Implications of the Little Higgs Dark Matter and T-odd fermions

We study the phenomenology of dark matter in the Littlest Higgs model with T-parity after the discovery of Higgs boson. We analyze the relic abundance of dark matter, focusing on the effects of coannihilaitons with T-odd fermions. After determining the parameter space that predicts the correct relic abundance measured by WMAP and Planck collaborations, we evaluate the elastic scattering cross section between dark matter and nucleon. In comparison with experimental results, we find that the lower mass of dark matter is constrained mildly by LUX 2013 while the future XENON experiment has potential to explore most of the parameter space for both T-odd lepton and T-odd quark coannihilation scenarios. We also study the collider signatures of T-odd fermion pair production at the LHC. Even though the production cross sections are large, it turns out very challenging to search for these T-odd fermions directly at the collider because the visible charged leptons or jets are very soft. Furthermore, we show that, with an extra hard jet radiated out from the initial state, the T-odd quark pair production can contribute significantly to mono-jet plus missing energy search at the LHC.


JHEP06(2014)074
We found that it is possible to explain the measurement of dark matter relic abundance with ν H dark matter. However, the direct search experiment excludes such a possibility. The reason is that the coupling strength of ν H to Z-boson is similar to that of Z-boson to SM fermions, therefore, the cross section of elastic scattering between ν H and nucleus is about 4 ∼ 5 order of magnitude larger than the current experimental search bound. This situation has been also noticed in the case of KK-neutrino dark matter in Universal Extradimensional model [37]. Hence, we will focus on A H dark matter in our study. To predict the relic abundance of dark matter, we should calculate the dark matter annihilation cross section. The pair annihilation of two A H dark matter is mainly through the Higgs boson in the s-channel to SM particle final states. With a 125 GeV Higgs boson, the annihilation cross section of a pair of A H is too small to fit the measurement of dark matter relic density (eq. (1.1)) unless the mass of A H is very close to half of the Higgs boson mass in order to have resonance enhancement. Such restriction can be lifted when various coannihilation channels are included [38].In this paper, we take coannihilation processes into account and explore the effects in direct search experiments and implications of LHC phenomenology.
The rest of paper is organized as follows. We begin in section 2 with a brief introduction of the Littlest Higgs model with T-parity. In section 3, we study the coannihilation with light T-odd fermions, separately for T-odd leptons and for T-odd quarks. We first determine the parameter space that can fit the current measurement of dark matter relic abundance, and then calculate the dark matter elastic scattering with nucleus. The LHC study is present in section 4. We calculate the production cross section of T-odd fermion. We point out that the T-odd fermion production contribute significantly to the mono-jet with missing energy search for dark matter at the LHC. Our conclusion appears in section 5.

Littlest Higgs model with T-parity
The Littlest Higgs model is a SU(5)/SO(5) nonlinear sigma model where Higgs boson is a pseudo Nambu-Goldstone boson. The global symmetry SU (5) is broken down to SO(5) by a 5 × 5 symmetric tensor Σ 0 at the scale f , where sector, the T-odd fermion is assigned as ψ H = (ψ 1 + ψ 2 )/ √ 2, while the SM fermion is ψ SM = (ψ 1 − ψ 2 )/ √ 2 that is even under T-parity transformation, where ψ 1,2 are doublet under SU (2) 1,2 . As a result, there is no mixing between heavy T-odd particles and SM particles and the all the tree-level contributions to electroweak precision tests are forbidden. The stringent constraints on scale f can be relaxed and the model is then testable at the LHC.
To cancel the one-loop quadratic divergence indued by SM top quark in Higgs boson mass correction, another set of singlet U 1 and U 2 is introduced in such a way that a new T-even particle T + = (U 1 − U 2 )/ √ 2 exactly cancels the contribution of SM top quark loop. And the other combination T − = (U 1 + U 2 )/ √ 2 is odd under T-parity. The masses of these new gauge bosons and fermions, including the corrections after SM electroweak symmetry is broken, are where g and g ′ are respectively the gauge couplings for SM SU (2) L and U (1) Y , κ q , κ ℓ , λ 1 and λ 2 , 1 are free parameters in the Lagrangian that generates masses for heavy fermions.
Here, u H (d H ) denote the T-odd partners of the SM up-(down-)type quarks, ν H are T-odd partners of the neutrinos and ℓ H are T-odd partners of the charged leptons. Note that the masses of heavy gauge bosons depend on f only, and T-odd fermions rely on an additional parameter κ q or κ ℓ . For simplicity, we assume the universal κ ℓ (κ q ) for T-odd partners of three generations of leptons (quarks) of SM unless otherwise specified. We can see that, when κ q is smaller than 0.11, the up-type T-odd quarks become the lightest T-odd particle and are stable, which conflicts with results of dark matter searches. For κ ℓ 0.11, T-odd ν H can be the dark matter candidate. However, such a possibility is excluded by the direct searches as we mentioned previously. Therefore, in the following studies, we take both κ q and κ ℓ to be larger than 0.11. We should refer readers who are interested in details of the Littlest Higgs model with T-parity to ref. [10][11][12].

Relic abundance and direct detection
In this section, we study the dark matter A H in Littlest Higgs Model with T-parity (LHT) in comparison with current dark matter experiments. We will first calculate the relic abundance and then the direct detections. All the calculations shown here are performed with MicrOMEGAs3.1 package [39]. The parameter space of (κ ℓ , m AH ) and also (m νH − m AH , m AH ) that predicts dark matter relic abundance that is consistent with observation eq. (1.1) within 1σ, 3σ, 5σ, shown respectively in black, blue and red lines. The thick black line shows lower limit of f from searches of heavy charged particles at LEP II. Right: The one-nucleon-normalized spin-independent elastic cross-section of heavy photon A H scattering off the nucleon together with the current experimental limits of XENON100 (2012) [49], LUX(2013) [46]. The projective limits of LUX(2014), XENON 1T [50], and XENON10T [51,52] are also shown.
The relic density of dark matter today is determined by its annihilation processes in the non-relativistic limit. In LHT, a pair of A H mainly annihilate through a Higgs boson as the mediator in s-channel to a pair of b quarks, W/Z bosons, Higgs bosons or top quarks, depending on the mass of A H [14,33,34]. When A H is lighter than W -boson, it annihilates to b quarks, however, when W -boson channel is open, a pair of W -boson final state is always dominant. With Higgs boson mass m H = 125 GeV, A H has to be heavier than m H /2 = 62.5 GeV so that the decay of Higgs boson into a pair of A H is kinematically forbidden. Otherwise, if the channel H → A H A H is opened, it always dominates the Higgs boson decay branching ratios and conflicts with the current limit of invisible decay branching ratio of Higgs boson [40][41][42][43][44][45]. In our study, we vary f between ∼ 480 GeV and ∼ 2 TeV which corresponds to 62.7 GeV m A H 312.4 GeV.
In the case that A H is much lighter than other new heavy particles, in order to have right pair annihilation cross section to fit the current dark matter relic abundance measurement, cf. eq. (1.1), the mass of A H is required to be just slightly heavier than half of Higgs boson mass m A H m H /2 = 62.5 GeV. However, the A H -nucleon elastic scattering cross section is about 10 −9 pb that is in tension with current result form LUX [46] and JHEP06(2014)074 will be certainly examined by projected LUX 2014 data. When A H becomes heavier, A H pair annihilation cross section drops quickly and the corresponding relic abundance will be too large to agree with the observation. One possible solution to enlarge the dark matter annihilation cross section is to include coannihilation effects with T-odd fermions [47]. For coannihilation to take place, T-odd fermions should be as light as dark matter A H . We will demonstrate the coannihilation of T-odd leptons and T-odd quarks separately.
When T-odd leptons participate in coannihilation processes, one more parameter κ ℓ needs to be considered in addition to f . We show in left panel of figure 1 the parameter space of κ ℓ and f that predicts the right dark matter relic abundance together with a mass difference between T-odd lepton ν H and dark matter A H for demonstration. The black, blue and red curves denote that the relic abundance is consistent with observation within 1σ, 3σ, and 5σ level. We also see that, when coannihilation occurs, the number density of T-odd lepton is about e − ∆m T ∼ 35% of that of dark matter A H when m A H = 300 GeV. Here, we take the freeze-out temperature T ∼ m A H /20. The value of κ ℓ is bounded from below by 0.11 below which the A H is no longer the LTP and can not serve as the dark matter candidate. In calculations, we set all of the T-odd quarks to be much heavier than A H and are irrelevant in annihilation cross section. The nearly vertical narrow band in the left part of the plot corresponds to the case that m A H is just slightly heavier than m H /2 where the resonance effect is significant. However, due to narrow width of Higgs boson, when A H becomes heavier, A H pair annihilation cross section drops quickly and significant contributions of coannihilation processes involving light T-odd leptons are needed to enlarge the total annihilation cross section of dark matter. When A H is heavier than W -boson, pair annihilation cross section becomes larger since A H A H → W + W − is opened. Therefore, the contribution from T-odd lepton coannihilation becomes less important than the previous case. As a result, we see that κ ℓ goes higher, meaning the mass gap between the T-odd lepton and A H is larger so that the coannihilation becomes less efficient. Furthermore, the nearly vertical black line situated at m A H ≃ 92 GeV (or f ∼ 650 GeV) is the lower mass limit of T-odd lepton set by the heavy charged lepton searches at LEP, below which the T-odd pair production is too large and contradicts the null result. 2 In the right panel of figure 1 we show the spin-independence scattering cross-section of heavy photon A H with nucleon using the parameter space of (κ ℓ , f ) which is compatible to the correct relic. The scattering process is dominated by Higgs t-channel mediated while the contributions from heavy T-odd quarks is small due to the small couplings of A H to T-odd quarks and the heaviness of T-odd quarks [33]. We also show the experimental limits of XENON in 2012 [49], LUX 2013 [46], LUX expected in 2014, projective XENON 1T [50], and projective XENON 10T [51,52] for comparison. As we see that the predicted SI cross-section monotonically decreases from ∼ 2 ×  The parameter space of (κ q , m AH ) and also (m uH − m AH , m AH ) that predicts dark matter relic abundance that is consistent with observation eq. (1.1) within 1σ, 3σ, 5σ, shown respectively in black, blue and red lines. Right: The one-nucleon-normalized spin-independent elastic cross-section of heavy photon A H scattering off the nucleon together with the current experimental limits of XENON100 (2012) [49], LUX(2013) [46]. The projective limits of LUX(2014), XENON 1T [50], and XENON10T [51,52] are also shown.
A H up to about 230 GeV and XENON 1T can test all the model parameter space we are interested in.
For the scenario of coannihilation with T-odd quarks, we notice that if T-odd quarks are degenerate, the decay width of T-odd partner of top quark t H is so narrow that it will leave a displaced vertex inside the detector, which contradicts with current data [53]. The reason is that the 2-body t H → A H t and 3-body t H → A H bW + decay modes are kinematically forbidden since the mass difference between t H and A H is much smaller than top quark mass and W -boson mass. Therefore the only decay channel for t H is t H → A H bff ′ , where f and f ′ are SM light fermions. Numerically, the decay width of t H is at the order of 10 −14 GeV in the coannihiliation region. Hence, we focus on the situation that the T-odd partners of third generation quarks are much heavier than the first two generations and are irrelevant in coannihilation processes. Also, we set all of the T-odd leptons to be much heavier than dark matter A H .
The left panel of figure 2 shows the parameter space of (κ q , f ) that agrees with the dark mater relic abundance together with a mass difference between T-odd quark u H and dark matter A H for demonstration. The region between black (blue, red) lines is consistent with measurement within 1(3, 5) σ level. Similar to the case of T-odd lepton coannihilation, as we explained previously, the sharp dropoff at m A H 62.5 GeV is due to the fact that the cross section of A H pair annihilation is dropping very quickly when A H is away the resonance of Higgs boson. Therefore, light T-odd quarks are needed to join the coannihilation to compensate. The rising is because the W -boson final state is opened in A H pair annihilation and enlarges the annihilation cross section. Then, the mass gap between T-odd quarks and A H should be larger to suppress the contributions of coannihilaiton processes. Similar to our estimate in the scenario of T-odd lepton coannihilation, the number density of T-odd quark is about 7% of that of dark matter A H when m A H = 300 GeV.
Shown in the right plot of figure 2 is the predicted spin-independent cross-section of A H scattering off nucleon, using the parameter space corresponding to the correct relic abundance of dark matter in the left panel of figure 2. In addition to the Higgs-boson-exchanged t-channel diagrams in A H -nucleon scattering, as we mentioned in previous T-odd lepton coannihilation case, the diagrams which involve T-odd quarks also play an important role since the T-odd quarks now can be as light as about 100 GeV [33]. Therefore, the effects of T-odd quarks can be as significant as the one involving Higgs boson. The amplitudes between diagrams with T-odd quark exchanged interference destructively between s-channel and t-channel and may become negative in some portions of parameter space. The sharp drop-off structure at m A H ∼ 80 GeV is due to the fact that the total amplitude of the T-odd quark diagrams is negative and destructs the amplitude of the Higgs boson diagram.
In  fermions are light, LHC can copiously produce them. First, we study the case of light T-odd leptons. As shown in figure 3 , we use CalcHEP [54] package with CTEQ6L [55] parton distribution functions (PDF) to calculate the total production cross sections of Todd lepton pairs at 8 TeV LHC. The cross section of ℓ ± H ν H can be as large as 13 pb when f is about 500 GeV, where ℓ = e, or µ or τ . The collider signature of ℓ ± H ν H is single charged lepton plus MET / E T , which is the same as generic W ′ search in leptonic decay modes. The current limit for W ′ in single charged lepton plus MET final state requires that the W ′ should be heavier than about 2.9 TeV when W ′ with SM couplings to SM fermions [56], which, in principle, can be used to interpret the constraints on T-odd lepton mass. However, since the mass gap between T-odd lepton ℓ ± H and dark matter A H is small, the charge lepton ℓ ± in the decay ℓ ± H → ℓ ± A H is soft. As shown in right plot of figure 4, the transverse momentum of ℓ ± peaks around 10 GeV. After imposing selection cuts in W ′ search [56], the signal of pp → ℓ ± H ν H → ℓ ± νA H A H is entirely cut off, especially the high transverse momentum cut on charged lepton p ℓ T > 40 GeV and hard transverse mass cut M T = 2p ℓ T · / E T · (1 − cos ∆φ ℓν ) > 1 TeV. For ℓ + H ℓ − H pair production at the LHC, the signal is ℓ + ℓ − A H A H after T-odd leptons decay. Such dilepton plus MET signal has been searched at the LHC for slepton or chargino pair production in supersymmetry [57]. However, due to high transverse momentum cuts on charged leptons p (20) GeV and high m T 2 cut (m T 2 > 90 GeV) [57], most of the signal of ℓ + H ℓ − H do not pass the event selection. Therefore, the current searches for heavy colorless charged particles in single charge lepton plus MET or in dilepton plus MET have no sensitivity to light T-odd lepton pair production in coannihilation region. It is quite challenging to directly search for them because of the soft charged lepton in the final state.
We now turn to study the coannihilation region where T-odd quark is light. The main production mechanism of T-odd quarks is though QCD interaction. After summing over  first two generations, the total cross section of pair production of T-odd quarks pp → q HqH at the LHC with 8 TeV center-of-mass energy is shown in figure 5. The cross section can be as large as about 2 × 10 4 pb when f ≃ 500 GeV and decreases down to ∼ 10 pb when f is 2 TeV. The collider signature of T-odd quark pairs at the LHC is 2 jets plus MET, which can mimic the signal of squark or gluino pairs in supersymmetry. One may expect that the JHEP06(2014)074 current data of dijet plus MET will testify this parameter space of LHT. However, similar to the charged lepton in T-odd lepton production, the jet in T-odd quark production at the LHC is very soft and most of the signals won't pass the jet (p leading T > 130 GeV) and MET (/ E T > 160 GeV) selection criteria in dijet plus MET search [58]. As a result, the current dijet plus MET search for new physics at the LHC has no constraint on light T-odd quark scenario in the region of coannihilation. Instead, we consider the situation that a hard jet radiated from the initial state pp → q HqH j. As we see in figure 6, the transverse momentum distribution of jet radiated from the initial state decreases slower than the one from the decay of T-odd quark. Furthermore, the MET also shifts to higher value when there exists an extra hard jet. Therefore, the process of a T-odd quark pair plus one jet can contribute to collider search for new physics, like dark matter, in mono-jet plus MET channel. The effects of initial state radiation in the search of T-odd quark that is nearly degenerate with A H at the LHC has also been studied in [38]. We calculate the T-odd pair with an extra jet from initial state using Madgraph 4 [59] and compare with results of monojet plus missing transverse momentum final states at the LHC by ATLAS collaboration [60]. The event selection criteria in the analysis are summarized as follows: • MET / E T > 120 GeV.
• If there are more than one jet, the azimuthal angle difference between second leading jet and MET ∆φ(j 2nd , / E T ) > 0.5.
Furthermore, four signal regions denoted as SR1, SR2, SR3, SR4 are defined with different cuts on p T > 120 GeV, 220 GeV, 350 GeV, 500 GeV and / E T > 120 GeV, 220 GeV, 350 GeV, 500 GeV. We find that the SR1 give the most stringent constraint, and the result is shown in figure 7. The black curve is the cross section of pp → q HqH j → qqA H A H j after imposing cuts in signal region of SR1. The grey shaded region represents the exclusion of mono-jet plus MET search. We can see that the f value below about 1.4 TeV is excluded, which corresponds to exclusion of m A H 215 GeV.

Conclusion
In this paper, we study the phenomenology of dark matter in the Littlest Higgs Model with T-parity, focusing on the coannhilation scenario and its implications at the LHC. We find that, even though the T-odd partner of neutrino ν H can be the lightest T-odd particle and satisfy the current measurement of dark matter relic abundance, it is ruled out by the direct dark matte search experiments because of its large cross section when scattering with nucleon. The T-odd heavy photon A H is therefore the only suitable dark The predicted cross section of mono-jet plus MET events from process of pp → q HqH j at the LHC, using the event selection criteria in signal region SR1. The grey shaded area is excluded by the ATLAS [60]. matter in the model. For pair annihilation cross section of A H to explain the observed dark matter relic abundance with the Higgs boson being 125 GeV, the mass of A H should be just slightly heavier than half of Higgs boson mass. However, the direct detection of dark matter of current data from LUX disfavors this case. When m A H becomes heavier, coannihilation contributions from T-odd fermions must be taken into account in order to fit the relic abundance measurement.
For the case of coannihilation with light T-odd leptons, the existing LEP limit for heavy charged lepton search excludes the region where f 650 GeV, which corresponds to m A H 92 GeV. The current dark matter direct search result from LUX 2013 requires m A H 110 GeV while the future projected sensitivity of XENON 1T has potential to examine the whole parameter space. Since the T-odd leptons are as light as dark matter, the production cross section of T-odd lepton pairs at the LHC can be as large as O(10) pb. However, the charged lepton in the final sate is very soft since the mass difference between T-odd lepton and dark matter is small. Therefore the direct searches at the LHC is very challenging and the current searches for heavy charged particles have no constraints on these light T-odd leptons. For the scenario of T-odd quark coannihilation, we have to consider the situation that T-odd partner of top quark t H is much heavier than the Todd partners of the first two generations of SM quarks and is irrelevant in all the results we have shown here. Otherwise, t H will be stable enough to generate displaced vertex signature at the LHC, which is inconsistent with data. When T-odd quarks are light, they contribute significantly to the A H -nucleon elastic scattering cross section. The current dark matter direct search result from LUX 2013 imposes a constraint on m A H 70 GeV while the projected result of LUX 2014 has the sensitivity for m A H 200 GeV. The future XENON 1T has potential to testify the whole parameter space except for m A H ∼ 80 GeV where a large destruction effect by T-odd quarks happens. Similar to the case of T-odd JHEP06(2014)074 leptons, even though the LHC can produce the light T-odd quarks copiously, with the total production cross section as large as 10 4 pb for f ∼ 500 GeV, the final state jets from T-odd quark decay are very soft, and therefore, it very difficult to directly search for these light T-odd quarks at the LHC. However, the production of a T-odd quark pair plus one jet contributes significantly to mono-jet plus missing energy signature and the current data from ATLAS set a stringent constraint that disfavors f 1.4 TeV, corresponding to m A H 215 GeV.
In summary, A H dark matter, the T-odd partner of photon, in Littlest Higgs Model with T-parity fits the measurement of dark matter relic abundance and can satisfy the direct search results well when coannihilaiton processes are considered. The current LHC mono-jet plus missing energy data sets a strong constraint on A H mass in the T-odd quark coannihilation scenario. For T-odd lepton coannihilaiton case, the LHC has no constraint while the direct detection of dark matter can be sensitive. Combining the future data from dark matter direct search experiments and from LHC, we should be able to fully testify the whole interesting parameter of the model.