Effective Theory of WIMP Dark Matter

[.3em] supplemented by Simplified Models:

[0.7em] Singlet-like Majorana fermion case

Shigeki Matsumoto, Satyanarayan Mukhopadhyay,

[.5em] and Yue-Lin Sming Tsai

Kavli IPMU (WPI), UTIAS, University of Tokyo, Kashiwa, 277-8583, Japan PITT-PACC, Department of Physics and Astronomy, University of Pittsburgh,                                       PA 15260, USA

We enumerate the set of simplified models which match onto the complete set of gauge invariant effective operators up to dimension six describing interactions of a singlet-like Majorana fermion dark matter with the standard model. Tree level matching conditions for each case are worked out in the large mediator mass limit, defining a one to one correspondence between the effective operator coefficients and the simplified model parameters for weakly interacting models. Utilizing such a mapping, we compute the dark matter annihilation rate in the early universe, as well as other low-energy observables like nuclear recoil rates using the effective operators, while the simplified models are used to compute the dark matter production rates at high energy colliders like LEP, LHC and future lepton colliders. Combining all relevant constraints with a profile likelihood analysis, we then discuss the currently allowed parameter regions and prospects for future searches in terms of the effective operator parameters, reducing the model dependence to a minimal level. In the parameter region where such a model-independent analysis is applicable, and leaving aside the special dark matter mass regions where the annihilation proceeds through an s-channel or Higgs boson pole, the current constraints allow effective operator suppression scales () of the order of a few hundred GeV for dark matter masses 20 GeV at C.L., while the maximum allowed scale is around TeV for . An estimate of the future reach of ton-scale direct detection experiments and planned electron-positron colliders show that most of the remaining regions can be probed, apart from dark matter masses near half of the Z-boson mass (with ) and those beyond the kinematic reach of the future lepton colliders.

1 Introduction

Dark matter candidates charged under the weak isospin can interact with the standard model (SM) particles via known gauge interactions, a fact that simplifies their phenomenology considerably. A gauge singlet scalar field can also have a renormalizable interaction with the SM Higgs doublet. However, in order to couple the SM sector to a gauge singlet fermion dark matter (stabilized by a postulated symmetry), one needs to introduce either additional bosonic degrees of freedom that couple to both the sectors, or additional fermionic degrees of freedom with electroweak charges which can mix with the singlet state after electroweak symmetry breaking. Possible frameworks to discuss the phenomenology of a singlet-like fermion WIMP (weakly interacting massive particle) candidate have been studied at length for decades, from specific ultra-violet (UV) complete models (e.g., the bino in the MSSM), to model independent setups with effective operators.

The three most relevant observables for a stable WIMP are it’s relic density obtained via thermal freeze out, it’s pair annihilation cross-section in the current epoch in dark matter dense regions, and the elastic scattering rate of WIMP’s with nuclei. All of these processes involve scatterings with an energy scale or momentum transfer comparable to the WIMP mass or much lower. Therefore, as long as the new states mediating the interactions are at least a few times heavier than the dark matter, we can work with a set of effective operators of leading dimension relevant to the process, and truncating the operator series would not lead to any inconsistency nor amount to large theoretical errors.

Pair production of WIMP’s are being searched for at high-energy colliders as well, and for certain types of interactions or for lower dark matter masses, they could be competitive with, or even have a larger reach than that of the direct or indirect detection probes. Collider searches for dark matter necessarily rely on a recoil of the dark matter pair against a hard radiation in the initial state, which can lead to a sub-process centre of mass energy () considerably higher than the dark matter pair mass. Therefore, in such a situation, the effective operator description would be accurate in the region where the suppression scale of the operators are higher than the involved, or we should discard events from the analysis with larger than the suppression scale of the effective field theory (EFT) [1]. An alternative approach would be to use simplified models including both the dark matter and the mediators as new states to interpret the collider searches, in which case inevitably the number of possibilities to study is large, and for comparison with other constraints on dark matter, we need to perform multiple sets of global analyses. On the other hand, if we focus on the parameter region where we can be model-independent as far as relic density, direct and indirect detections are concerned by using effective operators to describe these processes, the collider searches can be interpreted in terms of simplified models which match onto those effective operators, in the limit of very heavy mediators.

We propose to carry out such a programme for the singlet Majorana fermion dark matter in this study, the end goal being performing a complete likelihood analysis with a proper treatment of all data at hand. Once a one-to-one correspondence between the simplified model parameters and the EFT parameters are established, we can consistently compute the collider cross-sections within the simplified models, and rest of the observables with the corresponding effective operators. The final results will be presented in terms of the EFT parameters. Such an approach does not require any additional effort from the experimental collaborations of high energy colliders, an upper bound on the relevant cross-sections after a set of kinematical cuts can be interpreted within this framework without losing any information from the tail of the missing momentum distributions. At the same time, interpreting the results within the EFT parameters helps us create a global picture of WIMP’s without any prejudice to a particular new physics model.

The SM gauge invariant set of effective operators that describe the interactions of a Majorana fermion WIMP are well studied, and we provide a brief review on the operators in Sec. 2, following our previous study in Ref. [2]. Therefore, the first step in our approach is to write down all possible simplified models that would match onto each of the effective operators when the mediating particles are heavy enough. We carry out this exercise (parts of which have been reported in Ref. [3]) in Sec. 3. The first half of Sec. 4 is devoted to a discussion of all the relevant experimental constraints and the construction of the likelihood function for each. In the second half of this section we first address the question of direct search for the mediators versus the monojet plus missing momentum search at the LHC, and then go on to discuss the results of our likelihood analysis, where the latter include only the monojet constraints for reasons explained in that section. We summarize our study in Sec. 5, with a view towards the role of future experiments in probing the parameter space that survive all current constraints.

2 Effective operators in dark matter interactions

In this section, we briefly review the effective field theory of a Majorana fermion dark matter (DM) in which the low-energy degrees of freedom consist of the SM fields and the DM field [2]. The EFT is described by the following Lagrangian in general:


where is the renormalizable SM Lagrangian, is the bare DM mass parameter before the electroweak symmetry breaking, and represent operators of mass dimension describing interactions between the DM and the SM particles, with dimensionless coefficients . We assume that physics behind the EFT is described by a weakly interacting theory and restrict . For such theories, the suppression scale for a tree generated operator corresponds to the mass of a heavy intermediate particle generating .

The set of operators are required to respect the SM gauge symmetry, and an additional symmetry to ensure the stability of the DM particle, under which the DM field is odd while the SM fields are even. If the DM field is a singlet under the SM gauge symmetry, we have eight types of independent operators up to mass dimension-six; all of which are shown in Tab. 1 with , , , , and denoting the Higgs doublet, the quark doublets, the up-type quark singlets, the down-type quark singlets, the lepton doublets and the charged lepton singlets, respectively.1 Here, the subscripts ’’ and ’’ represent flavour indices, while is the covariant derivative acting on the Higgs field . Some comments on the SM gauge invariant operators in this table are in order below:

  • The operators constitute a complete set up to mass dimension six, so that other operators such as the anapole moment, , with being the field strength tensor of the hypercharge gauge boson, can be written as a linear combination of this operator basis using equations of motion.

  • All interactions between the DM and the SM particles are from higher dimensional operators because of the symmetry. Furthermore, all except preserve the CP symmetry.

  • Since the DM is a Majorana fermion, a factor of is included in front of the interaction terms, following standard normalization.

  • The DM particle is not necessarily a pure singlet under the SM gauge symmetry, though the DM field is a singlet. In fact, can mix with another odd SU(2) doublet field (which can be introduced in a simplified model) after the electroweak symmetry breaking, and it generates , and , as shown in the next section.

  • There is an implicit assumption that all suppression scales are of a similar order. We also assume that mass dimension 7 operators play a sub-leading role.

  • Generically, one needs to fix a scale defining the EFT Lagrangian (1), and consider RGE effects on the Wilson coefficients to calculate physical quantities [4]. Since we are assuming a weakly interacting theory behind the EFT, these effects do not give a sizeable contribution. We therefore neglect the RGE effects for simplicity, without inducing large errors for weakly coupled UV theories.

Dim. 5
Dim. 6
Table 1: SM gauge invariant operators for interactions between the DM and the SM particles.

It is convenient to rewrite the Lagrangian (1) using the lowest suppression scale , which makes our numerical analysis easier. The Lagrangian then reads


where the coefficients are defined by . Since the absolute value of is always smaller than that of , the complete model parameter space defined by the original effective Lagrangian (1) is covered by varying all ’s in the region .

3 Connection between the EFT and simplified models

When the physical mass of the DM (denoted by ) and the electroweak scale are much lower than the suppression scales , the EFT can be used to describe non-relativistic DM phenomena such as DM freeze-out in the early universe and DM annihilation in the present universe or its scattering with nuclei. On the other hand, when we consider relativistic DM phenomena such as DM production at the LHC, the EFT description is of limited applicability since a clear separation of scales no longer exists due to a variable sub-process collision energy. In particular, if the energy is close to or higher than the suppression scale of the operators, we cannot describe the process by a truncated effective operator series anymore.

We propose a programme to address this problem, which is based on a specific relation between the EFT operators and the corresponding simplified models which match onto them. The basic idea is to consider a tree-level process mediated by a new heavy particle whose mass is fixed to be the suppression scale , where the operator is generated in the large limit. This can be realized in a simplified model setting, with all the vertices being now from mass dimension-three or four terms. For each operator in Tab. 1, there are two different ways to introduce such a new heavy particle; in one case the new particle is even under the symmetry, and in the other case it’s -odd. The difference between the two corresponds to which process reproduces , an -channel or -channel process. In what follows, we consider each of these ways in further detail for each operator.

3.1 The four-Fermi operators

We first consider the four-Fermi operators with being SM fermions , , , and , as this is a simple case to intuitively understand the relation between the EFT and simplified models.

-even mediator

The four-Fermi operator describes, for example, the process , so that the new heavy -even particle must be bosonic, being exchanged in the -channel. Since the four-Fermi operator has the form of a current-current interaction, this particle should also be a massive vector. Furthermore, the vector particle directly couples to the dark matter current, and thus it must be a singlet under the SM gauge symmetry. We therefore introduce a real singlet Proca field whose mass is :


where the superscript indicates that is even under the symmetry, while the index corresponds to , , , or . The flavour index for the SM fermions has been suppressed for simplicity. The field strength tensor of the Proca field is denoted by , and at this stage, the coupling constants and can take arbitrary values, which will be fixed by the relation between the EFT and this simplified model.

After integrating out from the simplified model Lagrangian (3), we obtain the following effective Lagrangian which involves a non-local interaction term,2


where is the Green’s function (the two-point function) of in the coordinate space and has the following asymptotic form in the limit of large ,


The Lagrangian (4) generates the four-Fermi operator with the coefficient . Since we are interested in tree-level diagrams involving both the DM and the SM particles, physical quantities depend only on the combination of . Hence, each coupling constant can be fixed without loss of generality as


This defines the required relation between the EFT and the simplified model (3).

-odd mediator

The process is reproduced by the -channel exchanges of a new heavy -odd particle, and the new particle must to be bosonic. On the other hand, unlike the previous case, this new -odd bosonic particle can be either a massive scalar or a vector because of the nature of -channel diagrams. Moreover, since the new bosonic field couples to one DM field and one SM fermion field at each vertex, its quantum numbers must be the same as those of the SM fermion. We therefore introduce a complex scalar field () and a complex Proca field simultaneously, where their masses and the quantum numbers are fixed to be and those of , respectively:


The SM gauge indices for , and as well as the flavour index for have been suppressed for the sake of simplicity. The field strength tensor of the new vector field is given by with being the covariant derivative which is common for all of the three fields , and .

After integrating the heavy vector and scaler fields out from the above simplified model Lagrangian (7), we obtain the following effective Lagrangian,


where, as before, and are the Green’s functions of and , respectively. Here, we have neglected the gauge interaction terms of the massive fields to derive the above effective Lagrangian, because they contribute only to operators of mass dimension higher than six. Taking the large limit of the Green’s functions in Eq. (8), we obtain the following four-Fermi operators:


where Fierz transformations have been used to derive the limit.3 Regarding the sign of the equation, the ’’ sign should be used for , and , while the ’’ sign should be used for and . The introduction of both the massive Proca and scalar fields simultaneously is necessary to make the coefficient of the operator take both positive and negative values. Both the coupling constants and can be real, for their phases can be removed by the redefinitions of the fields and . Thus the relation between the EFT and the simplified model (7) can be expressed as


3.2 The operator

We next consider the operator, , involving the DM and the Higgs fields, which plays an important role in DM signals at colliders.

-even mediator

The operator describes, e.g., the process , so that the new heavy -even particle must be a boson connecting the DM current and the Higgs boson current , which is almost the same situation as the previous case. Hence, we introduce a real singlet Proca field with the mass of :


where is the SM covariant derivative acting on the Higgs field . The kinetic term of gives two interactions, and . Because the latter term contributes only to operators of mass dimension more than six, we shall drop it. After integrating the Proca field out from the above simplified model Lagrangian (11), we obtain the following effective Lagrangian,


Taking a large limit of the Green’s function, the operator is obtained with its coefficient being . With the same reasoning as before, the relation between the EFT and the simplified model (11) can be written as


-odd mediator

The process can take place via the -channel exchange of a new heavy -odd particle. Hence, the new particle must be a fermion having the same quantum numbers as those of the Higgs boson in this case. We therefore introduce a Dirac fermion field (denoted by ) with mass in the following simplified model:


where the covariant derivative acting on the new fermion field is exactly the same as the one acting on the Higgs field . After integrating out from the above simplified model Lagrangian (14), we obtain the following effective Lagrangian,


where and are the SU(2) and U(1) gauge fields, respectively, with and being their gauge couplings. Here again we have neglected other terms which contribute only to operators of mass dimension more than six. Then, the Green’s function has the following asymptotic form in the limit of large ,


As a result, the effective Lagrangian (15) has the following asymptotic form in the limit, which involves not only the operator , but also other ones:


where the equation of motion for the DM field, , has been used to obtain the above result. One of the phases of the coupling constants and can be removed by the redefinition of the Dirac field . By imposing the condition that the coefficients of the other operators and are zero, the relation between the EFT and the simplified model (14) is obtained as


The emergence of the operators in Eq. (17) can be interpreted as a consequence of the mixing between and . In fact, describes a vertex between the boson and two fields after the electroweak symmetry breaking (EWSB), and thus the gauge interaction of . The DM particle is no longer a pure singlet under the SM gauge symmetry, but acquires a small doublet component after the EWSB.

3.3 The scalar interaction operator

Here, we consider the dimension-five operator , which is relevant, in the context of colliders, to the decay of the Higgs boson into a pair of DM particles.

-even mediator

This operator describes the process again, and thus the new heavy -even particle must be bosonic. Since it connects the DM and the Higgs scalar operators instead of their currents, we introduce a real singlet scalar field with mass :


Other renormalizable terms in the Lagrangian, which are not relevant in deriving , are not shown here. After integrating the scalar field out from the above simplified model Lagrangian (19), we obtain the following effective Lagrangian:


Taking the large limit, the effective Lagrangian (20) generates the operator with its coefficient being . Thus, the relation between the EFT and the simplified model (19) can be obtained as:


-odd mediator

The process is generated by the -channel exchange of a new heavy -odd particle, and the new particle must be a fermion having the same quantum numbers as those of . We therefore introduce a Dirac fermion field () with mass :


where is the covariant derivative acting on the new fermion field . Since the simplified model (22) is exactly the same as the one in Eq. (14), it gives rise to the three higher dimensional operators , and after integrating the heavy field out from the simplified model Lagrangian (22) and taking a large limit. As a result, by imposing the coefficients of the latter two operators to be zero, the relation between the EFT and the simplified model (22) is obtained as


3.4 The pseudoscalar interaction operator

Finally, we consider the other (CP violating) dimension-five operator .

-even mediator

It is essentially the same as in the case of the scalar operator in the previous subsection. We therefore introduce a real singlet scalar field whose mass is fixed to be , and use the pseudoscalar interaction instead of the scalar one :


After integrating out from the Lagrangian above and taking a large limit, we obtain the dimension-five operator with coefficient . Thus, the relation between the EFT and the simplified model (24) is obtained as


-odd mediator

This is again similar to the case of the scalar operator , and we therefore introduce a Dirac fermion field () again with its mass being :


It generates the three higher dimensional operators in exactly the same manner as before. By imposing the coefficients of the operators and to be zero, the relation between the EFT and this simplified model is obtained as


3.5 Some caveats to using simplified models

In UV completions of models with vector mediators, one needs to consider the questions of gauge invariance, the origin of the vector boson mass, perturbative unitarity of scattering amplitudes and cancellation of anomalies. For an s-channel UV completion with a new gauge group (and a massive boson), a detailed analysis has been carried out in Ref. [6]. However, with our EFT assumption of , the process leading to strong unitarity constraint is kinematically forbidden, where refers to the longitudinally polarized state of . From unitarity of and similar 2-to-2 interactions between pairs of SM fermions () and , one again finds the constraint , both of which are satisfied for the region of our interest , 0.3 TeV] (see next section for details on the minimum value of ). The only constraint that applies therefore is that of gauge invariance under the new , when left and right handed SM fermions have different charges under the new gauge group. In this case, the SM Higgs doublet needs to be charged under as well, implying that if the SM quarks are charged, so are the SM leptons, leading to stronger constraints from LHC dilepton resonance searches. Such a conclusion can however be avoided if the SM charged lepton and quark masses are generated by two different Higgs doublets, thereby making their charges uncorrelated. In the spirit of this study, we would resort to such an UV completion, in order not to over-constrain the general picture for DM. To summarize, specific UV models can be more constrained by the requirements of gauge invariance or unitarity, however, in the domain of validity of the EFT, unitarity constraints are satisfied, and gauge invariance can be achieved with, for e.g., a non-minimal Higgs sector in order to avoid stronger dilepton constraints. Moreover, as is well known, unless we consider very specific charge assignments, there would be additional gauge anomalies in the theory, and to cancel such anomalies we have to introduce new heavy quarks which are vector-like under the SM gauge group, but are chiral under the new . Such new quarks can be taken to be heavier than , and their effect on DM phenomenology is not expected to be generically significant. For the -odd vector boson mediator, the UV completion needs more model building, where we either have to embed the DM particle and the SM fermions in a multiplet of a non-abelian gauge group, or look for an extra-dimensional or Little Higgs type model with an exact KK or T-parity. In the latter two cases, the -odd partner of a right-handed neutrino, for example, can be the fermionic DM candidate.

The simplified models with scalar mediators involve only new Yukawa-type interactions, and they therefore can be considered UV complete. It should however be ensured that the complete scalar potential, which is not relevant to this study, does not develop any charge or colour breaking minimum. Since we did not consider any mixing between the singlet and doublet scalars (which would induce dimension 7 operators), charge breaking minimum is not expected to develop.

Finally, certain SM observables are also sensitive to the UV completion of the DM simplified models we consider, one example being the production cross-section and partial decay widths to SM states of the Higgs boson. Even if the mediators introduced are sufficiently heavy compared to the electroweak scale, they can modify these quantities and only a global fit of all current LHC Higgs data can determine whether such deviations are permissible at present. This is not a generic effect though, as for example, the -even mediator for the scalar interaction operator in Sec. 3.3 can induce such modifications via the mixing of the singlet scalar with the neutral CP-even Higgs field after the electroweak symmetry breaking, but the -odd mediator mixing with the singlet DM state will not have any such effect at the leading order. Even for the -even mediator case, a UV completion can be designed to cancel such modifications predicted by the minimal DM simplified models. Therefore once again, in order not to over-constrain the generic DM picture, we assume the SM predictions for such processes are unaltered in the complete UV theory. The same approach also applies to the four-fermion SM operators generated in the simplified models for DM-SM fermion interactions (at leading order, in the s-channel models), which can furnish strong constraints especially for the first generation quark and charged lepton couplings of the DM, in the absence of accidental cancellations or symmetry protection.

4 Application to singlet-like Majorana fermion DM

We are in a position to apply the method in the previous section to a singlet-like Majorana fermion DM, in order to determine the viable regions, in the space spanned by the DM mass , the suppressions scales and the coupling constants . Even in the absence of flavour changing interactions of the form with , we have 37 parameters (19 parameters if we only use the EFT [7]) to deal with, which is beyond the scope of our numerical analysis. Hence, we need to impose the following simplifying assumptions before attempting a scan of the entire parameter space:

  • (unless otherwise stated): All the intermediate heavy particles introduced in the simplified models are assumed to have a common mass .

  • : In order to describe DM pair annihilation by the EFT, we must have (note that induces error in the theoretical prediction based only on dimension-6 operators for s-channel UV completions). Furthermore, to reduce the error in the prediction of the invisible decays of the or the Higgs boson using the leading effective operators, we need , where is the scale of electroweak symmetry breaking. Therefore, we choose the minimum value of being consistent with these two requirements.

  • : The four-Fermi operators have a flavour-blind structure, i.e., does not depend on the flavour indices. On the other hand, SM fermions belonging to different representations are allowed to have different couplings with the DM.

  • : This reflects the implicit assumption followed throughout this paper: the UV physics behind the EFT is described by a weakly coupled theory.

  • : CP symmetry is assumed to be preserved in the interactions of the DM particle with the SM particles.

Under these conditions, the number of free parameters reduces to 9 (, , , , , , , and ), which is within the scope of our analysis. Even though this is a limited scan of the most general parameter space, it includes sufficient degrees of freedom to grasp the broad picture of a singlet-like Majorana fermion DM scenario. We should mention that if the CP violating pseudoscalar coupling is non-zero, the phenomenology changes considerably as observed in our previous study [2], and in contrast to the CP-conserving case, indirect detection experiments become more relevant.

In order to explore the high probability density regions of the multi-dimensional parameter space, we employ the profile-likelihood method [8]. All the relevant experimental and observational constraints are incorporated in the likelihood function taking into account their statistical and systematic uncertainties. The likelihood function that we adopt in our analysis is composed of three parts:


where these components involve information obtained from DM cosmology, DM detection experiments and collider experiments, respectively. In the following subsections, we shall describe each component in detail. In particular, we carefully discuss how the relation between the EFT and the simplified models is applied to calculate the component . We adopt the MultiNest sampling algorithm [9], which is an efficient implementation of the Markov Chain Monte Carlo algorithm [10]. While describing our results in the relevant set of two-dimensional parameter space, e.g., the -space, we maximize the likelihood function along the directions of the other parameters. Our scan of the parameters spans the following ranges, 10 GeV 5 TeV and 100 TeV, which are determined based on our previous study [2]. We use a flat prior for all the operator coefficients, while both flat and log priors are used for different regions in and , in order to obtain as large a coverage of the whole parameter space as possible.

4.1 Constraints from DM cosmology

We adopt the following Gaussian likelihood function in our analysis: