# Quasiparticle transport in a two-dimensional boundary superfluid

S. Autti1<EMAIL_ADDRESS>R. P. Haley1 A. Jennings1 Current address:
RIKEN Center for Quantum Computing, RIKEN, Wako, 351-0198, Japan G. R.
Pickett1 M. Poole1 R. Schanen1 A. A. Soldatov2 V. Tsepelin1 J. Vonka1 Current
address: Paul Scherrer Institute, Forschungsstrasse 111, 5232 Villigen PSI,
Switzerland V. V. Zavjalov1 D. E. Zmeev1 1Department of Physics, Lancaster
University, Lancaster LA1 4YB, UK.  2P.L. Kapitza Institute for Physical
Problems of RAS, 119334 Moscow, Russia

The B phase of superfluid 3He can be cooled into the “pure” superfluid regime,
characterised by negligible thermal quasiparticle density. Here, the bulk
superfluid is bounded by a two-dimensional quantum well at the boundaries of
the container, where creating quasiparticles requires much less
energy.autti2020fundamental ; Nagato19981135 ; Nagai2008 ; Murakawa2011 ;
Okuda2012 ; Murakawa2011 ; Aoki2005 ; Zheng2016  In this Article, we carry out
experiments where we create a non-equilibrium state within the quantum well
and show that the induced quasiparticle currents flow diffusively in the two-
dimensional system. We conclude that the bulk of superfluid 3He is wrapped by
an independent two-dimensional superfluid that interacts with mechanical
probes instead of the bulk superfluid, only providing access to the bulk
superfluid if given a sudden burst of energy. That is, superfluid 3He at the
lowest temperatures and applied energies is thermo-mechanically two
dimensional. Our work opens this two-dimensional quantum condensate and the
interface it forms between the observer and the bulk superfluid for
exploration, and provides the possibility of creating two-dimensional
condensates of arbitrary topology.

Figure 1: The two-dimensional quasiparticle quantum well: (a) The superfluid
gap is suppressed as the container boundary is approached. This yields a
potential well in which bound quasiparticles (red halo) can exist to arbitrary
low energies. At low temperatures the density of quasiparticles in the bulk
superfluid is vanishingly low. (b) The probe wire (cross-section shown by the
grey disk) is surrounded by the bound quasiparticle potential well (red halo).
The thickness of the potential well $a\xi$ ($a\sim 1$, see Methods) determines
the bound quasiparticle mean free path as indicated by the double arrow. Here
$R$ is the radius of the wire cross bar. When the cross bar is moving, the
local superflow velocity around it depends on polar angle $\theta$. Green
notches indicate the span of $\theta=\pm 80^{\circ}$ where the quasiparticle
escape condition $v_{\rm fl}>v_{\rm c}$ is satisfied locally for the wire
velocity of $v$=$45\text{\,}\mathrm{mm}\text{\,}\mathrm{s}$-1.

At the lowest temperatures (and here in zero field and pressure) bulk
superfluid 3He exists in the B phase. Here, the minimum energy required to
create a quasiparticle, or one half of a broken Cooper pair, is the energy gap
$\varDelta\approx 1.6\,$mK. These quasiparticles are responsible for the
macroscopic transfer of momentum and energy in the superfluid, and the
quasiparticle density decreases exponentially with decreasing temperature.
Therefore, at temperatures below a quarter of the superfluid transition
temperature $\approx 1$ mK, the bulk superfluid only conducts heat efficiently
from sources hot enough to create new quasiparticles. However, within roughly
a coherence length $\xi\approx 80$ nm of the sample container walls, the
energy gap is suppressed, as shown in Fig. 1a. The coherence length is the
smallest length scale across which the superfluid wave function can change,
and therefore the gap suppression region is effectively two-dimensional. This
suppression gives rise to a quantum well allowing quasiparticles to exist at
arbitrarily small energies. We know that these bound quasiparticles can be
pushed to the bulk superfluid where they become regular thermal
quasiparticlesautti2020fundamental , but basic physics in the absence of bulk
emission such as transport within the quantum well remain unexplored. Thus, it
has not been clear whether the two-dimensional system is an independent
superfluid condensate. Here we argue that the bound quasiparticles are
decoupled both from phonons in the container wall and from thermal
quasiparticles in the bulk superfluid. They thus have their own mode of
transport by diffusion within the two-dimensional superfluid, forming the
primary system at low energies that a probe rod inserted in the superlfuid
interacts with.

In the simplest description, the bound quasiparticles have a linear dispersion
as a function of their in-plane momentum $p_{||}$,
$E=v_{\mathrm{L}}p_{||}$.silaev2014andreev ; Nagato19981135 ; Nagai2008 ;
Murakawa2011 ; Okuda2012 ; Aoki2005 ; Zheng2016  Here $v_{\mathrm{L}}=27$ mm/s
is the Landau critical velocity, which means that such bound quasiparticles
move at a uniform group velocity $v_{\mathrm{qp}}=v_{\mathrm{L}}$ in the plane
of the surfacevolovik2009fermion ; Murakawa2011 . Since this dispersion is
best approximated when the scattering from the containing wall is (partially)
specular Murakawa2011 , the primary experiments in this article were carried
out with approximately two monolayers of solid 4He coating all
surfacesautti2020fundamental , yielding a specularity in the range between 0.2
and 0.8, where one is full specularity and zero means diffuse surface
scattering. We can also measure the effect of removing the 4He
coatingdmitriev2005separation  so that specularity is approximately zero. In
this case the bound quasiparticles have a more complicated dispersion, but the
order of magnitude of $v_{\mathrm{qp}}$ remains the same Murakawa2011 ;
volovik2009fermion .

We are able to induce interaction between the two- and three-dimensional
superfluids when a probe inserted in the bulk superfluid is accelerated to a
velocity exceeding the critical velocity $v_{\mathrm{c}}=v_{\mathrm{L}}/3$ (in
this case, for a cylindrical rod), promoting the most energetic quasiparticles
bound on the probe surface to escape into the bulk. The probe we use is the
crossbar of a cylindrical goalpost-shaped wireBradley2016  (radius
$R=$63.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, crossbar 8 mm, legs 25 mm),
as shown in Fig. 6 in Methods. When the wire is moved, driven by the Laplace
force in a magnetic field, the superfluid flow velocity with respect to the
crossbar surface follows $v_{\rm fl}=2v\cos(\theta)$, where $v$ is the
velocity of the crossbar in the laboratory frame, and the polar angle $\theta$
is defined in Fig. 1b. The bound-state dispersion is Doppler shifted in energy
by $\pm v_{\rm fl}p_{\mathrm{F}}$ ($p_{\mathrm{F}}$ being the Fermi momentum).
By changing the direction and amplitude of the superflow along the surface we
can manipulate the bound quasiparticle dispersion curves, allowing us to
create a non-equilibrium state within the two-dimensional superfluid.

Figure 2: Schematic illustration of the cross bar motion: The wire can be
moved across 0.5 mm with a steady velocity, here
$45\text{\,}\mathrm{m}\mathrm{m}\,\mathrm{s}^{-1}$, and then paused (green
line). After a wait of $\Delta t$, the process can be repeated with a further
up movement (“up cycle” blue line), or reversed back to the starting point
(“down cycle”, red line). The combination of the green and blue lines is
denoted “up-up cycle” and that of the green and red lines “up-down cycle”. The
vertical bands of colour indicate where surface quasiparticles are emitted
from the wire during acceleration and deceleration. The emitted quasiparticles
increase the temperature of the bulk superfluid, which is detected using a
separate thermometer.

As the wire velocity $v$ is increased from zero beyond $v_{\mathrm{c}}$, the
most energetic bound quasiparticles escape to the bulk superfluid as
schematically shown in Fig. 2 and detailed in Fig. 8 in Methods. While the
wire is moving at a uniform velocityZmeev2014 , a new equilibrium of bound
quasiparticle distribution is established and no further quasiparticles are
released into the bulk. A similar process takes place during the deceleration
phase at the end of the motion, where a further pulse of bound quasiparticles
can escape.

Figure 3: A cartoon of the quasiparticle emission in an up-up cycle: (a) As
the superflow accelerates, the bound quasiparticle bands are Doppler-shifted
up for quasiparticles with momenta against the superfluid and down for momenta
along the flow direction (red pillars). This raises the most energetic bound
quasiparticles above the minimum energy in the bulk superfluid, allowing the
quasiparticles to escape into the bulk as a sudden burst (yellow star and
arrow). During the motion the quasiparticle deficit created on the right-hand
side band can only be filled slowly from transport of quasiparticles along the
potential well. When the motion ceases the bands return to their original
positions allowing another sudden burst of quasiparticles to escape. At the
end of the whole velocity cycle the left-hand side band is left with a
deficit. (b) If the initial up cycle is followed with no delay by another up
cycle, the second cycle will start with a quasiparticle deficit allowing no
quasiparticle emission during the acceleration. Therefore, only one pulse of
quasiparticles is emitted, that from the deceleration. Allowing for (partial)
recovery and measuring the amount of emitted quasiparticles thus allows us to
take snapshots of the bound quasiparticle diffusion.

The pulses of quasiparticles released into the bulk increase the temperature
of the bulk liquid. We can infer the dependence between the temperature
increase and the heat released by the motion of the wire. Moreover, because
each quasiparticle in the bulk liquid carries an energy very close to
$\varDelta$, this measured temperature rise yields the number of
quasiparticles that have escaped during the motion of the wire. We label the
direction of the initial acceleration-deceleration cycle “up”. The
quasiparticle release process in an up cycle is schematically illustrated in
Fig. 3a.

We can measure the heat released by such cycle of the cross bar by varying the
bulk quasiparticle density. That is, the crossbar also scatters bulk
quasiparticles while it moves, with the resulting drag force $F$ yielding the
heating $Q=Fd$ ($d$ is the distance travelled). This drag force can be varied
by changing the temperature of the bulk superfluid and measured independently
using a separate thermometer wire whose resonance width $\Delta f\propto F$.
The proportionality constant is calculated in Methods, yielding the black line
in Fig. 4a, in good agreement with the measured $Q$. The effect of $Q$ can
thus be removed by extrapolating the measured heat release linearly to $\Delta
f=0$. The heat released from the bound quasiparticles extracted this way from
a single acceleration is $q=12\pm 3~{}$pJ, which is in good agreement with the
theoretical estimate $\sim 10\,$pJ as detailed in Methods.

Figure 4: Direct measurement of bound quasiparticle heat release: The heat
$Q_{\mathrm{tot}}$ released in an up cycle (green points) depends linearly
(green dashed line) on the thermal bulk drag force, proportional to the
resonance width $\Delta f$ of the vibrating wire thermometer in the same
superfluid volume. The black line shows an estimate of the heating from the
drag force due to thermal bulk quasiparticles (Methods) in good agreement with
the slope of the green data. Extrapolation to zero $\Delta f=0$ yields the
bound state contribution $q=14\pm 3~{}$pJ (intersection with the y-axis is at
$2q$). For an up-up cycle with $\Delta t>25~{}$ms (blue points), the slope is
doubled because the distance travelled is twice longer (blue dashed line), and
the $\Delta f=0$ intersection yields $4q=60\pm 4~{}$pJ. These experiments were
carried out with 4He preplating. Up-up cycles measured without 4He preplating
(magenta points) show a 10% reduction in the slope and the linear fit (dashed
magenta line) extrapolates to $4q\approx 28~{}\mathrm{pJ}$.

Importantly for the current experiment, at the end of the up cycle, the
dispersion curves of the quasiparticles have returned to their zero-velocity
profile, but leaving on one branch a deficit where the highest energy
quasiparticles have energies well below zero (see Fig. 3a). Since at low
temperatures the equilibrium density of quasiparticles in the bulk liquid is
very low, the mechanism for the deficit of quasiparticles to be replaced by
quasiparticles coming from the bulk is too slow to be directly
observedautti2020fundamental . Therefore, equilibrium for these particles can
be reestablished only by the flow of bound quasiparticles along the potential
well around the cross bar.

Now we can progress to measuring bound quasiparticle transport in the surface
system. Referring back to the cycle of cross bar motion shown in Figure 3a,
the end state is not the same as the initial state as we are left with a
quasiparticle deficit in the left-hand branch. The key concept of the
experiment is to repeat the acceleration-deceleration cycle once more after a
wait of $\Delta t$ (up-up cycle). This can be done for two extrema. If we
repeat the cycle immediately after the first has ended, that is, setting
$\Delta t$ to zero then the second cycle follows that shown in Fig. 3b where
we start already with the full deficit in the left-hand branch. However, this
second process only yields one burst of quasiparticles, that on the
deceleration. Thus, the combined series of two cycles yields three bursts of
quasiparticles into the bulk. Alternatively, after the first cycle we can wait
for an “infinite” period, i.e. longer than the time taken for the deficit to
fill by flow of localised quasiparticles round the wire periphery, in which
case we have the same starting conditions as shown in Fig. 3a. Thus, a series
of two cycles with a long wait between will emit four equal bursts of bound
quasiparticles. Between these two extremes we can explore the situation with
intermediate values of the delay $\Delta t$ and can map out how rapidly the
deficit fills. That is the basis of the measurement which provides directly a
measure of the diffusion rate of the localised quasiparticles through the
potential well around the wire.

To interpret the data we now need to devise a model for diffusive
quasiparticle transport in the two-dimensional surface quantum well, which we
do following the theoretical lead of Refs. 13; 14; 15. This model is based on
three assumptions (all well justified). First, there is no interaction between
surface-bound and bulk quasiparticles, as has been observed across two orders
of magnitude in bulk quasiparticle densityautti2020fundamental . Secondly, the
thermal Kapitza resistance between phonons in the solid boundary material and
the superfluid quasiparticles is very large at the lowest temperatures. A
simple estimate for the exponential decay of the energy of the bound
quasiparticles into the material of the cross bar yields a time constant
$\tau_{\mathrm{R_{\mathrm{K}}}}>10$ s (Methods). These two assumptions imply
that energy in the bound quasiparticle system is conserved. Third, the
distribution of the bound quasiparticle system reflects the history of cross
bar motion as detailed below and in Ref. 1, meaning that also the bound
quasiparticle momentum distribution equilibrates with the time constant
$\tau$. Thus, quasiparticle transport acts as the equilibration mechanism.

The diffusion coefficient for quasiparticle transport can be estimated as
$D\sim v_{\mathrm{qp}}l_{||}$. Here $l_{||}$ is the mean free path which, in
the absence of quasiparticle-quasiparticle collisions (this assumption is
confirmed below), is determined by the thickness of the quantum well, that is,
the distance between the wire surface and the edge of the surface layer,
$\l_{||}\sim$$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (Fig. 1b and Methods).
When the cross bar is stopped, the density of bound quasiparticles carrying
the momentum imbalance, $n$, reflects the flow velocity along the cross bar
surface immediately before the wire was stopped. We estimate this distribution
as $n\propto v_{\mathrm{fl}}\propto\cos\theta$. Solving the diffusion equation
for this initial state yields exponential recovery of equilibrium $n$ with the
time constant $\tau=R^{2}/D\sim 10~{}$ms (see Methods). We emphasise that the
time constant is tied to the population gradient arising from the flow
velocity profile and, hence, to non-local diffusion.

We can probe the diffusive bound-state redistribution by varying the delay
$\Delta t$ between the two cycles of motion. Fewer quasiparticles are released
in the second cycle if the susceptible population has been depleted during the
first cycle. The replenishment of the deficit begins already during the
deceleration and continues until the velocity reaches zero velocity again.
According to the diffusion picture, the energy released in the second
acceleration is $q\left[1-\exp(-(\Delta t+t_{0})/\tau)\right]$, where
$t_{0}=6$ ms is the combined acceleration and deceleration time. The total
energy release from the up-up cycle is detailed in Methods. The measured
$\Delta t$ dependence, shown in Fig. 5, confirms the exponential equilibration
with the fitted value $\tau=6\pm 3$ msautti2020fundamental  in good agreement
with the theoretically estimated time constant.

Figure 5: Diffusion in the two-dimensional superfluid: (a) We record the total
energy released to the bulk superfluid as a function of the recovery time
$\Delta t$. Blue circles show the result for an up-up cycle at two different
bulk temperatures and red circles for corresponding up-down cycle. The fitted
exponential time dependencies describe the diffusion process that
redistributes quasiparticles until equilibrium is recovered. This allows
inferring the magnitude of the bound quasiparticle depletion (or excess for
the up-down cycles) that results from the first up cycle. Fitted parameter
values $q$ and $q_{\mathrm{down}}$ are indicated in the figure with errors
corresponding to 68% confidence intervals. Data in this panel was measured
with applied 4He coating on the cross bar surface. (b) The number of bound
quasiparticles released is approximately halved when the cross bar surface is
not coated with solid 4He. We vary the bulk temperature to show that $-q$
(blue points) and $q_{\mathrm{down}}$ (red points) are independent of the
quasiparticle density in the bulk. The top x-axis shows the thermometer wire
resonance width which is proportional to density of bulk quasiparticles.
Dashed horisontal lines are a guide to the eye corresponding to $-q=-6$ pJ
(blue line) and $q_{\mathrm{down}}=9$ pJ (red line). At temperatures higher
than 0.19 mK, $Q$ becomes larger than $q$ or $q_{\mathrm{down}}$, thus causing
difficulties in extracting $q$ and $q_{\mathrm{down}}$ reliably. Error bars
show the 68% confidence interval of the exponential fits. Additionally, the
crossing of the exponential tails seen for the 0.22 mK data in panel a is due
to a hysteresis effectautti2020fundamental , proportional to $\Delta f$. This
acts to increase the apparent $q_{\mathrm{down}}$ . The magnetic field was 136
mT and $v=45$ mm/s in both panels.

When the first motion cycle depletes the bound quasiparticles available for
bulk escape, none will be released if the cycle follows with no intermediate
recovery time. That is, the fitted $q$ should be consistent with that
extracted by temperature extrapolation above. The blue lines in Fig. 5 are
fits with $\tau=6~{}$ms at two different temperatures, with the average value
${q}=11\pm 5\,$pJ. This is in good agreement with $q=12\pm 3\,$pJ obtained
above with the temperature extrapolation. Combining this with the observation
detailed below that $q$ does not depend on bulk temperature yields two
immediate conclusions. First, starting the second cycle before the bound state
recovery is complete provides a quantitative snapshot of bound quasiparticles’
dynamics. Second, the bound state population available for bulk escape at this
velocity is fully depleted by the first cycle of motion.

We also note that none of the measured dissipation originates from direct
creation of bulk quasiparticles. That is, the temperature extrapolation tells
us the total magnitude of dissipation at zero bulk temperature, which matches
with the magnitude of the diffusion process. This is despite the fact that
also a large bulk superfluid volume flows at speeds exceeding the Landau
critical velocity when the wire is moved, and naively we might expect a direct
pair breaking mechanism to arise. This observation provides a new perspective
to experiments carried out in the polar phase of superfluid 3He, Mineev2016 ;
PolarDmitriev ; PhysRevLett.121.025303  where exceeding the bulk Landau
velocity in a large volume potentially causes no observable dissipation
either, contrary to theoretical expectationautti2020exceeding ; HQVs_prl .

Reversing the direction of motion for the second cycle (up-down cycle) results
in a temporary excess of bound quasiparticles available for escape during the
acceleration of the down cycle. This scenario is illustrated schematically in
Figure S1. The measured data shown in Figure 5 confirms this excess. Applying
the diffusion picture, we expect the excess bound quasiparticle emission to be
removed by diffusion as ${q}_{\mathrm{down}}\exp(-(\Delta t+t_{0})/\tau)$, in
good agreement with the data shown in Figure 5. The low-temperature fitted
$q_{\mathrm{down}}\approx 30$ pJ is consistent with a temporary population
excess that is removed by diffusion, as detailed in Methods. At higher
temperatures the hysteretic effect explained in Ref. 1 distorts the
measurement.

We can decrease the number of bound quasiparticles susceptible to the bulk
flow by removing the 4He coating of the surface and thus taking the
specularity close to zero Murakawa2011 . Reducing the surface specularity
moves the density of states towards states with lower momenta in the plane of
the wire surface. States with zero in-plane momentum gain no energy from
$v_{\mathrm{fl}}$ and therefore cannot escape to the bulk. Repeating the
extrapolation to $\Delta f=0$ as before yields $q\approx 7~{}$pJ (Fig. 4).
Assuming the escape process is equally effective with and without 4He, this
implies that the susceptible bound state density is roughly twice higher with
4He coating than without it. This is qualitatively in line with the
theoretical prediction in Ref. 4. The slope of $Q(T)$ is 10% larger with 4He
coating than without it. This is because surface specularity does not change
the bulk quasiparticle density at a given temperature, and the drag force $F$
at large velocities $v$ increases slowly with growing specularity
fisher1991microscopic ; enrico1996specular ; enrico_thesis . Change in the
surface specularity has no effect on the diffusion time constant process
within experimental uncertaintiesautti2020fundamental . That is,
quasiparticle-quasiparticle collisions remain negligible regardless of changes
in the density of states, substantiating the assumption that the mean free
path is determined geometrically by the thickness of the surface layer.

To add further evidence for the independence of the surface dynamics, we can
extract $q$ and $q_{\mathrm{down}}$ by varying $\Delta t$ at different bulk
temperatures (no 4He, Fig. 5b). The bulk quasiparticle density changes by two
orders of magnitude over the temperature range studied but, remarkably, the
bound state process remains undisturbed, indicated by the constant energy
release. This shows that the snapshot technique reliably probes the bound
quasiparticles’ dynamics, and that there is no coupling between the bound
quasiparticles and thermal bulk quasiparticles.

We conclude that the two-dimensional superfluid manifests independent
quasiparticle transport dynamics, carrying energy and momentum. At energies
below the superfluid gap, the surface provides a preferential path for heat
flow between hot and cold objects immersed in the superfluid and a direct
interaction channel between immersed mechanical probes. This means that zero-
temperature superfluid 3He is thermo-mechanically two dimensional.

Confining a fermion gas at a low temperature to a high-purity two-dimensional
solid-state system has led to the discovery of a variety of quantum Hall
phases and topological quantum states. Similarly, spontaneous formation of
ultra-cold superfluid 3He, an extremely pure fermionic system, into a two-
dimensional surface system is likely to yield a diversity of new physics. For
example, the bound qausiparticles’ possible interactions with the sub-gap
bosonic excitations or bulk topological defects span at least 7 orders of
magnitude in energy below the superfluid gap and some 18 degrees of
freedomVolovikBook ; vollhardt2013superfluid . Our results imply that devising
suitable nanoprobes that fit within the two-dimensional superfluid should
reveal long-range quasiparticle transport that can be studied in varying
topological configurations, such as across different bulk superfluid phases
and interfaces, via controlled confinement provided by engineered
nanostructures lotnyk2020thermal ; heikkinen2021fragility ;
todoshchenko2020mobile ; Levitin841 ; shook2020stabilized ;
PhysRevLett.122.085301 ; levitin2014study ; zhelev2017ab , or across the free
surfaceautti2021ac ; autti2021nonlinear . It may also be possible to access
these phenomena by engineering the topology of the mechanical
probesarrayas2021design ; arrayas2022progress . Finally, the surface layer is
also expected to host Majorana zero modes Murakawa2011 ; Shapiro2013  that
detailed transport measurements may reveal. These research avenues has the
potential to transform our understanding of this versatile macroscopic quantum
system.

## Author contributions

The measurements and data analysis were carried out by S.A., A.J., M.P., R.S.,
J.V., A.A.S., and D.E.Z. The experimental protocols were developed by R.S.,
J.V. and D.E.Z. Theoretical work, interpretation of the results, and
background investigations were done by S.A. with contributions from R.P.H.,
G.R.P., V.T., V.V.Z. and D.E.Z. The manuscript was prepared by S.A. with help
from G.R.P., while all authors provided comments and clarifications. D.E.Z.
supervised the experiments.

## Acknowledgements

We thank Grigory Volovik, Jaakko Nissinen and Petri Heikkinen for stimulating
discussions. This work was funded by UKRI EPSRC (EP/P024203/1, EP/X004597/1,
and EP/W015730/1) and STFC (ST/T006773/1), as well as EU H2020 European
Microkelvin Platform (Grant Agreement 824109). We acknowledge M.G. Ward and A.
Stokes for their excellent technical support. S.A. acknowledges financial
support from the Jenny and Antti Wihuri Foundation.

## Competing interests

The authors declare no competing interests

## Data availability

The data that supports the findings of this study are available in Lancaster
University data repository at [link/reference to be added]

## Code availability

No specialised software was used in analysing the data.

## Methods

All experimental data and parameter values in this Article are for the
saturated vapour pressure, which is vanishingly small at ultra-low
temperatures. The superfluid transition temperature at this pressure is
$T_{\mathrm{c}}\approx\,$$920\text{\,}\mathrm{\SIUnitSymbolMicro K}$. The
superfluid sample is contained in a box made from Stycast paper composite
containing sintered silver heat exchangers, surrounded by a guard cell also
filled with cold 3He.Sinters2020  The motion of the goal-post
wireBradley2011114  in the cell is illustrated schematically in Fig. 6.

Figure 6: The goal-post shaped wire: We can move the cross bar of the goal-
post wire at constant speed over a distance of several mm, enabling the
current experimentsZmeev2014 . The wire is surrounded by a volume of
superfluid used as a bolometer for detecting the heat released from the motion
of the wireautti2020fundamental . The detection coils are used for calibrating
the velocity of the wire cross bar.

### Vibrating wire thermometry and bolometry

The changes in the bulk thermal quasiparticle density can be measured using a
superconducting $4.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$-thick vibrating
wire, whose resonance width $\Delta f\propto\exp(-\varDelta/k_{B}T)\propto
F$.fisher1991microscopic ; PhysRevB.57.14381 ; Guenault1986  Here $\varDelta$
is the superfluid gap and $k_{B}$ the Boltzmann constant,
$F_{$4\text{\,}\mathrm{\SIUnitSymbolMicro m}$}$ is the drag force acting on
the vibrating wire resonator, $F$ is the drag force acting on the cross bar of
the goalpost-shaped wire, and $T$ is the superfluid temperature. This implies
that $Q(T)\propto\Delta f$ with the proportionality constant determined by the
ratio of the probe diameters and the thermal quasiparticle scattering
efficiency, which depends on probe velocity. For further confirmation we scale
the measured $F_{$4\text{\,}\mathrm{\SIUnitSymbolMicro m}$}$ by the ratio of
cross-sectional areas and according to Equation 17 in Ref. 21 (see also Ref.
23). Estimating $Q(T)$ this way yields the black line in Fig. 4, in good
agreement with measured $Q(T)+2q$ (green points and line) considering that no
fitting parameters were used and only the drag force experienced by the cross
bar is included in the estimate (wire legs are excluded).

The heat release due to ejected bound quasiparticles is monitored by using the
surrounding superfluid volume as a bolometer. Changes in temperature are
measured using the thermometer wire resonator. Typical bolometer data curves
are shown in Figure 7. The temperature is stable before the goalpost wire is
moved at $t=0$, and the peak of the bulk temperature increase occurs well
after the wire motion ends. This is because the bolometer readout time,
determined by the thermometer wire resonance width $\Delta f$ and the
thermalisation time of the bolometer,bib:Winkelmann:2007  is of the order of a
second. Comparing two data curves with the delay between the two cycles of
crossbar motion 0 and 30 ms. The temperature peak shifts according to delay
$\Delta t$ added between the two cycles.

The bolometer is calibrated using resonant AC measurements of the goalpost-
shaped wireMSkyba_phd , where the energy output can be directly recorded with
a four-point measurement. The data obtained is fitted with the BCS heat
capacityvollhardt2013superfluid  using the effective volume of the sample as a
fitting parameter. The fitted volume is $16~{}\mathrm{cm}^{3}$, which falls
between the free volume of the sample container, $15~{}\mathrm{cm}^{3}$, and
the total volume of the sample container including the volume within the heat
exchangers, $32~{}\mathrm{cm}^{3}$.

Figure 7: Temperature evolution of the bulk liquid after a double cycle: The
superfluid temperature is stable before the cross bar motion starts at $t=0$.
The cross bar motion takes less than $30$ ms for $\Delta t=0$, but the
superfluid bolometer reacts slowly to this sudden quasiparticle release. If we
look first at the $\Delta t=0$ ms up-up data, only three pulses of bound
quasiparticle are emitted, producing the smallest rise in temperature. For the
$\Delta t=30$ ms data, four pulses of quasiparticles are emitted giving a
larger temperature swing of the bulk liquid. For comparison, reversal of the
direction results in increased (as opposed to depleted) bound quasiparticle
release, as shown in the $\Delta t=0$ ms up-down data. We can use these curves
as transfer functions to infer the level of filling of the left-hand bound
state band after the first cycle as a function of the delay time. The right
y-axis is drawn so that the peak value of each curve corresponds to the total
energy release to the bulk liquid. The temperature of the bulk superfluid was
$0.22$ mK in this measurement.

### Escape condition

All fermionic quasiparticles in superfluid 3He move at the Fermi velocity
($\approx 50$ m/s). However, the bound quasiparticles are nearly perfectly
retroreflected from the edge of the surface quantum well owing to Andreev
reflection. If reflection from the wire surface is specular, the resulting
bound quasiparticle dispersion is the Dirac dispersion
$E=v_{\mathrm{L}}p_{||}$. The corresponding group velocity $v_{\mathrm{L}}$ is
the drift speed arising from the minute misalignment of the inbound and
outbound trajectories in the Andreev reflection processmizushima2015symmetry .

Let us assume that the bound quasiparticle system is in equilibrium at zero
temperature. This means that the quasiparticle dispersion bands are filled up
to the Fermi energy, selected to be equal to zero for simplicity in the
schematic Figures 3, 8 and S1. If the wire is accelerated instantaneously to
velocity $v$, the energetic escape condition for the highest-energy bound
quasiparticles in vector form is
$\mathbf{v}_{\mathrm{fl}}\cdot\mathbf{\hat{p}}_{\mathrm{in}}\geq\varDelta/p_{\mathrm{F}}+\mathbf{v}\cdot\mathbf{\hat{p}}_{\mathrm{out}}$
where $\mathbf{\hat{p}}_{\mathrm{in}}$ is a unit vector that corresponds to
the bound quasiparticle’s momentum during the increase in
$\mathbf{v}_{\mathrm{fl}}$ and $\mathbf{\hat{p}}_{\mathrm{out}}$ is the
direction of escape to bulk. The local flow velocity near the wire surface
follows $\mathbf{v}_{\mathrm{fl}}=2\cos(\theta)v\hat{\mathbf{\theta}}$, where
$\hat{\mathbf{\theta}}$ is the azimuthal unit vector perpendicular to the
cylinder radius.

For increasing $v$, the escape condition is first satisfied for
$\mathbf{\hat{p}}_{\mathrm{in}}$ and $\mathbf{v}_{\mathrm{fl}}$ in the same
direction while $\mathbf{\hat{p}}_{\mathrm{in}}$ is antiparallel with
$\mathbf{\hat{p}}_{\mathrm{out}}$ for quasiparticles at the wire vertices
($\theta=0$) where $v_{\mathrm{fl}}$ reaches its maximum. In this case we get
the well-known critical cross-bar velocity
$v_{\mathrm{c}}=\varDelta/(3p_{\mathrm{F}})=v_{\mathrm{L}}/3$. This process is
schematically illustrated in Fig. 8. We can use similar arguments for
deceleration from a steady state configuration at nonzero $v_{\mathrm{fl}}$,
obtaining the critical velocity
$v_{\mathrm{c}}^{\prime}=\varDelta/(2p_{\mathrm{F}})=v_{\mathrm{L}}/2$.

At velocities significantly higher than $v_{\mathrm{c}}$, the vector picture
is more complicated. For simplicity, the main text and Figs. 8 and S1 only
discuss scalar quantities along the direction of the external flow. At
$v=45~{}$mm/s, the bulk escape process concerns about 90% of the cross bar
surface (see Fig. 1b) , but the largest contribution of the bound state escape
originates from the vicinity of the vertices where $v_{\mathrm{fl}}$ is the
largest. Precise calculation of the distribution requires a three-dimensional
treatment of the system, and such numerical simulations are left for the
future.

We note that oscillatory motion has been speculated to overheat the bound
state system enough to result in observable bound state escape below
$v_{\mathrm{L}}/3$, LAMBERT1992294  but no such “pumping” is observed in our
experiments. The pumping of quasiparticles towards higher energies is unlikely
because the quasiparticle redistribution is governed by diffusion, as
discussed below.

Figure 8: Schematic presentation of the quasiparticle dispersion curves: The
panels represent dispersion curves for the bound quasiparticles and for
quasiparticles in the bulk superfluid during an acceleration, steady velocity,
and deceleration cycle with the indicated times referring to the initial “up”
sequence shown in Fig. 2. In panel (a) the wire is stationary. As we apply an
increasing superflow along the cross bar surface (in the wire frame) the
quasiparticle bands become tipped until as in (b) those bound quasiparticles
with energies above the minimum in the bulk liquid escape into the bulk. When
the acceleration ceases, no more bound quasiparticles can escape, and
equilibrium in the wire frame as shown in panel (c) is re-established by
diffusion. On deceleration, a second burst of bound quasiparticles escapes
into the bulk. Finally, with the superflow velocity again zero as shown in
panel (d), the bands return to their initial state, but leaving the bound
quasiparticles to redistribute via diffusion. A more detailed description of
the same process is shown in Fig. S1. Note that this figure is for
illustration purposes only. The local dispersion curves cannot be really
represented in this way, especially for the depletion situation, but it gives
the gist of the idea.

### Diffusion rate

Assuming there are no quasiparticle-quasiparticle collisions, we can estimate
the mean free path in the surface layer as the longest distance that a
quasiparticle can travel without changing the direction of the group velocity.
This distance is $l_{||}=2\sqrt{2Ra\xi}$, where $R$ is the cross bar radius,
$\xi$ is the coherence length, and $a\xi$ is the effective thickness of the
surface layer (see Fig.1b). If we assume $a=3$ as an estimate of the surface
layer edge where the most energetic quasiparticles (the ones that the
experiment is sensitive to) would be reflectedNagato19981135 , then
$l_{||}\approx$$12\text{\,}\mathrm{\SIUnitSymbolMicro m}$ at zero pressure.
This yields the diffusion constant $D\sim
l_{||}v_{\mathrm{qp}}=l_{||}v_{\mathrm{L}}$. Note that along the length of the
cross bar cylinder the system is homogeneous, and the diffusion experiment is
not sensitive to the mean free path in this direction.

We can now solve the diffusion equation $\partial_{t}n=D\nabla^{2}n$. Here $n$
stands for the quasiparticle population which is out of equilibrium,
$\nabla^{2}$ is the Laplace operator and $\partial_{t}$ stands for time
derivative. The Fermi sea seen in Fig. S1 at $t=0$ at energies below zero is
uniformly filled and there are no quasiparticles above zero energy. Thus, in
that case $n=0$. The population carrying the momentum imbalance when the wire
is stopped depends on the details of the dispersion relation, but we estimate
that it initially follows the local flow velocity so that
$n(\theta)\sim\cos\theta$. The resulting time dependence is
$n(t,\theta)=n(t=0,\theta)\exp(-tD/R^{2})$, yielding the diffusion time
constant $\tau=R^{2}/D$. Inserting
$R=\,$$63.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ gives $\tau\approx 13$ ms.
On the other hand, fitting $a$ to the experimentally observed time constant
gives $a\approx 10$, which exceeds the theoretical expectation by a factor of
$\sim 2$.Nagato19981135  The feasibility of these values of $a$ is discussed
in the next section.

### Heat release from bound quasiparticles

The total energy released during the measurement cycle “up-up” (green+blue
lines in Fig. 2) is $Q_{\mathrm{tot}}^{\mathrm{up-
up}}=[Q+2q]+[Q_{\mathrm{up}}+q_{\mathrm{up}}+q]$, where the brackets separate
the contributions from the first and second phases of cross bar motion. Here
$Q$ and $Q_{\mathrm{up}}$ are due to the drag force from collisions with
thermal bulk quasiparticles. The bulk thermal quasiparticle density increases
by less than 10% in the course of a typical up cycle, thus
$Q_{\mathrm{up}}\approx Q$. Note also that $Q_{\mathrm{up}}$ is independent of
$\Delta t$, which means that the small difference between $Q$ and
$Q_{\mathrm{up}}$ can be ignored in the analysis of the bound state dynamics.

The first acceleration phase and each deceleration phase releases
approximately the same amount of heat originating from the bound quasiparticle
system, denoted $q$. That is, the critical velocity for quasiparticle release
from the acceleration is $v_{\mathrm{L}}/3$ while that from the deceleration
is $v_{\mathrm{c}}^{\prime}=v_{\mathrm{L}}/2$. This means that the
deceleration should release less quasiparticles than the acceleration, but the
difference can be ignored if the wire moves at a speed much higher than
$v_{c}$ (here $5v_{c}$). In the above expression for
$Q_{\mathrm{tot}}^{\mathrm{up-up}}$ we have thus approximated that the heat
released from the deceleration is equal to that from the acceleration.

The heat release from the second acceleration phase is
$q_{\mathrm{up}}=q(1-\exp(-(\Delta t+t_{0})/\tau))$. That is, the bound state
population available for ejection during acceleration is depleted by the first
cycle. Full depletion of the bound quasiparticles that are available for bulk
escape implies $q_{\mathrm{up}}=0$, and after a long enough recovery time
$q_{\mathrm{up}}=q$. For intermediate values of $\Delta t$ the bound state
population available for bulk ejection recovers exponentially owing to the
diffusion with time constant $\tau$. Note that the bound state recovery starts
as soon as the deceleration at the end of the first up cycle begins, and
continues until the same velocity is reached again during the following
acceleration. This process is approximately accounted for by adding $t_{0}=6$
ms (sum of the acceleration and deceleration times) to the recovery time
$\Delta t$ in the exponential decay expression.

For an “up-down” cycle (green+red lines in Fig. 2), the energy release is
expected initially to be larger than for up-up cycles (see Fig. S1),
decreasing as a function of $\Delta t$ and reaching the same level as the “up-
up” cycles at large $\Delta t$. The time dependence is determined by the same
diffusion process as detailed above. The total energy released is therefore
$Q_{\mathrm{tot}}^{\mathrm{up-
down}}=[Q+2q]+[Q_{\mathrm{down}}+\bar{q}_{\mathrm{down}}+2q]$. With similar
arguments as above, $Q_{\mathrm{down}}\approx Q$. Note, however, that
$Q_{\mathrm{up}}>Q_{\mathrm{down}}$ owing to a hysteresis in the distance
covered by the cross barautti2020fundamental  . This effect is proportional to
the density of thermal bulk quasiparticles and therefore vanishes at the
lowest temperatures measured in this Article.

The diffusion picture does not determine the magnitude of the time dependent
excess of bound quasiparticles ejected by the acceleration in the up-down
cycle. Separating the asymptotic bound state contribution from the decaying
excess, the excess heat release from the second phase of motion is denoted
$\bar{q}_{\mathrm{down}}=q_{\mathrm{down}}\exp(-(\Delta t+t_{0})/\tau)$. The
number of bound quasiparticles released is a faster-than-linear function of
velocity $v$ at $v>v_{c}$ ,autti2020fundamental  which hints that
$q_{\mathrm{down}}>q$, as confirmed in the main text.

We can estimate the available quasiparticle energy release from the surface
layer as follows. The gap suppression region around the cross bar has the
volume $V=2\pi Ra\xi L$ ($L$ is the cross bar length). Taking the diffusion
calculation above literally yields the self-consistent thickness of the layer
$a\xi$ with $a\approx 10$. The Doppler shift energy bridges the bulk escape at
cross bar velocity $v=v_{\mathrm{c}}$. If we assume the surface layer is
populated according to the normal state density of states $N(0)$, then the
energy release from the entire cross bar surface is $q\sim
VN(0)\varDelta^{2}(v/v_{\mathrm{c}}-1)^{2}\approx 13~{}$pJ for $v=45~{}$mm/s,
in decent agreement with both the measured value with (12 pJ) and without (7
pJ) 4He preplating of the wire surface. Choosing a more conservative $a=3$
yields $q\approx 4~{}$pJ. We emphasise that this estimate neglects many
important contributions, such as the quasiparticle release from the legs of
the moving superconducting wire (inclusion of which would increase the energy)
and the fact that a significant part of the cross bar surface has a smaller
flow velocity than the maximum at the vertices, which corresponds to
$v_{\mathrm{c}}$ (thus, decreasing the released energy). Furthermore,
according to Ref. 46 page 296, the gap suppression near the wire is
significantly expanded by the increased density of quasiparticles when
$v>2v_{\mathrm{c}}$, which would act to increase the quasiparticle emission.
This effect may explain why the fitted parameter $a$ appears so large.

### Kapitza resistance and bound quasiparticle life time

The experimentally determined thermal Kapitza resistance between phonons in a
metal and superfluid 3He quasiparticles in the bulk superfluid is in the range
$R_{\mathrm{K}}\sim 10^{4}$ to
$10^{5}$$\text{\,}\mathrm{K}\mathrm{m}^{2}\mathrm{W}^{-1}$. Here the upper end
of the range corresponds to a 4He preplated and the lower end to pure 3He
interface between the fluid and the solid. These values are obtained by
extrapolation to $1\text{\,}\mathrm{mK}$ using the data in Ref. 47. The
temperature of the surface-bound quasiparticles is not above
$1\text{\,}\mathrm{mK}$ because otherwise they would escape to the bulk. For
lower temperatures the Kapitza resistance increases as $R_{\mathrm{K}}\sim
1/T$.Sinters2020  For a pessimistic estimate of the decoupling of the cross
bar phonons and the bound quasiparticles, we therefore take the measured pure
3He $R_{\mathrm{K}}$, and assume the bound quasiparticle system is at
$1\text{\,}\mathrm{mK}$ temperature during the decay described by $\tau$ in
the main text.

The decay of energy via the Kapitza resistance is exponential with the time
constant $\tau_{R_{\mathrm{K}}}=R_{\mathrm{K}}C$. Here $C$ is the heat
capacity of the body of heat equilibrating via $R_{\mathrm{K}}$. If we
approximate the bound quasiparticle system to be a layer of normal fluid of
thickness $a\xi$ at $1\text{\,}\mathrm{mK}$ right after the cross bar has
stopped (consistent with the temperature chosen above), the resulting
quasiparticle life time (decay rate of energy) in the bound quasiparticle
system is $\tau_{R_{\mathrm{K}}}\approx 12$ s for $a=3$. Thus, even in
pessimistic estimate the bound quasiparticles exchange a negligible amount of
energy with the wire phonons.

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## Extended data

Figure S1: Bound quasiparticle dynamics: The cross bar up-up cycle shown in
Fig. 2 starts at $t=0$ ms with an equilibrium bound state population (red and
blue disks illustrate quasiparticles). At zero temperature, no quasiparticles
are above the Fermi energy (here at $E=0$). After the acceleration
($t=3~{}$ms), the superflow $v_{\mathrm{fl}}$ along the surface has shifted
the states by $\pm p_{\mathrm{F}}v_{\mathrm{fl}}$. The quasiparticles
energetic enough to escape to bulk scatter away, and diffusion equilibrates
the rest of the spectrum. Here we have assumed this process is finished at
$t=10~{}$ms. The wire is then brought to a halt ($t=13~{}$ms). The diffusive
redistribution of the quasiparticles is determined by the time constant $\tau$
($t=19~{}$ms). Moving the wire soon thereafter in either the same direction or
the opposite direction ($t=23$ ms) and recording the resulting quasiparticle
emission reveals the population dynamics. At $t=35~{}$ms the wire is
stationary again. Note that the spectral evolution is shown here for a Dirac
spectrum at zero temperature, the effect of the bulk escape is neglected for
simplicity, and the direction of momentum is assumed to be uniformly along the
surface. These choices allow for compact visualisation, but they should not be
used for detailed calculations.


CTPU-PTC-22-05

$W$ boson mass and muon $g-2$
in a lepton portal dark matter model

Junichiro<EMAIL_ADDRESS>Shohei<EMAIL_ADDRESS>and
Yuji<EMAIL_ADDRESS>

1 Center for Theoretical Physics of the Universe, Institute for Basic Science
(IBS), Daejeon, 34051, Korea

2 Departament de Física Quàntica i Astrofísica, Institut de Ciències del
Cosmos (ICCUB), Universitat de Barcelona, Martí i Franquès 1, E-08028
Barcelona, Spain

3 Department of Physics, Kindai University, Higashi-Osaka, Osaka 577-8502,
Japan

We study a lepton portal dark matter model, motivated by the deviation of the
$W$ boson mass reported by the CDF collaboration. We introduce vector-like
leptons and a scalar dark matter (DM) which exclusively couples to the extra
leptons and muon. The one-loop corrections induced by the new particles can
shift the $W$ boson mass. Besides, the discrepancy in the muon anomalous
magnetic moment and the DM density can simultaneously be explained by this
setup, if the vector-like lepton is lighter than 200 GeV and nearly degenerate
with the DM particle. We also see that the constraints on such a light extra
lepton from the collider experiments can be evaded due to the existence of the
DM particle.

##  1 Introduction

The precision measurement of the electroweak (EW) interaction plays an
important role in tests of new physics. In the Standard Model (SM), a Higgs
doublet field develops a non-vanishing vacuum expectation value (VEV), and
then the $Z$ and $W$ gauge bosons acquire their masses from the VEV. The
mechanism can be tested by the prediction of the EW precision observables
(EWPOs), such as the $\rho$ parameter, defined as
$\rho:=m_{W}^{2}/(m_{Z}\cos\theta_{W})^{2}$, where $\theta_{W}$ is the weak
mixing angle. It is well known that new fields which contribute to the EW
symmetry breaking can, in general, affect to the $\rho$ parameter at the tree-
level, and hence it can constrain new physics models. Besides, there are loop
corrections to the EWPOs where new particles charged under the EW symmetry run
in the loops.

The CDF collaboration recently announced a new result of the $W$ boson mass
measurement, $m_{W}^{\mathrm{CDF}}=80.4335~{}(94)~{}\mathrm{GeV}$ [1]. This
value disagrees with the combinations of the other previous measurements,
$m_{W}^{\mathrm{PDG}}=80.379~{}(12)~{}\mathrm{GeV}$ [2] and the significance
is $7\sigma$. This new experimental result may suggest that the $W$ boson mass
deviates from the SM prediction due to the existence of new physics beyond the
SM. It has already been pointed out that the deviation can be interpreted by
the corrections to the oblique parameters [3, 4, 5]. In particular, the $T$
parameter is relevant to this deviation. In order to explain this discrepancy
in new physics models, there should be a new particle at the EW scale that is
not neutral under the EW symmetry, and/or mixes with SM particles [4, 6, 7, 8,
9, 3, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 5, 25, 26,
27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. Such a light
particle, however, tends to be excluded by the constraints from the LHC
experiments if it decays to detectable SM particles. A way to avoid the limits
is that an EW particle responsible for the $m_{W}$ anomaly decays to an
undetectable particle so that the signals are effectively invisible.

In this work, we discuss the $W$ boson mass in an extended SM where two
vector-like leptons and a real scalar dark matter (DM) are introduced. In this
model, the scalar DM couples to the SM leptons via Yukawa couplings involving
vector-like leptons. In Ref. [41], the authors show that the relic density of
DM and another recent anomaly in muon anomalous magnetic moment, $\Delta
a_{\mu}=2.51~{}(50)\times 10^{-9}$ [42, 43, 44, 45, 46, 47, 48, 49, 50, 51,
52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63], can be explained
simultaneously, when the DM specifically couples to muon. We shall show that
the discrepancy in the $W$ boson mass can be solved in this model as well.

The rest of this paper is organized as follows. The $W$ boson mass in the
lepton portal DM model is discussed in Sec. 2, and then the DM physics and its
relation to $\Delta a_{\mu}$ are studied in Sec. 3. Section 4 is devoted to
summarize this paper. Loop functions used in our analysis are shown in
Appendix A.

##  2 $W$ boson mass in lepton portal DM model

###  2.1 Model

We briefly review our model in this section. The more details are shown in
Ref. [41]. The terms relevant to new particles in the lagrangian are given by

$\displaystyle-\mathcal{L}_{\mathrm{NP}}=$ $\displaystyle\
\frac{1}{2}m_{X}^{2}X^{2}+m_{L}\overline{L}_{L}L_{R}+m_{E}\overline{E}_{L}E_{R}$
(1) $\displaystyle\
+\lambda_{L}\overline{\ell}_{L}XL_{R}+\lambda_{R}\overline{E}_{L}X\mu_{R}+\kappa\overline{L}_{L}HE_{R}+\overline{\kappa}\overline{E}_{L}i\sigma_{2}H^{\dagger}L_{R}+h.c.,$

where $L_{L,R}$ and $E_{L,R}$ are $SU(2)_{L}$ doublet and singlet vector-like
leptons, respectively. A real scalar DM field is denoted by $X$ which is a
singlet under the SM gauge group. $\ell_{L}=(\nu_{\mu},\mu_{L})$, $\mu_{R}$
and $H$ are the SM leptons in the second generation and the Higgs doublet,
respectively. We neglect couplings of the scalar DM to the SM Higgs boson in
the scalar potential. In our setup, we assume that the vector-like leptons
couple exclusively to the second generation leptons, so that the muon
anomalous magnetic moment is explained without lepton flavor violations. We
note that a $Z_{2}$ symmetry is assigned and only the vector-like leptons and
the DM particle $X$ are odd to ensure the stability of the DM.

After the EW symmetry breaking, the vector-like lepton masses are given by

$\displaystyle\mathcal{M}_{E}=\begin{pmatrix}m_{L}&\overline{\kappa}v_{H}\\\
\kappa v_{H}&m_{E}\end{pmatrix},\quad\mathcal{M}_{N}=m_{L},$ (2)

where $v_{H}=174~{}\mathrm{GeV}$ is the Higgs VEV. Note that there is no
mixing between the vector-like leptons and the SM leptons. We define the mass
eigenstates and diagonalizing matrices as

$\displaystyle\begin{pmatrix}E_{L}^{\prime}\\\
E_{L}\end{pmatrix}=U_{L}\begin{pmatrix}E_{L_{1}}\\\
E_{L_{2}}\end{pmatrix},\quad\begin{pmatrix}E_{R}^{\prime}\\\
E_{R}\end{pmatrix}=U_{R}\begin{pmatrix}E_{R_{1}}\\\
E_{R_{2}}\end{pmatrix},\quad
U_{L}^{\dagger}\mathcal{M}_{E}U_{R}=\mathrm{diag}\left(m_{E_{1}},m_{E_{2}}\right).$
(3)

and we parametrize the diagonalizing matrices as

$\displaystyle U_{L}=\begin{pmatrix}c_{L}&s_{L}\\\
-s_{L}&c_{L}\end{pmatrix},\quad U_{R}=\begin{pmatrix}c_{R}&s_{R}\\\
-s_{R}&c_{R}\end{pmatrix},$ (4)

where $c_{A}^{2}+s_{A}^{2}=1~{}(A=L,R)$. $E_{L,R}^{\prime}$ is a charged
component in the doublet $L_{L,R}$. The neutral component in the gauge basis,
$N_{L,R}$, is already in the mass basis $N_{1}$.

###  2.2 Muon anomalous magnetic moment

In this model, the new physics contribution to the muon anomalous magnetic
moment, $\Delta a_{\mu}$, is originated from the 1-loop effects induced by the
Yukawa couplings of muon with the vector-like leptons and $X$. $\Delta
a_{\mu}$ is evaluated as [64, 65, 66]

$\displaystyle\Delta a_{\mu}=\frac{m_{\mu}}{16\pi^{2}m_{X}^{2}}\Bigl{[}$
$\displaystyle\left(c_{R}^{2}|\lambda_{L}|^{2}+s^{2}_{L}|\lambda_{R}|^{2}\right)m_{\mu}F(x_{1})+c_{R}s_{L}\text{Re}\left(\lambda_{L}\lambda_{R}\right)m_{E_{1}}G(x_{1})$
(5)
$\displaystyle\quad+\left(s_{R}^{2}|\lambda_{L}|^{2}+c^{2}_{L}|\lambda_{R}|^{2}\right)m_{\mu}F(x_{2})-c_{L}s_{R}\text{Re}\left(\lambda_{L}\lambda_{R}\right)m_{E_{2}}G(x_{2})\Bigr{]},$

where $x_{i}=m_{E_{i}}^{2}/m_{X}^{2}~{}(i=1,2)$. The functions $F$ and $G$ are
defined in Appendix A. For
$m_{E_{i}}=\mathcal{O}\left(100\right)~{}\mathrm{GeV}$, $s_{L}\neq 0$ or
$s_{R}\neq 0$ is required to explain the discrepancy due to the chirality
enhanced effect. This means that both singlet and doublet vector-like leptons
are necessary to explain $\Delta a_{\mu}$.

###  2.3 Oblique parameters

New physics effects to the EWPOs are well described by the oblique parameters
[67, 68]. The one-loop contribution of the vector-like leptons, $E_{1}$,
$E_{2}$ and $N_{1}$, to the oblique parameter $T$ is given by [69],

$\displaystyle 16\pi s_{W}^{2}c_{W}^{2}T=$ $\displaystyle\
\left(c_{L}^{2}+c_{R}^{2}\right)\theta_{+}(x,y_{1})+2c_{L}c_{R}\theta_{-}(x,y_{1})$
(6) $\displaystyle\
+\left(s_{L}^{2}+s_{R}^{2}\right)\theta_{+}(x,y_{2})+2s_{L}s_{R}\theta_{-}(x,y_{2})$
$\displaystyle\
-\left(c_{L}^{2}s_{L}^{2}+c_{R}^{2}s_{R}^{2}\right)\theta_{+}(y_{1},y_{2})-2c_{L}s_{L}c_{R}s_{R}\theta_{-}(y_{1},y_{2}),$

where $x=m_{N_{1}}^{2}/m_{Z}^{2}$ and $y_{i}=m_{E_{i}}^{2}/m_{Z}^{2}$. Here,
$c_{W}=\cos\theta_{W}$ and $s_{W}=\sin\theta_{W}$. The formula for $-2\pi U$
is given by replacing the loop functions $\theta_{\pm}\to\chi_{\pm}$. The one-
loop contribution to the $S$ parameter is given by

$\displaystyle 2\pi S=$ $\displaystyle\
\left(c_{L}^{2}+c_{R}^{2}\right)\psi_{+}(x,y_{1})+2c_{L}c_{R}\psi_{-}(x,y_{1})$
(7) $\displaystyle\
+\left(s_{L}^{2}+s_{R}^{2}\right)\psi_{+}(x,y_{2})+2s_{L}s_{R}\psi_{-}(x,y_{2})$
$\displaystyle\
-\left(c_{L}^{2}s_{L}^{2}+c_{R}^{2}s_{R}^{2}\right)\chi_{+}(y_{1},y_{2})-2c_{L}s_{L}c_{R}s_{R}\chi_{-}(y_{1},y_{2}).$

The loop functions $\theta_{\pm}$, $\chi_{\pm}$ and $\psi_{\pm}$ are defined
in Appendix A.

Figure 1:  Values of $S$ and $T$ parameters for $\kappa=\overline{\kappa}=1.0$
with varying the vector-like masses.

Figure 1 shows the $S$ and $T$ parameters on the left and right panels,
respectively. The horizontal and vertical axes correspond to $m_{L}$ and
$m_{E}$, respectively. We take $\kappa=\overline{\kappa}=1.0$ in this figure.
The $W$ boson mass reported by the CDF collaboration is explained on the blue
line. Note that the deviation of the $W$ boson mass from the SM prediction is
related to the oblique parameters as [70, 71]

$\displaystyle\frac{\delta m_{W}^{2}}{m_{W}^{2}|_{\mathrm{SM}}}=$
$\displaystyle\
\frac{\alpha}{c_{W}^{2}-s_{W}^{2}}\left(-\frac{1}{2}S+c_{W}^{2}T+\frac{c_{W}^{2}-s_{W}^{2}}{4s_{W}^{2}}U\right)\sim-0.007S+0.011T+0.0087U,$
(8)

where $\alpha=1/128$ and $s_{W}^{2}=0.22337$ are used in the second equality.
The dot-dashed lines correspond to $1\sigma$ range of the CDF result. The
lightest vector-like lepton mass is less than 90 GeV in the gray region, and
is 200 GeV on the dashed line. As discussed later, the vector-like lepton
lighter than 90 GeV may be excluded by the LEP experiment. $T\gtrsim 0.15$ can
be realized by the light vector-like lepton, $m_{E_{1}}\lesssim
200~{}\mathrm{GeV}$. $S$ is positive in our parameter region. We also find
$U<0.03$ which is much smaller than the other oblique parameters.

We also compare our predictions of the oblique parameters with the results
based on the CDF measurement and the previous works in the PDG. In Ref. [3],
the EW-fit with the new CDF measurement and the PDG value are studied. The
favored values of the oblique parameters with $U=0$ are given by
$(S,T)=(0.15\pm 0.08,0.27\pm 0.06)$ with the correlation coefficient $0.93$
for the analysis with the CDF result and $(S,T)=(0.05\pm 0.08,0.09\pm 0.07)$
with the correlation coefficient $0.92$ for the analysis based on the PDG
data. Similar values are obtained in Refs. [4, 5]. In the SM, $S=T=U=0$, the
values of $\chi^{2}$ are 60 and 3.7 with the CDF and PDG data, respectively.
Thus, the new data strongly favors the existence of new physics.

Figure 2:  Values of $\chi$ from the vector-like leptons.  Table 1:  Benchmark points (A) and (B). $\kappa=\overline{\kappa}=1.0$. $\chi_{\mathrm{CDF}}$ and $\chi_{\mathrm{PDG}}$ are the values of $\chi$ with the new CDF and PDG results, respectively.  | A | B
---|---|---
$(m_{L},m_{E})~{}\mathrm{GeV}$ | $(250,400)$ | $(700,500)$
$(m_{E_{1}},m_{E_{2}})~{}\mathrm{GeV}$ | $(135.5,514.5)$ | $(399.3,800.7)$
$s_{L}=s_{R}$ | $0.549623$ | $0.86553$
$(S,T,U)$ | $(0.087,0.228,0.012)$ | $(0.019,0.050,0.0004)$
$m_{W}~{}\mathrm{GeV}$ | 80.4371 | 80.3740
$(\chi_{\mathrm{CDF}},\chi_{\mathrm{PDG}})$ | $(0.80,4.00)$ | $(6.03,0.65)$

Figure 2 shows values of $\chi$ from the vector-like leptons, $L$ and $E$, for
$\kappa=\overline{\kappa}=1.0$. In the (light) red region, $\chi<1~{}(2)$ for
the CDF data, while the blue region are those for the PDG data. We see that
the new CDF result favors the light vector-like leptons, and $\chi<2$ is
achieved where $m_{E_{1}}<200~{}\mathrm{GeV}$. Whereas the PDG result favors
heavier vector-like leptons. The red region almost coincides with the region
where the new $m_{W}$ is explained in Fig. 1. The values at the benchmark
points are shown in Table 1. The point (A) fits to the new CDF result and
$\chi=0.80$, while the point (B) fits to the combined PDG result and
$\chi=0.65$. These points are plotted as the dots in Fig. 2.

###  2.4 Limits from the collider experiments

The vector-like lepton should be lighter than 200 GeV to explain the new CDF
result. Such a light vector-like lepton may be excluded by direct searches in
the collider experiments. If there is no DM particle and a vector-like lepton
decays to a muon and a SM boson, the Run-1 data at the LHC [72] rules out most
cases [73]. Furthermore, the recent studies using the Run-2 data exclude
doublet-like vector-like leptons, favored to explain the $m_{W}$ anomaly; the
lower bound on the mass is about 800 GeV [74, 75], under the assumptions that
the lepton flavor is different in these analyses. Therefore the light vector-
like lepton is possibly excluded by the LHC searches.

In our setup, the vector-like lepton decays to a DM particle and muon. In this
case, the signal is $2\mu+E_{T}^{\mathrm{miss}}$, where
$E_{T}^{\mathrm{miss}}$ comes from the DM. This signal is studied in the
slepton searches at the LHC [76, 77, 78, 79, 80]. As we have already shown in
Ref. [41], the limit for the doublet vector-like lepton is about 900 GeV if
the DM mass is sufficiently light to produce energetic muons. The limit is,
however, much weaker if $\Delta m:=m_{E_{1}}-m_{X}\lesssim 100~{}\mathrm{GeV}$
since the muons become soft in this region. There are dedicated searches for
signals with soft leptons and $E_{T}^{\mathrm{miss}}$ associated with an
additional jet [77]. If we use the limit on degenerate sleptons [77] as a
conservative one, the limit is about 250 GeV for $\Delta m\sim
10~{}\mathrm{GeV}$, while the limit is less than 100 GeV for $\Delta m\lesssim
1$ or $\gtrsim 30~{}\mathrm{GeV}$.

The muons are too soft to be detected in the detector for even smaller $\Delta
m$, and hence there is no detectable signal from the vector-like lepton
decays. In this case, we refer to the limits from the higgsino searches since
the higgsino has the same quantum number as the doublet vector-like lepton. At
the LHC, the mono-jet searches [81, 82, 83] can not constrain the higgsinos
due to its large backgrounds 111 Recently, it has been proposed that
mono-$W/Z$ signal may cover light higgsinos $\sim 110~{}\mathrm{GeV}$ with the
full Run-2 data [84]. Therefore, the current lower limit of the vector-like
lepton may be about 90 GeV at the LEP experiment [85, 86].

To sum up, the vector-like lepton with $90<m_{E_{1}}<200~{}\mathrm{GeV}$ can
still be viable if the DM is nearly degenerate with the vector-like lepton.
Limits in the degenerate region are very sensitive to the mass difference. For
instance, there will be no limits stronger than $90~{}\mathrm{GeV}$ from the
degenerate slepton searches [77] for $\Delta m\lesssim 1~{}\mathrm{GeV}$ or
$\gtrsim 30~{}\mathrm{GeV}$. A dedicated study for the degenerate vector-like
lepton is an interesting subject but beyond the scope of this paper, and thus
we only take the LEP bound into account while bearing in mind that there may
be an exclusion limit on certain mass differences.

##  3 Dark matter physics and $\Delta a_{\mu}$

The vector-like leptons can participate in the DM thermal production via the
Yukawa couplings to the SM leptons and DM. With the couplings to muon, it also
contributes to $\Delta a_{\mu}$. We have figured out in Ref. [41] that $\Delta
a_{\mu}$ is explained consistently with the DM relic abundance if the double
and singlet vector-like leptons have a mass mixing. In this section, we
examine the compatibility of the DM production with the muon $g-2$ and $W$
boson mass anomalies on the two benchmark points in Table 1.

In the thermal freeze-out scenario, DM relic abundance is controlled by DM
pair annihilation cross section at the freeze-out temperature $T_{f}\simeq
m_{X}/20$, where DM particles can be considered as non-relativistic. In the
presence of the doublet-singlet mixing, the DM couples to both left and right-
handed muons and the pair annihilation $XX\to\mu\bar{\mu}$ has the $s$-wave
part,

$(\sigma
v)_{XX\to\mu\bar{\mu}}\simeq\frac{(\lambda_{L}\lambda_{R})^{2}}{\pi}\left(\frac{c_{R}s_{L}m_{E_{1}}}{m_{X}^{2}+m_{E_{1}}^{2}}-\frac{c_{L}s_{R}m_{E_{2}}}{m_{X}^{2}+m_{E_{2}}^{2}}\right)^{2}+{\cal
O}(v^{2}),$ (9)

where we neglect the muon mass and assume that $\lambda_{L}$ and $\lambda_{R}$
are real. Here, $v$ is relative velocity of DM particles. This process is
expressed in terms of a coupling combination $\lambda_{L}\lambda_{R}$ like the
chirality enhanced contribution in $\Delta a_{\mu}$. This suggests that the DM
abundance can be highly correlated to $\Delta a_{\mu}$ in this model. In fact,
as we discuss in Ref. [41] in detail, a large new physics contribution $\Delta
a_{\mu}\sim 10^{-8}$ is predicted in this model when the $s$-wave part is
mainly responsible for the DM production. Thus, one needs to make the $s$-wave
contribution sub-dominant in the DM annihilation by e.g. employing sizable co-
annihilation or introducing a large hierarchy between two coupling constants
$\lambda_{L,R}$ to invoke the velocity suppressed $d$-wave annihilation
instead of the $s$-wave.

Figure 3:  The predicted value of $\Delta a_{\mu}$ on the benchmark point A
(left) and B (right). The purple band represents the 2$\sigma$ favored value
of the muon $g-2$. For further details, see the text.

Figure 3 shows the predicted value of $\Delta a_{\mu}$ on the benchmark A
(left) and B (right). The black solid, dashed and dotted lines correspond to
$\lambda_{R}=\lambda_{L},\,0.01\lambda_{L}$ and $5\times 10^{-4}\lambda_{L}$
respectively, and the value of $\lambda_{L}$ is fixed to explain the observed
DM abundance in each case. The purple band represents the 2$\sigma$ range of
$\Delta a_{\mu}$. We employ micromegas_5_2_4 [87] to calculate the DM relic
abundance including all co-annihilation processes.

We see in Fig. 3 that $\Delta a_{\mu}$ and DM can be consistently explained on
both benchmarks only if the DM is degenerate to the lighter vector-like lepton
$E_{1}$. In this mass region, the DM pair annihilation is sub-dominant and the
dominant process is the co-annihilation of $E_{1}$; $E_{1}\bar{E}_{1}\to\gamma
h,WW,ZZ,f\bar{f}$, where $f$ is the SM fermions. Although all of the co-
annihilation processes are comparable, $E_{1}\bar{E}_{1}\to\gamma h$ gives the
largest contribution for our benchmark points due to the large $\kappa$ and
$\bar{\kappa}$ which are favored to deviate the $T$ parameter. Apart from the
co-annihilation region, a large $\Delta a_{\mu}$ is predicted due in part to a
large Yukawa coupling constant $\lambda_{L}$ being required for the DM
production, and as a result such mass region is strongly disfavored. It also
follows from Fig. 3 (left) that for $\lambda_{R}/\lambda_{L}\gtrsim 0.01$,
$\Delta m\simeq 10~{}\mathrm{GeV}$ is favored by $\Delta a_{\mu}$ and DM, and
the LHC result is likely to exclude the 135 GeV vector-like lepton. While, we
find $\Delta m\gtrsim 30~{}\mathrm{GeV}$ to explain $\Delta a_{\mu}$ if
$\lambda_{R}/\lambda_{L}\sim 5\times 10^{-4}$. In this case, the current LHC
limit can be evaded. It should be noted, however, that $\lambda_{L}$ is
sizable with a small $\lambda_{R}/\lambda_{L}$ and $\lambda_{L}\simeq 1.8$ is
required for the DM production when $m_{X}=100$ GeV and
$\lambda_{R}/\lambda_{L}\lesssim 0.01$. Otherwise, $\lambda_{L}={\cal
O}(0.01-0.1)$ is enough to fit the DM abundance. In the benchmark (B), the
value of $\lambda_{L}$ tends to be larger than that of the benchmark (A)
mainly because $m_{E_{1}}$ is heavier. This tendency is pronounced for a small
$\lambda_{R}/\lambda_{L}$ and, in fact, we find $\lambda_{L}$ exceeds the
perturbative value $\sqrt{4\pi}$ below $m_{X}\simeq 330$ GeV when
$\lambda_{R}/\lambda_{L}\lesssim 0.01$. That is why the black dashed and
dotted lines are interrupted in $m_{X}\lesssim 330$ GeV.

We briefly comment on direct and indirect detection constraints. The DM
candidate is a SM gauge singlet and couples only to the muon, so DM-nucleon
scattering arises at the loop level. Furthermore, it is known that in the
lepton portal models, a real scalar DM starts the scattering at two-loop level
via di-photon exchanging[88]. The resulting cross section is highly suppressed
and thus well below the reach of the direct detection experiments. As regards
the indirect detection, gamma-ray searches at Fermi-LAT [89, 90] and H.E.S.S
[91], and positron flux measurements at AMS [92, 93, 94] are important. In
particular, a real scalar DM with the lepton portal couplings predicts a sharp
photon flux via virtual internal bremsstrahlung and one-loop processes, and
can be effectively searched by the Fermi/H.E.S.S line gamma searches [95, 96,
97]. If there is no doublet-singlet mixing, the bound reads $m_{E_{1}}\lesssim
1.1~{}(1.2)~{}m_{X}$ at best for a purely singlet (doublet) vector-like lepton
[41]. Thus, the mass region of our interest may be within the reach of the
gamma-ray searches. One should note, however, that there is no detailed study
on the gamma-ray bound in the case with both left and right-handed couplings,
so that this Fermi/H.E.S.S limit cannot be applied directly to this model. For
the AMS positron measurements, the current conservative lower limit on DM mass
is 30 GeV [98], assuming the cross section of $XX\to\mu\bar{\mu}$ has the
canonical size for thermal production, i.e. $(\sigma
v)_{XX\to\mu\bar{\mu}}\simeq 3\times 10^{-26}\,{\rm cm^{3}/s}$. Hence, the AMS
measurements do not provide the relevant limit in this case.

##  4 Summary

In this paper, we studied corrections to the oblique parameters, which are
strongly correlated with the $W$ boson mass, in the extended SM with the real
scalar DM and the vector-like leptons. The sizable mixing between the singlet
and doublet vector-like leptons is crucial to deviate the oblique parameters.
Since the mixing is induced by the Higgs VEV, the vector-like leptons should
be close to the EW symmetry breaking scale. In fact, we found that the
lightest vector-like lepton has to be lighter than 200 GeV to explain the new
CDF result, see Fig. 2. The light vector-like lepton may be excluded by the
direct searches at the LHC if it decays to the SM particles. The new lepton,
however, can evade the limit if it decays to a DM particle whose mass is close
to the lepton, so that the signals are effectively invisible.

As discussed in Ref. [41], the Yukawa couplings involving muon, a real scalar
DM and vector-like leptons can resolve the discrepancy in the muon anomalous
magnetic moment, $\Delta a_{\mu}$. The simultaneous explanation of the DM and
$\Delta a_{\mu}$ requires the sizable mixing in the vector-like leptons and
the mass degeneracy of the lighter vector-like lepton and DM. Interestingly,
the former is required to achieve the shift in $W$ boson mass and the latter
is to avoid the LHC limits. The mass region favored by those anomalies would
be covered by the future gamma-ray searches utilizing the sharp photon flux
from the galactic DM annihilation.

## Acknowledgment

The work of J.K. is supported in part by the Institute for Basic Science
(IBS-R018-D1) and the Grant-in-Aid for Scientific Research from the Ministry
of Education, Science, Sports and Culture (MEXT), Japan No. 18K13534. S.O.
acknowledges financial support from the State Agency for Research of the
Spanish Ministry of Science and Innovation through the “Unit of Excellence
María de Maeztu 2020-2023” award to the Institute of Cosmos Sciences
(CEX2019-000918-M) and from PID2019-105614GB-C21 and 2017-SGR-929 grants. The
work of Y. O. is supported by Grant-in-Aid for Scientific research from the
MEXT, Japan, No. 19K03867.

##  Appendix A Loop functions

The loop functions for $\Delta a_{\mu}$ are defined as

$\displaystyle F(x)$ $\displaystyle\ =\frac{2+3x-6x^{2}+x^{3}+6x\log
x}{6(1-x)^{4}},\quad G(x)=\frac{3-4x+x^{2}+2\log x}{(1-x)^{3}}.$ (10)

The loop functions for the oblique parameters are defined as

$\displaystyle\theta_{+}(y_{1},y_{2})=y_{1}+y_{2}-\frac{2y_{1}y_{2}}{y_{1}-y_{2}}\log\frac{y_{1}}{y_{2}},\quad\theta_{-}(y_{1},y_{2})=2\sqrt{y_{1}y_{2}}\left(\frac{y_{1}+y_{2}}{y_{1}-y_{2}}\log\frac{y_{1}}{y_{2}}-2\right),$
(11)

and

$\displaystyle\chi_{+}(y_{1},y_{2})=$ $\displaystyle\
\frac{y_{1}+y_{2}}{2}-\frac{(y_{1}-y_{2})^{2}}{3}+\left(\frac{(y_{1}-y_{2})^{3}}{6}-\frac{1}{2}\frac{y_{1}^{2}+y_{2}^{2}}{y_{1}-y_{2}}\right)\log\frac{y_{1}}{y_{2}}$
(12) $\displaystyle+\frac{y_{1}-1}{6}$ $\displaystyle\
f(y_{1},y_{1})+\frac{y_{2}-1}{6}f(y_{2},y_{2})+\left(\frac{1}{3}-\frac{y_{1}+y_{2}}{6}-\frac{(y_{1}-y_{2})^{2}}{6}\right)f(y_{1},y_{2}),$
$\displaystyle\chi_{-}(y_{1},y_{2})=$ $\displaystyle\
-\sqrt{y_{1}y_{2}}\left[2+\left(y_{1}-y_{2}-\frac{y_{1}+y_{2}}{y_{1}-y_{2}}\right)\log\frac{y_{1}}{y_{2}}\right.$
(13) $\displaystyle\ \hskip
56.9055pt\left.+\frac{f(y_{1},y_{1})+f(y_{2},y_{2})}{2}-f(y_{1},y_{2})\right],$
$\displaystyle\psi_{+}(y_{1},y_{2})=$ $\displaystyle\
\frac{2y_{1}+10y_{2}}{3}+\frac{1}{3}\log\frac{y_{1}}{y_{2}}+\frac{y_{1}-1}{6}f(y_{1},y_{1})+\frac{5y_{2}+1}{6}f(y_{2},y_{2}),$
(14) $\displaystyle\psi_{-}(y_{1},y_{2})=$ $\displaystyle\
-\sqrt{y_{1}y_{2}}\left(4+\frac{f(y_{1},y_{1})+f(y_{2},y_{2})}{2}\right).$
(15)

Here,

$\displaystyle
f(y_{1},y_{2})=\begin{cases}\sqrt{d}\log\left|\dfrac{y_{1}+y_{2}-1+\sqrt{d}}{y_{1}+y_{2}-1-\sqrt{d}}\right|&d>0\\\
0&d=0\\\
-2\sqrt{\left|d\right|}\left[\tan^{-1}\dfrac{y_{1}-y_{2}+1}{\sqrt{\left|d\right|}}-\tan^{-1}\dfrac{y_{1}-y_{2}-1}{\sqrt{\left|d\right|}}\right]&d<0\end{cases}$
(16)

with $d:=(1+y_{1}-y_{2})^{2}-4y_{1}$. Note that $\psi_{\pm}$ are different
from those for vector-like quarks shown in Ref. [69].

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