How to prove that a E T excess at the LHC is not due to dark matter

If the LHC is able to produce dark matter particles, they would appear at the end of cascade decay chains, manifesting themselves as missing transverse energy. However, such “dark matter candidates” may themselves decay invisibly. We propose to test for this possibility by studying the effect of particle widths on the observable invariant mass distributions of the visible particles seen in the detector. We consider the simplest nontrivial case of a two-step two-body cascade decay and derive analytically the shapes of the invariant mass distributions, for generic values of the widths of the new particles. We demonstrate that the resulting distortion in the shape of the invariant mass distribution can be significant enough to measure the width of the dark matter “candidate,” ruling it out as the source of the cosmological dark matter.


I. INTRODUCTION
Events with missing transverse energy, = E T , at the Large Hadron Collider (LHC) at CERN could be indicative of the production of dark matter particles; the latter are stable and weakly interacting, and, once produced in the collision, will escape without leaving a trace inside the detector.This causes an imbalance in the transverse momentum of the event, ⃗ = P T , whose magnitude is colloquially known as the "missing transverse energy," = E T ¼ j ⃗ = P T j.However, the reverse statement is not so obvious: if we observe an excess of = E T events at the LHC, how can one be sure that what we are seeing is indeed the cosmological dark matter?
The question of proving that a = E T signal observed at the LHC is indeed due to dark matter, has attracted a lot of attention in the past [1][2][3][4][5][6][7][8].The basic idea was to test whether the newly discovered weakly interacting massive particle was consistent with being a thermal relic or not.The general approach was to assume a specific model, most often some version of low-energy supersymmetry, and then attempt to measure all relevant model parameters affecting the thermal relic density calculation.Unfortunately, such an approach is model dependent, applies only to thermal relics (for alternative nonthermal scenarios, see Refs.[9,10]), requires full understanding of the early cosmology, and typically demands a large number of additional measurements, possibly at future (or futuristic) facilities.
Given that proving the discovery of dark matter at the LHC is such a difficult task, perhaps one should focus on the opposite question: how to disprove that the newly found invisible particle is the cosmological dark matter.One possibility is to perform a precise measurement of its mass, and if the mass is consistent with zero, it may just be one of the Standard Model (SM) neutrinos instead of a brand new particle [11].However, this logic is not ironclad either: there exist examples where the dark matter particles are very light [12,13] and cannot be ruled out just on the basis of their small mass.
A much more direct approach would be to test whether the particle which is the source of the = E T is indeed stable; after all, we only know that it did not decay inside the detector.If its lifetime is relatively short, so that it does decay outside, but not too far from the detector, one could attempt to build a dedicated experiment to record such delayed decays.In the past, there were proposals to place such supplementary detectors near the D0 experiment at Fermilab [14] and near the LHC [15], and these ideas were recently revived in Ref. [16].However, any such experiment is doomed if the dark matter candidate decays invisibly, e.g., to hidden sector particles [17].
In this paper we address the worst-case scenario, when the dark matter candidate produced at the LHC is unstable and decays invisibly.For concreteness, we consider the standard new physics decay chain shown inside the solid box of Fig. 1: where v 1;2 are SM particles (assumed to be massless here for simplicity), while A, B, and C are new particles, with C being the dark matter candidate.The canonical example for the processes (1) is the neutralino decay χ0 2 → l e l Ã → l lχ 0 1 in supersymmetry [18], where χ0 2 (χ 0 1 ) is the second-lightest (lightest) neutralino, l ( lÃ ) is a charged (anti)slepton and l ( l) is a SM (anti)lepton.The masses of the particles A, B and C are denoted with m A , m B and m C , respectively, and in general all three particles will have corresponding widths Γ A , Γ B and Γ C .In particular, we shall pay special attention to the case when the dark matter "candidate" C is unstable and thus its decay width Γ C is strictly nonzero.In fact, Γ C could easily be surprisingly large, e.g., it can be enhanced by the multiplicity of available decay channels for C in the hidden sector.Our key idea here is to attempt a direct measurement of the new particle widths (including Γ C ) from the kinematic distributions of the visible decay products v 1 and v 2 .If one could unambiguously establish experimentally that Γ C > 0, then C will be ruled out as a dark matter candidate.Therefore, our first goal is to derive the effect of nonzero widths on the observable kinematics.In particular, we shall demonstrate as a proof of principle that nonzero particle widths can have an observable impact on the shape of the distribution of the invariant mass m ≡ m v 1 v 2 of the two visible particles v 1 and v 2 .A more detailed analysis of the exact particle width sensitivity is left for a future study.

II. PURE ON-SHELL CASE
In the purely on-shell case, where all three particles A, B and C are exactly on shell, the distribution dN=dm has the well-known "triangular" shape which extends up to the kinematic end point m max on The validity of Eq. ( 2) is ensured (at tree level) as long as the narrow-width approximation holds and there are no significant polarization effects.We shall now investigate how the result ( 2) is modified in the case of non-negligible widths Γ A , Γ B and, most importantly, Γ C (for studies in other contexts, see Refs.[19][20][21][22][23]).For simplicity, here we shall be turning on those widths one at a time, postponing the discussion of the simultaneous measurement of Γ A , Γ B , and Γ C to a future publication [24].

III. NON-NEGLIGIBLE Γ B
As a warm-up, we begin with the case when only B is relatively broad, Γ B ≠ 0. In that case, the narrow-width result (2) gets modified to [25] where In the limit of massless v 1 and v 2 , the lower end point of Eq. ( 4) is at m ¼ 0, while the upper end point, m max Γ B , is obtained by solving the equation s a result identical to the one for the direct three-body decay Note that in the narrow-width approximation limit of Γ B =m B → 0, the integrand in Eq. ( 4) becomes lim and we recover the purely on-shell result (2). Figure 2 illustrates the effect of a finite-width Γ B on the invariant mass distribution (4).In general, one should expect sizable effects whenever the width Γ B is comparable to a relevant mass splitting, e. mass distribution (4) for several different values of Γ B =m B , from 1% (red lines) all the way to 50% (purple lines).For comparison, the m distribution for the three-body decay (7) is shown by the magenta dashed curve.We see that initially, as the width Γ B is relatively small, the shape of the distribution still resembles the triangular shape (indicated by the dotted line) of Eq. ( 2), but there are a certain number of events which leak out beyond the nominal upper kinematic end point (3).As the width Γ B increases, so does the fraction of events which leak out, and very soon, for Γ B =m B ∼ 5-10%, no discernible end point is visible at all at the location predicted by Eq. ( 3).Instead, we obtain a relatively broad distribution which terminates at the new kinematic end point (6).Eventually, as the width Γ B further increases, the distribution asymptotes to the magenta dashed line corresponding to the case of the three-body decay (7).
Figure 2 demonstrates that the effect of a finite Γ B on the invariant mass distribution (4) can be quite significant; for one, all curves in the figure have shapes which are clearly different from the triangular shape (2) obtained in the limit of Γ B ¼ 0. At the same time, unless the B resonance is extremely broad (Γ B ∼ m B ), the obtained distribution is also distinguishable from that of a three-body decay (7).We thus conclude that the observation of a nontrivial invariant mass shape like the ones seen in Fig. 2 would not only suggest a finite value for Γ B , but will also allow its measurement with a decent precision.
Before we move on to the case of a non-negligible Γ C , let us briefly comment on the effect of spin correlations.Our previous results were obtained in the pure phase space limit, where the width dependence comes only from the B propagator.However, these results would be valid only if all involved particles are spin 0, which is unrealistic: the SM particles v 1 and v 2 are fermions (leptons or quarkinitiated jets).Therefore, some nontrivial chiralities are present in the interaction vertices, as shown in the left panel of Fig. 3, where for concreteness we have chosen the intermediate particle B to be a fermion.In general, the fermion couplings are arbitrary mixtures of left-handed and right-handed chiral couplings proportional to P L ≡ ð1 − γ 5 Þ=2 and P R ≡ ð1 þ γ 5 Þ=2, respectively.In Fig. 3, we contrast three special cases: (a) vectorlike couplings, (b) opposite chiralities at the two vertices and (c) the same chiralities at the two vertices.Then, the spin-averaged matrix element squared receives an additional contribution proportional to Therefore, the result for vectorlike couplings [Fig.3(a)] is simply the sum of these two cases (times a factor of 2 due to L ↔ R exchange) The chirality effects ( 9)-( 10) on the shape of the invariant mass distribution are illustrated in the right panel of Fig. 3, for a mass spectrum ðm A ; m B ; m C Þ ¼ ð1000; 970; 500Þ GeV and Γ B =m B ¼ 1%.For reference, the black dotted line shows the pure scalar theory result (4).The green dot-dashed and the red dashed lines represent the distributions obtained in the presence of spin correlations as in Fig. 3(b) and Fig. 3(c), respectively.The case of vectorlike couplings, Fig. 3(a), is then obtained by simply adding those two distributions (blue solid line).In the narrow-width approximation, for vectorlike couplings one would recover the phase space result (2), since the spin correlations from Fig. 3(b) and Fig. 3(c) would cancel exactly.However, in the presence of nontrivial width effects as in Eq. ( 9), the cancellation is incomplete and even the case of vector-like couplings is markedly different from the pure scalar theory result (compare the blue solid and black dotted lines in Fig. 3) [26,27].

IV. NON-NEGLIGIBLE Γ C
We now consider perhaps the most interesting case, when the dark matter candidate (particle C) has a nonvanishing width, Γ C ≠ 0, due to an invisible decay to two dark sector particles X and x, as shown in the right (dotdashed) boxed extension of Fig. 1.Under those circumstances, we find that the shape of the invariant mass distribution is given by where m X and m x are the respective masses of the hidden sector particles X and x and As before, the upper kinematic end point, m max Γ C , of the distribution ( 11) is found from s Comparing to Eq. ( 3), we notice that which is easily understood as the limit when C becomes extremely off shell.
In analogy to Fig. 3, the left panel in Fig. 4 illustrates the impact of the nonvanishing width Γ C on the shape of the invariant mass distribution (11).We take the mass spectrum to be ðm A ; m B ; m C Þ ¼ ð1000; 520; 500Þ GeV and again vary the dimensionless ratio Γ C =m C from 1% to 50% as indicated in the legend.For concreteness, we assume the hidden sector particles X and x to be massless, i.e., m X ¼ m x ¼ 0, in which case the distributions in Fig. 4 have a common upper kinematic end point 4 demonstrates that the effect of Γ C can be quite drastic.Even when the width Γ C is as small as 1% of the resonance mass m C , the shape of the distribution is visibly distorted from the standard triangular shape (2), and a sizable fraction of events are already leaking out beyond the expected kinematic end point (3), which is indicated with the vertical dashed line.Increasing the width to Γ C ∼ 0.05m C appears already sufficient to render the triangular shape unrecognizable and indicate the presence of off-shell effects.
In order to test the method in the presence of realistic detector resolution effects, in the right panel of Fig. 4 we plot the invariant mass distribution of 500 events for the worst-case scenario when particles v 1 and v 2 are jets.We apply Gaussian smearing of 12.5%, which is typical for jets, and contrast the cases of Γ C ¼ 0 (blue solid line) and Γ C =m C ¼ 2% for a single experiment (red solid line) or the average of 100 pseudoexperiments (red dashed line).It is clear that even in the presence of jet smearing, the two cases are easily distinguishable: applying a Kolmogorov-Smirnov test, we find a p-value of 10 −7 which is sufficient for a 5σ discovery of a nonvanishing Γ C .

V. NON-NEGLIGIBLE Γ A
Finally, for completeness we also consider the case where the decay width of particle A is non-negligible, Γ A ≠ 0. This case is a little bit more model dependent, since we must know how to sample the 4-momentum squared, p 2 A , of particle A. One simple possibility is that A is the decay product of a narrow resonance Y with mass m Y , Y → yA, as shown in the left (dashed) boxed extension of Fig. 1.Under those circumstances, the invariant mass distribution is given by where m Y and m y are the masses of the particles Y and y, respectively, while The upper kinematic end point, m max Γ A , of the distribution (15) is again found from s − ¼ s þ : and can be equivalently interpreted as Figure 5 shows the effect of a nonvanishing width Γ A on the shape of the invariant mass distribution (15).The mass spectrum is chosen as ðm Y ; m A ; m B ; m C Þ ¼ ð1500; 1000; 970; 500Þ GeV and the dimensionless ratio Γ A =m A is again varied from 1% to 50%, as indicated in the legend.For concreteness, we assume that the additional final-state particle y is massless, and thus all distributions in Fig. 5 have a common kinematic end point m max on ðm Y ; m B ; m C Þ ¼ 980 GeV, as predicted by Eq. ( 18).Once again, we observe that even a width of only 1% leads to a noticeable change in the expected triangular shape and an overflow of events beyond the nominal kinematic end point of 208.3 GeV predicted by Eq. ( 3) and denoted by the vertical dashed line.As the width is further increased, the shape distortion becomes quite significant, confirming the sensitivity to the value Γ A .

VI. SUMMARY AND OUTLOOK
We derived the effects of nonzero particle widths on the observable invariant mass distribution dN=dm in the case of the decay chain of Fig. 1.We showed that the shape of the distribution can be very sensitive to the widths and therefore can be used to perform a measurement of Γ A , Γ B and, most importantly, Γ C , thus directly probing the nature of the dark matter candidate C, which appears invisible in the detector.Our results for these three cases can be compactly summarized as where i ¼ fA; B; Cg, the integration limits s iAE are given by Eqs. ( 16), ( 5) and ( 12), respectively, while λ 1=2 ðm 2 Y ; m 2 y ; sÞ s for i ¼ A; The analysis presented here can be readily generalized to the case of massive SM particles v 1 and v 2 , to three-body decays, and to a simultaneous measurement of Γ A , Γ B and Γ C [24].
One should be mindful of the fact that there are other factors which also affect the shape of the invariant mass distribution dN=dm.On the theoretical side, there could be spin correlations [26,[28][29][30], interference [31,32] and higher-order effects [33,34].On the experimental side, the analysis cuts, the detector resolution, and the SM backgrounds will also play roles in this measurement.However, these effects are well known and under control, and can be readily accounted for (see, e.g., the kinematic end point measurements in Ref. [35]).Furthermore, the width measurement relies mostly on the events above the nominal kinematic end point (3), while all those effects impact mostly the softer part of the distribution dN=dm.We are therefore optimistic that such width measurements will be feasible, once a sufficiently strong and clean missing energy signal of new physics is observed at the LHC.1).The particle y is assumed massless, and may or may not be visible in the detector.
FIG. 1.The new physics decay chain under study.

FIG. 3 .
FIG. 3. Left panel: Three different fermion chirality structures for the boxed decay chain of Fig. 1: (a) vectorlike couplings, (b) opposite chiralities, and (c) the same chiralities at the neighboring fermion vertices.Right panel: Unit-normalized invariant mass distributions for those three cases, compared to the pure scalar theory result (4) (black dotted line), for ðm A ; m B ; m C Þ ¼ ð1000; 970; 500Þ GeV and Γ B =m B ¼ 1%.

FIG. 4 .
FIG. 4. Left: Unit-normalized invariant mass distributions for ðm A ; m B ; m C Þ ¼ ð1000; 520; 500Þ GeV and several different values of Γ C =m C as shown in the legend.We assume that particle C further decays invisibly to two massless particles X and x, C → Xx, as shown in the dot-dashed box of Fig. 1.Right: the invariant mass distribution of 500 events, after applying Gaussian smearing of 12.5% for the case of Γ C ¼ 0 (blue solid line) and Γ C =m C ¼ 2% (red solid line).The red dashed line represents the average of 100 pseudoexperiments.

FIG. 5 .
FIG. 5. Unit-normalized invariant mass distributions for ðm Y ; m A ; m B ; m C Þ ¼ ð1500; 1000; 970; 500Þ GeV and several different values of Γ A =m A as shown in the legend.We assume that A results from the decay of a parent particle Y, Y → yA (see the dashed box of Fig.1).The particle y is assumed massless, and may or may not be visible in the detector.