# Performance and Complexity of Sequential Decoding of PAC Codes

Mohsen Moradi, Amir Mozammel, Kangjian Qin, and Erdal Arıkan The authors are
with the Department of Electrical-Electronics Engineering, Bilkent University,
Ankara TR-06800, Turkey (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>arikan@ee.bilkent.edu.tr).

###### Abstract

Performance and complexity of sequential decoding of polarization-adjusted
convolutional (PAC) codes is studied. In particular, a performance and
computational complexity comparison of PAC codes with 5G polar codes and
convolutional codes is given. A method for bounding the complexity of
sequential decoding of PAC codes is proposed.

###### Index Terms:

PAC codes, polar codes, sequential decoding.

##  I Introduction

Polarization-adjusted convolutional (PAC) codes are a class of liner block
codes that combine ideas from channel polarization and convolutional coding
[1, 2]. PAC codes show remarkably good performance at short block lengths,
approaching theoretical limits in some instances. In this paper we carry out a
comparison of PAC with convolutional codes and polar codes in terms of
performance and complexity. We focus exclusively on sequential decoding of PAC
codes. PAC codes can also be decoded using fixed-complexity list-decoders as
discussed in [3] and [4]; however, such algorithms lie outside the scope of
the present paper.

We show that PAC codes under sequential decoding are capable of providing
better performance than some 5G polar codes. We develop methods for a
reasonable comparison of the computational complexity of sequential decoding
of PAC codes with the list decoding of 5G polar codes. Finally, we consider
sequential decoding under a strict limit on its search complexity so as to
eliminate implementation difficulties arising from the variable nature of
computation in sequential decoding. Our main conclusion is that PAC codes
under sequential decoding can be seen as a viable alternative to convolutional
and polar codes both in terms of performance and complexity.

The rest of this paper is organized as follows. Section II provides a brief
introduction to PAC coding and sequential decoding of PAC codes. Section III
compares sequential decoding of PAC codes with that of ordinary convolutional
codes. Section IV compares PAC codes with 5G polar codes. Section V studies
bounded-complexity sequential decoders and quantifies the loss in performance
as a result of bounding the complexity. Finally, Section VI concludes the
paper with a summary and suggestions for future work.

##  II PAC Coding

A PAC code is a linear block code over the binary field
${\mathbb{F}}_{2}=\\{0,1\\}$. Fig. 1 shows a flow chart of a PAC coding scheme
with block-length $N$ and rate $R=K/N$ for some integer $1\leq K\leq N$.

Figure 1: Flowchart of PAC coding scheme.

The input to the system is a data word $\mathbf{d}=(d_{1},\ldots,d_{K})$ which
is assumed to be chosen uniformly at random from ${\mathbb{F}}_{2}^{K}$. The
data insertion module maps the data word $\mathbf{d}$ to a data container
$\mathbf{v}\in{\mathbb{F}}_{2}^{N}$ by setting
$\mathbf{v}_{\mathcal{A}}=\mathbf{d}$ and $\mathbf{v}_{\mathcal{A}^{c}}=0$,
where $\mathcal{A}$ is a data index set of size $K$. In this paper, the data
index set $\mathcal{A}$ is chosen according to the Reed-Muller (RM) rule as
explained in [2]. Encoding continues by a convolution operation
$\mathbf{u}=\mathbf{v}\ast\mathbf{c}$ where
$\mathbf{c}=(c_{0},c_{1},\ldots,c_{m})$ is a generator sequence. Next, a polar
transform is applied on the convolution output to obtain the PAC codeword
$\mathbf{x}=\mathbf{u}\mathbf{F}^{\otimes n}$, where $\mathbf{F}^{\otimes n}$
is the $n$th Kronecker power of a kernel matrix
$\mathbf{F}=\begin{bmatrix}1&0\\\ 1&1\end{bmatrix}$ and where $n=\log_{2}N$.
The codeword $\mathbf{x}$ is sent through a binary-input additive white
Gaussian noise (BI-AWGN) channel and a channel output $\mathbf{y}$ is
received.

At the receiver side, a sequential decoder produces an estimate
$\hat{\mathbf{v}}\in{\mathbb{F}}_{2}^{N}$ of the data container $\mathbf{v}$
with the aid of a metric calculator. An estimate of the data word is extracted
by setting $\hat{\mathbf{d}}=\hat{\mathbf{v}}_{\mathcal{A}}$. The main
performance metric for the system is the frame error rate (FER), defined as
the probability $\text{P}(\hat{\mathbf{d}}\neq\mathbf{d})$.

To study the performance of PAC codes, we implemented a sequential decoder as
described in [2]. Specifically, we implemented a Fano decoder [5] with a
metric of the form

$\Gamma(\hat{\mathbf{u}}^{i};\mathbf{y})=\log_{2}\frac{\text{P}(\mathbf{y}|\hat{\mathbf{u}}^{i})}{\text{P}(\mathbf{y})}-\sum_{j=1}^{i}b_{j},$

where $\hat{\mathbf{u}}^{i}$ is a candidate path at level $i$ of a search tree
corresponding to the PAC code and $b_{j}=\rho$ for $j\in{\mathcal{A}}$ and
$b_{j}=0$ for $j\notin{\mathcal{A}}$ for some constant $\rho$. This metric is
calculated incrementally as a sum of branch metrics of the form

$\displaystyle\gamma(\hat{u}_{j};\mathbf{y},\hat{\mathbf{u}}^{j-1})$
$\displaystyle=\Gamma(\hat{\mathbf{u}}^{j};\mathbf{y})-\Gamma(\hat{\mathbf{u}}^{j-1};\mathbf{y})$
$\displaystyle=\log_{2}\frac{\text{P}(\mathbf{y},\hat{\mathbf{u}}^{j-1}|\hat{u}_{j})}{\text{P}(\mathbf{y},\hat{\mathbf{u}}^{j-1})}-b_{j}.$

In PAC code simulations presented below, we used a version of the Fano decoder
as described in Section 6.9 of [6]. The bias parameter $\rho$ for the Fano
metric and the threshold spacing $\Delta$ for the Fano decoder are specified
in the related section.

Sequential decoder carries out a search for the correct path in the code tree
corresponding to the PAC code. The search complexity is a random variable that
depends, among other things, on the severity of noise. In the following, we
will measure the complexity of sequential decoding by a random variable $Z$,
which counts the number of nodes in the decoding tree that are visited by the
Fano algorithm during a decoding session. Since the Fano algorithm has a
backtracking feature, it may visit some nodes more than once, and $Z$ counts
each such visit. The variable $Z$ only counts the nodes visited by a forward
move and ignores lateral or backward moves [6, p. 273]. (A version of
sequential decoding, called the stack algorithm [7, 8], visits each node at
most once but it uses more memory and requires sorting of nodes in a stack.)
We will be primarily interested in the expectation $E(Z)$, which we will call
average number of visits (ANV) and estimate it by computing the empirical mean
of $Z$ over a sufficiently large number of simulation runs.

##  III Comparison with Convolutional Codes

In this section, we compare sequential decoding of PAC codes with that of
ordinary convolutional codes. Fig. 2 shows the result of such a comparison
from the viewpoint of FER performance, with a favorable outcome for PAC codes.
Also shown in Fig. 2 is the dispersion approximation, which approximates the
best achievable performance by any code of a given length and rate [9]. PAC
code shows a near-optimal performance over a range of SNR ($E_{s}/N_{0}$ dB)
values.

Figure 2: FER performance of PAC and convolutional codes.

The convolutional code in Fig. 2 uses a pair of generator sequences
$\mathbf{c}=(133,171)$ (in octal notation) and encodes $K=64$ message bits
into $N=140$ codeword bits (with 12 termination bits), for an effective coding
rate of $R=64/140=0.457$. The corresponding parameters for the PAC code in
Fig. 2 are $\mathbf{c}=133$, $K=59$, $N=128$, and $R=0.461$. The bias and
threshold spacing parameters in sequential decoding of the PAC code are fixed
as $\rho=1.35$ and $\Delta=2$.

Next, we turn to a comparison of computational complexity between the two
codes. Fig. 3 displays the ANV for sequential decoding of the two codes. At
low SNR points, the PAC Fano decoder has a significantly larger average
complexity as measured by the ANV, but as SNR increases, the complexity gap
between the two sequential decoders decreases.

A fundamental result on sequential decoding of ordinary convolutional codes
states that $Z$ has a heavy-tailed Pareto distribution [10]. Further, for
rates $R$ above a cutoff rate $R_{0}$, the average computation $E(Z)$
increases exponentially in $N(R-R_{0})$ [10, 11]. If we ignore the
termination, the convolutional code here operates at rate $1/2$ and we have
$R_{0}=1/2$ at 2.46 dB SNR. Simulation results show a significant rise in
computation complexity at SNRs below 2.5 dB, as predicted by theory. We also
see that the ANV for the sequential decoding of the PAC code experiences a
sharp rise at SNR values below 2.5 dB. So, the PAC codes also suffer from a
cutoff rate phenomenon. This is not unexpected in view of a data processing
theorem [6, p. 149-150] that states that cutoff rate cannot be increased by
pre- and post-processing operations on a channel.

Figure 3: Average number of visits for PAC and convolutional Fano decoders.

We should point out that it is actually possible to change the performance
versus complexity tradeoff in sequential decoding of PAC codes by changing the
data index set $\mathcal{A}$. For example, in experiments (not reported here)
we observed that if $\mathcal{A}$ is chosen in accordance with a polar code
design rule, the ANV decreases; however, the reduction in ANV takes place at
the expense of a worsening of the FER performance.

##  IV Comparison with 5G Polar Codes

Fig. 4 compares the FER performance of PAC codes and 5G polar codes [12] for
three different code rates with $K\in\\{29,64,99\\}$ and $N=128$. The 5G polar
codes had a CRC of length 11. To decode the 5G polar codes, we used a CRC-
aided successive cancellation list (CA-SCL) decoder from MATLAB® 5G ToolboxTM
with a list size of $L=64$. All PAC codes in the comparison were constructed
using the RM design rule. Table I lists the generator sequences $\mathbf{c}$
for the PAC encoder and the bias $\rho$ and threshold spacing $\Delta$
parameters for the Fano decoder. The generator sequence $\mathbf{c}=3211$ is
obtained by an ad-hoc search method with the goal of maximizing FER
performance. As Fig. 4 shows, for the three specific cases considered here,
PAC codes perform better than 5G polar codes.

Figure 4: FER performance of PAC and 5G polar codes. TABLE I: PAC encoder and decoder parameters. $N$ | $K$ | $\mathbf{c}$ | $\rho$ | $\Delta$
---|---|---|---|---
128 | 29 | 3211 | 1.4 | 2
128 | 64 | 133 | 1.35 | 2
128 | 99 | 133 | 1.14 | 2

The ANV figures for the three PAC codes are listed in Table II. Sequential
decoder has a built-in mechanism for adjusting search complexity in response
to severity of channel noise, which explains why complexity becomes smaller as
the SNR is increased. The SNR values corresponding to $R_{0}$ being equal to
$29/128$, $64/128$, and $99/128$ are -1.63, 2.46, and 5.49 dB, respectively.
The computation in sequential decoding of PAC codes sees a sharp rise as the
SNR crosses the cutoff rate thresholds from above.

TABLE II: Average number of visits per codeword for PAC Fano decoder. $K=29$ | $K=64$ | $K=99$
---|---|---
SNR (dB) | AVN | SNR (dB) | AVN | SNR (dB) | AVN
-3 | 1301 | 0 | 219963 | 4 | 6569
-2.5 | 770 | 0.5 | 87166 | 4.5 | 2570
-2 | 509 | 1 | 32237 | 5 | 826
-1.5 | 344 | 1.5 | 9369 | 5.5 | 401
-1 | 256 | 2 | 2749 | 6 | 303
-0.5 | 213 | 2.5 | 736 | 6.5 | 279
0 | 191 | 3 | 368 | 7 | 270
|  | 3.5 | 287 |  |

We will measure the complexity of a CA-SCL decoder for polar codes by the
total number of nodes that the decoder processes. A list decoder begins with
an empty list and builds up a list of size $L$ after $\log_{2}L$ branching
points in the code tree for polar codes. Thereafter, the list decoder examines
$2L$ candidate nodes at each branching point. We take the number of branching
points in the code tree to be $K+C$ where $C$ is the number of parity or CRC
bits. We ignore the lower complexity of the first $\log_{2}L$ steps and
approximate the total number of nodes inspected by a list-of-$L$ decoder by
$2L(K+C)$. Note that this complexity estimate excludes the sorting complexity
of a list decoder, which actually may dominate the complexity for a large list
size, such as $L=64$.

For the 5G polar codes here, we have $L=64$, $C=11$, and $K\in\\{29,64,99\\}$.
Accordingly, we estimate the CA-SCL decoder complexity as 5120 for $K=29$,
9600 for $K=64$, and 14080 for $K=99$. Comparing these 5G polar complexity
figures with the PAC complexity figures in Table II, we see that, for message
lengths $K=29$ and $K=99$, decoding of PAC codes is significantly less complex
than that of 5G polar codes on average. For $K=64$, the decoding complexity of
the PAC code is smaller than that of the 5G polar code at SNR values greater
than 1.5 dB. Based on this comparison, we conclude that the PAC decoder has a
complexity comparable to that of CA-SCL decoder.

##  V Bounded-complexity PAC Fano Decoder

Sequential decoding is a variable-complexity decoding algorithm. In many
applications it is desirable to have a fixed- or bounded-complexity decoding
method. In this section, we consider imposing a strict limit on the complexity
of sequential decoding and study its performance under such a constraint. For
this we need to study the distribution of the complexity measure $Z$ rather
than ANV, which corresponds to the mean complexity $E(Z)$.

To this end, we performed a simulation study of sequential decoding of the
above $(128,64)$ PAC code and recorded the observed $Z$ values. We discarded
the $Z$ values when a decoding error occurred and kept the $Z$ values in
correct decodings. The results are shown in Table III for SNR equal to 2.5 dB
and 3.0 dB. The heavy-tailed nature of distribution of $Z$ is confirmed by the
data in Table III. A small fraction of decoding instances require a very large
decoding complexity. Motivated by this we consider stopping the sequential
decoder and declaring a decoder error when $Z$ exceeds a parameter
$Z_{\text{max}}$.

TABLE III: Frequency distribution of number of visits $Z$ Number of visits | 2.5 dB (%) | 3.0 dB (%)
---|---|---
$Z\leq 2^{10}$ | 92.8475 | 98.1415
$2^{10}<Z\leq 2^{11}$ | 3.6951 | 1.1643
$2^{11}<Z\leq 2^{12}$ | 1.7146 | 0.4120
$2^{12}<Z\leq 2^{13}$ | 0.8982 | 0.1707
$2^{13}<Z\leq 2^{14}$ | 0.4551 | 0.0679
$2^{14}<Z\leq 2^{15}$ | 0.2369 | 0.0279
$2^{15}<Z\leq 2^{16}$ | 0.1060 | 0.0115
$2^{16}<Z\leq 2^{17}$ | 0.0388 | 0.0032
$2^{17}<Z\leq 2^{18}$ | 0.0073 | 0.0009
$2^{18}<Z$ | 0.0005 | 0.0001

Fig. 5 displays the trade-off between FER and $Z_{\text{max}}$ at two
different SNR values. Such figures prove to be useful in deciding the value of
$Z_{\text{max}}$. For example, suppose the goal is to match the FER
performance of the (128,64) 5G polar code at 2.5 dB SNR. From Fig. 4, we read
the target FER value as $6\times 10^{-3}$. Fig. 5 indicates that this FER
target is achievable with $Z_{\text{max}}=2^{14}=16384$. Fig. 6 confirms that
the resulting bounded-complexity Fano decoder does indeed achieve the target
FER at 2.5 dB for a PAC$(128,64)$ code. In comparison, the CA-SCL decoder for
the 5G polar code in Fig. 4 has a complexity of 9600 node visits.

Figure 5: FER v. $Z_{\text{max}}$ for Fano decoder. Figure 6: Performance and
complexity of Fano decoder with $Z_{\text{max}}=16384$.

##  VI Conclusion

We showed that PAC codes under sequential decoding have the potential to
improve the performance of 5G polar codes. The main drawback of sequential
decoding is its variable complexity. We addressed this problem by introducing
a bound on the complexity of sequential decoding. In this connection, one may
consider limiting the total number of node visits by introducing per-level
limits on the number of node visits to reduce the computational complexity
still further without significantly degrading FER performance.

We considered PAC codes based only on the RM design rule for selection of the
data index set $\mathcal{A}$. The RM design rule maximizes the minimum
distance $d_{\text{min}}$ of PAC codes as shown in [13], which is important
for FER performance at high SNR. On the other hand, the selection of
$\mathcal{A}$ in accordance with the RM rule is not exactly compatible with
the channel polarization created by the polar transform, which results in high
computational complexity. Another subject for future study is to compare
alternative design rules for $\mathcal{A}$ with respect to performance and
complexity.

The connection sequence $\mathbf{c}$ also has a significant impact on the
performance of PAC codes through its effect on the multiplicity
$A_{d_{\text{min}}}$ of minimum-distance codewords. The generator sequence
$\mathbf{c}=3211$ used in the PAC code with dimension $K=29$ was obtained by
an ad-hoc search method with the goal of minimizing $A_{d_{\text{min}}}$. The
choice $\mathbf{c}=3211$ improved the FER performance significantly compared
to our default generator sequence $\mathbf{c}=133$. Identifying good generator
sequences in connection with other design parameters of PAC codes remains
another subject for future study.

## References

  * [1] E. Arıkan, “Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels,” _IEEE Trans. Inf. Theory_ , vol. 55, no. 7, pp. 3051–3073, 2009.
  * [2] ——, “From sequential decoding to channel polarization and back again,” _arXiv preprint arXiv:1908.09594_ , 2019.
  * [3] M. Rowshan, A. Burg, and E. Viterbo, “Polarization-adjusted convolutional (pac) codes: Fano decoding vs list decoding,” _arXiv preprint arXiv:2002.06805_ , 2020.
  * [4] H. Yao, A. Fazeli, and A. Vardy, “List decoding of arikan’s PAC codes,” _arXiv preprint arXiv: 2005.13711_ , 2020.
  * [5] R. Fano, “A heuristic discussion of probabilistic decoding,” _IEEE Trans. Inf. Theory_ , vol. 9, p. 64–74, 1963.
  * [6] R. G. Gallager, _Information Theory and Reliable Communication_. New York: Wiley, 1968\.
  * [7] K. Zigangirov, “Some sequential decoding procedures,” _Problemy Peredachi Informatsii_ , vol. 2, no. 4, pp. 13–25, 1966.
  * [8] F. Jelinek, “Fast sequential decoding algorithm using a stack,” _IBM journal of research and development_ , vol. 13, no. 6, pp. 675–685, 1969.
  * [9] Y. Polyanskiy, H. Poor, and S. Verdu, “Channel coding rate in the finite blocklength regime,” _IEEE Transactions on Information Theory_ , vol. 56, no. 5, pp. 2307–2359, May 2010, 00443.
  * [10] I. Jacobs and E. Berlekamp, “A lower bound to the distribution of computation for sequential decoding,” _IEEE Transactions on Information Theory_ , vol. 13, no. 2, pp. 167–174, Apr. 1967.
  * [11] E. Arıkan, “An inequality on guessing and its application to sequential decoding,” _IEEE Transactions on Information Theory_ , vol. 42, no. 1, pp. 99–105, Jan. 1996.
  * [12] 3GPP, “NR; Multiplexing and Channel Coding,” _http://www.3gpp.org/DynaReport/38-series.htm, Tech. Rep. TS 38.212, June 2018, Rel. 15._
  * [13] B. Li, H. Zhang, and J. Gu, “On pre-transformed polar codes,” _arXiv preprint arXiv:1912.06359_ , 2019.


LUPM 22-024

BONN-TH-2022-20

$M_{W}$, Dark Matter and $a_{\mu}$ in the NMSSM

Florian Domingoa, Ulrich Ellwangerb, Cyril Hugoniec
a Bethe Center for Theoretical Physics $\&$ Physikalisches Institut der
Universität Bonn, Nussallee 12, D-53115 Bonn, Germany
<EMAIL_ADDRESS>
b IJCLab, CNRS/IN2P3, University Paris-Saclay, F-91405 Orsay, France
<EMAIL_ADDRESS>
c LUPM, UMR 5299, CNRS/IN2P3, Université de Montpellier, F-34095 Montpellier,
France
<EMAIL_ADDRESS>
We study regions in the parameter space of the NMSSM which are able to
simultaneously explain the current measured values for the $W$ mass $M_{W}$
and the muon anomalous magnetic moment $a_{\mu}$, and provide a dark matter
relic density consistent with the observations as well as constraints from
detection experiments. The corresponding regions feature light charginos and
sleptons in the 100 GeV$-$1 TeV range, at least some of them with masses below
175 GeV such that the electroweakly-interacting SUSY particles generate
sufficiently large contributions to $M_{W}$. The LSP is always singlino-like
with a mass below 150 GeV, and could possibly remain invisible even at future
detection experiments. Decays of electroweak sparticles proceed through
cascades via staus and/or sleptons which makes their detection challenging. We
propose benchmark points for future searches of such sparticles. The lightest
CP-even scalar may have a mass in the 95$-$98 GeV range with, however, modest
signal rates in view of the mild excesses reported in this range at LEP and by
CMS at the LHC.

##  1 Introduction

An updated measurement of the $W$ boson mass by the CDF collaboration at the
Tevatron [1] returned the result $M_{W}=80.4335\pm 0.0094$ GeV, with
substantially reduced uncertainties compared to earlier measurements and
averages. Combining this recent CDF result with former measurements at LEP,
the Tevatron and the LHC, the new world average becomes [2]

$M_{W}^{\text{Exp}}=80.4133\pm 0.0080\ \text{GeV}\;.$ (1.1)

Of course, the origin of the tension between the new CDF result and the
average of the former measurements from LEP, the Tevatron and the LHC remains
to be analysed before any strong implications are drawn concerning the
viability of models of particle physics.

On the other hand, a fit of electroweak precision data within the Standard
Model (SM) gives [3]

$M_{W}^{\text{SM}}=80.3545\pm 0.0057\ \text{GeV}\;.$ (1.2)

If of merely statistical nature, the difference $\Delta M_{W}$ between
$M_{W}^{\text{Exp}}$ and $M_{W}^{\text{SM}}$ settles at $\sim 6\,\sigma$ and
calls (if taken at face value) for some contribution from physics beyond the
Standard Model (BSM) to the $W$ boson mass.

Theories beyond the SM are more particularly motivated if they simultaneously
explain additional phenomena that are also impossible or difficult to explain
within the SM. The presence of dark matter – compatible with present
constraints from direct detection experiments – is a striking (perhaps the
most striking) phenomenon of this kind. Another example is the anomalous
magnetic moment of the muon $a_{\mu}\equiv(g_{\mu}-2)/2$, where the measured
value (combining the BNL E821 [4] and the Fermilab Muon g-2 [5] experiments)
deviates by $4.2\,\sigma$ from the SM.

Supersymmetric (SUSY) extensions of the SM – in particular non-minimal
versions thereof – can address the latter two deviations from the SM
phenomenology, while being in addition motivated by an alleviation of the
hierarchy problem of the SM. (Existing lower bounds on masses of
supersymmetric particles from fruitless searches prevent SUSY from addressing
the hierarchy problem in a fully exhaustive way; some fine tuning at least in
the percent range would thus persist.)

R-parity conserving SUSY nearly automatically leads to the existence of
weakly- or feebly-interacting dark matter in the form of the LSP (lightest
SUSY particle), usually a mixture of binos, winos or higgsinos within the
Minimal Supersymmetric extension of the SM (MSSM), or in the form of singlinos
within the Non-Minimal Supersymmetric extension of the SM (NMSSM). However, in
the MSSM, it is difficult to reconcile the observed relic density $\Omega
h^{2}=0.1200\pm 0.0012$ [6] with upper limits from [7, 8, 9, 10, 11, 12] on
direct detection cross sections [13] (though not impossible; see e.g. [14, 15,
16].)

On the other hand, the observed relic density and the upper limits on direct
detection cross sections can be explained without contradiction in the NMSSM
when considering the scenario of singlino-dominated dark matter [17, 18, 19,
20, 21, 22, 23, 25, 24, 26, 27, 28, 29, 30]. In fact, also the computed
anomalous magnetic moment $a_{\mu}$ of the muon can be brought near the
corresponding measured value in the NMSSM [28, 30, 31, 32]. ($a_{\mu}$ can
also be explained in the MSSM, see e.g. [34, 35, 33, 37, 36], at least as long
as one allows for a dark matter abundance below the observed relic density.)

Furthermore, the extended SUSY spectrum also affects the predictions for the
$W$ boson mass. Corresponding calculations have been performed in [38, 39, 40,
41, 42, 43, 37, 33, 36] and references therein for the MSSM, and for the NMSSM
in [44, 45, 46]. The possibility to describe the new world average for $M_{W}$
together with the anomalous magnetic moment of the muon and constraints on the
relic density was discussed in [36] for the MSSM, and in [30] for the NMSSM.

Using FlexibleSUSY [46] for the calculation of $M_{W}$ the authors in [30]
conclude that viable points in the parameter space of the NMSSM which
simultaneously explain the new world average for $M_{W}$, $a_{\mu}$ and
$\Omega h^{2}$ feature a LSP $\chi^{0}_{1}$ with a mass of $50-60$ GeV or
$150-300$ GeV with a spin-independent direct detection cross section on
protons $\sigma^{SI}_{p}$ with $\sigma^{SI}_{p}\;\raise
1.29167pt\hbox{$>$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}\;10^{-11}$ pb,
which can thus be tested by the future direct detection experiments
XENONnT[47], LZ [48] and DARWIN [49].

In the present paper we explore the NMSSM in the light of the new value for
$M_{W}^{\text{Exp}}$. We use an implementation of the calculation of $M_{W}$
in the code NMSSMTools [50, 51, 52] that was described in [44] (see Section
2). The dark matter relic density, direct detection cross sections and
comparisons to upper limits (dominant are the ones on $\sigma^{SI}_{p}$ from
[12]) are computed with the help of the program micrOMEGAs [53]. The
calculation of $a_{\mu}$ provided within NMSSMTools follows the description in
[54].

We find new viable points in the parameter space of the NMSSM which
simultaneously explain $M_{W}$, $a_{\mu}$ and $\Omega h^{2}$ (without
obviously violating existing collider constraints), many of which have a
direct detection cross section $\sigma^{SI}_{p}$ below the reach of future
direct detection experiments [47, 48, 49] and even below the neutrino floor
[55]. The masses of at least some of the electroweak sparticles (sleptons,
staus, charginos) are in the 100-150 GeV range. Due to the dominantly singlino
nature of the LSP (hence its reduced couplings to electroweakly-charged
(s)particles), practically all sparticle decays proceed via the NLSP, often
the tau sneutrino $\tilde{\nu}_{\tau}$. The corresponding cascade decays, and
the small mass difference between the electroweak sparticles and the NLSP,
reduce the impact of typical electroweakino (= chargino, neutralino and
slepton) searches at the LHC (or LEP) on such spectra, which we discuss in
Section 3. Limits from LHC runs at 8 and 13 TeV are verified using
SModelS-2.2.0 [56, 57, 58, 59]. In Section 4 we present summaries of viable
points and the corresponding decay cascades for some benchmark points, and
conclude in Section 5.

##  2 $\Delta M_{W}$ in the NMSSM

For the particle content and the Lagrangian of the NMSSM we refer the reader
to the reviews in [60, 61]. Compared to the MSSM, the NMSSM contains a 5th
neutralino degree of freedom (the singlino) and extra mostly $SU(2)$-singlet-
like CP-even and CP-odd scalars. The mixing angles of these singlet-like
states with their MSSM-like counterparts are proportional to a dimensionless
coupling $\lambda$, i.e. these states decouple for $\lambda\to 0$. This
feature is particularly relevant for the neutralino sector where the singlino,
if it is the LSP, represents a good dark matter candidate: its relic density
can coincide with the observed one using co-annihilation or annihilation via a
resonance (e.g. a singlet-dominated scalar) in the s-channel. On the other
hand, its direct detection cross section can be very small for reduced
higgsino, wino and bino components, or for destructive interferences between
$SU(2)$-singlet and SM-like contributions.

In this paper, we employ the calculation of $M_{W}$ in the NMSSM that was
described in [44] and which is incorporated within NMSSMTools-6.0.0 [50, 51,
52] as the subroutine MWNMSSM.f. We briefly summarize the main features of
this implementation below.

First, we remind the reader that the structure of the electroweak theory
correlates observables such that the fine-structure constant in the Thomson
limit $\alpha$, the Fermi constant $G_{\mu}$, extracted from the muon decay
$\mu\to e\nu_{\mu}\bar{\nu}_{e}$, and the masses $M_{W,Z}$ of the massive
gauge bosons cannot be chosen independently from one another. Selecting
$M_{W}$ as the observable that should be derived from the other three, this
correlation may be expressed as:

$M_{W}^{2}=\frac{M_{Z}^{2}}{2}\left\\{1+\left[1-\tfrac{2\sqrt{2}\pi\alpha}{G_{\mu}M_{Z}^{2}}(1+\Delta
r)\right]^{1/2}\right\\}$ (2.1)

where $\Delta r$ encodes the (non-QED) radiative contributions to the muon
decay, consisting of self-energy, vertex and box corrections (and itself
implicitly dependent on $M_{W}$).

In the SM, contributions to $\Delta r$ are known up to leading four-loop
order, for an estimated higher-order uncertainty reducing to a few MeV at the
level of $M_{W}$ [62]. This SM prediction is conveniently parametrized in a
fit formula [62, 63].

Beyond the SM, it is convenient to take advantage of the higher accuracy
achieved in contributions to $\Delta r$ of SM-type via the splitting

$\Delta r^{\text{BSM}}=\Delta r^{\text{SM}}+\Delta r^{\text{NP}}\;,\quad\Delta
r^{\text{NP}}=\Delta r^{\text{BSM}}-\Delta r^{\text{SM}}\,.$ (2.2)

Here, the first term of the right-hand side of the first equation is given by
the four-loop prediction in the SM, while $\Delta r^{\text{NP}}$ is
diagrammatically evaluated at a less precise order (typically one-loop), with
many diagrams canceling out due to the spectrum overlap of the new-physics
model with the SM (in any phenomenologically realistic BSM theory). In the
(N)MSSM, $\Delta r^{\text{NP}}$ receives one-loop contributions from the
modified Higgs sector and, more significantly, from the additional SUSY
sector: explicit expressions are provided in e.g. [38, 64]. Partial two-loop
order corrections are also included in MWNMSSM.f via the parameter
$\Delta\rho=\frac{\Sigma^{Z}(0)}{M_{Z}^{2}}-\frac{\Sigma^{W}(0)}{M_{W}^{2}}$
(where $\Sigma^{Z,W}(0)$ denote the transverse parts of the $Z$ and $W$ boson
self-energies at vanishing momenta, respectively, including terms mixing SM
and BSM contributions or gluonic corrections to squark loops): we refer the
reader to the discussion in [44] and references therein.

In fact, we specialize below on spectra with a massive squark sector (as
hinted by the constraints from the LHC): from dimensional analysis, $\Delta
r^{\text{NP}}\approx\tfrac{M_{W}^{2}}{16\pi^{2}\Lambda_{\text{NP}}^{2}}$,
where $\Delta r^{\text{NP}}$ represents new-physics contributions intervening
at various scales $\Lambda_{\text{NP}}$. The effect targeted by the difference
between Eqs.(1.1) and (1.2) is of order $\Delta r^{\text{NP}}\approx 5\cdot
10^{-3}$ and it is obvious that particles at the TeV scale (and beyond) cannot
provide a significant contribution. Since contributions from the extended
Higgs sectors in SUSY models are also severely constrained, the new physics
responsible for the shift in $\Delta r$ needs to intervene at the electroweak
scale and consists of particles interacting in a purely electroweak fashion.

In view of Eq.(2.2), we need to combine the uncertainty associated with
$\Delta r^{\text{SM}}$, of higher-order and parametric natures, and that of
the BSM evaluation, of essentially higher-order type. Here, for simplicity, we
will combine the uncertainty on the experimental average in Eq.(1.1), that on
the SM evaluation (compatible with Eq.(1.2)) and a $10\%$ fluctuation at the
level of the BSM contribution in quadrature: for a BSM effect shifting the
$M_{W}$ prediction to the central value of the experimental average, this
corresponds to a global uncertainty of order $12$ MeV for $\Delta M_{W}\equiv
M_{W}^{\text{Exp}}-M_{W}^{\text{BSM}}$.

At this point, we stress the necessity of a consistent control over
electroweak-symmetry breaking effects in the diagrammatic calculation.
Contributions to $M_{W}$ should indeed vanish in the $SU(2)_{L}$-conserving
limit. In the case of squark loops, this implies a fine cancellation between
diagrams involving the components of the same $SU(2)_{L}$-doublet:

$\Delta
r^{\tilde{Q}}\sim\sum_{i}\big{[}R^{2}_{\tilde{t}}(i,1)f(M_{\tilde{t}_{i}}^{2})-R^{2}_{\tilde{b}}(i,1)f(M_{\tilde{b}_{i}}^{2})\big{]}\,,$
(2.3)

where $R_{\tilde{Q}}(i,1)$ denotes the $\tilde{Q}_{L}$ component contained in
the mass eigenstate $\tilde{Q}_{i}$ with mass $M^{2}_{\tilde{Q}_{i}}$ while
$f$ represents a loop function (depending on further parameters and masses).
Then, if one employs parameters originating from two different renormalization
schemes, such as pole-corrected squark masses together with
$\overline{\text{DR}}$ mixing angles (which formally introduces a mismatch of
higher (two-loop) order at the level of $\Delta r$), it may produce large
shifts of $M_{W}$ that are associated to a spurious breakdown of the
electroweak symmetry at high-energy.

One measurement of this artificial symmetry-violation is provided by the
squared mass splitting $\Delta
M^{2}_{LL}=\sum_{i}\big{(}R^{2}_{\tilde{t}}(i,1)M_{\tilde{t}_{i}}^{2}-R^{2}_{\tilde{b}}(i,1)M_{\tilde{b}_{i}}^{2}\big{)}$:
a departure of this quantity from its expected size at
$\mathcal{O}(M_{Z}^{2})$ – notably for squark masses in the (multi-)TeV range
– would indeed indicate an inordinate breaking of $SU(2)_{L}$ which is bound
to affect Eq.(2.3) as well111As the SLHA output of NMSSMTools provides squark
masses including QCD pole corrections while mixing matrices remain
$\overline{\text{DR}}$, such problems could typically occur if one were to
compute $\Delta M_{W}$ externally from the on-shell spectrum. As it is,
MWNMSSM.f runs internally, with access to strict $\overline{\text{DR}}$
parameters. On the other hand, we suspect an artifact of this type to be at
the origin of the results of [30] where a comparatively heavy electroweakino
spectrum seems able to explain the deviation in $M_{W}$.. As we explained
before, dimensional analysis indicates that squarks at the TeV scale cannot
significantly contribute to $\Delta M_{W}$.

To conclude, the shift $\Delta M_{W}$ can only originate from a comparatively
light electroweak sector (electroweakinos $\equiv$ charginos, neutralinos,
left-handed sleptons and staus) with masses in the $100-200$ GeV range, a
feature previously observed in the MSSM in [43].

##  3 Experimental Constraints

First of all, masses of light electroweakinos have to satisfy lower bounds
from LEP [65, 66]. Lower bounds on the chargino mass depend on the decay modes
of this particle: assuming $\chi^{\pm}_{1}\to W^{(*)}+\chi^{0}_{1}$ to be
dominant, constraints from LEP imply $M_{\chi^{\pm}_{1}}>103.5$ GeV [65].
However, for most of the NMSSM points considered here, charginos decay through
a different channel (see below) in which case we impose
$M_{\chi^{\pm}_{1}}>100$ GeV. The same constraint is imposed on slepton
$\tilde{\ell}$ masses (selectrons and smuons are assumed degenerate); in the
case of stau ($\tilde{\tau}$) masses we require $M_{\tilde{\tau}_{1}}>93.2$
GeV [66].

Constraints from searches at the LHC are verified using SModels-2.2.0 [56, 57,
58, 59]. Most searches for charginos $\chi^{\pm}_{1}$ and neutralinos
$\chi^{0}_{i}$ ($i\geq 2$) at the LHC assume dominant decays
$\chi^{\pm}_{1}\to W^{(*)}+\chi^{0}_{1}$ and $\chi^{0}_{i}\to
Z^{(*)}/H_{SM}+\chi^{0}_{1}$. However, given the presence of light lepton-
sneutrinos, tau-sneutrinos, a light mostly singlet-like Higgs state $H_{1}$
and the small couplings of the mostly singlet-like $\chi^{0}_{1}$ for the
parameter points on which we focus, the branching fractions for
$\chi^{\pm}_{1}\to W^{(*)}+\chi^{0}_{1}$ and/or $\chi^{0}_{i}\to
Z^{(*)}/H_{SM}+\chi^{0}_{1}$ are always small, resulting in limited
sensitivity of the “classical” electroweakino searches.

We find that the following searches for electroweakinos implemented in
SModels-2.2.0 play a dominant role in the scenario described above:

– Searches for charginos and sleptons at 8 TeV by ATLAS in [67, 68];

– Searches for charginos and sleptons at 13 TeV by ATLAS in [69, 70, 71, 72,
73, 74] and by CMS in [75, 76, 77, 78, 79];

– Searches for staus by ATLAS in [80].

Further potentially relevant searches for light electroweakinos at the LHC –
not implemented in SModels-2.2.0 – have been performed by CMS in [81, 82, 83,
84] and by ATLAS in [85, 86]. (For a recent review of searches for
electroweakinos at the LHC see [87].) For a variety of reasons many of the
obtained limits do not apply to the light electroweakinos considered here:
assumed $\chi^{0}_{1}$ masses in [81, 86] are smaller, assumed chargino masses
in [83] are larger, assumed branching fractions for
$\chi^{0}_{i}\to\chi^{0}_{1}+Z/H_{125}$ in [85, 82] are larger, and assumed
branching fractions for $\chi^{0}_{i}\to\ell/\tau+\tilde{\ell}/\tilde{\tau}$
for $\tilde{\ell},\tilde{\tau}$ searches in [82] are larger than the values
obtained here after imposing the constraints implemented in SModels-2.2.0.
Remaining relevant limits on purely left-handed stau-pairs from Fig. 7b in
[84] as function of $M_{\chi^{0}_{1}}$, and on purely left-handed slepton
pairs from Fig. 16 in [85] and Fig. 7 in [86] are checked separately. We found
that notably the constraints on purely left-handed sleptons from Fig. 7 in
[86] eliminate some points in the NMSSM parameter region under discussion. But
again, limits on left-handed sleptons do not apply if these undergo dominantly
cascade decays e.g. via lighter staus or charginos which reduce the
$E_{\text{T}}^{\text{miss}}$ of the events.

$\delta a_{\mu}\equiv a_{\mu}^{\text{Exp}}-a_{\mu}^{\text{SM}}$ is computed
following [54]. The corresponding experimental uncertainty is $\Delta
a_{\mu}^{\text{Exp}}=\pm 0.41\times 10^{-9}$ [5], while the uncertainty from
its determination within the SM is $\Delta a_{\mu}^{\text{SM}}=\pm 0.43\times
10^{-9}$ [88]. We add an uncertainty from (parameter dependent) NMSSM specific
contributions to $\delta a_{\mu}$; the combination of uncertainties amounts to
about $\pm 1.55\times 10^{-9}$ at the $2\,\sigma$ level.

The presence of light electroweakinos also implies possible effects in the
precise properties of SM particles, such as the $Z$\- or the observed Higgs
bosons. Nevertheless, the leptonic branching ratios of the $Z$-bosons measured
at LEP [89] or the Higgs couplings to leptons deduced from the LHC [90, 91]
measurements come with large experimental uncertainties as compared to the
magnitude of the effects discussed here, for $M_{W}$ or $a_{\mu}$: the impact
of these observables is thus currently negligible.

Finally, we also require that viable points satisfy (at the $2\,\sigma$ level)
all the phenomenological requirements implemented within
NMSSMTools222NMSSMTools implements a collection of constraints from Higgs and
SUSY searches (LEP, Tevatron, LHC) and low-energy observables (flavor,
quarkonia, electroweak) which are listed on the website [52]., in particular
on the properties of the SM-like Higgs, flavor observables [92, 93], the dark
matter relic density as measured by Planck [6], and constraints on direct
detection cross sections using micrOMEGAs [53].

##  4 Results and Benchmark Points

We scanned the NMSSM parameter space using dedicated Monte Carlo programs that
lead to several hundred thousand points satisfying all constraints (at the
$2\sigma$ level, as explained in the previous sections), only suitable subsets
thereof are shown in the figures below. The ranges of the NMSSM specific
couplings, soft SUSY breaking terms and masses (with focus on the electroweak
sector) are shown in Table 1. (Soft SUSY breaking terms and masses are given
in GeV. For the Lagrangian of the NMSSM see [60, 61].)

$\lambda$ | $\kappa$ | $\tan\beta$ | $A_{\lambda}$ | $A_{\kappa}$
---|---|---|---|---
$0.013-0.35$ | $0.001-0.019$ | $4.5-36$ | $15-4000$ | -$196-0$
$\mu_{\text{eff}}$ | $M_{\text{bino}}$ | $M_{\text{wino}}$ | $M_{\text{gluino}}$ | $A_{\text{top}}$
$118-625$ | $69-4000$ | $96-4000$ | $100-4000$ | -$4000\,-$ -1400
$A_{\tau}$ | $A_{\mu}$ | $m_{\ell_{L}}$ | $m_{\tau_{L}}$ | $m_{\tau_{R}}$
-$7-4000$ | -$2-4000$ | $89-1130$ | $83-4000$ | $83-4000$
$m_{Q3/U3}$ | $m_{Q2/U2}$ | $m_{D2/D3}$ | $m_{E2/E3}$ |
$12-4000$ | $36-4000$ | $0-4000$ | $83-4000$ |
Table 1: The ranges of the NMSSM specific couplings, soft SUSY breaking terms
and masses (soft SUSY breaking terms and masses in GeV) leading to points
satisfying all constraints.

The LSP is always a mostly singlino-like neutralino with a mass $\sim 20-150$
GeV, and a singlino component $>0.98$. Consequently its spin-independent
direct detection cross sections on protons $\sigma^{SI}_{p}$ for dark matter
experiments are small, possibly below $2.5\times 10^{-13}$ pb, i.e. below the
sensitivities of the future experiments XENONnT[47], LZ [48] and DARWIN [49]
as well as the neutrino floor. Dark matter annihilation dominantly proceeds
via $Z$ or the SM Higgs boson in the s-channel, or via sneutrino co-
annihilation. These cases are clearly identifiable in Fig. 1 showing
$\sigma^{SI}_{p}$ as function of the mass of the LSP $\chi^{0}_{1}$, where
$M_{\chi^{0}_{1}}$ equals about half of the mass of an s-channel resonance, or
more in the case of co-annihilation.333We note in passing that the dark matter
phenomenology significantly contrasts with that identified in [30]. We believe
that the scanning procedure is at the origin of these divergent results, our
scanning algorithm allowing for a better coverage of the NMSSM parameter
space.

Figure 1: Direct detection cross section $\sigma^{SI}_{p}$ as function of the mass of the LSP $\chi^{0}_{1}$. |
---|---
Fig. 2a |  Fig. 2b
Figure 2: $\delta a_{\mu}$ (left) and $M_{W}$ (right) as function of the left-
handed smuon mass $m_{\tilde{\mu}_{L}}$. The colors indicate the dominant
decay modes of the chargino $\chi^{\pm}_{1}$: blue for
$\ell+\tilde{\nu}_{\ell}$, red for $\tau+\tilde{\nu}_{\tau}$, green for
$\chi^{0}_{1}+W^{(*)}$, yellow for $\chi^{0}_{2}+W^{(*)}$, orange for
$\nu_{\ell}+\tilde{\ell}$ and black for $\nu_{\tau}+\tilde{\tau}$.

In Figs. 2 we show $\delta a_{\mu}\equiv
a_{\mu}^{\text{Exp}}-a_{\mu}^{\text{SM}}$ and $M_{W}$ as function of the left-
handed smuon mass $m_{\tilde{\mu}_{L}}$ for viable points. (Recall that we
assume degenerate selectrons and smuons.) Fig. 2a is consistent with the idea
that a lower limit on $\delta a_{\mu}$ (depending on the combined experimental
and theoretical uncertainties) results in an upper bound on
$m_{\tilde{\mu}_{L}}$. Similarly, from Fig. 2b one infers that only a
$2\,\sigma$ uncertainty on $M_{W}$ larger than 7 MeV allows for
$m_{\tilde{\mu}_{L}}>$ 700 GeV.

Points in the parameter space of the NMSSM which satisfy all the above
constraints still differ in the dominant decay modes of the lightest chargino
$\chi^{\pm}_{1}$ among which we count $\ell+\tilde{\nu}_{\ell}$,
$\tau+\tilde{\nu}_{\tau}$, $\chi^{0}_{1,2}+W^{(*)}$, $\nu_{\tau}+\tilde{\tau}$
and only occasionally $\nu_{\ell}+\tilde{\ell}$. In view of future
investigations at the LHC (addressing searches of the chargino or of heavier
particles decaying through it), the identification of the dominant decay modes
of $\chi^{\pm}_{1}$ will play an important role. Therefore we specify these
decay modes with distinct colors in the Figures 2-3: blue for
$\ell+\tilde{\nu}_{\ell}$, red for $\tau+\tilde{\nu}_{\tau}$, green for
$\chi^{0}_{1}+W^{(*)}$, yellow for $\chi^{0}_{2}+W^{(*)}$, orange for
$\nu_{\ell}+\tilde{\ell}$ and black for $\nu_{\tau}+\tilde{\tau}$.

In Figs. 3 we show the masses of the chargino $\chi^{\pm}_{1}$ and the left-
handed stau $\tilde{\tau}_{L}$ as function of the left-handed smuon mass for
viable points. We see that $\chi^{\pm}_{1}$ and smuon masses, as well as
$\tilde{\tau}_{L}$ and smuon masses, cannot be simultaneously be large
($\;\raise 1.29167pt\hbox{$>$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}\;300$
GeV and $\;\raise
1.29167pt\hbox{$>$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}\;630$ GeV,
respectively). The reason is that at least some electroweakinos must be light
in order to generate sufficiently large radiative corrections to $M_{W}$. It
is remarkable that the possible existence of such comparatively light
electroweakly-interacting exotic particles is compatible with the current
collider limits.

|
---|---
Fig. 3a |  Fig. 3b
Figure 3: Masses of the chargino $\chi^{\pm}_{1}$ (Fig. 3a) and the left-
handed stau $\tilde{\tau}_{L}$ (Fig. 3b) as function of the left-handed smuon
mass. The color code is as in Fig. 2.

The viable points in the parameter space of the NMSSM obtained here always
feature a mostly singlet-like Higgs scalar $H_{1}$ with a mass below 125 GeV.
It is checked in NMSSMTools that its direct production rate satisfies
constraints from LEP and the LHC, and that the branching fractions of the SM
Higgs to $H_{1}+H_{1}$ satisfy constraints from ATLAS and CMS. In Fig. 4 we
show the magnitude of the singlet component squared $S_{13}^{2}$ as function
of its mass and observe that, for the viable points, $S_{13}^{2}$ is always
above 0.93, i.e. the doublet component squared (needed in its direct
production of this state in any process at LEP and the LHC) is always below
0.07, implying a challenging detectability at colliders.

Figure 4: $S_{13}^{2}$ as function of $M_{H_{1}}$. The colors indicate the
dominant decay modes of $H_{1}$: $b\bar{b}$ for red points, $A_{1}A_{1}$ for
blue points, 2 jets (gluon gluon or $c\bar{c}$) for green points, $WW^{*}$ for
yellow points and $\tau\bar{\tau}$ for black points (for $M_{H_{1}}$ below the
$b\bar{b}$ threshold).

Interestingly enough, the mass of $H_{1}$ possibly falls in the $95-98$ GeV
range. A mild excess in the $b\bar{b}$ channel in this mass range was observed
at LEP [94, 95], corresponding to a signal strength of $\sim 10\%$ in units of
the signal strength of a SM Higgs in the same channel. Moreover, the CMS
collaboration has observed a mild excess in the $\gamma\gamma$ channel in this
mass range both at 8 and 13 TeV [96]. Locally, the excess corresponds to a
signal strength of $\sim 0.5\pm 0.18$ in units of the signal strength of a SM-
like Higgs boson in the same channel. Within the NMSSM, the excess at LEP was
first discussed in [97]. A possible description of the excess at CMS within
the NMSSM was proposed in [98, 99, 100, 101, 102, 103]. Clearly, the
production of a mostly singlet-like $H_{1}$ at LEP or the LHC requires some
singlet-doublet mixing. Such a mixing involving the SM-like Higgs boson at 125
GeV reduces its couplings $\kappa_{W,Z}$ to the electroweak gauge bosons. Such
a reduction is severely constrained by the recent measurements of properties
of the SM Higgs boson by ATLAS [104] and CMS [105] which, taken together,
imply $\kappa_{W,Z}>0.966$ at the $2\,\sigma$ level. Applying this constraint,
we find that $H_{1}$ in the $95-98$ GeV range can have a reduced signal
strength of $\sim 6\%$ in the $b\bar{b}$ channel at LEP and, simultaneously, a
reduced signal strength of $\sim 8\%$ in the $\gamma\gamma$ channel at the
LHC. (The constraint on the $H_{1}-H_{125}$ mixing angle also explains why
$S_{13}^{2}$ is bounded from below by $\sim 0.93$ in Fig. 4.) On the other
hand, we could not discover points leading to a sizable signal in the
$\tau^{+}\tau^{-}$ channel, as motivated in the same mass range by the
searches from CMS [106].

Stop decay cascades are initiated by decays into either $b$ quarks + charginos
or top quarks + neutralinos $\chi^{0}_{2,3}$, but charginos and neutralinos
$\chi^{0}_{2,3}$ often undergo lepton-rich or tau-rich subsequent decays.
(Cascade decays via the stau sector have also been found in [36] for viable
points in the MSSM where, however, the relic density is below the observed
one.) The extra MSSM-like neutral and charged Higgs bosons also have masses in
the $1-4$ TeV range.

In order to illustrate the various decay modes in the electroweak sector we
show the masses and branching fractions for four benchmark points BP1-BP4 in
the Tables below. These points differ in the dominant decay modes of the
lighter chargino $\chi^{\pm}_{1}$. (BP2 and BP4 are invisible even in future
dark matter detection experiments [47, 48, 49].)

| $m_{\chi^{\pm}_{1}}$ | $m_{\chi^{\pm}_{2}}$ | $m_{\chi^{0}_{1}}$ | $m_{\chi^{0}_{2}}$ | $m_{\chi^{0}_{3}}$ | $m_{\tilde{\ell}_{L}}$ | $m_{\tilde{\nu}_{\ell}}$ | $m_{\tilde{\tau}_{1}}$ | $m_{\tilde{\nu}_{\tau}}$
---|---|---|---|---|---|---|---|---|---
BP1 | 107 | 307 | 58 | 95 | 114 | 147 | 125 | 97 | 105
BP2 | 139 | 949 | 58 | 75 | 147 | 113 | 83 | 100 | 95
BP3 | 109 | 258 | 44 | 107 | 167 | 140 | 116 | 210 | 196
BP4 | 114 | 387 | 44 | 84 | 116 | 176 | 158 | 110 | 79
| $M_{H_{1}}$ | $M_{A_{1}}$ | $\lambda$ | $\kappa$ | $A_{\lambda}$ | $A_{\kappa}$ | $\sigma^{SI}_{p}$ | $\delta a_{\mu}$ | $M_{W}$
BP1 | 54 | 11 | 0.089 | 0.0095 | 2526 | -0.6 | $1.87\times 10^{-12}$ | $2.98\times 10^{-9}$ | 80.402
BP2 | 53 | 27 | 0.036 | 0.0075 | 3310 | -8.5 | $2.18\times 10^{-15}$ | $2.04\times 10^{-9}$ | 80.406
BP3 | 41 | 29 | 0.123 | 0.013 | 2245 | -10.5 | $2.77\times 10^{-11}$ | $2.99\times 10^{-9}$ | 80.404
BP4 | 47 | 56 | 0.211 | 0.013 | 387 | -42.5 | $3.90\times 10^{-16}$ | $2.40\times 10^{-9}$ | 80.401
Table 2: Sparticle and mostly singlet-like Higgs boson masses (in GeV), NMSSM
specific parameters, the spin-independent direct detection cross section of
the LSP on protons (in pb), $\delta a_{\mu}$ and $M_{W}$ for four benchmark
points. Leptons $\ell$ denote both electrons and muons.

BP1 corresponds to a case where $\chi^{+}_{1}$ and $\chi^{0}_{2}$ have
sizeable wino components 0.9 and 0.5, respectively. The dominant decay of
$\chi^{+}_{1}$ is $\chi^{+}_{1}\to\nu_{\tau}+\tilde{\tau}^{+}$ with a
branching ratio of $72\%$, $\tilde{\tau}^{+}$ decays to
$\tau^{+}+\chi^{0}_{1}$ with a branching ratio of $100\%$. $\chi^{0}_{2}$
decays dominantly into 3-body channels
$\nu_{\tau}+\bar{\nu}_{\tau}+\chi^{0}_{1}$ (to $41\%$) and
$\tau^{+}+\tau^{-}+\chi^{0}_{1}$ (to $39\%$). Hence BP1 is difficult to detect
in standard search channels for charginos/neutralinos. Given the
$\tilde{\tau}_{1}$ and $\chi^{0}_{1}$ masses shown in Table 2, the dedicated
searches for staus by ATLAS in [80] and CMS in [81, 84] are also not sensitive
to BP1.

In the case of BP2, the mostly higgsino-like $\chi^{+}_{1}$ decays into
sleptons, the tau sneutrino and stau with branching fractions shown in Table
3. The same holds for the higgsino-like neutralinos $\chi^{0}_{3,4}$ which are
dominantly produced in association with $\chi^{+}_{1}$. Sleptons, tau
sneutrino and stau decay into the bino-like neutralino $\chi^{0}_{2}$ which
finally decays into $\chi^{0}_{1}+W^{*}$. Such lengthy cascade decays into
lepton/tau-rich final states are challenging to detect. Interestingly, mild
(non-significant) excesses have been observed in search channels for $2-3$
leptons plus $E_{T}^{\text{miss}}$ by ATLAS in [85, 74] and by CMS in [79].

$BR(\chi^{+}_{1}\to\ell^{+}+\tilde{\nu}_{\ell})$ | $BR(\chi^{+}_{1}\to\tau^{+}+\tilde{\nu}_{\tau})$ | $BR(\chi^{+}_{1}\to\nu_{\tau}+\tilde{\tau}^{+})$ | $BR(\chi^{0}_{3}\to\tau^{+}+\tilde{\tau}^{-})$
---|---|---|---
0.172 | 0.419 | 0.212 | 0.149
$BR(\chi^{0}_{3}\to\nu_{\ell}+\tilde{\nu}_{\ell})$ | $BR(\chi^{0}_{4}\to\tau^{+}+\tilde{\tau}^{-})$ | $BR(\chi^{0}_{4}\to\ell^{+}+\tilde{\ell}^{-})$ | $BR(\chi^{0}_{4}\to\nu_{\ell}+\tilde{\nu}_{\ell})$
0.203 | 0.201 | 0.058 | 0.094
$BR(\tilde{\ell}\to\ell+\chi^{0}_{2})$ | $BR(\tilde{\tau}\to\tau+\chi^{0}_{2})$ | $BR(\tilde{\nu}_{\ell}\to\nu_{\ell}+\chi^{0}_{2})$ | $BR(\tilde{\nu}_{\tau}\to\nu_{\tau}+\chi^{0}_{2})$
0.95 | 0.98 | 0.88 | 0.95
Table 3: Branching fractions for the relevant electroweak sparticles of BP2.
$\tilde{\ell}$ denote left-handed selectrons or smuons. The neutralino
$\chi^{0}_{2}$ decays to $100\%$ to $W^{*}+\chi^{0}_{1}$.

In the case of BP3 the mostly wino-like chargino $\chi^{+}_{1}$ decays into
$\chi^{0}_{1}+W^{*}$ as it is assumed in most searches. However, the mostly
wino-like neutralino $\chi^{0}_{2}$ decays to $\chi^{0}_{1}+H_{1}$ with a
branching fraction of $97\%$. $H_{1}$ with its mass of 41 GeV decays to
$b\bar{b}$ with a branching fraction of $91\%$.

BP4 corresponds to a mostly wino-like chargino $\chi^{+}_{1}$ decaying with a
branching fraction of $98\%$ to $\tau+\tilde{\nu}_{\tau}$. $\chi^{0}_{2}$ is
bino-like, and the wino-like $\chi^{0}_{3}$ decays invisibly to
$\nu_{\tau}+\tilde{\nu}_{\tau}$ since stau sneutrinos decay to
$\nu_{\tau}+\chi^{0}_{1}$.

The properties of these benchmark points clarify why many scenarios in the
NMSSM with light electroweakinos, but a singlino-like LSP, would have remained
undetected at the LHC so far. It is of course desirable that dedicated
searches investigate these unconstrained (and motivated) regions in parameter
space.

##  5 Conclusions and Outlook

In the present paper we have shown which sparticle spectra in the NMSSM can
simultaneously describe $\Delta M_{W}$, dark matter and $a_{\mu}$. They are
characterized by light staus, charginos and sleptons. It is interesting that
light charginos and sleptons, required for a SUSY explanation of $a_{\mu}$,
can also lead to sizable contributions to $\Delta M_{W}$. Still, additional
contributions from light staus are helpful if one requires that $\Delta M_{W}$
satisfies eq. (1.1).

In principle, these features could also be fulfilled in the MSSM. However,
only the presence of the extra singlino-like LSP in the NMSSM and tau- or
lepton-sneutrinos as NLSPs makes this scenario phenomenologically viable given
the present constraints from the LHC (and Dark Matter): The couplings of all
MSSM-like sparticles to the singlino-like LSP are so small that they prefer to
decay via the NLSP (staus, sleptons or sneutrinos) leading to extra steps in
the sparticle decay cascades and tau- or lepton-rich final states with reduced
$E_{T}^{\text{miss}}$. These require dedicated searches and limits from more
standard search channels for sparticles are alleviated.

Moreover the extra singlino-like LSP in the NMSSM is a dark matter candidate
allowing to describe the dark matter relic density without conflict with
constraints from present, and possibly even from future direct detection
experiments. In fact, $\sigma^{SI}_{p}$ may fall below the neutrino floor.
Hence future tests of these scenarios can and have to rely on searches for
electroweakly interacting (s)particles in the $100-200$ GeV range.

## Acknowledgements

We like to thank Tian-Peng Tang and Yue-Lin Sming Tsai for discussions and
information on their paper [30]. Our numerical results have been obtained
thanks to the cloud computing infrastructure at LUPM, founded by OCEVU Labex,
and France-Grilles. We thank Sabine Kraml, Andre Lessa and Wolfgang
Waltenberger for help with SModelS.

## References

  * [1] T. Aaltonen et al. (CDF collaboration), Science 376, 170 (2022).
  * [2] J. de Blas, M. Pierini, L. Reina and L. Silvestrini, [arXiv:2204.04204 [hep-ph]].
  * [3] J. de Blas, M. Ciuchini, E. Franco, A. Goncalves, S. Mishima, M. Pierini, L. Reina and L. Silvestrini, [arXiv:2112.07274 [hep-ph]].
  * [4] G. W. Bennett et al. [Muon g-2], Phys. Rev. D 73 (2006), 072003 [arXiv:hep-ex/0602035 [hep-ex]].
  * [5] B. Abi et al. [Muon g-2], Phys. Rev. Lett. 126 (2021) no.14, 141801 [arXiv:2104.03281 [hep-ex]].
  * [6] N. Aghanim et al. [Planck], Astron. Astrophys. 641 (2020), A6 [erratum: Astron. Astrophys. 652 (2021), C4] [arXiv:1807.06209 [astro-ph.CO]].
  * [7] G. Angloher et al. [CRESST], Eur. Phys. J. C 76 (2016) no.1, 25 [arXiv:1509.01515 [astro-ph.CO]].
  * [8] P. Agnes et al. [DarkSide], Phys. Rev. Lett. 121 (2018) no.8, 081307 [arXiv:1802.06994 [astro-ph.HE]].
  * [9] E. Aprile et al. [XENON], Phys. Rev. Lett. 121 (2018) no.11, 111302 [arXiv:1805.12562 [astro-ph.CO]].
  * [10] E. Aprile et al. [XENON], Phys. Rev. Lett. 122 (2019) no.14, 141301 [arXiv:1902.03234 [astro-ph.CO]].
  * [11] C. Amole et al. [PICO], Phys. Rev. D 100 (2019) no.2, 022001 [arXiv:1902.04031 [astro-ph.CO]].
  * [12] Y. Meng et al. [PandaX-4T], Phys. Rev. Lett. 127 (2021) no.26, 261802 [arXiv:2107.13438 [hep-ex]].
  * [13] R. K. Barman, G. Bélanger, B. Bhattacherjee, R. M. Godbole and R. Sengupta, “Is the light neutralino thermal dark matter in the MSSM ruled out?,” [arXiv:2207.06238 [hep-ph]].
  * [14] M. Endo, K. Hamaguchi, S. Iwamoto and T. Kitahara, JHEP 07 (2021), 075 [arXiv:2104.03217 [hep-ph]].
  * [15] M. Chakraborti, S. Heinemeyer and I. Saha, [arXiv:2105.06408 [hep-ph]].
  * [16] J. Ellis, K. A. Olive, V. C. Spanos and I. D. Stamou, [arXiv:2210.16337 [hep-ph]].
  * [17] U. Ellwanger and C. Hugonie, Eur. Phys. J. C 78 (2018) no.9, 735 [arXiv:1806.09478 [hep-ph]].
  * [18] J. Cao, Y. He, L. Shang, Y. Zhang and P. Zhu, Phys. Rev. D 99 (2019) no.7, 075020 [arXiv:1810.09143 [hep-ph]].
  * [19] F. Domingo, J. S. Kim, V. M. Lozano, P. Martin-Ramiro and R. Ruiz de Austri, Phys. Rev. D 101 (2020) no.7, 075010 [arXiv:1812.05186 [hep-ph]].
  * [20] W. Abdallah, A. Chatterjee and A. Datta, JHEP 09 (2019), 095 [arXiv:1907.06270 [hep-ph]].
  * [21] J. Cao, L. Meng, Y. Yue, H. Zhou and P. Zhu, Phys. Rev. D 101 (2020) no.7, 075003 [arXiv:1910.14317 [hep-ph]].
  * [22] K. Wang and J. Zhu, Phys. Rev. D 101 (2020) no.9, 095028 [arXiv:2003.01662 [hep-ph]].
  * [23] M. Guchait and A. Roy, Phys. Rev. D 102 (2020) no.7, 075023 [arXiv:2005.05190 [hep-ph]].
  * [24] R. K. Barman, G. Bélanger, B. Bhattacherjee, R. Godbole, D. Sengupta and X. Tata, Phys. Rev. D 103 (2021) no.1, 015029 [arXiv:2006.07854 [hep-ph]].
  * [25] R. Kumar Barman, G. Belanger and R. M. Godbole, Eur. Phys. J. ST 229 (2020) no.21, 3159-3185 [arXiv:2010.11674 [hep-ph]].
  * [26] H. Zhou, J. Cao, J. Lian and D. Zhang, Phys. Rev. D 104 (2021) no.1, 015017 [arXiv:2102.05309 [hep-ph]].
  * [27] M. Abdughani, Y. Z. Fan, L. Feng, Y. L. S. Tsai, L. Wu and Q. Yuan, Sci. Bull. 66 (2021), 2170-2174 [arXiv:2104.03274 [hep-ph]].
  * [28] J. Cao, J. Lian, Y. Pan, D. Zhang and P. Zhu, JHEP 09 (2021), 175 [arXiv:2104.03284 [hep-ph]].
  * [29] W. Wang, K. P. Xie, W. L. Xu and J. M. Yang, [arXiv:2204.01928 [hep-ph]].
  * [30] T. P. Tang, M. Abdughani, L. Feng, Y. L. S. Tsai and Y. Z. Fan, [arXiv:2204.04356 [hep-ph]].
  * [31] J. Cao, J. Lian, Y. Pan, Y. Yue and D. Zhang, JHEP 03 (2022), 203 [arXiv:2201.11490 [hep-ph]].
  * [32] J. Cao, F. Li, J. Lian, Y. Pan and D. Zhang, Sci. China Phys. Mech. Astron. 65 (2022) no.9, 291012 [arXiv:2204.04710 [hep-ph]].
  * [33] M. Chakraborti, S. Heinemeyer and I. Saha, Eur. Phys. J. C 81 (2021) no.12, 1069 [arXiv:2103.13403 [hep-ph]].
  * [34] P. Athron, C. Balázs, D. H. J. Jacob, W. Kotlarski, D. Stöckinger and H. Stöckinger-Kim, JHEP 09 (2021), 080 [arXiv:2104.03691 [hep-ph]].
  * [35] S. Baum, M. Carena, N. R. Shah and C. E. M. Wagner, JHEP 01 (2022), 025 [arXiv:2104.03302 [hep-ph]].
  * [36] J. M. Yang and Y. Zhang, [arXiv:2204.04202 [hep-ph]].
  * [37] E. Bagnaschi, M. Chakraborti, S. Heinemeyer, I. Saha and G. Weiglein, Eur. Phys. J. C 82 (2022) no.5, 474 [arXiv:2203.15710 [hep-ph]].
  * [38] D. M. Pierce, J. A. Bagger, K. T. Matchev and R. j. Zhang, Nucl. Phys. B 491 (1997), 3-67 [arXiv:hep-ph/9606211 [hep-ph]].
  * [39] S. Heinemeyer and G. Weiglein, JHEP 10 (2002), 072 [arXiv:hep-ph/0209305 [hep-ph]].
  * [40] S. Heinemeyer, W. Hollik and G. Weiglein, Phys. Rept. 425 (2006), 265-368 [arXiv:hep-ph/0412214 [hep-ph]].
  * [41] S. Heinemeyer, W. Hollik, D. Stockinger, A. M. Weber and G. Weiglein, JHEP 08 (2006), 052 [arXiv:hep-ph/0604147 [hep-ph]].
  * [42] S. Heinemeyer, W. Hollik, A. M. Weber and G. Weiglein, JHEP 04 (2008), 039 [arXiv:0710.2972 [hep-ph]].
  * [43] S. Heinemeyer, W. Hollik, G. Weiglein and L. Zeune, JHEP 12 (2013), 084
[arXiv:1311.1663 [hep-ph]].

  * [44] F. Domingo and T. Lenz, JHEP 07 (2011), 101 [arXiv:1101.4758 [hep-ph]].
  * [45] O. Stål, G. Weiglein and L. Zeune, JHEP 09 (2015), 158 [arXiv:1506.07465 [hep-ph]].
  * [46] P. Athron, M. Bach, D. H. J. Jacob, W. Kotlarski, D. Stöckinger and A. Voigt, “Precise calculation of the W boson pole mass beyond the Standard Model with FlexibleSUSY,” [arXiv:2204.05285 [hep-ph]].
  * [47] E. Aprile et al. [XENON], JCAP 11 (2020), 031 [arXiv:2007.08796 [physics.ins-det]].
  * [48] D. S. Akerib et al. [LZ], “LUX-ZEPLIN (LZ) Conceptual Design Report,”
[arXiv:1509.02910 [physics.ins-det]].

  * [49] M. Schumann, L. Baudis, L. Bütikofer, A. Kish and M. Selvi, JCAP 10 (2015), 016 [arXiv:1506.08309 [physics.ins-det]].
  * [50] U. Ellwanger, J. F. Gunion and C. Hugonie, JHEP 02 (2005), 066 [arXiv:hep-ph/0406215 [hep-ph]].
  * [51] U. Ellwanger and C. Hugonie, Comput. Phys. Commun. 175 (2006), 290-303 [arXiv:hep-ph/0508022 [hep-ph]].
  * [52] https://www.lupm.in2p3.fr/users/nmssm/index.html
  * [53] G. Belanger, F. Boudjema, A. Pukhov and A. Semenov, Comput. Phys. Commun. 185 (2014), 960-985 [arXiv:1305.0237 [hep-ph]].
  * [54] F. Domingo and U. Ellwanger, JHEP 07 (2008), 079 [arXiv:0806.0733 [hep-ph]].
  * [55] J. Billard, L. Strigari and E. Figueroa-Feliciano, Phys. Rev. D 89 (2014) no.2, 023524 [arXiv:1307.5458 [hep-ph]].
  * [56] S. Kraml, S. Kulkarni, U. Laa, A. Lessa, W. Magerl, D. Proschofsky-Spindler and W. Waltenberger, Eur. Phys. J. C 74 (2014), 2868 [arXiv:1312.4175 [hep-ph]].
  * [57] J. Dutta, S. Kraml, A. Lessa and W. Waltenberger, LHEP 1 (2018) no.1, 5-12 [arXiv:1803.02204 [hep-ph]].
  * [58] C. K. Khosa, S. Kraml, A. Lessa, P. Neuhuber and W. Waltenberger, LHEP 2020 (2020), 158 [arXiv:2005.00555 [hep-ph]].
  * [59] G. Alguero, J. Heisig, C. Khosa, S. Kraml, S. Kulkarni, A. Lessa, H. Reyes-González, W. Waltenberger and A. Wongel, “Constraining new physics with SModelS version 2,” [arXiv:2112.00769 [hep-ph]].
  * [60] M. Maniatis, Int. J. Mod. Phys. A 25 (2010), 3505-3602 [arXiv:0906.0777 [hep-ph]].
  * [61] U. Ellwanger, C. Hugonie and A. M. Teixeira, Phys. Rept. 496 (2010) 1
[arXiv:0910.1785 [hep-ph]].

  * [62] M. Awramik, M. Czakon, A. Freitas and G. Weiglein, Phys. Rev. D 69 (2004), 053006 [arXiv:hep-ph/0311148 [hep-ph]].
  * [63] A. B. Arbuzov, M. Awramik, M. Czakon, A. Freitas, M. W. Grunewald, K. Monig, S. Riemann and T. Riemann, Comput. Phys. Commun. 174 (2006), 728-758 [arXiv:hep-ph/0507146 [hep-ph]].
  * [64] J. Cao and J. M. Yang, JHEP 12 (2008), 006 [arXiv:0810.0751 [hep-ph]].
  * [65] ALEPH, DELPHI, L3, OPAL Experiments, Combined LEP Chargino Results, up to
208 GeV for low DM, LEPSUSYWG/02-04.1, 2002, url:
http://lepsusy.web.cern.ch/lepsusy/www/inoslowdmsummer02/charginolowdm_pub.html.

  * [66] ALEPH, DELPHI, L3, OPAL Experiments, Combined LEP Selectron/Smuon/Stau
Results, 183-208 GeV, LEPSUSYWG/04-01.1, 2004, url:
http://lepsusy.web.cern.ch/lepsusy/www/sleptons_summer04/slep_final.html.

  * [67] G. Aad et al. [ATLAS], JHEP 04 (2014), 169 [arXiv:1402.7029 [hep-ex]].
  * [68] G. Aad et al. [ATLAS], JHEP 05 (2014), 071 [arXiv:1403.5294 [hep-ex]].
  * [69] M. Aaboud et al. [ATLAS], Eur. Phys. J. C 78 (2018) no.12, 995 [arXiv:1803.02762 [hep-ex]].
  * [70] M. Aaboud et al. [ATLAS], Phys. Rev. D 98 (2018) no.9, 092012 [arXiv:1806.02293 [hep-ex]].
  * [71] G. Aad et al. [ATLAS], Eur. Phys. J. C 80 (2020) no.2, 123 [arXiv:1908.08215 [hep-ex]].
  * [72] G. Aad et al. [ATLAS], Eur. Phys. J. C 80 (2020) no.8, 691 [arXiv:1909.09226 [hep-ex]].
  * [73] G. Aad et al. [ATLAS], Phys. Rev. D 101 (2020) no.7, 072001 [arXiv:1912.08479 [hep-ex]].
  * [74] G. Aad et al. [ATLAS], Eur. Phys. J. C 81 (2021) no.12, 1118 [arXiv:2106.01676 [hep-ex]].
  * [75] A. M. Sirunyan et al. [CMS], Phys. Lett. B 782 (2018), 440-467 [arXiv:1801.01846 [hep-ex]].
  * [76] A. M. Sirunyan et al. [CMS], JHEP 03 (2018), 160 [arXiv:1801.03957 [hep-ex]].
  * [77] A. M. Sirunyan et al. [CMS], Phys. Lett. B 790 (2019), 140-166
[arXiv:1806.05264 [hep-ex]].

  * [78] A. M. Sirunyan et al. [CMS], JHEP 04 (2021), 123 [arXiv:2012.08600 [hep-ex]].
  * [79] A. Tumasyan et al. [CMS], JHEP 04 (2022), 091 [arXiv:2111.06296 [hep-ex]].
  * [80] G. Aad et al. [ATLAS], Phys. Rev. D 101 (2020) no.3, 032009 [arXiv:1911.06660 [hep-ex]].
  * [81] A. M. Sirunyan et al. [CMS], Eur. Phys. J. C 80 (2020) no.3, 189 [arXiv:1907.13179 [hep-ex]].
  * [82] A. Tumasyan et al. [CMS], JHEP 04 (2022), 147 [arXiv:2106.14246 [hep-ex]].
  * [83] [CMS], “Search for electroweak production of charginos and neutralinos at $\sqrt{s}$ =13 TeV in final states containing hadronic decays of WW, WZ, or WH and missing transverse momentum,” [arXiv:2205.09597 [hep-ex]].
  * [84] [CMS], “Search for direct pair production of supersymmetric partners of $\tau$ leptons in the final state with two hadronically decaying $\tau$ leptons and missing transverse momentum in proton-proton collisions at $\sqrt{s}$ = 13 TeV,” [arXiv:2207.02254 [hep-ex]].
  * [85] G. Aad et al. [ATLAS], Phys. Rev. D 101 (2020) no.5, 052005 [arXiv:1911.12606 [hep-ex]].
  * [86] ATLAS collaboration, “Search for direct pair production of sleptons and charginos decaying to two leptons and neutralinos with mass splittings near the W boson mass in $\sqrt{s}=$ TeV $pp$ collisions with the ATLAS detector”, ATLAS-CONF-2022-006
  * [87] W. Adam and I. Vivarelli, Int. J. Mod. Phys. A 37 (2022) no.02, 2130022 [arXiv:2111.10180 [hep-ex]].
  * [88] T. Aoyama, N. Asmussen, M. Benayoun, J. Bijnens, T. Blum, M. Bruno, I. Caprini, C. M. Carloni Calame, M. Cè and G. Colangelo, et al. Phys. Rept. 887 (2020), 1-166 [arXiv:2006.04822 [hep-ph]].
  * [89] S. Schael et al. [ALEPH, DELPHI, L3, OPAL, SLD, LEP Electroweak Working Group, SLD Electroweak Group and SLD Heavy Flavour Group], Phys. Rept. 427 (2006), 257-454 [arXiv:hep-ex/0509008 [hep-ex]].
  * [90] A. M. Sirunyan et al. [CMS], JHEP 01 (2021), 148 [arXiv:2009.04363 [hep-ex]].
  * [91] G. Aad et al. [ATLAS], Phys. Lett. B 812 (2021), 135980 [arXiv:2007.07830 [hep-ex]].
  * [92] F. Domingo and U. Ellwanger, JHEP 12 (2007), 090 [arXiv:0710.3714 [hep-ph]].
  * [93] F. Domingo, Eur. Phys. J. C 76 (2016) no.8, 452 [arXiv:1512.02091 [hep-ph]].
  * [94] R. Barate et al. [LEP Working Group for Higgs boson searches, ALEPH, DELPHI, L3 and OPAL], Phys. Lett. B 565 (2003), 61-75 [arXiv:hep-ex/0306033 [hep-ex]].
  * [95] S. Schael et al. [ALEPH, DELPHI, L3, OPAL and LEP Working Group for Higgs Boson Searches], Eur. Phys. J. C 47 (2006), 547-587 [arXiv:hep-ex/0602042 [hep-ex]].
  * [96] A. M. Sirunyan et al. [CMS], Phys. Lett. B 793 (2019), 320-347 [arXiv:1811.08459 [hep-ex]].
  * [97] G. Belanger, U. Ellwanger, J. F. Gunion, Y. Jiang, S. Kraml and J. H. Schwarz, JHEP 01 (2013), 069 [arXiv:1210.1976 [hep-ph]].
  * [98] J. Cao, X. Guo, Y. He, P. Wu and Y. Zhang, Phys. Rev. D 95 (2017) no.11, 116001 [arXiv:1612.08522 [hep-ph]].
  * [99] K. Wang, F. Wang, J. Zhu and Q. Jie, Chin. Phys. C 42 (2018) no.10, 103109-103109 [arXiv:1811.04435 [hep-ph]].
  * [100] K. Choi, S. H. Im, K. S. Jeong and C. B. Park, Eur. Phys. J. C 79 (2019) no.11, 956 [arXiv:1906.03389 [hep-ph]].
  * [101] J. Cao, X. Jia, Y. Yue, H. Zhou and P. Zhu, Phys. Rev. D 101 (2020) no.5, 055008 [arXiv:1908.07206 [hep-ph]].
  * [102] W. G. Hollik, C. Li, G. Moortgat-Pick and S. Paasch, Eur. Phys. J. C 81 (2021) no.2, 141 [arXiv:2004.14852 [hep-ph]].
  * [103] T. Biekötter, A. Grohsjean, S. Heinemeyer, C. Schwanenberger and G. Weiglein, Eur. Phys. J. C 82 (2022) no.2, 178 [arXiv:2109.01128 [hep-ph]].
  * [104] [ATLAS], Nature 607 (2022) no.7917, 52-59 [arXiv:2207.00092 [hep-ex]].
  * [105] [CMS], Nature 607 (2022) no.7917, 60-68 [arXiv:2207.00043 [hep-ex]].
  * [106] [CMS], “Searches for additional Higgs bosons and for vector leptoquarks in $\tau\tau$ final states in proton-proton collisions at $\sqrt{s}$ = 13 TeV,” [arXiv:2208.02717 [hep-ex]].