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.


I. INTRODUCTION
Recently, a new scalar with mass of about 125 GeV has been discovered at the Large Hadron Collider (LHC) [1,2], it is still unclear at this moment whether or not this new particle is the Higgs boson in the Standard Model (SM). Due to the large radiative corrections in Higgs boson mass parameter in the SM, a very precise cancelation must take place in order to have the Higgs boson mass in the electroweak scale if the SM is valid up to Planck scale.
This so-called naturalness problem serves a driving force for theorists to propose many solutions, the leading new physics beyond the SM is supersymmetry. Another elegant idea is Little Higgs mechanism, in which the light Higgs boson is realized as a pseudo-Goldston boson. With collective symmetry breaking mechanism [3] (also see [4,5] for review), the global symmetry breaking scale f can be at O(1 TeV) without a fine tuning. The one-loop quadratic divergences induced by the SM particles are exactly cancelled by new fermions, gauge bosons and scalars. An economical model is Littlest Higgs model [6]. However, in order to satisfy the electroweak precision measurements, the scale f is required to be greater than 4 TeV [7][8][9] and a fine tuning to the light Higgs boson mass is reintroduced. One of the solutions is to embed a discrete symmetry, called T-parity, into the model [10][11][12] so that no mixing between new particles which are assigned with T-parity odd and the SM particles which are T-parity even. The corrections to electroweak observables are therefore all loop-induced. As a result, the scale f can be as low as about 500 GeV [13] and the LHC has great potential to examine the model.The phenomenology of the Littlest Higgs model with T-parity has been studied in the literature [14,15] . Moreover, the T-parity also ensures the stability of the lightest T-odd particle (LTP) that is naturally can be the candidate of dark matter if it is charge-neutral and colorless. There are two possible candidates of dark matter in Littlest Higgs model with T-parity: T-odd partner of the hypercharge gauge boson A H [14,16,17] and the T-odd partner of neutrino ν H .
The current dark matter relic abundance in our universe has been measured by WMAP [18] and recently by Planck [19] with the combined value Ω DM h 2 = 0.1199 ± 0.0027. (1) 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 [20]. Hence, we will focus on A H dark matter in our study.
A subgroup of SU (5), 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. 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].

III. 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 [21].
The relic density of dark matter today is determined by its annihilation processes in the  [22,23]. In our study, we vary f between ∼ 480 GeV and ∼ 2 TeV 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, c.f. Eq.(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 [24] and 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 [25]. 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 Fig. 1 the parameter space of κ ℓ and f that predicts the right dark matter relic abundance. The black, blue and red curves denote that the relic abundance is consistent with observation within 1σ, 3σ, and 5σ level. 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 Fig.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 [16]. We also show the experimental limits of XENON in 2012 [27], LUX 2013 [24], LUX expected in 2014, projective XENON 1T [28], and projective XENON 10T [29]  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 Fig. 2 shows the parameter space of (κ q , f ) that agrees with the dark mater relic abundance. 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 com-

pensate. 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.
Shown in the right plot of Fig.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 Fig. 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 [16]. 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 TeV when W ′ with SM couplings to SM fermions [33], 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 Fig. 4, the transverse momentum of ℓ ± peaks around 10 GeV. After imposing selection cuts in W ′ search [33], 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 [34]. However, due to high transverse momentum cuts on charged leptons 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 Fig. 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 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 [35]. 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 Fig. 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. We calculate the T-odd pair with an extra jet from initial state using Madgraph 4 [36] and compare with results of monojet plus missing transverse momentum final states at the LHC by ATLAS collaboration [37]. 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.
We find that the SR1 give the most stringent constraint, and the result is shown in Fig. 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.

V. 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