Characterizing dark matter at the LHC in Drell-Yan events

Spectral features in LHC dileptonic events may signal radiative corrections coming from new degrees of freedom, notably dark matter and mediators. Using simplified models, we show how these features can reveal the fundamental properties of the dark sector, such as self-conjugation, spin and mass of dark matter, and the quantum numbers of the mediator. Distributions of both the invariant mass $m_{\ell \ell}$ and the Collins-Soper scattering angle $cos\theta_{CS}$ are studied to pinpoint these properties. We derive constraints on the models from LHC measurements of $m_{\ell \ell}$ and $cos\theta_{CS}$, which are competitive with direct detection and jets + Missing Energy searches. We find that in certain scenarios the $cos\theta_{CS}$ spectrum provides the strongest bounds, underlying the importance of scattering angle measurements for non-resonant new physics.


I. INTRODUCTION
Astrophysical evidence for dark matter (dm) abounds, but its fundamental properties remain elusive. Key puzzles that remain unsolved are: • Is dm its own antiparticle?
• Does it carry spin?
• What is its mass?
• How does it couple to the Standard Model, if at all?
These properties result in qualitatively diverse signals in direct detection searches (such as whether scattering is spin-independent, spin-dependent and/or momentumdependent) and indirect detection (such as whether annihilation is s-wave or p-wave). Thus, tests may devised by which these properties may be marked out [1][2][3][4][5][6][7]. In this paper, we ask if the same can be done at a collider, compelled by the fact that Run 2 of the Large Hadron Collider (lhc) is well underway. We find a surprising lack in the literature of lhc-related work addressing the questions of self-conjugation, spin and coupling structure, perhaps because the primary focus of most collider searches is to extract the mass of dm and possibly that of a mediator that couples the dm particle to the sm. Two exceptions are Ref. [8], where spin-1/2 and spin-1 dm were distinguished using distributions of / E T , jet rapidity, and dm invariant mass, and Ref. [9], where dm properties were distinguished by decomposing the missing energy spectrum into basis functions. These studies make use of / E T , the most striking feature of dm directly produced on-shell at the lhc.
In this work, we will focus on collider signals that can potentially address these questions, but take an approach that is not / E T -based. Instead, we ask if event distributions of fully visible final states can hold the key. A dark sector can leave its imprint in visible spectra if it induces loop processes interfering with Standard Model (sm) amplitudes; in particular, threshold effects may generate distinct signal features. As shown in Ref. [10], such non-resonant signals are best discernible in + − production at the lhc: the backgrounds are simple and intelligible, the rates are high, and the events are precisely reconstructed 1 . Indeed, the channel is so clean that in some regions it turns out to be more sensitive to the dark sector parameters than conventional jets + / E T and direct detection searches. Note that the collider signals are agnostic to the dm abundance, and can be relevant for models that populate a fraction < 1 of dm via thermal freezeout.
In the current paper we extend the program of Ref. [10] to study how well dileptonic information can shed light on the quantum properties of the dark sector. As in [10] we will construct "simplified models" where dm couples to both quarks and leptons, for which we introduce mediators charged appropriately. The simplified frameworks are renormalizable effective theories characterized by a minimal set of inputs, usually no more than the sm-dm coupling, the masses of dm and the mediator, and specifications of dm spin and the mediator's quantum numbers [20][21][22][23][24][25][26][27][28][29][30][31]. While Ref. [10] focused on Dirac dm that only coupled to right handed sm fermions, we will survey and compare several scenarios: dm that is self-conjugate and not, dm with spin 0 and spin 1/2, dm that couples to right-handed fermions and left-handed. We make full use of the information available in dileptonic events at the lhc, meaning we study spectra of the invariant mass (m ) and scattering angle. To our knowledge, this is the first paper availing lhc measurements of dilepton angular spectra to probe dm and its mediators. In fact, we find that angular spectra may provide the strongest constraints. This statement is not entirely surprising; nonresonant new physics must produce comparable effects on both the invariant mass and angular distributions, and so it is reasonable to expect the latter to sometimes have more sensitivity [19].
Our paper is laid out as follows. In Sec. II, we introduce the dm models we will use for our study and state our simplifying assumptions. In Sec. III, we show the various signals produced by our set-up and discuss how their features may help distinguish between our models. Next, in Sec. IV, we discuss all the relevant constraints on our models, comparing the dileptonic probes with jets + / E T , relic density and direct detection constraints. We also show the future prospects of our set-up at the lhc at high luminosity. In Sec. V we summarize our findings and conclude.

II. THE MODELS
Our study focuses on simplified models in which dm χ has renormalizable Yukawa interactions with sm fermions f through a partner fieldF , with the interaction schematically given by L ⊃ χF f . These are sometimes called "tchannel" simplified models in reference to the t-channel exchange ofF in dm annihilation. It is usually assumed that a Z 2 symmetry under which all non-sm fields are charged odd (and sm fields charged even) is responsible for dm stability.
We consider models comprising two sm singlets χ A,B , motivated by the possibility that their mass parameters may be tuned to interpolate between a limit of selfconjugacy, i.e. dm is Majorana or real scalar, and a limit where dm is Dirac or complex scalar. We also introduce a colored fieldQ to mediate the singlets' interactions with quarks, and an uncolored field oneL to mediate their interactions with leptons. If χ A,B are fermions, the me-diatorsQ andL are complex scalars, while if χ A,B are real scalars the mediators are fermions. We consider the following interaction Lagrangian involving these fields: where we have suppressed indices denoting fermion chirality and flavor. For spin-1/2 dm, the most general dm mass Lagrangian is given by A similar-looking (squared) mass matrix may be written down for spin-0 dm in terms of the field φ χ ≡ ( The fields χ A,B mix to give mass eigenstates χ 1,2 , with the lighter species χ 1 serving as dm. In our analysis we will refer to this dm field as simply χ. The mediator masses are free parameters that need not originate from symmetry breaking. For instance, they may arise from the scalar potential if the mediator is spin-0, or could be vector-like if dm is spin-1/2.
We now make the following assumptions that simplify our analysis: 1. We assume a common mass m φ for the colored and uncolored mediators, and equal dm couplings to quarks and leptons, λ ≡ λQ = λL.
2. We assume that dm couples to only a single chirality of sm fermions. This restricts the number of mediator species, since otherwise one would need to introduce mediators that are both singlet and doublet under SU (2) W . We will consider couplings to both left-handed and right-handed leptons, but only couplings to right-handed quarks. We do not consider couplings to left-handed quarks because, due to SU (2) W invariance, they will lead to new physics (np) signals at once from both up-type and down-type quarks in the initial state. These contributions affect proton-level cross sections in nontrivial ways due to differences in parton densities between up and down quarks, which is a complication we wish to avoid in our analysis. For further simplicity, we only consider couplings to electrons and muons, and to either the right-handed up quark or the right-handed down quark. This can be arranged by a special flavor structure, which we spell out next.
3. In order to avoid flavor changing neutral currents, we assume the existence of three generations of mediators with their couplings aligned with the sm Yukawa couplings such that, in the mass basis, each mediator generation couples only to a single generation of sm fermions. In order for dm to couple solely to the up/down quark, or to the electron/muon, we assume that mediators of the other generations are heavy.
4. As manifest in Eq. (1), we assume that only χ B interacts with the sm fermions. This assumption captures all the qualitative features of our results; allowing both χ A and χ B to interact tends to only rescale the couplings required to produce similar signal rates.
5. Setting δm = 0 and varying δm, we can interpolate between Majorana and Dirac (or real and complex scalar) scenarios. Specifically, the Majorana (or real scalar) limit is achieved by tuning δm, with δm → ∞ [10,32], while δm → 0 renders spin-1/2 dm Dirac and spin-0 dm a complex scalar. Pure Dirac/complex scalar dm notoriously has a large spin-independent cross section scattering off nuclei, and is excluded by direct detection experiments for the range of dm masses and couplings of interest. Therefore, in our study we will never truly take δm → 0, setting δm = 1 MeV as the lower limit. As discussed in Ref. [10], for splittings of this size dm behaves like a Majorana fermion (if spin-1/2, real scalar if spin-0) in direct Model χ spinQ,L spinQ under GSML under GSM The simplified models studied in this paper. dm could be either spin 1/2 or 0, which fixes the spin of the colored and uncolored mediators. We assume that dm couples to only right-handed quarks, and but couple to either right-handed or left-handed leptons. This choice picks the transformations of the mediators under the sm gauge group  Table I for more details. detection experiments, since the heavier state is kinematically inaccessible given the local dm velocity ∼ 10 −3 . Majorana/real scalar dm typically has a much smaller scattering cross section than the Dirac/complex scalar case and hence is much more viable [20] (see Sec. IV). Meanwhile, O(MeV) mass splitting is well below the lhc detector resolution, hence χ 1,2 are indistinguishable at colliders and dm will appear as a Dirac or complex scalar particle in our collider study. Thus, for a fixed dm mass, varying δm ≥ 1 MeV will have no effect on how dm appears in direct detection as all scenarios will interact as Majorana/real scalars. However, as we will see, δm will dramatically change how dm appears in dilepton distributions. For the remainder of this paper, we will refer to the δm ≥ 1 MeV regime as "pseudo-Dirac" for spin-1/2 dm and "pseudo-complex-scalar" for spin-0 dm.
6. We assume that cp-violating phases in the masses and couplings vanish.
7. We neglect quartic couplings involving new scalars introduced in our set-up, as they have little impact on our dilepton signals.
To summarize, we assume that dm couples to either right-handed up or down quarks and to electrons or muons of either chirality, with dm itself having spin 0 or 1/2.
Thus we may classify our set-up into eight models, which we dub pD u RR , pD u RL , pCS u RR , pCS u RL , pD d RR , pD d RL , pCS d RR , and pCS d RL . The superscript denotes the quark to which dm couples, and the first (second) subscript the chirality of the quark (lepton), while "pD" and "pCS" denote whether dm is pseudo-Dirac or a pseudo-complex scalar. The field content of these models is summarized in Table I.

III. DISCRIMINATING SIGNALS
In this section we illustrate the various effects of radiative corrections from the dark sector on pp → + − spectra, and how these may help distinguish the properties of χ. We will go about this task by contrasting the signals produced by mutually exclusive cases of a single property, keeping everything else the same, e.g. we will compare signals of pD u RR and pCS u RR while keeping all masses and self-conjugation properties the same.
Assuming massless quarks and leptons, and denoting by θ the centre-of-momentum scattering angle between the incoming quark and outgoing lepton, the parton level leading order (lo) Drell-Yan double differential cross section is given by with where N c = 3 is the number of qcd colors, M SM = M γ +M Z is the sm amplitude for the tree-level Feynman diagram in Fig. 1. As our np effects enter at loop level, care must be taken to ensure that all effects at a given coupling order are consistently included. Additionally, purely sm loop (mainly qcd) effects must be accounted for. These issues give rise to the following considerations: • One-loop sm effects enter at the amplitude level at O(g 2 g 2 s ), where g and g s are the qed and qcd couplings, while np effects enter at O(g 2 λ 2 ) for vertex corrections and O(λ 4 ) for the box diagrams. The net result of the purely sm loop effects is to replace dσ SM in Eq. (4) by the sm cross section at next to leading order (in qcd), dσ SM,NLO .
• Interference between sm and np loops (dσ int ) results in contributions to dσ tot of O(g 4 λ 2 ) and O(g 2 λ 4 ), where the former involve vertex corrections and the latter involve box diagrams. 2 Comparing these terms, we find the box contributions significantly larger when λ ∼ 1, which is also the regime of couplings where the np effects have enough statistical significance for lhc bounds to apply. This happens not only due to the difference in power-counting the couplings, but also because the box diagrams generate more pronounced threshold effects. Moreover there is a partial cancelation between triangle diagrams with a photon and with a Z, as they have opposite signs.
Note also that the np effects do not interfere with the entire sm amplitude. As our models involve couplings to a specific set of fermion chiralities, interference proceeds only with the part of the sm amplitude involving the same set of fermion chiralities. For instance, the np pieces in pD u RL only interfere with q RqR → Z/γ * → L¯ L .
• The dσ χ term involves the square of box and vertex corrections. These are, in principle, the same order in perturbation theory as the interference between the tree-level and np two-loop amplitudes. As we have only calculated np effects at one loop, most terms in dσ χ cannot be consistently included in the calculation 3 . An important exception that can be included consistently is the square of the np box diagrams, which is the only O(λ 8 ) contribution to the cross section at any order.
• As the box diagrams dominate the interference term, in the following discussion we will drop the vertex correction entirely and use "M χ " as a loose notation to describe the box amplitude. The resulting cross section expressions dσ int and dσ χ are provided in Appendix B.
As our focus in this section is on the qualitative differences between various dm models, rather than between the sm and dm, we will work with dσ SM,LO for now. We will return to nlo sm effects and the considerations here itemized in Sec. IV, when we use the dilepton distributions to derive limits.
The most unique feature of dσ tot occurs at √ŝ 2m χ , when χ goes on-shell in the box diagrams in Fig. 1, and M χ develops an imaginary part Im(M χ ) determined by the optical theorem. According to the optical theorem, Im(M χ ) is proportional to the product of the amplitudes of the tree-level diagrams (with χ's and fermions as external legs) obtained from "cutting" the box diagram vertically. This imaginary part feeds into the real part Re(M χ ) through dispersion relations, causing the amplitude to rapidly rise near the threshold. At √ŝ 2m χ , Re(M χ ) falls away while Im(M χ ) takes over as the dominant contributor to |M χ | 2 . The net effect of this takeover at √ŝ 2m χ is no more than the addition of a new channel of dilepton production, hence dσ tot will be separated from dσ SM by some offset. All these effects are reviewed in detail in Ref. [10], where the shape of the new physics spectrum was identified as a "monocline". (See also Ref. [34], which comprehensively reviews dispersion relations.) In the following, we show that the above effects also carry the imprint of dm's microscopic properties, leading to diverse features in dilepton spectra.
For our m spectra, we integrate the cross sections in Eq. 3 over cos θ, and for our angular spectra we integrate them over 400 GeV ≤ m ≤ 4500 GeV, the range used by the 8 TeV atlas analysis [35]. The angular spectra are computed in the Collins-Soper reference frame [36], in which the directional ambiguity of the initial state quark/antiquark in a pp collider is resolved by boosting to the dilepton center-of-momentum frame and then assuming that the quark originated in the boost direction. This assumption leads to an m -dependent probability of initial-quark misidentification, in principle determinable using information of the parton densities (see Appendix A of [19]). The scattering angle in this frame is given by where Q is the net momentum of the dilepton system with Q z (Q T ) the longitudinal (transverse) piece, and the momentum of the lepton (anti-lepton). Neglecting Q T at high longitudinal momenta, the above may be re-written as where ∆η = η 1 − η 2 is the difference in the lepton and anti-lepton pseudo-rapidities. It is often useful to characterize the angular spectrum as a forward-backward asymmetry, or a center-edge asymmetry, which marks out how much scattering occurs in central regions.
We now apply the above discussions to our various models. All spectra are shown by convolving partonlevel cross sections with MSTW2008NLO parton distribution functions (PDFs) [33] at √ s = 13 TeV. For our illustrative plots here, we approximate lhc dilepton production with the Drell-Yan process qq → + − . The treatment of secondary processes that also contribute to dilepton production, such as diboson, tt, dijet and W +jets, will become important when we set constraints in Sec. IV. We will also show only tree-level sm cross sections, treating qcd corrections more carefully in Sec. IV.
As our current analysis is qualitative, we only show here the behavior of models in which our dm couples to up quarks, but the broad conclusions we draw hold also for dm coupling to down quarks. See Appendix A for the signals arising from the latter scenario. We pick an illustrative benchmark point with the coupling λ fixed to 2.0, and masses m χ = 500 GeV and m φ = 550 GeV. As elaborated in Ref. [10], varying λ has the effect of raising or lowering dσ int and dσ χ . Depending on the sign of dσ int , this could enhance or diminish the dm signal. Moreover, increasing (decreasing) m φ enhances (diminishes) the bump feature near m 2m χ . Thus these variations affect the signal significance at the lhc, a point to which we will return when finding our constraints in Sec. IV. Here we note that the spectrum chosen here, being a "compressed" one, is illustrative of a point where our dilepton probes are expected to outperform jets + / E T searches, which suffer from low signal acceptance in these regions.
Finally, in computing our dilepton distributions we impose the following kinematic cuts: We now sketch and contrast the spectral features induced by various dm species. We also elucidate why differences arise between mutually exclusive cases (e.g. spin-0 vs spin-1/2 dm), and explain how these differences can help us to sort out the properties of dm and the mediators. Such a sorting exercise can be successfully carried out at the lhc if our dm signals are uncovered with sufficient statistical significance. Conversely, if the signal-to-background ratios in the event distributions (in our case dσ tot /dσ SM ) are inadequate, the dm properties that can be disentangled could only be a few, or none. This may happen if our couplings are small or the mass scales large so as to suppress the effects of new physics amplitudes.

III.1. Self-conjugation
As explained in Sec II, we may interpolate between the Dirac (complex) and Majorana (real) limits of dm by tuning δm. These limits are readily distinguished by the monocline signature, as shown in Fig. 2, where we have plotted dσ tot in the non-self-conjugate limit δm → 0 for the models pD u RR (solid orange), pD u RL (dashed orange), pCS u RR (solid green) and pCS u RL (dashed green), as well as at the self-conjugate limit δm → ∞ for the models pD u RR (solid red) and pCS u RR (solid purple). In the self-conjugate limit, a subdued signal is produced, while non-self-conjugate dm can produce large, detectable signals. Also, in the self-conjugate limit pD u RL (pCS u RL ) gives near-identical cross sections as pD u RR (pCS u RL ). One may compare all these signals with the blue curve, which corresponds to dσ SM .
In the Majorana limit, where χ 2 is completely decoupled and only diagrams with χ 1 contribute to the signal, the models pD u RR and pD u RL produce suppressed signals due to destructive interference between the standard box and crossed box amplitudes. This arises from a relative minus sign due to an odd permutation of spinors. As explained in [10], this can also be understood in terms of the intricate pattern of interferences between the four standard and four crossed boxes that makes them cancel out one another. The monocline feature is inferred to appear at m χ + (m χ + δm) → ∞, so that dσ tot remains close to dσ SM across m .
The suppression of rates in the real scalar limit of pCS u RR and pCS u RL occurs for subtler reasons. Due to our choice of coupling to a single fermion chirality, the projection operators pick only the momentum piece in the numerator of the propagator ofL. The momentum flow in this propagator in the crossed box diagram is reversed with respect to the standard box (while the fermion flows are the same); consequently, a relative minus sign between the two amplitudes appears, giving rise to the rate suppression.
In the limit m φ , m χ ŝ, where the loops can be shrunk to contact operators, the suppressions in the selfconjugate limit are consistent with the loop functions given in the effective theory treatment of Ref. [37]. Since the suppressed rates are a result of a modest addition to M SM from the dark sector in the self-conjugate limit, no sizable signals appear in the angular spectra either. Finally, we re-emphasize that our "non-self-conjugate limit" does not correspond to Dirac or complex scalar dm, but only to the limit where δm is small enough to be irresolvable at colliders while remaining large enough to evade direct detection constraints 4 .

III.2. Spin
Distinguishing the spin of dm is more challenging than the self-conjugation property, but some headway can be made. In Fig. 2, we see a pronounced "kick" in the signal rates at m 2m χ for fermionic dm (pD u RR , pD u RL ), while the rise in rates appears gentle for scalar dm (pCS u RR , pCS u RL ). This may be understood from the fact that near threshold, the box amplitude is determined by Im(M χ ), which, as mentioned above, is in turn determined by the tree-level amplitudes for ff → χχ. The pair-production of complex scalar χ is more phase-space suppressed at threshold than a Dirac χ, ultimately resulting in a subdued slope of the rise in dσ/dm ee for spin-0 dm. In any case, even this difference fades for larger mass splittings between the mediators and dm, where the kick feature is not as pronounced.
One would naively expect the angular distributions to discern the spin of dm, on the strength of their ability to clearly distinguish the spin of mediators in the s-channel [38] and t-channel [19]. However, this does not turn out to be entirely true. While angular spectra are capable of picking up the spin of new particles interfering with the sm via tree-level amplitudes, the angular spectrum resulting from interference with a loop amplitude is nontrivial. Moreover, as the loop consists of particles with multiple spins, one expects information on the spins to be washed away in the spectrum. To illustrate this, in Fig. 3 we have shown the angular spectra of our models following the color code of Fig 2 and using the same benchmark points. No visible difference in the spectral shape exists between pD u RR and pCS u RR ; somewhat fewer events populate the cos θ CS < 0 region for pD u RL than for pCS u RL , but this does not amount to a qualitative difference. We notice a smaller net deviation from the sm background for spin-1/2 dm, which can be understood from their m spectra in Fig. 2. Due to negative interference between the tree and box amplitudes, we see a deficit in cross sections with respect to the sm for m < 2m χ , while an excess appears at m > 2m χ from the squared box amplitude and threshold effects overwhelming the interference terms. These deviations are however washed away when integrating over m , as done for obtaining the cos θ CS spectrum. No such washing away occurs for spin-0 dm as the tree-box interference is always constructive, giving only an excess of events in the m spectrum. No such washing away would occur for dm coupling to down quarks either, as the tree-box interference is constructive here as well. Moreover, the magnitude of the net deviation from background in all cases is sensitive to the m window over which cross sections are integrated. For these reasons the scattering angle is not a reliable tool to determine the spin of dm.

III.3. Mass
From the previous sub-section, it is apparent that the mass of dm may be readily cornered if dm is a fermion and if its mass is not much separated from the media-tor's. In that case, the pronounced kick feature in the m signal appears at an invariant mass of 2m χ . As this feature is a result of amplitude-level deviations, it must also be reflected in some way in angular observables plotted as a function of m . For instance, one would see it in the A FB , defined in Eq. (6), plotted at the parton level (for illustration) in the left panel of Fig. 4 using the same color code as above. The behavior of the A FB as a function of m with respect to the sm is in accord with the behavior of the m spectrum -the telltale imprint of interference effects. Consequently, an abrupt change of slope is visible in the orange curves at m 2m χ . One would also see the kick feature in A CE (defined in Eq. 7) plotted at the partonic level in the right panel of Fig. 4, where the choice cos θ 0 = 0.596 sets the sm value to zero. Once again the abrupt change of slope at m 2m χ may be seen in the orange curves.
In principle, the dm mass is resolvable for all our models if the mediator mass is of the same order, a task achievable with sufficiently high statistics, by shapefitting signals from both m and cos θ CS spectra to various hypotheses.

III.4. Chirality
The relative chirality between the quarks and leptons in the new physics amplitude, i.e. whether the model is RR or RL, shows up in dilepton spectra in quite interesting ways. (RR and LL, and separately RL and LR, yield similar spectra.) We see a difference in the pD u RL and pD u RR signals in the m spectrum in Fig. 2, though both exhibit similar shapes. This is due to the difference in projection operators in the fermion chains: in pD u RL (pD u RR ), the combination of P R and P L (P R and P R ) picks the mass (momentum) piece from the numerator of the χ 1 propagator. When we turn to the pCS u RL and pCS u RR signals, however, we find negligible difference. This is because the fermion chains are now different, always coming with the combination P R and P L (that now picks the momentum piece from the numerator of the mediator propagators).
Much more revealing differences appear in the angular spectrum; we know from Z boson physics and from contact operator analyses (such as in [35]) that the chiral nature of new states has an impact on the scattering angle. Such an impact is seen in the cases where the np amplitude interferes constructively with the tree-level one, i.e. in pCS u RL and pCS u RR (also in pD d RL and pD d RR in Appendix A). In Fig. 3, more forward (cos θ CS > 0) events are produced by pCS u RR versus pCS u RL , and pCS u RL produces visibly more backward (cos θ CS < 0) events than pCS u RR . These differences are best seen by plotting the A FB , as in the left panel of Fig. 4. The A FB neatly separates the cases of RR and RL, putting them above and below the sm value at large m . Further, the A CE in the right panel of Fig. 4 clearly signals the chirality combination by putting RR (RL) above (below) the sm value.
To summarize this section, we have shown that dilepton measurements can carry information on dm's selfconjugation, spin and mass, and the chiral structure of dm's interactions with sm fermions. We did this by picking a benchmark point with an interaction strength large enough and dm/ mediator mass scale small enough to produce clear signal features in dilepton spectra. Whether these features can be actually discerned at the lhc will depend on the viability and signal significance of each point in parameter space. Thus, in the next section we will derive constraints on our models from available lhc data and project our sensitivity at the high luminosities of the 13 TeV run. In Sec. V we will point out interesting regions that are currently viable and can be probed by the future lhc, which is indicative of regions where the above analysis would apply.

IV. CONSTRAINTS AND PROSPECTS
Having illustrated that dilepton distributions may help distinguish the properties of dm, we now show that dm could in fact reveal itself first in lhc measurements of dilepton events. In this section we will derive constraints on our models from the available lhc data on m ee and cos θ CS , and compare them to constraints from conventional dm searches such as jets + met, direct detection, and relic density measurements. In Appendix C we discuss a few other probes that set much weaker constraints. We will find that current cos θ CS measurements can outdo all other bounds; this is one of the main results of our paper.
We show the above limits in the plane of the Yukawa coupling λ versus dm mass m χ in Figs 5 and 6, which correspond respectively to spin-1/2 and spin-0 dm; the left-hand (right-hand) panels depict dm coupling to up (down) quarks. As mentioned in the previous section, these limits must depend on the hierarchy of mediator and dm masses. Thus we pick two benchmark spectra for illustrating the constraints, one where the spectrum is "compressed" with m φ = 1.1 m χ , and one where it is "uncompressed" with m φ = 2 m χ . These correspond to the top and bottom row respectively. Throughout our analysis here we fix δm = 1 MeV. As explained in Sec. II, this gives our dm safety from direct detection constraints while masquerading as a Dirac/ complex scalar particle at the lhc.
We begin our discussion with dilepton constraints. The orange and blue curves in Figs 5 and 6 show the 95% c.l. limits from the lhc Run 1 (8 TeV, 20 fb −1 ) measurements of dilepton spectra, corresponding to the RR and RL models respectively. The solid (dashed) curves correspond to m ee (cos θ CS ) measurements by atlas [35,39] in e + e − production. Due to the similarity of results, we expect similar limits from cms data [40,41] and from dimuon production. We perform a ∆χ 2 fit as done in [10], but considerably improve on the treatment to obtain realistic bounds.
Broadly speaking, the recasting of dilepton measurements into bounds is performed by comparing between three sets of events across m or cos θ CS bins (labelled by i): the data N di , the background N bi , and the signal N si .We take N di from atlas [35,39]. It is useful to divide the background into its dominant and subdominant components, N bi = N dom bi + N sub bi . The former comprises of the Drell-Yan s-channel process in Fig. 1, while the latter (which we also take from atlas) comprises of the reducible backgrounds of diboson, top, dijet, and W + jets production.
To obtain an N dom bi that is as accurate as possible, we impose the cuts described in Sec III, and obtain Drell-Yan events at nlo-qcd (i.e. at O(α s )) using MCFM 8.0 [42] with MSTW2008NLO PDFs [43], and a renormalization and factorization scale of m . Then we account for the efficiency of lepton reconstruction by scaling our events by a global factor that best matches the Drell-Yan background provided by atlas. At √ s = 8 TeV, this factor is 0.74 (0.67) for the m ee (cos θ CS ) distribution.
Obtaining the signal events is a subtler process. First, we obtain the parton level total cross section dσ toti defined in Eq. (3). For reasons explained in Sec. III, we neglect two terms: the interference between the sm O(α s ) corrections and M χ , and all terms involving triangle diagrams. Next, we convolve dσ SMi and dσ toti with MSTW2008NLO PDFs to obtain the hadron-level cross sections dσ SMi and dσ toti . The N si are now obtained by first scaling the dominant background by dσ tot /dσ SM , and then adding the result to the subdominant background: Using all the above information, we compute and locate the 95% c.l. bound at ∆χ 2 ≡ χ 2 s −χ 2 b = 5.99. Here the systematic errors δ sys i are taken from [35,39].
Our central findings are best understood by directly comparing the right-and left-hand panels of Figs 5 and 6. The relative behavior of these bounds is dictated by two ingredients -(i) the PDFs: as the up quark has higher parton densities in the proton than the down quark, one expects stronger dilepton bounds for dm coupling to up quarks for dm coupling to down quarks, and (ii) interfer-  6. The bounds on our models at two different hierarchies between m φ and mχ. Bounds at 95% c.l. from the lhc are obtained from measurements at 8 TeV and 20 fb −1 ; the orange (blue) curves depict dilepton bounds on the RR (RL) models, and are solid (dashed) for mee (cos θCS) bounds; the magenta curves depict jets + met constraints. The green curves are 90% c.l. Xenon1T constraints on spin-independent scattering, and the red region leads to dm overabundance through freeze-out. See text for further details.
ence effects, or more precisely, the signal contribution of the interference versus the squared box, i.e. dσ int versus dσ χ in Eq. (4). For example, for the models pD u RR and pD u RL , the tree-level and box diagrams interfere destructively, resulting in a deficit of events with respect to the sm for m < 2m χ ; this may be seen in Fig 2. (On the other hand, the relative sign of the down quark's electric charge with respect to the up quark ensures that tree-box interference in the case of dm coupling to down quarks is constructive.) As this deficit occurs in the low m ee bins, where the event population is high, its contribu-tion to the signal χ 2 is considerable 5 . This explains why the m ee bound for pD u RR (and to some extent pD u RL ) is so much stronger on the left-hand than on the right-hand panels of Fig. 5. At the same time, the cos θ CS bounds do not show this hierarchy since the effects of the deficit below and excess above m ee 2m χ are washed out by the integration over m ee bins.
In all four plots, we find the RR models more constrained than the RL models. In the spin-1/2 dm models we understand this from the observation made in Sec. III, that pD u RL gives smaller cross sections than pD u RR due to differences in how interference proceeds between the standard and crossed boxes. As for the spin-0 dm models, we see from Fig. 2 that pCS u RR yields slightly larger cross sections than pCS u RL and is thus subject to slightly stronger constraints. Also, as discussed in Sec. III, our dilepton signal rates decline with m φ /m χ due to propagator suppression in the loop. This results in the weaker limits in the m φ = 2m χ plots in comparison to the m φ = 1.1 m χ plots: in fact, the pD u RL limits are so weak as to disappear from the parametric range displayed.
We now compare our dilepton results with conventional dm probes. In addition to modifying dilepton spectra, our models also have the following effects.
(a) They can pair-produce colored mediators both through qcd and through exchanging χ in the tchannel of a qq-initiated process, and these mediators can decay to a quark and dm. Thus, our models confront constraints from dedicated searches for the mediators using jets plus missing energy signatures, (b) dm can annihilate into quarks and leptons through t-channel exchange of mediators and freeze out in the early universe, confronting the relic density measurement by Planck, (c) dm can scatter against nucleons through s-channel exchange of the mediatorQ, confronting underground direct detection searches.
Bounds derived as a result of (a) -(c) can be seen in Figs 5 and 6 along with the dilepton bounds explained earlier.
Turning first to the jets + / E T bounds, the purple regions are excluded at 95% c.l. by the cms Run 1 search [44]. To determine this bound, we reinterpreted the T2qq bounds that assumes squark production through qcd (that is, the gluino is decoupled) followed by prompt decay to light quark + lsp. Specifically, we generated leading-order cross sections forQ pair production using MadGraph5 [45] and CTEQ6L1 parton distribution functions [46], and matched them with exclusion cross sections provided by cms. We assume here that the detector acceptances of our models are similar to the T2qq model of cms. Since this constraint is agnostic to the chirality of the lepton in our models (with both spin-0 and spin-1/2 dm), we do not distinguish between RR and LR.
The red curve in the plots corresponds to the thermal line where Ω χ h 2 = 0.12, with dm being overproduced in the red shaded region below. This curve was obtained using MicrOmegas4.3 [47], and takes into account coannihilation between dm and mediators, which becomes important in the compressed region m φ 1.1 m χ 6 . Leptonic modes constitute only a small fraction of the annihilation cross section σv , as opposed to quark modes that come with a color factor of 3. Therefore, though the RL models must give a slightly higher σv than the RR models due to neutrino final states, there is no visible difference in the red curves. It may also be seen that, barring a non-standard thermal history of the universe, the dm in our models make up a fraction of the total dm population in regions where most of our dileptonic bounds apply.
Finally, the green curves show 90% c.l. bounds from spin-independent scattering at Xenon1T [48]. (The current spin-dependent limits are consistently weaker and are not shown.) To obtain these bounds, we assume that the density fraction of dm at freeze-out equals the density fraction in the galactic halo today, i.e. Ω χ h 2 /(0.12) = ρ χ /(0.3 GeV cm −3 ), which effectively scales the exclusion cross sections by 0.12/Ω χ h 2 . Our annihilation and scattering cross sections are provided in Appendix B.
Our dileptonic probes are highly complementary to jets + / E T searches. At m φ = 1.1m χ the latter are generally weak, since in this compressed region only a small fraction of events pass the tight cuts applied on missing energy. Consequently, the dileptonic limits for spin-1/2 dm are seen to generally surpass the jets + / E T limits. This is true of pCS u RR as well, except the bound on the coupling now rapidly tightens at m χ 330 GeV. This happens because the production rate of the fermionic mediator (in pCS u RR , pCS u RL ) is higher than the scalar mediator (in pD u RR , pD u RL ), and we are able to saturate the cms exclusion cross section with pure qcd production (λ → 0) in this region. As for pCS d RR , the dilepton bounds are weaker due to small down quark PDFs. At m φ = 2m χ the jets + / E T limits are comparable to those at m φ = 1.1m χ . This is because, though the signal / E T acceptance improves in the uncompressed region, the mediator production rates fall with m φ . As the dilepton signal is diminished in this region, it complements jets + / E T in a model-dependent fashion: for spin-1/2 dm, the m ee (cos θ CS ) bound outdoes jets + / E T for pD u RR (pD d RR ), while for spin-0 dm, no dilepton bound surpasses jets + / E T . The relic density constraint on dm overproduction is generally stronger for spin-0 dm than spin-1/2 dm. This is because the s-wave piece of complex scalar dm annihilation is chirality suppressed, whereas that of Dirac dm is not [4,20]. At m φ = 1.1 m χ and m χ ≤ 450 GeV, spin-0 dm gives weaker bounds since the efficient selfannihilation of the colored fermion mediator drives the co-annihilation mechanism in this region.
Our dilepton probes greatly complement direct detection searches as well. Limits from the latter are generally strong when the mediator is near-degenerate with dm in mass, as seen in the m φ = 1.1 m χ plots. This is due to the factor of (m 2 φ − m 2 χ ) −k in the cross sections, where k = 4 (2) for spin-1/2 (spin-0) dm. The limit on spin-1/2 dm is mostly insensitive to λ at m χ 600 GeV due to our scaling of the scattering cross sections with Ω χ h 2 /0.12 ∝ σv -the former ∝ λ 4 and the latter ∝ λ −4 at large λ, where co-annihilations with the mediators are unimportant. The limit does vary with the coupling at small λ, where co-annihilations dominate. This asymp- totic behavior of the limit with respect to λ allows our dilepton probes to constrain our set-up better than direct detection at m χ 600 GeV. On the other hand, the direct detection limit does not asymptote as quickly for spin-0 dm. This is because, as just mentioned, dm annihilations are chirality-suppressed in the s-wave, allowing co-annihilations to influence freeze-out even at large λ. As a result, direct detection limits dwarf all other constraints for spin-0 dm at m φ = 1.1m χ . The potency of dilepton probes is better at higher m φ . Due to the m −k φ scaling, the limits weaken with m φ so much as to disappear from the m φ = 2m χ plots, allowing dilepons to probe this region better.
Finally, in Figs 7 and 8 we show the future 95% c.l. sensitivity of our dilepton probes at the 13 TeV lhc with a luminosity of L = 300 fb −1 . The color code is as in Fig. 5. To obtain these sensitivities, we performed a χ 2 fit to the background, i.e. we set N di → N bi in Eq. (9), assuming a systematic error of 2% and reusing our 8 TeV lepton reconstruction efficiencies. Our sensitivities im-prove with respect to the 8 TeV measurements with 20 fb −1 luminosity.
This results not only from the usual effect of obtaining better statistics from increase in luminosity, but also crucially, from the increase in collider energy as well. The latter effect originates in the PDFs of the qq initial state; at a given m ee , their parton luminosity increases with an increase in √ s. From this follows the otherwise surprising result that, at the 13 TeV lhc, m ee measurements surpass cos θ CS in sensitivity for all our models. By populating the m ee spectrum with more events, the increase in parton luminosities magnifies the interference signals seen in Fig. 2 and contributes more to the χ 2 of the m ee spectrum, while the same effects need not be apparent in the cos θ CS spectrum, which integrates over a wide range of invariant masses, 400 GeV ≤ m ee ≤ 4.5 TeV 7 .
In the left-hand panel of Fig. 7 the blue dots populate the exclusion regions of the dashed blue curves. These "islands" of exclusion form because the model pD u RL allows for destructive interference between the dm box and sm tree amplitudes, which affects the cos θ CS spectrum in peculiar ways. At large λ the signal spectrum is higher than the background, but as we dial λ down the signal spectrum approaches the background and eventually crosses it, giving a deficit in events. The same behavior is seen as we dial the dm mass up. As a result, the χ 2 bound first gets weaker and then stronger again as we scan from top to bottom in λ or left to right in m χ .
An interesting prospect emerges for pCS u RR at 400 GeV ≤ m χ ≤ 600 GeV. It can be seen from the bottom left panel of Fig. 8 that in this mass range the m sensitivity roughly coincides with the thermal line. This implies that, should this scenario be realized in nature, the lhc is poised to find all of the cosmological dm in the Drell-Yan process.

V. CONCLUSIONS
In this work we have shown that dark matter may be characterized using invariant mass and scattering angle spectra of dilepton distributions at the lhc. If dm coupled to both quarks and leptons through t-channel mediators, radiative corrections from this dark sector, in combination with threshold effects, produce unique spectral features that may single out dm properties. We have shown these features in Figs. 2, 3 and 4. Our findings can be summarized thus: 1. Finding a dileptonic signal as sketched in Fig. 2 would imply that dm is not its own anti-particle.
2. The spin of dm is determinable from the m spectrum. In the region where the signal cross section rises quickly, its slope must be inspected; at higher m , one must check whether the event ratio of signal over background grows rapidly or settles to a steady value. Signals would be visible in the angular spectrum, but untangling the spin is more challenging.
3. If dm is a fermion, the signal cross section rises abruptly near twice the dm mass, thus revealing the mass of dm. This feature must be reflected as an abrupt change of slope at the same m in angular asymmetries, such as the forward-backward (A FB ) or center-edge asymmetry (A CE ).
4. The angular spectrum pinpoints the chirality of the fermions to which dm couples, an effect best seen in the angular asymmetries A FB and A CE . The m spectrum picks out this difference poorly.
Having analyzed the signal features, we then placed constraints on the couplings and masses that we introduced using lhc dilepton data, and contrasted them against bounds from multijets + / E T searches, relic density measurements, and direct detection. We found that angular distributions sometimes gave better constraints than m distributions, significantly updating the conclusions of Ref. [10]. We also found that dileptonic measurements in general are complementary to conventional dm searches. This is especially true of dm coupling to up quarks, where these probes often set the strongest collider bound on the RR model. It must be remembered that, just like jets + / E T , the Drell-Yan process sets bounds not only on dm but on any analogous neutral particle that lives longer than collider time scales.
The dilepton sensitivities to our models would increase in future lhc runs. The 13 TeV lhc with a luminosity of 300 fb −1 is poised to cover couplings down to λ 1 for m χ ≥ 300 GeV, and dm masses up to m χ = 1400 GeV for λ ≤ 2. A potential hint with sufficient signal significance in these regions would enable us to infer some subset of the dm properties discussed above. This could be valuable to complementary experiments such as direct detection and met-based collider searches.
For the sake of illustration, we only considered dm couplings to right-handed quarks. Should one of our dilepton signals arise in forthcoming runs of the lhc, we must also entertain interpretations of dm coupling to other flavors and chiralities of quarks, and consider a wider range of splittings δm when shape-fitting. Disentangling the exact Lagrangian structure would be a challenging task, and may involve deeper scrutiny of all available spectral information.
All in all, we look forward to the amusing prospect of visible particle production educating us on dark matter.

Direct detection
At direct detection experiments, our dm behaves like a Majorana or real scalar particle (see Sec. II). Here we provide the appropriate spin-independent per-nucleon scattering cross sections, σ SI . We have where µ χN is the dm-nucleon (N = p, n) reduced mass, and effective coupling f N is given by [20] f N m N = f q f Tu + 3 4 (q 2 +q 2 )g q − 8π 9α s f T G f G , with the Wilson coefficients and the coefficients f Tu (proton) = 0.023, f Tu (neutron) = 0.017, f T d (proton) = 0.032, f T d (neutron) = 0.041, u 2 = 0.22,ū 2 = 0.034, d 2 = 0.11,d 2 = 0.036, g q = 4f q , f T G (proton) = 0.925, f T G (neutron) = 0.922 [55,57]. Here The coefficients on the right-hand side are the same as before.