Exploring Fermionic Dark Matter via Higgs Precision Measurements at the Circular Electron Positron Collider
Abstract
We study the impact of fermionic dark matter (DM) on projected Higgs precision measurements at the Circular Electron Positron Collider (CEPC), including the oneloop effects on the cross section and the Higgs boson diphoton decay, as well as the treelevel effects on the Higgs boson invisible decay. As illuminating examples, we discuss two UVcomplete DM models, whose dark sector contains electroweak multiplets that interact with the Higgs boson via Yukawa couplings. The CEPC sensitivity to these models and current constraints from DM detection and collider experiments are investigated. We find that there exit some parameter regions where the Higgs measurements at the CEPC will be complementary to current DM searches.
pacs:
12.15.Lk,12.60.Cn,13.66.Jn1.2
Contents
I Introduction
The discovery of the Higgs boson at the Large Hadron Collider (LHC) Aad:2012tfa ; Chatrchyan:2012xdj confirms the particle content of the standard model (SM). However, the existence of dark matter (DM) Jungman:1995df ; Bertone:2004pz ; Feng:2010gw undoubtedly implies the new physics beyond the SM (BSM). While searches for new particles at the LHC will continue in the coming years, an alternative way to probe new physics is by studying its loop effects via high precision observables at colliders.
Several electronpositron colliders have been currently proposed, including the Circular Electron Positron Collider (CEPC) CEPCSPPCStudyGroup:2015csa , the Future Circular Collider with collisions (FCCee) GomezCeballos:2013zzn , and the International Linear Collider (ILC) Baer:2013cma . These machines are planned to serve as “Higgs factories” for precisely measuring the properties of the Higgs boson. In particular, CEPC will run at a centerofmass energy of , which maximizes the production, over ten years to collect a data set of .
Exploiting the physics potential of the CEPC has attracted many interests. Recent works for probing anomalous couplings include studies on the anomalous and couplings through the measurement McCullough:2013rea ; Shen:2015pha ; Huang:2015izx ; Kobakhidze:2016mfx , the anomalous and couplings through the measurement Cao:2015iua ; Hu:2014eia , and the anomalous coupling Gori:2015nqa , and high order effective operators Fedderke:2015txa ; Ge:2016zro . Other CEPC researches about new physics models involve studies on natural supersymmetry Fan:2014axa ; Cao:2016uwt ; Wu:2017kgr , DM models Yu:2013aca ; Yu:2014ula ; Harigaya:2015yaa ; Cao:2016qgc ; Xiang:2016jni and electroweak oblique parameters Fedderke:2015txa ; Cai:2016sjz ; Cai:2017wdu , and so on Cao:2014ita .
In this work, we mainly study the impact of fermionic DM on the Higgs physics at the CEPC. Particularly, we focus on the loop effects on the production cross section, whose relative precision will be pinned down to CEPCSPPCStudyGroup:2015csa . For this purpose, the DM particle should couple to both the Higgs and bosons and modify the coupling at oneloop level. This requirement can be fulfilled by introducing a dark sector consisting of electroweak multiplets, which is a simple, UVcomplete extension to the SM. Such a dark sector would provide an attractive DM candidate that naturally satisfies the observed relic abundance. Related Model buildings typically involve one multiplet, which leads to the socalled minimal DM models Cirelli:2005uq ; Cirelli:2007xd ; Cirelli:2009uv ; Hambye:2009pw ; Cai:2012kt ; Ostdiek:2015aga ; Cai:2015kpa , or more than one multiplet Mahbubani:2005pt ; DEramo:2007anh ; Enberg:2007rp ; Cohen:2011ec ; Fischer:2013hwa ; Cheung:2013dua ; Dedes:2014hga ; Fedderke:2015txa ; Calibbi:2015nha ; Freitas:2015hsa ; Yaguna:2015mva ; Tait:2016qbg ; Horiuchi:2016tqw ; Banerjee:2016hsk ; Cai:2016sjz ; Abe:2017glm ; Cai:2017wdu ; Arcadi:2017kky ; Maru:2017otg ; Liu:2017gfg ; EganaUgrinovic:2017jib . As we would like to discuss fermionic DM, more than one multiplet is needed for allowing renormalizable couplings to the Higgs boson with respect to the gauge invariance.
We calculate oneloop corrections to contributed by the dark sector. For the purpose of illustration, we study two simple models with additional fermionic multiplets:

Singletdoublet Fermionic Dark Matter (SDFDM) model: the dark sector involves one singlet Weyl spinor and two doublet Weyl spinors;

Doublettriplet Fermionic Dark Matter (DTFDM) model: the dark sector involves two doublet Weyl spinor and one triplet Weyl spinors.
These spinors are assumed to be vectorlike, in order to cancel gauge anomalies. This means that the two doublets should have opposite hypercharges, while the singlet or the triplet should have zero hypercharge.
Models  Gauge eigenstates  Mass eigenstates 

SingletDoublet  , ,  
DoubletTriplet  , , 
After electroweak symmetrybreaking (EWSB), the vacuum expectation value (VEV) of the Higgs doublet provides Dirac mass terms to the dark multiplets, leading to state mixings. Field contents in the gauge and mass bases for the two models are denoted in Table 1. The lightest neutral eigenstate () in the dark sector serves as a Majorana DM candidate. For ensuring the stability of , we need to impose a symmetry, under which all SM particles are even and dark sector particles are odd. These models can be regarded as the generalizations of some electroweak sectors in supersymmetric models. For instance, the SDFDM model is similar to the binoHiggsino sector, while the DTFDM model is similar to the Higgsinowino sector.
Serving as a DM candidate, should be consistent with the observed DM relic abundance Planck:2015xua . The couplings to the and Higgs bosons could induce spindependent and spinindependent scatterings between nuclei and DM, respectively. They would be constrained by direct detection experiments Tan:2016zwf ; Fu:2016ega . Besides, there are bounds from colliders experiments, such as bounds from the invisible decay of the boson ALEPH:2005ab , from searches for charged particles at the LEP, and from the monojet searches at the LHC Aad:2015zva . Moreover, dark sector particles may affect the invisible and diphoton decays of the Higgs boson, which will be precisely determined by CEPC CEPCSPPCStudyGroup:2015csa . In this work, we investigate both the CEPC prospect and current experimental constraints for the two DM models.
The paper is outlined as follows. In Sec. II we give a brief description of the SDFDM model, identify the parameter regions that could be explored by Higgs measurements at the CEPC, and study current constraints from DM detection and collider experiments. In Sec. III, we repeat the calculations, but for the DTFDM model. Sec. IV contains our conclusions and discussions.
Ii SingletDoublet Fermionic Dark Matter
ii.1 Model details
In the SDFDM model Mahbubani:2005pt ; DEramo:2007anh ; Enberg:2007rp ; Cohen:2011ec ; Cheung:2013dua ; Calibbi:2015nha ; Horiuchi:2016tqw ; Banerjee:2016hsk ; Cai:2016sjz ; Abe:2017glm , we introduce a dark sector with one Weyl singlet and two Weyl doublets obeying the gauge transformations:
(1) 
Here, the assignment of opposite hypercharges to the two doublets is essential to cancel the gauge anomalies. We can write down the following gauge invariant Lagrangians:
(2)  
(3) 
where , with the generators expressed by the Pauli matrices . More specifically, gauge interactions of the doublets are given by
(4)  
where and are related to the Weinberg angle . The dark sector fields interact with the SM Higgs doublet through the Yukawa couplings
(5) 
After the EWSB, dark sector fermions obtain Dirac mass terms through the Higgs mechanism. In the unitary gauge, with the VEV . The mass terms in the model can be expressed as
(6) 
where , , and . The mass matrix of the neutral states and the corresponding mixing matrix to diagonalize it are given by
(7) 
Thus, the dark sector contains one charged Dirac fermion and three Majorana fermions , with the lightest neutral fermion serving as the DM particle.
This model is totally determined by four parameters, , , , and . In principle, all of them could be complex and induce CP violation. However, three phases can be eliminated by redefinition of the fields, leaving only one independent CP violation phase. The effects of this CP violation phase on electric dipole moments and on DM direct detection have been studied by several groups Mahbubani:2005pt ; DEramo:2007anh ; Abe:2017glm . We do not discuss these effects further, and take all parameters to be real below.
In Fig. 1 we show the masses of the dark sector fermions as functions of with for two typical cases, and . If , is singletdominated, with a mass close to when and are small; and are doubletdominated, with masses close to for small Yukawa couplings. On the other hand, if , and are doubletdominated, while is singletdominated. When , we have or due to a custodial symmetry.
It is instructive to reform the interaction terms with fourcomponent spinors. Defining Dirac spinor and Majorana spinors () as
(8) 
we have
(9)  
where and . The couplings to and are given by
(10)  
(11) 
It is obvious to find that , due to the Majorana nature of . Since and are real parameters, the CPviolating couplings also vanish. For DM phenomenology, the and couplings are particularly important, inducing spindependent (SD) and spinindependent (SI) DMnucleon scattering, respectively. Therefore, they could be probed in direct detection experiments.
When , there is a custodial global symmetry resulting and a vanishing SD scattering cross section. Besides, if , the condition also leads to and a vanishing SI cross section Cai:2016sjz . It would be useful to explore other conditions that give rise to , which implies blind spots in direct detection experiments Cohen:2011ec ; Cheung:2012qy ; Cheung:2013dua ; Abe:2017glm . According to the lowenergy Higgs theorems Ellis:1975ap ; Shifman:1979eb , the couplings of the neutral fermions to the Higgs boson can be derived by the replacement in the DM candidate mass :
(12) 
which means Cohen:2011ec ; Cheung:2011aa .
satisfies the characteristic equation , which is just
(13) 
Differentiating its lefthand side with respect to and imposing , one obtain the condition that leads to is
(14) 
Plugging this condition into Eq. (13), one obtains
(15) 
Thus, the latter equation could also induce when .
ii.2 Higgs Precision Measurements at the CEPC
ii.2.1 Corrections to the associated production
The associated production is the primary Higgs production process in a Higgs factory with . For the measurement of its cross section, a relative precision of is expected to be achieved at the CEPC with an integrated luminosity of CEPCSPPCStudyGroup:2015csa . Below we discuss the impact of the SDFDM model on this cross section at oneloop level.
Neglecting the extremely small coupling, the only treelevel Feynman diagram for in the SM is shown in Fig. 2. It involves the coupling, whose precise strength is a chief goal of a Higgs factory. BSM particles that couple to both the and Higgs bosons, such as the Majorana fermions , are presumed to modify this coupling via triangle loops, as demonstrated in Fig. 3(a). Besides, Figs. 3(b) and 3(c) show that dark sector fermions in the SDFDM model can also affect the propagator in the diagram at oneloop level. Moreover, the dark sector contributes to the selfenergies of the Higgs boson and the electroweak gauge bosons, and hence influences the determination of the related renormalization constants. In practice, these contributions must be included to cancel the ultraviolet divergences from Fig. 3.
Formally, the cross section can be split into two parts:
(16) 
where is the SM prediction, while is the contribution due to BSM physics, which, in our case, is the dark sector multiplets. The nexttoleading corrections to in the SM have been calculated two decades ago Fleischer:1982af ; Kniehl:1991hk ; Denner:1992bc ; DENNER1992263 , while the mixed electroweakQCD () corrections have been studied in 2016 Gong:2016jys ; Sun:2016bel . Here we calculate with oneloop corrections except for the virtual photon correction. Thus, we would not need to involve the real photon radiation process for dealing with soft and collinear divergences. This treatment should be sufficient for our purpose, as we are only interested in the relative deviation of the cross section due to the dark sector.
We utilize the packages FeynArts 3.9 Hahn:2000kx , FormCalc 9.4 Hahn:1998yk , and LoopTools 2.13 vanOldenborgh:1990yc to calculate oneloop corrections from the SM and from the SDFDM model at . The onshell renormalization scheme is adopted to fix the renormalization constants. Fig. 5 shows the relative deviation of the cross section as a function of . Other parameters are chosen to be and , leading to . The deviation could be either positive or negative, depending on the parameters. As increases to the TeV scale, the deviation becomes very small, because the dark sector basically decouples.
When the dark sector fermions in the loops are able to close to their mass shells, their contributions could vary dramatically. In the lower frame of Fig. 5 shows the sums of fermion masses in order to demonstrate the mass threshold effects with , , , and . For instance, would allow a new decay process, ; this means that the boson selfenergy develops a new imaginary part, which is absent for . As a result, reaches a dip at . Similarly, we have threshold effects with and . In addition, the threshold effect with is caused by the triangle loop in Fig. 3(a), because also leads to a imaginary part in the amplitude of the triangle loop.
In Fig. 6, we show heat maps for the absolute relative deviation in the SDFDM model with two parameters fixed. The regions with colors have sufficient deviations that could be explored by the CEPC measurement of the cross section, while the gray regions are beyond its capability. The complicated behaviors of these heat maps can be attributed to mass threshold effects, as shown in Fig. 5.
ii.2.2 Higgs boson invisible decay
If the dark sector fermions are sufficient light, the Higgs boson and the boson would be able to decay into them. When such decay processes are kinematically allowed, their widths are given by ( in the expressions below)
(17)  
(18)  
(19)  
(20)  
(21) 
where .
Since cannot be directly probed by detectors in collider experiments, the decay processes and are invisible. On the other hand, if and decay into other dark sector fermions, the symmetry will force them subsequently decay into associated with SM particles in final states. Such and decays may also be invisible due to . Moreover, when these decay processes are allowed, the SM products would probably be very soft, as the related mass spectrum in the dark sector should be compressed. As a result, they could be effectively invisible. Therefore, the invisible decays of and provide another promising approach to reveal the dark sector.
With an integrated luminosity of , CEPC is expected to constrain the branching ratio of the invisible decay down to at 95% CL CEPCSPPCStudyGroup:2015csa . As the Higgs boson width in the SM is 4.08 MeV for Heinemeyer:2013tqa , this means that the expected constraint on the Higgs invisible decay width is . On the other hand, LEP experiments have put an upper bound on the invisible width, which is at CL ALEPH:2005ab .
In Fig. 7 we present the expected CEPC constraint and the LEP constraint from the invisible decays of the Higgs boson and the boson, respectively. We have included all allowed decay channels into the dark sector as invisible decays for the reasons we mentioned above. Although this treatment overestimates the invisible decay widths, it actually closes to the most conservative estimation that only takes into account and , because in most of the parameter regions we are interested in only one or a few of these decay channels would open. From Fig. 7, we can see that the expected CEPC constraints from the Higgs invisible decay are basically stronger than the LEP constraint from the invisible decay. Exceptions happen mostly when . In such a region, the decay is allowed, while the coupling for could be small, or even vanishes if .
ii.3 Current experimental constraints
In this subsection, we investigate current experimental constraints on the SDFDM model. Relevant bounds come from the observation of DM relic abundance, DM direct detection experiment, LHC monojet searches, and LEP searches for charged particles. Below we discuss them one by one.
ii.3.1 Relic abundance
The observed cold DM relic density reported by the Planck collaboration is Planck:2015xua . Assuming DM particles were thermally produced in the early Universe, the relic density is determined by their thermally averaged annihilation cross section into SM particles when they decoupled. If the annihilation cross section is too small, DM would be overproduced, contradicting the observation.
The freezeout temperature is controlled by the DM particle mass, which is in the SDFDM model. However, other dark sector fermions may have masses similar to . For instance, could lead to a doubletdominated , whose mass can be very close to and . As a result, coannihilation processes among the dark sector fermions could be important and significantly influence the DM relic abundance. For this reason, we take into account the coannihilation effect when the mass differences are within . We adopt MadDM Backovic:2013dpa , which is based on MadGraph 5 Alwall:2014hca , to calculate the relic density involving all annihilation and coannihilation channels. The model is implemented with FeynRules 2 Alloul:2013bka .
The parameter regions where DM is overproduced are indicated by red color in Fig. 8. For a DM candidate purely from the doublets, the observed relic abundance corresponds to a DM particle mass of Cirelli:2005uq . The mixing with the singlet complicates the situation. Nonetheless, Figs. 8(a) and 8(b) still show that the observation favors . Annihilation through a or resonance would significantly increase the cross section and hence reduce the relic density. This effect results in the bands of underproduction among the overproduction regions in Figs. 8(a) and 8(b).
Fig. 8(a) also has a overproduction region with , due to lacking of effective annihilation mechanisms. In this region, forbids the annihilation into weak gauge bosons, while the annihilation into SM fermions is helicitysuppressed and the coannihilation effect with is insufficient. A similar region dose not show up in Fig. 8(b), because in this case leads to a significant coannihilation effect. Figs. 8(c) and 8(d) demonstrate the complicate overproduction regions depending on the Yukawa couplings for specified mass parameters of the dark sector.
ii.3.2 DM direct detection
The and couplings could induce spindependent (SD) and spinindependent (SI) DMnucleon scattering, respectively. Therefore, the model is testable in direct detection experiments. MadDM Backovic:2015cra is used to calculate the DMnucleon scattering cross sections. We also present the results in Fig. 8, with blue and orange regions excluded at 90% CL by the PandaX experiment for SI interactions Tan:2016zwf and for SD interactions Fu:2016ega , respectively.
As in this model SI and SD interactions have different origins, their effects are comparable and complementary in direct detection experiments, as shown in Fig. 8(a), 8(c), and 8(d). When , the coupling vanishes, and thus there is no SD exclusion region in Fig. 8(b). Moreover, as and lead to a vanishing coupling, no SI constraint is available in the related regions of Figs. 8(b) and Fig. 8(d).
In Fig. 8(c), the model is severely constrained by DM direct detection. Exceptions occur when the coupling happens to vanish. For and , from Eq. 15 we know vanishes when or . This explains a region free from SI direct detection in Fig. 8(c). However, taking into account the constraints from SD direct detection and from the relic abundance, however, there is no blind spot left.
ii.3.3 LHC and LEP searches
Searching for direct production of dark sector fermions at high energy colliders, like LHC, is another way to reveal the SDFDM model. Due to the symmetry, dark sector fermions must be produced in pairs and those other than eventually decay into . Consequently, a large missing transverse energy () is a typical signature for such production processes. The channel could effectively probe the pair production associated with one or two hard jets from the initial state radiation. Other dark sector pair production processes could also contribute to the final state if the mass spectrum is compressed. Therefore, we should consider the following electroweak production processes for the monojet searches at the LHC:
(22) 
We utilize MadGraph 5 Alwall:2014hca to simulate these production processes. PYTHIA 6 Sjostrand:2006za is adopted to deal with particle decay, parton shower, and hadronization processes. Delphes 3 deFavereau:2013fsa is used to carry out a fast detector simulation with a setup for the ATLAS detector. The same cut conditions as in the ATLAS analysis with of data at Aad:2015zva are applied to the above production signals in the SDFDM model. By this way we reinterpret the experimental result to constrain the model.
In Fig. 8, the green regions are excluded by the search, based on our reinterpretation. Figs. 8(a) and 8(b) show that the monojet search can exclude the parameter space up to . The exclusion regions hardly show dependence on , as the singlet components in do not contribute to the production processes mediated by electroweak gauge bosons. In Fig. 8(c) with , the monojet search only rules out four tiny parameter regions, because in this case is singletdominated, leading to a very low production rate for .
The charge fermion has similar properties as the charginos in supersymmetric models. For a rough estimation, we treat the LEP bound on the chargino mass, Abdallah:2003xe , as a bound on . As a result, the pink regions with in Figs. 8(a) and 8(b) are excluded. It seems that this constraint is stronger than the monojet search at the 8 TeV LHC.
Iii DoubletTriplet Fermionic Dark Matter
In the previous section, we find that current constraints on the SDFDM model are quit severe. As a result, most of the CEPC sensitive region has already been excluded. Actually, the singlet does not have electroweak gauge interactions, so the modification to the cross section would not be very significant. This observation inspires us to replace the singlet with a triplet, leading to the DTFDM model. This model should be more capable to affect the cross section. In this section, we discuss its impact on Higgs measurements at the CEPC and current constraints on its parameter space.
iii.1 Model details
In the DTFDM model, two Weyl doublets and one Weyl triplet are introduced Dedes:2014hga ; Cai:2016sjz :
(23) 
We have the following gauge invariant Lagrangians:
(24)  
(25) 
where the constants render the gauge invariance of the term. can be derived from ClebschGordan coefficients multiplied by a factor to normalize mass terms for the components of . The nonzero values are
(26) 
Since the hypercharge of the triplet is zero, its covariant derivative is , where are generators of the representation for the group that are chosen as
(27) 
Any irreducible representation is real, in the sense that it is equivalent to its conjugate. This equivalence means that one can find an invertible matrix satisfying . For the generators we choose, is defined as
(28) 
We can use the charge conjugation matrix to define the conjugate of the triplet as , which transforms as a vector in , rather than in . In this work, we would like to study a real triplet, which means that . This is the reason why there is a minus sign in front of the third component of in Eq. (23).
The gauge interactions of the doublets have been explicitly listed in Eq. (4), while the gauge interactions of the triplet are given by