#  12th Workshop on the CKM Unitarity Triangle
Santiago de Compostela, 18-22 September 2023
Radiative $B$ decays at Belle and Belle II

Rahul Tiwary on behalf of the Belle and Belle II Collaborations  Tata
Institute of Fundamental Research, Mumbai 400005, India

###### Abstract

Rare decays of $B$ mesons to radiative final states serve as an ideal ground
to search for New Physics effects from short range contributions. Observed by
CLEO in the early 1990s as one of the first transitions of the electroweak
penguin family, these decays continue to play a key role in testing
predictions of the Standard Model. We present the latest results from Belle
and Belle II for the radiative penguin transitions.

##  1 Introduction

In the Standard Model (SM), transitions involving Flavour Changing Neutral
Currents (FCNC) are forbidden at the tree level [1], and occur through
electroweak loop diagrams. Beyond-the-SM (BSM) physics can either contribute
to the loop diagrams or directly appear at tree level, which would alter
various physics observables from their SM predictions.

A typical FCNC transition is the radiative decay of a $B$ meson to the
inclusive final state $X_{s}\gamma$, where $X_{s}$ includes all final states
with net strangeness [2]. This decay offers excellent sensitivity to BSM
effects [3]. The photon-energy spectra can also provide insights into SM
parameters, such as the $b$ quark mass and its motion within the $B$ meson [4,
5]. BSM searches using radiative decays of $B$ mesons to exclusive final
states, like $B\to K^{*}\gamma$ and $B\to\rho\gamma$, are also promising. The
$b\to s/d\gamma$ operator is the dominant contributor to the decay, making it
easier to tell the difference between SM and BSM physics contributions [6].

The remainder of this document is arranged as follows: Section 2 provides a
brief description of Belle and Belle II detectors; Section 3 summarizes the
measurement of photon energy spectrum obtained for the inclusive $B\to
X_{s}\gamma$ channel using Belle II data; and Section 4 provides a summary of
the measurement of observables of exclusive final state $B\to\rho\gamma$ using
the combined Belle and Belle II dataset.

##  2 The Belle and Belle II detectors

The Belle detector [7, 8] was a large-solid-angle spectrometer that operated
at the KEKB asymmetric-energy $e^{+}e^{-}$ collider [10, 11]. The energies of
the electron and positron beams were 8.0 GeV and 3.5 GeV, respectively. The
detector consisted of a silicon-strip vertex detector, a central drift
chamber, an array of aerogel Cherenkov counters and time-of-flight
scintillation counters for identification of charged particles, and a
CsI(Tl)-based electromagnetic calorimeter (ECL), all of which were surrounded
by a superconducting solenoid coil providing a magnetic field of 1.5 T. An
iron flux return yoke located outside the coil instrumented with resistive-
plate chambers to facilitate the detection of $K^{0}_{\rm L}$ mesons and to
identify muons.

Belle II [12] is an upgraded version of Belle and located at the SuperKEKB
[13] $e^{+}e^{-}$ collider. The energies of electron and positron beams are
7.0 GeV and 4.0 GeV, respectively. The Belle II detector includes two layers
of silicon pixel sensors, four layers of double-sided silicon-strip vertex
detectors [14] and an upgraded 56-layer central drift chamber. Two types of
Cherenkov-light detector systems surround the drift chamber: an azimuthal
array of time-of-propagation detectors for the barrel region and an aerogel
ring-imaging Cherenkov detector for the forward endcap region. Belle II reuses
the ECL of Belle along with its solenoid and the iron flux return yoke; the
latter is equipped with both resistive-plate chamber and plastic scintillator
modules to detect $K^{0}_{\rm L}$ mesons and muons. The $z$ axis of the
laboratory frame is defined as the solenoid axis, where the positive direction
is along the electron beam. This convention applies both to Belle and Belle
II.

##  3 Inclusive measurement of $B\to X_{s}\gamma$ at Belle II

This section summarizes a measurement of inclusive $B\to X_{s}\gamma$ decays
using hadronic tagging for the partner $B$ meson ($B$ tag candidate)
reconstruction, performed at Belle II [15]. The measurement is based on
$189\,\rm{fb}^{-1}$ of data collected by the Belle II experiment. This
approach is complementary to the untagged or lepton-tagged [16] and sum-of-
exclusive [17] methods. The hadronic tag ensures higher signal purity, whereas
the kinematic constraints from the $B$ tag candidate allows us to measure
observables pertaining to the signal-side $B$ meson.

To reconstruct a $B\to X_{s}\gamma$ candidate, we pair the highest energy
photon in the event, satisfying $E^{B}_{\gamma}>1.4$ GeV, with the $B$ tag
candidate reconstructed using the full-event-interpretation (FEI) algorithm
[18]. Here, the photon energy in the rest frame of the signal-side $B$ meson
is denoted as $E^{B}_{\gamma}$. The FEI algorithm reconstructs hadronic $B$
decays from numerous sub-decay chains. We use a stochastic gradient-boosted
decision tree (BDT) [19], trained on Zernike moments [20] to separate high-
energy photons from $K^{0}_{\rm L}$ clusters.

The sources of background include high-energy or hard photons coming from
$\pi^{0}/\eta\to\gamma\gamma$ decays, where a photon emitted along the boost
direction of $\pi^{0}/\eta$ can be misidentified as the signal-side photon.
Another contribution comes from $e^{+}e^{-}\rightarrow q\overline{q}$ decays,
where $q$ represents $u,d,s,$ and $c$ quarks. The latter is also referred to
as continuum background. To mitigate the background from $\pi^{0}$ and $\eta$
photons, the signal-side hard photon is paired with all low-energy photons in
the event. Events having pairs consistent with $\pi^{0}/\eta\to\gamma\gamma$
decays are then vetoed using a dedicated BDT trained on kinematic variables
such as the diphoton invariant mass, helicity, and properties of the low-
energy photon, including its energy, polar angle, and smallest cluster-to-
track distance. Another dedicated BDT trained to suppress continuum background
is employed. The BDT utilizes input features that are known to have good
separation power between $B$ decays and continuum background; these features
include the modified Fox-Wolfram moments [21], CLEO cones [22], thrust, etc.

A fit to the $M_{\rm bc}(\equiv\sqrt{(E^{\ast}_{\rm
beam})^{2}-(\vec{p}^{\,\ast}_{B})^{2}})$ distribution of $B$ tag candidates in
bins of $E^{B}_{\gamma}$ for the region $E^{B}_{\gamma}>1.8$ GeV is performed
to extract the signal yield. Here, $E^{\ast}_{\rm beam}$ and $E^{\ast}_{B}$
are the beam energy and $B$-meson energy in the center-of-mass frame. The
selections and fit procedures are validated in a control region
$1.4<E^{B}_{\gamma}<1.8$ GeV. Since this inclusive analysis does not
differentiate between $b\to d\gamma$ and $b\to s\gamma$ processes, we subtract
the smaller $b\to d\gamma$ contribution based on simulation. The measured
$B\rightarrow X_{s}\gamma$ spectrum undergoes correction (unfolding) for
smearing effects. The unfolding procedure uses bin-by-bin multiplicative
factors, obtained as the ratios between the expected number of events in the
generated spectrum and those in the reconstructed spectrum within a given
$E^{B}_{\gamma}$ interval. The integrated branching fractions for various
$E^{B}_{\gamma}$ thresholds are listed in Table 1, the results are consistent
with world average values [9].

Table 1: The integrated partial branching fractions for three $E^{B}_{\gamma}$ thresholds. $E^{B}_{\gamma}$thres. (GeV) | $\mathcal{B}(B\to X_{s}\gamma)\times(10^{-4})$
---|---
1.8 | 3.54 $\pm$ 0.78 (stat.) $\pm$ 0.83 (syst.)
2.0 | 3.06 $\pm$ 0.56 (stat.) $\pm$ 0.47 (syst.)
2.1 | 2.49 $\pm$ 0.46 (stat.) $\pm$ 0.35 (syst.)

##  4 Exclusive measurement of $B\to\rho\gamma$ at Belle and Belle II

This section summarizes the most precise measurement of observables for
exclusive $B\to\rho\gamma$ decays, based on a combined data sample of the
Belle ($711\,\rm{fb}^{-1}$) and Belle II ($362\,\rm{fb}^{-1}$) experiments.
The exclusive radiative decay of $B$ mesons to the $\rho\gamma$ final state
allows for an independent search of BSM, complementary to the $b\to s\gamma$
modes. The $B\to\rho\gamma$ decays, which is a $b\to d$ quark-level
transition, has a branching fraction an order of magnitude smaller than the
radiative $B$ decays involving a $b\to s$ transition. Owing to a significant
difference in the branching fractions of $b\to d\gamma$ and $b\to s\gamma$
transitions, one needs good particle identification detectors to cut down the
charged kaon contamination from $B\to X_{s}\gamma$ decays. The branching
fractions give weak constraints on BSM parameters, since their SM predictions
suffer from large uncertainties (around 20%) due to the form factors [24]. One
can instead study observables such as $C\\!P$ ($\mathcal{A}_{C\\!P}$) and
isospin asymmetry ($\mathcal{A}_{I}$), which are theoretically cleaner due to
the cancellation of such effects. Precision measurement of the
$\mathcal{A}_{I}$ in $B\to\rho\gamma$ is particularly interesting since the
current world average [9] is in slight tension with the SM [23].

The reconstruction of $B\to\rho\gamma$ decay follows a hierarchical approach,
starting with the final-state particles. Hard photon candidates exhibiting a
shower shape consistent with that of an isolated photon are selected within
the energy range of 1.8 to 2.8 GeV. Tracks produced near the $e^{+}e^{-}$
interaction point are selected based on the requirements ${|d_{r}|<0.5~{}\rm
cm}$ and ${|d_{z}|<2.0~{}\rm cm}$, where $d_{r}$ ($d_{z}$) denotes the track’s
transverse (longitudinal) impact parameters. A likelihood-based particle
selector combining information from various detectors of Belle [25] or Belle
II is used to identify charged tracks.

The $\pi^{0}$ candidates are reconstructed in the diphoton invariant-mass
range of $119<M_{\gamma\gamma}<151\,{\rm MeV}\\!/c^{2}$. The photons are
further required to satisfy various energy thresholds depending on the
detector (Belle or Belle II) and the region of the ECL where the photon is
detected. Subsequently, we reconstruct the $\rho$ mesons via
$\rho^{0}\rightarrow\pi^{+}\pi^{-}$ and $\rho^{+}\rightarrow\pi^{+}\pi^{0}$
modes with the selection 0.64 (0.65) $<M_{\pi\pi}<$ 0.89 (0.90)$\,{\rm
GeV}\\!/c^{2}$ for Belle (Belle II). We reconstruct the $B$ meson by combining
a high-energy photon with the pion pair. Further selection criteria are
applied to the variables $M_{\rm bc}>5.2\,\rm GeV\\!/c^{2}$ and $|\Delta
E(\equiv E^{\ast}_{B}-E^{\ast}_{\rm beam})|<0.3\,{\rm GeV}$. For the neutral
mode, the momentum of the $B$ meson in the center-of-mass frame is calculated
as
$\vec{p}^{\,\ast}_{B^{0}}=\vec{p}^{\,\ast}_{\rho^{0}}+\frac{\vec{p}^{\,\ast}_{\gamma}}{|\vec{p}^{\,\ast}_{\gamma}|}\times(E^{\ast}_{\rm
beam}-E^{*}_{\rho^{0}})$, to improve the resolution of $M_{\rm bc}$.

The sources of background, akin to the $B\to X_{s}\gamma$ channel, are hard
photons from $\pi^{0}\\!/\eta$ decays, and combinatorial background from
$e^{+}e^{-}\rightarrow q\overline{q}$ events. The background suppression
strategy is to use a dedicated BDT classifier to suppress each kind of
background, the same as $B\to X_{s}\gamma$. $B^{+}\to
D^{0}[K^{-}\pi^{+}]\pi^{+}$, $B^{0}\to D^{-}[K^{+}\pi^{-}\pi^{-}]\pi^{+}$,
$B^{0}\to K^{*0}[K^{+}\pi^{-}]\gamma$, and $B^{+}\to
K^{*+}[K^{+}\pi^{0}]\gamma$ control channels are studied to assess quality of
the simulation and assign systematics for the BDT classifiers. These control
channels have a similar final state as the $B\to\rho\gamma$ modes but with
significantly higher statistics, and relatively low background contamination.
The invariant mass regions for $D^{0}$ $(D^{-})$ and $K^{*}$ particles are
1.85 (1.86) $<M_{K^{-}\pi^{+}}(M_{K^{+}\pi^{-}\pi^{-}})<1.88\,{\rm
GeV}\\!/c^{2}$, and $0.817<M_{K\pi}<0.967\,{\rm GeV}\\!/c^{2}$, respectively.

The physics observables of $B\to\rho\gamma$ decay are obtained from an
extended maximum-likelihood fit to unbinned $M_{\rm bc}$, $\Delta E$, and
$M_{K\pi}$ distributions, performed simultaneously for six independent
datasets: $B^{+}$, $B^{-}$, and $B^{0}$ in Belle and Belle II. Here,
$M_{K\pi}$ is the invariant mass calculated assuming a $\pi^{+}$ to be a
$K^{+}$. Using $M_{K\pi}$ instead of $M_{\pi\pi}$ aids in better separation of
the $B\rightarrow K^{*}\gamma$ background. The measured observables for
$B\to\rho\gamma$ decays from combined Belle and Belle II datasets are as
follows:

$\mathcal{B}(B^{+}\to\rho^{+}\gamma)=(13.1^{+2.0+1.3}_{-1.9-1.2})\times
10^{-7},$ (4.1) $\mathcal{B}(B^{0}\to\rho^{0}\gamma)=(7.5\pm
1.3^{+1.0}_{-0.8})\times 10^{-7},$ (4.2)
$\mathcal{A}_{C\\!P}(B\to\rho\gamma)=-8.2\pm 15.2^{+1.6}_{-1.2}\%,$ (4.3)
$\mathcal{A}_{I}(B\to\rho\gamma)=10.9^{+11.2+6.8+3.8}_{-11.7-6.2-3.9}\%.$
(4.4)

The third uncertainty appearing for $\mathcal{A}_{I}$ measurement is due to
the ratio of branching fraction of $\Upsilon(4\text{S})$ to charged and
neutral $B$ meson pairs. These are the most precise measurements of
$B\to\rho\gamma$ observables to date and supersede the previous measurements
performed by Belle [26].

##  5 Acknowledgment

The author expresses gratitude to the Belle II Collaboration for the
opportunity to present the talk, and commends organizers of the 12th Workshop
on the CKM Unitarity Triangle for hosting a successful conference.

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NMSSM with correct relic density and an additional 95 GeV Higgs boson

Ulrich Ellwangera and Cyril Hugonieb
a IJCLab, CNRS/IN2P3, University Paris-Saclay, 91405 Orsay, France
<EMAIL_ADDRESS>
b LUPM, UMR 5299, CNRS/IN2P3, Université de Montpellier, 34095 Montpellier,
France
<EMAIL_ADDRESS>
We investigate whether it is possible within the NMSSM to describe
simultaneously a correct dark matter relic density complying with the latest
null results from the LZ experiment, an extra Higgs boson with a mass of $\sim
95$ GeV visible in the bb channel at LEP and the diphoton channel at the LHC,
and the deviation of the anomalous magnetic moment of the muon from its value
within the Standard Model. We find that this is still possible for a variety
of dark matter annihilation mechanisms, for singlino-like but also bino-like
lightest supersymmetric particles. We show the signal rates of the extra Higgs
boson and the dark matter detection rates as function of the dark matter mass
and annihilation mechanism. The dark matter direct detection cross section may
well fall below the neutrino floor. The masses of the lightest electroweakly
interacting supersymmetric particles are typically not far above 100 GeV, but
not excluded due to unconventional decays.

##  1 Introduction

Supersymmetric extensions of the Standard Model may explain the dark matter of
the universe. The lightest supersymmetric partner particle is stable, provided
R-parity is conserved. It can well be neutral and freeze out into the desired
relic density $\Omega_{DM}h^{2}=0.1187$ (we add $\pm 10\%$ theoretical
uncertainty) in agreement with WMAP/Planck results [1, 2]. However, it also
has to comply with the absence of signals in direct detection experiments in
spin-independent channels by the CRESST [3], PandaX-II [4], LUX [5], XENON1T
[6, 8], DarkSide [7] and LZ [9] collaborations, and in spin-dependent channels
by the PICO-2L [10], LUX [11] and PandaX-II [12], Xenon1T [6, 8] and LZ [9]
collaborations.

Within the Minimal Supersymmetric extension of the Standard Model (MSSM) it is
no longer easy to comply with these constraints [13]; in [14] it has been
argued that additional contributions to the dark matter relic density would be
required, but co-annihilation of higgsinos with charginos, with masses beyond
1 TeV, may still be a possibility [15]. In [16] it has been observed that co-
annihilation with charginos or sleptons can allow for dark matter made out of
a Lightest Supersymmetric Particle (LSP) consisting of a bino/wino-mixture,
however at the price of some fine tuning [17, 18, 19] (see also [20, 21, 22,
23, 24, 25, 26, 27, 28, 29, 30, 31, 32]). Staus degenerate with the LSP seem
also possible [33]. Given the lower bounds on chargino and slepton masses
already from LEP, the mass of the LSP has then to be above 100 GeV. In [34] it
is shown that, considering the constraints from LZ [9] and the LHC, even if
additional contributions to the dark matter relic density are allowed the MSSM
requires heavy higgsinos, or narrow regions in parameter space with light
higgsinos or light staus.

Such constraints are alleviated within the Next-to-Minimal extension of the
Standard Model (NMSSM) [35] where the LSP can have at least a strong singlino-
component and correspond to the fermionic superpartner of the gauge singlet
superfield. A pure singlino has no couplings to quarks and leptons, hence such
an LSP has very small direct detection cross sections which can fall below the
projections for the exclusion reach of the XENON-nT [36] and LZ [37], or even
below the neutrino floor [38]. Hence the singlino is a quite attractive dark
matter candidate [39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
54, 55, 56, 58, 62, 63, 57, 59, 60, 64, 61, 65, 66, 67, 68, 69, 71, 70, 72,
73, 74, 75, 76, 77, 79, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91,
92, 93, 94, 95, 96, 97] which can remain consistent with the absence of
signals in direct detection experiments.

Actually hints exist for additional Higgs bosons as present in the NMSSM. The
combination of searches for the SM Higgs boson by the ALEPH, DELPHI, L3 and
OPAL experiments at LEP [98] showed some mild excess of events in the
$Z^{*}\to Z+b\bar{b}$ channel in the mass region $95-100$ GeV. Searches for
Beyond-the-Standard Model (BSM) Higgs bosons at the LHC in the diphoton
channel were performed by CMS and ATLAS. A search at run 1 by CMS showed a
$\sim 2\,\sigma$ excess at 97 GeV [99], which was confirmed by CMS later in
[100] and in [101] for a mass hypothesis of 95.4 GeV. A somewhat less
sensitive search in the diphoton channel by ATLAS in [102] lead to an upper
limit on the fiducial cross section which did not contradict the possible
excess observed by CMS, a recent analysis by ATLAS in the diphoton channel in
[103] showed a mild excess of $1.7\,\sigma$ at 95 GeV. A search for BSM Higgs
bosons in the di-tau channel by CMS in [104] showed an excess of $2.6\,\sigma$
(local) for a mass of $95-100$ GeV.

The $\sim 2\,\sigma$ excess at LEP was quantified in [105]. Let us denote the
extra (lighter) Higgs boson by $H_{95}$, with a reduced coupling to vector
bosons $W^{\pm},Z$ (relative to the coupling of a SM-like Higgs boson of
corresponding mass) given by $C_{V}(H_{95})$. Then the authors in [105] define

$\mu^{LEP}_{bb}\equiv C_{V}(H_{95})^{2}\times BR(H_{95}\to
b\bar{b})/BR(H_{SM}^{95}\to b\bar{b})=0.117\pm 0.057$ (1.1)

where $H_{SM}^{95}$ denotes a fictitious SM-like Higgs boson with a mass of 95
GeV.

The best fits for a diphoton signal of $H_{95}$ in CMS and ATLAS were combined
in [106]. The authors in [106] obtain

$\mu_{\gamma\gamma}^{LHC}=\frac{\sigma(gg\to
H_{95}\to\gamma\gamma)}{\sigma(gg\to
H_{SM}^{95}\to\gamma\gamma)}={0.24^{+0.09}_{-0.08}}\;.$ (1.2)

Again, $H_{SM}^{95}$ denotes a SM-like Higgs boson with a mass of $\sim 95$
GeV.

The best fit for the excess in the di-tau channel at 95 GeV observed by CMS in
[104] corresponds to a cross section times branching fraction

$\sigma(gg\to H_{95}\to\tau\tau)=7.8^{+3.9}_{-3.1}\text{pb}\;.$ (1.3)

Relative to $H_{SM}^{95}$ we obtain

$\mu_{\tau\tau}^{LHC}=\frac{\sigma(gg\to H_{95}\to\tau\tau)}{\sigma(gg\to
H_{SM}^{95}\to\tau\tau)}=1.38^{+0.69}_{-0.55}\;.$ (1.4)

Such an additional Higgs boson with a mass of $\sim 95$ GeV has been discussed
within the NMSSM in [107, 105, 108, 110, 109, 111, 113, 112, 114, 115, 116,
117, 106, 118, 119, 120]; mostly, however, allowing for a too small dark
matter relic density (assuming additional contributions), and/or before the
consideration of constraints from LZ [9]. It is the aim of the present paper
to verify whether attractive scenarios within the NMSSM exist with a correct
relic density, satisfying the LZ-constraints, and an additional Higgs boson
with a mass of $\sim 95$ GeV. To this end we scan the parameter space over the
NMSSM-parameters $\lambda$, $\kappa$, $A_{\lambda}$, $A_{\kappa}$,
$\mu_{\text{eff}}$, $\tan\beta$ in the Higgs sector and non universal soft
supersymmetry breaking terms in the gaugino/sfermion sector; for their
definitions we refer to review in [35]. We employ the codes NMSSMTools-6.0.2
[121, 122, 123] and micrOMEGAs$\\_$3 [124].

We impose constraints from b-physics and constraints from searches for BSM
Higgs bosons by ATLAS and CMS as implemented in NMSSMTools-6.0.2. The
references to constraints from BSM Higgs-boson searches and b-physics (of
little relevance here) are listed on the web page
https://www.lupm.in2p3.fr/users/nmssm/history.html. Constraints from the
absence of a Landau singularity for the Yukawa couplings below the GUT scale
confine values of the NMSSM-specific coupling $\lambda$ to $\lambda\;\raise
1.29167pt\hbox{$<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}\;0.7$ (see [35]
for the the meaning of NMSSM specific parameters).

Constraints from the anomalous magnetic moment of the muon $a_{\mu}$ [125,
126] as in [127] can always be satisfied by choosing the soft supersymmetry
breaking smuon, bino and/or wino masses small enough, and the trilinear
coupling $A_{\mu}$ large enough. These parameters have little impact on the
quantities of relevance here which are shown in the Figures in the next
Section, hence constraints from $a_{\mu}$ are not imposed on most of the
points in parameter space. However, for the benchmark points in the next
Section, constraints from the $a_{\mu}$ are satisfied. The constraint on
$M_{W}$ as applied in [127] is not used since it relies on a single
experimental result which differs significantly from many others. All soft
supersymmetry breaking terms are taken below 5 TeV.

Constraints on the sparticle spectrum from the absence of signals at the LHC
are taken into account using the code SModels-2.2.0 [128, 129, 130, 131].
Constraints on light electroweakly interacting sparticles (in the $\sim
100-200$ GeV range) from the LHC rely on production cross sections for
higgsinos or winos, and their decays into W- or Z-bosons, see e.g. the recent
ATLAS combination in [132]. In the NMSSM, both production cross sections of
light electroweakly interacting sparticles as well as corresponding decay
rates can be suppressed and the constraints from the LHC become weak.
Moreover, in the presence of several light electroweakly interacting
sparticles, heavy sparticles (squarks, gluino) can undergo several distinct
cascade decays such that the probability for each of them is reduced. As a
consequence such scenarios are difficult to rule out.

We require that the singlet-like scalar has a mass in the range $95.4\pm 3$
GeV (allowing for a theoretical uncertainty of 3 GeV), $\mu^{LEP}_{bb}$ in the
$2\,\sigma$ range of (1.1), and $\mu_{\gamma\gamma}^{LHC}$ in the $2\,\sigma$
range of (1.2). For the SM-like Higgs boson we require a mass within $125.2\pm
3$ GeV (allowing for theoretical uncertainties), and that the couplings in the
$\kappa$-framework satisfy combined limits of CMS [133] and ATLAS [134]. These
limits require that the reduced couplings $C_{V}(H_{SM}$) of the SM-like Higgs
boson to gauge fields satisfy $C_{V}(H_{SM}$) $>$$\sim$ 0.96, hence the mixing
angle between $H_{95}$ and $H_{SM}$ is limited such that the reduced coupling
of $H_{95}$ to $\tau\tau$ is bounded to below $25\%$. Then the (reduced) cross
section $\times$ BR($H_{95}\to\tau\tau$) can hardly exceed 0.1 in contrast to
what is desired within the $2\,\sigma$ range for the di-tau excess
$\mu_{\tau\tau}^{LHC}$ in (1.4); its description within the $2\,\sigma$ range
is therefore left aside since impossible for the type II Yukawa couplings
present in the (N)MSSM.

In the next Section we present our results in the form of Figures and
benchmark points, Section 3 is devoted to conclusions.

##  2 Results

Various mechanisms exist within the NMSSM for a sufficiently fast annihilation
of dark matter in the form of a singlino-like LSP:

  1. (a)

Annihilation via a pseudoscalar in the s-channel. Then $M_{A_{1}}$ should be
about $2\times M_{\chi_{1}^{0}}$ such that the annihilation cross section is
enhanced by the s-channel pole. We find that this annihilation mechanism is
possible in the NMSSM for $M_{\chi_{1}^{0}}\;\raise
1.29167pt\hbox{$<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}\;24$ GeV, i.e.
$M_{A_{1}}\;\raise
1.29167pt\hbox{$<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}\;50$ GeV. $A_{1}$
is always mostly singlet-like, and decays into $b\bar{b}$ for
$M_{A_{1}}\;\raise
1.29167pt\hbox{$>$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}\;9$ GeV, into
$\tau\tau$ otherwise.

  2. (b)

Annihilation via the Standard Model-like Higgs boson $H_{SM}$ or the mostly
singlet-like Higgs boson $H_{95}$ in the s-channel if the singlino mass is
about half the corresponding mass. (With $M_{H_{95}}\sim 95$ GeV, the
$Z$-boson plays little role in the s-channel since the couplings of an LSP
with $M_{LSP}\sim M_{Z}/2$ to $H_{95}$ are typically much stronger.)

  3. (c)

Co-annihilation with charginos, sleptons, or the next-to-lightest neutralinos.
(Subsequently we include staus and their sneutrinos in the notion sleptons.)
Often several of these sparticles intervene simultaneously with different
strengths; then these sparticles must be close in mass (within $\sim 10$ GeV)
to the singlino. In most cases, the lightest charged or electroweakly
interacting sparticle must be heavier than $\sim 100$ GeV from constraints
from LEP. Then a lower bound of $\sim 90$ GeV holds for the singlino, and the
properties of the valid points shown in the figures below depend little on the
nature of the sparticle(s) which intervene in co-annihilation. An exception
are sneutrinos which can be lighter, as light as 55 GeV if they annihilate
through the $H_{SM}$ funnel.

  4. (d)

A bino-like LSP as in the MSSM under the latest constraints [34] (or in the
NMSSM e.g. in [135, 136]). For a practically pure bino, LEP and LHC
constraints allow for masses up to 330 GeV down to $\sim 20$ GeV. Compared to
singlino-like LSPs, pair-annihilation of binos via a (pseudo-)scalar in the
s-channel is relatively less important. The reason is that singlino-singlino-
Higgs Yukawa couplings $\sim\lambda$ exist in the NMSSM for the singlet-like
(pseudo-)scalar at tree level, and other scalars through mixing. However,
bino-bino-Higgs Yukawa couplings do not exist for pure binos since the bino is
a gaugino of an abelian gauge symmetry. The dominant annihilation channelis
therefore co-annihilation with sleptons.

All mechanisms allow for a dark matter relic density in agreement with the
WMAP/Planck result $\Omega_{DM}h^{2}=0.1187\pm 10\%$.

It is interesting to compare the possible values of the signal strengths
$\mu^{LEP}_{bb}$, $\mu_{\gamma\gamma}^{LHC}$ and $\mu_{\tau\tau}^{LHC}$ as
well as the dark matter cross sections (spin independent on protons
$\sigma_{p}^{SI}$, and spin dependent on neutrons $\sigma_{n}^{SD}$) for the
various possible dark matter masses associated to various annihilation
processes, as function of the mass of the LSP.

In Figs. 1 we show in how far points in the NMSSM can accomodate an additional
Higgs boson near 95 GeV with signal rates $\mu^{LEP}_{bb}$ at LEP as in eq.
(1.1). (Only points within the $2\,\sigma$ range of (1.1) are shown.) Here and
in the subsequent figures, points on the left hand side refer to a singlino-
like LSP. Points corresponding to annihilation via a pseudoscalar $A_{1}$ (A1
funnel) are shown in violet, annihilation via $H_{95}$ or $H_{SM}$ in orange,
co-annihilation with charginos, sleptons or next-to-lightest neutralinos in
blue. Note that, for a LSP mass near $H_{SM}/2$ or $H_{95}/2$, the dominant
dark matter annihilation mechanism can be either via the corresponding Higgs
funnel or via co-annihilation. Points with a bino-like LSP are shown on the
right hand side in black. The red dots denote benchmark points whose details
are given in the Table 2-4 below. Comparing the right hand side and the left
hand side of Figs. 1 one observes that the possible values of $\mu^{LEP}_{bb}$
are generally somewhat larger for bino LSPs compared to singlino LSPs.

|
---|---
Figure 1: $\mu^{LEP}_{bb}$ for points in the NMSSM satisfying all constraints.
Left hand side: Points with a singlino-like LSP. Points corresponding to
annihilation via a pseudoscalar $A_{1}$ (A1 funnel) are shown in violet,
annihilation via $H_{95}$ or $H_{SM}$ in orange, co-annihilation with
charginos, sleptons, next-to-lightest neutralinos in blue. Right hand side:
Points with a bino-like LSP. The red dots denote benchmark points whose
details are given in the Tables 2-4 below.

The contributing range of parameters depends somewhat on the annihilation
mechanism. In Table 1 we give these ranges for the NMSSM specific parameters
and the soft supersymmetry breaking masses $M_{1}$, $M_{2}$ and $M_{3}$ where
the latter refer to the bino, wino and gluino masses, respectively. The soft
supersymmetry breaking mass terms for scalars vary from 83 GeV (for sleptons)
and 320 GeV (for squarks) up to 5 TeV, the soft supersymmetry breaking
trilinear couplings from $-5$ TeV to 5 TeV. (The upper limits of 5 TeV are
imposed by hand.) Given the possibly complicated squark decay cascades,
squarks can be relatively light without contradicting constraints from
searches at the LHC. In any case, due to radiative corrections the pole squark
masses can differ considerably from the soft supersymmetry breaking mass
terms.

|  $A_{1}$-funnel |  $H_{95}$\- or $H_{SM}$-funnels | co-annihilation | bino LSP
---|---|---|---|---
$\lambda$ | 0.17$-$0.52 | 0.18$-$0.32 | 0.019$-$0.34 | 0.019$-$0.22
$\kappa$ | -(6.3$-$0.36)$\times 10^{-3}$ | -(1.48$-$0.59)$\times 10^{-2}$ | -0.011$-$0.024 | 0.0019$-$0.020
$A_{\lambda}$ | 2862$-$5000 | 3486$-$5000 | -5000$-$4320 | -(5000$-$1503)
$A_{\kappa}$ | 12$-$65 | -122$-$175 | -3.4$-$551 | -1.0$-$585
$\mu_{\rm eff}$ | 350$-$1388 | 575$-$946 | -1326$-$136 | -(1226$-$361)
$\tan\beta$ | 3.3$-$11 | 4.0$-$7.7 | 3.2$-$14 | 3.8$-$19
$M_{1}$ | 15$-$5000 | -227$-$5000 | 56$-$5000 | 25$-$329
$M_{2}$ | 89$-$5000 | 91$-$5000 | -5000$-$680 | -(5000$-$89)
$M_{3}$ | 746$-$5000 | 729$-$5000 | 722$-$5000 | 721$-$5000
Table 1: Range of NMSSM specific input parameters and bino ($M_{1}$), wino
($M_{2}$) and gluino ($M_{3}$) masses for the various dark matter annihilation
mechanisms. All dimensionful parameters are given in GeV.

In Fig. 2 we show $\mu_{\gamma\gamma}^{LHC}$ for points within the $2\,\sigma$
range of the signal rate (1.2) for an additional Higgs boson near 95 GeV in
the $\gamma\gamma$ channel at the LHC. The colors denote the dark matter
annihilation mechanisms as in Fig. 1.

|
---|---
Figure 2: $\mu^{LHC}_{\gamma\gamma}$ for an additional Higgs boson near 95 GeV for points in the NMSSM satisfying all constraints. The colors denote the dark matter annihilation mechanisms as in Fig. 1. |
---|---
Figure 3: $\mu^{LHC}_{\tau\tau}$ for an additional Higgs boson near 95 GeV for
points in the NMSSM satisfying all constraints. The colors denote the dark
matter annihilation mechanisms as in Fig. 1.

Fig. 3 shows the signal rate $\mu^{LHC}_{\tau\tau}$ for an additional Higgs
boson near 95 GeV for points in the NMSSM within the $2\,\sigma$ ranges of the
signal rates (1.1) and (1.2), and all other constraints. Actually Fig. 3 looks
similar to Fig. 1, the reason being that the couplings of $H_{95}$ to b-quarks
and $\tau$-leptons are related in the NMSSM with its type 2 Yukawa couplings
of $H_{SM}$; the Yukawa couplings of $H_{95}$ are inherited from those of
$H_{SM}$ through mixing. A larger mixing angle would enhance the couplings of
$H_{95}$, but reduce simultaneously those of $H_{SM}$. The latter are limited
from below by the latest measurements of ATLAS [134] and CMS [133]. Hence it
is difficult in the NMSSM to obtain large couplings of $H_{95}$, i.e. larger
signal rates than those shown in Fig. 3 which are below the $2\,\sigma$ range
of 0.28 for $\mu^{LHC}_{\tau\tau}$ in (1.4).

In Fig. 4 we show the spin-independent (SI) and spin-dependent (SD) direct
detection cross section. We also show the present exclusion of the LZ
experiment [9], projections for the LZ exclusion reach from [37], and the
neutrino floor from [38]. We see that the SI direct detection cross section
can well fall below the neutrino floor, in which case direct detection
experiments are not able to verify this form of dark matter.

|
---|---
|
Figure 4: $\sigma_{p}^{SI}$ and $\sigma_{n}^{SD}$ for points in the NMSSM with
the correct relic density and satisfying all constraints. The colors denote
the dark matter annihilation mechanisms as in Fig. 1. The full green line
indicates the present upper limits from LZ [9], the dashed green line the
future prospects on upper limits from LZ from [37], the dotted green line the
neutrino floor from [38].

In Fig. 5 we show the mass $M_{A1}$ of the mostly singlet-like pseudoscalar,
which is $\sim 2\times M_{LSP}$ for points in the $A_{1}$-funnel. For
$M_{A1}\;\raise
1.29167pt\hbox{$<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}\;60$ GeV, $A_{1}$
may be visible in decays of $H_{SM}$, see the latest study by CMS in [137].
Accordingly $A_{1}$ may be visible (but not necessarily) for all dark matter
annihilation mechanisms.

|
---|---
Figure 5: $M_{A1}$ for points in the NMSSM satisfying all constraints. The
colors denote the dark matter annihilation mechanisms as in Fig. 1.

In Tables 2-4 we show the properties of some benchmark points (BPs) with
distinct dark matter annihilation processes: Input parameters in Table 2, some
Higgs masses and the nature of light electroweakinos in Table 3, and the
properties shown in Figs. 1-5 in Table 4.

BP1 corresponds to dark matter annihilation via the $A_{1}$ funnel,
accordingly the mass of the singlino-like LSP is close to $M_{A1}/2$. BP2 and
BP3 correspond to dark matter annihilation via the $H_{95}$ and $H_{SM}$
funnels, respectively. Accordingly the masses of the singlino-like LSP are
close to $M_{H_{95}}/2$ and $M_{H_{SM}}/2$, respectively. BP4 corresponds to
co-annihilation with a slepton and/or the second-lightest bino-like
neutralino; in fact several possibilities are included here. $\sigma_{p}^{SI}$
for BP4 is below the neutrino floor. Finally BP5 corresponds to a bino-like
LSP which is possible in the NMSSM as well as in the MSSM.

| BP1 | BP2 | BP3 | BP4 | BP5
---|---|---|---|---|---
$\lambda$ | $0.458$ | $0.242$ | $0.205$ | $0.0223$ | $0.0590$
$\kappa$ | $-5.74\times 10^{-3}$ | $-7.26\times 10^{-3}$ | $-8.06\times 10^{-3}$ | $9.63\times 10^{-3}$ | $7.75\times 10^{-3}$
$A_{\lambda}$ | $3970$ | $4999$ | $4544$ | $-3390$ | $-4995$
$A_{\kappa}$ | $34.9$ | $11.0$ | $11.0$ | $183$ | $252$
$\mu_{\rm eff}$ | $404$ | $736$ | $794$ | $-179$ | $-681$
$\tan\beta$ | $9.76$ | $6.25$ | $5.13$ | $7.01$ | $8.94$
$M_{1}$ | $163$ | $2065$ | $76.6$ | $196$ | $45.2$
$M_{2}$ | $111$ | $100$ | $96.7$ | $-502$ | $-106$
$M_{3}$ | $2735$ | $4429$ | $1201$ | $3505$ | $825$
Table 2: NMSSM specific input parameters and bino ($M_{1}$), wino ($M_{2}$) and gluino ($M_{3}$) masses for the five benchmark points. All dimensionful parameters are given in GeV. | BP1 | BP2 | BP3 | BP4 | BP5
---|---|---|---|---|---
$M_{H3}$ | $3966$ | $4852$ | $4367$ | $2069$ | $5592$
$M_{A1}$ | $21$ | $18$ | $31$ | $206$ | $260$
LSP | singl. | singl. | singl. | singl. | bino
$M_{\text{LSP}}$ | $9.0$ | $43.9$ | $62.9$ | $157$ | $44.1$
NLSP | wino± | wino± | bino | bino | wino0
$M_{\text{NLSP}}$ | $115$ | $107$ | $74.8$ | $164$ | $113$
Slepton | $\tilde{\nu}_{\tau}$ | $\tilde{\nu}_{e}$ | $\tilde{e}_{R.h.}$ | $\tilde{\nu}_{e}$ | $\tilde{\nu}_{\tau}$
$M_{\text{Slepton}}$ | $140$ | $171$ | $255$ | $165$ | $93.4$
Table 3: Some BSM Higgs masses, the nature of the LSP, NLSP, the lightest slepton, and the corresponding masses for five BMpoints. All dimensionful parameters are given in GeV. | BP1 | BP2 | BP3 | BP4 | BP5
---|---|---|---|---|---
$\mu^{LEP}_{bb}$ | $1.18\times 10^{-2}$ | $0.115$ | $0.106$ | $0.105$ | $0.116$
$\mu^{LHC}_{\gamma\gamma}$ | $0.331$ | $8.01\times 10^{-2}$ | $8.01\times 10^{-2}$ | $8.75\times 10^{-2}$ | $0.130$
$\mu^{LHC}_{\tau\tau}$ | $1.28\times 10^{-2}$ | $0.112$ | $0.102$ | $9.34\times 10^{-2}$ | $0.117$
$\sigma_{p}^{SI}$ | $3.43\times 10^{-11}$ | $4.98\times 10^{-12}$ | $1.59\times 10^{-12}$ | $1.49\times 10^{-13}$ | $8.90\times 10^{-12}$
$\sigma_{n}^{SD}$ | $3.66\times 10^{-5}$ | $3.12\times 10^{-7}$ | $1.15\times 10^{-7}$ | $1.06\times 10^{-7}$ | $4.40\times 10^{-7}$
Table 4: $\mu^{LEP}_{bb}$, $\mu^{LHC}_{\gamma\gamma}$, $\mu^{LHC}_{\tau\tau}$,
$\sigma_{p}^{SI}$ and $\sigma_{n}^{SD}$ (the latter in pb) for the five
benchmark points. $\sigma_{p}^{SI}$ for BP4 is below the neutrino floor.

##  3 Conclusions

The NMSSM allows to describe simultaneously three phenomena related to physics
beyond the Standard Model: A dark matter relic density of the amount observed
in cosmology, but compatible with the absence of direct detection signals; a
deviation of the muon anomalous magnetic moment by 4.2$\,\sigma$ from its
value in the Standard Model; anomalies which indicate the presence of an extra
scalar of a mass near 95 GeV in the $bb$ channel at LEP, and in $\gamma\gamma$
and $\tau\tau$ channels at the LHC. The latter would be impossible to describe
within the MSSM. The LSP may have a mass well below 100 GeV, and at least part
of the electroweak sparticle sector (sleptons, winos, bino) is typically
relatively light. Such sparticles are difficult to rule out at the LHC, and
generate complicated decay cascades also for the heavier strongly interacting
sparticles. Their verification may then be a demanding task for the future.

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