Effective Field Theories (EFTs) are a useful tool across a wide range of DM searches, including LHC searches and direct detection. Given the current lack of indications about the nature of the DM particle and its interactions, a model independent interpretation of the collider bounds appears mandatory, especially in complementarity with the reinterpretation of the exclusion limits within a choice

of simplified models, which cannot exhaust the set of possible completions of an effective Lagrangian. However EFTs must be used with caution at LHC energies, where the energy scale of the interaction is at a scale where the EFT approximation can no longer be assumed to be valid. Here we introduce some tools that allow the validity of the EFT approximation to be quantified, and provide case studies for

two operators. We also show a technique that allows EFT constraints from collider searches to be made substantially more robust, even at large center-of-mass energies. This allows EFT constraints from different classes of experiment to be compared in a much more robust manner

Jacques, T.

English

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DOI 10.1393/ncc/i2015-15119-9Colloquia: LaThuile15IL NUOVO CIMENTO 38 C (2015) 119Dark Matter: Collider vs. direct searchesT. JacquesSection de Physique, Universite´ de Gene`ve - 24 quai E. Ansermet, CH-1211 Geneva, Switzerlandreceived 2 October 2015Summary. — Effective Field Theories (EFTs) are a useful tool across a wide rangeof DM searches, including LHC searches and direct detection. Given the currentlack of indications about the nature of the DM particle and its interactions, a modelindependent interpretation of the collider bounds appears mandatory, especially incomplementarity with the reinterpretation of the exclusion limits within a choiceof simplified models, which cannot exhaust the set of possible completions of aneffective Lagrangian. However EFTs must be used with caution at LHC energies,where the energy scale of the interaction is at a scale where the EFT approximationcan no longer be assumed to be valid. Here we introduce some tools that allow thevalidity of the EFT approximation to be quantified, and provide case studies fortwo operators. We also show a technique that allows EFT constraints from collidersearches to be made substantially more robust, even at large center-of-mass energies.This allows EFT constraints from different classes of experiment to be compared ina much more robust manner.PACS 95.35.+d – Dark matter.1. – IntroductionThe LHC is searching for direct DM production at unprecedented energies, yet it hasproven difficult to constrain the WIMP sector in a model-independent way. One potentialsolution is the use of Effective Field Theories (EFTs), where a DM-SM interaction iswritten as a single effective operator, integrating out the mediator. This has the satisfyingfeature of reducing the parameter space to a single mass (mDM) and an energy scale (Λ,also known as M∗ in the literature), and reducing the number of WIMP models down toa small basis set.Another advantage is that the EFT formalism makes it easy to compare the strengthof constraints placed on DM by different experiments. Since each operator is describedby just two parameters, it is easy to convert constraints on, say, the production crosssection at the LHC into constraints on the DM-nucleon cross section, with conversionformulae given in, e.g., refs. [1, 2]. However, it is important to remember that colliderand direct searches for DM operate with completely different signals and energy scales,Creative Commons Attribution 4.0 License (http://creativecommons.org/licenses/by/4.0) 12 T. JACQUESso that any comparison must be done carefully and bearing in mind the different rangeof applicability of the constraints.One crucial difference is that the operators relevant to direct detection are the non-relativistic (NR) limits of the operators used for collider searches. There is not a one-to-one correspondence between the two sets of operators; several collider operators reduce tothe same NR operator, and some collider operators correspond to a linear combination ofNR operators. One consequence of this is that constraints on the usual spin-independent(SI) or spin-independent (SD) scattering cross sections apply to several collider operators,and that no single collider constraint on an operator covers the same model-space as aconstraint on the SI or SD scattering rate. The translation between relativistic andnon-relativistic operators has been studied in detail, e.g. [3, 4].EFTs are inherently an approximation to a full UV-complete theory, and hence mustbe used with caution. Whilst the EFT approximation is clearly valid for the keV-scaleenergy transfers of direct detection experiments, the LHC is operating at a much largerenergy scale where it is important to ensure that constraints on EFTs are internallyconsistent and fall in a region where the EFT approximation is valid. This issue has beenknown about since the early days of EFT studies. For example, refs. [5,6] have comparedconstraints on some EFTs to those on simplified models, and found that constraints onΛ using UV complete models can either be substantially stronger or substantially weakerthan those on EFTs, depending on the choice of parameters. This is especially importantwhen comparing LHC EFT constraints with those from direct or indirect detection, asit can give a misleading impression of the relative strengths of the different classes ofexperiment if the LHC EFT constraints are not robust and presented with clear caveatson the range of validity.References [7,8] have proposed a method to quantify the validity of the EFT approx-imation, applying the approach to s-channel type operators. This has been extended tothe t-channel in ref. [9] by considering a model where Dirac DM couples to SM quarksvia t-channel exchange of a coloured scalar mediator.The goal here is to determine the regions of parameter space where the EFT approachis a valid description of a given model. To do this, first we need to consider what theEFT approximation physically means. The approximation is made by integrating out themediator, and combining the remaining free parameters into a single energy scale. For atree-level interaction between DM and the Standard Model (SM) via some mediator withmass M , this corresponds to expanding the propagator in powers of Q2tr/M2, truncatingat lowest order, and combining the remaining parameters into a single parameter Λ. Foran example scenario with a Z ′-type mediator this corresponds to setting(1)g2effQ2tr −M2= − g2effM2(1 +Q2trM2+O(Q4trM4)) − g2effM2≡ − 1Λ2,where Qtr is the momentum carried by the mediator, g2eff ≡ gDM gq, and gDM, gq are theDM-mediator and quark-mediator couplings respectively. Similar expressions exist forother operators.There is a necessary kinematic condition for the validity of the EFT, derived inref. [10], which has been used in the past as a guideline for the validity of the EFTapproximation. In the s-channel, the mediator must carry at least enough energy toproduce the DM at rest, Qtr > 2mDM. For the EFT to be valid, M > Qtr, and so werequire M > 2mDM. Taking the couplings as large as possible while still remaining inthe perturbative regime, g2eff  4π, the relationship g2effM2 ≡ 1Λ2 becomes Λ > mDM2π .DARK MATTER: COLLIDER VS. DIRECT SEARCHES 3Clearly this is not a sufficient condition for the approximation to be valid. Examiningeq. (1), we see that instead we must satisfy the condition(2) Q2tr M2 = g2effΛ2.Unfortunately this condition is impossible to test in the true EFT limit, since M hasbeen combined with geff to form Λ. Instead, an assumption about geff or choice of Mmust be made. There is no lower limit to the coupling strength, so regardless of the scaleof Λ, it is always possible that M is small enough that the EFT approximation does notapply.Alternatively, the most optimistic choice is to assume that geff  4π, the maximumpossible coupling strength such that the model could still lie in the perturbative regime.As a middle ground, we test whether the EFT approximation is valid for values of geff > 1,a natural scale for the coupling in the absence of any other information. In this case, thecondition for the validity of the EFT approximation becomes Q2tr < Λ2, which we willadopt in the following to assess the validity of the use of EFT at LHC for DM searches.2. – Measuring the Validity of the EFT approximationThe validity of the EFT approximation has been studied for a broad, representativerange of operators in refs. [7-9]. Here we will show results for two examples. First weconsider the operator describing a vector coupling between Dirac DM χ and quarks q,(3) OV = 1Λ2 (χ¯γμχ) (q¯γμq).This corresponds to the EFT limit of a simplified Z ′ model with pure vector (i.e. noaxial-vector) couplings, described by the interaction Lagrangian(4) Lint = −∑qZ ′μgqgDM[q¯γμq]− Z ′μ [χ¯γμχ],where the kinetic and gauge terms have been omitted, and the sum is over all quarkflavours of choice. We also consider the following effective operator describing the inter-actions between Dirac dark matter and left-handed quarks,(5) Ot = 1Λ2 (χ¯PLq) (q¯PRχ).For this operator, only the coupling between dark matter and the first generation ofquarks is considered. This operator can be viewed as the low-energy limit of a simplifiedmodel describing a quark doublet QL coupling to DM, via t-channel exchange of a scalarmediator SQ,(6) Lint = g χ¯QLS∗Q + h.c.and integrating out the mediator. This model is popular as an example of a simpleDM model with t-channel couplings and a coloured mediator, which exist also in well-motivated models such as supersymmetry where the mediator particle is identified as asquark, although the DM is a Majorana particle in this case.4 T. JACQUESThe standard search channel for these scenarios is missing energy plus one or morejets, although there are other promising complementary search channels. The dominantprocess contributing to the missing energy plus jet signal is qq¯ → χχ¯g, for which thedifferential cross section has been calculated.To test whether the EFT approximation is valid in jet searches, a ratio is definedof the cross section truncated so that all events pass the condition, to the total crosssection:(7) RΛ ≡ σ|Qtr<Λσ=∫ pmaxTpminTdpT∫ 2−2 dηd2σdpTdη∣∣∣Qtr<Λ∫ pmaxTpminTdpT∫ 2−2 dηd2σdpTdη.The integration limits on these quantities are chosen to be comparable to thoseused in standard searches for WIMP DM by the LHC Collaborations (see, for instance,refs. [11-13]).Figures 1, 2 show isocontours of four fixed values of RΛ as a function of both mDMand Λ, for the OV and Ot operators respectively. Contrasted with OV , the ratio forthe t-channel operator has less DM mass dependence, being even smaller than in thes-channel case at low DM masses and larger at large DM masses, without becominglarge enough to save EFTs.Fig. 1. – Contours of the fraction of events for which the EFT approximation is valid, for theeffective operator with a vector-type coupling, at√s = 8TeV (left) and 14TeV (right). Thegrey shaded region indicates where the EFT approximation has necessarily fully broken downfor kinematic reasons. From ref. [8].s 8 TeV500GeV  pT 1 TeV, 2R 75R 50R 25R 10DM h2=0.1210 102 10301000200030004000mDM GeVGeVs =14TeV500GeV  pT 2 TeV, 2R 75R 50R 25R 10DM h2=0.1210 102 103010002000300040005000mDM GeVGeVFig. 2. – Contours of the fraction of events for which the EFT approximation is valid, for the Otoperator described in the text, at√s = 8TeV (left) and 14TeV (right). The black solid curvesindicates the correct relic abundance. From ref. [9].DARK MATTER: COLLIDER VS. DIRECT SEARCHES 5s 8 TeV500GeV  pT 1 TeV, 2R 50R4 50mDM210 102 103100200500100020005000mDM GeVGeVs =14TeV500GeV  pT 2 TeV, 2R 50R4 50mDM210 102 103100200500100020005000mDM GeVGeVFig. 3. – Contours for RΛ for the Ot operator at √s = 8TeV (left) and 14TeV (right), varyingthe cutoff Qtr < Λ and Qtr < 4πΛ. The grey shaded area shows Λ < mDM/(2π), often used asa benchmark for the validity of the EFT. From ref. [9].In fig. 2 we also see the curves corresponding to the correct DM relic density. Forgiven mDM, larger Λ leads to a smaller self-annihilation cross section and therefore tolarger relic abundance. It is evident that the large-Λ region where the EFT is validtypically leads to an unacceptably large DM density.In the most optimistic scenario for EFTs, the coupling strength g takes the maximumvalue (4π) such that the model remains in the perturbative regime. To demonstrate howthese results depend on the coupling strength, fig. 3 shows isocontours for R = 50% forOt, for two cases: 1) the standard requirement that Q2tr < Λ2, equivalent to requiringg  1, and 2) requiring Q2tr < (4πΛ)2, equivalent to requiring g  4π.The grey shaded area indicates the region where Λ < mDM/(2π). As discussed, thisis sometimes used as a benchmark for the validity of the EFT approximation, since inthe s-channel, Qtr is kinematically forced to be greater than 2mDM.3. – Rescaling EFT constraintsWe have seen in the previous section that the EFT approximation is not generallyvalid at LHC energies. In this section we discuss a technique that can make EFT con-straints robust at the cost of a weakened constraint. Recall that for a tree-level interactionbetween DM and the Standard Model (SM) via a Z ′-type mediator, the EFT approxima-tion corresponds to assuming eq. (1) holds. Similar expressions exist for other operators.Clearly the condition that must be satisfied for this approximation to be valid is eq. (2).The condition “” is poorly defined, and so we instead use Q2tr < g2effΛ2 as a reason-able criteria with which to measure the validity of the EFT approximation for a giveninteraction.We can use this condition to enforce the validity of the EFT approximation by re-stricting the signal (after the imposition of the cuts of the analysis) to events for whichQ2tr < M2. This truncated signal can then be used to derive the new, truncated limit onΛ as a function of (mDM, geff).For the operators we consider here, σ ∝ Λ−4, and so there is a simple rule for convert-ing a rescaled cross section into a rescaled constraint on Λ if the original limit is basedon a simple cut-and-count procedure. Defining σcutEFT as the cross section truncated suchthat all events pass the condition geffΛrescaled > Qtr, we have(8) Λrescaled =(σEFTσcutEFT)1/4Λoriginal,6 T. JACQUESDMgSMg1 1.5 2 2.5 3M* [TeV]00.20.40.60.81alidvM* > 400 GeVmissTEe pxM* > 600 GeVmissTE > 800 GeVmissTEATLAS Simulation Preliminary = 8 TeVs 1 - L = 20 fb∫ = 50 GeVχmDMgSMg1 1.5 2 2.5 3M* [TeV]00.20.40.60.811.21.41.61.82alidvM* > 400 GeVmissTEe pxM* > 600 GeVmissTE > 800 GeVmissTEATLAS Simulation Preliminary = 14 TeVs 1 - L = 25 fb∫ = 50 GeVχmFig. 4. – Rescaled simulated missing energy + jet limits on the OV operator at 8TeV (left) and14TeV (right), from ref. [13]. Note that M∗ is equivalent to Λ.Fig. 5. – Comparison of direct detection constraints with rescaled ATLAS constraints, fromref. [12].which can be solved for Λrescaled via either iteration or a scan (note that Λrescaled appearson both the LHS and RHS of the equation, via σcutEFT). Similar relations exist for a givenUV completion of each operator. The details and application of this procedure by ATLASto their EFT constraints can be found in refs. [12,13] for a range of operators. Since thismethod uses the physical couplings and energy scale Qtr, it gives the strongest possibleconstraints in the EFT limit while remaining robust by ensuring the validity of the EFTapproximation.An example of this rescaling is shown in fig. 4, for the OV operator, for simulatedmissing energy + jet constraints at 8 and 14TeV. We see that the EFT constraintis substantially weaker than the “naive” constraint except for relatively large couplingstrengths. One interesting feature is that the range of validity is larger at√s = 14TeVthan 8TeV. This is slightly counter-intuitive, as we would expect the larger energy scaleto lead to a larger Qtr, and hence less validity. However, this is balanced by the factthat the baseline “naive” constraint is also much larger at 14TeV, and hence the fractionof events passing the condition Qtr < geffΛ is greater at the larger energy scale for agiven geff.This truncation procedure allows a much fairer comparison with other experimentalconstraints. For a given geff, the collider EFT constraints are now robust and self-consistent, allowing comparison with direct detection results using the usual translationformulae [1, 2]. An example of this is given in fig. 5, from ref. [12], which comparesDARK MATTER: COLLIDER VS. DIRECT SEARCHES 7ATLAS and CMS constraints on a range of operators with direct constraints on the SI(left) and SD (right) scattering cross section from a variety of experiments. The ATLASconstraints are shown for two values of the coupling strength: geff = 1 as a standardbenchmark, and the maximum coupling that remains in the perturbative regime as abest-case-scenario for the ATLAS constraints.4. – ConclusionsEffective operators are a useful tool to constrain dark matter with as few free param-eters as possible, however care must be taken with the assumptions involved, especiallywhen comparing with other experiments. In this article we have introduced some mea-surements of the validity of the EFT approach for DM searches at colliders, by examiningtwo reference operators, OV and Ot. It is clear that even for relatively modest couplingstrengths and mediator masses, the validity of the EFT approximation at LHC energiescan not be assumed, reinforcing the need to go beyond the EFT at the LHC when lookingfor DM signals.However, it is not necessary to abandon the EFT approach entirely. We have revieweda technique that allows constraints on Λ to be rescaled such that only events for whichthe EFT approximation is valid are utilised. This allows for a more robust compari-son with other experimental constraints, for example direct detection, reinforcing thecomplementarity of the two approaches to DM searches.∗ ∗ ∗This article is primarily based on ref. [9], by Giorgio Busoni, Andrea De Simone, TDJ,Enrico Morgante and Antonio Riotto, and ref. [12], by the ATLAS Collaboration withAndrea De Simone, TDJ and Antonio Riotto. I thank the organisers of the conferencefor the invitation and their generous support, and congratulate them on an excellentmeeting.REFERENCES[1] Bai Y., Fox P. J. and Harnik R., JHEP, 12 (2010) 048 (arXiv:1005.3797 [hep-ph]).[2] Goodman J., Ibe M., Rajaraman A., Shepherd W., Tait T. M. P. and Yu H. B.,Phys. Rev. D, 82 (2010) 116010 (arXiv:1008.1783 [hep-ph]).[3] Fitzpatrick A. L., Haxton W., Katz E., Lubbers N. and Xu Y., JCAP, 02 (2013)004 (arXiv:1203.3542 [hep-ph]).[4] Cirelli M., Del Nobile E. and Panci P., JCAP, 10 (2013) 019 (arXiv:1307.5955 [hep-ph]).[5] Fox P. J., Harnik R., Kopp J. and Tsai Y., Phys. Rev. D, 85 (2012) 056011(arXiv:1109.4398 [hep-ph]).[6] Buchmueller O., Dolan M. J. and McCabe C., JHEP, 01 (2014) 025 (arXiv:1308.6799[hep-ph]).[7] Busoni G., De Simone A., Morgante E. and Riotto A., Phys. Lett. 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Dark Matter: Collider vs. direct searches

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Dark Matter: Collider vs. direct searches

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