# Effective mass overshoot in single degree of freedom mechanical systems with
a particle damper

Martín Sánchez  Luis A. Pugnaloni<EMAIL_ADDRESS>Instituto de
Física de Líquidos y Sistemas Biológicos (CONICET La Plata, UNLP), Calle 59
Nro 789, 1900 La Plata, Argentina.  Departamento de Ingeniería Mecánica,
Facultad Regional La Plata, Universidad Tecnológica Nacional, 60 esq. 124 S/N,
1900 La Plata, Argentina.

###### Abstract

We study the response of a single degree of freedom mechanical system composed
of a primary mass, $M$, a linear spring, a viscous damper and a particle
damper. The particle damper consists in a prismatic enclosure of variable
height that contains spherical grains (total mass $m_{\mathrm{p}}$). Contrary
to what it has been discussed in previous experimental and simulation studies,
we show that, for small containers, the system does not approach the fully
detuned mass limit in a monotonous way. Rather, the system increases its
effective mass up and above $M+m_{\mathrm{p}}$ before reaching this expected
limiting value (which is associated with the immobilization of the particles
due to a very restrictive container). Moreover, we show that a similar effect
appears in the tall container limit where the system reaches effective masses
below the expected asymptotic value $M$. We present a discussion on the origin
of these overshoot responses and the consequences for industrial applications.

###### keywords:

Particle dampers , Granular materials , Effective mass

††journal: Journal of Sound and Vibration

##  1 Introduction

Most mechanical systems, such as rotating machinery and aeronautic or
aerospace structures, achieve damping through viscoelastic materials or
viscous fluids. In general, viscoelastic materials and viscous fluids are very
effective at moderate temperatures (less than 250∘C), but the performance of
these is poor at low and high temperatures. Moreover, these materials degrade
over time and lose their effectiveness.

In recent years, particle dampers (PD) have been studied extensively for use
in harsh environments where other types of damping are not efficient. A PD is
an element that increases the structural damping by inserting dissipative
particles in a box attached to the primary system or by embedding grains
within holes in a vibrating structure [1]. The grains absorb the kinetic
energy of the primary system and convert it into heat through inelastic
collisions and friction between the particles and between the particles and
the walls of the box or hole. This results in a highly nonlinear mechanical
system. PD are effective over a wide frequency range [1]. Moreover, PD are
durable, inexpensive, easy to maintain and have great potential for vibration
and noise suppression for many applications (see e.g. [2] and [3]).

Parameters such as the size and shape of the particles, density, coefficient
of restitution, size and shape of the enclosure, and the type of excitation of
the primary system, among many other features, are important in damping
performance [4]. Thus, appropriate treatment of the PD in a given structure
requires careful analysis and design.

PD use a large number of small grains; therefore, its behavior is directly
related to cooperative movements of those inside the cavity. The theoretical
models derived from single particle systems [5] are not applicable to predict
the performance of multi-particle systems. For more than 15 years, particle
dynamics simulations have been used as a powerful tool for investigating the
behavior of these types of granular systems [6, 7, 8, 9].

In previous works, particle dampers composed of containers of various sizes
have been considered. In all of these works, the resonant frequency of the
Single-Degree-of-Freedom (SDoF) system falls with respect to the undamped
system. This is generally attributed to the addition of the mass of the
particles, $m_{\mathrm{p}}$. At very low excitation amplitudes, the system
behaves as if the entire mass of the particles was attached to the primary
mass, $M$, of the container (i.e. $M+m_{\mathrm{p}}$). If the excitation level
is increased, the resonant frequency gradually increases (and the damping
performance increases). Eventually, the resonant frequency tends to the
resonant frequency of the undamped system. This overall behavior has been
discussed in various papers [9, 10].

Yang [11] has studied, experimentally, particle dampers under different
conditions of excitation, different frequencies and variable gap size. The gap
size is the free space left between the granular bed and the enclosure ceiling
when the system is at rest. He has found that, under some conditions, the
system may display effective masses above $M+m_{\mathrm{p}}$ or below $M$.
However, a careful analysis of this phenomenon has not been carried out yet.

In this paper, we discuss results, obtained through simulations via a Discrete
Elements Method (DEM), on the resonant frequency shift of SDoF mechanical
systems with granular damping. We show that, contrary to what has been
discussed previously, for small containers, the system does not approach the
fully detuned mass limit in a monotonous way. Rather, the system increases its
effective mass up and above $M+m_{\mathrm{p}}$ before reaching the expected
limiting value. Moreover, there is a similar effect in the tall enclosure
limit, where the system reaches effective masses below the expected asymptotic
value $M$.

##  2 Discrete Elements Method

In order to simulate the motion of the particles in the enclosure of a PD we
use a DEM. This scheme, first used by Cundall and Strack [12], is widely used
for numerical simulations of granular media [13]. We implement a velocity
Verlet algorithm [14] to update the positions (orientations) and velocities of
the particles. Orientations are represented through quaternions to prevent
numerical singularities [15].

We consider spherical soft particles. If $R_{i}$ and $R_{j}$ are the radii of
two colliding particles, $\alpha=R_{i}+R_{j}-d_{ij}$ is the normal
displacement or virtual overlap between the spheres, where $d_{ij}$ is the
distance between the centers of the two spheres. Under these conditions, the
interaction force $F_{\mathrm{n}}$ in the normal direction is based on the
Hertz–Kuwabara–Kono model [16, 17].

$F_{\mathrm{n}}=-k_{\mathrm{n}}\alpha^{3/2}-\gamma_{\mathrm{n}}\upsilon_{\mathrm{n}}\sqrt{\alpha}$
(1)

where $k_{\mathrm{n}}=\frac{2}{3}E\sqrt{\frac{R}{2}}(1-\upsilon^{2})^{-1}$ is
the normal stiffness (with $E$ the Young’s modulus, $\upsilon$ the Poisson’s
ratio and $R^{-1}=R_{i}^{-1}+R_{j}^{-1}$), $\gamma_{\mathrm{n}}$ the normal
damping coefficient of the viscoelastic contact, and $\upsilon_{\mathrm{n}}$
the relative normal velocity.

On the other hand, the tangential force $F_{\mathrm{s}}$ is based on Coulomb’s
law of friction [16, 18]. We used a simplified model in which the friction
force takes the minimum value between the shear damping force and the dynamic
friction.

$F_{\mathrm{s}}=-\min\left(\left|\gamma_{\mathrm{s}}\upsilon_{\mathrm{s}}\sqrt{\alpha}\right|,\left|\mu_{\mathrm{d}}F_{\mathrm{n}}\right|\right)\rm{sgn}\left(\upsilon_{\mathrm{s}}\right)$
(2)

where $\gamma_{\mathrm{s}}$ is the shear damping coefficient,
$\upsilon_{\mathrm{s}}$ the relative tangential velocity between the two
spheres in contact and $\mu_{\mathrm{d}}$ the dynamic friction coefficient.
The sign function indicates that the friction force always opposes the
direction of the relative tangential velocity.

The particles are enclosed in a prismatic container (the box or enclosure)
built up of six flat walls with the same material properties as the particles
defined through the parameters in Eqs. (1) and (2).

##  3 The SDoF model

Figure 1 shows the model of our SDoF system with PD which is assumed to move
only along the direction of the vertical $z$-axis. The primary system consists
of a mass $M=2.37$ kg, a spring $K=21500$ Nm-1 and a viscous damper with
damping constant $C$. We have used two different values for viscous damping,
$C=7.6$ and $26.3$ Nsm-1. The undamped ($C=0$ and no PD) natural frequency of
the primary system is $f_{0}=15.16$ Hz.

Figure 1: (a) Model of the SDoF system with a particle damper. (b) Snapshots
of the particles in the enclosure during a typical simulation.

The PD is modeled as $N=250$ spherical grains in a prismatic enclosure of
lateral side $L_{x}=L_{y}=0.03675$ m and different heights $L_{z}$. The
material properties of the particles (and walls) and the simulation parameters
are listed in Table 1. The gravitational field $g=9.8$ ms-2 is considered in
the negative vertical direction. Although the SDoF system can only move in the
vertical direction, the particles move freely inside the enclosure.

The system is excited by the harmonic displacement of the base to which the
spring and viscous damper are attached (see Fig. 1). Let $u(t)$ and $z(t)$ be
the displacement of the base and the primary mass, respectively. Then, the
equation of motion for the system is given by

$M\ddot{z}(t)+C\dot{z}(t)+K{z}(t)=C\dot{u}(t)+Ku(t)+F_{\mathrm{part}}(t),\ \ \
\ \ \ u(t)=U\cos(\omega t),$ (3)

where $F_{\mathrm{part}}(t)$ is the $z$-component of the force resulting from
all the interactions (normal and tangential) of the enclosure walls with the
particles. The amplitude, $U$, and the angular frequency, $\omega$, of the
harmonic vibrating base, are control parameters.

Property | Value
---|---
Young’s modulus $E$ |  $2.03\times 10^{11}$ Nm-2
Density | 8030 kgm-3
Poisson’s ratio $\upsilon$ | 0.28
Friction coefficient $\mu_{\mathrm{d}}$ | 0.3
Normal damping coefficient $\gamma_{\mathrm{n}}$ |  $3.660\times 10^{3}$ kgs-1m-1/2
Shear damping coefficient $\gamma_{\mathrm{s}}$ |  $1.098\times 10^{4}$ kg$s^{-1}$m-1/2
Excitation amplitude $U$ | 0.0045 m
Time step $\delta t$ |  $8.75\times 10^{-8}$ s
Time of simulation | 13.12 s
Particle radius | 0.003 m
Total particle mass $m_{\mathrm{p}}$ | 0.227 kg
Table 1: Material properties of the particles and simulation parameters.

We have obtained the frequency response function (FRF) for different enclosure
heights $L_{z}$. The initial condition for each simulation consists of a
simple deposition of the particles inside the enclosure, starting from a
dilute random arrangement, before applying the base excitation.

##  4 Data analysis

As shown in Table 1, we have simulated the vibration of the system for $13.12$
s. After an initial transient, the system reaches a steady state. This steady
state can display either regular or chaotic behavior, depending on the
excitation frequency, box height, etc. In all cases, the final 10% of the time
of the simulations has been used for the data analysis, which has proved to be
sufficient to ensure that the steady state has been reached. We have studied
frequency excitations in the range ($0.5-30.0$ Hz).

We carry out a simple evaluation of the effective mass and effective damping
of the PD by fitting the FRF to a SDoF system with no particles in the
enclosure. Other approximate methods such as the power flow method used by
Yang [11] present numerical instabilities and can yield negative effective
masses (see e.g. [19]).

Figure 2: Examples of the FRF of the SDoF system with PD for different
$L_{z}$. Green squares: $L_{z}=0.057$ m, blue circles: $L_{z}=0.1225$ m and
red diamonds: $L_{z}=0.372$ m. These results correspond to the simulations
with $C=7.6$ Nsm-1. The solid lines are fits of the FRF with an equivalent
mass–spring–dashpot model without PD [see Eq. (4)].

The amplitude of the response $X$ of a system with no PD is given by

$X=U\left[\frac{K^{2}+(C_{\mathrm{eff}}\omega)^{2}}{(K-M_{\mathrm{eff}}\omega^{2})^{2}+(C_{\mathrm{eff}}\omega)^{2}}\right]^{1/2}$
(4)

We carry out a least-squares curve fitting of the DEM data with Eq. (4). The
values of $K$ and $U$ are fixed to the corresponding values in our simulations
and $C_{\mathrm{eff}}$ and $M_{\mathrm{eff}}$ are fitting parameters.

It is important to mention that for certain enclosure heights, the FRF is not
a smooth function as the one corresponding to Eq. (4). However, the overall
shape is well described by this fit. For small and large enclosures, the shape
of the FRF of the PD is rather smooth and the fits are warranted. Examples of
the quality of the fits are shown in Fig. 2 for three different $L_{z}$. An
improvement of the fits for the FRF at intermediate $L_{z}$ for which some
fluctuations are present could be achieved by using models with extra degrees
of freedoms (e.g. a tuned mass damper). However, the characteristic
frequencies of the PD are not well separated and the equivalent
mass–spring–dashpot model constitute a good first order approximation.

##  5 Results

Figure 3: Frequency response function for the SDoF system with PD. Each curve
corresponds to a different box height $L_{z}$. Red squares: $L_{z}=0.057$ m,
green circles: $L_{z}=0.1225$ m and blue diamonds: $L_{z}=0.372$ m. The black
continuous line corresponds to the SDoF system without particles in the
enclosure. The black dashed line corresponds to an equivalent system with an
added mass $m_{\mathrm{p}}$ equal to the mass of the particles. These results
correspond to the simulations with $C=7.6$ Nsm-1. The green circles correspond
to value of $L_{z}$ for which the maximum effective damping is obtained.

Examples of the FRF of the PD can been seen in Fig. 3 for a few box sizes
$L_{z}$. It is important to note that, in general, the gap size and not
$L_{z}$ controls the dynamics. In this work all simulations are carried out
with the same number of particles and particle size. Hence, the gap size is
directly obtained as $L_{z}-h$; with $h=0.039$ m the approximate height of the
granular bed at rest. For a study on the dependence of the dynamics on the
number of particles and particle size see Ref. [7, 20].

The general trends observed in Fig. 3 are consistent with previous
experimental and simulation works (see e.g. [7], [9] and [10]). For small gaps
($L_{z}<0.087$ m) the FRF tends to the response of a system without particles
but with an added mass equivalent to the mass $m_{\mathrm{p}}$ of the
particles. For very tall boxes ($L_{z}>0.222$ m), the response follows the one
expected for an empty container. For intermediate values of $L_{z}$,
significant granular damping is obtained with damped resonant frequencies
intermediate between the two asymptotic cases. However, these intermediate
values of $L_{z}$ yield more complex FRFs with the presence of more than one
peak [10].

###  5.1 Effective damping and effective mass

A simple evaluation of the effective damping yield by the particles can be
done by fitting the FRF to a SDoF system including only a viscous damper as
discussed in Section 4. The effective damping, $C_{\mathrm{eff}}$ is plotted
as a function of $L_{z}$ in Fig. 4. A clear optimum value of $L_{z}$ is
predicted as in several previous works [7, 20]. Notice that an increase of the
viscous damping $C$ leads to a less important influence of the PD on the
effective damping $C_{\mathrm{eff}}$. This is due to the fact that the damping
due to the dashpot of the SDoF system reduces the transfer of energy to the
particles inside the enclosure.

Figure 4: Effective damping $C_{\mathrm{eff}}$ relative to the viscous damping
$C$ as a function of $L_{z}$. The blue line corresponds to the SDoF system
with $C=7.6$ Nsm-1 and the green line to the system with $C=26.365$ Nsm-1. The
error bars correspond to the asymptotic standard error of the
$C_{\mathrm{eff}}$ best fit.

The two limiting cases (small $L_{z}$ and large $L_{z}$) have been explained
[9, 11, 21] in terms of the proportion of the time that particles spend in
contact with the enclosure. For small $L_{z}$, particles are essentially fixed
in their positions since the constraint imposed by the walls impede their
relative motion. Therefore, the particles behave as a simple mass added to the
primary mass $M$ of the system and provide no extra damping (in this limit the
effective damping equals the viscous damping). Conversely, if the enclosure
leaves sufficient room for the motion of the particles, the granular bed will
reach a gas-like state (at the large excitation levels achieved near
resonance) in which most of the time particles are in the air and only
occasionally collide against the floor and ceiling of the box. In this case,
the particles will barely influence the motion of the primary system and
$C_{\mathrm{eff}}$ falls back to the baseline, $C$, imposed by the viscous
damper.

In Fig. 5 we plot the effective mass, $M_{\mathrm{eff}}$, obtained from the
fits. The expected limiting masses $M$ and $M+m_{\mathrm{p}}$ are shown as
horizontal lines for reference. Our data surveys a larger number of box
heights in comparison with previous studies. We can see that, in contrast with
the suggestion of previous studies, the two limiting cases are not approached
in a monotonous way. The system reaches effective masses above
$M+m_{\mathrm{p}}$ as $L_{z}$ is decreased and finally falls towards the limit
value. Similarly, for large $L_{z}$, an increase of the box height leads to
effective masses below $M$ before the limit value is approached. It is worth
mentioning that a similar behavior has been observed in some experiments [11].
The author reports that the resonant frequency (which is simply related to our
effective mass by $M_{\mathrm{eff}}=k/(2\pi f_{0})^{2}$) can reach values
above the expected for an empty enclosure.

Figure 5: Effective mass as a function of $L_{z}$. The blue line corresponds
to the SDoF system with $C=7.6$ Nsm-1 and the green line to the system with
$C=26.365$ Nsm-1. The error bars correspond to the asymptotic standard error
of the $M_{\mathrm{eff}}$ best fit.

The presence of these overshoots when approaching the two limiting cases (zero
gap size and infinite gap size) has been overlooked or received little
attention in the past and is somewhat against intuition. In general, studies
of PD are focused on the region of gap sizes where the maximum effective
damping is observed. There, the effective mass is, as expected, intermediate
between $M$ and $M+m_{\mathrm{p}}$. However, design constraints may require a
PD to work at gap sizes off the optimum damping and in the region of effective
mass overshoot. In what follows, we discuss the origin of such behavior in the
response of the PD by considering the internal motion of the granular bed.

###  5.2 Internal motion of the granular bed

In Fig. 6 we plot the trajectory of the floor and ceiling of the enclosure
over a few periods of excitation in the steady state regime. We have chosen
frequencies close to the resonant frequency for each $L_{z}$ considered. This
is because the effective values of mass and damping obtained by fitting are
determined to a large extent by the frequencies with larger amplitude of
motion. In Fig. 6, we have indicated the position of the granular bed inside
the enclosure by a band limited by the $z$-coordinates of the uppermost and
lowermost particle at any given time. Notice that such representation lets us
have an indication of the density of the granular bed and the approximate time
of impact with the box top and bottom walls. However, if the granular bed is
somewhat dilute, the time of impact of the uppermost or lowermost particle
does not coincide with the time at which the most substantial momentum
exchange happens between the grains and the enclosure. For this reason we also
plot in Fig. 6 the total force exerted by the grains on the enclosure in the
$z$-direction (i.e. $F_{\mathrm{part}}$, see Eq. (3)) which not only provides
a more precise assessment of the time of impact but of the intensity of such
impacts.

We recall here that the effective mass of an equivalent mass–spring-dashpot
model is associated with the response of the PD in phase with the spring
force, whereas the effective damping will be associated with the response in
phase with the viscous force.

As we mentioned, for very small gaps, the response of the system is similar to
the response of an equivalent SDoF system where the total mass of the
particles is simply added to the primary mass (i.e.,
$M_{\mathrm{eff}}=M+m_{\mathrm{p}}$). This is due to the fact that the
particles are not able to move in such reduced enclosures. As we slightly
increase $L_{z}$, particles behave as a dense pack that travels between the
floor and ceiling of the enclosure (see Fig. 6(a)). However, within an
oscillation, the granular bed is in full contact with the ceiling or the floor
during important portions of the time. During such periods, the granular bed
can be considered essentially as an added mass $m_{\mathrm{p}}$. Notice that
after leaving the ceiling (floor) the grains hit the floor (ceiling) before
the primary system has reached its maximum displacement. The particles
transfer momentum to the enclosure against the direction of the spring force.
As a consequence, the effective inertia of the system during the impact is
equivalent to a sudden mass increase. This effective mass increase exceeds the
loss due to the short periods in which the granular bed is detached from the
floor (ceiling) (see the zero force segments in Fig. 6(a)). The overall result
is an effective mass $M_{\mathrm{eff}}$ above $M+m_{\mathrm{p}}$.

If we further increase $L_{z}$, the period of detachment within an oscillation
increases, but the transfer of momentum at impact also increases leading to an
overall increase of $M_{\mathrm{eff}}$ (see Fig. 6(b)). Eventually, the
transfer of momentum at impact is exactly balanced by the loss of added mass
due to the detachment periods. This makes the system render again
$M_{\mathrm{eff}}=M+m_{\mathrm{p}}$ (see Fig. 6(c)). Interestingly, this
crossover occurs when the granular bed hits the floor (ceiling) at the point
of maximum displacement.

Figure 6: Displacement and force of particles against the enclosure for
different $L_{z}$ with $C=7.6$ Nsm-1. The solid lines show the position of
floor and ceiling of the enclosure and the colored area indicates the limits
of the granular bed defined as the position of the uppermost and lowermost
particle. (a) $L_{z}=0.042$ m and $f=14.5$ Hz. (b) $L_{z}=0.057$ m and
$f=14.5$ Hz. (c) $L_{z}=0.087$ m and $f=14.5$ Hz. (d) $L_{z}=0.1095$ m and
$f=15.0$ Hz. (e) $L_{z}=0.1225$ m and $f=15.0$ Hz. (f) $L_{z}=0.147$ m and
$f=15.5$ Hz. (g) $L_{z}=0.372$ m and $f=15.0$ Hz. (h) Same as in Fig. 5 with
arrows indicating the values of $L_{z}$ corresponding to each panel (a)-(g).

There exist a range of values of $L_{z}$ for which
$M<M_{\mathrm{eff}}<M+m_{\mathrm{p}}$. In such cases the effective damping is
rather high. This is mainly due to the fact that the granular bed hits the
enclosure out of phase. In particular, the grains hit the base when both the
primary mass is moving upward and the spring force is pulling upward (see Fig.
6(d)). This results in a strong reduction of the maximum displacement of the
system and in an effective added mass with respect to $M$. However, since the
periods of detachment from the enclosure are significantly long, the average
added mass due to the impacts is smaller than $m_{\mathrm{p}}$.

If the impacts happens to be always at the time where no spring force is
applied to the primary mass, then the transfer of momentum will not result in
an effective added mass. This happens when the primary mass pass through the
equilibrium point of the spring (i.e., zero displacement). Indeed, we find
that there exists a particular value of $L_{z}$ at which the granular bed hits
the floor (ceiling) at this point and the effective mass obtained by fitting
corresponds to $M$ (i.e., no added mass, see Fig. 6(e)). It is important to
realize that this is also the value of $L_{z}$ at which the maximum effective
damping is obtained (see Fig. 4 and Ref.[11, 22, 23]). Therefore, at the
optimum $L_{z}$ where maximum damping is achieved, the effective mass
coincides with the primary mass. This means that the addition of particles to
create the PD do not affect the resonant frequency of the system if the
optimum $L_{z}$ is chosen, which implies that under such conditions there is
no need for compensation of the mass of the particles during design.

As expected, an increase of $L_{z}$ beyond the optimum damping leads the
granular bed to hit the enclosure in phase with the spring force (see Fig.
6(f)). That is, the grains hit the floor when the spring pulls the system
downward. The effect in the apparent inertial response is as if the system
suffered a sudden mass decrease. Therefore, the effective mass is smaller than
the primary mass (notice that here the granular bed is only in contact with
the enclosure during the impacts).

Much larger gap sizes lead the granular bed to expand significantly. The
granular sample enters a gas-like state with only a few particles colliding
with the enclosure in each oscillation. This transfers little momentum to the
primary mass and the system presents an $M_{\mathrm{eff}}$ which is close to
$M$ (see Fig. 6(g)).

##  6 Conclusions

We have studied a PD by means of simulations via a DEM. We have considered the
effective mass and effective damping of the entire SDoF system by fitting the
FRF to a simple mass–spring–dashpot system. In particular, we study the effect
of the height $L_{z}$ of the enclosure.

We have observed that the effective mass of the system reaches the two limits
described in the literature for small and large enclosures. However, those
limits are not approached in a monotonous way and clear overshoots appear. For
small gap sizes, the system presents effective masses above the direct sum of
the primary mass $M$ and the particle mass $m_{\mathrm{p}}$. For large
enclosures, the effective masses fall below $M$.

We have observed that such behavior can be explained by considering both the
period of time over which the granular bed is in full contact with the
enclosure and the inertial effects due to the grains hitting the floor or
ceiling in or out of phase with the spring force.

Interestingly, we found that the value of $L_{z}$ at which $M_{\mathrm{eff}}$
crosses $M$ coincides with the optimum damping value described in the
literature. Since the optimum $L_{z}$ should be simple to interpolate from the
intersection with the horizontal $M$ level in a plot of $M_{\mathrm{eff}}$ vs
$L_{z}$, we suggest that such estimation can be a more suitable approach than
the search for a maximum in a plot of $C_{\mathrm{eff}}$ versus $L_{z}$.

The overshoot effects described are present outside the range of maximal
damping performance and might be considered of secondary interest in
industrial applications at first sight. However, design constrains may require
a PD to work off the optimum damping and in one of the overshoot regions. In
particular, for enclosures somewhat taller than the one corresponding to the
optimum damping, one achieves effective masses only slightly below the primary
mass. This implies that the resonant frequency is almost unaltered upon
addition of the particles. Moreover, such values of $L_{z}$ achieve a
remarkable damping (although not maximal) while still presenting a FRF with a
shape very similar to the one observed for a simple SDoF mass–spring–dashpot
system. This may simplify the prediction of the behavior of the PD under such
conditions. On the other hand, if the primary system has more degrees of
freedom, the off-design resonant frequencies may fall in either overshoot
regime and the side effects should be taken into consideration.

Although we have studied a PD driven in the direction of gravity, similar
results are expected if a horizontal setup is considered. It has been shown
that near the resonant frequency the response of an impact damper does not
depend on the relative direction between the motion of the system and the
gravity [5]. Since the effective mass is largely determined by the response
near the resonant frequency, the same general trends should be found in
horizontally driven PD.

## Acknowledgments

LAP acknowledges financial support from CONICET (Argentina).

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#  $W$ boson mass shift, dark matter and $(g-2)_{\ell}$ in a scotogenic-Zee
model

Ritu Dcruz<EMAIL_ADDRESS>Department of Physics, Oklahoma State
University, Stillwater, OK 74078, USA  Anil Thapa<EMAIL_ADDRESS>Department of Physics, University of Virginia, Charlottesville, Virginia
22904-4714, USA

###### Abstract

We present a singly charged scalar extension of the Scotogenic model,
ScotoZee, which resolves the recently reported deviations in $W$ boson mass as
well as lepton $g-2$. The model admits a scalar or a fermionic dark matter
while realizing naturally small radiative neutrino masses. The mass splitting
of $\sim 100$ GeV, required by the shift in $W$ boson mass, among the inert
doublets fields can be evaded by its mixing with the singlet scalar, which is
also key to resolving $(g-2)_{\ell}$ anomaly within $1\sigma$. We establish
the consistency of this framework with dark matter relic abundance while
satisfying constraints from charged lepton flavor violation, direct detection
as well as collider bounds. The model gives predictions for the lepton flavor
violating $\tau\to\ell\gamma$ processes testable in upcoming experiments.

##  I Introduction

The CDF collaboration at Fermilab CDF:2022hxs  reported a precision
measurement of $W$ boson mass, $M_{W}^{\textrm{CDF}}=(80.4335\pm
0.0094)\;\textrm{GeV}$, which is in tension with the Standard Model (SM)
prediction, $M_{W}^{\textrm{SM}}=(80.357\pm 0.004)\;\textrm{GeV}$
Awramik:2003rn , with an excess at $7\sigma$ level, which may be an indication
of new physics (NP) beyond the Standard Model (SM). The new result from CDF
collaboration, with a much reduced uncertainty, has a higher precision than
the PDG world average of $M_{W}^{\textrm{PDG}}=(80.377\pm
0.012)\;\textrm{GeV}$ ParticleDataGroup:2022pth  which takes into account the
$W$ mass measurements from LEP ALEPH:2013dgf , Tevatron (CDF CDF:2012gpf  and
D0 D0:2012kms ) CDF:2013dpa  and LHCb collaboration LHCb:2021abm . The PDG
average, which is in agreement with the SM prediction, disagrees with the CDF
Run-II result. It has been shown that the improvement in parton density
functions Gao:2022wxk  and perturbative matrix elements Martin:2022qiv ;
Isaacson:2022rts  cannot account for this discrepancy. However, the
discrepancy may be due to high-twist power corrections within the SM that are
normally not considered in perturbative calculations ELLIS19821 ; Ellis:1982cd
. This work proceeds under the assumption that the new CDF measurement will be
validated as the right result of $W$ boson mass.

Some possible explanations to the $W$ boson mass shift can arise at tree-level
Strumia:2022qkt ; Asadi:2022xiy ; Bagnaschi:2022whn ; DiLuzio:2022ziu ;
Chen:2022ocr ; PerezFileviez:2022gmy ; Ghoshal:2022vzo ; Borah:2022zim ;
Athron:2022isz ; Almeida:2022lcs ; Addazi:2022fbj ; Heeck:2022fvl ; Du:2022fqv
; Zeng:2022lkk ; Cheng:2022aau ; Cai:2022cti , or at loop level Liu:2022jdq ;
Song:2022xts ; Biekotter:2022abc ; Crivellin:2022fdf ; Heo:2022dey ;
Bahl:2022xzi ; Ahn:2022xeq ; Carpenter:2022oyg ; Popov:2022ldh ;
Ghorbani:2022vtv ; Lu:2022bgw ; Han:2022juu ; Heckman:2022the ; Cao:2022mif ;
Lee:2022gyf ; Abouabid:2022lpg ; Benbrik:2022dja , along with the prospect of
reconciling one or more discrepancies Baek:2022agi ; Bhaskar:2022vgk ;
Chowdhury:2022moc ; Cheung:2022zsb ; Babu:2022pdn ; Lee:2022nqz ;
Athron:2022qpo ; Yang:2022gvz ; Du:2022pbp ; Zhang:2022nnh ; Zhu:2022scj ;
Cacciapaglia:2022xih ; Fan:2022dck ; Zhu:2022tpr ; Batra:2022org ;
Arcadi:2022dmt ; Nagao:2022oin ; Kawamura:2022uft ; Tang:2022pxh ;
Zhou:2022cql ; Botella:2022rte ; Kim:2022xuo ; Kim:2022hvh  such as flavor
anomalies and dark matter. Several other papers deBlas:2022hdk ;
Campagnari:2022vzx ; Blennow:2022yfm ; Sakurai:2022hwh ; Fan:2022yly ; Arias-
Aragon:2022ats ; Paul:2022dds ; Gu:2022htv ; DiLuzio:2022xns ; Cheng:2022jyi ;
Endo:2022kiw ; Du:2022brr ; Balkin:2022glu ; Krasnikov:2022xsi ; Zheng:2022irz
; Sun:2022zbq ; Peli:2022ybi ; Kanemura:2022ahw ; Mondal:2022xdy ; Tan:2022bip
; Cirigliano:2022qdm ; Borah:2022obi ; Yang:2022qgs ; Rahaman:2022dwp ;
Pellen:2022fom ; Dermisek:2022xal ; Chen:2022ntw ; Perez:2022uil ;
Gupta:2022lrt ; Isaacson:2022rts ; Yuan:2022cpw  also examine the consequence
of the CDF $M_{W}$ anomaly on new physics scenarios.

Independently, muon $(g-2)$ collaboration at Fermilab Abi:2021gix  has
confirmed the long standing discrepancy in the anomalous magnetic moment (AMM)
of muon measurement at BNL in 2006 Bennett:2006fi  at a combined
$4.2\sigma$111The recent results from the BMW collaboration Borsanyi:2020mff
agrees with the experimental measurement within $1.6\sigma$. deviation,
$\Delta a_{\mu}^{\textrm{exp}}=(2.51\pm 0.59)\times 10^{-9}$, from the SM
prediction (see Ref. Aoyama:2020ynm  and the references therein). In addition
to these recent anomalies, astrophysical and cosmological observations
Planck:2013pxb ; WMAP:2010qai ; Planck:2018vyg  present a compelling evidence
for the existence of dark matter (DM), for which the SM fails to provide an
explanation. Moreover, one of the major shortcomings of the SM is its
inability to explain the origin of non-zero neutrino mass substantiated by
several experiments ParticleDataGroup:2020ssz .

In this work we show that a simple extension of the Scotogenic model Ma:2006km
with a charged singlet (ScotoZee model) can simultaneously address all the
puzzles previously mentioned. Our novel ScotoZee model222The scalar content is
the same as the Inert Zee model Longas:2015cnr ; Longas:2015sxk  with only
right-handed neutrinos in contrast to vector-like singlets and doublets. The
ScotoSinglet model Beniwal:2020hjc  is a neutral scalar extension of the
Scotogenic model. Neither models can resolve the discrepancy in $(g-2)_{\mu}$
Gaviria:2018cwb . is the simplest model that furnish a direct link between
neutrino mass generation, dark matter, AMM of muon and also provide an upward
mass shift in $W$ boson in agreement with the CDF measurement. Additionally,
the presumed anomaly in the AMM of electron Hanneke:2008tm ; Parker:2018vye ;
Aoyama:2017uqe  can also be addressed within the same framework. We explore
the parameter space of the ScotoZee model spanned by both the bosonic and
fermionic DM candidates, while being consistent with the current experimental
constraints.

The rest of the paper is organized as follows. In Sec. II we give a brief
description of the ScotoZee model, discussing the neutrino mass generation and
the scalar sector. In Sec. III and Sec. IV we respectively introduce and
examine a resolution to $W$ mass shift and AMM phenomenology in the model.
Sec. V discusses DM phenomenology for both scalar and fermionic DM candidates.
Lastly, we integrate the three puzzles ($W$-mass shift, lepton $g-2$, and DM)
with neutrino mass generation and LFV constraints invoking a highly predictive
flavor texture in Sec. VI before concluding in Sec. VII.

##  II Model

The proposed ScotoZee model is a simple charged singlet $S^{+}~{}(1,1;-)$
extension of the Scotogenic Ma:2006km  model, which contains Majorana singlet
fermions $N_{R_{i}}~{}(1,0;-)$ and the scalar doublet
$(\eta^{+},\eta^{0})\equiv\eta~{}(2,1/2;-)$, under the gauge group
$SU(2)_{L}\times U(1)_{Y}\times\mathbb{Z}_{2}$. All the new particles are odd
under $\mathbb{Z}_{2}$ while the SM particles are even, guaranteeing the
stability of the DM candidate; the lightest among the new neutral
$\mathbb{Z}_{2}$-odd particles. The charged scalar singlet $S^{+}$ not only
gives corrections to anomalous magnetic moment of muon and electron through
the mixing with charged doublet, but also serves as a portal to generate
correct relic abundance for fermionic DM.

The effective Yukawa Lagrangian in the extended model can be written as

$-\mathcal{L}_{Y}\supset
Y_{ij}\overline{L}_{L_{i}}\widetilde{\eta}N_{R_{j}}+f_{ij}\overline{\ell}_{R_{i}}S^{-}\overline{N}_{R_{j}}+\text{h.c.}.$
(1)

The $\mathbb{Z}_{2}$ symmetry, being exact, prevents $\eta^{0}$ from obtaining
a non-zero vacuum expectation value (VEV) and neutrinos remain massless at
tree-level. Moreover, the SM Higgs $h$ is decoupled from the new
$\mathcal{CP}$-even ($Re(\eta^{0})\approx H$) and -odd ($Im(\eta^{0})\approx
A$) scalars. The tree-level scalar potential of the model is given by

$\displaystyle V$
$\displaystyle=\mu_{h}^{2}\phi^{\dagger}\phi+\mu_{S}^{2}S^{-}S^{+}+\mu_{\eta}^{2}\eta^{\dagger}\eta+\frac{\lambda_{1}}{2}(\phi^{\dagger}\phi)^{2}+\frac{\lambda_{2}}{2}(\eta^{\dagger}\eta)^{2}$
$\displaystyle+\lambda_{3}(\phi^{\dagger}\phi)(\eta^{\dagger}\eta)+\lambda_{4}(\phi^{\dagger}\eta)(\eta^{\dagger}\phi)+\frac{\lambda_{5}}{2}\\{(\phi^{\dagger}\eta)^{2}+\text{h.c.}\\}$
$\displaystyle+\frac{\lambda_{6}}{2}(S^{-}S^{+})^{2}+\lambda_{7}(\phi^{\dagger}\phi)(S^{-}S^{+})+\lambda_{8}(\eta^{\dagger}\eta)(S^{-}S^{+})$
$\displaystyle+\frac{\mu}{2}\\{\varepsilon_{\alpha\beta}\phi^{\alpha}\eta^{\beta}S^{-}+\text{h.c.}\\}.$
(2)

The charged scalars $\\{\eta^{+},S^{+}\\}$ mix giving rise to mass eigenstates
$\\{H_{1}^{+},H_{2}^{+}\\}$. The masses of the scalar fields in the physical
basis are given by

$\displaystyle m^{2}_{h}=\lambda_{1}v^{2},\,\hskip
11.38109ptm^{2}_{H(A)}=\mu_{\eta}^{2}+\frac{v^{2}}{2}\left(\lambda_{3}+\lambda_{4}\pm\lambda_{5}\right),$
$\displaystyle
m_{H^{+}_{i}}^{2}=\frac{1}{2}\left(\mu_{2}+\mu_{3}\pm\sqrt{(\mu_{2}-\mu_{3})^{2}+2\mu^{2}v^{2}}\right),$
(3)

where $\mu_{2}=\mu_{\eta}^{2}+\frac{\lambda_{3}}{2}v^{2}$,
$\mu_{3}=\mu_{S}^{2}+\frac{\lambda_{7}}{2}v^{2}$. Here $\mu_{\eta,S}$,
$\lambda_{i}$, and $\mu$ are the bare-mass terms, quartic couplings, and cubic
coupling, respectively. The mixing angle between the charged scalar fields is
defined as

$\sin 2\theta=\frac{-\sqrt{2}\mu v}{m_{H_{1}^{+}}^{2}-m_{H_{2}^{+}}^{2}}\,,$
(4)

with the VEV, $v\simeq 246$ GeV. In this work, we comply with the perturbative
and vacuum stability conditions Deshpande:1977rw ; Kannike:2012pe
constraining the scalar couplings. We also have ensured that our parameter
space does not drive the the mass parameters $\mu_{\eta}^{2}$ and
$\mu_{S}^{2}$ to negative values; negative bare mass terms can lead the inert
doublet or the charged singlet to attain a non-zero VEV, thereby breaking the
$\mathbb{Z}_{2}$ symmetry Merle:2015gea . Such $\mathbb{Z}_{2}$ parity
violation not only destabilizes the DM candidates but will also allow
unacceptably large tree-level neutrino masses. These arise from the RG
evolution of the bare mass terms and might pose a serious problem when the
coupling of the inert scalars to heavy neutrinos are of $\mathcal{O}(1)$. As
will be discussed in the following sections, the coupling of $\eta$ to
$N_{R_{k}}$ are small enough to avoid breaking the $\mathbb{Z}_{2}$ symmetry.
On the other hand, the coupling between $S^{+}$ and $N_{R_{k}}$ are taken to
be $\mathcal{O}(1)$. Thus, in order for the model to be valid above the EW
scale (such that the $\mathbb{Z}_{2}$ symmetry is preserved), it is important
to check that the RG evolution does not only preserve $\mu_{\eta}^{2}>0$ but
also $\mu_{S}^{2}>0$. The trilinear coupling $\mu$ gives positive
contributions to the RG evolution of the new scalar mass-squared parameters,
thereby helping prevent the breaking of the model symmetries. Note that it is
also essential to ensure that $\mu$ is not much larger than the scalar masses
as it can result in a deeper minimum than the SM one Barroso:2005hc ;
Babu:2019mfe . The Majorana mass term $\frac{1}{2}M_{N_{i}}N_{i}N_{i}$ along
with the scalar quartic term
$\frac{\lambda_{5}}{2}\\{(\phi^{\dagger}\eta)^{2}+\text{h.c.}\\}$ breaks the
lepton number by two units generating one-loop neutrino mass (c.f. Fig. 1
(top)) $\mathcal{M}_{\nu}$ expressed as

$\displaystyle\left(\mathcal{M}_{\nu}\right)_{ij}=\sum_{k}Y_{ik}\Lambda_{k}Y^{*}_{kj}\,,$
$\displaystyle\Lambda_{k}=\frac{M_{N_{k}}}{16\pi^{2}}\Bigg{[}\frac{m_{H}^{2}}{m_{H}^{2}-M_{N_{k}}^{2}}\log\frac{m_{H}^{2}}{M_{N_{k}}^{2}}-(m_{H}\leftrightarrow
m_{A})\Bigg{]}.$ (5)

Figure 1: Radiative neutrino mass generation at one-loop (top). The dominant
correction to AMMs arising through the chiral enhancement (bottom). The cross
($\times$) represents mass insertion whereas $\ell=e~{}(\mu)$ for electron
(muon) AMM.

Here the lightest mass eigenstates $\\{H,A\\}$ and $N_{i}$ can serve as a
viable bosonic and fermionic DM candidates, respectively. It is important to
point out that unlike the Scotogenic model, where $M_{N}$ can be at canonical
seesaw scale of $10^{9}$ GeV or the Yukawa coupling $Y$ arbitrarily small, the
$(g-2)_{\mu}$ in the model requires the scale to be in the (sub) TeV range
along with $\mathcal{O}(0.1-1.0)$ Yukawa coupling. Thus, a successful
explanation of $m_{\nu}\sim 0.1$ eV would naturally require $m_{H}$ to be
nearly degenerate with $m_{A}$.

##  III Correction to $W$ boson mass

The shift in $W$ boson mass Grimus:2008nb  can be evaluated as a function of
the oblique parameters, $S,\,T$ and $U$ Peskin:1990zt ; Peskin:1991sw  that
quantify the deviation of a new physics model from the SM through radiative
corrections arising from shifts in gauge boson self energies. The oblique
parameters in our model get corrections from the extended Higgs sector which
is same as in the Zee model Zee:1980ai  except for the $\mathbb{Z}_{2}$ charge
preventing the mixing with the SM Higgs doublet. Therefore, we use the
expressions for $S$, $T$ and $U$ given in Ref. Herrero-Garcia:2017xdu  under
the alignment limit Haber:2010bw . Note that the corrections to $U$ at one-
loop level is suppressed compared to $S$ and $T$.

With the new precision measurement of $M_{W}$ by CDF, some electroweak (EW)
observables are expected to suffer from this deviation. We incorporate the
global EW fit Asadi:2022xiy  with the new CDF data to quote the $2\sigma$
allowed ranges of oblique parameters. We confirm the necessity of mass
splitting in 2HDM Ahn:2022xeq ; Bahl:2022xzi  to accommodate the recent CDF
results and show that the introduction of the charged singlet scalar allows
the components of the doublet field to be degenerate in certain regions for a
specific choice of $\sin\theta$ and $m_{H_{2}^{+}}$, as can be seen from Fig.
2 (top) (for more detail see Fig. 5). The splitting
$\delta_{H+}=m_{H_{1}^{+}}-m_{H}$ depends on the mixing angle, for instance,
it can be at most $\sim 140$ GeV for $\sin\theta=0.2$. In spite of explaining
the CDF $W$ mass shift, the proposed model remains consistent with the
previous experiments resulting in the PDG world average of $W$ boson mass. It
can be seen from Fig 2 (bottom) that region consistent with the older
experiments opens up more parameter space of the model, even allowing for
degeneracy between the doublet fields in the whole range of allowed parameter
space for particular values of $\sin\theta$ and $m_{H_{2}^{+}}$.

Figure 2: Top figure shows the mass splitting between the components of
doublet scalar required by the new CDF measurement of $W$ boson mass using
$2\sigma$ ranges for $S$ and $T$ from Ref. Asadi:2022xiy . Bottom figure shows
the $2\sigma$ band allowed by the CDF measurement in contrast with the PDG
world average ParticleDataGroup:2022pth ; Lu:2022bgw .

##  IV Anomalous Magnetic Moment

The charged scalar contributions to anomalous magnetic moment at one-loop
Leveille:1977rc  as shown in Fig. 1 (bottom) is

$\displaystyle\Delta a_{\ell}^{H^{+}_{1}}=$
$\displaystyle\frac{m_{\ell}^{2}}{16\pi^{2}}\Big{(}\left(|Y_{\ell
i}|^{2}\sin^{2}\theta+|f_{\ell i}|^{2}\cos^{2}\theta\right)\
G[m_{H_{1}^{+}},2]$ $\displaystyle+\frac{M_{N_{i}}}{m_{\ell}}\
\textrm{Re}(Y_{\ell i}f^{*}_{\ell i})\sin 2\theta\ G[m_{H^{+}_{1}},1]\Big{)},$
(6)

where,

$\displaystyle G[M,\varepsilon]=\int_{0}^{1}\frac{x^{\varepsilon}(x-1)\
\,dx}{m_{\ell}^{2}x^{2}+(M^{2}-m_{\ell}^{2})x+M_{N_{i}}^{2}(1-x)},$ (7)

and $\Delta a_{\ell}^{H^{+}_{2}}=\Delta
a_{\ell}^{H^{+}_{1}}(\theta\to\frac{\pi}{2}+\theta)$. The dominant
contribution to $\Delta a_{\mu}$ comes from the Majorana neutrino mass
enhancement aided by the mixing of the charged scalar mediators as shown in
Fig. 1 (bottom). The sign of the product of Yukawa couplings and the mixing
angle can be chosen independently. This in turn allows for the simultaneous
explanations of $\Delta a_{\ell}$ ($\ell=e,\,\mu$). Moreover, $\Delta a_{\mu}$
provides an upper limit on the mass of Majorana neutrino (charged scalar) of
order 15 (6.5) TeV with $f,Y\sim\mathcal{O}(1)$. The mass limit is relaxed in
the case of $\Delta a_{e}$.

Note that the Yukawa couplings and the masses of charged scalars are severely
restricted by the charged lepton flavor violating (cLFV) processes such as
radiative decay $\ell_{i}\to\ell_{j}\gamma$ Lavoura:2003xp ; such processes
are enhanced in our model by the mass insertion of Majorana neutrinos.
Moreover, although trilepton decays such as $\mu\to 3e$ do not occur at the
tree-level, they arise at the one-loop with large branching ratios
Toma:2013zsa . The same is also true for $\mu-e$ conversion in the nuclei. We
impose these constraints in our parameter scan.

##  V DM Phenomenology

Figure 3: The allowed parameter space for fermion (top) and neutral scalar
(bottom) dark matter candidate to simultaneously explain the AMM of muon and
$W$ boson mass shift. The green (orange) band satisfies the correct relic
abundance ($(g-2)_{\mu}$) for fixed $\kappa=Y^{*}f\sin\theta$. The choice of
scalar masses provide an upward $W$ boson mass shift as expected from CDF
measurement. The purple band (top figure) is the exclusion region region from
$pp\to\ell^{+}\ell^{-}+\not{E}_{T}$ signature obtained by recasting
supersymmetric slepton searches at the LHC CMS:2018eqb  and from chargino
production searches at LEP OPAL:2003wxm ; OPAL:2003nhx  (bottom figure). The
blue dashed-dotted lines (top) shows the direct detection cross section lower
bounds Hisano:2015bma ; Herrero-Garcia:2018koq  from the LZ experiment
LZ:2022ufs  for different quartic coulings $\lambda_{7}$ and our choice of
Yukawa coupling $f=1.5$.

In addition to explaining $W$ boson mass shift and $\Delta a_{\ell}$, the
proposed model can easily accommodate both the scalar (lightest of $H$ and
$A$) and fermionic (lightest among $N_{i}$) dark matter candidates ($\chi$).
We consider both scenarios and analyze the parameter space by implementing the
model in SARAH Staub:2008uz  and numerically evaluating the relic abundance
using the software MicrOMEGAs Belanger:2006is . The relic density of DM is
achieved through standard thermal freeze-out mechanism.

For the case of Majorana fermion as a DM ($\chi\equiv N$) candidate, the
annihilation channel which determines the observed relic abundance are DM
self-(co-)annihilation into charged leptons
$\ell_{\alpha}^{-}\ell_{\beta}^{+}$ (light neutrinos
$\nu_{\alpha}\bar{\nu}_{\beta}$) through $t$-channel processes mediated by the
$\mathbb{Z}_{2}$-odd scalars, $H_{i}^{+}$ ($H,A$) via Yukawa couplings $Y$
and/or $f$. The neutrino oscillation data determines the flavor structure of Y
making it natural to select relatively small $Y$ and heavy doublet scalar
$\eta$ $\sim\mathcal{O}\ ({\rm TeV})$ such that the LFV constraints are
relaxed. Thus, we choose $f_{ii}\sim\mathcal{O}(1)\ (i=1,2)$ and degenerate
$N_{i}$ to maximize the contribution to annihilation mode
$\chi\chi\to\ell\ell$ via $S^{+}$; the allowed parameter space in the mass
plane can be seen in Fig. 3 (top) along with the region resolving muon AMM for
a specific choice of $\kappa=Y^{*}f\sin\theta=0.015$.

Note that in Fig. 3, the effects of co-annihilation are not taken into account
for fermionic DM. The co-annihilation between fermionic DM and the charged
scalar $H_{2}^{\pm}$ becomes important when the mass difference is small;
$(m_{H_{2}^{\pm}}-m_{N})/m_{N}<0.1$ Jueid:2020yfj . This dominates over the
annihilation processes mainly when $f<1$, a parameter space not favorable for
our work to incorporate $(g-2)_{\ell}$ and satisfy LFV. Thus, we simply avoid
co-annihilation by taking the mass splitting larger.

Although fermionic DM does not contribute to tree-level direct detection cross
section, it can arise at one-loop order via photon, $Z$, and Higgs penguin
diagrams involving charged scalars and charged leptons in the loop Herrero-
Garcia:2018koq . The relevant dimension-6 operators for Majorana DM direct
detection are the ones with bilinears $\overline{\chi}\chi$,
$\overline{\chi}\gamma_{5}\chi$, and
$\overline{\chi}\gamma^{\mu}\gamma_{5}\chi$. In the model presented here, both
Yukawa couplings $f$ and $Y$ give rise to such diagrams. As previously
mentioned the Yukawa coupling $Y$ needs to be small. Thus the dominant
contribution comes from the coupling $f\sim\mathcal{O}(1)$ of Eq. (1) with
right-handed charged lepton $\ell_{R}$ and charged scalar $S^{+}$ in the loop.
In this case, the $Z$ penguin contribution is suppressed due to the absence of
axial-vector coupling to $\ell_{R}$. Photon mediated processes can lead to
anapole operator Ho:2012bg  of the form
$\mathcal{O}^{q}_{AV}=(\overline{\chi}\gamma^{\mu}\gamma_{5}\chi)(\overline{q}\gamma_{\mu}q)$,
where $q$ represent the quark flavor eigenstates. Anapole contributions,
however, are momentum and velocity-suppressed in the non-relativistic limit
Gresham:2014vja . We find that the cross section from anapole contribution for
the DM masses in our framework is at most of $\mathcal{O}(10^{-55})\
\text{cm}^{2}$ Bai:2014osa ; Herrero-Garcia:2018koq , well beyond the current
sensitivity. The dominant contribution, therefore, appears from the Higgs
penguin diagram leading to spin-independent direct detection from the operator
$\mathcal{O}^{q}_{SS}=m_{q}(\overline{\chi}\chi)(\overline{q}q)$. The
corresponding Wilson coefficient is dependent not only on the Yukawa coupling
$f$ but also on the quartic coupling $\lambda_{7}$. This contribution can
therefore be suppressed by choosing $\lambda_{7}$ small, allowing for $f$
coupling to be $\gg 1$. The allowed parameter space with the current bounds
LZ:2022ufs  for direct detection cross sections Hisano:2015bma ; Herrero-
Garcia:2018koq  is shown in Fig. 3 (top) for different choices of
$\lambda_{7}$ and a fixed Yukawa coupling $f=1.5$.

In the case of scalar dark matter, which we choose to be the
$\mathcal{CP}$-even $H\equiv\chi$ (nearly degenerate333The mass splitting is
of order $\mathcal{O}(100)$ keV Arina:2009um  to evade direct detection. with
$A$ and $\lambda_{5}<0$), pair of DM can annihilate to $W^{+}W^{-}$, $ZZ$,
$\nu_{\alpha}\nu_{\beta}$, $hh$, $\bar{\ell}\ell$, and $\bar{q}q$. The low
mass regime suffer a strong constraint form LEP OPAL:2003wxm ; OPAL:2003nhx ;
Pierce:2007ut  which can be satisfied if one assumes $m_{\chi}>M_{Z}/2$,
$m_{H_{1}^{+}}>M_{W}/2$ and $m_{H_{1}^{+}}+m_{\chi}>M_{W}$. For larger DM
mass, it predominantly annihilates to a pair of $W^{+}W^{-}$ and $ZZ$, for
which the allowed region is $m_{\chi}\gtrsim 500$ GeV and mass splitting
$\delta_{H^{+}}=m_{H_{1}^{+}}-m_{\chi}\lesssim 30$ GeV as shown in Fig. 3
(bottom). This can be relaxed by making the Higgs quartic coupling $\gtrsim
1$, a choice strongly constrained by direct detection bound Lehnert:2018vzs ;
LUX-ZEPLIN:2018poe ; XENON:2018voc ; CDMS-II:2009ktb ; LZ:2022ufs .

In this work we take the quartic couplings
$\lambda_{3}+\lambda_{4}+\lambda_{5}\ll 1$ to automatically satisfy direct
detection bound obtained from the scalar DM interacting with nucleus at the
tree-level through the SM Higgs boson. Moreover, it is favored to take the
couplings $Y_{ij}$ small and $M_{N}\sim\mathcal{O}$(TeV) to be consistent with
neutrino fit, which implies that the DM analysis is indistinguishable from the
known inert doublet model (IDM). It turns out that the CDF measurement
requires mass splitting among the inert doublet fields of $\mathcal{O}(100)$
GeV, disfavouring the scalar DM candidates in the Scotogenic/IDM. However, the
mixing between the charged scalars in this model allows the components of the
doublet field to be degenerate (c.f. Fig 2), thereby admitting the
$\mathcal{CP}$-even $H$ to be a viable DM candidate, as shown in Fig. 3
(bottom).

The direct detection cross section can arise at one-loop from vertices like
$HHZZ$, $HHWW$, $HZA$, and $HWH_{1}^{\pm}$ driven by the gauge coupling. These
contributions can be as important as the tree-level processes. In the model
presented here, the quartic couplings that give rise to tree-level direct
detection is chosen to be small, thus its contribution in comparison to loop
effect is negligible. For the masses $m_{\chi}\geq 500$ GeV required to
explain the total observed relic density within the model (see Figure 3
(bottom)), we find that the one-loop cross section is $\sim 1.1\times
10^{-46}$ cm2 (see Ref. Cirelli:2005uq ; Klasen:2013btp ; Banerjee:2019luv
for details). This evades the current experimental bound Lehnert:2018vzs ;
LUX-ZEPLIN:2018poe ; XENON:2018voc ; CDMS-II:2009ktb ; LZ:2022ufs , but will
be sensitive to upcoming experiments XENON:2020kmp ; Church:2020env .

##  VI Neutrino Fit/ Lepton Flavor violation

Figure 4: Scattered plot assuming the fermionic DM with the same parameter
space given in Fig. 3 (top). Colored shaded regions are the current exclusion
limits BaBar:2009hkt , whereas dashed-dotted lines represent the future
projected sensitivities Hayasaka:2013dsa . The orange (green) dots correspond
to solutions that resolve $\Delta a_{\mu}$ ($\Delta a_{e,\mu}$), satisfy the
observed relic density as well as neutrino oscillation observables within
their $2\sigma$ measured values Esteban:2020cvm .

The neutrino mass formula of Eq. (5), lepton $g-2$ and the dark matter
analysis have close-knit correlation through Yukawa couplings, Majorana
fermions, and new scalars. As previously stated, $(g-2)_{\ell}$ sets an upper
bound on the masses of the Majorana fermions and the charged scalars with
$f,Y\sim\mathcal{O}(1)$. Moreover, the maximum splitting among the doublet
fields is restricted by the shift in $W$-boson mass, thereby forcing the
parameter space to the region $m_{A}\simeq m_{H}$, crucial in explaining the
observed neutrino oscillation data.

In order to check the consistency with the neutrino oscillation data and
efficiently probe the model with LFV observables, we adopt Casas-Ibarra
parametrization Casas:2001sr  to rewrite the Yukawa matrix $Y$ of Eq. (5) in
terms of neutrino mass parameters

$Y=\ U\sqrt{\mathcal{M}_{\nu}^{\rm diag}}R^{\dagger}\,\sqrt{\Lambda}^{-1}\,,$
(8)

where $R$ is an arbitrary complex orthogonal matrix. The neutrino oscillation
parameters are scanned within the $2\sigma$ allowed ranges Esteban:2020cvm  to
obtain the Yukawa matrix.

As mentioned earlier, the product of Yukawa couplings $Y_{\ell i}f_{\ell
i}^{*}$ can explain $(g-2)_{\ell}$; however these couplings are constrained by
LFV processes. To facilitate a direct correlation between the observed relic
density and both muon and electron $(g-2)$, we consider two (degenerate)
stable DM candidates with their respecting Yukawa couplings of
$\mathcal{O}(1)$. This also allows more parameter space for the DM mass
$m_{\chi}$ and mediator mass $m_{H_{2}^{+}}$ as shown in Fig 3 (top). With
such a choice of large $f_{ii}(i=1,2)$, the mass enhancement to
$\ell_{i}\to\ell_{j}\gamma$ severely restricts the parameter space. Such
chirally enhanced contribution can be suppressed with a suitable choice of
Yukawa couplings and masses of the Majorana fermions. For instance, chirally
enhanced $\mu\to e\gamma$ can be evaded with the choice of $Y_{12}=Y_{21}\sim
0$ or $Y_{21}f_{11}\simeq-Y_{12}f_{22}$ for $M_{N_{1}}=M_{N_{2}}(=m_{\chi})$.
We then check the consistency of our fit by computing the branching fractions
of $\ell_{i}\to\ell_{j}\gamma$ and $\ell_{i}\to 3\ell_{j}$ process at one-loop
level and make testable predictions for fermionic DM (see Fig. 4). In the case
of scalar DM, since the Yukawa coupling $f$ does not play any role in relic
abundance, there is more freedom in the choice of parameters and yield no
sizeable predictions. It is important to note that a single Majorana fermionic
DM candidate by itself can successfully resolve $(g-2)_{\mu}$ with $f_{\mu
i}\sim\mathcal{O}(1)$. This would open up the parameter space with much weaker
constraints arising from LFV processes. However, such a choice would not lead
to direct correlation between $(g-2)_{e}$ and DM, and also does not lead to
LFV prediction.

##  VII Conclusions

In the light of recent experimental results confirming a $4.2\sigma$
discrepancy in the measurement of $(g-2)_{\mu}$ and a possible $7\sigma$
excess in the mass of $W$ boson it is imperative to investigate new physics
contributions for clarification. We propose the ScotoZee model, a simple
charged singlet extension of the Scotogenic model, to show a direct
correlation between these anomalies and the observed neutrino oscillation data
as well as dark matter relic abundance. We explore the parameter space spanned
by both the bosonic and fermionic dark matter candidates and provide a
coherent resolution to electron and muon AMMs and $M_{W}$ anomaly while
evading dangerous LFV processes like $\mu\to e\gamma$ and $\mu\to 3e$. In
contrast to the IDM/Scotogenic models where the small mass splitting among the
doublet fields required for the observed relic density is disfavored by the
CDF measurement, the scalar DM candidate in our model survives due to the
presence of the extra charged singlet which is essential in resolving $\Delta
a_{\ell}$. This model predicts large rates for LFV processes
$\tau\to\ell\gamma$ which can be tested in the future experiments.

Acknowledgements: We would like to thank Julian Heeck, K.S. Babu, Vishnu P.K.,
and D. Raut for useful discussions. RD thanks the U.S. Department of Energy
for the financial support, under grant number DE-SC 0016013.

## Appendix: Oblique parameters

Here we provide additional plots in the model to show the allowed parameter
space consistent with the upward CDF W mass shift.

Figure 5: Mixing angle $\theta$ as a function of charged scalar mass
$m_{H_{2}^{+}}$ for different mass splitting,
$\delta_{H^{+}}=m_{H_{1}^{+}}-m_{H}$ (left) and the mass splitting
$\delta_{H^{+}}$ as a function of neutral scalars for different choices of
charged singlet scalar mass (right) explaining the upward shift in $M_{W}$
reported by CDF measurement, consistent with the $2\sigma$ ranges of $S$ and
$T$.

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$\displaystyle+3\omega^{2}|1000100\rangle+3\omega|1001110\rangle+4\omega|1001111\rangle+3\omega^{3}|1010110\rangle-4\omega^{3}|1010111\rangle-4\omega|1011011\rangle$
$\displaystyle-3|1011100\rangle-3\omega^{3}|1100110\rangle-4\omega^{3}|1100111\rangle\rangle-4\omega|1101011\rangle+3|1101100\rangle+4\omega^{3}|1110011\rangle$
$\displaystyle-3\omega^{2}|1110100\rangle-3\omega|1111110\rangle-4\omega|1111111\rangle),$
$\displaystyle|\psi_{2}\rangle=$
$\displaystyle\frac{1}{20}(4\omega|0000011\rangle-3|0000100\rangle+3\omega^{3}|0001110\rangle+4\omega^{3}|0001111\rangle+3\omega|0010110\rangle+4\omega|0010111\rangle$
(149)
$\displaystyle-4\omega^{3}|0011011\rangle+3\omega^{2}|0011100\rangle-3\omega|0100110\rangle-4\omega|0100111\rangle+4\omega^{3}|0101011\rangle-3\omega^{2}|0101100\rangle$
$\displaystyle-4\omega|0110011\rangle+3|0110100\rangle-3\omega^{3}|0111110\rangle-4\omega^{3}|0111111\rangle-4\omega|1000011\rangle+3|1000100\rangle$
$\displaystyle+3\omega^{3}|1001110\rangle+4\omega^{3}|1001111\rangle-3\omega|1010110\rangle-4\omega|1010111\rangle-4\omega^{3}|1011011\rangle+3\omega^{2}|1011100\rangle$
$\displaystyle-3\omega|1100110\rangle-4\omega|1100111\rangle-4\omega^{3}|1101011\rangle+3\omega^{2}|1101100\rangle-4\omega|1110011\rangle+3|1110100\rangle$
$\displaystyle+3\omega^{3}|1111110\rangle-4\omega^{3}|1111111\rangle),$

where $\omega=\exp(i\pi/4)$. Its weight enumerators are

$\displaystyle A(z)=$ $\displaystyle
1+\frac{106}{125}z+z^{2}+\frac{144}{125}z^{3}+\frac{1299}{125}z^{4}$ (150)
$\displaystyle+\frac{3318}{125}z^{5}+\frac{2451}{125}z^{6}+\frac{432}{125}z^{7},\mkern
50.0mu$ $\displaystyle B(z)=$ $\displaystyle
1+\frac{106}{125}z+z^{2}++\frac{606}{25}z^{3}+\frac{7071}{125}z^{4}$ (151)
$\displaystyle+\frac{9318}{125}z^{5}+\frac{8679}{125}z^{6}+\frac{3546}{125}z^{7}.$

##  Appendix F Quantum Weight Enumerators

This section lists the quantum weight enumerators of some QECCs
discovered/rediscovered by VarQEC.

###  F.1 Symmetric codes

$\displaystyle A^{\\{5,6,2\\}}(z)=1+1.667z^{4}+2.667z^{5},$ (152)
$\displaystyle B^{\\{5,6,2\\}}(z)=1+20z^{2}+50z^{3}+75z^{4}+46z^{5};$ (153)
$\displaystyle A^{\\{5,2,3\\}}(z)=1+15z^{4},$ (154) $\displaystyle
B^{\\{5,2,3\\}}(z)=1+30z^{3}+15z^{4}+18z^{5};$ (155)

$\displaystyle\begin{aligned}
A^{\\{6,2,3\\}}(z)=&1+0.267z+0.732z^{2}+12.070z^{4}\\\
&+15.732z^{5}+2.197z^{6},\end{aligned}$ (156) $\displaystyle\begin{aligned}
B^{\\{6,2,3\\}}(z)=&1+0.267z+0.732z^{2}+25.605z^{3}\\\
&+37.676z^{4}+38.126z^{5}+24.591z^{6};\end{aligned}$ (157)

$\displaystyle A^{\\{7,8,2\\}}(z)=1+5z^{4}+2z^{5}+2z^{6}+6z^{7},$ (158)
$\displaystyle\begin{aligned}
B^{\\{7,8,2\\}}(z)=&1+17z^{2}+40z^{3}+195z^{4}+328z^{5}\\\
&+299z^{6}+144z^{7};\end{aligned}$ (159)

Non-degenerate $((7,2,3))_{2}$:

$\displaystyle\begin{aligned}
A^{\\{7,2,3\\}}(z)=&1+1.437z^{2}+18.125z^{4}+43.437z^{6},\end{aligned}$ (160)
$\displaystyle\begin{aligned}
B^{\\{7,2,3\\}}(z)=&1+1.437z^{2}+25.311z^{3}+18.125z^{4}\\\
&+117.377z^{5}+43.437z^{6}+49.311z^{7};\end{aligned}$ (161)

Degenerate $((7,2,3))_{2}$:

$\displaystyle\begin{aligned}
A^{\\{7,2,3\\}}(z)=1+5z^{2}+11z^{4}+47z^{6},\end{aligned}$ (162)
$\displaystyle\begin{aligned}
B^{\\{7,2,3\\}}(z)=&1+5z^{2}+36z^{3}+11z^{4}+96z^{5}\\\
&+47z^{6}+60z^{7};\end{aligned}$ (163)

$\displaystyle\begin{aligned}
A^{\\{8,2,3\\}}(z)=&1+0.015z^{3}+13.924z^{4}+24.091z^{5}\\\
&+40.030z^{6}+39.893z^{7}+9.046z^{8},\end{aligned}$ (164)
$\displaystyle\begin{aligned} B^{\\{8,2,3\\}}(z)=&1+11.970z^{3}+38.152z^{4}\\\
&+119.817z^{5}+159.939z^{6}\\\ &+124.213z^{7}+56.909z^{8};\end{aligned}$ (165)

$\displaystyle A^{\\{8,8,3\\}}(z)=1+28z^{6}+3z^{8},$ (166)
$\displaystyle\begin{aligned}
B^{\\{8,8,3\\}}(z)=&1+56z^{3}+210z^{4}+336z^{5}+728z^{6}\\\
&+504z^{7}+213z^{8};\end{aligned}$ (167)

$\displaystyle\begin{aligned}
A^{\\{9,8,3\\}}(z)=&1+0.042z^{4}+5.875z^{5}+16.083z^{6}\\\
&+24.083z^{7}+14.875z^{8}+2.042z^{9},\end{aligned}$ (168)
$\displaystyle\begin{aligned}
B^{\\{9,8,3\\}}(z)=&1+40z^{3}+162.332z^{4}+479.004z^{5}\\\
&+952.664z^{6}+1224.664z^{7}\\\ &+932.004z^{8}+304.332z^{9};\end{aligned}$
(169)

$\displaystyle\begin{aligned}
A^{\\{10,16,3\\}}(z)=&1+0.127z^{5}+8.525z^{6}+20.443z^{7}\\\
&+21.253z^{8}+11.430z^{9}+1.221z^{10},\end{aligned}$ (170)
$\displaystyle\begin{aligned}
B^{\\{10,16,3\\}}(z)=&1+55.810z^{3}+275.961z^{4}\\\
&+954.241z^{5}+2366.014z^{6}\\\ &+4120.948z^{7}+4622.227z^{8}\\\
&+3061.001z^{9}+926.797z^{10};\end{aligned}$ (171)

$\displaystyle A^{\\{10,4,4\\}}(z)=1+90z^{6}+135z^{8}+30z^{10},$ (172)
$\displaystyle\begin{aligned}
B^{\\{10,4,4\\}}(z)=&1+90z^{4}+216z^{5}+720z^{6}+720z^{7}\\\
&+1485z^{8}+600z^{9}+264z^{10}.\end{aligned}$ (173)

$\displaystyle\begin{aligned}
A^{\\{11,32,3\\}}(z)=&1+0.160z^{6}+12.522z^{7}+23.316z^{8}\\\
&+18.319z^{9}+7.524z^{10}+1.158z^{11}\end{aligned}$ (174)
$\displaystyle\begin{aligned}
B^{\\{11,32,3\\}}(z)=&1+75.003z^{3}+443.260z^{4}\\\
&+1729.654z^{5}+5219.456z^{6}\\\ &+11343.613z^{7}+16918.654z^{8}\\\
&+16859.456z^{9}+10185.630z^{10}\\\ &+2760.274z^{11}\end{aligned}$ (175)

$\displaystyle\begin{aligned}
A^{\\{12,64,3\\}}(z)=&1+2z^{7}+15z^{8}+24z^{9}+16z^{10}\\\
&+6z^{11},\end{aligned}$ (176) $\displaystyle\begin{aligned}
B^{\\{12,64,3\\}}(z)=&1+104z^{3}+649z^{4}+2976z^{5}\\\
&+10472z^{6}+27184z^{7}+50691z^{8}\\\ &+67616z^{9}+60952z^{10}+33192z^{11}\\\
&+8307z^{12}.\end{aligned}$ (177)

$\displaystyle\begin{aligned}
A^{\\{13,128,3\\}}(z)=&1+5z^{8}+16z^{9}+24z^{10}+16z^{11}\\\
&+2z^{12},\end{aligned}$ (178) $\displaystyle\begin{aligned}
B^{\\{13,128,3\\}}(z)=&1+138z^{3}+929z^{4}+4814z^{5}\\\
&+19592z^{6}+58628z^{7}+131987z^{8}\\\ &+219836z^{9}+263864z^{10}\\\
&+215954z^{11}+107925z^{12}\\\ &+24918z^{13}.\end{aligned}$ (179)

$\displaystyle\begin{aligned}
A^{\\{14,256,3\\}}(z)=&1+2z^{8}+4z^{9}+18z^{10}+28z^{11}\\\
&+11z^{12},\end{aligned}$ (180) $\displaystyle\begin{aligned}
B^{\\{14,256,3\\}}(z)=&1+180z^{3}+1295z^{4}+7436z^{5}\\\
&+34418z^{6}+117320z^{7}\\\ &+307391z^{8}+616280z^{9}\\\
&+923372z^{10}+1007300z^{11}\\\ &+755921z^{12}+348636z^{13}\\\
&+74754z^{14}.\end{aligned}$ (181)

###  F.2 Asymmetric codes

$\displaystyle\begin{aligned}
A^{\\{6,2,d_{e}(\frac{1}{2})=2\\}}(z)=1+6z^{2}+9z^{4}+16z^{6},\end{aligned}$
(182) $\displaystyle\begin{aligned}
B^{\\{6,2,d_{e}(\frac{1}{2})=2\\}}(z)=&1+15z^{2}+8z^{3}+39z^{4}\\\
&+24z^{5}+41z^{6};\end{aligned}$ (183)

$\displaystyle\begin{aligned}
A^{\\{7,3,d_{e}(\frac{1}{2})=2\\}}(z)=&1+1.111z^{2}+2.667z^{3}\\\
&+4.778z^{4}+13.333z^{5}\\\ &+15.444z^{6}+53.333z^{7},\end{aligned}$ (184)
$\displaystyle\begin{aligned}
B^{\\{7,3,d_{e}(\frac{1}{2})=2\\}}(z)=&1+3.667z^{2}+24z^{3}+61.667z^{4}\\\
&+120z^{5}+125.667z^{6}+48z^{7};\end{aligned}$ (185)

$\displaystyle\begin{aligned}
A^{\\{5,2,d_{e}(2)=3\\}}(z)=1+4z^{2}+3z^{4}+8z^{5},\end{aligned}$ (186)
$\displaystyle\begin{aligned}
B^{\\{5,2,d_{e}(2)=3\\}}(z)=&1+12z^{2}+10z^{3}+19z^{4}\\\
&+22z^{5};\end{aligned}$ (187)

$\displaystyle\begin{aligned}
A^{\\{6,4,d_{e}(2)=3\\}}(z)=1+z^{3}+4z^{4}+7z^{5}+3z^{6},\end{aligned}$ (188)
$\displaystyle\begin{aligned}
B^{\\{6,4,d_{e}(2)=3\\}}(z)=&1+7z^{2}+36z^{3}+75z^{4}+92z^{5}\\\
&+45z^{6};\end{aligned}$ (189)

$\displaystyle\begin{aligned}
A^{\\{7,8,d_{e}(2)=3\\}}(z)=1+z^{4}+6z^{5}+6z^{6}+2z^{7},\end{aligned}$ (190)
$\displaystyle\begin{aligned}
B^{\\{7,8,d_{e}(2)=3\\}}(z)=&1+9z^{2}+64z^{3}+179z^{4}\\\
&+312z^{5}+323z^{6}+136z^{7};\end{aligned}$ (191)

$\displaystyle\begin{aligned}
A^{\\{6,2,d_{e}(2)=4\\}}(z)=1+z^{2}+11z^{4}+16z^{5}+3z^{6},\end{aligned}$
(192) $\displaystyle\begin{aligned}
B^{\\{6,2,d_{e}(2)=4\\}}(z)=&1+z^{2}+24z^{3}+35z^{4}+40z^{5}\\\
&+27z^{6};\end{aligned}$ (193)

$\displaystyle\begin{aligned}
A^{\\{8,3,d_{e}(2)=4\\}}(z)=&1+0.224z^{2}+1.770z^{3}\\\
&+4.919z^{4}+17.794z^{5}\\\ &+28.155z^{6}+23.103z^{7}\\\
&+8.368z^{8},\end{aligned}$ (194) $\displaystyle\begin{aligned}
B^{\\{8,3,d_{e}(2)=4\\}}(z)=&1+2.021z^{2}+18.612z^{3}\\\
&+68.016z^{4}+154.775z^{5}\\\ &+237.904z^{6}+210.612z^{7}\\\
&+75.058z^{8}.\end{aligned}$ (195)

###  F.3 Channel-adaptive codes

####  F.3.1 Nearest-neighbor collective amplitude damping

This part lists the quantum weight enumerators of the channel-adaptive codes
for the one-dimensional nearest-neighbor collective amplitude damping errors
discussed in Sec. 5.3.1.

$\displaystyle\begin{aligned}
A^{\\{4,3\\}}(z)=&1+0.111z+1.222z^{2}+0.778z^{3}\\\
&+2.222z^{4},\end{aligned}$ (196) $\displaystyle\begin{aligned}
B^{\\{4,3\\}}(z)=&1+1.667z+11.667z^{2}+17z^{3}\\\ &+16.667z^{4};\end{aligned}$
(197)

$\displaystyle\begin{aligned}
A^{\\{5,2\\}}(z)=1+z+2z^{2}+2z^{3}+5z^{4}+5z^{5},\end{aligned}$ (198)
$\displaystyle\begin{aligned}
B^{\\{5,2\\}}(z)=1+z+10z^{2}+18z^{3}+21z^{4}+13z^{5};\end{aligned}$ (199)

$\displaystyle\begin{aligned}
A^{\\{6,5\\}}(z)=&1+1.307z^{2}+0.032z^{3}+3.890z^{4}\\\
&+0.097z^{5}+6.474z^{6},\end{aligned}$ (200) $\displaystyle\begin{aligned}
B^{\\{6,5\\}}(z)=&1+18.533z^{2}+32.162z^{3}+103.449z^{4}\\\
&+96.485z^{5}+68.371z^{6};\end{aligned}$ (201)

$\displaystyle\begin{aligned}
A^{\\{7,8\\}}(z)=1+z^{4}+6z^{5}+6z^{6}+2z^{7},\end{aligned}$ (202)
$\displaystyle\begin{aligned}
B^{\\{7,8\\}}(z)=&1+9z^{2}+64z^{3}+179z^{4}+312z^{5}\\\
&+323z^{6}+136z^{7};\end{aligned}$ (203)

$\displaystyle\begin{aligned}
A^{\\{8,9\\}}(z)=&1+0.038z^{2}+0.124z^{3}+0.827z^{4}\\\
&+5.282z^{5}+10.035z^{6}+8.816z^{7}\\\ &+2.322z^{8},\end{aligned}$ (204)
$\displaystyle\begin{aligned}
B^{\\{8,9\\}}(z)=&1+6.119z^{2}+57.119z^{3}+200.778z^{4}\\\
&+475.985z^{5}+713.867z^{6}+618.896z^{7}\\\ &+230.237z^{8};\end{aligned}$
(205)

$\displaystyle\begin{aligned}
A^{\\{9,16\\}}(z)=&1+z^{5}+8z^{6}+14z^{7}+7z^{8}+z^{9},\end{aligned}$ (206)
$\displaystyle\begin{aligned}
B^{\\{9,16\\}}(z)=&1+4z^{2}+80z^{3}+326z^{4}+936z^{5}\\\
&+1924z^{6}+2464z^{7}+1841z^{8}\\\ &+616z^{9}.\end{aligned}$ (207)

####  F.3.2 Nearest-neighbor collective phase-flips

This part lists the quantum weight enumerators of the channel-adaptive codes
for the combined noise channel $\mathcal{N}$ (Eq. (79)) with hardware
connectivity graphs shown in Fig. 7.

$\displaystyle A^{\\{a\\}}(z)=1+z^{2}+11z^{4}+16z^{5}+3z^{6},$ (208)

$\displaystyle B^{\\{a\\}}(z)=1+z^{2}+24z^{3}+35z^{4}+40z^{5}+27z^{6};$ (209)
$\displaystyle A^{\\{b\\}}(z)=1+z^{2}+11z^{4}+16z^{5}+3z^{6},$ (210)
$\displaystyle B^{\\{b\\}}(z)=1+z^{2}+24z^{3}+35z^{4}+40z^{5}+27z^{6};$ (211)
$\displaystyle A^{\\{c\\}}(z)=1+2z^{3}+9z^{4}+24z^{5}+22z^{6}+6z^{7},$ (212)
$\displaystyle B^{\\{c\\}}(z)=1+17z^{3}+45z^{4}+78z^{5}+82z^{6}+33z^{7};$
(213) $\displaystyle A^{\\{d\\}}(z)=1+2z^{3}+9z^{4}+24z^{5}+22z^{6}+6z^{7},$
(214) $\displaystyle
B^{\\{d\\}}(z)=1+17z^{3}+45z^{4}+78z^{5}+82z^{6}+33z^{7};$ (215)
$\displaystyle\begin{aligned}
A^{\\{e\\}}(z)=&1+3.944z^{4}+12.112z^{5}+24z^{6}\\\
&+19.888z^{7}+3.056z^{8},\end{aligned}$ (216) $\displaystyle\begin{aligned}
B^{\\{e\\}}(z)=&1+27.888z^{3}+86.336z^{4}+215.776z^{5}\\\
&+319.776z^{6}+268.336z^{7}+104.888z^{8};\end{aligned}$ (217)

$\displaystyle\begin{aligned}
A^{\\{f\\}}(z)=&1+3.983z^{4}+12.033z^{5}+24z^{6}\\\
&+19.967z^{7}+3.017z^{8},\end{aligned}$ (218)

$\displaystyle\begin{aligned}
B^{\\{f\\}}(z)=&1+27.967z^{3}+86.100z^{4}+215.933z^{5}\\\
&+319.933z^{6}+268.100z^{7}+104.967z^{8};\end{aligned}$ (219)

$\displaystyle\begin{aligned}
A^{\\{g\\}}(z)=&1+0.886z^{3}+3.282z^{4}+11.604z^{5}\\\
&+30.352z^{6}+45.220z^{7}+29.366z^{8}\\\ &+6.290z^{9},\end{aligned}$ (220)
$\displaystyle\begin{aligned}
B^{\\{g\\}}(z)=&1+21.648z^{3}+78.704z^{4}+232.520z^{5}\\\
&+482.255z^{6}+618.352z^{7}+462.041z^{8}\\\ &+151.480z^{9};\end{aligned}$
(221) $\displaystyle\begin{aligned}
A^{\\{h\\}}(z)=&1+0.004z^{3}+3.996z^{4}+10.009z^{5}\\\
&+35.974z^{6}+44.004z^{7}+23.030z^{8}\\\ &+9.983z^{9},\end{aligned}$ (222)
$\displaystyle\begin{aligned}
B^{\\{h\\}}(z)=&1+16.026z^{3}+89.948z^{4}+224.044z^{5}\\\
&+487.965z^{6}+623.974z^{7}+445.087z^{8}\\\ &+159.956z^{9}.\end{aligned}$
(223)