aainstitutetext: Department of Mathematical Sciences, University of Liverpool,
Liverpool L69 3BX, United Kingdombbinstitutetext: Deutsches Elektronen-
Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germanyccinstitutetext: Faculty
of Physics and Technology, Karaganda Buketov University, 100028 Karaganda,
Kazakhstan

#  $\mathbf{Z}^{\prime}$-mediated Majorana dark matter: suppressed direct-
detection rate and complementarity of LHC searches

T. Alanne b F. Bishara a J. Fiaschi a O. Fischer a M. Gorbahn a,c and U.
Moldanazarova<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the direct-detection rate for axial-vectorial dark matter scattering
off nuclei in an $\mathrm{SU}(2)\times\mathrm{U}(1)$ invariant effective
theory and compare it against the LHC reach. Current constraints from direct
detection experiments are already bounding the mediator mass to be well into
the TeV range for WIMP-like scenarios. This motivates a consistent and
systematic exploration of the parameter space to map out possible regions
where the rates could be suppressed. We do indeed find such regions and
proceed to construct consistent UV models that generate the relevant effective
theory. We then discuss the corresponding constraints from both collider and
direct-detection experiments on the same parameter space. We find a benchmark
scenario, where even for future XENONnT experiment, LHC constraints will have
a greater sensitivity to the mediator mass.

††preprint: LTH 1298, DESY-22-022

##  1 Introduction

While weakly interacting massive particles (WIMPs) remain attractive
candidates for explaining the dark-matter (DM) content in the Universe, the
null results from the leading direct-detection experiments LUX LUX:2016ggv ;
LUX:2017ree  and XENON1T XENON:2018voc ; XENON:2019rxp  severely constrain the
interaction rates between WIMPs and the Standard Model (SM) particles. It is
possible however to retain sizeable WIMP–quark couplings while suppressing the
direct-detection rates by tuning the up and down quark interaction strengths
in order to cancel the coherent spin-independent contributions of protons and
neutrons in a particular isotope, as was shown e.g. in Ref. Kopp:2010su  for
the case of vectorial DM couplings.

In this paper, we focus on axial-vectorial DM interactions, where the direct-
detection rate is suppressed either due to the absence of coherent enhancement
or via dependence on the velocity of the DM in the halo or on the momentum
exchange between the DM and the nucleus Jungman:1995df . The continuously
tightening constraints led by XENON1T make this scenario phenomenologically
relevant Blanco:2019hah , and if the null results persist in the future,
further suppression will be required to justify the absence of signals.
However, in the case of an axial-vectorial DM coupling, the suppression by
isospin-breaking interactions is more complicated since axial-vectorial and
vectorial quark currents contribute equally to the scattering cross section as
we will show. Furthermore, the $\mathrm{SU}(2)\times\mathrm{U}(1)$ gauge
invariance of the SM implies that the $V-A$ (vectorial minus axial) couplings
of up and down quarks have equal strength, i.e. the effective dimension-6
interactions,

$(\bar{\chi}\chi)_{A}(\bar{u}u)_{V-A}\leftrightarrow(\bar{\chi}\chi)_{A}(\bar{d}d)_{V-A}$
(1)

are directly related. Consequently, it is not clear how efficiently the
isospin suppression can occur for axial-vectorial DM currents.

To consistently incorporate the SM gauge invariance, we use the effective
theory above the electroweak (EW) scale from Ref. Bishara:2018vix  that
couples SU(2)$\times$U(1) invariant SM fields with axial-vectorial DM
currents. Such a theory naturally arises in models where the DM candidate is a
Weyl fermion and the coupling between the dark and visible sectors is mediated
by a tree-level neutral vector-boson exchange, typically referred to as
$Z^{\prime}$ and often related to an additional U$(1)^{\prime}$ gauge symmetry
that is spontaneously broken at a scale $M_{*}\gg M_{\textsc{w}}$. This setup
gives rise to a Majorana fermion in the broken phase below $M_{*}$. In such
models, the vectorial couplings between the DM candidate and the $Z^{\prime}$
vanish because Majorana fermions are self-conjugate under charge conjugation,
while the vectorial current is odd. In addition, the extra U$(1)^{\prime}$
gauge symmetry in these models imposes constraints from anomaly cancellation
similar to those studied in Ref. Ellis:2017tkh  for Dirac DM. The UV
completion is also needed to cure the breakdown of perturbative unitarity
above the cutoff of the effective theory Kahlhoefer:2015bea ; Englert:2016joy
; Jacques:2016dqz ; Cui:2017juz .

Experimentally, this scenario can be tested both at colliders and direct-
detection experiments. At the LHC, this is primarily done by searching for an
excess in the monojet final state, and projecting the excluded cross section
onto the parameter space of simplified benchmark models ATLAS:2021kxv ;
CMS:2021far , selected based on recommendations from both the theory and
experimental communities Buchmueller:2014yoa ; Abdallah:2015ter ;
Abercrombie:2015wmb ; Boveia:2016mrp . The analysis of other final states,
especially dijets ATLAS:2019fgd ; ATLAS:2018qto  and dileptons ATLAS:2015rbx ;
CMS:2016xbv ; ATLAS:2019erb ; CMS:2020ulv ; CMS:2021ctt , can lead to even
tighter, albeit more model-dependent, constraints on the couplings and mass of
the $Z^{\prime}$ mediator. The non-observation of signals in these searches
generally requires mediator masses around the TeV scale.

In direct-detection experiments, the large separation between the $Z^{\prime}$
mass and the momentum transferred in the scattering of DM off nuclei justifies
the adoption of an EFT description of the interaction between the DM and the
baryons and mesons.

This paper is organised as follows. In Sec. 2 we identify the relevant
operators for the EFT of axial-vectorial DM current and study the possible
suppression of direct-detection rates. In Sec. 3 we consider the constraints
arising from the embedding of the low-energy theory into a consistent UV
completion, with particular focus on the gauge-anomaly-cancellation
requirement. We then identify minimal UV-consistent benchmark models featuring
near maximal direct-detection suppression. In Sec. 4 we outline the technical
details for the calculation of the experimental constraints, and in Sec. 5 we
compare the sensitivities of current and future direct-detection experiments
against the ones from collider searches within the selected benchmark
scenarios. Finally, we conclude in Sec. 6. We refer to Appendix A for details
about the anomaly cancellation and Appendix B outlines an example UV-complete
model and details how the UV model with a single Weyl fermion results in low-
energy Majorana DM model with axial-vectorial couplings.

##  2 Effective Field Theory for Direct Detection of Axial Vector Dark Matter

We consider an axial-vectorial DM current coupled to the SM at the dimension-6
level in the unbroken EW phase. The resulting dark-matter-EFT (DMEFT)
Lagrangian can be written as Bishara:2018vix ,

$\mathcal{L}_{\textsc{dmeft}}=\sum_{i,d}{C}_{i}^{(d)}Q^{(d)}_{i}\equiv\frac{\hat{c}_{i}^{(d)}}{\Lambda^{d-4}}Q^{(d)}_{i}\xrightarrow{\textrm{EWSB}}\sum_{i,d}\mathscr{C}_{i}^{(d)}\mathcal{Q}_{i}^{(d)}\,,$
(2)

where we used a curly-script notation for the operators and their coefficients
below the EW scale to distinguish them from the ones above it. The lower-case
hatted coefficients are dimensionless (in natural units) while the upper-case
un-hatted ones are dimensionful.

To be concrete, if $\chi$ is an $\textrm{SU}(2)$-singlet Majorana fermion, the
following three operators coupling $\chi$ to the SM quarks will be generated
above the EW scale (following the notation of Ref. Bishara:2018vix ),

$Q_{6,i}^{(6)}=(\bar{\chi}\gamma_{\mu}\gamma_{5}\chi)(\bar{Q}_{L}^{i}\gamma^{\mu}Q_{L}^{i})\,,\quad
Q_{7,i}^{(6)}=(\bar{\chi}\gamma_{\mu}\gamma_{5}\chi)(\bar{u}_{R}^{i}\gamma^{\mu}u_{R}^{i})\,,\quad
Q_{8,i}^{(6)}=(\bar{\chi}\gamma_{\mu}\gamma_{5}\chi)(\bar{d}_{R}^{i}\gamma^{\mu}d_{R}^{i})\,,$
(3)

where $i=1,2,3$ denotes the quark generation. In principle, we should also
include the operator involving the Higgs-current,
$iH^{\dagger}\stackrel{{\scriptstyle\leftrightarrow}}{{D}}_{\mu}H$, since it
mixes with the operators in Eq. (3) above the EW scale and matches onto both
operators in Eq. (4) below it Bishara:2018vix ; Crivellin:2014qxa ;
DEramo:2014nmf ; Brod:2018ust . However, this effect is not relevant in our
setup because both quark currents appear anyway. Furthermore, the vectorial
quark-current does not lead to an enhanced direct-detection rate since we are
only concerned with the axial-vectorial DM current. After EW symmetry
breaking, the three operators in Eq. (3) match onto the following two:

$\displaystyle{\cal Q}_{2,q}^{(6)}$
$\displaystyle=(\bar{\chi}\gamma_{\mu}\gamma_{5}\chi)(\bar{q}\gamma^{\mu}q)\,,$
$\displaystyle{\cal Q}_{4,q}^{(6)}$
$\displaystyle=(\bar{\chi}\gamma_{\mu}\gamma_{5}\chi)(\bar{q}\gamma^{\mu}\gamma_{5}q)\,.$
(4)

The matching conditions for operators involving first-generation quarks are,

$\begin{split}(A\otimes
V)_{u}:\quad\mathscr{C}_{2,u}^{(6)}&={C}_{7,1}^{(6)}+{C}_{6,1}^{(6)}\,,\qquad\qquad(A\otimes
V)_{d}:\quad\mathscr{C}_{2,d}^{(6)}={C}_{8,1}^{(6)}+{C}_{6,1}^{(6)}\,,\\\
(A\otimes
A)_{u}:\quad\mathscr{C}_{4,u}^{(6)}&={C}_{7,1}^{(6)}-{C}_{6,1}^{(6)}\,,\qquad\qquad(A\otimes
A)_{d}:\quad\mathscr{C}_{4,d}^{(6)}={C}_{8,1}^{(6)}-{C}_{6,1}^{(6)}\,,\end{split}$
(5)

where the shorthand notation on the left-hand side of the colon gives the
Lorentz structure of the operator as a product of DM and SM currents,
respectively, $(A)V$ stands for the (axial-)vectorial current, and the
subscript denotes the quark flavour of the operator.

Unlike in the spin-independent case, however, if the coupling to the DM is
purely axial, both the $A\otimes V$ and $A\otimes A$ operators can contribute
equally to the direct-detection cross section for heavy nuclei (i.e. with mass
number $\mathscr{A}\gtrsim 100$). Consequently, it is not trivial to obtain a
similar suppression as in the spin-independent case. It is already known that
in the $V\otimes V$ interaction, the cancellation can be effected by tuning
the non-relativistic coefficients of the proton and neutron operator, i.e. by
breaking isospin symmetry. For a single isotope, this cancellation can be
complete only in the zero momentum transfer limit, $q\to 0$. For the
interactions of concern here, the mechanism is the same. The complication
arises when both vectorial and axial SM currents contribute equally to the
cross section. The reason is that the amount, and more importantly the sign,
of the isospin-breaking ratio required to cancel each interaction is
different. This is because the vectorial currents count the number of up and
down quarks in the nucleus, therefore, the coefficients of the operators
containing up and down currents have to have opposite signs in order for the
interference term to be negative. For the axial current, on the other hand,
the contribution of the up and down quarks to the spin of the nucleon have
opposite signs to begin with. Thus, the coefficients of the operators
containing axial up and down currents have to have the same sign for the
interference term to be, again, negative. The ability to align the minima of
both the $A\otimes A$ and $A\otimes V$ stems from the fact that the former is
much more sensitive to which combination of the two right-handed quark
currents (up vs. down) implements isospin breaking.

The matching conditions in Eq. (5) present us with different possibilities as
follows.

  1. 1.

We can entirely eliminate either the vectorial, $V$, or axial-vectorial, $A$,
currents on the SM side (i.e. for all flavours). This can be accomplished with
the choice,

${C}_{6,1}^{(6)}=\mp{C}_{7,1}^{(6)}=\mp{C}_{8,1}^{(6)}\,,$ (6)

which automatically enforces isospin-symmetric coefficients in the EFT.

  2. 2.

If isospin breaking is desired, neither the vectorial nor the axial-vectorial
currents can be completely eliminated, and one is left with a mixture of both,
see Eq. (5). As we will show, this makes suppressing the direct detection rate
more complicated since only special regions in parameter space allow for the
simultaneous suppression of the contributions from both currents – see, e.g.,
Figs. 1(b) – 1(d).

To simplify the analysis, we adopt the following parametrisation for the
coefficients,

$C_{6,1}^{(6)}\to\frac{g^{\prime\,2}}{\Lambda^{2}}\cos\theta\,,\qquad
C_{7,1}^{(6)}\to\frac{g^{\prime\,2}}{\Lambda^{2}}\sin\theta\cos\phi\,,\qquad
C_{8,1}^{(6)}\to\frac{g^{\prime\,2}}{\Lambda^{2}}\sin\theta\sin\phi\,,$ (7)

such that the sum of their squares equals $g^{\prime\,4}/\Lambda^{4}$. This
parametrisation removes one variable and allows us to project the relative
direct-detection rate, in a model with only first-generation quarks coupled to
DM, as function of the overall value of the Wilson coefficients in the
$\theta-\phi$ plane.

The contours in Fig. 1(a) represent the relative direct-detection rates for a
reference DM mass $m_{\chi}$ = 100 GeV. The figure shows that for specific
choices of the parameters a factor of roughly $10^{-2}$ suppression can be
achieved. For heavier DM masses, while the overall event rate falls as
$1/m_{\chi}$, there is no qualitative change in the arguments presented here.
The isospin-symmetric limit is realized on two horizontal lines in the
$\theta-\phi$ plane with $\phi=\pi/4,5\pi/4$. These lines are shown in Fig.
1(a) as thick grey lines labelled ‘isospin limit’.

Figure 1(b) shows the relative rate with the normalisation of Fig. 1(a) along
the curve $\theta=\pi/2$ corresponding to scenarios where the coefficient
$C_{6,1}^{(6)}=0$, thus where only first-generation right-handed quarks are
coupled to the DM current. The $A\otimes V$ and $A\otimes A$ contributions are
shown separately in dashed blue and dotted red curves, respectively, and the
combination of these two, i.e. the total relative rate corresponding to the
contours in Fig. 1(a), in solid gray. The gray dots denote the isospin limit
while the blue and orange stars, also reported in Fig. 1(b), correspond to the
benchmark scenarios BM1 and BM2 that will be analysed in Sec. 5. The scenario
BM1 corresponds to the case where the $A\otimes V$ rate is significantly
suppressed, and naive treatment without considering also the $A\otimes A$
contribution would lead to a wrong conclusion since, in fact, only
$\mathcal{O}(1)$ suppression of the direct-detection rate can be obtained. The
benchmark BM2 has instead the minimal direct-detection rate in this
construction where the first-generation doublet does not couple to the DM
current.

Figure 1(c) depicts the relative rate with the same normalisation in the
isospin limit showing explicitly that significant suppression is not possible
by cancelling separately the $A\otimes V$ or the $A\otimes A$ contribution.
Finally, Fig. 1(d) illustrates the rate along the $\phi\approx 0.02$ line
passing through the global minimum in the $\theta-\phi$ plane showing that the
largest suppression can be achieved in the parameter space point where both
$A\otimes V$ and $A\otimes A$ simultaneously have a minimum.

(a) Contours of the relative direct-detection event rate in the $\theta-\phi$
plane using the parameterisation of Eq. (7). The thick horizontal lines
correspond to the isospin limit.

(b) Relative event rate along $\theta=\pi/2$ corresponding to
$C_{6,1}^{(6)}=0$.

(c) Relative event rate along $\phi=\pi/4$, i.e. the isospin limit.

(d) Relative event rate along $\phi=0.02$ that passes throught the global
minimum.

Figure 1: The direct detection rate for $m_{\chi}=100$ GeV using the
parameterisation of Eq. (7) with normalisation corresponding to the overall
size of the Wilson coefficients. The blue (orange) star shows the benchmark
point BM1 (BM2). The solid grey curve in the upper-right panel, (1(b)), and
both lower panels, (1(c)) & (1(d)), show the combined relative rate with the
normalisation of panel (2(a)) along different lines in the two-dimensional
plot, while the dashed (blue) and dotted (red) curves show the sole $A\otimes
V$ and $A\otimes A$ contributions along the same line, respectively.

##  3 UV completions and anomaly cancellation

A straightforward path to UV completing the EFT setup in Sec. 2 is to augment
the SM gauge group by a $\textrm{U}(1)^{\prime}$ group which couples to the SM
fermions and the DM candidate via a heavy $Z^{\prime}$, $m_{Z^{\prime}}\gg
m_{Z}$. For the $Z^{\prime}$ to couple axial-vectorially to the DM, the latter
has to be chiral under the $\textrm{U}(1)^{\prime}$. Apart from the need to
Higgs this gauge symmetry in order to generate a mass for the mediator, the DM
and SM fermion $\textrm{U}(1)^{\prime}$ charges must be chosen such that pure
and mixed gauge anomalies cancel. In the following, we briefly discuss this
and other general aspects that arise from considering possible UV completions
of the EFT setup, and we refer to Appendix A for general anomaly equations
involving first- and second-generation SM quarks, and to Appendix B for an
explicit construction of a one-generation model that addresses (most) of these
general points.

  1. (i)

_Anomaly cancellation._ Anomaly-free DM models were discussed in Ref.
Ellis:2017tkh ; Cui:2017juz ; FileviezPerez:2019jju ; Perez:2020jyg ;
Costa:2019zzy  where, however, the minimal model with the additional matter-
field content consisting of only one Weyl fermion was not discussed. The
general anomaly equations that must be satisfied are given in Appendix A and
their solution in this case requires that the SM fermions to be charged under
$\textrm{U}(1)^{\prime}$.

  2. (ii)

_Couplings of the $Z^{\prime}$ to leptons._ A feature of the mixed anomaly
equations is that charges of the SM fermions are, in general, a linear
combination of their hypercharge $Y$ and $B-L$ where $B(L)$ are the
baryon(lepton) numbers which are $\pm\tfrac{1}{3}(\pm 1)$. This general
statement has a significant consequence, namely that coupling the $Z^{\prime}$
to the SM leptons is unavoidable, and thus observables involving leptons must
be taken into account.

  3. (iii)

_SM Yukawa couplings._ The $\textrm{U}(1)^{\prime}$ gauge invariance forbids
some SM Yukawa couplings. We show a possible mechanism for generating the
Yukawa couplings in the model with only one generation of SM quarks carrying
U$(1)^{\prime}$ charge in Appendix B. This construction includes a dark Higgs,
$S$, charged under $\textrm{U}(1)^{\prime}$ and generating the effective
Yukawa couplings upon spontaneous breaking of the symmetry. Another option is
to add Higgs doublets that also carry a $\textrm{U}(1)^{\prime}$ charge
Ko:2012hd .

  4. (iv)

_Tree-level-induced spin-independent contributions._ Depending on the scalar
sector of the theory, the tree-level exchange of physical scalars could induce
a sizeable spin-independent cross section. However, we stress that this is
model dependent and could be suppressed, for example, in the following two
ways. First, with one dark Higgs, a small U$(1)^{\prime}$ gauge coupling
allows the mass of the scalar to be significantly above that of the
$Z^{\prime}$. Second, it could be tuned to zero at the tree level by extending
the scalar sector such that no single state couples to both the quarks and the
DM sector at the same time. In this case, the mixing between the scalars would
still generate this interaction though it would be additionally suppressed by
at least a loop factor.

  5. (v)

_Loop-induced spin-independent contributions._ While spin-independent
scattering is suppressed at the tree level, it can be induced at the one-loop
level depending on the UV completion via either two insertions of the axial-
vectorial coupling or by a potential dark Higgs penguin with a $Z^{\prime}$ in
the loop. Using naive dimensional analysis, we estimate the relative size of
the loop-induced scalar-scalar interaction via internal vector-boson exchange
in comparison with the tree-level current-current interaction:

$\frac{\sigma_{\textsc{si}}}{\sigma_{\textsc{sd}}}\sim
A^{2}\frac{g^{\prime\,4}}{(4\pi)^{4}}\frac{m_{N}^{2}m\chi^{2}}{m_{Z^{\prime}}^{4}}\sim\mathcal{O}(10^{-11})\,,$
(8)

where we used $g^{\prime}=0.1$, $m_{Z^{\prime}}=1$ TeV,
$m_{\chi}=m_{Z^{\prime}}/2$, and $m_{N}=1$ GeV is the nucleon mass. We also
took the mass number of the atomic nucleus (xenon, for example) to be
$\mathscr{A}=100$. The dependence on the DM and nucleon masses arises from the
required chirality flips in the scalar-scalar operator. Whether one should
insert the nucleon mass or $\Lambda_{\textsc{qcd}}$ is irrelevant to the
estimate, but we note that the respective form factors would suppress the
spin-independent contribution even further. The heavy suppression of the spin-
independent scattering rate and its model-dependent origin justifies our
choice of neglecting this contribution

  6. (vi)

_Flavour violation._ In models where only the right-handed quarks are charged
under the U$(1)^{\prime}$, the mass- and gauge-bases can always be aligned
using the freedom in choosing the right-handed rotation matrices. However, in
models where the three generations of left-handed quark doublets carry non-
universal U$(1)^{\prime}$ charges, and working the down-basis, flavour
violation in the up quark sector is unavoidable. As a result, the $Z^{\prime}$
gauge boson mediates tree-level flavour-changing neutral currents. Thus,
neutral $D$ meson mixing provides strong constraints on the breaking of the
first- and second-generation U$(2)$ flavour symmetry which bounds the
$Z^{\prime}$ mass to be $m_{Z^{\prime}}\gtrsim 50$ TeV for the model in Table
1, see Ref. UTfit:2007eik . Retaining the U$(2)$ symmetry can lower this bound
significantly.

(a) The dotted, dashed, and dash-dotted curves correspond to gauge-anomaly-
free scenarios S1, S2 and S3, respectively. The thick horizontal lines
correspond to the isospin limit.

(b) Relative event rate along $\tan\theta=2/(\sin\phi+\cos\phi)$.

(c) Relative event rate along $\phi=0$.

Figure 2: The direct-detection rate for $m_{\chi}=100$ GeV using the
parameterisation of Eq. (7) with normalisation corresponding to the overall
size of the Wilson coefficients. The green (red) star shows the benchmark
point BM3 (BM4)in Table 2. The solid grey curve in upper right and lower
panels shows the combined relative rate with the normalisation of panel (2(a))
along $\phi=0$, while the dashed (blue) and dotted (red) curves show the sole
$A\otimes V$ and $A\otimes A$ contributions along the same curve,
respectively.

We use the parameterisation of Eq. (7) to study the potential suppression of
the direct-detection rate in minimal anomaly-free UV completions of the
scenario. With only one dark Weyl fermion and only first-generation quarks
charged under U$(1)^{\prime}$ (or in the case of equal charges for all
generations), we are restricted to lie on the dotted gray line of Fig. 2(a),
which corresponds to the choice $\tan\theta=2/(\sin\phi+\cos\phi)$. The
projection of the direct-detection rate with the same normalisation as before
is shown in Fig. 2(b). Here the green star represents the choice with the
minimal relative rate for this one-generation scenario, confirming that only
$\mathcal{O}(1)$ suppression is achievable. A more detailed discussion about
the construction of the corresponding UV complete model of Eq. (12) is given
in Appendix B.

To reach near maximal suppression in the $\theta-\phi$ plane, we need to
augment the minimal one-generation scenario with generation-dependent charges.
The dot-dashed (dashed) horizontal gray lines in Fig. 2(a) show the anomaly-
free solutions with second-generation up(down)-type quark carrying an
independent U$(1)^{\prime}$ charge, respectively. This solution is referred to
as S2 (S3) in Appendix A. We checked explicitly that the contribution of the
strange-quark current to the direct detection is sub-leading and does not
affect the positions of the minima as expected.

Figure 2(c) shows the relative event rate using the same arbitrary
normalization factor along the $\phi=0$ curve. Along this projection, we
identify the BM3 (red star) whose direct-detection event rate is very close to
the achievable minimum, and which will be analysed in Sec. 5. This model is
realised by charging the first-generation left- and right-handed quarks and
the right-handed strange quark under U$(1)^{\prime}$, while we choose to
couple only the third-generation leptons in order to alleviate the stringent
bounds from dilepton searches. The specific charge assignments for the BM3
benchmark are summarised in Table 1.

| $Q_{L,1}^{c}$ | $u_{R,1}$ | $d_{R,1}$ | $u_{R,2}$ | $d_{R,2}$ | $L_{L,3}^{c}$ | $e_{R,3}$ | $\chi_{R}$
---|---|---|---|---|---|---|---|---
BM3 | $+\frac{1}{2}$ | +1 | 0 | 0 | -2 | $-\frac{3}{2}$ | 0 | 3
Table 1: The benchmark model BM3 with only one addtional Weyl fermion and
first-generation quarks, right-handed second-generation down quarks and third-
generation left-handed leptons carrying U$(1)^{\prime}$ charge.

In the next sections, we study the experimental constraints for the benchmark
models, and compare the current and future sensitivities at direct-detection
experiments against up-to-date collider exclusions.

##  4 Technical details for experimental constraints

###  4.1 Direct detection

The DM direct-detection experiments search for signals from DM scattering off
atomic nuclei in shielded detectors. For concreteness, we consider the XENON1T
experiment with an exposure of 278.8 days $\times$ 1300 kg XENON:2018voc , and
also the projected exposure of 20 ton $\times$ year for the XENONnT
XENON:2020kmp . The scattering rate $\mathcal{R}$, which is the expected
number of events per detector mass per unit time, can be expressed
differentially with respect to the recoil energy as Lewin:1995rx ,

$\frac{d\mathcal{R}}{dE_{\rm
R}}=\frac{\rho_{\chi}}{m_{A}m_{\chi}}\int_{v_{\text{min}}}\frac{d\sigma}{dE_{\rm
R}}vf_{\oplus}(\vec{v})d^{3}\vec{v},$ (9)

where $E_{\rm R}$ is the recoil energy of the nucleus, $m_{A}$ is the mass of
the nucleus, and $\rho_{\chi}$ is the local DM density. We approximate the DM
velocity in the halo, $f_{\oplus}(\vec{v})$, with a Boltzmann distribution and
integrate over recoil energies in the range $E_{R}\in[3,40]$ keV XENON:2018voc
to approximate the detector efficiency. The exclusion curves in Fig. 3 were
obtained (naively) using Poisson statistics assuming zero events in the signal
region. The coefficients of the Galilean-invariant effective theory
Fitzpatrick:2012ix  were computed using DirectDM bishara2017directdm  and the
nuclear responses and direct-detection rates were obtained with DMFormFactor
anand2013model .

###  4.2 Monojet searches at the LHC

Model-independent searches for DM at the LHC are primarily performed via
monojet ATLAS:2021kxv ; CMS:2021far  and monophoton analyses ATLAS:2020uiq ;
CMS:2017qyo , which can be interpreted in the context of simplified DM models,
see Refs. Buchmueller:2014yoa ; Abdallah:2015ter ; Abercrombie:2015wmb , with
the monojet final state typically providing the stronger limits. Here we
consider the recent DM analysis by the ATLAS collaboration ATLAS:2021kxv
which uses 139 fb-1 data and where monojet events with large missing energy
were used to constrain a simplified model with a vector mediator,
$Z^{\prime}$, coupling to the DM, $\chi$, and the quark, $q$, excluding
$Z^{\prime}$ with masses around 2 TeV. The quoted limits on axial mediators
are very similar.

We were able to reproduce the exclusion limits with good approximation for the
simplified DM benchmark model adopted in the experimental analyses
Buchmueller:2014yoa ; Abdallah:2015ter ; Abercrombie:2015wmb ; Boveia:2016mrp
, where the couplings of the vector mediator $Z^{\prime}$ to quarks and the DM
are fixed to $g_{q}$ = 0.25 and $g_{\chi}$ = 1. We scanned over several
configurations of DM and mediator masses by importing the simplified DM model
from Ref. Backovic:2015soa  into MadGraph5_aMC@NLO Alwall:2014hca  to generate
the WIMP $s$-channel process $pp\to jV\to j\bar{\chi}\chi$ where $j$ is a jet
from initial state radiation, and the DM particle pair, $\bar{\chi}\chi$,
gives rise to missing transverse energy, $E_{T}^{\rm miss}$, in the detector.
The process is implemented at LO in the strong coupling constant. We adopted
the NNPDF3.0_LO PDF set NNPDF:2014otw , and for each event the factorization
and renormalisation scales were set to $H_{T}/2$, with the total hadronic
transverse energy $H_{T}=\sqrt{m_{\chi\chi}^{2}+p_{T,j}^{2}}+p_{T,j}$ where
$m_{\chi\chi}$ is the invariant mass of the DM pair, and $p_{T,j}$ is the
transverse momentum of the parton-level jet. Events are hadronised using
Pythia8 Sjostrand:2007gs , and a fast detector simulation is carried out using
Delphes deFavereau:2013fsa . We apply the kinematic cuts from Ref.
ATLAS:2021kxv , which are as follows: $E_{T}^{\textrm{miss}}>$ 200 GeV;
leading jet with $p_{T}>$ 150 GeV and $|\eta|<$ 2.4; no more than three
additional jets with $p_{T}>$ 30 GeV and $|\eta|<$ 2.8; separation between
missing transverse momentum and each of the jets
$\Delta\phi(\textrm{jet},p_{T}^{\textrm{miss}})>$ 0.4 (0.6) for events with
$E_{T}^{\textrm{miss}}>$ 250 GeV (200 GeV $<E_{T}^{\textrm{miss}}<$ 250 GeV).
The remaining simulated events were binned in thirteen exclusive signal
regions as in Ref. ATLAS:2021kxv  according to their missing transverse
energy. We simulated a sufficient number of events such that, after the
selection cuts, we still obtain a statistically significant sample in all the
bins. Finally we excluded parameter space points where the fiducial cross
section of the signal in any bin is bigger than its uncertainty at 95%
confidence, which is evaluated by adding the total systematic uncertainty of
the signal quadratically to the statistical uncertainty of the signal and the
overall uncertainty of the background (statistical and systematic111The
systematic uncertainty of the signal is obtained by combining the relative
uncertainties from Ref. ATLAS:2021kxv : luminosity uncertainty 1.7%; cross
section scale uncertainty 10%; a PDF uncertainty 5%; PDF choice 10%; 1% to 7%
for the jet $E_{T}^{\textrm{miss}}$ reconstruction, energy scale and
resolution; modelling initial and final state radiation 3% to 6%. Scale
uncertainty of the signal is neglected. The systematic uncertainties are added
linearly, overall systematic and statistical uncertainties are added in
quadrature.) from Ref. ATLAS:2021kxv .

We validated our analysis by reproducing the existing exclusion limits and by
comparing with results from Ref. Lozano:2021zbu , we explored the current LHC
monojet sensitivity in the three BMs introduced in the previous sections. The
exclusion limits we obtain will be presented and discussed in Sec. 5.

###  4.3 Dijets and dileptons

Heavy mediators that couple to quarks can be detected at the LHC via their
decays into quarks and leptons. The most recent analysis by the ATLAS
collaboration searching for heavy resonances in dijet final states uses 139
fb-1 data for state-of-the art constraints ATLAS:2019fgd  for mediator masses
above 2 TeV. For lower masses above 700 GeV we use the results presented in
Ref. ATLAS:2018qto  that are based on 29.3 fb-1 of data. For the decay into
taus we used the constraints on the cross section of the combined hadronic and
leptonic channels ATLAS:2015rbx , which cover mediator masses between 0.5 and
2.5 TeV and are based on an integrated luminosity of $19.5-20.3$ fb-1. We
recast these limits into the considered model parameter space, and estimate a
lower limit on the $Z^{\prime}$ mass of about 2 TeV using the code developed
in Ref. Bishara:2018sgl  except that we use the NNPDF3.0_LO PDF set.

##  5 Results

Figure 3: Exclusion limits for axial mediator couplings to DM from XENON
experiment for 1 ton $\times$ year exposure (red solid) and for 20 ton
$\times$ year exposure (red dashed), from LHC monojet (blue) and dijet (green)
analyses with 139 fb-1 integrated luminosity. The results are shown in the
plane with the DM mass on the $x$ axis and inverse of the square root of the
Wilson coefficient $|C^{(6)}_{7,1}|$ on the $y$ axis for the BM1 (left) and
BM2 (right).

In this section we compare the exclusion limits from the XENON experiment and
from the LHC for the introduced BMs. In Fig. 3, we show the results for BM1
and BM2. The two benchmarks represent realisations of models where only the
first generation of quarks is coupled to the DM current through a vector
mediator, $Z^{\prime}$. They both feature a null charge to the left-handed
quarks, i.e. $|C^{(6)}_{6,1}|$ = 0, corresponding to the choice $\theta=\pi/2$
in the parametrisation of Eq. (7). The exclusion limits from collider monojet
and dijet searches and from the current XENON1T and projected XENON20T
sensitivities are presented in the plane with the DM mass (in GeV) on the
horizontal axis and the (inverse of the square root of the) Wilson coefficient
$|C^{(6)}_{7,1}|$ on the vertical axis. The magnitude of the couplings to
right-handed quarks has been fixed in order to match the interaction strength
of the experimental benchmarks of Ref. Buchmueller:2014yoa ; Abdallah:2015ter
; Abercrombie:2015wmb ; Boveia:2016mrp , such that a direct comparison with
the results in the literature is straightforward.

The benchmark BM1 (left panel of Fig. 3) probes the case where the $A\otimes
V$ contribution to the direct-detection rate is at minimum (blue star of Fig.
1(b)) while the $A\otimes A$ contribution is still substantial. This case
corresponds to the choice $\phi=2.4$ and gives the couplings $g_{u_{R}}\simeq
0.372$ and $g_{d_{R}}\simeq-0.334$. The second benchmark, BM2, (right panel of
Fig. 3) probes the case where the one-generation model has the minimum overall
direct-detection rate ($A\otimes V$ \+ $A\otimes A$) (orange star of Fig.
1(b)). This case corresponds to the choice $\phi=0.14$ and gives the couplings
$g_{u_{R}}\simeq 0.495$ and $g_{d_{R}}\simeq 0.069$.

The limits from the monojet analysis (blue curves) depend mainly on the
overall magnitude of the interaction, which have been kept fixed in the two BM
scenarios in order to be directly comparable with the experimental benchmark
Buchmueller:2014yoa ; Abdallah:2015ter ; Abercrombie:2015wmb ; Boveia:2016mrp
. The very marginal differences in the sensitivity curves for our choices of
the chiral structure of the couplings arise when the latter are convoluted
with the respective PDF weights. The conservative strongest exclusion of the
monojet analyses rules out $Z^{\prime}$ masses below 1.6 TeV, and it
translates into an upper limit reach of about 800 GeV DM, which corresponds to
the on-shell limit of the mediator decaying into DM $m_{\rm DM}\leq
m_{Z^{\prime}}/2$. The limits from dijet searches for mediator masses above
700 GeV (green curves) are largely independent on the DM mass and provide with
comparable exclusion to the monojet for light DM. The rather small increase in
sensitivity of the dijet analysis appearing at DM masses of 400 (600) GeV for
the BM1 (BM2) occurs at the transition between analyses optimised for
intermediate and heavy mediator, as explained in Sec. 4.3. The current XENON
sensitivity (solid red curves) is somewhat weaker than the collider
exclusions, especially for heavy DM, albeit being able to probe an extended DM
mass interval and lighter mediator masses. On the other hand, the projection
for the XENON experiment with 20 ton $\times$ year exposure (dashed red
curves) will be able to test a larger parameter space region in comparison
with the current collider reach.

Figure 4: Exclusion limits for the model of Table 1 from XENON experiment for
1 ton $\times$ year exposure (red solid) and for 20 ton $\times$ year exposure
(red dashed), from LHC monojet (blue) and dijet (green) analyses with 139 fb-1
integrated luminosity and ditau (yellow) analyses with $19.5-20.3$ fb-1
integrated luminosity. The results are shown in the plane with the DM mass on
the $x$ axis and the inverse of the square root of the Wilson coefficient
$|C^{(6)}_{7,1}|$ on the $y$ axis.

Figure 4 shows the exclusion limits for the anomaly-free BM3. The interaction
strengths of this model are given by the U$(1)^{\prime}$ charges of SM and DM
particles, reported in Table 1, multiplied by an overall gauge coupling that
we fix to $g^{\prime}=0.1$. As already mentioned, this construction features
an almost minimal direct-detection rate, thus the constraints on heavy
mediators from the XENON experiment, both current (solid red line) and future
(dashed red line), are outperformed by the collider sensitivity. Smaller
mediator masses, between 300 and 700 GeV, are constrained by neither dijet nor
direct-detection searches, see also Blanco:2019hah . Because of the smaller
couplings to quarks and DM, the monojet searches (blue curve) can probe only
up to 1.2 TeV $Z^{\prime}$ masses. As the complete model features other decay
channels for the mediator that increase its decay width, the reach of the
monojet analysis in terms of DM mass is reduced, and it can only test masses
up to about 300 GeV, well below the on-shell limit. On the other hand the
additional decay channels lead to other testable signatures and they can be
searched for via dijets (green line) and ditaus (yellow line), which have a
stronger sensitivity compared to the monojet searches and are mostly
independent of the DM mass.

##  6 Summary and Outlook

In this work we considered a WIMP-like Majorana dark matter candidate that
mainly interacts through an axial-vectorial current with the visible sector.
The direct-detection rate is dominated by the interaction with light quarks in
this scenario, and we consider the three $\mathrm{SU}(2)\times\mathrm{U}(1)$
invariant operators that couple the first generation of quarks at dimension 6.
Varying the relative magnitude of the three respective Wilson coefficients, we
identify regions in the parameter space — as shown in Fig. 2(a) — where the
direct-detection rate for a xenon target is significantly suppressed.

In models where these Wilson coefficients are generated via $Z^{\prime}$
exchange, we find that collider and direct-detection experiments have
comparable sensitivity for typical choices of the couplings. For parameter
points that are chosen to be comparable to current experimental benchmark
scenarios, we find that future direct-detection experiments will test large
parts of the parameter space, see e.g. Fig. 3, that are not accessible at the
LHC.

Yet, in a UV-consistent $Z^{\prime}$ model, anomaly conditions further
constrain the allowed parameter space of the Wilson coefficients, and the
direct-detection rate can only be suppressed by considering generation-
dependent charges. By charging the right-handed strange quark instead of the
right-handed down quark, we find an anomaly-free charge assignment that
suppresses the direct-detection rate for xenon targets. If realised in Nature,
such a scenario would have to be tested via collider searches, which could
close the window of smaller mediator masses and smaller gauge couplings.

## Acknowledgements

We would like to thank Joachim Brod, Uli Haisch, Ed Hardy, and Jure Zupan for
helpful discussions. The work of JF and MG has been supported by STFC under
the Consolidated Grant ST/T000988/1. This work was also supported by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy - EXC 2121 “Quantum Universe” - 390833306 and
the Aspen Center for Physics, which is supported by National Science
Foundation grant PHY-1607611.

##  Appendix A Anomaly conditions

We construct here the anomaly-free conditions with only one additional Weyl
fermion, and first-generation SM quarks plus potentially one quark flavour
from the second-generation. In addition, one generation of leptons must also
carry a U$(1)^{\prime}$ charge in this setup; choosing the third generation
helps to evade to some extent the stringent constraints from dilepton
searches. The generic U$(1)^{\prime}$ charge assignment of the right-handed
matter fields is given by

$Q_{L,1}^{c}$ | $u_{R,1}$ | $d_{R,1}$ | $L_{L,3}^{c}$ | $e_{R,3}$ | $\chi_{R}$ |  $u_{R,2}$ or $d_{R,2}$
---|---|---|---|---|---|---
$a$ | $b$ | $c$ | $d$ | $e$ | $x$ | $z$
(10)

and leads to the following anomaly equations (see Sec. 22.4 of Ref.
Weinberg:1996kr ):

$\begin{array}[]{c
l}\text{SU}(3)\times\text{SU}(3)\times\text{U}(1)^{\prime}:&2a+b+c+z=0\\\
\text{Grav}\times\text{Grav}\times\text{U}(1)^{\prime}:&6a+3b+3c+2d+e+x+3z=0\\\
\text{SU}(2)\times\text{SU}(2)\times\text{U}(1)^{\prime}:&3a+d=0\\\
\text{U}(1)_{Y}\times\text{U}(1)_{Y}\times\text{U}(1)^{\prime}:&6a+48b+12c+18d+36e+108\,Q_{u(d)}^{2}\,z=0\\\
\text{U}(1)_{Y}\times\text{U}(1)^{\prime}\times\text{U}(1)^{\prime}:&-6a^{2}+12b^{2}-6c^{2}+6d^{2}-6e^{2}+18\,Q_{u(d)}\,z^{2}=0\\\
\text{U}(1)^{\prime}\times\text{U}(1)^{\prime}\times\text{U}(1)^{\prime}:&6a^{3}+3b^{3}+3c^{3}+2d^{3}+e^{3}+x^{3}+3z^{3}=0\\\
\end{array}$ (11)

| $Q_{L,1}^{c}$ | $u_{R,1}$ | $d_{R,1}$ | $u_{R,2}$ | $d_{R,2}$ | $L_{L}^{c}$ | $e_{R}$ | $\chi_{R}$
---|---|---|---|---|---|---|---|---
S1 | $\frac{1}{6}(e+x)$ | $\frac{1}{3}(x-2e)$ | $\frac{1}{3}(e-2x)$ | $0$ | $0$ | $-\frac{1}{2}(e+x)$ | $e$ | $x$
S2 | $\frac{1}{4}(x-z)$ | $0$ | $-\frac{1}{2}(x+z)$ | $z\neq 0$ | $0$ | $-\frac{3}{4}(x-z)$ | $\frac{1}{2}(x-3z)$ | $x$
S3 | $\frac{1}{2}(x+z)$ | $-x-2z$ | $0$ | $0$ | $z\neq 0$ | $-\frac{3}{2}(x+z)$ | $2x+3z$ | $x$
Table 2: Anomaly-free models where only first-generation quarks or one full
generation plus either a right-handed second-generation down or up quarks
carry U$(1)^{\prime}$ carges. Only the left-handed third-generation lepton
doublet carries a U$(1)^{\prime}$ charge to avoid the most stringent bounds
from dilepton searches.

Here, $Q_{u(d)}$ is the electric charge, $+2/3(-1/3)$, of the quark to which
the $U(1)^{\prime}$ charge, $z$, is assigned in Eq. (10). There is one
solution with only one-generation, S1, and one solution with an additional
second generation up(down)-type quark carrying an independent charge, S2 (S3).
These solutions are shown in Table 2. We denote the charge of the new Weyl
fermion, $\chi_{R}$, by $x$, the charge of the right-handed lepton in S1 by
$e$, and the charge of the second generation quarks by $z$ in S2 and S3.

##  Appendix B A possible UV completion

We want a DM candidate with purely axial-vectorial coupling to a spin-1
mediator. Since the most minimal additional matter field content — one Weyl
fermion charged under a spontaneously broken $\textrm{U}(1)^{\prime}$ gauge
group — gives rise to a Majorana fermion after spontaneous symmetry breaking,
the desired axial-vectorial coupling is automatically guaranteed. Hence, we
extend the SM gauge group by an additional $\textrm{U}(1)^{\prime}$ which is
spontaneously broken by the vacuum expectation value of a scalar field, $S$,
and add one Weyl fermion, $\chi_{R}$, that is charged under this
$\textrm{U}(1)^{\prime}$ and is additionally odd under a $\mathbb{Z}_{2}$
symmetry that remains exact. This fermion is neutral under the SM gauge group.

Since the DM candidate $\chi_{R}$ is chiral under the
$\textrm{U}(1)^{\prime}$, the gauge symmetry is anomalous. One simple solution
to make it anomaly free is to also charge one generation of right-handed SM
fermions under the $\textrm{U}(1)^{\prime}$:

$u_{R}$ | $d_{R}$ | $e_{R}$ | $\chi_{R}$
---|---|---|---
$-1$ | $+1$ | $+1$ | $-1$
(12)

This assignment is sufficient to cancel all mixed and pure anomalies, and it
corresponds to the solution S1 in Appendix A with $e=1,x=-1$. However,
charging the right-handed SM fermions under $\textrm{U}(1)^{\prime}$ forbids
their Yukawa terms at dimension four. To write these terms, one needs to
include powers of the $\textrm{U}(1)^{\prime}$ Higgs, $S$, suppressed by the
same power of the scale $M_{*}$ where the interaction is generated.

Writing in terms of Weyl spinor fields transforming under the
$(0,\tfrac{1}{2})$ representation of the Lorentz group and following the
conventions of Ref. Dreiner:2008tw , the Lagrangian describing the fields
charged under the $\textrm{U}(1)^{\prime}$ is given by

$\mathcal{L}_{\textrm{U}(1)^{\prime}}=\sum_{f=u,d,e,\chi}\,f_{R}^{\dagger}\,iD_{\mu}\sigma^{\mu}\,f_{R}+\left(D_{\mu}S\right)^{\dagger}D^{\mu}S-\left[\frac{1}{2}y_{\chi}\,\chi_{R}\chi_{R}\,S+h.c.\right]\,,$
(13)

where $D_{\mu}=\partial_{\mu}+i\,g\,q^{\prime}\,Z^{\prime}_{\mu}$ is the
$\textrm{U}(1)^{\prime}$ covariant derivative and
$\sigma_{\mu}\equiv(\mathbb{1}_{2\times 2};\vec{\sigma})$ where
$\sigma_{i}\;\forall\;i\in\\{1,2,3\\}$ are the Pauli matrices, see Ref.
Dreiner:2008tw  and references therein. In order to be able to write the
Yukawa term in the square bracket, the $\textrm{U}(1)^{\prime}$ charge of the
Higgs field $S$ must be $+2$. However, such a choice would require one
additional $\textrm{U}(1)^{\prime}$ charged scalar with charge $+1$ to allow
for the SM Yukawa terms given the charge assignment (12). Thus, a solution
that would allow all Yukakwa interactions with only one
$\textrm{U}(1)^{\prime}$-charged scalar, $S$, forces us to assign it a charge
$+1$. This choice forbids the Yukawa term in Eq. (13) at the renormalizable
level and all Yukawa terms now arise at dimension five in the following way
(again, writing in terms of fields that transform under the $(0,\tfrac{1}{2})$
representation of the Lorentz group as before)

$\mathcal{L}_{\text{U}(1)^{\prime}}^{\text{Yukawa}}=-y_{u}\,\widetilde{H}\cdot
Q_{L}^{\dagger}u_{R}\,\frac{S}{M_{*}}-y_{d}\,H\cdot
Q_{L}^{\dagger}d_{R}\,\frac{S^{\dagger}}{M_{*}}-y_{e}\,H\cdot
L_{L}^{\dagger}e_{R}\,\frac{S^{\dagger}}{M_{*}}-y_{\chi}\,\chi_{R}\chi_{R}\,\frac{S^{2}}{M_{*}^{2}}+\text{h.c.}\,,$
(14)

where $\widetilde{H}^{a}=\epsilon^{ab}H^{\dagger}_{b}$ and $a,b$ are
$\textrm{SU}(2)_{L}$ indices which are made explicit here for clarity while
they are suppressed in the equation above where their contraction via the
$\delta_{a}^{b}$ invariant tensor is denoted by $X\cdot
Y\equiv\delta_{a}^{b}\,X^{a}Y_{b}$ with $X$ and $Y$ transforming under
(formally) conjugate representations. These higher dimensional operators can
be generated at the scale $M_{*}$ via vector-like fermions with masses of
$\mathcal{O}(M_{*})$ as shown in Fig. 5. For each of the fermions of (12), we
require one pair of vector-like Weyl fermions which are neutral under
$\textrm{U}(1)^{\prime}$ but are otherwise charged under $\textrm{SU}(3)_{C}$
or $\textrm{U}(1)_{Y}$ as necessary. The vector-like fermion $X$ corresponding
to the DM candidate $\chi_{R}$ is completely neutral under the SM and the
$\textrm{U}(1)^{\prime}$ gauge group. However, it must also be odd under
$\mathbb{Z}_{2}$ in order for the DM Yukawa term to respect it.

$u_{L}^{\dagger}$$u_{R}$$\widetilde{H}$$S$$U^{\dagger}_{L}$$U^{\phantom{\dagger}}_{R}$

$d_{L}^{\dagger}$$d_{R}$$H$$S^{\dagger}$$D^{\dagger}_{L}$$D^{\phantom{\dagger}}_{R}$

$\chi_{R}$$\chi_{R}$$S$$S$$X^{\dagger}_{L}$$X^{\phantom{\dagger}}_{R}$

Figure 5: Generating the dimension-5 Yukawa interactions via vector-like-
fermions. The fermion flow reflects the fact that we work with
$(0,\tfrac{1}{2})$-representation fermions, see Dreiner:2008tw . The diagram
for the electron is omitted but it can be obtained from the down-quark one by
the replacement $d\to e$ and $D\to E$. The vector-like fermions, $U,D,E,X$ are
neutral under $\textrm{U}(1)^{\prime}$.

After spontaneous symmetry breaking of the $\rm{U}(1)^{\prime}$, the DM
candidate $\chi_{R}$ transforms only under the $\mathbb{Z}_{2}$ symmetry as
$\chi_{R}\to-\chi_{R}$ but does not carry any additional conserved charges.
Thus, we can construct a left-handed, $(\tfrac{1}{2},0)$, fermion
$\epsilon_{\alpha\beta}(\chi_{R}^{\dagger})^{\beta}$ with the same quantum
numbers as $\chi_{R}$ and, consequently, we can construct a four-component
Majorana spinor as

$\displaystyle\chi_{M}=\begin{pmatrix}\big{[}\chi_{R}^{\dagger}\big{]}_{\alpha}\\\
\big{[}\chi_{R}\big{]}^{\dot{\alpha}}\end{pmatrix}\,,$ (15)

which explicitly satisfies the Majorana “reality” condition
$\chi_{M}^{c}=\chi_{M}$, though it is manifestly obvious that this must be so
since the four-component spinor is constructed from only one Weyl fermion. The
Lagrangian of this Majorana DM is given by

$\mathcal{L}_{M}=\frac{1}{2}\bar{\chi}_{M}\,i\not{\partial}\,\chi_{M}+\frac{1}{2}\bar{\chi}_{M}\,\gamma^{\mu}\gamma_{5}\,\chi_{M}\,Z^{\prime}_{\mu}-\frac{1}{2}m\,\bar{\chi}_{M}\,\chi_{M}\,.$
(16)

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