DDARK MATTER AND LONG-LIVED PARTICLES AT THE LHC
JAN HEISIG
Institute for Theoretical Particle Physics and Cosmology, RWTH Aachen University,Sommerfeldstraße 16, D-52056 Aachen, Germany
While the paradigm of a weakly interacting massive particle (WIMP) has guided our searchstrategies for dark matter in the past decades, their null-results have stimulated growinginterest in alternative explanations pointing towards non-standard signatures. In this articlewe discuss the phenomenology of dark matter models that predict long-lived particle at theLHC. We focus on models with a Z -odd dark sector where – in decreasing order of thedark matter coupling – a coannihilation, conversion-driven freeze-out or superWIMP/freeze-in scenario could be realized. Pinpointing the nature of dark matter is among the key scientific goals of the LHC. So farsearches for dark matter have strongly focussed on missing energy signatures following thewidely studied paradigm of WIMP dark matter. However, the LHC has not found any hintfor a corresponding signal yet, proceeding to strengthen the constraints on WIMP dark mattermodels. At the same time, if the WIMP scenario is not realized in nature, a potential signal ofdark matter related physics might hide in other places. It is therefore of utmost importance toinvestigate alternative ideas of dark matter genesis that could point towards new signatures.Motivated by a variety of theories beyond the standard model long-lived particle (LLP)searches have recently attracted growing interest. LLPs provide a wide range of possible signa-tures: highly ionizing, disappearing or kinked tracks (charged LLPs), displaced vertices (chargedor neutral LLPs) as well as trackless jets or displaced leptons (neutral LLPs). While many LLPsignatures provide very promising search prospects exploiting extremely low (and often solelyinstrumental) backgrounds they are typically hard to trigger on. Specialized trigger settings –often relying on additional activity in the event – are needed. This makes it a timely enterpriseto investigate a comprehensive search program for LLPs in order to fully exploit the immensecapability of the LHC to illuminate physics beyond the standard model.This article constitutes a contribution to this effort by investigating dark matter scenariosthat predict LLP signatures at the LHC. We focus on models with a Z -odd dark sector. Theprime example is the well-known coannihilation scenario , in which the coannihilating particlemight or might not appear long-lived at collider time scales depending on the mass splitting be-tween the coannihilating partner and dark matter. While the canonical coannihilation scenarioassumes relative chemical equilibrium in the dark sector during dark matter freeze-out otherviable possibilities exist with couplings significantly weaker than the weak force. In conversion-driven freeze-out , (or co-scattering ) the decoupling from relative chemical equilibrium (facili-tated by conversion processes) governs the dark matter abundance. For even smaller dark mattercouplings chemical equilibrium of dark matter might never even have been established leadingto a superWIMP (or freeze-in ) scenario. Interestingly, for particles in the GeV to TeV range a a r X i v : . [ h e p - ph ] M a y eparture from relative chemical equilibrium during freeze-out implies macroscopic decay lengthat the LHC – an intriguing coincidence that renders the LHC to be a powerful tool to explorethese scenarios. We discuss the appearance of LLPs in coannihilation and conversion-drivenfreeze-out scenarios in Sec. 2 while focussing on a concrete realization of the latter in Sec. 3. InSec. 4 we comment on thermally decoupled dark matter before finally concluding in Sec. 5. A Z symmetry (or a larger symmetry with a Z subgroup) is commonly imposed in theoriesbeyond the standard model in order to stabilize dark matter. UV-complete models often comewith an entire Z -odd sector. This opens up the possibility for interesting phenomena regardingthe evolution of its particle densities in the early Universe. A prime example in this concern isthe well-know coannihilation scenario. , , For small relative mass splittings between a heavier Z -odd state, χ , and dark matter, χ , ∆ m/m χ ≡ ( m χ − m χ ) /m χ (cid:46) a the relativenumber density of the heavier state could still be significant during dark matter freeze-out: n eq χ /n eq χ ∝ e − ∆ m/T f (cid:39) e −
25 ∆ m/m χ . (1)In the last expression we inserted T f (cid:39) m χ /
25 as the typical freeze-out temperature. Conse-quently, χ participates in the freeze-out processes providing additional annihilation channelsthat can deplete the number density in the dark sector and, hence, due to conversion processeswithin the dark sector, the number density of dark matter. Since the Z symmetry forces the heavier states to decay into lighter states of the dark sec-tor, a small relative mass splitting potentially leads to a kinematic suppression of the decaywidth. Therefore coannihilation scenarios can provide LLPs. A prominent example is the staucoannihilation strip of the constrained MSSM b , parts of which are most strongly constrainedby searches for heavy stable charged particles. , , Another example concerns minimal darkmatter which extends the standard model by an electroweak multiplet. Here, a naturally smallmass splitting between the neutral and charges states of the multiplet (of order 100 MeV) arisesfrom electroweak corrections. For the fermion triplet, which corresponds to a wino dark matterscenario in supersymmetry, the strongest LHC constraints arise from searches for disappearingtracks excluding masses up to 430 GeV. However, so far much stronger limits arise from in-direct detection experiments. Other coannihilation scenarios that naturally provide LLPs are e.g. pseudo-Dirac dark matter models , and models with colored dark sectors. , , In the standard treatment of coannihilation conversion processes within the dark sector are as-sumed to be efficient during freeze-out and relative chemical equilibrium is maintained, n χ i /n eq χ i = n χ j /n eq χ j . c In this case annihilations in the dark sector can be described by an effective, thermallyaveraged cross section (cid:104) σv (cid:105) eff = (cid:88) i,j (cid:104) σv (cid:105) ij n eq χ i n eq n eq χ j n eq , (2)where n eq = (cid:80) i n eq i . For very small couplings of the dark matter to the standard model (cid:104) σv (cid:105) ij can be negligible for all channels containing dark matter in the initial state. The dilution of a The exact value depends on the hierarchy between the involved couplings. In extreme cases coannihilationcan be important for much larger mass splittings. b Minimal supersymmetric standard model. c This assumption is commonly made in numerical relic density calculators. , , �� - � �� - � ���� � ��� �� � �� � �� - � �� - � ���������� � �� � / H (1 TeV) /T p r o m p t m e t a - s t a b l e d e t ec t o r - s t a b l e ⇠ H Figure 1 – Ratio between decay rate and Hubble rate as a function of the inverse temperature. the number density of the dark sector is then driven entirely by annihilations of heavier statesand not by dark matter annihilations. In this case the relic density becomes independent ofthe coupling strength of dark matter. However, this conclusion is only true for couplings thatare still large enough to maintain relative chemical equilibrium. d For even smaller couplingsrelative chemical equilibrium breaks down. In this case conversion processes are responsible forthe chemical decoupling of dark matter and hence set the relic density. This conversion-drivenfreeze-out mechanism is phenomenologically distinct and opens up a new region in parameterspace where coannihilation would lead to under-abundant dark matter, if relative chemicalequilibrium would hold.
The departure from relative chemical equilibrium has an immediate consequence for the possibledecay length of the heavier states. As the decay contributes to the conversions, requiring theirrate to become inefficient necessarily requiresΓ dec (cid:46)
H . (3)In the radiation dominated Universe H = (cid:112) g ∗ / πT /M Pl , where M Pl (cid:39) . × GeV is thereduced Planck mass. We can translates the inverse Hubble rate into a length. Using g ∗ = 100,the inequality (3) then reads cτ (cid:38) H − (cid:39) . (cid:18) (100 GeV) T (cid:19) . (4)This is an important results which states that for particles in the GeV to TeV range a departurefrom relative chemical equilibrium during freeze-out ( T (cid:39) m χ /
30) implies macroscopic decaylength at the LHC – an intriguing coincidence that renders the LHC to be a powerful tool toexplore these scenarios. Figure 1 illustrates the prompt, meta-stable and detector-stable regimein the plane spanned by the inverse temperature and Γ dec /H . In this section we discuss a realization of conversion-driven freeze-out within a simplified darkmatter model. We consider an extension of the standard model by a neutral Majorana fermion χ and a colored scalar particle ˜ q that acts as a ( t -channel) mediator of the dark matter interactionswith the standard model quarks q : L int = | D µ ˜ q | + λ χ ˜ q ¯ q − γ χ + h.c. . (5) d Note that conversion rates are enhance compared to annihilations by a Boltzmann factor of order e m χ /T . ere D µ is the usual covariant derivative and λ χ is a coupling strength of the dark matterinteraction. For a certain choice of λ χ this model resembles a subset of the MSSM, whilesmaller couplings could be realized in extensions of the MSSM. However, we do not refer toany particular UV-complete theory here considering λ χ as a free parameter.Imposing a Z symmetry under which all standard model particles are even while χ → − χ and ˜ q → − ˜ q are odd, the Majorana fermion χ provides a viable dark matter candidate for m χ < m ˜ q . We consider the cases of a bottom- and top-philic model, q = b, t , providing a distinctphenomenology. While the mass of the bottom is mostly small compared to the energies of therelevant processes the mass of the top is sizable leading to additional suppressions e.g. of themediator decay. Without the assumption of chemical equilibrium between dark matter and the mediator thecomputation of the relic density requires the solution of the full coupled set of Boltzmannequations explicitly including conversion processes in the dark sector. We take into account theleading decay process (2- and 4-body decay for the bottom- and top-philic model, respectively)and all 2 → → , The cosmologically viable parameter space (Ω h = 0 . is shown in Fig. 2 for thebottom- (left) and top-philic (right) model in the plane spanned by the dark matter mass andthe mass difference ∆ χ ˜ q = m ˜ q − m χ .Above the black thick curve relative chemical equilibrium holds resembling a standardWIMP/coannihilation scenario while below this curve solutions for conversion-driven freeze-out exist where λ χ is in the range 10 − –10 − and 10 − –10 − for the bottom- and top-philicmodel, respectively. At the curve itself the measured relic density can be obtained for a widerange of λ χ that provide a negligible dark matter annihilation cross section but still sizableconversion rates maintaining chemical equilibrium in the dark sector. For illustration, Fig. 3shows the respective solution for λ χ as a function of the dark matter mass for the top-philicmodel and for a fixed mass splitting of ∆ m χ ˜ t = 20 GeV. At the transition from the WIMP tothe conversion-driven freeze-out region the coupling drops by several orders of magnitude.As discussed in Sec. 2.3 conversion-driven freeze-out predicts macroscopic decay length of theheavier Z -odd state. For the bottom-partner the 2-body decay of the mediator is open renderingboth conversion via decays and scatterings to be similarly important and hence Γ dec ∼ H duringfreeze-out. As a consequence the decay length is of the order of several cm, see gray dottedcurves in the left panel of Fig. 2. For the top-philic model, in the parameter region of interest,the leading decay channel is a 4-body decay. Accordingly, conversions during freeze-out aremediated solely by scatterings while the decay becomes efficient only well after freeze-out. Notethat mediator lifetimes above O (1 s) are subject to constraints from big bang nucleosynthesis(BBN) indicated by the red shaded region in the right panel of Fig. 2. At the LHC mediators pairs could be copiously produced. Being a colored state the mediator isexpected to hadronize and form R -hadron bound states. Depending on its decay length, it willtypically decay inside or traverse the detector. Accordingly, for the case of the bottom-partnerthe signatures of kinked or disappearing tracks (depending on the detectability of the radiated b -quark) provide promising search channels. Although similar searches have been performed forsupersymmetric models , , these cannot be reinterpreted within the present model withoutadditional information provided by the collaborations. For instance, searches for disappearingtracks are performed under the assumption of purely electrically charged particle while R -hadronundergo a more complicated traverse through the detector being able to flip charge or becomeneutral through interactions with the detector material. Therefore their applicability is unclear.
00 1000 1500 200005101520253035 m e b [ G e V ] m [ GeV ] ⇣ ⇥ ⌘ c m c m c m . c m " W I M P r e g i o n ⌦ h = 0 . bottom-philic
500 1000 1500 200005101520253035 m e t [ G e V ] m [ GeV ] ⌦ h = 0 . " W I M P r e g i o n s · · · · · · . · . · top-philic Figure 2 – Cosmologically viable parameter space (Ω h = 0 .
12) in the conversion-driven freeze-out region (belowblack thick curve) for the bottom- (left) and top-partner (right) mediator. , Contours of constant λ χ are shown ingreen ( × − in the left plot). Contours of constant mediator decay length (left) and lifetimes (right) are shownas gray dotted curves. The displayed lifetimes range from 10 − s to 10 s in steps of an order of magnitude (thecurve for 1 s is highlighted in red for better readability). The 95% C.L. exclusion regions from R -hadron searchesat the 8 and 13 TeV LHC are shown in dark and light blue, respectively. The red shaded region bordered by thered dot-dot-dashed curve in the left plot denotes the constraint from monojet searches at the 13 TeV LHC whereasthe light red shaded region in the right plot denotes constraints from BBN. Below the horizontal gray dashed line( ∼ Searches for detector-stable R -hadrons can, however, be reinterpreted for finite decay lengths using the signature efficiencies provided for heavy stable charge particle searches released byCMS. The resulting constraints obtained from the 8 TeV and 13 TeV LHC data are shownin Fig. 2 as the dark and light blue shaded regions, respectively. Note that the model can also beconstrained by mono-jet searches exploiting a large missing energy from initial state radiation inthe mediator production process. Here we show the limit from ATLAS using 3 . − of 13 TeVdata, see the red shaded region in the left panel of Fig. 2.While a dedicated search is expected to be able to significantly increase the sensitivity to thebottom-philic model, the top-philic model – featuring a detector-stable mediator – is already verywell constraints from R -hadron searches. Dedicated searches for meta-stable top-partners could,however, fill some gaps in the sensitivity outside the conversion-driven freeze-out region wherethe mediator tends to have intermediate lifetimes, cf. Fig. 3. Note that the entire cosmologicallyallowed conversion-driven freeze-out region of the top-philic model is expected to be probed by R -hadron searches at 13 TeV with an integrated luminosity of approximately 300 fb − . So far we have considered scenarios where dark matter undergoes a phase of thermalization andfreeze-out. Another possibility is that dark matter never reaches chemical equilibrium with thestandard model bath being produced through out-of-equilibrium processes. While we cannothope to observe an entirely thermally decoupled dark sector, a partly thermalized dark sectorcan provide promising prospects to be explored at the LHC. There are two main scenariosfor non-thermalized dark matter genesis in the literature: the freeze-in and the superWIMP scenario. While freeze-in mediated by a Z -even mediator in general does not provide promisingprospects for the LHC, e both scenarios may be observable, if dark matter production is mediated e A valuable exception can, however, arise from a non-standard cosmological history.
100 200 500 1000 2000 5000 1 ¥ - m [ GeV ] top-philic m ˜ t = 20 GeV ⌦ h = . ( ± % t h e o . e r r . ) R - h a d r o n s T e V R - h a d r o n s T e V L Z p r o j . X e n o n T A M S - ¯ p F e r m i d w . L H C s t o p II L EP LHC stop I
Figure 3 – Constraints on the coupling λ χ as a function of the dark matter mass for ∆ m χ ˜ t = 20 GeV in thetop-philic model. The green curve and green shaded band shows the coupling that provides Ω h = 0 .
12 and itstheoretical uncertainty, respectively, assuming a relative error of 10% on the prediction for Ω h . The 95% C.L.exclusion regions from R -hadron searches at the 8 and 13 TeV LHC are shown in dark and light blue, respectively.For comparison, we show limits from canonical WIMP searches, i.e.
95% C.L. upper limits on λ χ from indirectdetection searches from Fermi-LAT dwarfs (light red curves) and AMS-02 antiprotons (dark red curves) as well as90% C.L. direct detection upper limits from Xenon1T 2017 (purple solid curve) and direct detection projectionsfor the LZ experiment (purple dashed curves). Additionally, we show 95% C.L. exclusion regions from searchesfor supersymmetric top-partners at the LHC and LEP. Further details can be found in Ref. 4. by a Z -odd state. f In the simplified model introduced in Eq. (5) the leading contributions to dark matterproduction would be the conversion processes ˜ qi ↔ χj and ˜ q ↔ χj , where i, j denote standardmodel particles. Depending on the masses, coupling strength and the model under consideration( q = b, t ) the dominant contribution to dark matter production occurs around or after the freeze-out of ˜ q . The first case constitutes a realization of freeze-in while the second case resembles asuperWIMP scenario. However, in general both contributions are present. Furthermore, as themodel is sensitive to the initial conditions – which are not washed out by thermalization – afurther contribution might stem from physics relevant at earlier times that are not captured bythe (low-energy) simplified model considered here, e.g. a contribution from reheating.In Fig. 4 we plot the cosmologically viable viable parameter space (Ω h = 0 .
12) for thesuperWIMP scenario for the top-philic model ( q = t ). We assume a sufficiently small couplingso that the mediator decay occurs well after its freeze-out and plot the curve for which(Ω h ) χ = m χ m (cid:101) t (Ω h ) (cid:101) t + (Ω h ) init χ = 0 . , (6)where (Ω h ) (cid:101) t is the freeze-out abundance of the mediator in the absence of any coupling todark matter while (Ω h ) init χ is the initial dark matter abundance prior to the mediator decay.We show the result for three different choices of (Ω h ) init χ : a vanishing abundance as well as afraction of 0.7 and 0.9, respectively, of the total ( i.e. final) abundance. This initial abundancerepresents a possible contribution from the very early Universe ( e.g. reheating phase) or fromfreeze-in. In the model under consideration the latter would arise from conversion via scatteringand could be computed for a given λ χ .The red shaded region in Fig. 4 denotes a mass splitting below the top mass, introducingan additional suppression of the mediator decay leading to large lifetimes that are potentially f Here we concentrate on dark matter with masses in the GeV to TeV range, typically providing detector-stablemediators in the regime of thermally decoupled dark matter while lighter dark matter can also provide decayswithin the LHC detector, e.g. displayed vertices.
100 200 500 1000 2000 50001050100500100050001 ¥ m ˜ t [ G e V ] m [ GeV ] ⌦ h = 0 . top-philic ! ⌦ i n i t = R - h a d r o n s T e V . ⌦ t o t . ⌦ t o t Figure 4 – Cosmologically viable parameter space (Ω h = 0 .
12) for a superWIMP scenario ( λ χ → i.e. final) abundance. The light blue shaded area denotes the 95% C.L. exclusion regionfrom R -hadron searches at the 13 TeV LHC. For comparison, the dotted black curve shows the border of theconversion-driven freeze-out region. in conflict with BBN for the very small couplings considered here. However, outside this regionwhere the 2-body decay of the mediator is open we find that consistency with BBN can easilybe achieved in the limit of a dominant superWIMP scenario, e.g. τ < λ χ (cid:38) − for which conversion rates are entirely inefficient, Γ conv (cid:28) H , until well after the freeze-out ofthe mediator. The blue shaded region shows the respective limit for detector-stable mediatorsfrom R -hadron searches at the 13 TeV LHC. It constraints part of the parameter space withrelatively large (Ω h ) init χ where the mediator freeze-out abundance, (Ω h ) (cid:101) t , is required to besmall. For comparison we show the boundary of the conversion-driven freeze-out as a dottedblack curve. In this article we discussed dark matter scenarios that predict long-lived particles at the LHC.We focussed on models with a Z -odd dark sector providing three distinct regions characterizedby a decreasing coupling strength: The well-know coannihilation scenario, conversion-drivenfreeze-out and the superWIMP/freeze-in scenario. The latter two cases exploit an intrinsicconnection between the dynamics of dark matter genesis and the involved decay rates of aheavier Z -odd state. For particles in the GeV to TeV range a departure from relative chemicalequilibrium during its freeze-out implies macroscopic decay length at the LHC – an intriguingcoincidence that renders the LHC to be a powerful tool to explore these scenarios.We presented realizations of conversion-driven freeze-out within the framework of simpli-fied dark matter with a top- and bottom-partner mediator. While the former model predictsdetector-stable R -hadrons that are well constrained by existing searches at the LHC, the latterprovides decay length of the order of several cm. These are relatively poorly constraint as exist-ing searches for meta-stable colored states cannot be straightforwardly reinterpreted within themodel. Dedicated searches are required for a full exploration of the model. Despite the smallcouplings required by conversion-driven freeze-out ranging between 10 − and 10 − it provides anefficient thermalization of dark matter prior to freeze-out. This is contrast to the superWIMPscenario where dark matter never reaches thermal equilibrium with the standard model bath.The cosmologically viable parameter space, hence, depends on a possible contribution of darkatter production from the very early Universe as well as from freeze-in around the time of themediator freeze-out and contains a wide range of yet unexplored masses. Parts of the parameterspace can be probed with R -hadron searches at the 13 TeV LHC. Acknowledgements
This work is supported by the German Research Foundation DFG through the research unit“New physics at the LHC”.
References
1. K. Griest and D. Seckel,
Phys. Rev.
D43 , 3191 (1991).2. J. Edsjo and P. Gondolo,
Phys. Rev.
D56 , 1879 (1997), hep-ph/9704361.3. M. Garny, J. Heisig, B. L¨ulf, and S. Vogl,
Phys. Rev.
D96 , 103521 (2017), 1705.09292.4. M. Garny, J. Heisig, M. Hufnagel, and B. L¨ulf,
Phys. Rev.
D97 , 075002 (2018), 1802.00814.5. R. T. D’Agnolo, D. Pappadopulo, and J. T. Ruderman,
Phys. Rev. Lett. , 061102(2017), 1705.08450.6. J. L. Feng, A. Rajaraman, and F. Takayama,
Phys. Rev.
D68 , 063504 (2003), hep-ph/0306024.7. L. J. Hall, K. Jedamzik, J. March-Russell, and S. M. West,
JHEP , 080 (2010),0911.1120.8. M. J. Baker et al. , JHEP , 120 (2015), 1510.03434.9. R. T. D’Agnolo, C. Mondino, J. T. Ruderman, and P.-J. Wang, (2018), 1803.02901.10. M. Citron et al. , Phys. Rev.
D87 , 036012 (2013), 1212.2886.11. N. Desai, J. Ellis, F. Luo, and J. Marrouche,
Phys. Rev.
D90 , 055031 (2014), 1404.5061.12. J. Heisig, A. Lessa, and L. Quertenmont,
JHEP , 087 (2015), 1509.00473.13. M. Cirelli, N. Fornengo, and A. Strumia, Nucl. Phys.
B753 , 178 (2006), hep-ph/0512090.14. ATLAS, M. Aaboud et al. , Report No. ATLAS-CONF-2017-017, 2017.15. A. Cuoco, J. Heisig, M. Korsmeier, and M. Kr¨amer,
JCAP , 004 (2018), 1711.05274.16. A. De Simone, V. Sanz, and H. P. Sato,
Phys. Rev. Lett. , 121802 (2010), 1004.1567.17. A. Davoli, A. De Simone, T. Jacques, and V. Sanz,
JHEP , 025 (2017), 1706.08985.18. J. Ellis, F. Luo, and K. A. Olive, JHEP , 127 (2015), 1503.07142.19. S. El Hedri, A. Kaminska, M. de Vries, and J. Zurita, JHEP , 118 (2017), 1703.00452.20. A. Davoli, A. De Simone, T. Jacques, and A. Morandini, (2018), 1803.02861.21. G. Blanger, F. Boudjema, A. Goudelis, A. Pukhov, and B. Zaldivar, (2018), 1801.03509.22. F. Ambrogi et al. , (2018), 1804.00044.23. T. Bringmann, J. Edsj¨o, P. Gondolo, P. Ullio, and L. Bergstr¨om, (2018), 1802.03399.24. G. Belanger, F. Boudjema, C. Hugonie, A. Pukhov, and A. Semenov, JCAP , 001(2005), hep-ph/0505142.25. Planck, P. A. R. Ade et al. , Astron. Astrophys. , A13 (2016), 1502.01589.26. K. Jedamzik,
Phys. Rev.
D74 , 103509 (2006), hep-ph/0604251.27. ATLAS, G. Aad et al. , Phys. Rev.
D92 , 072004 (2015), 1504.05162.28. ATLAS, G. Aad et al. , Phys. Rev.
D88 , 112006 (2013), 1310.3675.29. CMS, V. Khachatryan et al. , JHEP , 096 (2015), 1411.6006.30. CMS, V. Khachatryan et al. , Eur. Phys. J.
C75 , 325 (2015), 1502.02522.31. CMS, S. Chatrchyan et al. , JHEP , 122 (2013), 1305.0491.32. CMS, V. Khachatryan et al. , Report No. CMS-PAS-EXO-16-036, 2016.33. ATLAS, M. Aaboud et al. , Phys. Rev.
D94 , 032005 (2016), 1604.07773.34. F. Kahlhoefer,
Phys. Lett.
B779 , 388 (2018), 1801.07621.35. R. T. Co, F. D’Eramo, L. J. Hall, and D. Pappadopulo,
JCAP12