The windows for kinetically mixed Z'-mediated dark matter and the galactic center gamma ray excess

One of the simplest hidden sectors with signatures in the visible sector is fermionic dark matter $\chi$ coupled to a $Z'$ gauge boson that has purely kinetic mixing with the standard model hypercharge. We consider the combined constraints from relic density, direct detection and collider experiments on such models in which the dark matter is either a Dirac or a Majorana fermion. We point out sensitivity to details of the UV completion for the Majorana model. For kinetic mixing parameter $\epsilon \le 0.01$, only relic density and direct detection are relevant, while for larger $\epsilon$, electroweak precision, LHC dilepton, and missing energy constraints become important. We identify regions of the parameter space of $m_\chi$, $m_{Z'}$, dark gauge coupling and $\epsilon$ that are most promising for discovery through these experimental probes. We study the compatibility of the models with the galactic center gamma ray excess, finding agreement at the 2-3$\sigma$ level for the Dirac model.


INTRODUCTION
A popular paradigm for dark matter (DM) models is that there exists a hidden sector [1,2], including the dark matter particle and possibly many others, connected to the visible sector (the standard model, SM) by some weak "portal" interactions [3,4]. Fermionic dark matter is theoretically attractive because its mass is protected by chiral symmetry and so does not introduce any new hierarchies of scale. It is natural to suppose that it has some gauge interactions in the hidden sector, of which the simplest possibility is U(1) (where the prime distinguishes it from the SM weak hypercharge). The portal is gauge kinetic mixing between the U(1) field strengthZ µν and the SM hypercharge Y µν [5]: One is then led to a simple and predictive model where there are only four essential parameters: , the U(1) gauge coupling g , and the masses m χ , m Z of the dark matter χ and the U(1) gauge boson Z . Although there may be additional particles at a similar scale, such as a dark Higgs boson to give mass to the Z , it is not necessary to assume that they play an essential role, and it is consistent to consider the model with only four parameters. These can be constrained to a great extent by assuming a thermal origin for the DM relic density, and imposing constraints from direct searches for the DM and collider searches for the Z , as well as precision electroweak constraints. The above statements are strictly true when the DM couples vectorially to the Z . Another possibility is to have axial vector couplings, and so we consider both cases Z µ J μ Z = g χγ µZ µ χ, 1 2 g χγ µ γ 5Z µ χ, where χ is assumed to be a Dirac particle in the first case, and Majorana in the second. This is motivated by the fact that a Majorana fermion could have couplings only of the second type (though a Dirac fermion could have couplings of both types). We will refer to these two models as "Dirac" and "Majorana" dark matter. In the Majorana model we are obliged to also consider dependence upon the mass of the dark Higgs that is responsible for spontaneous breaking of the U(1) , as will be explained.
This work aims to synthesize the most important constraints on kinetically mixed Z -mediated dark matter models. Some aspects of our study are similar to previous ones [6]- [12], but with the exception of ref. [8], these papers study Z models that are not just kinetically mixed but have additional interactions with the standard model. Ref. [8] focuses on electroweak precision constraints, while we incorporate in addition the constraints from relic density, direct detection and collider physics. Our analysis is distinctive in identifying the allowed parameter space in the well-motivated and economical hidden sector models where the mediation to the standard model is purely through gauge kinetic mixing.
We start in sect. 2 with a description of the models under consideration and a discussion of the extent to which they can be considered complete without reference to physics at higher scales. In sect. 3 the couplings of the Z to standard model particles and to the DM are specified, as well as the visible and invisible decay widths of the Z . Here we also briefly discuss electroweak precision constraints on the finely tuned region of parameter space where m Z ∼ = m Z . Sect. 4 presents constraints from the relic density assuming that the DM is thermally produced. In sect. 5 we derive constraints coming from direct detection, while sect. 6 deals with those coming from dilepton searches at the LHC and precision electroweak studies. Sensitivity of missing energy signals (monojets) is also discussed. We synthesize the results in sect. 7, giving a summary of the regions of parameter space that are still allowed, as well as which experimental probes are most promising for discovery. In sect. 8 we discuss the potential for these models to address the galactic center gamma ray excess that has attracted attention recently. Conclusions are drawn in sect. 9, and details of cross section calculations are given in the appendices.

MODELS
At the phenomenological level, the Dirac DM model is the simplest because the U(1) gauge symmetry does not prevent giving a mass to χ that is unrelated to spontaneous symmetry breaking. Moreover there need not be a Higgs field associated with the Z mass; one can use the Stueckelberg mechanism [13] to directly give the Z a mass. Hence it makes sense to consider the Dirac DM model as depending upon only the four parameters , g , m χ , m Z . One indication of the consistency of this procedure is the fact that the DM annihilation cross section for χχ → Z Z has unitary behavior at large center of mass energy even if there is only χ exchange in the tchannel, with no need for Higgs exchange. The complete theory can be specified by the kinetic mixing (1) and the usual terms where D µ = ∂ µ − ig Z µ is the covariant derivative. However for the Majorana DM model, it is not possible to have a bare mass term for χ consistent with the gauge symmetry; the Stueckelberg mechanism by itself would imply m χ = 0. To avoid this, we are obliged to consider spontaneous symmetry breaking, in which the dark Higgs boson h cannot be much heavier than χ or Z unless its self-coupling λ is much greater than g or the Yukawa coupling y that gives rise to m χ = y h . A consequence of this is that the cross section for χχ → Z Z violates unitarity at high energy unless the h exchange diagram is included.
An ultraviolet complete version of the Majorana model is given by where the two Majorana fermions have charge ±g γ 5 to allow for anomaly cancellation, the scalar has charge 2g , and P L,R = 1 2 (1 ∓ γ 5 ). A bare Dirac mass termχ 1 χ 2 can be forbidden by the discrete symmetry χ 1 → χ 1 , χ 2 → −χ 2 . Then after spontaneous symmetry breaking, we can consider the lighter of the two mass eigenstates χ 1,2 to be the principal dark matter particle, while the heavier one (also stable) is subdominant, as we will verify when computing the relic density. This justifies the neglect of the extra DM component in our treatment.

COUPLINGS AND DECAYS OF Z
The couplings of the Z to standard model particles, via kinetic mixing, determine the visible contributions to the width of the Z and the DM annihilation cross section, while the respective processes Z → χχ or χχ → Z Z give the invisible contributions, if they are kinematically allowed. We distinguished (using the tilde) the interaction eigenstateZ µ of the U(1) boson that appears in eqs. (1,2) from the corresponding mass eigenstate Z µ . Assuming that there is no mass mixing between Z and Z other than that induced by , the interaction Lagrangians for the physical Z and Z are given by [7] where c W = cos θ W , s W = sin θ W , c ζ = cos ζ, s ζ = sin ζ, and we have assumed 1. The mass mixing angle ζ is given by where m Z represents the SM prediction for the Z boson mass.
In the Z models considered here, the predicted value of m Z gets shifted away from the SM value by an amount δm 2 Z = (m 2 Z − m 2 Z ) tan 2 (2ζ), which is constrained by precision electroweak data, namely the deviation δρ in the ρ parameter from its SM prediction ρ = 1. This leads to the constraint where |δρ| < 10 −3 , conservatively. The maximum allowed value of tan(ζ) is then of order δρ/ . In the following we will focus on ≥ 0.01, for which ζ must therefore be small. For m Z > m Z , it is then often adequate to approximate c ζ = 1, s ζ = 0. For smaller values of this approximation can break down, but only in a finely-tuned situation where m Z is very close to m Z . We will ignore this possibility in what follows. There are however a few situations where it is important to keep track of ζ more acccurately. One is when m Z m Z . In this regime, ζ → s W and the coefficient ( s W c ζ − s ζ ) in (5) that couples Z to the Z current J µ Z is highly suppressed. We will see that this leads to a strong suppression of the spin-dependent cross section for scattering of Majorana DM on nucleons. A second such situation is the annihilation χχ → ff through the Z in the s-channel, where we keep sin ζ = 0 since the smallness of ζ can be compensated by the Z being nearly on shell in case of the accidental degeneracy m χ ∼ = m Z /2, leading to resonant enhancement of the annihilation cross section. Parametrizing the couplings of the Z and Z to SM fermions as where Q i is the electric charge and T 3,i is the weak isospin. We have ignored corrections of O( 2 ) here. If m Z m t and ζ 1, we can approximate the width of the Z decaying into SM particles as The contribution from the top quark should be corrected by the factor (1 + 7 m Z , as explained in the previous paragraph, we cannot approximate ζ ∼ = 0 because of the suppressed coupling of Z to J µ Z (due to the factor s W c ζ − s ζ ). In that regime, Z couples to SM fermions only through the electromagnetic current, and we find that Γ SM is smaller by the factor 4c 4 W /3 relative to (10).
The invisible width due to Z → χχ is given by assuming that m Z > 2m χ .

RELIC DENSITY
There are two potentially important processes for determining the DM thermal relic density: χχ → ff , where f is any SM fermion coupling to Z (the contribution from W + W − final states turns out to be negligible), and χχ → Z Z in the case where m χ > m Z . The corresponding processes χχ → ff ff or χχ → ff Z , where one or both of the Z s is off-shell, turn out to give negligible contributions to the annihilation. Annihilations into Z bosons can only be important where s ζ is so large that electroweak precision constraints are violated. We give details of the cross section calculations in appendices A-C.
To determine the relic density we have solved the full Boltzmann equation as well as using the accurate approximation described in ref. [19]. We find that a faster and accurate enough method is to compute the thermally averaged cross section σv rel at the temperature T = m χ /20 and compare it to the standard value (σv) th needed for getting the right relic density. This quantity has been accurately determined as a function of m χ in ref. [20]. Then the ratio of the χ relic density to that measured by WMAP7 (Ω CDM h 2 = 0.112±0.006) is given by where g χ = 4(2) for Dirac (Majorana) DM.
We display contours for f rel = 1 in the m χ -m Z plane for the two models (Dirac and Majorana DM) in figures 1 and 2, for a range of g and . In nearly all cases, the observational uncertainty in Ω CDM does not exceed the widths of the curves. There are generally two regions where σv rel has the desired value: one near m Z ∼ = 2m χ , where χχ → ff is resonantly enhanced, and the second (visible for large enough values of g ) where m Z ∼ = m χ so that χχ → Z Z is suppressed by lack of phase space. For Dirac DM, this second branch becomes vertical in the m χ -m Z plane at a sufficiently large value of m χ , beyond which the cross section becomes too small (because of the suppression from the intermediate χ propagator in the t channel). However for Majorana DM, the cross section falls much more slowly as a function of m χ , and so the lower branch continues to large values of m χ in this model. This is related to the different behavior at large s (the Mandelstam invariant) in the two models, that was described in section 2.
The slow fall-off of σv rel with s in the Majorana model necessitates doing the full thermal average to find σv rel , rather than simply evaluating it at s = 4m 2 χ (the threshold approximation). In fig. 3 we give an example (with g = 0.1, = 0.03) showing that the latter is a very bad approximation when m χ starts to exceed a certain (g -dependent) value. Similarly, the cross section for χχ → Z Z is somewhat sensitive to the mass of the dark Higgs boson, since its contribution to the scattering is necessary for getting physically sensible results. Whereas we fixed R φ = m φ /m χ = 1 in fig. 2, in fig. 3 we display the dependence upon R φ . There is a marked increase in the cross section starting at R φ = 2, since the dark Higgs can be produced resonantly in that case. It is worth noting that in this model, the Yukawa coupling y 1 that enters into the scattering matrix element is related to the gauge coupling by y 1 /g = 2m χ /m Z since both χ and Z get their mass from the VEV of φ.
As a point of consistency for the Majorana model, we require that the heavier of the two fermions (which was required for anomaly cancellation) makes a subdominant contribution to the overall relic density. The contributions to σv rel from the longitudinal polarizations of the Z bosons scale as m 2 χ /m 4 Z , so that the relative abundance of the heavier species is suppressed by (m χ1 /m χ2 ) 2 . We numerically verify this expectation in the high-m χ , lowm Z parts of the relic density contours that are associated with χχ → Z Z .

DIRECT DETECTION
The cross section for spin-independent (SI) scattering of Dirac DM χ to scatter on nucleons at zero velocity is given by is the reduced mass, and we have averaged over protons and neutrons to account for coherence, using the charge Z N and atomic number A N of the nucleus. The vector couplings of the Z and Z to the proton and neutron are given by (9), which is also valid for nucleons because of the conserved vector current. The corresponding couplings to χ are v χ,Z = s ζ g and v χ,Z = c ζ g . Numerically, we find that the cross section is fit to a good approximation by for xenon. However we use the more exact formula (13) to obtain the limits presented below. For Majorana DM there is a SI contribution to the scattering due to the vector current at the nucleon, which is suppressed by the relative velocity, and has different mass dependence: where m n is the nucleon mass. There is in addition a spin-dependent (SD) contribution for Majorana DM. We define an effective averaged cross section on nucleons as The axial vector couplings are not simply related to those of the constituent quarks, instead being given by where g A = 1.27 is the axial-vector coupling for neutron decay and g s = 0.19 is the strange quark contribution, while a χ,i = −v χ,i . The actual SD cross section σ N on xenon nuclei depends upon a different linear combination of a p,i and a n,i , as described in appendix D; the combination |a p,i | + |a n,i | is just a normalization factor in the definition of (17) that divides out in the physical σ N . This procedure is consistent because of the fact that a p,i /a n,i = −(g A + g s )/(g A − g s ) regardless of i, a con-straint we have imposed when computing the bound on σ SD,M . The LUX direct detection limit can be applied directly to σ SI,D ; however we allow for the possibility for χ to be a subdominant component of the total dark matter by weakening the constraint according to (where σ SI,LUX is the experimental upper limit) in regions of parameter space where f rel < 1, since the signal is expected to be reduced by this factor. 1 The corresponding constraints on m Z are shown in fig. 1 as the dashed (blue) curves. The use of (18) rather than the more common criterion σ SI,D < σ SI,LUX that assumes f rel = 1 has the virtue that our exclusion curves indicate the true po-  : LUX limits on spin-independent, spin-dependent, and velocity-dependent nucleon cross sections, given respectively by eqs. (13,15,17). For the velocity-dependent case, σSI,M is the constrained quantity.
tential for direct detectability throughout the parameter space, rather than overestimating it. For the velocity-and spin-dependent cross sections we must determine the limits onσ SI,M ≡ σ SI,M /v 2 rel and σ SD,M ourselves, by computing the corresponding cross sections on the Xe 131 nucleus and comparing to the LUX data. Details are given in appendix D. The results are shown in fig. 4.
For the Majorana DM model, we find that the limit onσ SI,M gives more stringent constraints than that on σ SD,M , despite the velocity suppression in the former. 2 2 Stronger limits on SD scattering on protons in the sun have been obtained by neutrino detection experiments [21,22]. These depend upon the efficiency of getting neutrinos from the decays of This happens because the coefficient ( s W c ζ −s ζ ) appearing in (17) is approximately zero for small m Z , making a j,Z ∼ = 0. For heavier m Z , the Z -mediated contribution to the cross section is suppressed by 1/m 4 Z . (Although ( s W c ζ − s ζ ) is also small in the SI cross section for the Dirac model, v p,Z has an unsupressed contribution from the −c W c ζ Q p term.) The corresponding limits on m Z in the Majorana DM model are given by the dashed (blue) curves in fig. 2, with dark (blue) shading indicating the excluded regions.

COLLIDER CONSTRAINTS
There are constraints on the coupling of Z to leptons from the processes pp → Z → e + e − , µ + µ − [18]. These were derived for other Z models than the one considered here, so we have reanalyzed the ATLAS data to constrain the purely kinetically mixed Z , as described in appendix E. In fig. 5(a) we show the limits on σBR for Z → e + e − and Z → µ + µ − , where BR denotes the branching ratio for Z to decay into these final states. Assuming that there are no invisible decays, the predicted values of σBR for models with a given value of are also shown there. This allows us to derive the upper bound 0 (m Z ) as a function of Z , assuming that Z decays only into SM fermions with the width Γ SM given by (10). The function 0 (m Z ) is shown in fig. 5(b).
In general, the above limit must be corrected for the invisible decays Z → χχ through the branching ratio BR SM = Γ SM /Γ tot , where Γ tot = Γ SM + Γ inv , with Γ inv given by (12). The general constraint is then given by which depends upon both m Z and m χ . The ATLAS limit extends only down to m Z = 166 GeV. At lower masses, upper bounds on exist from electroweak precision data (EWPD) constraints [8]. We combine these with (19) to cover the range down to m Z = 10 GeV. Generically, the dilepton and EWPD considerations are only relevant for 0.01, with slightly more stringent constraints applying near m Z = m Z and other narrow mass regions in the case where Γ inv is small. We adopt the "wide" Z limit of ref. [8], replotted here in fig.  6. For comparison our limit 0 is also plotted there. It should be kept in mind that even though 0 is lower than the EWPD limit in the region where they overlap, EWPD can be more stringent if BR SM is sufficiently small.
A third collider signal for dark matter models such as those considered here is missing transverse energy which final state particles from χχ annihilation. We have checked that even with the most sensitive channels, the SD limits obtained are not competitive with the LUX SI limit on our Majorana DM model.  could occur in the on-shell production of the Z if it decays invisibly into χχ. Initial state radiation from the incoming quarks could lead to monophotons or monojets. The ultimate sensitivity of LHC to Z models similar to ours has been estimated in ref. [23], where projected constraints on the couplings of the Z have been computed as a function of m Z for m χ = 100 and 1000 GeV. In particular, the effective coupling g Z = g g q is bounded, where g is the coupling of Z to χ, and g q is its coupling to quarks. For our purposes, we take g q ∼ = c W (2e/3) corresponding to the up quark coupling; then g Z ∼ = (0.175 g ) 1/2 . In fig. 7(left), we reproduce the projected limits of [23] for the LHC at 14 TeV center-of-mass energy and Larger values of m Z can be probed than those currently constrained by the dilepton and EWPD studies. The hatched region for m χ < 100 GeV is an extrapolation of the results taken from [23].

ALLOWED WINDOWS
In figs. 1 and 2 we plot the contours for the relic density along with upper limits on m Z from null direct detection searches, and the regions ruled out by dilepton and EWPD constraints. As has been noted in previous literature [11], the Dirac DM model ( fig. 1) is more highly constrained because of its typically larger cross section on nuclei. For small values of g , the only allowed regions are the ones where χχ annihilation into SM fermions is resonantly enhanced due to the accidental tuning of masses m χ ∼ = m Z /2. For g 5 × 10 −5 , the direct detection constraint falls below the relic density curve along m χ ∼ = m Z /2, leaving all such models currently viable.
In the Dirac DM model, only for large values of the U(1) coupling g ∼ 1 does the competing channel χχ → Z Z become strong enough to provide an alternative for satisfying both relic density and direct detection constraints. This window is largest for 0.01, below which direct detection and collider constraints are weakest. But it survives even for nearly as large as 0.1, at m χ ∼ = 1.8 TeV, m Z ∼ = 1.4 TeV. For > 0.03, the collider/EWPD constraints become stronger than those from direct detection.
The Majorana DM model is less constrained because its cross section on nucleons is either spin-dependent or velocity suppressed. We found that the SI (but vdependent) interaction gives the stronger limit. Even so, it hardly excludes any of the regions favored by the relic density. Only for g ∼ 1 and m χ ∼ m Z ∼ 10 GeV is there significant overlap of the direct detection and relic density curves. Like in the Dirac model, the relic density can be achieved either through χχ → ff (for m χ ∼ = m Z /2) or χχ → Z Z . But in contrast, the relic density contour due to the latter process extends to higher m χ , due to the relatively larger contributions to the annihilation cross section from the emission of longitudinal gauge bosons. For 0.01 the collider/EWPD bounds are more important that those for direct detection, giving the most promising means of discovery. For g ∼ 1, allowed regions with m χ ∼ m Z ∼ several TeV exist even for as large as ∼ 0.1.

GALACTIC CENTER GAMMA RAY EXCESS
Evidence from the Fermi Telescope has been found for excess 1-10 GeV gamma rays emanating from the galactic center (GC). Although millisecond pulsars may be a plausible source [24,25], the possibility of dark matter annihilation has been vigorously pursued; for a recent discussion with references see [26]. Analyses of the data indicate that 40 GeV dark matter annihilating into bb provide a good fit to the signal [24].
Ref. [27] studied vector and axial-vector mediators in the s-channel, assuming only couplings to dark matter and to b quarks, showing that they are nearly ruled out as an explanation for the GC excess, by constraints from LUX direct detection and from CMS sbottom searches. On the other hand, refs. [28,29] pointed out that these constraints are alleviated if m Z < m χ so that χχ → Z Z → 4f (where f is a SM fermion) can proceed through on-shell Z bosons in the GC. The coupling of Z to ff can be much smaller in this case, since the onshell Z need only decay eventually into SM particles. Primarily g , m χ and m Z determine the strength of the GC signal, while the branching ratios of the decays into different final states affect the shape of the gamma ray spectrum.
We undertake a similar study here for the case where Z couples to the SM through gauge kinetic mixing (this possibility was also considered in [28]). Since the models that give the best fit to the GC excess spectrum have light Z , the couplings of Z to fermions are to a good approximation given by the − c W c ζ eQ i term in (9), i.e., the Z couples to their charges. We have generated the final photon spectrum using the Pythia-based results provided by ref. [30], which mainly considers the processes χχ → ff where each fermion has energy m χ . To approximate the effect of 4-body final states, we convolve the photon spectra from a monoenergetic source with a box distribution, where δm ≡ m 2 χ − m 2 Z and dNγ dEγ (m) is the spectrum from a 2-body annihilation of particles with mass m. (The factor of 2 accounts for the decays of both Z s.) To relate the spectrum to the observed gamma-ray flux from the GC, we use the fact that in the galaxy the DM velocity is small, so that the zero temperature cross section (B4) is applicable. The flux is given by where the J factor is the integral along the line of sight and σv 0 is the annihilation cross section at the kinematic threshold. We take for the local density at the sun ρ = 0.3 GeV/cm 3 and r = 8.5 kpc. We compare our theoretical prediction for the flux to the observed values reported in ref. [26], varying m χ and m Z which affect the shape of the spectrum, and adjusting g at each (m χ , m Z ) to obtain the best fit. We take to be negligibly small so that annihilations to Z Z dominate over ff final states and direct detection and collider constraints are unimportant. The data and our model's fit to the spectral shape are shown in fig. 8. The resulting best-fit regions in the m χ -m Z plane are shown in fig. 9, along with contours of the corresponding values of g (left) and of the relic density fraction for the Dirac DM model f relic (right). The best-fit point has m χ ∼ = m Z ∼ = 28 GeV, but the 3σ confidence region extends to low values of m Z ∼ 10 GeV and m χ ∼ 26 GeV. The relic density is too low by a factor of ∼ 6 at the best-fit point, but consistent with the observed value at the lower values of m Z ∼ 15 GeV. (For the Majorana DM model, not shown here, the tension between  the GC signal and the relic density is greater, due to the larger thermal annihilation cross section at the time of freeze-out, even though at threshold the two models have equal annihilation cross sections.) The discrepancy between f rel and the parameters preferred for the GC excess may be ameliorated by taking into account astrophysical uncertainties [29], especially the possibility of a more concentrated DM halo profile, or accounting for part of the signal through millisecond pulsar emissions. Our allowed regions are similar to those found in ref. [31], though somewhat lower in the masses of χ and Z .

CONCLUSIONS
We have systematically studied the constraints from relic density, direct detection and collider experiments (dilepton production and electroweak precision data) on a simple dark sector, consisting of Dirac or Majorana dark matter, connected to the standard model by a kinetically mixed massive Z gauge boson. The Dirac model can be considered to be UV (ultraviolet) complete, while the Majorana model is somewhat sensitive to details of the complete theory, such as the mass of the Higgs boson that spontaneously breaks the U(1) gauge symmetry, or the presence of an additional, heavier, subdominant DM component.
We have shown that the Dirac DM model requires the coincidence m χ ∼ = m Z /2 to get the right relic density if χ, and small values of g to evade direct detection, if m χ 300 GeV. For heavier DM, there exist allowed models with larger values of g where χχ → Z Z determines the relic density, and χ could be discovered in future searches for scattering on nuclei or at colliders. About the Majorana model, although it has some dependence upon extra parameters, the qualitative picture is clear: it much more easily escapes direct detection constraints except for strong couplings g ∼ 1 and small masses m χ ∼ m Z ∼ 10 GeV. At large masses, only collider probes are sensitive, and then only for relatively large values of the kinetic mixing, 0.01. In this regime, models with resonantly enhanced annihilation (m χ ∼ = m Z /2) are more likely to be compatible with the constraints, unless g 0.3, in which case the more generic χχ → Z Z branch of the relic-density-allowed regions (with lower values of m Z ) can also be viable. This region may be discoverable not only through searches for dileptons but also monojets in the upcoming run of LHC.
These cross sections at threshold are the same for Dirac and Majorana DM in the models under consideration: However we find that the thermally averaged values can differ significantly from the threshold values. This is especially true for the Majorana model, as described in section 4 (see fig. 3.) a wide range of DM masses, and at large m χ the momentum dependence makes an essential correction to the cross section, the two nuclear form factors Φ i (q) for Xe 131 and Xe 129 are taken into account here [15]. Following ref.
[16], we take the form factor for each element to be The result is plotted in fig. 10 as a function of u ≡ q 2 b 2 /2, where q = √ 2m N E R is the momentum transfer and b = 2.2853 fm (2.2905 fm) for Xe 129 (Xe 131 ).
For a given DM model, the predicted number of events is computed by integrating the recoil rate over the recoil energy from 3 keV nr to 38 keV nr . The upper limit of the DM cross section is derived by comparing the predicted number of events with the expected signal events, which ranges from 2.4 to 5.3 for different dark matter masses.
The rate for spin-independent (SI) scattering is also given by an expression of the form (D1). The only difference relative to standard SI scattering in the case of the Majorana model is the extra dependence on v 2 rel of (15), appearing in the phase space integral in (D1).

Appendix E: Dilepton production cross section
The predicted cross section for dilepton production at the LHC is given by where M is the invariant mass of the lepton pair, √ s = 8 TeV is the LHC hadronic centre of mass energy, for the relevant ATLAS constraints we consider, f q,q (x) are the parton distribution functions, and τ = M 2 /s. The sum over quarks is implicit. We include a K-factor to account for next-to-leading-order corrections, which we take as K = 1.5 for the purposes of our analysis. The parton level cross section for the process, which proceeds via s-channel exchange of γ, Z, or Z , is given byσ (v 2 q,Z + a 2 q,Z )(v 2 l,Z + a 2 l,Z ) (v q,Z v q,Z + a q,Z a q,Z )(v l,Z v l,Z + a l,Z a l,Z ) − 4e 2 Q q v q,Z v l,Z + 1 2π The couplings of the Z and Z to SM fermions, v f,X and a f,X , are as given in eq. 9. The Z width, Γ Z is taken to be the decay width to SM particles, as given by eq. 10. We determine the branching ratio to leptons, using the partial width where l = e or µ.
We determine the quantity σBR(Z → l + l − ) as a function of the Z mass, for several choices of the kinetic mixing parameter, . Our result is shown in fig. 5. From this constraint, we further determine an upper limit on as a function of m Z , equating our predicted cross section to the expected ATLAS limit, in the combined channel e + e − + µ + µ − . The result is shown in fig. 6.