How to prove that the LHC did not discover dark matter
CCERN-TH-2017-279
How to prove that the LHC did not discover dark matter
Doojin Kim and Konstantin T. Matchev Theoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland Physics Department, University of Florida, Gainesville, FL 32611, USA
If the LHC is able to produce dark matter particles, they would appear at the end of cascadedecay chains, manifesting themselves as missing transverse energy. However, such “dark mattercandidates” may decay invisibly later on. We propose to test for this possibility by studying theeffect of particle widths on the observable invariant mass distributions of the visible particles seenin the detector. We consider the simplest non-trivial case of a two-step two-body cascade decay andderive analytically the shapes of the invariant mass distributions, for generic values of the widths ofthe new particles. We demonstrate that the resulting distortion in the shape of the invariant massdistribution can be significant enough to measure the width of the dark matter “candidate”, rulingit out as the source of the cosmological dark matter.
PACS numbers: 95.35.+d, 14.80.-j, 13.85.Qk
Introduction. /E T events at the Large Hadron Col-lider (LHC) at CERN are motivated by the dark mat-ter problem — the dark matter particles are stable andweakly interacting, and, once produced in the LHC col-lisions, will escape without leaving a trace inside the de-tector. This will lead to an imbalance in the transversemomentum of the event, known as “missing transversemomentum” (cid:126)/P T .[29] However, the reverse statement isnot so obvious — if we observe an excess of /E T events atthe LHC, how can one be sure that what we are seeingis indeed the cosmological dark matter?The question of proving that a /E T signal observed atthe LHC is indeed due to dark matter, has attracted a lotof attention in the past [1–8]. The basic idea was to testwhether the newly discovered weakly interacting mas-sive particle (WIMP) was consistent with being a ther-mal relic or not. The general approach was to assumea specific model, most often some version of low-energysupersymmetry, and then attempt to measure all rele-vant model parameters affecting the thermal relic densitycalculation. Unfortunately, such an approach is model-dependent; applies only to thermal relics (for alternativenon-thermal scenarios, see [9, 10]); requires full under-standing of the early cosmology; and typically demandsa large number of additional measurements, possibly atfuture (or futuristic) facilities.Given that proving the discovery of dark matter at theLHC is such a difficult task, perhaps one should focus onthe opposite question — how to disprove that the newlyfound invisible particle is the cosmological dark matter.One possibility is to perform a precise measurement of itsmass, and if the mass is consistent with zero, it may justbe one of the Standard Model (SM) neutrinos insteadof a brand new particle [11]. However, this logic is notironclad either — there exist examples where the darkmatter particles are very light [12, 13] and cannot beruled out just on the basis of their small mass.A much more direct approach would be to test whetherthe particle which is the source of the /E T is indeed sta-ble — after all, we only know that it did not decay insidethe detector. If its lifetime is relatively short, so that it FIG. 1: The new physics decay chain under study. does decay outside, but not too far from the detector, onecould attempt to build a dedicated experiment to recordsuch delayed decays. In the past, there were proposalsto place such supplementary detectors near the D0 ex-periment at Fermilab [14] and near the LHC [15], andthese ideas were recently revived in [16]. However, anysuch experiment is doomed if the dark matter candidatedecays invisibly, e.g., to hidden sector particles [17].In this letter we address the worst case scenario, whenthe dark matter candidate produced at the LHC is un-stable and decays invisibly sufficiently quickly. For con-creteness, we consider the standard new physics decaychain shown inside the solid box of Fig. 1: A → v B → v v C , (1)where v , are massless SM particles, while A , B , and C are new particles, with C being the dark matter can-didate. The canonical example for the processes (1) isthe neutralino decay ˜ χ → (cid:96) ˜ (cid:96) ∗ → (cid:96) ¯ (cid:96) ˜ χ in supersymmetry[18], where ˜ χ ( ˜ χ ) is the second-lightest (lightest) neu-tralino, ˜ (cid:96) (˜ (cid:96) ∗ ) is a charged (anti-)slepton and (cid:96) (¯ (cid:96) ) 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 ingeneral all three particles will have corresponding widthsΓ A , Γ B and Γ C . In particular, we shall pay special atten-tion to the case when the dark matter “candidate” C isunstable and thus its decay width Γ C is strictly non-zero.Our key idea here is to attempt a direct measurementof the new particle widths (including Γ C ) from the kine-matic distributions of the visible decay products v and a r X i v : . [ h e p - ph ] D ec v . If one could unambiguously establish experimentallythat Γ C >
0, then C will be ruled out as a dark mattercandidate. Therefore, our first goal is to derive the effectof non-zero widths on the observable kinematics. Pure on-shell case.
In what follows, we shall beinvestigating the distribution of the invariant mass m ≡ m v v of the two visible particles v and v . In the purelyon-shell case, where all three particles A , B and C areexactly on-shell, the unit-normalized distribution dN/dm has the well-known “triangular” shape dNdm = m π m A m B Γ B , (2)which extends up to the kinematic endpoint m maxon m maxon ( m A , m B , m C ) ≡ (cid:113) ( m A − m B )( m B − m C ) /m B . (3)The validity of (2) is ensured (at tree-level) as long asthe narrow width approximation holds and there are nosignificant polarization effects. We shall now investigatehow the result (2) is modified in the case of non-negligiblewidths Γ A , Γ B and, most importantly, Γ C . For simplic-ity, we shall be turning on those widths one at a time. Non-negligible Γ B . As a warm-up, we begin withthe case when only B is relatively broad, Γ B (cid:54) = 0. In thatcase, the narrow-width result (2) gets modified to [19] dNdm = m π m A (cid:90) s + s − ds ( s − m B ) + m B Γ B , (4)where s ± ≡ (cid:104) m A + m C − m ± λ / ( m A , m C , m ) (cid:105) , (5)and λ ( x, y, z ) ≡ x + y + z − xy − yz − xz . In thelimit of massless v and v , the lower endpoint of (4) isat m = 0, while the upper endpoint, m maxΓ B , is obtainedby solving the equation s − = s + , which results in m maxΓ B = m A − m B , (6)a result identical to the one for the direct three-bodydecay A → v v C. (7)Note that in the narrow width approximation limit ofΓ B /m B →
0, the integrand in (4) becomeslim Γ BmB → s − m B ) + m B Γ B = πm B Γ B δ (cid:18) sm B − (cid:19) (8)and we recover the purely on-shell result (2).Fig. 2 illustrates the effect of a finite width Γ B on theinvariant mass distribution (4). In general, one shouldexpect sizable effects whenever the width Γ B is compa-rable to a relevant mass splitting,[30] e.g., m A − m B (leftpanel) or m B − m C (right panel). The solid lines depict FIG. 2: The solid lines represent unit-normalized invariantmass distributions (4) for ( m A , m B , m C ) = (1000 , , m A , m B , m C ) = (1000 , , A and Γ C and several differentchoices of Γ B /m B as shown in the legends. The magentadashed curve corresponds to the case of a pure three-bodydecay (e.g., m B (cid:29) m A ). the invariant mass distribution (4) for several differentvalues of Γ B /m B , from 1% (red lines) all the way to 50%(purple lines). For comparison, the m distribution forthe three-body decay (7) is shown by the magenta dashedcurve. We see that initially, as the width Γ B is relativelysmall, the shape of the distribution still resembles thetriangular shape of (2), but there are a certain numberof events which leak out beyond the nominal upper kine-matic endpoint (3). As the width Γ B increases, so doesthe fraction of events which leak out, and very soon, forΓ B /m B ∼ − B further increases, the distributionasymptotes to the magenta dashed line corresponding tothe case of the three-body decay (7).Fig. 2 demonstrates that the effect of a finite Γ B on theinvariant mass distribution (4) can be quite significant— for one, all curves in the figure have shapes which areclearly different from the triangular shape (2) obtainedin the limit of Γ B = 0. At the same time, unless the B resonance is extremely broad (Γ B ∼ m B ), the obtaineddistribution is also distinguishable from that of a three-body decay (7). We thus conclude that the observationof a non-trivial invariant mass shape like the ones seen inFig. 2 would not only suggest a finite value for Γ B , butwill 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 phasespace limit, where the width dependence comes only fromthe B propagator. However, these results would be validonly if all involved particles are spin 0, which is unreal-istic — the SM particles v and v are fermions (leptonsor quark-initiated jets). Therefore, some non-trivial chi-ralities are present in the interaction vertices, as shownin the left panel of Fig. 3, where for concreteness we havechosen the intermediate particle B to be a fermion.[31]In general, the fermion couplings are arbitrary mixturesof left-handed and right-handed chiral couplings propor-tional to P L ≡ (1 − γ ) / P R ≡ (1 + γ ) /
2, respec-
FIG. 3: Left panel: Three different fermion chirality struc-tures for the boxed decay chain of Fig. 1: (a) vectorlike cou-plings, (b) opposite chiralities, and (c) same chiralities at theneighboring fermion vertices. Right panel: Unit-normalizedinvariant mass distributions for those three cases, comparedto the pure scalar theory result (4) (black dotted line), for( m A , m B , m C ) = (1000 , , B /m B = 1%. tively. In Fig. 3, we contrast three special cases: (a)vectorlike couplings, (b) opposite chiralities at the twovertices and (c) the same chiralities at the two vertices.Then, the spin-averaged matrix element squared receivesan additional contribution proportional to |M| ∼ ( m A − s )( s − m C ) − m s, for Fig. 3(b) m (cid:16) Γ B + m B (cid:17) , for Fig. 3(c) . (9)Therefore, the result for vectorlike couplings (Fig. 3(a))is simply the sum of these two cases (times a factor of 2due to L ↔ R exchange) |M| ∼ m A − s )( s − m C ) + 2 m (cid:18) Γ B m B − s (cid:19) . (10)The chirality effects (9,10) on the shape of the in-variant mass distribution are illustrated in the rightpanel of Fig. 3, for a mass spectrum ( m A , m B , m C ) =(1000 , , B /m B = 1%. For reference,the black dotted line shows the pure scalar theory re-sult (4). The green dot-dashed and the red dashed linesrepresent the distributions obtained in the presence ofspin correlations as in Fig. 3(b) and Fig. 3(c), respec-tively. The case of vectorlike couplings, Fig. 3(a), is thenobtained by simply adding those two distributions (bluesolid line). In the narrow width approximation, for vec-torlike couplings one would recover the phase space re-sult (2), since the spin correlations from Fig. 3(b) andFig. 3(c) would cancel exactly. However, in the pres-ence of non-trivial width effects as in (9), the cancellationis incomplete and even the case of vector-like couplingsis markedly different from the pure scalar theory result(compare the blue solid and black dotted lines in Fig. 3)[20]. Non-negligible Γ C . We now consider perhaps themost interesting case, when the dark matter candidate(particle C ) has a non-vanishing width, Γ C (cid:54) = 0, due toan invisible decay to two dark sector particles X and FIG. 4: Unit-normalized invariant mass distributions for( m A , m B , m C ) = (1000 , , C /m C as shown in the legend. We assume thatparticle C further decays invisibly to two massless particles X and x , C → Xx , as shown in the dot-dashed box of Fig. 1. x , as shown in the right (dot-dashed) boxed extensionof Fig. 1. Under those circumstances, we find that theshape of the invariant mass distribution is given by dNdm = m π m A m B Γ B (cid:90) s + s − dss λ / ( s, m X , m x )( s − m C ) + m C Γ C , (11)where m X and m x are the respective masses of the hiddensector particles X and x and s − ≡ ( m X + m x ) , s + ≡ m B (cid:18) − m m A − m B (cid:19) . (12)As before, the upper kinematic endpoint, m maxΓ C , of thedistribution (11) is found from s − = s + , which yields m maxΓ C = (cid:113) ( m A − m B ) { m B − ( m X + m x ) } /m B . (13)Comparing to (3), we notice that m maxΓ C = m maxon ( m A , m B , m X + m x ) , (14)which is easily understood as the limit when C becomesextremely off-shell.In analogy to Fig. 3, Fig. 4 illustrates the impact ofthe non-vanishing width Γ C on the shape of the invari-ant mass distribution (11). We take the mass spectrumto be ( m A , m B , m C ) = (1000 , , C /m C from 1% to 50%as indicated in the legend. For concreteness, we as-sume the hidden sector particles X and x to be mass-less, i.e., m X = m x = 0, in which case the distribu-tions in Fig. 4 have a common upper kinematic endpoint m maxΓ C = (cid:112) m A − m B = 854 GeV.Fig. 4 demonstrates that the effect of Γ C can be quitedrastic. Even when the width Γ C is as small as 1% ofthe resonance mass m C , the shape of the distributionis visibly distorted from the standard triangular shape(2), and a sizable fraction of events are already leakingout beyond the expected kinematic endpoint (3), whichis indicated with the vertical dashed line. Increasing thewidth to Γ C ∼ . m C appears already sufficient to ren-der the triangular shape unrecognizable and indicate thepresence of off-shell effects. FIG. 5: Unit-normalized invariant mass distributions for( m Y , m A , m B , m C ) = (1500 , , , A /m A as shown in the legend. We assumethat A results from the decay of a parent particle Y , Y → yA (see the dashed box of Fig. 1). The particle y is assumedmassless, and may or may not be visible in the detector. Non-negligible Γ A . Finally, for completeness wealso consider the case where the decay width of parti-cle A is non-negligible, Γ A (cid:54) = 0. This case is a little bitmore model-dependent, since we must know how to sam-ple the 4-momentum squared, p A , of particle A . Onesimple possibility is that A is the decay product of a nar-row resonance Y with mass m Y , Y → yA , as shown inthe left (dashed) boxed extension of Fig. 1. Under thosecircumstances, the invariant mass distribution is given by dNdm = m π m Y m B Γ B (cid:90) s + s − dss λ / ( m Y , m y , s )( s − m A ) + m A Γ A , (15)where m Y and m y are the masses of the particles Y and y , respectively, while s − ≡ m B (cid:18) m m B − m C (cid:19) , s + ≡ ( m Y − m y ) . (16)The upper kinematic endpoint, m maxΓ A , of the distribution(15) is again found from s − = s + : m maxΓ A = (cid:113) { ( m Y − m y ) − m B )( m B − m C ) /m B , (17)and can be equivalently interpreted as m maxΓ A = m maxon ( m Y − m y , m B , m C ) . (18)Fig. 5 shows the effect of a non-vanishing width Γ A on the shape of the invariant mass distribution (15).The mass spectrum is chosen as ( m Y , m A , m B , m C ) =(1500 , , , A /m A is again varied from 1% to 50%, as indicatedin the legend. For concreteness, we assume that the ad-ditional final state particle y is massless, then all dis-tributions in Fig. 5 have a common kinematic endpoint m maxon ( m Y , m B , m C ) = 980 GeV, as predicted by (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 kine-matic endpoint of 208.3 GeV predicted by (3) and de-noted by the vertical dashed line. As the width is furtherincreased, the shape distortion becomes quite significant,confirming the sensitivity to the value of Γ A . Summary and outlook.
We derived the effects ofnon-zero particle widths on the observable invariant massdistribution dN/dm in the case of the decay chain ofFig. 1. We showed that the shape of the distribution canbe very sensitive to the widths and therefore can be usedto perform a measurement of Γ A , Γ B and, most impor-tantly, Γ C , thus directly probing the nature of the darkmatter candidate C , which appears invisible in the detec-tor. Our results for these three cases can be compactlysummarized as dNdm ∼ m (cid:90) s i + s i − ds s − m i ) + m i Γ i F i ( s ) , (19)where i = { A, B, C } , the integration limits s i ± are givenby eqs. (16), (5) and (12), respectively, while F i ( s ) = λ / ( m Y ,m y ,s ) s , for i = A ;1 , for i = B ; λ / ( s,m X ,m x ) s , for i = C. (20)One should be mindful of the fact that there are otherfactors which also affect the shape of the invariant massdistribution dN/dm . On the theoretical side, there couldbe spin correlations [20–23], interference [24, 25] andhigher order effects [26, 27]. On the experimental side,the cuts and the detector resolution will also play a role inthis measurement. However, these effects are well knownand under control, and can be readily accounted for (see,e.g., the kinematic endpoint measurements in [28]). Fur-thermore, the width measurement relies mostly on theevents above the nominal kinematic endpoint (3), whileall those effects impact mostly the softer part of the dis-tribution dN/dm . We are therefore optimistic that suchwidth measurements will be feasible, once a sufficientlystrong and clean missing energy signal of new physics isobserved at the LHC. Acknowledgments