Isospin-Violating Dark Matter in the U ( 1 ) ′ Model with E 6 Origin

We propose a U(1) model from E6 which has an isospin-violation dark matter. By choosing a proper linear combination of two extra U(1) gauge symmetries in E6, it is natural to realize the ratio fn/fp = −0.7 so as to maximally relax the constraints from the Xenon based direct detection experiments. We study the sensitivities of the dark matter direct and indirect detection experiments, and identify the parameter spaces that can give the observed relic density. We also study the sensitivities of the future colliders with center mass energy √ s= 33/50/100 TeV, and compare the different detection methods. We show that in some parameter spaces the future colliders can give much stronger limits. PACS numbers: 12.10.-g, 12.60.-i, 95.35.+d ∗ tli@itp.ac.cn † xiangqf@pku.edu.cn ‡ yanqishu@ucas.ac.cn § zhangxianhui16@mails.ucas.ac.cn ¶ zhouhan@itp.ac.cn


I. INTRODUCTION
The observations of astrophysics and cosmology reveal that the main component of matter in the Universe is Dark Matter (DM). However, till now, all evidence for DM is through its gravitational effects, and the nature of DM particles remains a mystery. Determining the fundamental nature of the dark matter particle is one of the most important problems in particle and astro-particle physics. Great efforts have been taken to identify dark matter, including direct detection, indirect detection, and collider searches, while the answer is still unclear.
DM would be detectable through their elastic scattering with nuclei in terrestrial particle detectors. The most remarkable DM signals is the one claimed by the DAMA Collaboration (including DAMA/NaI and DAMA/LIBRA experiment) [1][2][3][4][5], which uses a NaI-based scintillation detector. With data collected over 14 annual cycles, the statistical significance of DAMA/LIBRA-phase2 has reached 12.9σ [5]. The CoGeNT experiment, using Germanium as target, also found an irreducible excess [6] and annual modulation [7]. The low energy excesses in the CaWO 4 based experiment CRESST-II have been reported as well [8]. However, these observations are challenged by the null results of the other experiments, such as PandaX-II (2017) [9], LUX (2017) [10], and XENON1T (2018) [11].
The Isospin-Violating Dark Matter (IVDM), in which DM couples differently to protons and neutrons, has been proposed to reconcile the tensions among the different direct detection experimental results [12]. Recently, the COSINE-100 experiment, that also uses the same NaI crystal as target, observes no signal excess in the first 59.5 days of data [13]. This observation makes it difficult to explain all the direct detection observations, especially the observations of DAMA. In some particular models, such as the proton-philic spin-dependent inelastic Dark Matter (pSIDM), one could still explain DAMA modulation amplitude consistent with the constraints from other experiments [14]. Here we only focus on the concept of how to realize the isospin violation in a UV complete model, instead of trying to explain all experimental observations. Nowadays the most stringent constraint on the DM-nucleus scattering cross sections is from the Xenon based experiments [9][10][11]. In this work, we will maximally relax these constraints by naturally realizing [12] Several IVDM models have been proposed in recent years. For scalar dark matter, Ref.
[15] proposed a model with colored mediators and Ref. [16] considered a two-Higgs doublet model. For Dirac dark matter, an effective Z ′ model was proposed in Ref. [17], a double portal scenario was considered in Ref. [18], and a string-theory inspired UV model was studied in [19]. Within the framework of supersymmetry, different realizations were examined [20][21][22] . In this work, we propose a U(1) ′ Model with E 6 origin. E 6 is of particular interesting in the sense that it is anomaly free, and its fundamental representation is chiral representation. II.
The representation 16 contains the 15 SM fermions, as well as a right-handed neutrino.
The 10 representation under SU(5) decomposes as The 5 contains a color triplet and a SU(2) L doublet, whereas5 contains a color antitriplet and another SU(2) doublet, and the 1 is a SM singlet. The gauge boson is contained in the adjoint 78 representation of E 6 . The particle content of the 27 representation, which contains the SM fermions as well as extra fermions, are shown in the first two columns of Table I. The SM has three generations of fermion, so we use three such 27.
The E 6 gauge symmetry can be broken as follows [34,35] The U(1) ψ and U(1) χ charges for the E 6 fundamental 27 representation are also given in Table I.
The U(1) ′ attracting us is one linear combination of the U(1) χ and U(1) ψ The other U(1) gauge symmetry from the orthogonal linear combination as well as the SU(5) is broken at a high scale. This allows us to have a large doublet-triplet splitting scale, which prevents rapid proton decay if the E 6 Yukawa relations were enforced. This will need either two pairs of (27,27) or one pair of (27,27), 78, in addition to one pair of (351 ′ , 351 ′ ) dimensional Higgs representations (Detailed studies of E 6 theories with broken Yukawa relations can be found in [36,37].) For our model, the unbroken symmetry at the In our model we introduce three fermionic 27s, one scalar Higgs doublet field H u from the doublet of 5 of SU (5) Table II.
By choosing  it is natural to realize IVDM with f n /f p = −0.7.
A SM singlet Higgs field Φ with U(1) ′ charge −14 are introduced to generate the masses of right-handed neutrinos. In order to break all the global symmetries in the Higgs potential and avoid the massless Nambu-Glodstone boson, we introduce another SM singlet Higgs field Moreover, to introduce a dark matter candidate, we introduce a SM singlet fermion χ with U(1) ′ charge −27/2. Thus, χ ′ cannot decay due to the residual discrete Z 2 gauge symmetry after U(1) ′ gauge symmetry breaking. For details, please see Table II as well.
The Higgs potential for the U(1) ′ gauge symmetry breaking is Note that without the A 1 , and λ terms to break three global U(1) symmetries. Then we are left with only one global symmetry in the above potential, which is the extra U(1) ′ gauge symmetry. Therefore, after S, T , Φ, and T ′ acquire the VEVs, the U(1) ′ gauge symmetry is broken. Also, S, T , Φ, and T ′ will mix with each other via the quartic and trilinear terms. In addition, the U(1) ′ symmetry breaking Higgs fields S, T , Φ, and T ′ and the electroweak symmetry breaking Higgs fields H u and H d can be mixed via the quartic terms as well, for example, |S| 2 |H u | 2 , etc, which can be written down easily.
The Yukawa couplings in our models are where i = 1, 2, 3. Thus, after S and T obtain VEVs or after U(1) ′ gauge symmetry breaking, After diagonalizing their mass matrices, we obtain the mixings between XD c i and D c i , and the mixings between XL i and L i . The discussion of the Higgs potential for electroweak symmetry breaking is similar to the Type II two Higgs doublet model, so we will not repeat it here.
At low energy, the relevant degrees of freedom are SM particles, Z ′ , and DM χ. The interactions can be expressed as The ratio of different U(1) ′ couplings are determined by its U(1) ′ charge tabulated in Table II.
Since our model is isospin-violated, u and d quarks couple different with Z ′ . After a brief combination, we get For example, if we set g u = 0.1 , then g d = −0.889 , g χ = −0.15 , g uA = 0 , g dA = −1.889.
Particularly, we do not have axial vector terms for u-quark in our model.

III. CONSTRAINTS FROM DARK MATTER EXPERIMENTS
Generally, DM direct detection experiments assume DM couples the same to proton and neutron, and then report their limits for cross sections per nucleon. In the more general framework of IVDM, the cross sections per nucleon σ Z N is defined as where A i refers to different isotopes and η i is corresponding fractional number abundance.
Ifσ is the limit reported by an experiment, then F Zσ is the limit for IVDM. It is obvious that the DM elastic scattering off nucleus will have coherent effect between σ p and σ n , which can leads to a strongly destructive effect with particular f n /f p . Xenon1T (2018) [11].
Previously, the IVDM with new experimental data has been studied in Ref. [38]. Here we update some experiment results and apply this bounds to our model. Shown in the left panel of Fig. 1 are F Z for three kinds of materials with isotopy effects taken into account.
For the case of Xenon, F Z get its maximum at f n /f p = −0.7. In the right panel of Fig. 1 we present the rescaled limits for three kinds of direct detection experiments. It is obvious that constraints of these Xenon based experiments could be relaxed by a factor of about 10 −4 .

IV. CONSTRAINTS FROM FUTURE COLLIDER
Another powerful methods to explore the nature of DM is collider search. In our model DM interacts directly with quarks, and can be copiously produced at hadron colliders such as the LHC and proposed LHC-hh [46] and SppC [47]. Once DM are produced, they will escape the detectors undetected, so another additional radiation is needed to trace these events. In this section we study the sensitivities of future colliders for this model, and compare them with those obtained from DM direct and indirect experiments. The techniques of collider research closely follow Ref. [42].
In this study, we focus on the monojet signal process pp → Z ′( * ) → χχ+ jets. The TeV. Events with more than two jets with p T > 100 GeV and |η| < 4 are rejected. The DM production process may involve more than one jet from initial state radiation. In order to keep more signal events, a second jet(j 2 ) is allowed if it satisfies the condition ∆φ(j 1 , j 2 ) < 2.5. The cut on ∆φ(j 1 , j 2 ) is necessary to suppress the QCD multijet background, where large fake / E T may come from inefficient measurement of one of the jets. Furthermore, in order to reduce other backgrounds, such as W (→ lv) + jets, Z(→ l + l − )+ jets, andtt+ jets with leptonic top decays, the events containing isolated electrons, muons, taus, or photons with p T > 20 GeV and |η| < 2.5 are discarded. We then count the events and present the exclusion limits at 95% C.L. in Fig. 4.
It is obvious from Fig. 4 that the sensitivity of collider strongly depends on whether Z ′ is on shell or not. When m χ < 1 2 m ′ Z , Z ′ is on shell produced and the cross section is resonantly enhanced. In this case the DM production cross sections and collider sensitivities are almost independent of its mass. When m χ > 1 2 m ′ Z , Z ′ is off shell produced, the DM production cross section is proportional to [g q g χ /(Q 2 − m 2 Z ′ )] 2 (Q 2 is the typical momentum transfer to the DM pair) and is suppressed by 1/Q 2 . Particularly, for the case m 2 Z ′ ≪ Q 2 , the DM Compared to direct and indirect detections, the collider search would have stronger capability for the region m χ < 1 2 m ′ Z . Direct detection will be sensitive for m χ > 10 GeV, while indirect detection will be sensitive for m χ > 100 GeV, they could probe different mass regions and are complementary to each other.

V. CONCLUSIONS
We constructed a U(1) ′ model from E 6 which has the isospin-violation dark matter.
After a few steps of gauge symmetry breaking, the unbroken gauge symmetry at TeV scale For the purpose of phenomenological study, we introduced some new particles to this model. Especially, due to the residual Z 2 symmetry, an SM singlet fermion χ with U(1) ′ charge −27/2 is absolutely stable and then a DM candidate.
By choosing a proper linear combination of two extra U(1) gauge symmetries in E 6 , we naturally obtained the ratio f n /f p = −0.7 so as to maximally relax the constraints from the Xenon based direct detection experiments. Compared to isospin-conservation case, the constraints from the Xenon based experiments are relaxed by a factor of about O(10 4 ). We studied the sensitivities of dark matter direct and indirect detection experiments, and found the parameter spaces that have the observed relic density. For m χ ∼ 1 2 m Z ′ , the constraints from indirect detection experiments are enhanced due to resonance effects.
We then studied the sensitivities of the future colliders with center mass energy √ s= 33/50/100 TeV. The sensitivities of the collider searches are highly dependent on whether Z ′ is on-shell or not. Moreover, we compared the different detection methods, and showed that the future colliders will provide the much better searches in our model, especially for the region m χ < 1 2 m ′ Z .

ACKNOWLEDGMENTS
where s is the squared center-of-mass energy of a DM particle pair and color factor c q = 3.
The width of Z ′ can be expressed as with The particle explanation of Z ′ is Γ Z ′ < m Z ′ , which in turn roughly require g u < 0.89.
In order to study DM relic density and indirect detection signals, we need to calculate the thermally averaged annihilation cross section is the Moller velocity. However, instead of calculating σ ann v M directly, it is more convenient to calculate σ ann v rel in the laboratory frame, which means one of the initial particles is at rest, and get the same result. Here v rel is the relative velocity between them.
In the laboratory frame, when DM is non-relativistic, s can be expanded as 4m 2 χ +m 2 To get relic density, we can use an approximate function instead of solving the Boltzmann equation numerically Ω χ h 2 = 2 × 1.04 × 10 9 GeV −1 ( T 0 2.725 K ) 3 x f M pl g * (x f )(a + 3b where x f ≡ mχ T f ∼ O(10) , T f is the DM freeze-out temperature, T 0 = 2.725 ± 0.002K is the present CMB temperature, and g * (x f ) is the effective relativistic degrees of freedom at the