# Compacton existence and spin-orbit density dependence in Bose-Einstein
condensates

F.Kh. Abdullaev1, M. S. A. Hadi2, B. Umarov2 and L. A. Taib2, and Mario
Salerno3 1 Physical-Technical Institute, Uzbekistan Academy of Sciences,
Tashkent, Bodomzor yuli, 2-b, Uzbekistan  2 Department of Computational and
Theoretical Sciences,, Kulliyyah of Science, International Islamic University
Malaysia, 25200 Kuantan, Pahang, Malaysia  3 Dipartimento di Fisica “E.R.
Caianiello”, CNISM and INFN \- Gruppo Collegato di Salerno, Universitï¿œ di
Salerno, Via Giovanni Paolo II, 84084 Fisciano (SA), Italy

###### Abstract

We demonstrate the existence of compactons matter waves in binary mixtures of
Bose-Einstein condensates (BEC) trapped in deep optical lattices (OL)
subjected to equal contributions of intra-species Rashba and Dresselhaus spin-
orbit coupling (SOC) under periodic time modulations of the intra-species
scattering length. We show that these modulations lead to a rescaling of the
SOC parameters that involves the density imbalance of the two components. This
gives rise to a density dependent SOC parameters that strongly influence the
existence and the stability of compacton matter waves. The stability of SOC-
compactons is investigated both by linear stability analysis and by time
integrations of the coupled Gross-Pitaevskii equations. We find that SOC
restricts the parameter ranges for stable stationary SOC-compacton existence
but, on the other side, it gives a more stringent signature of their
occurrence. In particular, SOC-compactons should appear when the intra-species
interactions and the number of atoms in the two components are perfectly
balanced (or close to be balanced for metastable case). The possibility to use
SOC-compactons as a tool for indirect measurements of the number of atoms
and/or the intra-species interactions, is also suggested.

###### pacs:

03.75.Nt, 05.30.Jp

##  I Introduction

Recently theoretical and experimental investigations of Bose-Einstein
condensates (BEC) based on Floquet engineering (FE) of the different
parameters have been performed (see for example the review Bukov ). The FE
usually involves linear optical lattices, periodically shaken in time either
through frequency or amplitude Lin ; Galitski , or nonlinear optical lattices
in which FE is realized through the modulation of the interactions Abdullaev1
. Periodic modulations of the scattering lengths can also be used to emulate
synthetic dimensions and to create density-dependent gauge fields Greschner ,
this being a research area of rapidly growing interest connected with
interesting physical phenomena, including pair superfluidity, exactly defect-
free Mott insulator states Rapp , etc. In the case of BEC with modulated
interactions in double-well potential, the investigation shows that the
tunneling transition amplitude between two wells depends on the relative
imbalance of the atomic population between wells. This phenomenon leads to the
suppression of the tunneling between wells for specific values of the
imbalance Gong , a prediction that was recently experimentally confirmed in
Meinert .

For Bose-Einstein condensates trapped in a deep optical lattice (OL) the
phenomenon of tunneling suppression leads to the existence of a new form the
localized matter waves, also called BEC compactons, in which the density is
strictly localized on a compact domain without any exponential decay tail at
the boundary Abdullaev1 . BEC compactons can exist not only in ordinary (e.g.
single component) BEC but also in binary BEC mixtures Abdullaev2 ; Abdullaev3
as well as in multidimensional contexts multidim . In the case of a deep
modulation of the potential, the problem can be reduced to the nonlinear
lattice described by a discrete nonlinear Schrödinger (DNLS) model with
modulated nonlinearity. In this case the maximum localization is achieved with
one-site compactons, i.e., excitations in which all the matter remains
localized in a single well of the OL.

In binary BEC mixtures, however, another type of coupling, besides the one
induced by the two-body inter- and intra-species interactions, is also
possible, namely the spin-orbit–coupling (SOC). BEC with spin-orbit coupling
(SOC-BEC) is presently receiving a lot of attention as with the ultra-cold
atoms variety of synthetic SOC can be engineered and controlled by external
laser fields. The experimental realization of SOC-BEC for the case of binary
mixtures was reported in Lin ; Galitski , while nonlinear excitations of
intra-SOC and inter-SOC, i.e., with the spatial derivative of the SOC term
acting inside each species or between the species, respectively, were
theoretically investigated in Belicev . For SOC-BEC trapped in deep OL, the
existence of strongly localized excitation in the form of discrete breathers
was theoretically investigated in Salerno1  and the SOC tunability induced by
periodic time modulations of the Zeeman term was demonstrated in Salerno2 .

Very recently the nonlinearity management of the spin-orbit coupled BEC has
been investigated in SM  where it has been shown that for slow time periodic
modulations of the nonlinearity, resonances between the frequencies of the
modulation and of the intrinsic nonlinear modes (solitons) can lead to the
appearance of instability and stability tongues for the solitons.

We remark that only a few studies exist for the case of SOC-BEC in deep OL
subjected to nonlinear management Lin . This case is rather interesting
because, compared to previous studies, the presence of the SOC term may affect
the compacton matter waves excitations found in absence of SOC Abdullaev1 ;
Abdullaev2 ; Abdullaev3 . In particular, the spatial derivative of the SOC
term may interfere with the tunneling of atoms between adjacent wells, making
it unclear whether SOC-compactons may still be possible. At the best of our
knowledge, this problem is completely uninvestigated.

The aim of the present paper is to provide an answer to the above problem,
i.e., we investigate the formation of binary matter-wave compactons in the
presence of SOC. For this, we consider an intra-SOC-BEC mixture in a deep OL
subjected to strong nonlinear management (SNM) consisting in time-periodic
modulations of the inter-species scattering length. Using the averaging
method, we show that the SNM induces a rescaling of the inter-well tunneling
constant and of the SOC parameters that depend on the density imbalance
between neighboring sites, this making the conditions for the compacton
existence and stability much more involved.

In particular, we show that the interplay between SOC and inter-well tunneling
produce a big effect on the stability of the compactons which appears very
much reduced by the presence of SOC. This is true even for the case of a
single site compacton for which the existence range is completely independent
on SOC parameters. A detailed linear stability analysis shows that single site
compactons are stable for a wide range of inter-well tunneling when the SOC
parameter is relatively small. The SOC range, however, can be made wider by
increasing the Rabi frequency. This is true also for compactons localized on
more than one site.

It is worth to note that the possibility to induce by means of SNM a local
density imbalance dependence on the SOC parameters is a fact that persists
independently on compacton existence and, as far as we know, was not reported
before. Very recently a density-dependent SOC was also introduced in ultracold
Bose and Fermi gases in terms of interaction assisted Raman processes Xu2021 .
We remark that our approach is different from this, being completely
independent on indirect Raman processes (the density dependence of the SOC in
our case is only due to scattering length modulations).

The paper is organized as follows. In Section II, we introduce the model
equations for the system and show in detail the averaging procedure to derive
the effective equations that are valid in the limit of strong modulations of
the interspecies scattering length. In Sections III and IV we study the SOC-
compacton stationary and existence conditions, respectively. In Section V we
present numerical results for the cases of one, two and three sites compactons
and investigate their stability both by linear analysis and by direct time
integrations of the equations of motion. Finally, in Section VI the main
results of this paper are briefly summarized.

##  II Model equations and averaging

As a model for a quasi-one-dimensional BEC mixture trapped in an OL in the
presence of SOC, we consider the following coupled Gross-Pitaevskii equations
(CGPE)

$\displaystyle i\hbar\frac{\partial\Psi}{\partial t}$ $\displaystyle=$
$\displaystyle H\Psi\equiv\left[H_{0}+H_{nl}\right]\Psi,$ (1) $\displaystyle
H_{0}$ $\displaystyle\equiv$
$\displaystyle\frac{p_{x}^{2}}{2m}+\frac{\hbar\kappa}{m}p_{x}\sigma_{z}+V_{ol}(x)+\hbar\bar{\Omega}\sigma_{x},$
$\displaystyle H_{nl}$ $\displaystyle\equiv$ $\displaystyle
2\hbar\omega\left(\begin{array}[]{cc}\sum_{j}a_{1j}\left|\Psi_{j}\right|^{2}&0\\\
0&\sum_{j}a_{j2}\left|\Psi_{j}\right|^{2}\end{array}\right),$ (4)

where $\sigma_{x,z}$ are usual Pauli matrices and
$\Psi=\left(\Psi_{1},\Psi_{2}\right)^{T}$ denotes the two-component wave
function normalized to the total number of atoms. This quasi-1D model can be
derived from a more general three-dimensional setting by considering a
trapping potential with the transversal frequency $\omega_{\perp}$ much larger
than the longitudinal one, $\omega_{\perp}\gg\omega_{\parallel}$. The linear
part, $H_{0}$, of the Hamiltonian $H$ includes the kinetic energy
$p_{x}^{2}/(2m)$ of the condensate, the diagonal SOC term of strength
$\kappa$, the off-diagonal Rabi oscillation term of frequency $\Omega$, and an
optical lattice in the $x$-direction represented by a periodic potential
$V_{ol}(x)\sim\sin^{2}\left(k_{L}x\right)$ where $k_{L}$ is the lattice
wavenumber. Note that the SOC term is diagonal in the two components, meaning
that we are considering the intra-SOC case (a similar analysis can be done for
the inter-SOC case with an off-diagonal $\kappa$ term and a diagonal Zeeman
term). The nonlinear part, $H_{nl}$, includes contact interactions, with
$a_{jj}\,\,(j=1,2)$ and $a_{12}$ the two-body intra-species and inter-species
scattering lengths, respectively. By rescaling variables and component
wavefunctions according to

$\displaystyle x$ $\displaystyle\rightarrow$
$\displaystyle\frac{x}{k_{L}},\,\,\,t\rightarrow\omega_{R}t,\,\,\,\mathrm{where}\,\,\omega_{R}\equiv\frac{E_{R}}{\hbar}\equiv\frac{\hbar
k_{L}^{2}}{2m};$ $\displaystyle V_{ol}(x)$ $\displaystyle\rightarrow$
$\displaystyle E_{R}V(x)=E_{R}V_{0}\cos(2x),$ (5) $\displaystyle\Psi_{j}$
$\displaystyle\equiv$
$\displaystyle\sqrt{\frac{\omega_{R}}{2\omega_{\perp}a_{0}}}\psi_{j}(x,t),$

the Eq. (1) can be put in the dimensionless form:

$\displaystyle i\frac{\partial\psi_{j}}{\partial t}$ $\displaystyle=$
$\displaystyle\left[-\frac{\partial^{2}}{\partial
x^{2}}+V\left(x\right)-\left(-\right)^{j}i\frac{2\kappa}{k_{L}}\frac{\partial}{\partial
x}\right]\psi_{j}$ $\displaystyle+$
$\displaystyle\left(\frac{a_{jj}}{a_{0}}\left|\psi_{j}\right|^{2}+\frac{a_{j,3-j}}{a_{0}}\left|\psi_{3-j}\right|^{2}\right)\psi_{j}$
$\displaystyle+$
$\displaystyle\frac{\bar{\Omega}}{\omega_{R}}\psi_{3-j},\;\;\;j=1,2,$

with $a_{0}$ denoting the background scattering length and $E_{R}$ the lattice
recoil energy.

By expanding the two component fields as:
$\psi_{1}(x,t)=\sum_{n}u_{n}(t)w(x-na)$,
$\psi_{2}(x,t)=\sum_{n}v_{n}(t)w(x-na)$ alfimov , with $w(x-na)$ the lower
band Wannier functions of the underlying linear system, one obtains in the
tight-binding approximation appropriate for deep optical lattices, the
following discrete nonlinear Schrödinger system Salerno1

$\displaystyle iu_{n,t}=$
$\displaystyle-\varGamma(u_{n+1}+u_{n-1})+i\sigma(u_{n+1}-u_{n-1})$ (7)
$\displaystyle+\Omega
v_{n}+[\gamma_{1}\left|u_{n}\right|^{2}+\gamma\left|v_{n}\right|^{2}]u_{n},$
$\displaystyle iv_{n,t}=$
$\displaystyle-\varGamma(v_{n+1}+v_{n-1})-i\sigma(v_{n+1}-v_{n-1})$
$\displaystyle+\Omega{u}_{n}+[\gamma\left|u_{n}\right|^{2}+\gamma_{2}\left|v_{n}\right|^{2}]v_{n}.$

Here $n$ denotes the lattice site at position $na$, $a$ is the lattice
constant, below fixed to $a=1$ without any loss of generality, $\varGamma$ is
the inter-site hopping coefficient, $\sigma$ the rescaled SOC strength,
$\gamma_{1},\gamma_{2}$ and $\gamma$ are the coefficients of intra and inter-
species interactions, respectively. The system (7) is of Hamiltonian form with
the Hamiltonian given by:

$\displaystyle H=$
$\displaystyle\sum_{n}\Big{[}-\varGamma(u_{n+1}u_{n}^{*}+u_{n+1}^{*}u_{n})$
(8) $\displaystyle-\varGamma(v_{n+1}v_{n}^{*}+v_{n+1}^{*}v_{n})$
$\displaystyle+i\sigma(u_{n+1}u_{n}^{*}-u_{n+1}^{*}u_{n})$
$\displaystyle-i\sigma(v_{n+1}v_{n}^{*}-v_{n+1}^{*}v_{n})$
$\displaystyle+\Omega(u_{n}v_{n}^{*}+u_{n}^{*}v_{n})$
$\displaystyle+\frac{1}{2}(\gamma_{1}|u_{n}|^{4}+\gamma_{2}|v_{n}|^{4})+\gamma(t)|u_{n}|^{2}|v_{n}|^{2}\Big{]}.$

In the absence of nonlinearity, i.e. $\gamma_{1}=\gamma_{2}=\gamma=0$, the
dispersion relation can be obtained by substituting
$u_{n}=A\exp\left[i\left(kn-\mu{t}\right)\right]$ and
$v_{n}=B\exp\left[i\left(kn-\mu{t}\right)\right]$ into Eq. (7), this giving
the following lower and upper branches for the chemical potentials of the two
components:

$\mu_{\pm}\left(k\right)=-2\varGamma\cos\left(k\right)\pm\sqrt{\Omega^{2}+\sigma^{2}\sin^{2}\left(k\right)}.$
(9)

In the following we shall investigate the possibility of compacton formation
via an inter-species NM, this being more simple than the intra-species case
since it involves the modulation of just a single parameter, i.e. the inter-
species scattering length. Specifically, we keep the intra-species nonlinear
coefficients, $\gamma_{i},i=1,2$, in Eq. (8) to be constants and assume the
inter-species nonlinearity coefficient $\gamma$ to be a time periodic function
of the form:

$\gamma(t)=\gamma^{(0)}+\frac{\gamma^{(1)}}{\epsilon}\cos\left(\frac{\omega
t}{\epsilon}\right)$

with $\epsilon$ a real parameter which allows to control the strength of the
management and to separate the fast and slow time scales (see below). Our
analysis will be valid in the limit of strong NM which requires $\epsilon\ll
1$. In this case it is possible to eliminate the explicit time dependence from
Eq. (8) by means of the transformation:

$u_{n}=U_{n}e^{-i\Lambda(\tau)|{V}_{n}|^{2}},\quad
v_{n}=V_{n}e^{-i\Lambda(\tau)|{U}_{n}|^{2}},$ (10)

with

$\Lambda(\tau)=\alpha\sin(\omega\tau),\quad\alpha=\frac{\gamma^{(1)}}{\omega},\quad\tau=\frac{t}{\epsilon},$
(11)

and to apply the averaging method to eliminate the fast time scale. The
resulting averaged equations for the amplitudes $U_{n},V_{n}$ are given in the
Appendix (see Eqs.(25) and (26)).

It turns out that also the averaged equations have Hamiltonian structure (see
Appendix) with the same Hamiltonian (8) of the unmodulated system but with
$u_{n},v_{n}$ replaced by $U_{n},V_{n}$, respectively, and
$\varGamma,\sigma,\Omega$ rescaled according to:

$\displaystyle\varGamma\rightarrow\tilde{\varGamma}_{i}\equiv\varGamma
J_{0}(\theta_{3-i}^{+}),$
$\displaystyle\sigma\rightarrow\tilde{\sigma}_{i}\equiv\sigma
J_{0}(\theta_{3-i}^{+}),$ (12)
$\displaystyle\Omega\rightarrow\tilde{\Omega}\equiv\Omega J_{0}(\theta_{0}).$

Here $i=1,2$ refer to first and second component respectively, $J_{0}$ is the
Bessel function of order zero and $\theta_{1}^{\pm}=|U_{n\pm
1}|^{2}-|U_{n}|^{2}$, $\theta_{2}^{\pm}=|V_{n\pm 1}|^{2}-|V_{n}|^{2}$ and
$\theta_{0}=|U_{n}|^{2}-|V_{n}|^{2}$. From this we see that the effect of the
modulation is to introduce a dependence in the parameters $\varGamma,\sigma$
of the unmodulated system, on the density imbalance between adjacent sites of
the same component, and a dependence on the density imbalance between the two
components on the same site, for $\Omega$. Obviously the condition for
compacton existence, i.e. the complete suppression of the tunneling at the
compacton edges, involves also the SOC parameters and will have more
restrictive conditions to be satisfied.

This can be also intuitively understood by observing that the term
proportional to $\sigma_{z}$ in the Hamiltonian changes the kinetic energy of
both components while the term proportional to $\sigma_{x}$ corresponds to
Rabi-coupling leading to oscillations between the two BEC components. On the
other hand, the variation of the kinetic energy due to the $\sigma$ part of
the SOC interferes with the natural dispersion of the system, i.e. with the
intra-well tunneling $\Gamma$, while the Rabi term interferes with the
stationarity condition of compacton. From this it is clear that the conditions
on parameters for the SOC-compacton existence are expected to be more
stringent than those in the absence of SOC.

Finally, we remark that while in the density-dependent SOC approach considered
in Ref. Xu2021  the rescaling of the parameter $\Omega$ involves both the
zeroth and the first-order Bessel functions, in our case, due to the absence
of indirect Raman processes, only the function $J_{0}$ is involved.

Figure 1: (a),(b) Binary one-site compactons (16) localized at $n=0$, for the
case $\sigma=0,\Omega=0$ and with $A_{0}=B_{0}=a+ib$ with
$a=0.5\sqrt{\xi_{1}/\alpha}$ and $b=\sqrt{\xi_{1}/\alpha-a^{2}}$. Other
parameters are fixed as $\varGamma=1,\gamma_{1}=\gamma_{2}=-1$,
$\gamma^{(0)}=-0.5$, $\gamma^{(1)}=0.5$, $\omega=1.0$. (c) The eigenvalue
spectrum at $\varGamma=1$, (d) Gain $G$ versus $\varGamma$.

##  III Stationary conditions for SOC-compactons

We look for stationary SOC-compactons of the form:

$U_{n}=A_{n}e^{-i\mu_{1}t},\quad V_{n}=B_{n}e^{-i\mu_{2}t},$ (13)

with $A_{n}$, $B_{n}$ complex amplitudes and $\mu_{1}$, $\mu_{2}$ chemical
potentials for the two BEC components. By substituting these expressions into
the averaged equations for $U_{n}$, $V_{n}$ given in the Appendix, one finds
that all the time dependent exponential factors can be cancelled out except
the ones that are proportional to $\Omega$, these being of the form:

$\displaystyle\Omega e^{-i(\mu_{1}+\mu_{2})t}\Big{\\{}\alpha
A_{n}^{2}B_{n}^{*}e^{2i\mu_{2}t}J_{1}(\alpha\theta_{0})\;\;\;\;\;\;\;\;\;\;\;\;$
(14)
$\displaystyle\;\;\;\;\;\;\;\;-e^{2i\mu_{1}t}B_{n}\Big{[}J_{0}(\alpha\theta_{0})-\alpha|A_{n}|^{2}J_{1}(\alpha\theta_{0})\Big{]}\Big{\\}}$

for Eq. (27), and similar expression for Eq. (28) but with the exchanges
$A\leftrightarrow B$ and $\mu_{1}\leftrightarrow\mu_{2}$. From these
expressions it is clear that stationary SOC-compactons are possible only if
$\Omega=0$ or if the chemical potentials of the two species are equal, i.e.
$\mu_{1}=\mu_{2}\equiv\mu$. The corresponding steady-state equations then are:

$\mu_{1}A_{n}=F_{1}|_{\Omega=0},\,\,\mu_{2}B_{n}=F_{2}|_{\Omega=0},$ (15)

for case $\Omega=0$, and

$\mu A_{n}=F_{1},\,\,\,\mu B_{n}=F_{2},$ (16)

for case $\mu_{1}=\mu_{2}=\mu$. Here $F_{1}$ and $F_{2}$ denote the right hand
side of Eqs.(27) and (28), respectively, but with the replacements:
$U_{n}\rightarrow A_{n}$, $V_{n}\rightarrow B_{n}$. We remark that although
the above stationary conditions are specific of the intra-SOC system, similar
conditions would appear also for the inter-SOC system Note2 .

As we shall see in the next section, the condition $\Omega=0$ is too
restrictive and can be satisfied only for one-site compactons. In this case,
however, although the stability of the solutions will be influenced by the
SOC, the analytical expression of the solutions will be the same as for the
non-SOC case.

Note that Eq. (16) implies the equality of the number of atoms $N_{1},N_{2}$
in the two components, this being true in general also for multi-site SOC-
compactons. This can be easily understood from the fact that for $N_{1}\neq
N_{2}$ and $\Omega\neq 0$, there will be oscillations in the number of atoms
that will make the zero tunneling conditions at the compacton edges impossible
to satisfy. The only way to avoid this is to equilibrate the two components.
This, however, requires the additional restriction of the equality of the
intra-species interactions (see below).

##  IV Existence conditions for SOC-compactons

The conditions for the existence of stationary compactons are obtained from
Eqs.(15), once the following compacton ansatz is adopted

$\begin{gathered}A_{n},\,B_{n}\neq 0\;\mbox{\;\;\;\;\; if\;\;$n_{0}-s\leq
n\leq n_{0}+s$},\\\
A_{n},\,B_{n}=0\;\;\;\;\;\;\mbox{otherwise,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{gathered}$
(17)

with $n_{0}-s$ and $n_{0}+s$ denoting the left and right edge of the
compacton, respectively, and $s+1$ its width. Since the coupling in the
stationary equations involves only first next neighbors, the substitution of
the ansatz $\eqref{eq:013}$ into either Eq. (15) or Eq. (16) will give
$2(s+3)$ nontrivial equations in correspondence of sites
$n_{0}-s-1,...,n_{0},...,n_{0}+s+1,$, with all the others, $n\leq n_{0}-s-2$
and $n\geq n_{0}+s+2$, automatically satisfied. The compacton existence relies
on the possibility to solve these $2(s+3)$ equations for the nonzero
amplitudes by achieving both the tunneling suppression at the compacton edges
and the fixing of the chemical potentials.

Figure 2: Parameters for panels (a) and (b) are fixed as in panels (c) and (d)
of Fig. 1 except $\sigma=1$ and for panels (c) and (d), the parameters are
fixed as in (a) and (b) except $\Omega=-2$. Figure 3: Linear stability
analysis for the one-site compactons , (a) The eigenvalue spectrum for case
$\varGamma=1,\gamma_{1}=\gamma_{2}=-1$, $\gamma^{(0)}=-1$, $\gamma^{(1)}=0.2$,
$\omega=1.0$, $\Omega=-0.4,\sigma=1$, (b) Gain $G$ versus $\varGamma$ with
parameters as in (a), (c) The eigenvalue spectrum for case
$\varGamma=1,\gamma_{1}=\gamma_{2}=-1$, $\gamma^{(0)}=-0.2$, $\gamma^{(1)}=1$,
$\omega=1$ , $\Omega=-0.4,\sigma=1$, (d) Gain $G$ versus $\varGamma$ with
parameters as in (c). Figure 4: Space-time evolution of the on-site compactons
by solving Eq. (7) with $\epsilon=0.01$ , (a) $\left|u_{n}\right|$ and (b)
$\left|v_{n}\right|$ for stable case as in Fig.3(a) while (c)
$\left|u_{n}\right|$ and (d) $\left|v_{n}\right|$ for unstable case as in Fig.
3(c).

###  IV.1 One-site SOC-compactons

For a one-site compacton the condition $\Omega=0$ implies existence conditions
completely independent on the SOC parameters $\sigma,\Omega$ and therefore
will be similar to the case of BEC mixtures in absence of SOC considered in
previous papers Abdullaev2 ; Abdullaev3 . On the other hand, for
$N_{1}=N_{2}=N/2$ the compacton existence will depend on $\Omega$ but not on
$\sigma$, this is because the $\sigma$ term always involves as a factor a zero
amplitude on a site $n_{0}\pm 1$ different to the site $n_{0}$. In both cases,
however, the SOC terms will strongly influence the stability, therefore it is
worth considering in more detail the existence conditions for these two cases.

Case $\Omega=0.$

We take $s=0$ in Eq. (17) and fix the amplitudes at site $n_{0}$ as
$A_{n_{0}}=a+ib$, $B_{n_{0}}=c+id$, with $a,b,c,d$ reals and by substituting
into Eqs. (15), one obtains in correspondence to sites $n_{0}\pm 1$ four
equations that are automatically satisfied if $J_{0}(\alpha(c^{2}+d^{2}))=0$
and $J_{0}(\alpha(a^{2}+b^{2}))=0$, e.g., if $a,b,c,d$ are taken as

$a^{2}+b^{2}=N_{1}=\frac{\xi_{m}}{\alpha},\;\;c^{2}+d^{2}=N_{2}=\frac{\xi_{l}}{\alpha},$
(18)

with $\xi_{m},\xi_{l}$ two different zeros of the Bessel function $J_{0}$. One
can then check that the other two equations, e.g., the ones for the site
$n_{0}$, give the chemical potentials as

$\mu_{1}=N_{2}\gamma^{(0)}+N_{1}\gamma_{1},\quad\mu_{2}=N_{1}\gamma^{(0)}+N_{2}\gamma_{2}.$
(19)

Case ${\bf\Omega\neq 0.}$

In this case, we must have $N_{1}=N_{2}=N/2$, so we fix $s=0$ in Eq. (17) and
take $A_{n_{0}}=B_{n_{0}}=a+ib$, with $a,b$ reals. One can check that the
equations in correspondence to the edges $n_{0}\pm 1$ are automatically
satisfied if $J_{0}(\alpha(a^{2}+b^{2}))=0$, e.g., if
$a^{2}+b^{2}=N/2=\xi_{l}/\alpha$, with $\xi_{l}$ a zeros of $J_{0}$. The other
two equations for the site $n_{0}$ can be solved for the chemical potential
only if $\gamma_{1}=\gamma_{2}\equiv\tilde{\gamma}$, this giving:

$\mu=\frac{N}{2}(\gamma^{(0)}+\tilde{\gamma})+\Omega.$ (20)

Note that in this case $\Omega$ enters the $\mu$ expressions simply as a
shift.

###  IV.2 Two-site SOC-compactons

For a two-site compacton, $\Omega=0$ will not provide real solutions for the
chemical potentials due to the presence of imaginary terms proportional to
$\sigma$. It is possible however to eliminate such terms by considering some
suitable ansatzes for the amplitudes. For this, let us fix $s=1$ in Eq. (17)
and take $A_{n_{0}}=B_{n_{0}+1}=a+ib$, $B_{n_{0}}=A_{n_{0}+1}=a-ib$, with
$a,b$ reals (in-phase solution). From Eqs. (16) one then obtains eight
equations in correspondence of sites $n_{0}-1,n_{0},n_{0}+1,n_{0}+2$. One can
show that these equations lead to the following solution for the chemical
potential

$\mu=\left(a^{2}+b^{2}\right)\left(\frac{\Omega-\varGamma}{a^{2}-b^{2}}+\gamma+\gamma^{(0)}\right),$
(21)

with $a,b$ given by

$\begin{gathered}a^{2}=\frac{\xi_{l}}{2\alpha}\left(1+\frac{\Omega-\varGamma}{\sqrt{(\varGamma-\Omega)^{2}+\sigma^{2}}}\right),\\\
b^{2}=\frac{\xi_{l}}{2\alpha}\left(1-\frac{\Omega-\varGamma}{\sqrt{(\varGamma-\Omega)^{2}+\sigma^{2}}}\right),\end{gathered}$
(22)

and $\xi_{l}$ denoting a zero of $J_{0}$. Using Eqs. (22) one can rewrite Eqs.
(21) in a more explicit form

$\mu=\frac{N}{4}\left(\gamma+\gamma^{(0)}\right)+\sqrt{\sigma^{2}+(\varGamma-\Omega)^{2}},$
(23)

showing the full dependence on parameters $\varGamma,\sigma,\Omega$ and on the
total numbers of atoms $N$.

Similarly, one can obtain out-of-phase two-site compactons solutions by
assuming the ansatz $A_{n_{0}}=-B_{n_{0}+1}=a+ib$,
$B_{n_{0}}=-A_{n_{0}+1}=a-ib$. In this case, the expressions for $\mu,a,b$ are
the same as in Eqs. (21) - (23), except for the replacement
$\varGamma\rightarrow-\varGamma$ in all equations and for the change of sign
in front of the fractions appearing inside the parenthesis in Eqs. (22).

Figure 5: Two-site in-phase compacton for case $\gamma_{1}=\gamma_{2}=1$,
$\gamma^{(0)}=4$, $\gamma^{(1)}=1$, $\omega=1$, $\Omega=4$ and $\sigma=1$

.

Figure 6: Two-site out-phase compacton for case $\gamma_{1}=\gamma_{2}=-1$,
$\gamma^{(0)}=-4$,$\gamma^{(1)}=1$, $\omega=1$, $\Omega=-4$ and $\sigma=1$
Figure 7: Two-site in-phase compacton for case $\gamma_{1}=\gamma_{2}=1$,
$\gamma^{(0)}=4$, $\gamma^{(1)}=8.0$, $\omega=1$, $\Omega=4$ and $\sigma=1$
Figure 8: Space-time evolution of the two-site compactons obtained from
numerical integrations of Eq. (7) with $\epsilon=0.01$: (a)
$\left|u_{n}\right|$ and (b) $\left|v_{n}\right|$ for the stable case at
$\Gamma=1$ in Fig.5; (c) $\left|u_{n}\right|$ and (d) $\left|v_{n}\right|$ for
unstable case at $\Gamma=1$ in Fig.7.

###  IV.3 Multi-site SOC-compactons

In principle, the above analysis, being completely general, can be applied to
SOC-compactons of arbitrary size. The equations to solve, however, in these
cases become much more involved for an exact analysis and one must recourse to
numerical methods. An example of this is given for a three-site compacton in
the next section.

##  V Numerical results

In order to check the above results, we will use both linear stability
analysis and direct numerical integrations of the original discrete CGPE
equations in Eq. (7). For the linear stability analysis, the standard
procedure is employed by considering the ansatz with perturbation of the form:

$\begin{gathered}U_{n}=[A_{n}+\varepsilon(a_{n}e^{-i\lambda
t}+b_{n}e^{i\lambda^{*}t})]e^{-i\mu t},\\\
V_{n}=[B_{n}+\varepsilon(c_{n}e^{-i\lambda t}+d_{n}e^{i\lambda^{*}t})]e^{-i\mu
t},\end{gathered}$ (24)

where $\lambda=\lambda_{r}+i\lambda_{i}$ denotes the linearization eigenvalue
and $\varepsilon\ll{1}$. Substituting Eq. (24) in Eqs. (27), (28) and taking
only the terms with $O\left(\varepsilon\right)$ (linear terms of
$\varepsilon$), one obtains an eigenvalue problem for $\lambda$ that can be
solved numerically. Ideally, the perturbation part in Eq. (24) will remain
small (stable) if all imaginary eigenvalues $\lambda_{i}$ are zeros or
negligibly small, for gain $G=\max\left(\left|\lambda_{i}\right|\right)\simeq
0$.

Typically, the results of the linear stability analysis of SOC-compactons are
further checked by direct integrations of both the averaged and original
equations Eq. (7). Since they are found in agreement, however, results in the
following will be reported only for the original equations.

###  V.1 One-site SOC-compactons

As discussed above, the existence condition for one-site SOC-compactons, apart
possible shifts of the chemical potential by $\Omega$, are the same as in the
absence of SOC. This, however, does not imply that there is no effects of the
SOC on compactons because both $\sigma$ and $\Omega$ can strongly influence
their stability.

We show this by comparing results of the linear stability analysis in the case
of absence of SOC , i.e. $\sigma=0,\Omega=0$, (see Fig. 1) with the ones
obtained in the presence of SOC (see Figs.2 and 3). From Fig. 1 we see that
the linear spectrum has a gain $G$ that is practically zero for all
$\varGamma\in\left[0,1\right]$, meaning that the solutions are stable, a fact
that is already known from previous results Abdullaev3 . By increasing
$\sigma$ away from zero but retaining $\Omega=0$, we find that the solution
becomes unstable (see panels a, b of Fig. 2) for the case $\sigma=1,\Omega=0$.
This instability, however, is physically not very significative because the
SOC implementation usually requires both $\sigma,\Omega$ to be nonzero. By
increasing also $|\Omega|$ away from zero, the stability can be restored
almost for the whole range in $\varGamma$ of the zero SOC case, as one can see
from panels (c, d) of Fig. 2, for the case $\sigma=1,\Omega=-2$. Similar
results are obtained for different choices of the system parameters and for
different compacton amplitudes. In general, we have that stable one-site SOC-
compactons can exist in wide regions of the parameter space.

We have also investigated the effects of the nonlinearity management
parameters $\gamma^{(0)}$, $\gamma^{(1)}$, on the stability of one-site SOC-
compactons. Note from Eq. (18) that the compacton amplitude is inversely
proportional to the value of $\gamma^{(1)}$, despite $\gamma^{(1)}$ alone does
not affect the stability. Remarkably, the solutions become significantly
stable when $\left|\gamma^{(0)}\right|$ is much larger than $\gamma^{(1)}$ as
demonstrated in Fig. 3(a, b) and, oppositely, the solutions become unstable
when $\left|\gamma^{(0)}\right|$ is much lower than $\gamma^{(1)}$, as
portrayed in Fig. 3 (c, d). The above linear stability results are further
confirmed by nonlinear stability analysis achieved via numerical time-
integrations of the nonlinear system in Eq. (7), as one can see from Fig. 4.

###  V.2 Two-site SOC-compactons

For multi-site compactons, we must necessarily to have the condition
$\Omega\neq 0$, i.e. equal chemical potentials for the two BEC components and
this implies equal intra-species interactions. Let us consider first the case
of an in-phase SOC-compacton with amplitudes
$A_{0}=B_{1}=a+ib,A_{1}=B_{0}=a-ib,$ with $a=\sqrt{\xi_{1}/2\alpha}$,
$b=\sqrt{\xi_{1}/\alpha-a^{2}}$. We fix parameters as:
$\gamma_{1}=\gamma_{2}=1$, $\gamma^{(0)}=4$, $\gamma^{(1)}=1$, $\omega=1$,
$\Omega=4$, $\sigma=1$. Note that all the nonlinear parameters here are
positive, i.e interactions are repulsive. In panels a,b of Fig. 5 we show the
real and imaginary parts versus $n$ of the first and second component
$A_{n},B_{n}$ , respectively, of a two-site in-phase compactons, while in the
bottom panels (c,d) their stability analysis is reported as a function of the
parameter $\Gamma$. From panel d) it is clear that for $\Gamma$ less than
$\approx 0.6$ the gain $G$ is positive and the solution is unstable, while for
all $\Gamma\geq 0.6$ the gain reduces to zero and the solution becomes stable.
The panel c of the figure just shows the stability in terms of the linear
eigenvalues at $\Gamma=1$ from which we see that they are all real numbers.

Similarly, for an out-of-phase two-site compacton we fix the amplitudes as:
$A_{0}=-B_{1}=a+ib$, $B_{0}=-A_{1}=a-ib$, with the same $a$, $b$ of the
previous case. Parameters are fixed as $\gamma_{1}=\gamma_{2}=-1$,
$\gamma^{(0)}=-4$, $\gamma^{(1)}=1$, $\omega=1$, $\Omega=-4$, $\sigma=1$.
Results, are reported in Fig. 6, where panels (a, b) depict the solutions of
typical out-of-phase profiles and panels (c,d) show the results from the
stability analysis. Notice that despite the type of interactions is changed
from repulsive to attractive, the range of stability is the same as for the
in-phase case. Thus, independently on the sign of the interaction, in the
interval $0.6\leq\varGamma\leq 1$ both the in-phase and the out-of phase
compactons appear to be stable. This is quite interesting because it shows
SOC-compacton bistability in contrast with usual inter-site breathers (i.e.
localized out-of-phase solutions with tails) which are known to be unstable in
conventional (i.e. in absence of SNM) SOC-BEC mixtures Belicev .

The stability of the solution also depends on the ratio
$\eta=\gamma^{(0)}<\gamma^{(1)}$ between the dc and ac strengths of the inter-
site interaction, with the two-site SOC-compacton becoming unstable for
$\eta<1$. This is shown in Fig.7 for the case of the same two-site in-phase
compacton depicted in Fig. 5 but with $\eta=0.5$.

Finally, we find the results of the linear stability analysis to be in very
good agreement with those obtained from direct numerical integrations of Eq.
(7). An example of this is shown in Fig.8 where the stable and unstable time
evolution of the two-site in-phase compactons of Fig.s 5 and 7 at $\Gamma=1$,
are reported.

Figure 9: Three-site compacton for case $\gamma_{1}=\gamma_{2}=1$,
$\gamma^{(0)}=10$, $\gamma^{(1)}=1$, $\omega=1$, $\Omega=4$ and $\sigma=1$.
Figure 10: Three-site compacton for case $\gamma_{1}=\gamma_{2}=1$,
$\gamma^{(0)}=10$, $\gamma^{(1)}=6.0$, $\omega=1$, $\Omega=4$ and $\sigma=1$.
Figure 11: Space-time evolution of the three-site compactons obtained from
numerical integrations of Eq. (7) with $\epsilon=0.01$, (a)
$\left|u_{n}\right|$ and (b) $\left|v_{n}\right|$ for the stable case at
$\Gamma=1$ in Fig.9; (c) $\left|u_{n}\right|$ and (d) $\left|v_{n}\right|$ for
unstable case at $\Gamma=1$ in Fig.10.

###  V.3 Three-site SOC-compactons

Three-site SOC-compacton solutions are obtained from Eq. (17) with $s=2$ by
taking $\mu_{1}=\mu_{2}=\mu$ and assuming the following ansatz for the
solution: $A_{n_{0}}=a+ib$, $B_{n_{0}}=a-ib$, $A_{n_{0}\pm 1}=c+id$ and
$B_{n_{0}\pm 1}=c-id$, with $a,b,c,d$ real numbers. Direct substitution of the
ansatz into Eq. (16) provides ten equations which are reduced to six (for
instance only those for sites $n_{0}-2,n_{0}-1,n_{0}$) thanks to the symmetry
of the solution around the central site $n_{0}$. From the equations for the
site $n_{0}-2$ one obtains the zero tunneling conditions for $A_{n_{0}\pm
2},B_{n_{0}\pm 2}$ at the edges of the compacton in terms of roots of the
Bessel function as in Eq. (18). Substituting these conditions into the
equation relative to sites $n_{0}-1$ one obtains two equations that can be
combined into the following expression for the chemical potential:

$\displaystyle\mu$ $\displaystyle=$
$\displaystyle-\frac{1}{Re\left(w_{2}\right)}\left[\alpha\varGamma\left(Re\left(w_{2}\bar{z}_{1}w_{1}\right)\right.\right.$
$\displaystyle\left.+Re\left(w_{2}z_{1}\bar{w}_{1}\right)\right)J_{1}\left(\alpha\left(\left|z_{2}\right|^{2}-\left|w_{2}\right|^{2}\right)\right)$
$\displaystyle+\varGamma
Re\left(z_{2}\right)J_{0}\left(\alpha\left(\left|z_{1}\right|^{2}-\left|w_{1}\right|^{2}\right)\right)$
$\displaystyle-\Omega
Re\left(w_{1}\right)-\gamma\left|w_{1}\right|^{2}Re\left(w_{2}\right)$
$\displaystyle-\gamma^{\left(0\right)}\left|w_{1}\right|^{2}Re\left(w_{2}\right)$
$\displaystyle-\alpha\sigma
J_{1}\left(\alpha\left(\left|z_{2}\right|^{2}-\left|w_{2}\right|^{2}\right)\right)\cdot$
$\displaystyle\left(Im\left(w_{2}\bar{z}_{1}w_{1}\right)-Im\left(w_{2}z_{1}\bar{w}_{1}\right)\right)$
$\displaystyle-\sigma
J_{0}\left(\alpha\left(\left|z_{1}\right|^{2}-\left|w_{1}\right|^{2}\right)\right)\left.Im\left(z_{2}\right)\right].$

Here we used $\gamma_{1}=\gamma_{2}=\gamma$ and denoted
$z_{1}=a+ib,z_{2}=a-ib,w_{1}=c+id,w_{2}=c-id$. Lastly, from the equations for
site $n_{0}$ one gets:

$\displaystyle\mu z_{j+1}+2\alpha\varGamma
z_{j+1}\left(\bar{w}_{2-j}z_{2-j}+w_{2-j}\bar{z}_{2-j}\right)\cdot$
$\displaystyle
J_{1}\left(\alpha\left(\left|w_{j+1}\right|^{2}-\left|z_{j+1}\right|^{2}\right)\right)$
$\displaystyle+\alpha\varOmega\left(z_{2-j}\left|z_{j+1}\right|^{2}+z_{2-j}z_{j+1}^{2}\right)\cdot$
$\displaystyle
J_{1}\left(\alpha\left(\left|z_{j+1}\right|^{2}-\left|z_{2-j}\right|^{2}\right)\right)$
$\displaystyle+2\varGamma
w_{j+1}J_{0}\left(\alpha\left(\left|w_{2-j}\right|^{2}-\left|z_{2-j}\right|^{2}\right)\right)$
$\displaystyle-\varOmega
z_{2-j}J_{0}\left(\alpha\left(\left|z_{j+1}\right|^{2}-\left|z_{2-j}\right|^{2}\right)\right)$
$\displaystyle-\gamma\left|z_{j+1}\right|^{2}z_{j+1}-\gamma^{\left(0\right)}\left|z_{2-j}\right|^{2}z_{j+1}$
$\displaystyle=$ $\displaystyle 0,$

with $j=0,1$. These nonlinear equations together with the previous ones should
be solved numerically for $a,b,c,d$ giving the the compacton amplitudes of the
two components at sites $n_{0},n_{0}\pm 1$.

Stability patterns for three-site compactons are in general similar to the
ones previously investigated but for $\sigma=1$ or larger it is more difficult
to obtain stable solutions unless $\varGamma\geq 1$, as shown in Fig. 9.
Another distinctive feature in this case is the absence of stable solutions
$\gamma^{(0)}<\gamma^{(1)}$, and even for $\gamma^{(0)}>\gamma^{(1)}$, if the
difference is not sufficiently large, the solution might become unstable as
depicted in Fig. 10. Lastly, in the top and bottom panels of Fig. 11 we report
the time evolution obtained from numerical integrations of Eq. (7) by initial
conditions the solutions at $\Gamma=1$ depicted in Fig. 9 and Fig. 10, for
which the linear stability analysis predicts stability and instability,
respectively. As one can see from panels (a,b) and (c,d) of Fig. 11 these
predictions are fully confirmed.

##  VI Conclusions

In this paper we have investigated the existence and stability properties of
compactons matter waves excitations of binary BEC mixtures with spin-orbit
coupling trapped in an OL, subjected to SNM. We considered for simplicity an
inter-SOC system (i.e. with an off-diagonal Rabi term) and assumed both the OL
to be deep enough to justify the tight-binding approximation and the strength
and the frequency of the modulation to be large enough to use the averaging
method.

Within this model, we derived effective averaged equations of motion for the
matter wave complex amplitudes in the form of a two coupled discrete NLS and
demonstrated that the SNM introduces a local density imbalance dependence not
only in the tunneling parameter $\Gamma$ but also in the SOC parameters
$\sigma$ and $\Omega$. This density dependence of the SOC parameters persists
in the system independently on the SOC-compacton existence.

The dependence of both $\sigma$ and $\Omega$ on the local density imbalance,
however, give rise to conditions for stable SOC-compactons existence that are
more restrictive than the ones obtained in the absence of SOC. In general, the
stationarity condition requires the equality of the chemical potentials of the
two components, a fact that can be satisfied when the number of atoms of the
two species are equal as well as their intra-species nonlinearities. We have
also investigated the stability of SOC-compactons versus the system
parameters, i.e. the SOC strength $\sigma$, the Rabi frequency $\Omega$, the
coupling or hopping constant $\varGamma$ and the ratio between the nonlinear
coefficients $\gamma^{(0)},\gamma^{(1)}$. This has been done both using a
linear stability analysis of the averaged equations and by direct numerical
time integrations of the original nonlinear PDE equations. Numerical
simulations show that by keeping the other parameters fixed, an enlargement of
the stability region of SOC-compactons can be achieved by decreasing $\sigma$
or by increasing $\Omega$. Stability seems also more easily obtainable for
$\left|\gamma^{(0)}\right|>\left|\gamma^{(1)}\right|$ and large values of
$\varGamma$.

From this, we conclude that while on one side the SOC restricts the parameter
range for stationary SOC-compacton existence, on the other side it gives a
more stringent signature of their occurrence. Indeed, they should appear when
the intra-species interactions and the number of atoms in the two components
are perfectly balanced (or close to being balanced for metastable cases). This
suggests SOC-compactons as possible tools for indirect measurements of the
number of atoms, and/or the intra-species interactions, in experiments.

## ACKNOWLEDGMENTS

M.S.A. H. and L. A. T. acknowledge support from Ministry of Higher Education,
Malaysia under research grant no. FRGS 16-013-0512. MS wishes to acknowledge
the International Islamic University of Malaysia in Kuantan for partial
support and for the hospitality received during the period this work was
started.

##  VII APPENDIX: Averaged equations and Hamiltonian structure

Below we provide the derivation of the averaged system Eq. (13). By
substituting Eq. (10) into Eq. (7) we get:

$\displaystyle iU_{n,t}=$ $\displaystyle-i\varGamma
U_{n}(\langle\Lambda\bar{X}_{1}^{+}\rangle V_{n}-\langle\Lambda
X_{1}^{+}\rangle V_{n}^{*})$ $\displaystyle-\sigma
U_{n}(\langle\Lambda\bar{X}_{1}^{-}\rangle V_{n}+\langle\Lambda
X_{1}^{-}\rangle V_{n}^{*})$
$\displaystyle+i\Omega(\langle\Lambda\bar{X}_{0}^{+}\rangle
V_{n}|U_{n}|^{2}-\langle\Lambda X_{0}^{-}\rangle V_{n}^{*}U_{n}^{2})$
$\displaystyle-\varGamma\langle X_{2}^{+}\rangle+i\sigma\langle
X_{2}^{-}\rangle+\Omega\langle X_{0}^{-}\rangle V_{n}$
$\displaystyle+\gamma_{1}|U_{n}|^{2}U_{n}+\gamma^{(0)}|V_{n}|^{2}U_{n},$ (25)
$\displaystyle iV_{n,t}=$ $\displaystyle-i\varGamma
V_{n}(\langle\Lambda\bar{X}_{2}^{+}\rangle U_{n}-\langle\Lambda
X_{2}^{+}\rangle U_{n}^{*})$ $\displaystyle+\sigma
V_{n}(\langle\Lambda\bar{X}_{2}^{-}\rangle U_{n}+\langle\Lambda
X_{2}^{-}\rangle U_{n}^{*})$
$\displaystyle+i\Omega(\langle\Lambda\bar{X}_{0}^{+}\rangle
U_{n}|V_{n}|^{2}-\langle\Lambda X_{0}^{-}\rangle U_{n}^{*}V_{n}^{2})$
$\displaystyle-\varGamma\langle X_{1}^{+}\rangle-i\sigma\langle
X_{1}^{-}\rangle+\Omega\langle X_{0}^{+}\rangle U_{n}$
$\displaystyle+\gamma_{2}|V_{n}|^{2}V_{n}+\gamma^{(0)}|U_{n}|^{2}V_{n}.$ (26)

where

$\displaystyle X_{2}^{\pm}$
$\displaystyle=U_{n+1}e^{i\Lambda(|{V_{n+1}}|^{2}-|{V_{n}}|^{2})}\pm
U_{n-1}e^{i\Lambda(|{V_{n-1}}|^{2}-|{V_{n}}|^{2})},$ $\displaystyle
X_{1}^{\pm}$
$\displaystyle=V_{n+1}e^{i\Lambda(|{U_{n+1}}|^{2}-|{U_{n}}|^{2})}\pm
V_{n-1}e^{i\Lambda(|{U_{n-1}}|^{2}-|{U_{n}}|^{2})},$ $\displaystyle
X_{0}^{\pm}$ $\displaystyle=e^{\pm i\Lambda(|{U_{n}}|^{2}-|{V_{n}}|^{2})}.$

and $\langle\cdot\rangle$ is the rapid modulated term to be averaged. The
averaged terms can be evaluated as:

$\displaystyle\langle\Lambda X_{1}^{\pm}\rangle=V_{n+1}\langle\Lambda
e^{i\Lambda\theta_{1}^{+}}\rangle\pm V_{n-1}\langle\Lambda
e^{i\Lambda\theta_{1}^{-}}\rangle,$ $\displaystyle\langle\Lambda
X_{2}^{\pm}\rangle=U_{n+1}\langle\Lambda e^{i\Lambda\theta_{2}^{+}}\rangle\pm
U_{n-1}\langle\Lambda e^{i\Lambda\theta_{2}^{-}}\rangle,$
$\displaystyle\langle X_{1}^{\pm}\rangle=V_{n+1}\langle
e^{i\Lambda\theta_{1}^{+}}\rangle\pm V_{n-1}\langle
e^{i\Lambda\theta_{1}^{-}}\rangle,$ $\displaystyle\langle
X_{2}^{\pm}\rangle=U_{n+1}\langle e^{i\Lambda\theta_{2}^{+}}\rangle\pm
U_{n-1}\langle e^{i\Lambda\theta_{2}^{-}}\rangle,$
$\displaystyle\langle\Lambda X_{0}^{\pm}\rangle=\langle\Lambda
e^{\pm{i}\Lambda\theta_{0}}\rangle,\quad\langle X_{0}^{\pm}\rangle=\langle
e^{\pm{i}\Lambda\theta_{0}}\rangle,$

with $\theta_{1}^{\pm}=|U_{n\pm 1}|^{2}-|U_{n}|^{2}$,
$\theta_{2}^{\pm}=|V_{n\pm 1}|^{2}-|V_{n}|^{2}$ and
$\theta_{0}=|U_{n}|^{2}-|V_{n}|^{2}$ and furthermore

$\displaystyle\langle\Lambda e^{\pm{i}\Lambda\theta_{j}^{\pm}}\rangle$
$\displaystyle=\frac{\omega}{2\pi}\int_{0}^{2\pi/\omega}\frac{\gamma^{(1)}}{\omega}\sin(\omega\tau)e^{\pm{i}\frac{\gamma^{(1)}}{\omega}\theta_{j}^{\pm}\sin(\omega\tau)}\
d\tau$ $\displaystyle=\pm{i}\alpha
J_{1}(\alpha\theta_{j}^{\pm}),\;\;\;\;\;\langle
e^{\pm{i}\Lambda\theta_{j}^{\pm}}\rangle=J_{0}(\alpha\theta_{j}^{\pm}),$
$\displaystyle\langle\Lambda e^{\pm{i}\Lambda\theta_{0}}\rangle$
$\displaystyle=\pm i\alpha{J_{1}}(\alpha\theta_{0}),\;\;\;\;\;\langle
e^{\pm{i}\Lambda\theta_{0}}\rangle=J_{1}(\alpha\theta_{0}),$

where $\alpha=\gamma^{(1)}/\omega$ and $J_{0}$, $J_{1}$ are the Bessel
functions of order 0 and 1 respectively. Substituting the averages into
Eqs.(25) and (26) yields:

$\displaystyle i$ $\displaystyle U_{n,t}=-\varGamma\Big{\\{}\alpha
U_{n}\Big{[}(V_{n+1}^{*}V_{n}+V_{n+1}V_{n}^{*})J_{1}(\alpha\theta_{1}^{+})+$
$\displaystyle(V_{n-1}^{*}V_{n}+V_{n-1}V_{n}^{*})J_{1}(\alpha\theta_{1}^{-})\Big{]}+U_{n+1}J_{0}(\alpha\theta_{2}^{+})+$
$\displaystyle
U_{n-1}J_{0}(\alpha\theta_{2}^{-})\Big{\\}}+\Big{\\{}\gamma_{1}|U_{n}|^{2}+\gamma^{(0)}|V_{n}|^{2}\Big{\\}}U_{n}+$
(27) $\displaystyle i\sigma\Big{\\{}\alpha
U_{n}\Big{[}(V_{n+1}^{*}V_{n}-V_{n+1}V_{n}^{*})J_{1}(\alpha\theta_{1}^{+})-(V_{n-1}^{*}V_{n}-$
$\displaystyle
V_{n-1}V_{n}^{*})J_{1}(\alpha\theta_{1}^{-})\Big{]}+U_{n+1}J_{0}(\alpha\theta_{2}^{+})-U_{n-1}J_{0}(\alpha\theta_{2}^{-})\Big{\\}}$
$\displaystyle-\Omega\Big{\\{}\alpha
J_{1}(\alpha\theta_{0})(V_{n}|U_{n}|^{2}+V_{n}^{*}U_{n}^{2})-V_{n}J_{0}(\alpha\theta_{0})\Big{\\}},$
$\displaystyle i$ $\displaystyle V_{n,t}=-\varGamma\Big{\\{}\alpha
V_{n}\Big{[}(U_{n+1}^{*}U_{n}+U_{n+1}U_{n}^{*})J_{1}(\alpha\theta_{2}^{+})+$
$\displaystyle(U_{n-1}^{*}U_{n}+U_{n-1}U_{n}^{*})J_{1}(\alpha\theta_{2}^{-})\Big{]}+V_{n+1}J_{0}(\alpha\theta_{1}^{+})+$
$\displaystyle
V_{n-1}J_{0}(\alpha\theta_{1}^{-})\Big{\\}}+\Big{\\{}\gamma_{2}|V_{n}|^{2}+\gamma^{(0)}|U_{n}|^{2}\Big{\\}}V_{n}-$
(28) $\displaystyle i\sigma\Big{\\{}\alpha
V_{n}\Big{[}(U_{n+1}^{*}U_{n}-U_{n+1}U_{n}^{*})J_{1}(\alpha\theta_{2}^{+})-(U_{n-1}^{*}U_{n}-$
$\displaystyle
U_{n-1}U_{n}^{*})J_{1}(\alpha\theta_{2}^{-})\Big{]}+V_{n+1}J_{0}(\alpha\theta_{1}^{+})-V_{n-1}J_{0}(\alpha\theta_{1}^{-})\Big{\\}}$
$\displaystyle+\Omega\Big{\\{}\alpha
J_{1}(\alpha\theta_{0})(U_{n}|V_{n}|^{2}+U_{n}^{*}V_{n}^{2})+U_{n}J_{0}(\alpha\theta_{0})\Big{\\}}.$

Furthermore, the pdes Eqs. (27) and (28) can be written in terms of
$iU_{n,t}=\delta H_{av}/\delta U_{n}^{*}$ and $iV_{n,t}=\delta H_{av}/\delta
V_{n}^{*}$ respectively, where the averaged Hamiltonian is:

$\displaystyle H_{av}=$ $\displaystyle\sum_{n}\Big{[}-\varGamma
J_{0}(\alpha\theta_{2}^{+})(U_{n+1}U_{n}^{*}+U_{n+1}^{*}U_{n})$ (29)
$\displaystyle-\varGamma
J_{0}(\alpha\theta_{1}^{+})(V_{n+1}V_{n}^{*}+V_{n+1}^{*}V_{n})$
$\displaystyle+i\sigma
J_{0}(\alpha\theta_{2}^{+})(U_{n+1}U_{n}^{*}-U_{n+1}^{*}U_{n})$
$\displaystyle-i\sigma
J_{0}(\alpha\theta_{1}^{+})(V_{n+1}V_{n}^{*}-V_{n+1}^{*}V_{n})$
$\displaystyle+\Omega J_{0}(\alpha\theta_{0})(U_{n}V_{n}^{*}+U_{n}^{*}V_{n})$
$\displaystyle+\frac{1}{2}(\gamma_{1}|U_{n}|^{4}+\gamma_{2}|V_{n}|^{2})+\gamma^{(0)}|U_{n}|^{2}|V_{n}|^{2}\Big{]}.$

By comparing the averaged Hamiltonian in Eq. (29) with the original
Hamiltonian Eq. (8), their terms coincide if one rescales the tunneling or
hopping constant $\varGamma$, the SOC parameter $\sigma$ and the Rabi
frequency $\Omega$ according to Eq. (12). It should be noted here that the
averaged equations only valid for $t\leq 1/\epsilon$ with accuracy of
$\mathfrak{\mathcal{O}}\left(\epsilon\right)$ Sanders .

## References

  * (1) M. Bukov, L. D’Alessio, A. Polkovnikov, Advances in Physics, 64, 139226 (2015).
  * (2) Y.J. Lin, K. Jiménez-García and I. B. Spielman, Nature, 471, 83–86, (2010).
  * (3) V. Galitski and I. B. Spielman, Nature 494, 49–54 (2013).
  * (4) F. Kh. Abdullaev, P. G. Kevrekidis, and M. Salerno, Phys. Rev.Lett. 105, 113901 (2010).
  * (5) S. Greschner, G. Sun, D. Poletti, and L. Santos, Phys. Rev. Lett. 113, 215303 (2014).
  * (6) A. Rapp, X. Deng, and L. Santos, Phys. Rev. Lett. 109, 203005 (2012).
  * (7) J. Gong, L. Morales-Molina, and P. Hänggi, Phys. Rev. Lett. 103, 133002 (2009).
  * (8) F Meinert, MJ Mark, K Lauber, AJ Daley, HC Nägerl, Phys. Rev. Lett. 116, 205301 (2020).
  * (9) F. Kh. Abdullaev, M. S. A. Hadi, M. Salerno, and B. Umarov, Phys. Rev. A 90, 063637 (2014).
  * (10) F. Kh. Abdullaev, M. S. A. Hadi, M. Salerno, and B. Umarov, J. Phys. B 50, 16 (2017).
  * (11) J. D’Ambroise, M. Salerno, P. G. Kevrekidis, and F. Kh. Abdullaev, Phys. Rev. A 92, 053621 (2015).
  * (12) P.P. Belicev, G. Gligoric, J. Petrovic, A. Maluckov, L. Hadievski, B.A. Malomed, J. Phys. B 48, 065301 (2015).
  * (13) M. Salerno and F. Kh. Abdullaev, Phys. Lett. A 379, 2252 (2015).
  * (14) M. Salerno, F. Kh. Abdullaev, A. Gammal, and Lauro Tomio, Phys. Rev. A 94, 043602 (2016).
  * (15) P. Xu, T. Deng, W. Zheng, H. Zhai, Phys Rev. A 103, L061302 (2021).
  * (16) G.L. Alfimov, P.G. Kevrekidis, V.V. Konotop, M. Salerno, Phys. Rev. E 66 (2002) 046608.
  * (17) S. Sakaguchi, and B. A. Malomed, Symmetry, 11, 388 (2019).
  * (18) J.A. Sanders, F. Verhulst, and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems (Springer Science+Business Media, LLC, 2007).
  * (19) In the inter-SOC case $\Omega$ is a diagonal Zeeman term while $\sigma$ term is off-diagonal. In this case the terms proportional to $\sigma$ in the averaged equations, will have time dependent factors similar to the ones in (III) but with the roles of $\sigma$ and $\Omega$ interchanged.


# Probing Electroweak Dark Matter at 14 TeV LHC

Shuai Xu1<EMAIL_ADDRESS>Sibo Zheng1,2<EMAIL_ADDRESS>1Department of Physics, Chongqing University, Chongqing 401331, China
2Department of Physics, Harvard University, Cambridge, MA 02138, USA

###### Abstract

Well-motivated electroweak dark matter is often hosted by an extended
electroweak sector which also contains new lepton pairs with masses near the
weak scale. In this paper, we explore such electroweak dark matter via
combining dark matter direct detections and high-luminosity LHC probes of new
lepton pairs. Using $Z$\- and $W$-associated electroweak processes with two or
three lepton final states, we show that dependent on the overall coupling
constant, dark matter mass up to $170-210$ GeV can be excluded at $2\sigma$
level and up to $175-205$ GeV can be discovered at $5\sigma$ level at the 14
TeV LHC with integrated luminosities 300 fb-1 and 3000 fb-1, respectively.

##  I Introduction

Dark Matter (DM) with mass of a few hundred GeVs, which is known as the weakly
interacting massive particle (WIMP), is one of leading motivations to the
weak-scale new physics beyond standard model (SM). Any such new physics, if
exists, should play an important role at the LHC. When a WIMP-like DM is in
the reach of DM direct-detection experiments, the LHC is useful for a double
check. What is even more interesting is that if the WIMP-like DM is beyond the
reach of DM direct detections, the LHC serves as an alternative discovery
platform. This paper is devoted to address the question - what is the
potential of high-luminosity (HL) 1310.8361 ; 1812.07831 ; 1902.10229  LHC
probe of general WIMP-like DM.

WIMP-like DM can be hosted in a variety of weak-scale new physics. For
instance, it can be identified as the neutral fermion of the fourth-generation
lepton model 9903387 ; 9804359 , where there are mixing effects between the
new leptons and SM leptons, the lightest neutralino of the minimal
supersymmetric standard model (MSSM) 9709356 , of next-to-minimal
supersymmetric standard model (NMSSM) 0910.1785 , or the lightest neutral
fermion of singlet-doublet (SD) 0510064 ; 0705.4493 ; 1109.2604  and vector-
like (VL) lepton model 0910.2732 ; 1706.01071 ; 1711.05362 ; 1904.10145 ,
where there are no direct mixing with SM leptons. In the various contexts as
above, there are a diversity of interactions and model parameters. As a
result, it seems difficult to work out universal predictions on such WIMP-like
DM. To address this question, the first task is to find a framework viable for
most of sophisticated WIMP-like DM models.

The structure of this paper is organized as follows. In Sec.II, we will employ
a framework which can describe three benchmark WIMP-like DM models above,
where model parameters will be introduced. In Sec.III, we uncover the DM
parameter space which satisfies the DM relic density and survives in the
latest DM direction-detection limits. Sec. IV is devoted to explore the
exclusion as well as discovery potentials on the DM mass ranges in the light
of LHC probes of lepton pairs in the $Z$\- and $W$-associated electroweak
processes such as $pp\to\ell^{\pm}\ell^{\mp}+E_{T}^{\rm miss}$,
$pp\to\ell^{\pm}\ell^{\mp}jj+E_{T}^{\rm miss}$ and
$pp\to\ell^{\pm}\ell^{\mp}\ell^{\pm}+E_{T}^{\rm miss}$ at the HL-LHC, where
$\ell$, $j$ and $E_{T}^{\rm miss}$ refer to lepton, jet and missing energy,
respectively, and the samples used for event simulations are directly
extracted from the parameter space of electroweak DM. Finally, we conclude in
Sec.V.

##  II Extended Electroweak Sector

In various contexts of weak-scale new physics, the SM electroweak sector is
extended by a couple of electroweak doublets $E^{\pm}$

$\displaystyle{}E^{+}=\left(\begin{array}[]{c}\eta^{+}\\\ \eta^{0}\\\
\end{array}\right),~{}~{}E^{-}=\left(\begin{array}[]{c}\tilde{\eta}^{0}\\\
\eta^{-}\\\ \end{array}\right),$ (5)

for which the effective Lagrangian at the weak scale can be described as

$\displaystyle{}\mathcal{L}\supset\frac{i}{2}\overline{E^{+}}\not{D}E^{+}+\frac{i}{2}\overline{E^{-}}\not{D}E^{-}-m_{E}E^{+}E^{-},$
(6)

with $m_{E}$ a vectorlike (VL) mass. If there are no other mass sources for
the doublets, the charged fermion $\chi^{\pm}$ composed of
$(\eta^{+},\overline{\eta^{-}})$ and the neutral fermion $\chi^{0}$ composed
of $(\eta^{0},\tilde{\eta}^{0})$ will have nearly degenerate masses
$m_{\chi^{\pm}}\simeq m_{\chi^{0}}$ up to small radiative correction 9804359 .
This mass degeneracy kinematically suppresses the decay
$\chi^{\pm}\rightarrow\chi^{0}W^{\pm}$, and reduces the signal of the lepton
pair $\chi^{+}\chi^{-}$.

The mass degeneracy disappears whenever there are moderate or large mixing
effects. The mixing effects can be classified into two different types. In the
first type, $E^{\pm}$ directly mix with SM leptons as in the fourth-generation
lepton models. In the second type, $E^{\pm}$ mix with some new fermion singlet
$N$ as in the MSSM, NMSSM, SD and VL lepton models, which gives rise to
sufficient splitting between the charged and neutral fermion masses. The scope
of this paper is restricted to the large mass splitting driven by the
electroweak singlet fermion.

In this case, the Lagrangian $\mathcal{L}$ in Eq.(6) is modified by

$\displaystyle{}\delta\mathcal{L}\supset\frac{i}{2}N\not{\partial}N-\frac{m_{N}}{2}N^{2}-(y_{1}N\overline{E^{+}}+y_{2}NE^{-})H+\text{h.c},$

where $m_{N}$ is the singlet mass, and $H$ refers to the SM Higgs doublet with
a vacuum expectation value (vev) $\upsilon=174$ GeV. With the new Yukawa
interactions in Eq.(II), the neutral fermions are now composed of
$(N,\eta^{0},\tilde{\eta}^{0})$, whose mass matrix $M_{\chi^{0}}$ reads as,

$\displaystyle{}\mathcal{M}_{\chi^{0}}=\left(\begin{array}[]{ccc}m_{N}&y_{1}\upsilon&y_{2}\upsilon\\\
&0&m_{E}\\\ &*&0\\\ \end{array}\right).$ (11)

The mixing effects in Eq.(11) are responsible for the mass splitting between
the three neutral fermion mass $m_{\chi^{0}_{i}}$ ($i=1-3$) and charged
fermion mass $m_{\chi^{\pm}}$.

Under the basis of mass eigenvalues, the Lagrangian is rewritten as

$\displaystyle{}\mathcal{L}\supset$ $\displaystyle-$ $\displaystyle
h\overline{\chi^{0}_{i}}\left(c_{h,ij}P_{L}+c^{*}_{h,ij}P_{R}\right)\chi^{0}_{j}$
(12) $\displaystyle-$ $\displaystyle
Z_{\mu}\overline{\chi^{0}_{i}}\gamma^{\mu}\left(c_{Z,ij}P_{L}-c^{*}_{Z,ij}P_{R}\right)\chi^{0}_{j}$
$\displaystyle-$
$\displaystyle\left[\frac{g}{\sqrt{2}}W_{\mu}^{-}\overline{\chi^{0}_{i}}\gamma^{\mu}(N_{i3}P_{L}-N_{i2}P_{R})\chi^{+}+\text{h.c}\right]$

with

$\displaystyle{}c_{Z,ij}$ $\displaystyle=$
$\displaystyle\frac{g}{4c_{W}}\left(U_{i3}U^{*}_{j3}-U_{i2}U^{*}_{j2}\right),$
$\displaystyle c_{h,ij}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(y_{2}U_{i3}U^{*}_{j1}+y_{1}U_{i2}U^{*}_{j1}\right),$
(13)

where $c_{W}$ and $g$ denote the weak mixing angle and weak gauge coupling
constant respectively, and the unitary matrix $U$ is introduced to diagonalize
$\mathcal{M}_{\chi^{0}}$. We have written fermions in the 4-component
notation, with $\chi^{+}=(\eta^{+},\overline{\eta^{-}})$ and
$\chi^{0}_{i}=(\chi^{0}_{i},\overline{\chi^{0}_{i}})$. In this setting, the
lepton pair is a couple of $\chi^{\pm}$ and $\chi^{0}_{i}$ ($i=2,3$), with
their couplings to the SM sector given in Eq.(12).

##  III Parameter Space of Electroweak DM

With the lightest neutral state $\chi^{0}_{1}$ identified as the thermal
electroweak DM, we can work out the parameter space of DM relic density from
Eq.(11) and Eq.(12). In the framework above, there are four parameters
$y_{1}$, $y_{2}$, $m_{N}$ and $m_{E}$. Since electroweak DM favors coupling
$y=\sqrt{y^{2}_{1}+y^{2}_{2}}$ of order weak interaction $\sim 0.3-1.0$ and DM
mass of order weak scale $\sim 100-1000$ GeV, we consider the following
parameter regions,

$\displaystyle{}0.3\leq$ $\displaystyle y$ $\displaystyle\leq 1.2,$
$\displaystyle 100~{}\rm{GeV}\leq$ $\displaystyle m_{E}$ $\displaystyle\leq
1500~{}\rm{GeV},$ (14) $\displaystyle 100~{}\rm{GeV}\leq$ $\displaystyle
m_{N}$ $\displaystyle\leq 1500~{}\rm{GeV},$

over which we perform random scans, with $\rm{sign}(y_{2}/y_{1})$ either
positive or negative.

$\rm{model}$ |  $\rm{fields}$ | $y$ | $\rm{sign}(y_{2}/y_{1})$
---|---|---|---
$\rm{MSSM}$ |  $\tilde{B}^{0}$, $\tilde{h}_{u}^{0}$, $\tilde{h}_{d}^{0}$ | $0.35$ | $-$
|  $\tilde{W}^{0}$, $\tilde{h}_{u}^{0}$, $\tilde{h}_{d}^{0}$ | $0.63$ | $-$
$\rm{SD}$ |  $N$, $E^{0}$, $\tilde{E}^{0}$ | $0.7-1.0$ | $\pm$
$\rm{VL}$ |  $N$, $E^{0}$, $\tilde{E}^{0}$ | $1.0-1.2$ | $\pm$
Table 1: The matter content, the magnitude of $y$, and
$\rm{sign}(y_{2}/y_{1})$ in various benchmark WIMP-like DM models. We focus on
the parameter regions in the SD model which differ from the MSSM, and those in
the VL model which are favored by the SM Higgs mass constraint.

We employ the code micrOMEGAs 1407.6129  to calculate the relic density of the
lightest neutral fermion $\chi^{0}_{1}$, and record the spin-independent (SI)
and spin-dependent (SD) $\chi^{0}_{1}$-nucleon scattering cross sections.
Shown in Fig.1 are the parameter spaces of the electroweak DM with
$\text{sign}(y_{2}/y_{1})<0$ (_left_) and $\text{sign}(y_{2}/y_{1})>0$
(_right_), respectively. The two types of plots therein clearly indicate that
the DM relic density is sensitive to both the magnitude of $y$ and
$\rm{sign}(y_{2}/y_{1})$. We refer the reader to Table.1 about
$\rm{sign}(y_{2}/y_{1})$ in the benchmark electroweak DM models.

Figure 1: The _left_ and _right_ plots correspond to
$\text{sign}(y_{1}/y_{2})<0$ and $\text{sign}(y_{1}/y_{2})>0$, respectively,
where we show the parameter space of DM relic density
$\Omega\rm{h}^{2}=0.12\pm 0.005$ (top), the SI $\chi^{0}_{1}$-nucleon
scattering cross section (middle) and the SD $\chi^{0}_{1}$-neutron scattering
cross section (bottom) as function of DM mass $m_{\text{DM}}$ and coupling
$y$, with the Xenon1T Xenon1T , LUX LUXSD  and the future LZ LZ  limits shown
for comparison.

A few comments are in order regarding the parameter space of electroweak DM.
Firstly, the left plots with $\rm{sign}(y_{2}/y_{1})<0$ cover simplified
MSSM1211.4873 ; 1306.1567 ; 1412.5952 ; 1506.07177 ; 1701.05869 , SD 1311.5896
; 1505.03867  and VL 1904.10145  models.

  * •

For the simplified MSSM, the coupling $y=e/\cos\theta_{W}$ and
$y=e/\sin\theta_{W}$ for simplified bino-higgsino and wino-higgsino systems,
respectively, with $\theta_{W}$ the weak mixing angle. In both simplified
models, $y_{2}/y_{1}\rightarrow-\tan\beta$, where $\tan\beta$ is the ratio of
two vacuum expectation values (vevs) of Higgs doublets. The left middle and
bottom plots show that a portion of samples with $y=0.35$ 1701.05869  or
$y=0.63$ 1506.07177  survives.

  * •

For either SD or VL model, the parameter space with coupling $y>1$ is nearly
excluded for DM mass beneath $1$ TeV if not all, in the light of the Xenon1T
limit as shown in the left middle plot. This plot suggests that both the SD
and VL models tend to survive in the small $y$ region, which is consistent
with previous observations on the SD model in 1311.5896 ; 1505.03867  and the
VL model in 1904.10145 , respectively.

Secondly, the right plots with $\rm{sign}(y_{2}/y_{1})>0$ cover the SD
1311.5896 ; 1505.03867  and VL 1904.10145  models.

  * •

Comparing the two middle plots in Fig.1 shows that the Xenon1T limit is more
stringent in the situation with $\rm{sign}(y_{2}/y_{1})>0$, although the
number of samples is relatively larger in this case. It turns out that DM mass
between $1$ TeV is almost excluded by the Xenon1T limit in this case, which
strengthens the earlier results in ref.1505.03867 .

  * •

The importance of $\rm{sign}(y_{2}/y_{1})$ on the SI or SD cross section can
be understood from earlier analysis on the blind spots in ref.1211.4873 ,
under which either the SI or SD cross section is dramatically suppressed. From
our setup in Eq.(III), the condition of blind spots for the SI cross section
favors a negative $\rm{sign}(y_{2}/y_{1})$.

Finally, we would like to mention that our analysis not only reproduces, but
also expands the parameter space beyond the three benchmark electroweak DM
models as shown in Table.1. Some of the new parameter regions may be useful
for other models of new physics around the weak scale. Moreover, one observes
that a portion of samples is even below the sensitivity of future LZ
experiment. However, they may be in the reach of future LHC, which is the
subject as what follows.

${\rm lepton~{}pair}$ | ${\rm SM~{}background}$ |  ${\rm refs}$
---|---|---
$\chi^{+}\chi^{-}$ | $pp\to\ell_{\alpha/\beta}^{\pm}\ell_{\alpha/\beta}^{\mp}+E_{T}^{\rm miss}$ |  1403.5294 ; 2019008 ; 1908.08215
$\chi^{\pm}\chi^{0}_{2}$ |  $pp\to\ell_{\alpha/\beta}^{\pm}\ell_{\alpha/\beta}^{\mp}jj^{\prime}+E_{T}^{\rm miss}$ |  1403.5294 ; 1803.02762 ; 1806.02293
|  $pp\to\ell_{\alpha}^{\pm}\ell_{\alpha}^{\mp}\ell_{\alpha/\beta}^{\pm}jj+E_{T}^{\rm miss}$ |  1403.5294 ; 1803.02762 ; 1806.02293
$\chi^{\pm}\chi^{0}_{2}$ | $pp\to\ell_{\alpha}^{\pm}\ell_{\alpha}^{\mp}\ell_{\alpha/\beta}^{\pm}+E_{T}^{\rm miss}$ |  1803.02762 ; 1806.02293
Table 2: SM backgrounds for the electroweak productions of lepton pairs
$\chi^{+}\chi^{-}$ and $\chi^{\pm}\chi^{0}_{2}$, with subsequent electroweak
decays to $\chi^{0}_{1}$, where $j,j^{\prime}$ and $\alpha,\,\beta$ refer to
jets and charged leptons $\ell=\\{e,\mu\\}$. For example, in the first case
the two leptons can have either the same or different flavors with opposite
signs.

##  IV Lepton Pairs at the LHC

According to the effective Lagrangian in Eqs.(6)-(II), the productions of new
lepton pairs at the LHC as shown in Tabel 2, for which the Feynman diagrams
can be found in Fig.2, mainly depend on the electroweak interactions with SM
gauge bosons and Higgs scalar. Here, we focus on the electroweak productions
of lepton pairs, which subsequently decays into on-shell $Z$ and/or $W$. The
direct combination of DM direct detections and LHC probes of lepton pairs
makes our analysis intuitive, which differs from either $i)$ the experimental
analysis 1403.5294 ; 1803.02762 ; 1806.02293 ; 2019008 ; 1908.08215  on the
electroweakinos reported by ATLAS and CMS collaborations, where the samples
were not extracted from the electroweak DM parameter space, or $ii)$ the DM
direct-detection predictions on the electroweak DM as above, where the
collider probes are not clear. Since we focus on the parameter space of
electroweak DM as shown in Fig.3, where the final lepton states are standard
rather than soft due to large mass splitting between the charged/heavier
neutral states and DM, our analysis is also different from $iii)$ the studies
on the electroweak DM in the parameter space with small mass splitting such as
the disappearing tracks 1804.07321 ; 1712.02118 ; 1703.09675 ; 1805.00015 ,
mono-jets 1805.00015 ; CMS12048 ; 1312.7350 ; 1311.7641 ; 1506.02148  and soft
lepton final states 1310.3675 ; 1307.5952 ; 1404.0682 ; 0909.4549 .

Figure 2: Feynman diagrams for the signal channels of new lepton pairs in
Table 2. Figure 3: An illustration of samples extracted from the parameter
space of electroweak DM in the left plots of Fig.1 in this work.

Shown in Table 2 are the SM backgrounds referring to the electroweak
productions of lepton pairs $\chi^{+}\chi^{-}$ and $\chi^{\pm}\chi^{0}_{2}$ at
the LHC.

  * •

The electroweak production of lepton pair $\chi^{+}\chi^{-}$ is dominated by
$pp\rightarrow Z^{*}\rightarrow\chi^{+}\chi^{-}$, which decays as
$\chi^{\pm}\rightarrow W^{\pm}\chi^{0}_{1}$, with $W\rightarrow\ell\nu$. The
SM background for this process is composed of two charged leptons with missing
energy $E_{T}^{\rm miss}$.

  * •

The electroweak production of lepton pair $\chi^{\pm}\chi^{0}_{2}$ is similar
to that of $\chi^{+}\chi^{-}$, which decay as $\chi^{\pm}\rightarrow
W^{\pm}\chi^{0}_{1}\rightarrow jj\chi^{0}_{1}$ or $\chi^{\pm}\rightarrow
W^{\pm}\chi^{0}_{1}\rightarrow\ell\nu\chi^{0}_{1}$ and
$\chi^{0}_{2}\rightarrow Z\chi^{0}_{1}\rightarrow\ell\ell\chi^{0}_{1}$,
respectively. The SM backgrounds are composed of either two leptons and two
jets with $E_{T}^{\rm miss}$ or three leptons with $E_{T}^{\rm miss}$.

For numerical calculation, we firstly use the package FeynRules 1310.1921  to
prepare the model files for MadGraph5 1405.0301 , which contains package
Pythia 8 Sjostrand:2014zea  for parton showering and hadronization, and
package Delphes 3 1307.6346  for fast detector simulation. Then, we use
Madgraph5 to generate $2000$ events for each sample which is directly
extracted from the parameter space of electroweak DM. The total number of
samples from the left plots in Fig.1 is about $\sim 3660$, some of which can
yield the on-shell decays $\chi^{\pm}\rightarrow\chi^{0}_{1}W^{\pm}$ and/or
$\chi^{0}_{2}\rightarrow\chi^{0}_{1}Z$, and finally reveal both the exclusion
and discovery limits at the 14 TeV LHC with the integrated luminosities 300
fb-1 and 3000 fb-1 respectively.

| $\chi^{+}\chi^{-}$ |  $\chi^{\pm}\chi^{0}_{2}$ (2$\ell$ 2$j$) |  $\chi^{\pm}\chi^{0}_{2}$ (3$\ell$)
---|---|---|---
$p_{T}^{l_{1(2)}}$ | $>25$ | $>25$ | $>25$
$\eta_{e(\mu)}$ | $>2.47(2.7)$ | $>2.47(2.7)$ | $>2.47(2.7)$
$m_{l_{1}l_{2}}$ | $>100(121.2)$ | 81-101 | 81.2-101.2
$E_{T}^{\rm miss}$ | $>110$ | $>100$ | $>170$
$m_{T}$ | $>100$ | - | $>110$
$n_{\text{non-b}}$ | 0 | 2 | 0
$n_{\text{b}}$ | 0 | 0 | 0
$m_{jj}$ | - | 70-90 | -
$E_{T,s}^{\rm miss}$ | $>10$ | - | -
Table 3: The main cuts used for event selections of both signals and
backgrounds as shown in Table.2, where mass parameters are in unit of GeV. See
text for definitions and explanations about these cuts.

For selections of events, we use the 13-TeV cuts reported by the ATLAS
Collaboration in ref.1803.02762  and ref.1908.08215  for lepton pair
$\chi^{\pm}\chi^{0}_{2}$ with SM backgrounds $pp\to 2\ell 2j+E_{T}^{\rm miss}$
or $pp\to 3\ell+E_{T}^{\rm miss}$ and lepton pair $\chi^{+}\chi^{-}$ with SM
background $pp\to 2\ell+E_{T}^{\rm miss}$, respectively. The effects on the
event numbers due to deviation from the 14-TeV cuts are expected to be in
percent level. We summarize the details on the cuts in Table.3, where
${p_{T}}^{l_{1(2)}}$ is the transverse momentum of the first (second) leading
lepton $\ell=\\{e,\mu\\}$, $\eta_{e(\mu)}$ is pseudo-rapidity of $e$($\mu$),
$m_{l_{1}l_{2}}$ is the invariant mass of the two leptons with same (or
different) flavor for the lepton pair $\chi^{+}\chi^{-}$ or the shell mass of
same lepton flavor for the lepton pair $\chi^{\pm}\chi^{0}_{2}$. In this
Table, the transverse mass is defined as
$m_{\textup{T}}=\sqrt{2\times|\mathbf{p}_{\mathrm{T,1}}|\times|\mathbf{p}_{\mathrm{T,2}}|\times(1-\mathrm{cos}(\Delta\phi))}$,
where $\Delta\phi$ is the difference in azimuthal angle between the particles
with transverse momenta $\mathbf{p}_{\mathrm{T,1}}$ and
$\mathbf{p}_{\mathrm{T,2}}$, $n_{\rm{b}}$ is the number of b-tagging jets,
variable $n_{\text{non-b}}$ refers to the number of jets with $p_{T}>30$ GeV
that do not satisfy the $b$-tagging criteria, $m_{jj}$ is the invariant mass
of the two leading jets, and significance of missing energy $E_{T,s}^{\rm
miss}$ is linked to the $E_{T}^{\rm miss}$ by $E_{T}^{\rm miss}/\sqrt{H}$,
with $H$ the transverse momenta of all final states.

For $pp\to\ell_{\alpha}^{\pm}\ell_{\beta}^{\pm}jj+E_{T}^{\rm miss}$ in the
second column in Table.3, we need additional cuts as follows:
$\Delta\phi_{(p_{T}^{miss},Z)}<0.8$, $\Delta\phi_{(p_{T}^{miss},W)}>1.5$,
$0.6<E_{T}^{\rm miss}/p^{Z}_{\rm T}<1.6$ and $E_{T}^{\rm miss}/p^{W}_{\rm
T}<0.8$, in which the symbols $W$ and $Z$ correspond to the reconstructed $W$
and $Z$ bosons in the final state. Because the $Z$ boson is always
reconstructed from the two leptons, whereas the $W$ boson is reconstructed
from the two jets. Here, $p^{Z}_{\rm T}$ and $p^{W}_{\rm T}$ are the
transverse momenta of dileptons and dijets respectively, with
$\Delta\phi_{(p_{T}^{miss},Z)}$ or $\Delta\phi_{(p_{T}^{miss},W)}$ referring
to the azimuthal angle between the dilepton or dijets transverse momentum and
$p_{T}^{miss}$.

Figure 4: The _left_ and _right_ plots refer to 2$\sigma$ exclusion and
5$\sigma$ discovery at the 14 TeV LHC with the integrated luminosity $L=300$
fb-1, where the lepton pair $\chi^{+}\chi^{-}$ and $\chi^{\pm}\chi^{0}_{2}$
are shown in the top as well as middle and bottom plots, respectively. All the
samples used for event simulations are extracted from the DM parameter space
of the _left_ plots in Fig.1, which satisfy the DM relic density and survive
in the DM direct detections simultaneously.

Figure 5: The same as Fig.4 with $L=3000$ fb-1 instead. $\rm{channel}$ |  $\sigma_{\rm{LO}}\rm(fb)$ | $N_{\rm{tot}}$ | $N_{\rm{re}}$
---|---|---|---
$pp\to W^{+}W^{-}\to 2\ell+E_{T}^{\rm miss}$ |  $3300$ | $10^{5}$ | $51$
$pp\to WZjj\to 3\ell+2j+E_{T}^{\rm miss}$ |  $540$ | $10^{5}$ | $7$
$pp\to ZZjj\to 2\ell+2j+E_{T}^{\rm miss}$ |  $100$ | $10^{5}$ | $12$
$pp\to WZ\to 3\ell+E_{T}^{\rm miss}$ |  $800$ | $10^{6}$ | $10$
Table 4: Raw data of the simulations of SM backgrounds in Table 2, where
$\sigma_{\rm{LO}}$ is the leading-order (LO) cross section, and the
efficiencies on the SM backgrounds after cuts imposed are given by
$\epsilon=N_{\rm{re}}/N_{\rm{tot}}$, with $N_{\rm{tot}}$ and $N_{\rm{re}}$
referring to the numbers of events before and after cuts imposed,
respectively.

In terms of the cuts above, we can analysis the signal significances. We
firstly show in Table 4 the simulations of events of the SM backgrounds. Then,
we show in Fig.4 and Fig.5 both the 2$\sigma$ exclusion (_left_) and 5$\sigma$
discovery (_right_) at the 14 TeV LHC with the integrated luminosity $L=300$
fb-1 and $3000$ fb-1, respectively. In these figures, the top, middle and
bottom plots refer to lepton pairs $\chi^{+}\chi^{-}$ and
$\chi^{\pm}\chi^{0}_{2}$, respectively, where the efficiencies among the three
channels are comparable. The combination of individual results yields our
final observations:

  * •

In the coupling range $y\sim 0.35-0.5$ as referred by the dark blue points,
which is viable for the bino-higgsino system, DM mass range between $\sim
170-210$ GeV can be excluded at 2$\sigma$ level with with $L=300$ fb-1, and DM
mass range between $\sim 175-205$ GeV can be discovered at 5$\sigma$ level
with $L=3000$ fb-1. These limits are comparable with those 1404.0682  of soft
lepton final states, where the exclusion and discovery limits are $\sim
200-280$ GeV and $\sim 130$ GeV, respectively. Earlier discussions about
simplified MSSM at the LHC, see refs.9706509 ; 1409.6322 ; 1510.03460 .

  * •

In the coupling range $y\sim 0.5-0.65$ as illustrated by the blue points,
which can be applied to the wino-higgsino system, DM mass range between $\sim
172-210$ GeV can be excluded at 2$\sigma$ level with $L=300$ fb-1, and DM mass
range between $\sim 180-200$ GeV can be discovered at 5$\sigma$ level with
$L=3000$ fb-1.

  * •

In the coupling range $y\sim 0.65-1.0$ as referred by the green and orange
points, which is viable for SD and VL model, all of samples with DM mass above
$\sim 250$ GeV give rise to very small significances. Thus, they are beyond
the reach of LHC. Earlier discussion on this point, see ref.0706.0918 .

Our analysis above are subject to both the Monte Carlo (MC) based
uncertainties and the systematic uncertainties. For the MC uncertainties, the
LO cross sections for both the signals and their SM backgrounds can be
enhanced by the higher-order QCD effects Campbell:1999ah ; Campbell:2011bn .
The K-factors $k_{b}$ respect to the WW and WZ channel in Table.4 are of order
$\sim 1.67$ and $\sim 1.85$, respectively, which imply that for the inferred
K-factor $k_{s}$ of order $\sim 1.2-1.3$ for the signals, the significance
measures in Fig.4 and Fig.5 are corrected by a factor of order $\sim 6\%$ and
$\sim 9\%$, respectively. For the systematic uncertainties, even although
those in the lepton reconstruction efficiency and the b-tagging efficiency etc
are small 1803.02762 , the uncertainties related to the jet energy scale and
resolution aren’t negligible TheATLAScollaboration:2015hmp .

##  V Conclusions

In this work, we have revisited general electroweak DM through the combined
searches of DM direct detections and LHC probe of new lepton pairs which
appear together with DM in the new physics of weak scale. Compared to earlier
works in the literature, this analysis is more intuitive. On the other hand,
we have utilized a general framework, which can effectively describe three
well-known extensions on the electroweak sector that contains both DM and new
lepton pairs simultaneously.

The outcomes are two-fold. First of all, instead of those separate studies
either from DM direct detection or LHC probe, the combination of the two
direct detections allows us to explore the real status of electroweak DM.
Second, the general description enables us to uncover more parameter regions
unexplored yet. Utilizing three $Z$\- and $W$-associated electroweak processes
at the 14 TeV LHC, we have illustrated that in the parameter space with the
lepton pairs decaying to on-shell $Z$ or $W$: $i)$ for $y\sim 0.35-0.5$ DM
mass up to $\sim 170-210$ GeV can be excluded at $2\sigma$ level with $L=300$
fb-1 and up to $\sim 175-205$ GeV can be discovered at $5\sigma$ level with
$L=3000$ fb-1; $ii)$ for $y\sim 0.5-0.65$ DM mass up to $\sim 172-210$ GeV can
be excluded at $2\sigma$ level with $L=300$ fb-1 and up to $\sim 180-200$ GeV
can be discovered at $5\sigma$ level with $L=3000$ fb-1; $iii)$ for $y\sim
0.65-1.0$ DM mass above $250$ GeV is totally beyond the reach of LHC.

The low reaches on the electroweak DM mass in the situation with large mass
splitting, together with similar trends in the small mass splitting, suggest
that new search strategies are needed in order to examine higher electroweak
DM mass ranges. There are a few directions as follows. Firstly, the ability of
detection on the electroweak DM may be improved in other channels that are
rarely considered. Moreover, compared to the sophisticated data analysis
adopted here, novel methods such as machine learning may provide alternative
views, see, e.g. 1709.04464 ; 1712.04793 . Finally, put aside colliders, one
may carefully consider astrophysical probes of the electroweak DM, which may
be stringent under certain circumstances.

###### Acknowledgements.

The authors would like to thank Lorenzo Calibbi, Huayong Han, Yang Zhang and
Pengxuan Zhu for helpful discussions. This research is supported in part by
the National Natural Science Foundation of China with Grant No.11775039, the
Chinese Scholarship Council, and the Fundamental Research Funds for the
Central Universities at Chongqing University with Grant No.cqu2017hbrc1B05.

## References

  * (1) S. Dawson et al., “Working Group Report: Higgs Boson,” [arXiv:1310.8361 [hep-ex]].
  * (2) X. Cid Vidal et al. [Working Group 3], “Beyond the Standard Model Physics at the HL-LHC and HE-LHC,” [arXiv:1812.07831 [hep-ph]].
  * (3) ATLAS and CMS Collaborations [ATLAS and CMS Collaborations], “Report on the Physics at the HL-LHC and Perspectives for the HE-LHC,” [arXiv:1902.10229 [hep-ex]].
  * (4) P. H. Frampton, P. Q. Hung and M. Sher, “Quarks and leptons beyond the third generation,” Phys. Rept. 330, 263 (2000), [hep-ph/9903387].
  * (5) S. D. Thomas and J. D. Wells, “Phenomenology of Massive Vectorlike Doublet Leptons,” Phys. Rev. Lett. 81, 34 (1998)， [hep-ph/9804359].
  * (6) S. P. Martin, “A Supersymmetry primer,” Adv. Ser. Direct. High Energy Phys. 21, 1 (2010), [Adv. Ser. Direct. High Energy Phys. 18, 1 (1998)], [hep-ph/9709356].
  * (7) U. Ellwanger, C. Hugonie and A. M. Teixeira, “The Next-to-Minimal Supersymmetric Standard Model,” Phys. Rept. 496, 1 (2010), [arXiv:0910.1785 [hep-ph]].
  * (8) R. Mahbubani and L. Senatore, “The Minimal model for dark matter and unification,” Phys. Rev. D 73, 043510 (2006), [hep-ph/0510064].
  * (9) F. D’Eramo, “Dark matter and Higgs boson physics,” Phys. Rev. D 76, 083522 (2007), [arXiv:0705.4493 [hep-ph]].
  * (10) T. Cohen, J. Kearney, A. Pierce and D. Tucker-Smith, “Singlet-Doublet Dark Matter,” Phys. Rev. D 85, 075003 (2012), [arXiv:1109.2604 [hep-ph]].
  * (11) S. P. Martin, “Extra vector-like matter and the lightest Higgs scalar boson mass in low-energy supersymmetry,” Phys. Rev. D 81, 035004 (2010), [arXiv:0910.2732 [hep-ph]].
  * (12) S. Zheng, “Effective Higgs Theories in Supersymmetric Grand Unification,” Eur. Phys. J. C 77, no. 9, 588 (2017), [arXiv:1706.01071 [hep-ph]].
  * (13) S. Zheng, “Extended MSSM in supersymmetric $\rm{SO}(10)$ grand unification,” Phys. Rev. D 98, no. 3, 035028 (2018), [arXiv:1711.05362 [hep-ph]].
  * (14) S. Zheng, “Minimal Vectorlike Model in Supersymmetric Unification,” Eur. Phys. J. C 80, no.3, 273 (2020), [arXiv:1904.10145 [hep-ph]].
  * (15) G. Belanger, F. Boudjema, A. Pukhov and A. Semenov, “micrOMEGAs4.1: two dark matter candidates,” Comput. Phys. Commun. 192, 322 (2015), arXiv:1407.6129 [hep-ph].
  * (16) E. Aprile et al. [XENON Collaboration], “Dark Matter Search Results from a One Ton-Year Exposure of XENON1T,” Phys. Rev. Lett. 121, no. 11, 111302 (2018), [arXiv:1805.12562 [astro-ph.CO]].
  * (17) D. S. Akerib et al. [LUX Collaboration], “Limits on spin-dependent WIMP-nucleon cross section obtained from the complete LUX exposure,” Phys. Rev. Lett. 118, no. 25, 251302 (2017), [arXiv:1705.03380 [astro-ph.CO]].
  * (18) D. S. Akerib et al. [LUX-ZEPLIN Collaboration], “Projected WIMP Sensitivity of the LUX-ZEPLIN (LZ) Dark Matter Experiment,” [arXiv:1802.06039 [astro-ph.IM]].
  * (19) C. Cheung, L. J. Hall, D. Pinner and J. T. Ruderman, “Prospects and Blind Spots for Neutralino Dark Matter,” JHEP 1305, 100 (2013), [arXiv:1211.4873 [hep-ph]].
  * (20) A. Fowlie, K. Kowalska, L. Roszkowski, E. M. Sessolo and Y. L. S. Tsai, “Dark matter and collider signatures of the MSSM,” Phys. Rev. D 88, 055012 (2013). [arXiv:1306.1567 [hep-ph]].
  * (21) G. Grilli di Cortona, “Hunting electroweakinos at future hadron colliders and direct detection experiments,” JHEP 1505, 035 (2015), [arXiv:1412.5952 [hep-ph]].
  * (22) M. Badziak, A. Delgado, M. Olechowski, S. Pokorski and K. Sakurai, “Detecting underabundant neutralinos,” JHEP 1511, 053 (2015), [arXiv:1506.07177 [hep-ph]].
  * (23) M. Badziak, M. Olechowski and P. Szczerbiak, “Is well-tempered neutralino in MSSM still alive after 2016 LUX results?,” Phys. Lett. B 770, 226 (2017), [arXiv:1701.05869 [hep-ph]].
  * (24) C. Cheung and D. Sanford, “Simplified Models of Mixed Dark Matter,” JCAP 1402, 011 (2014), [arXiv:1311.5896 [hep-ph]].
  * (25) L. Calibbi, A. Mariotti and P. Tziveloglou, “Singlet-Doublet Model: Dark matter searches and LHC constraints,” JHEP 1510, 116 (2015), [arXiv:1505.03867 [hep-ph]].
  * (26) A. Alloul, et al., “FeynRules 2.0 - A complete toolbox for tree-level phenomenology,” Comput. Phys. Commun. 185, 2250 (2014) [arXiv:1310.1921 [hep-ph]].
  * (27) J. Alwall et al., “The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations,” JHEP 1407, 079 (2014), [arXiv:1405.0301 [hep-ph]].
  * (28) T. Sjöstrand et al., “An Introduction to PYTHIA 8.2,” Comput. Phys. Commun. 191, 159 (2015), [arXiv:1410.3012 [hep-ph]]
  * (29) J. de Favereau et al. [DELPHES 3 Collaboration], “DELPHES 3, A modular framework for fast simulation of a generic collider experiment,” JHEP 1402, 057 (2014) [arXiv:1307.6346 [hep-ex]].
  * (30) G. Aad et al. [ATLAS Collaboration], “Search for direct production of charginos, neutralinos and sleptons in final states with two leptons and missing transverse momentum in $pp$ collisions at $\sqrt{s}=$ 8 TeV with the ATLAS detector,” JHEP 1405, 071 (2014)， [arXiv:1403.5294 [hep-ex]].
  * (31) M. Aaboud et al. [ATLAS Collaboration], “Search for electroweak production of supersymmetric particles in final states with two or three leptons at $\sqrt{s}=13\,$TeV with the ATLAS detector,” Eur. Phys. J. C 78, no. 12, 995 (2018), [arXiv:1803.02762 [hep-ex]].
  * (32) M. Aaboud et al. [ATLAS Collaboration], “Search for chargino-neutralino production using recursive jigsaw reconstruction in final states with two or three charged leptons in proton-proton collisions at $\sqrt{s}=13$ TeV with the ATLAS detector,” Phys. Rev. D 98, no. 9, 092012 (2018), [arXiv:1806.02293 [hep-ex]].
  * (33) The ATLAS collaboration, “Search for electroweak production of charginos and sleptons decaying in final states with two leptons and missing transverse momentum in $\sqrt{s}=13$ TeV $pp$ collisions using the ATLAS detector,” ATLAS-CONF-2019-008.
  * (34) G. Aad et al. [ATLAS Collaboration], “Search for electroweak production of charginos and sleptons decaying into final states with two leptons and missing transverse momentum in $\sqrt{s}=13$ TeV $pp$ collisions using the ATLAS detector,” [arXiv:1908.08215 [hep-ex]].
  * (35) A. M. Sirunyan et al. [CMS Collaboration], “Search for disappearing tracks as a signature of new long-lived particles in proton-proton collisions at $\sqrt{s}=$ 13 TeV,” JHEP 1808, 016 (2018), [arXiv:1804.07321 [hep-ex]].
  * (36) M. Aaboud et al. [ATLAS Collaboration], “Search for long-lived charginos based on a disappearing-track signature in pp collisions at $\sqrt{s}=13$ TeV with the ATLAS detector,” JHEP 1806, 022 (2018), [arXiv:1712.02118 [hep-ex]].
  * (37) H. Fukuda, N. Nagata, H. Otono and S. Shirai, “Higgsino Dark Matter or Not: Role of Disappearing Track Searches at the LHC and Future Colliders,” Phys. Lett. B 781, 306 (2018), [arXiv:1703.09675 [hep-ph]].
  * (38) T. Han, S. Mukhopadhyay and X. Wang, “Electroweak Dark Matter at Future Hadron Colliders,” Phys. Rev. D 98, no. 3, 035026 (2018), [arXiv:1805.00015 [hep-ph]].
  * (39) [CMS Collaboration], “Search for new physics in monojet events in pp collisions at sqrt(s)= 8 TeV,” CMS-PAS-EXO-12-048.
  * (40) P. Schwaller and J. Zurita, “Compressed electroweakino spectra at the LHC,” JHEP 1403, 060 (2014), [arXiv:1312.7350 [hep-ph]].
  * (41) A. Arbey, M. Battaglia and F. Mahmoudi, “Combining monojet, supersymmetry, and dark matter searches,” Phys. Rev. D 89, no. 7, 077701 (2014), [arXiv:1311.7641 [hep-ph]].
  * (42) A. Arbey, M. Battaglia and F. Mahmoudi, “Monojet Searches for MSSM Simplified Models,” Phys. Rev. D 94, no. 5, 055015 (2016), [arXiv:1506.02148 [hep-ph]].
  * (43) G. Aad et al. [ATLAS Collaboration], “Search for charginos nearly mass degenerate with the lightest neutralino based on a disappearing-track signature in pp collisions at $\sqrt{(}s)$=8 TeV with the ATLAS detector,” Phys. Rev. D 88, no. 11, 112006 (2013), [arXiv:1310.3675 [hep-ex]].
  * (44) S. Gori, S. Jung and L. T. Wang, “Cornering electroweakinos at the LHC,” JHEP 1310, 191 (2013), [arXiv:1307.5952 [hep-ph]].
  * (45) M. Low and L. T. Wang, “Neutralino dark matter at 14 TeV and 100 TeV,” JHEP 1408, 161 (2014), [arXiv:1404.0682 [hep-ph]].
  * (46) M. R. Buckley, L. Randall and B. Shuve, JHEP 1105, 097 (2011), [arXiv:0909.4549 [hep-ph]].
  * (47) H. Baer and M. Brhlik, “Neutralino dark matter in minimal supergravity: Direct detection versus collider searches,” Phys. Rev. D 57, 567 (1998), [hep-ph/9706509].
  * (48) T. A. W. Martin and D. Morrissey, “Electroweakino constraints from LHC data,” JHEP 1412, 168 (2014), [arXiv:1409.6322 [hep-ph]].
  * (49) J. Bramante, N. Desai, P. Fox, A. Martin, B. Ostdiek and T. Plehn, “Towards the Final Word on Neutralino Dark Matter,” Phys. Rev. D 93, no. 6, 063525 (2016), [arXiv:1510.03460 [hep-ph]].
  * (50) R. Enberg, P. J. Fox, L. J. Hall, A. Y. Papaioannou and M. Papucci, “LHC and dark matter signals of improved naturalness,” JHEP 0711, 014 (2007), [arXiv:0706.0918 [hep-ph]].
  * (51) J. M. Campbell and R. K. Ellis, “An Update on vector boson pair production at hadron colliders,” Phys. Rev. D 60, 113006 (1999), [hep-ph/9905386].
  * (52) J. M. Campbell, R. K. Ellis and C. Williams, “Vector boson pair production at the LHC,” JHEP 1107, 018 (2011), [arXiv:1105.0020 [hep-ph]].
  * (53) The ATLAS collaboration, “Expected performance of missing transverse momentum reconstruction for the ATLAS detector at $\sqrt{s}=13~{}\text{TeV}$,” ATL-PHYS-PUB-2015-023.
  * (54) A. J. Larkoski, I. Moult and B. Nachman, “Jet Substructure at the Large Hadron Collider: A Review of Recent Advances in Theory and Machine Learning,” [arXiv:1709.04464 [hep-ph]].
  * (55) G. Bertone, N. Bozorgnia, J. S. Kim, S. Liem, C. McCabe, S. Otten and R. Ruiz de Austri, “Identifying WIMP dark matter from particle and astroparticle data,” JCAP 1803, 026 (2018), [arXiv:1712.04793 [hep-ph]].