gamma_{\text{conv}}^{i\to
j}}{s}\left(Y_{i}-Y_{j}\frac{Y_{i}^{\mathrm{eq}}}{Y_{j}^{\mathrm{eq}}}\right)\right],$

where $i,j\in\\{H,A,{H^{+}},N_{1},N_{2},N_{3}\\}$. Here, $Y_{i}$ denotes the
comoving number density of species $i$, $x=m_{N_{1}}/T$ is the temperature
parameter (with $T$ being the temperature of the SM thermal bath),
$\mathcal{H}$ the Hubble rate, and $s$ the entropy density. Furthermore,
$\big{\langle}\sigma_{ij\to\mathrm{SM}}v\big{\rangle}$ denotes the thermally
averaged annihilation cross section times Møller velocity,
$\Gamma_{\text{conv}}^{i\to j}$ the sum of the conversion rates for
scatterings and (inverse) decays, and $Y_{i}^{\mathrm{eq}}$ the equilibrium
density, see _e.g._ Garny:2017rxs ; Edsjo:1997bg  for explicit expressions for
these quantities. We have neglected annihilations within the
$\mathbb{Z}_{2}$-odd sector as they are of higher order in the involved
couplings and are negligible compared to the conversions in the region of
parameter space where chemical equilibrium within the $\mathbb{Z}_{2}$-odd
sector is questionable.

In the scotogenic model, we can simplify Eq. (16). The sizeable couplings
within the scalar doublet $\eta$ due to its gauge interactions render the
conversions amongst them efficient, and we can assume

$\frac{Y_{i}}{Y_{i}^{\mathrm{eq}}}=\frac{Y_{j}}{Y_{j}^{\mathrm{eq}}}$ (17)

for $i,j\in\\{H,A,{H^{+}}\\}$, _i.e._ , apply the well-known coannihilation
approximation Edsjo:1997bg . We are left with the set of $1+3$ Boltzmann
equations, one for $Y_{\eta}=\sum_{i}Y_{i},\;i\in\\{H,A,{H^{+}}\\}$ and one
for each $Y_{N_{k}}$:

$\displaystyle\begin{split}\frac{{\rm d}Y_{\eta}}{{\rm
d}x}&=\frac{1}{3\mathcal{H}}\frac{{\rm d}s}{{\rm
d}x}\left[\,\big{\langle}\sigma
v\big{\rangle}_{\mathrm{eff}}\left(Y_{\eta}^{2}-(Y_{\eta}^{\mathrm{eq}})^{2}\right)\right.\\\
&\quad+\left.\sum_{i}\frac{\Gamma_{\text{conv,\,eff}}^{\eta\to{N_{i}}}}{s}\left(Y_{\eta}-Y_{N_{i}}\frac{Y_{\eta}^{\mathrm{eq}}}{Y_{N_{i}}^{\mathrm{eq}}}\right)\right],\end{split}$
(18) $\displaystyle\frac{{\rm d}Y_{N_{i}}}{{\rm d}x}$
$\displaystyle=-\frac{1}{3s\mathcal{H}}\frac{{\rm d}s}{{\rm
d}x}\,\Gamma_{\text{conv,\,eff}}^{\eta\to{N_{i}}}\left(Y_{\eta}-Y_{N_{i}}\frac{Y_{\eta}^{\mathrm{eq}}}{Y_{N_{i}}^{\mathrm{eq}}}\right),$
(19)

where we employed the commonly used effective cross section

$\big{\langle}\sigma
v\big{\rangle}_{\mathrm{eff}}\equiv\sum_{i,j\in\\{H,A,{H^{+}}\\}}\big{\langle}\sigma_{ij\to\mathrm{SM}}v\big{\rangle}\frac{Y_{i}^{\mathrm{eq}}Y_{j}^{\mathrm{eq}}}{(Y_{\eta}^{\mathrm{eq}})^{2}}\,.$
(20)

As we cannot assume efficient conversions between the doublet states and
$N_{k}$, the respective terms do not cancel out when summing up the Boltzmann
equations for the scalars. Similar to the effective cross section, they can,
however, be expressed as an effective conversion rate:

$\Gamma_{\text{conv,\,eff}}^{\eta\to
N_{k}}\equiv\sum_{j\in\\{H,A,{H^{+}}\\}}\Gamma_{\text{conv}}^{j\to
N_{k}}\frac{Y_{j}^{\mathrm{eq}}}{Y_{\eta}^{\mathrm{eq}}}\,.$ (21)

Note that the annihilation rate of $N_{k}$ can safely be neglected for the
small Yukawa couplings compatible with conversion-driven freeze-out.

The decay width of the charged scalar $H^{+}$ into a charged lepton $\ell$ and
a Majorana fermion $N$ reads as

$\displaystyle\begin{split}\Gamma^{H^{+}\to\ell_{i}N_{k}}&=\frac{|y_{ik}|^{2}\
m_{H^{+}}}{16\pi}\left(1-\frac{m_{N_{k}}^{2}}{m_{H^{+}}^{2}}-\frac{m_{\ell_{i}}^{2}}{m_{H^{+}}^{2}}\right)\\\
&\quad\times
F\left[\frac{m_{N_{k}}^{2}}{m_{H^{+}}^{2}},\frac{m_{\ell_{i}}^{2}}{m_{H^{+}}^{2}}\right]^{1/2}\end{split}$
(22)

with the function

$F[a,b]\equiv 1+a^{2}+b^{2}-2ab-2a-2b\,.$ (23)

The decay widths for the neutral scalars $\\{H,A\\}$ into $\nu_{i}N_{k}$ are
obtained by simply replacing the respective masses in the above expression,
$m_{H^{+}}\rightarrow m_{H,A}$ and $m_{\ell}\rightarrow m_{\nu}$.

For the numerical solution of the Boltzmann equations (18) and (19), we
compute $\big{\langle}\sigma v\big{\rangle}_{\mathrm{eff}}$ with micrOMEGAs
Belanger:2014vza , employing CalcHep Pukhov:2004ca  and the implementation of
the inert-doublet model included in the program package LopezHonorez:2010eeh .
For the computation of the conversion rate, we only take into account the
contribution of (inverse) decays following Eq. (22).

In general, conversions via scattering can be important, in particular for
small $x$, _i.e._ at an early state of the freeze-out process Garny:2017rxs .
However, as we will show in section V, in the region of interest, the mass
splitting between $N_{k}$ and the lightest doublet state is small compared to
the mass splitting within the doublet. Accordingly, the considerably stronger
phase-space suppression of the decay of the lightest inert state typically
renders the respective rate several orders of magnitude smaller than the one
for the heavier states. Hence, according to Eq. (21), the contribution to
$\Gamma_{\text{conv,\,eff}}^{\eta\to N_{k}}$ from decays of heavier states
dominates over the one of the lightest state at small $x$, _i.e._ for $x$
compatible with or smaller than the inverse of the relative mass splitting
within the doublet. As a result, scatterings are significantly less relevant
at an early state of the freeze-out process where they could potentially
dominate. In fact, they can safely be neglected for the bulk of the parameter
space of interest as we have checked by explicitly taking into account their
leading contributions for several benchmark points.111We have performed two
independent checks. On the one hand, we have computed the scattering cross
sections with MadGraph5_aMC@NLO/MadDM Alwall:2014hca ; Ambrogi:2018jqj  and
solved the Boltzmann equations (18) and (19) accordingly. On the other hand,
the most recent version of micrOMEGAs Alguero:2022inz  directly allows for the
computation of the relic density in the case of conversion-driven freeze-out.
Note, however, that the latter has provided no or obviously incorrect results
(due to double counting of decays and scatterings in the presence of
resonances) in some cases. For both checks, we have used SARAH Staub:2008uz ;
Staub:2009bi  for the implementation of the scotogenic model in the
appropriate format.  Scatterings can, however, become important for very small
mass splittings $\Delta m$, below $\sim 1\,$GeV for $m_{H^{+}}<m_{H},m_{A}$
and considerably smaller values for $m_{H}<m_{H^{+}},m_{A}$.

Further corrections that can affect our result quantitatively come from non-
perturbative effects such as Sommerfeld enhancement and bound state effects.
In particular, the latter can significantly enlarge the viable region of
conversion-driven freeze-out for a strongly interacting coannihilator as
recently found in Garny:2021qsr . We expect a qualitatively similar but
smaller effect here. Their study is, however, beyond the scope of this work.

In writing down Eq. (16), we have also implicitly assumed that the DM momentum
distribution is sufficiently close to the thermal distribution with
temperature $T$. While kinetic equilibrium is not guaranteed to be maintained
through efficient elastic scatterings, this assumption is, nevertheless,
expected to hold. Due to the small mass splitting between DM and the
coannihilating particles, the thermal distribution of the latter is inherited
by DM, as discussed in the appendix of Garny:2017rxs . In that study, the
unintegrated Boltzmann equations have been solved for conversion-driven
freeze-out with similar mass scales, implying a theoretical error of $\lesssim
10\%$ induced by the above assumption.

##  V Results and Discussion

As explained in section IV, the viable region of conversion-driven freeze out
requires underabundant DM if chemical equilibrium was maintained within the
$\mathbb{Z}_{2}$-odd sector. We can hence compute the boundary of that region
by the requirement to saturate the relic density under the assumption of
chemical equilibrium but still neglecting annihilation channels with $N_{k}$
in the initial states. The result for $m_{N_{1}}<m_{H}<m_{A}<m_{H^{+}}$ is
shown in the left panel of figure 3, where we consider four setups regarding
the parameters of the scalar sector:

  1. (i)

CDF preferred, Higgs-philic:

$\lambda_{5}=1$, $m_{H},m_{H^{+}}$ according to $2\sigma$-band in figure 2,
$\lambda_{\text{L}}=-1$ (blue line)

  2. (ii)

CDF preferred, Higgs-phobic:

$\lambda_{5}=1$, $m_{H},m_{H^{+}}$ according to $2\sigma$-band in figure 2,
$\lambda_{\text{L}}=0$ (red line)

  3. (iii)

partly mass-degenerate case:

$m_{A}=m_{H^{+}}$, $m_{H}-m_{A}$ sizeable, due to $\lambda_{5}=1$,
$\lambda_{\text{L}}=0$ (green line)

  4. (iv)

nearly mass-degenerate case:

$m_{A}=m_{H^{+}}=m_{H}+5\,$GeV, $\lambda_{\text{L}}=0$ (purple line)

In scenario (i), the lightest doublet scalar, $H$, has a large coupling
$\lambda_{L}=-1$ to the SM Higgs $h$, which we denote as Higgs-philic; in
scenario (ii), the coupling of $H$ to $h$ is turned off, rendering it Higgs-
phobic. The solid curves denote the case where only $N_{1}$ is lighter than
$H$ and thus the DM, whereas $N_{2,3}$ are chosen sufficiently heavy to not
affect the relic density.222For definiteness, we choose $N_{2,3}=m_{A}$, see
also discussion in section II. However, for the considered magnitude of Yukawa
couplings, $N_{2,3}$ are basically decoupled once their mass is about 10%
larger than the lightest scalar. The short dashed curves denote the case of
mass-degenerate fermions, so all three $N_{k}$ form DM [not present for
scenario (iv)]. Note that the boundary for any other configuration, for
instance, $m_{N_{1}}=m_{N_{2}}$ and $m_{N_{3}}$ heavy, or
$m_{N_{1}}<m_{N_{2,3}}\lesssim m_{H}$, lies between these two limiting cases.
The region _below_ the curves in figure 3 (left) would lead to underabundant
DM if conversion processes would remain efficient,
$\Gamma_{\text{conv}}\gg\mathcal{H}$.

Figure 3:  Left: Boundary of the conversion-driven freeze-out region for the
case $m_{N_{1}}<m_{H}<m_{A},m_{H^{+}}$ for four different slices in the
parameter space as a function of DM mass $m_{N_{1}}$ and mass splitting
$\Delta m=m_{H}-m_{N_{1}}$.
Right: Relic density as a function of the lightest active neutrino mass
$m_{1}$ for the benchmark point shown as a red star in the left panel for the
choice of one light $N$ only, $m_{N_{2,3}}=m_{A}$ (solid curve), and three
mass-degenerate $N$, $m_{N_{2,3}}=m_{N_{1}}$ (dashed).

Figure 4:  Evolution of the DM abundance $Y_{N}$ (blue) and the doublet
abundance $Y_{\eta}$ (red) for the benchmark point considered in the right
panel of figure 3, taking $m_{1}$ to match the measured relic density, $\Omega
h^{2}=0.12$. The left panel represents the case of three mass-degenerate $N$.
For the right panel, only $N_{1}$ is assumed to be light, while
$m_{N_{2}}=m_{N_{3}}=m_{A}$. The purple curves in the right panel show the
abundance of the heavier $N$. The parameters of the doublet are
$m_{H}=300\,$GeV, $m_{A}\simeq 388\,$GeV, $m_{H^{+}}\simeq 442\,$GeV, and
$\lambda_{\text{L}}=0$. The dashed lines show the equilibrium abundances,
solid lines show the solution of the coupled Boltzmann equations. Only
conversions via (inverse) decays are taken into account here, see text for
details.

Conversion driven freeze-out evidently requires a moderately small mass
splitting between DM and the next-to-lightest $\mathbb{Z}_{2}$-odd scalar, $H$
in figure 3 (left). We make the interesting observation that sizeable mass
splittings within the doublet (due to $\lambda_{5}=1$ and – to larger extent –
due to additionally fitting the CDF measurement) significantly enlarge the
viable region for conversion-driven freeze-out. This is due to destructive
interferences among the various diagrams of annihilation processes of the
scalars that suppress their cross section [and overcompensate the competing
Boltzmann suppression of heavier states in Eq. (20)] see _e.g._ discussion in
Refs. Lundstrom:2008ai ; LopezHonorez:2010tb ; Eiteneuer:2017hoh . The
relatively large mass splitting between $H$ and ${H^{+}}$ for the setups (i)
and (ii) preferred by the CDF measurement require a very large $\lambda_{4}$
for large masses, challenging the perturbativity. The steep drop of the
corresponding curves towards the right in figure 3 (left) stems from imposing
Eq. (10). Note that vacuum stability does not impose further constraints as
long as we allow for $\lambda_{2}\gtrsim 2$.

In the right panel of figure 3, we consider the benchmark point with
$m_{H}=300\,$GeV and $\Delta m=5\,$GeV (denoted by an asterisk in the left
panel of the same figure) that belongs to the setup (ii). We use the mass of
the lightest active neutrino, $m_{1}$, to parametrize the Yukawa couplings
following Eq. (13) and the prescription below that formula; see section II for
further details. We assume normal hierarchy. Solving the set of Boltzmann
equations (18) and (19), we compute $\Omega h^{2}$ as a function of $m_{1}$
for the cases light $N_{1}$ and mass degenerate $N_{1,2,3}$. The respective
points that yield the observed value $\Omega h^{2}=0.12$ Planck:2018vyg  are
considered in figure 4 where we show the evolution of the comoving number
densities during freeze out. We also display the corresponding Yukawa
matrices. The first column governs the interaction of $N_{1}$ and is somewhat
smaller than the one for $N_{2,3}$. As we assumed $R=\mathbb{1}$, here, we
have a direct relation between these Yukawas and $m_{1}$. Sizeable off-
diagonal elements in $R$, however, would lead to potentially larger entries in
the first column and could easily lead to chemical equilibrium and, hence,
$\Omega h^{2}<0.12$ for the chosen point inside the conversion-driven freeze-
out region. Therefore, the requirement of saturating the relic density
constraint for a given point inside the conversion-driven freeze-out region
(_i.e._ for a point below the curves in the left panel of figure 3) sets an
upper limit on $m_{1}$, since $R\neq\mathbb{1}$ can only lower the relic
density. Typical values for $m_{1}$ lie between $\sim 1\,$meV (for the
smallest considered masses and three degenerate sterile neutrinos) and $\sim
0.03\,$meV (for a DM mass around $1\,$TeV and the case of one light $N_{1}$
only).

The Yukawa matrices displayed in figure 4 contain values that span roughly 1.5
orders of magnitude. For smaller (larger) $m_{H}$ and otherwise compatible
parameters, the spread among the entries of $|y|$ become smaller (larger)
requiring $\Omega h^{2}=0.12$. Conversion-driven freeze out thus requires
small Yukawas – similar to that of the electron – but very little hierarchy in
the matrix. This conclusion also holds for inverted neutrino hierarchy.

So far, we have focused on the mass hierarchy
$m_{N_{1}}<m_{H}<m_{A}<m_{H^{+}}$, let us now briefly consider
$m_{N_{1}}<m_{H^{+}}<m_{H}<m_{A}$. The boundaries for conversion-driven
freeze-out for this case are shown in the top panel of figure 5. They are
qualitatively similar to the ones shown in figure 3. To reduce clutter, we
therefore only display the curves for the two setups preferred by the CDF
measurements:

  1. (v)

CDF preferred, Higgs-phobic:

$\lambda_{5}=1$, $m_{H},m_{H^{+}}$ according to $2\sigma$-band in figure 2,
$\lambda_{\text{3}}=0$ (red line)

  2. (vi)

CDF preferred, Higgs-philic:

$\lambda_{5}=1$, $m_{H},m_{H^{+}}$ according to $2\sigma$-band in figure 2,
$\lambda_{\text{3}}=-1$ (blue line)

Note that the $2\sigma$-band of the CDF anomaly (figure 2) extends to smaller
masses for this hierarchy, allowing us to explore masses down to 100 GeV. In
particular, we observe a bump in $\Delta m$ around $m_{H^{+}}\sim m_{h}$ due
to the threshold of the annihilation channel $H^{-}{H^{+}}\to hh$. As
$\lambda_{3}$ governs the Higgs-portal coupling of ${H^{+}}$, this bump only
occurs for the blue curve.

Figure 5:  Results for the scenario $m_{H^{+}}<m_{H},m_{A}$.
Top: Boundary of the conversion-driven freeze-out region for one light $N$
(solid lines) and degenerate $N$ (short dashed).
Bottom: LHC constraints from long-lived particle searches for disappearing
tracks (DT search) and heavy stable charged particles (HSCP searches). The red
symbols in both panels denote four benchmark points with $\lambda_{3}=0$ and
one light $N$. We chose $m_{H^{+}}=160$ GeV (accordingly, $m_{H}\simeq
218\,$GeV, $m_{A}\simeq 329\,$GeV), $R=\mathbb{1}$, and $m_{1}$ such as to
match $\Omega h^{2}=0.12$.

We would like to stress that the viable realization of conversion-driven
freeze-out with Yukawa couplings of the order of around $10^{-6}$ is not
restricted to the region preferred by the CDF measurement, although we focus
on these regions for the choice of our concrete examples. In fact,
qualitatively similar results can be found for the partly mass-degenerate
doublet case (iii). A further region worth mentioning is the Higgs resonant
region where $m_{H}\simeq m_{h}/2$. For sizeable $\lambda_{\text{L}}$,
resonant $H$ pair annihilation provides a large cross section allowing for
conversion-driven freeze-out solutions for a sizeable range of $\Delta m$.

###  V.1 LHC constraints

To derive constraints on the scotogenic model from searches for new physics at
the LHC, we use SModelS 2.2.1 Alguero:2021dig , employing the interface of
micrOMEGAs Barducci:2016pcb  for the computation of cross sections and decay
tables. Since the $N_{k}$ are gauge singlets and have small Yukawas, their
production rates are heavily suppressed compared to the doublet scalars. For
the case $m_{H}<m_{A},m_{H^{+}}$, the signature of the model is missing
transverse energy since the heavier scalars promptly decay into the neutral
$H$. As the lightest doublet scalar $H$ predominantly decays into a pair of
neutral particles (an active neutrino and $N$) its decay length does not
affect its signature. Such missing-energy signatures are typical for a wide
variety of DM models. Interestingly, this part of the parameter space of
interest is not challenged by any search implemented in SModelS. The most
relevant searches are Refs. ATLAS:2021moa ; ATLAS:2019lff ; ATLAS:2019wgx ;
CMS:2021edw ; CMS:2017moi .

In contrast, the scenario $m_{H^{+}}<m_{A},m_{H}$ can lead to striking
signatures of long-lived particles at the LHC, as $H^{+}$ can only decay
through the suppressed Yukawa interactions. With the small Yukawas and mass
splittings required for conversion-driven freeze-out, the lifetime of $H^{+}$
can become very long. For decay lengths larger than the size of the detector,
$H^{+}$ is likely to traverse the entire detector leading to the signature of
highly ionizing tracks and anomalous time-of-flight measurements, commonly
denoted as heavy stable charged particles (HSCPs). For decay lengths
compatible with the size of the inner detector, disappearing track or
displaced lepton searches are most sensitive. Figure 5 (bottom) shows the
respective exclusion at 95% CL, obtained from SModelS. The two isolated
exclusion regions above and below $\sim 1\,$m stem from HSCP ATLAS:2019gqq ;
CMS:2015lsu ; CMS:2013czn  and disappearing track ATLAS:2017oal  searches,
respectively. In addition, we display the decay length of the four benchmark
points in the conversion-driven freeze-out region shown in the top panel of
figure 5 spanning values of $\Delta m$ between 1 and 8 GeV, the latter of
which is very close to the boundary of the conversion-driven freeze-out
region. As can be seen, intermediate values for $\Delta m$ provide a
observable signal in disappearing track searches, already excluding a small
portion of the parameter.

Towards larger $\Delta m$ – and thus smaller lifetimes – disappearing-track
searches are less sensitive, requiring a hit in a minimal number of layers in
the inner detector. Displaced lepton searches are more promising to tackle
this region of parameter space. However, the ATLAS displaced lepton search,
Ref. ATLAS:2020wjh , implemented in SModelS, does not provide sensitivity,
being interpreted for a large mass splitting between the mother particle and
DM and hence for relatively hard leptons. In our considered scenario, the
displaced lepton from $H^{-}\to N\ell$ is rather soft due to the small mass
splitting $\Delta m$.

While very large decay lengths (relevant for very small $\Delta m$ only) are
well constraint by HSCP searches, there is a considerable gap between the
disappearing track and HSCP search, again, leaving the entire mass range
unconstrained. Dedicated searches for the particular scenario taking into
account the highly-ionizing nature of $H^{+}$ as well as the displaced decay
into a lepton and missing energy are expected to greatly enhance the
sensitivity.

###  V.2 Other constraints and predictions

Conversion driven freeze-out in the scotogenic model requires fermionic DM, so
DM interacts with the SM exclusively through the Yukawa couplings $y$. The
couplings $|y_{\alpha k}|$ of the $N_{k}$ that forms DM have to be of order
$10^{-7}$–$10^{-6}$ for conversion to work. Couplings of this magnitude and
masses in the 100 GeV to TeV region automatically suppress any and all DM
direct and indirect detection signatures.

Since $|y_{\alpha k}|$ for $N_{k}$ below $10^{-6}$ require a lightest neutrino
mass around or below meV, this is a prediction of our scenario. This upper
limit on the lightest Majorana neutrino mass also implies a lower limit on the
rate for neutrinoless double-beta decay, seeing as that rate can only vanish
for $m_{1}\in(1,10)\,\mathrm{meV}$ under normal ordering Rodejohann:2011mu .

The heavier $N_{k}$ could still have an impact, for example on lepton flavor
violation, if their Yukawa couplings were large. This is in principle allowed
in the conversion-driven region, but would imply a large hierarchy in the $y$
matrix columns, only to be constrained by lepton flavor violation. The more
natural region has all $y$ entries of order $10^{-6}$, which then suppresses
lepton flavor violation and all other signatures involving the neutral
fermions.

In a special region of parameter space the $N_{k}$ are almost degenerate,
leading to one or two long-lived fermions that decay via
$N\to\ell^{+}\ell^{-}N_{\text{DM}}$, $N\to\nu\nu N_{\text{DM}}$, or
$N\to\gamma N_{\text{DM}}$. Such scenarios could be constrained by Big Bang
nucleosynthesis if the lifetimes are long enough, but shall not be discussed
further here.

##  VI Conclusions

The scotogenic model is a well-motivated extension of the SM that
simultaneously addresses DM and the smallness of neutrino masses. The included
scalar doublet also allows for an amelioration of the CDF anomaly by
contributing positively to the $W$ mass. DM production via freeze-out and
freeze-in has been discussed in the literature at length; here we showed for
the first time that the scotogenic model also allows for DM production via
conversion-driven freeze-out. This requires fermionic DM with Yukawa couplings
in-between the typical freeze-out and freeze-in values. In particular, the
necessary Yukawa matrix is neither tiny nor hierarchical, and the same is true
for the parameters of the scalar potential. Conversion-driven freeze-out is
hence quite natural.

The lightest active-neutrino mass plays a crucial role and is typically
between 0.03 and 1 meV. Such a hierarchical active-neutrino spectrum is
becoming increasingly more probable to satisfy cosmological limits on the sum
of neutrino masses. Parts of the parameter space are furthermore testable at
colliders, notably through displaced-vertex decays of the charged scalar.

## Acknowledgements

This work was supported in part by the National Science Foundation under Grant
PHY-2210428. J. Heisig acknowledges support by the Alexander von Humboldt
foundation via the Feodor Lynen Research Fellowship for Experienced
Researchers.

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