Leptophilic dark matter with $Z^{\prime }$ interactions

We consider a scenario where dark matter (DM) interacts exclusively with Standard Model leptons at tree level. Because of the absence of tree-level couplings to quarks, the constraints on leptophilic dark matter arising from direct detection and hadron collider experiments are weaker than those for a generic weakly interacting massive particle. We study a simple model in which interactions of DM with Standard Model leptons are mediated by a leptophilic $Z^{\prime }$ boson, and determine constraints on this scenario arising from relic density, direct detection, and other experiments. We then determine current LHC limits and project the future discovery reach. We show that, despite the absence of direct interactions with quarks, this scenario can be strongly constrained.

Figure 1

Loop-suppressed direct detection signal for leptophilic dark matter.

Figure 2

DM annihilation processes, which determine the relic density at freeze-out.

Figure 3

Dominant direct detection process with a leptophilic $Z^{\prime }$ as SM-DM mediator.

Figure 4

Lower bounds on the cutoff, $\mathrm{\Lambda }$, for fermionic DM coupling to leptons via a four-fermion effective contact operator, using exclusion limits from LUX. Limits are shown for electrons (solid, blue), muons (dashed, purple) and taus (dotted, green) for both ${\mathrm{\Gamma }}_{\chi }\otimes {\mathrm{\Gamma }}_{\ell }=A\otimes V$ and $V\otimes V$. We see that the cutoff required for $V\otimes V$ is much higher than that for $A\otimes V$.

Figure 5

Constraints on the $g_{\ell }=g_{\chi }$ and $m_{Z^{\prime }}$ parameter space when ${\mathrm{\Gamma }}_{\chi }\otimes {\mathrm{\Gamma }}_{\ell }=V\otimes V$. The relic density curves are shown for $m_{\chi }=10$, 50, and 100 GeV (black, solid), while the shaded region is excluded by the LUX direct detection results (for $m_{\chi }=20–100\mathrm{GeV}$). Also shown is the approximate future LHC reach at $\sqrt{s}=14\mathrm{TeV}$ (see Sec. 6) for coupling to $e$ or $\mu$. It is clear that, for a ${\mathrm{\Gamma }}_{\chi }\otimes {\mathrm{\Gamma }}_{\ell }=V\otimes V$ operator, the current direct detection results already exclude most of the parameter space for which the correct relic density is obtained.

Figure 6

Kinetic mixing of the $Z^{\prime }$ and SM photon.

Figure 7

Production of the $Z^{\prime }$ at a hadron collider. The $Z^{\prime }$ is radiated from a lepton in the Drell–Yan process, and subsequently decays to either leptons or DM.

Figure 8

$Z^{\prime }$ production cross section at the 14 TeV LHC, via the process $pp\to {\ell }^{+}{\ell }^{-}{Z}^{\prime }$. We have set $g_{\ell }=0.1$.

Figure 9

Invariant mass for four muons (left) and transverse momentum $p_T$ for leading in $p_T$ muon (right) for $pp\to 4\mu$ in the SM and $Z^{\prime }$ model (with $m_{Z^{\prime }}=60\mathrm{GeV}$, $m_{\chi }=10\mathrm{GeV}$, $g_{\mu }=g_{\chi }=0.1$), at $\sqrt{s}=14\mathrm{TeV}$ and $\mathcal{L}=300{\mathrm{fb}}^{-1}$. The peak in the 4-muon invariant mass spectrum is a reconstruction of the $Z$ mass.

Figure 10

Invariant mass for first and second leading muons in $p_T$ (left) and third and fourth leading muons in $p_T$ (right) for $pp\to 4\mu$ in the SM and $Z^{\prime }$ model (with $m_{Z^{\prime }}=60\mathrm{GeV}$, $m_{\chi }=10\mathrm{GeV}$, $g_{\mu }=g_{\chi }=0.1$), at $\sqrt{s}=14\mathrm{TeV}$ and $\mathcal{L}=300{\mathrm{fb}}^{-1}$. The mass of the $Z^{\prime }$ can be seen clearly as the resonance at $m_{Z^{\prime }}=60\mathrm{GeV}$ in the invariant mass spectrum $M_{12}$.

Figure 11

Invariant mass for first and second leading muons in $p_T$ (left) and transverse momentum $p_T$ for $p_T$ leading muon (right) both before cuts, for $pp\to 4\mu$ in the SM and $Z^{\prime }$ model (with $m_{Z^{\prime }}=150\mathrm{GeV}$, $m_{\chi }=10\mathrm{GeV}$ and $g_{\mu }=g_{\chi }=0.19$), at $\sqrt{s}=14\mathrm{TeV}$ and $\mathcal{L}=3000{\mathrm{fb}}^{-1}$.

Figure 12

Invariant mass of third and fourth leading in $p_T$ muons before cuts (left) and after cuts (right), for $pp\to 4\mu$ in the SM and $Z^{\prime }$ model (with $m_{Z^{\prime }}=150\mathrm{GeV}$, $m_{\chi }=10\mathrm{GeV}$ and $g_{\mu }=g_{\chi }=0.19$), at $\sqrt{s}=14\mathrm{TeV}$ and $\mathcal{L}=3000{\mathrm{fb}}^{-1}$.

Figure 13

Parameter space for $g_{\mu }$ vs $m_{Z^{\prime }}$ with $g_{\chi }=g_{\mu }$. Shown are excluded regions from $(g-2{)}_{\mu }$, as well as two excluded regions from LUX direct detection corresponding to two different DM mass ranges: ${\mathrm{LUX}}^a$ is for approximately $m_{\chi }=10$, 1000 GeV, and ${\mathrm{LUX}}^b$ is for the range $m_{\chi }=20–100\mathrm{GeV}$. Exclusions from ${\mathrm{LUX}}^a$ and $(g-2{)}_{\mu }$ are overlapping in this plot. Relic density curves are shown in black for $m_{\chi }=10$, 50, 100 GeV. Dashed lines are ATLAS exclusions and reaches: the top dashed curve is ruled out by ATLAS data at $\sqrt{s}=8\mathrm{TeV}$ and $\mathcal{L}=20.7{\mathrm{fb}}^{-1}$, the middle is the ATLAS discovery reach at $\sqrt{s}=14\mathrm{TeV}$ and $\mathcal{L}=300{\mathrm{fb}}^{-1}$, and the bottom dashed curve is the ATLAS discovery reach at $\sqrt{s}=14\mathrm{TeV}$ and $\mathcal{L}=3000{\mathrm{fb}}^{-1}$. The dotted line shows the ATLAS reach at $3000{\mathrm{fb}}^{-1}$ for a high-mass $Z^{\prime }$. The LHC limits all assume $m_{\chi }=10\mathrm{GeV}$.

Figure 14

Parameter space for $g_{\mu }$ vs $m_{Z^{\prime }}$, where $g_{\chi }=4g_{\mu }$ (left) and $g_{\chi }=8g_{\mu }$ (right). Shown are excluded regions from $(g-2{)}_{\mu }$, and two excluded regions from LUX direct detection are shown corresponding to two different DM mass ranges: ${\mathrm{LUX}}^a$ is for approximately $m_{\chi }=10$, 1000 GeV, and ${\mathrm{LUX}}^b$ is for the range $m_{\chi }=20–100\mathrm{GeV}$. Relic density curves are shown in black for $m_{\chi }=10$, 50, 100 GeV. Dashed lines are ATLAS exclusions and reaches: the top is ruled out by ATLAS data at $\sqrt{s}=8\mathrm{TeV}$ and $\mathcal{L}=20.7{\mathrm{fb}}^{-1}$, the middle is the ATLAS discovery reach at $\sqrt{s}=14\mathrm{TeV}$ and $\mathcal{L}=300{\mathrm{fb}}^{-1}$, and the bottom is the ATLAS discovery reach at $\sqrt{s}=14\mathrm{TeV}$ and $\mathcal{L}=3000{\mathrm{fb}}^{-1}$. These exclusion/discovery curves are the same as for the $g_{\mu }=g_{\chi }$ case when $m_{Z^{\prime }}<2m_{\chi }$. The LHC limits all assume $m_{\chi }=10\mathrm{GeV}$.

Figure 15

Parameter space for $g_e$ vs $m_{Z^{\prime }}$ with $g_{\chi }=g_e$ (left) and $g_{\tau }$ vs $m_{Z^{\prime }}$, with $g_{\chi }=g_{\tau }$ (right). Two excluded regions from LUX direct detection are shown corresponding to two different DM mass ranges: ${\mathrm{LUX}}^a$ is for approximately $m_{\chi }=10$, 1000 GeV, and ${\mathrm{LUX}}^b$ is for the range $m_{\chi }=20–100\mathrm{GeV}$. Relic density curves are shown in black for $m_{\chi }=10$, 50, 100 GeV. Exclusions from LEP-II (electrons only) shown on the left. Dashed lines on the electron plot (left) are ATLAS exclusions and reaches: the top is ruled out by ATLAS data at $\sqrt{s}=8\mathrm{TeV}$ and $\mathcal{L}=20.7{\mathrm{fb}}^{-1}$, the middle is the ATLAS discovery reach at $\sqrt{s}=14\mathrm{TeV}$ and $\mathcal{L}=300{\mathrm{fb}}^{-1}$, and the bottom is the ATLAS discovery reach at $\sqrt{s}=14\mathrm{TeV}$ and $\mathcal{L}=3000{\mathrm{fb}}^{-1}$. The LHC limits all assume $m_{\chi }=10\mathrm{GeV}$. For the tau plot (right), there are no collider limits due to low efficiencies for reconstructing a four tau final state. Instead, dashed lines show contours of $Z^{\prime }$ production cross sections of 1, 10 and 100  fb.