# NEW SIGNATURES OF DIRAC NEUTRALINO DARK MATTER

M. GOODSELL1 S. KRAML2 H. REYES-GONZÁLEZ3,4 S. WILLIAMSON5 1Laboratoire de
Physique Théorique et Hautes Energies (LPTHE), UMR 7589, Sorbonne Université
et CNRS, 4 place Jussieu, 75252 Paris Cedex 05, France
2Laboratoire de Physique Subatomique et de Cosmologie (LPSC), Université
Grenoble-Alpes, CNRS/IN2P3, 53 Avenue des Martyrs, F-38026 Grenoble, France
3Department of Physics, University of Genova, Via Dodecaneso 33, 16146 Genova,
Italy
4INFN, Sezione di Genova, Via Dodenasco 33, I-16146 Genova, Italy
5Institute for Theoretical Physics, Karlsruhe Institute of Technology, 76128
Karlsruhe, Germany

###### Abstract

Supersymmetric dark matter has been studied extensively in the context of the
MSSM, where gauginos have Majorana masses. Introducing Dirac gaugino masses,
we obtain an enriched phenomenology from which considerable differences in,
e.g., LHC signatures can be expected. Concretely, in the Minimal Dirac Gaugino
Model (MDGSSM) we have an electroweakino sector extended by two extra
neutralinos and one extra chargino. The bino- and wino-like states bring about
small mass splittings leading to the frequent presence of scenarios with Long
Lived Particles (LLPs). In this contribution, we delineate the parameter space
of the electroweakino sector of the MDGSSM, where the lightest neutralino is a
viable dark matter candidate that escapes current dark matter direct
detection. We then focus on the allowed regions that contain LLPs and confront
them against the corresponding LHC searches. Finally, we discuss the
predominant case of long-lived neutralinos, to which no search is currently
sensitive.

##  1 Introduction.

Supersymmetric (SUSY) models with Dirac instead of Majorana gaugino masses can
have very distinctive features. In particular, the phenomenology of
neutralinos and charginos (electroweakinos or EW-inos) in Dirac gaugino models
is quite different from that of the Minimal Supersymmetric Standard Model
(MSSM). The EW-ino sector of the Minimal Dirac Gaugino Supersymmetric Standard
Model (MDGSSM) comprises six neutralinos and three charginos, as compared to
four and two, respectively, in the MSSM. More concretely, one obtains pairs of
bino-like, wino-like and higgsino-like neutralinos, with small mass splittings
within the bino (wino) pairs induced by the couplings $\lambda_{S}$
($\lambda_{T}$) between the singlet (triplet) fermions with the Higgs and
higgsino fields. This small mass splittings can lead to long-lived charginos
and/or neutralinos.

The MDGSSM can be defined as the minimal extension of the MSSM allowing for
Dirac gaugino masses. We add one adjoint chiral superfield for each gauge
group: one singlet, one triplet and one octet. We also assume that there is an
underlying R-symmetry that prevents R-symmetry-violating couplings in the
superpotential and supersymmetry-breaking sector, _except_ for an explicit
breaking in the Higgs sector through a (small) $B_{\mu}$ term. Here, we
concentrate on the EW-ino sector of the MDGSSM. After electroweak symmetry
breaking, we obtain 6 neutralino and 3 chargino mass eigenstates. The
neutralino mass matrix ${\cal M}_{N}$ in the basis
$(\tilde{B}^{\prime},\tilde{B},\tilde{W}^{\prime
0},\tilde{W}^{0},\tilde{H}_{d}^{0},\tilde{H}_{u}^{0})$ is given by

$\begin{array}[]{rc@{\,}c@{\,}l}{\cal M}_{N}=\left(\begin{array}[]{c c c c c
c}0&m_{DY}&0&0&-\frac{\sqrt{2}\lambda_{S}}{g_{Y}}z_{ss}&-\frac{\sqrt{2}\lambda_{S}}{g_{Y}}z_{sc}\\!\\!\\\
m_{DY}&0&0&0&-z_{sc}&z_{ss}\\!\\!\\\
0&0&0&m_{D2}&-\frac{\sqrt{2}\lambda_{T}}{g_{2}}z_{cs}&-\frac{\sqrt{2}\lambda_{T}}{g_{2}}z_{cc}\\\
0&0&m_{D2}&0&z_{cc}&-z_{cs}\\\
\\!\\!-\frac{\sqrt{2}\lambda_{S}}{g_{Y}}z_{ss}&-z_{sc}&\\!\\!-\frac{\sqrt{2}\lambda_{T}}{g_{2}}z_{cs}&z_{cc}&0&-\mu\\\
\\!\\!-\frac{\sqrt{2}\lambda_{S}}{g_{Y}}z_{sc}&z_{ss}&\\!\\!-\frac{\sqrt{2}\lambda_{T}}{g_{2}}z_{cc}&-z_{cs}&-\mu&0\\\
\end{array}\right),\\!\\!\\!\end{array}$ (1)

where $z_{ss}=m_{Z}\sin\theta_{W}\sin\beta$,
$z_{sc}=m_{Z}\sin\theta_{W}\cos\beta$, $z_{cc}=m_{Z}\cos\theta_{W}\cos\beta$,
$z_{cs}=m_{Z}\cos\theta_{W}\sin\beta$; $\tan\beta=v_{u}/v_{d}$ is the ratio of
the Higgs vevs; $m_{DY}$ and $m_{D2}$ are the ‘bino’ and ‘wino’ Dirac mass
parameters; $\mu$ is the higgsino mass term, and $\lambda_{S}$ and
$\lambda_{T}$ are the couplings between the singlet and triplet with the Higgs
and higgsino fields respectively, corresponding to the new terms in the
superpotential: $\lambda_{S}\mathbf{S}\,\mathbf{H_{u}}\cdot\mathbf{H_{d}}\
\mathrm{and}\ 2\lambda_{T}\,\mathbf{H_{d}}\cdot\mathbf{T}\mathbf{H_{u}}$.

By diagonalising eq. (1), one obtains pairs of bino-like, wino-like and
higgsino-like neutralinos,111For simplicity, we refer to the mostly bino/U(1)
adjoint states collectively as binos, and to the mostly wino/SU(2) adjoint
ones as winos. with small mass splittings within the bino or wino pairs
induced by $\lambda_{S}$ or $\lambda_{T}$, respectively. Turning to the
charged EW-inos, the chargino mass matrix in the basis
$v^{+}=(\tilde{W}^{\prime+},\tilde{W}^{+},\tilde{H}^{+}_{u})$,
$v^{-}=(\tilde{W}^{\prime-},\tilde{W}^{-},\tilde{H}^{-}_{d})$ is given by:

$\begin{array}[]{rc@{\,}c@{\,}l}{\cal M}_{C}=\left(\begin{array}[]{c c
c}0&m_{D2}&\frac{{2}\lambda_{T}}{g_{2}}m_{W}c_{\beta}\\\
m_{D2}&0&\sqrt{2}m_{W}s_{\beta}\\\
-\frac{{2}\lambda_{T}}{g_{2}}m_{W}s_{\beta}&\sqrt{2}m_{W}c_{\beta}&\mu\\\
\end{array}\right)\,,\end{array}$ (2)

where $s_{\beta}=\sin\beta$ and $c_{\beta}=\cos\beta$. Analogously to the
neutralino case, we now have wino-like $\tilde{\chi}^{\pm}$ pairs with a small
splitting driven by $\lambda_{T}$.

In the following, we will explore the parameter space of the EW-ino sector of
the MDGSSM where the lightest neutralino $\tilde{\chi}^{0}_{1}$ is a viable
dark matter (DM) candidate. Regarding the LHC phenomenology, we will focus on
the regions which include long-lived particles (LLPs). The complete study,
upon which this contribution is based, can be found in [1].

##  2 Dark matter viable scenarios.

We performed a Markov Chain Monte Carlo scan over the EW-ino parameters
defined above to find the regions where the Lightest Supersymmetric Particle
(LSP), the neutralino $\tilde{\chi}^{0}_{1}$, constitutes at least some
percentage of the observed DM content of the Universe and escapes direct
detection from the XENON1T experiment. The computation of the relic density
and other constraints was performed with MicrOMEGAs [2]; for details of the
setup see [1].

The resulting parameter space is shown in the left panel of Figure 1. The axes
correspond to the bino, wino and higgsino composition of the LSP, while the
color denotes its mass value. We see that cases where the
$\tilde{\chi}^{0}_{1}$ is a mixture of all states (bino, wino and higgsino)
are excluded, while cases where it is a mixture of only two states, with one
component being dominant, can satisfy all constraints. Also noteworthy is that
there are plenty of points in the low-mass region, $m_{\rm LSP}<400$ GeV.
Interestingly, the mass differences between the LSP and the next-to-LSP (NLSP)
are often so small that the NLSP (and sometimes even the next-to-next-to-LSP)
becomes long-lived on collider scales, i.e. it has a potentially visible decay
length of $c\tau>1$ mm. Concretely, about 20% of our DM-friendly parameter
points contain LLPs. The corresponding scenarios are illustrated in the right
panel of Figure 1, which shows the mean decay length of the LLPs as a function
of their mass difference with the LSP.

Figure 1: Left: bino, wino and higgsino admixtures of the LSP in the region
where it makes up for at least a part of the DM abundance. The colour denotes
the mass of the LSP. Right: mean decay length $c\tau$ as a function of the
mass difference with the LSP, for all points with long-lived EW-inos
($c\tau>1$ mm); blue points have a neutralino and orange points a chargino
LLP.

##  3 Long-lived Dirac electroweakinos at the LHC.

It is well known that current electroweak SUSY searches yield relatively weak
bounds on general SUSY models and their constraining power on the MDGSSM is
not an exception [1]. However, the sensitivity of LLP searches on new physics
provides promising prospects for testing models with long-lived EW-inos. As
seen above, such long-lived EW-inos occur naturally in the MDGSSM.

Long-lived charginos can be constrained by Heavy Stable Charged Particle
(HSCP) and Disappearing Track (DT) searches. For the HSCP constraints we used
SModelS v1.2 [3] to evaluate the bounds from the 8 TeV and early 13 TeV (13
fb-1) CMS analyses [4, 5]. For the DT case, we performed an interpolation of
the 95% CL upper limits on $\sigma\times{\rm BR}$ as a function of the decay
length provided by the 13 TeV ATLAS and CMS analyses [6, 7] for 36 fb-1 and
the CMS 140 fb-1 analysis [8]. Here, $\sigma\times{\rm BR}$ stands for the
direct production cross section of charginos,
$\sigma(\tilde{\chi}^{\pm}_{1}\tilde{\chi}^{\mp}_{1})+\sigma(\tilde{\chi}^{\pm}_{1}\tilde{\chi}^{0}_{1}$),
times $\mathrm{BR}(\chi^{\pm}_{1}\rightarrow\chi^{0}_{1}\pi^{\pm})$. From this
we computed the ratio $r$ of the predicted signal over the observed upper
limit in a similar fashion as SModelS, considering points with $r\geq 1$ as
excluded.222DT and other lifetime dependent constraints can be treated
directly in SModelS as of v2.0.0, released in March 2021.

The results for points with only long-lived charginos are shown in the left
panel of Figure 2 in the plane of chargino mass vs. mean decay length. Red
points are excluded by the HSCP searches, while orange points are excluded by
DT searches. The HSCP limits essentially eliminate all long-lived charginos
with $c\tau_{\tilde{\chi}^{\pm}}>1$ m up to about 1 TeV chargino mass. The
exclusion by the DT searches covers $10\leavevmode\nobreak\ {\rm
mm}<c\tau_{\tilde{\chi}^{\pm}_{1}}<1$ m and $m_{\tilde{\chi}^{\pm}_{1}}$ up to
about 600 GeV. The white band in-between $c\tau\approx 10^{3}$–$10^{4}$ mm
corresponds to $m_{\tilde{\chi}^{\pm}_{1}}-m_{\tilde{\chi}^{0}_{1}}\approx
m_{\pi^{\pm}}$: the chargino lifetime changes abruptly when decays into pions
(as well as those into muons) become kinematically forbidden, leaving decays
into electrons as the only possible channel.

Figure 2: Left: Simplified-model limits for points with long-lived charginos.
Red points are excluded by the HSCP searches, orange points are excluded by DT
searches; the latter are plotted as circles if excluded at $36$ fb-1 and as
triangles if excluded at $140$ fb-1. Non-excluded points are shown in blue.
Right: points with long-lived neutralinos in the plane
$m_{\tilde{\chi}_{1}^{0}}$ vs.
$m_{\tilde{\chi}_{2}^{0}}-m_{\tilde{\chi}_{1}^{0}}$; in colour the branching
ratio of the loop decay
$\tilde{\chi}_{2}^{0}\rightarrow\tilde{\chi}_{1}^{0}\gamma$.

Long-lived neutralinos could potentially lead to displaced vertices. However,
given the small mass differences involved, the decay products of the latter
will be very soft. Furthermore, in these compressed regions, the long-lived
neutralinos predominantly decays into photons via loop-induced processes
($\tilde{\chi}^{0}_{2}\rightarrow\tilde{\chi}^{0}_{1}+\gamma$). The right
panel in Figure 2 shows the importance of this radiative decay in the plane of
$\tilde{\chi}^{0}_{1}$ mass vs. $\tilde{\chi}^{0}_{2}$–$\tilde{\chi}^{0}_{1}$
mass difference. As can be seen, decays into soft photons are clearly
dominant. While this signature would be interesting for model discrimination,
e.g. MDGSSM vs. MSSM, there is currently no experimental analysis which would
be sensitive to it.

##  4 Summary.

We explored the parameter space of the electroweakino sector of the MDGSSM
where the $\tilde{\chi}_{1}^{0}$ is a DM candidate in agreement with relic
density and direct detection constraints. We found that a significant
percentage of the viable scenarios contain long-lived neutralinos and/or
charginos. For the case of long-lived charginos, we derived strong limits by
re-interpreting HSCP and DT searches from ATLAS and CMS. However, for the case
of long-lived neutralinos, we found that they predominantly yield very soft
photons, which are extremely difficult to detect.

## Acknowledgments

The work of HRG was funded by the Consejo Nacional de Ciencia y Tecnología,
CONACyT, scholarship no. 291169. HRG is currently funded by the Italian PRIN
grant 20172LNEEZ.

## References

## References

  * [1] Mark D. Goodsell, Sabine Kraml, Humberto Reyes-González, and Sophie L. Williamson.  Constraining Electroweakinos in the Minimal Dirac Gaugino Model.  SciPost Phys., 9(4):047, 2020.
  * [2] D. Barducci, G. Belanger, J. Bernon, F. Boudjema, J. Da Silva, S. Kraml, U. Laa, and A. Pukhov.  Collider limits on new physics within micrOMEGAs_4.3.  Comput. Phys. Commun., 222:327–338, 2018.
  * [3] Federico Ambrogi et al.  SModelS v1.2: long-lived particles, combination of signal regions, and other novelties.  Comput. Phys. Commun., 251:106848, 2020.
  * [4] Serguei Chatrchyan et al.  Searches for Long-Lived Charged Particles in $pp$ Collisions at $\sqrt{s}$=7 and 8 TeV.  JHEP, 07:122, 2013.
  * [5] Search for heavy stable charged particles with $12.9\leavevmode\nobreak\ \mathrm{fb}^{-1}$ of 2016 data.  Technical Report CMS-PAS-EXO-16-036, CERN, Geneva, 2016.
  * [6] Morad Aaboud et al.  Search for long-lived charginos based on a disappearing-track signature in pp collisions at $\sqrt{s}=13$ TeV with the ATLAS detector.  JHEP, 06:022, 2018.
  * [7] Albert M Sirunyan et al.  Search for disappearing tracks as a signature of new long-lived particles in proton-proton collisions at $\sqrt{s}=$ 13 TeV.  JHEP, 08:016, 2018.
  * [8] Albert M Sirunyan et al.  Search for disappearing tracks in proton-proton collisions at $\sqrt{s}=$ 13 TeV.  Phys. Lett. B, 806:135502, 2020.


nuclear norm minimization (Problem (85) with $D_{1},D_{2}$ being the
identity).

##### Experiment setup

We set $d=d_{1}=d_{2}=25$ and $m=1000$. We generate the underlying unit
Frobenius norm ground truth matrix $M_{\natural}$ randomly with rank
$r_{\natural}=\\{1,2,3,\dots,10\\}$. We vary the noise level
$\sigma=\\{0.1,0.2,\dots,1.3,1.4,1.5\\}$. For each rank $r_{\natural}$, we
generate the sensing Gaussian ensemble $\mathcal{A}$ with $m=1000$ and use the
same one for different noise levels. Then for each noise level, we generate 25
realization of the noise $e$ following $N(0,\sigma^{2}I)$, and solve the
corresponding Problem (85) and the nuclear norm minimization problem. We then
average the error
$\mathopen{}\mathclose{{}\left\|D_{1}^{-1}\hat{X}D_{2}^{-1}-M_{\natural}}\right\|_{\mbox{\tiny{F}}}$
and
$\mathopen{}\mathclose{{}\left\|\hat{X}-M_{\natural}}\right\|_{\mbox{\tiny{F}}}$
over the $25$ trials for each configuration of $r_{\natural}$ and $\sigma$.

##### Recovery Performance

We plot the error in Figure 6. The white color indicates small error and the
dark color indicates large error. It can be seen that it is hard to
differentiate the performance of the solution to the nuclear norm minimization
problem and the solution to Problem (85). This result validates our
theoretical results in Theorem A.1.

Figure 6: Matrix Sensing with nuclear norm minimization (left) and the convex
relaxation (or $k=d$) of Problem (85) (right) for different configurations of
$(r_{\natural},\sigma)$.