Electroweak interacting dark matter with a singlet scalar portal

We investigate an electroweak interacting dark matter (DM) model in which the DM is the neutral component of the SU$(2)_L$ triplet fermion that couples to the standard model (SM) Higgs sector via an SM singlet Higgs boson. In this setup, the DM can have a CP-violating coupling to the singlet Higgs boson at the renormalizable level. As long as the nonzero Higgs portal coupling (singlet-doublet Higgs boson mixing) exists, we can probe CP violation of the DM via the electric dipole moment of the electron. Assuming the $\mathcal{O}(1)$ CP-violating phase in magnitude, we investigate the relationship between the electron EDM and the singlet-like Higgs boson mass and coupling. It is found that for moderate values of the Higgs portal couplings, current experimental EDM bound is not able to exclude the wide parameter space due to a cancellation mechanism at work. We also study the spin-independent cross section of the DM in this model. It is found that although a similar cancellation mechanism may diminish the leading-order correction, as often occurs in the ordinary Higgs portal DM scenarios, the residual higher-order effects leave an $\mathcal{O}(10^{-47})~{\rm cm}^2$ correction in the cancellation region. It is shown that our benchmark scenarios would be fully tested by combining all future experiments of the electron EDM, DM direct detection and Higgs physics.


I. INTRODUCTION
The existence of dark matter (DM) in the Universe is firmly established by cosmological and astronomical observations, with its relic abundance measured by the comic microwave background being [1] Ω CDM h 2 = 0.1198 ± 0.0026 , where h is the reduced Hubble constant. In spite of the undoubted existence, we still do not know where to put the DM in the particle spectrum due to the lack of solid evidence from direct searches and identification of its quantum numbers.
Although the standard model (SM) is very successful in explaining most empirical observations in particle physics, one of its shortcomings is the absence of a DM candidate. To amend this, there have been many proposals to extend the SM with a dark sector, in which the lightest member, serving as a DM, cannot decay into SM particles due to some dark charge.
Weak-interacting massive particles (WIMP's) has attracted much attention as candidates for the DM because it is naturally accommodated in the TeV-scale physics. For example, non-singlet DM's under the SU(2) L ×U(1) Y emerge in supersymmetric (SUSY) models such as the minimal supersymmetric SM (MSSM) (see, e.g., Ref. [2] for a review). On the other hand, isospin singlet DM's commonly appear in the context of the Higgs portal scenarios in which the DM's can communicate with the SM particles only via the Higgs sector [3][4][5][6][7][8][9][10].
A lot of work have been done based on effective field theories or on specific renormalizable models, with both approaches complementary to each other. The former has a strong power in probing the dark sector in a model-independent way. However, some phenomena such as accidental cancellations due to light particles are often improperly described within this framework, and the latter is more appropriate to address such issues.
One of the unknown properties of the DM is its CP nature. In renormalizable fermionic DM Higgs portal scenarios, it is possible for the DM to have both scalar and pseudoscalar couplings (denoted by g S and g P , respectively). Explicitly, one may have Sχ c g S + iγ 5 g P χ + h.c. , where S an isospin singlet scalar playing the role of messenger between the dark sector and Higgs sector, and the phase of fermionic DM field χ is already rotated so that its mass is real. If χ is a singlet under the SM gauge symmetry, it will be hard to probe CP violation in the dark sector as its effect appears only at loop levels in the Higgs sector. If χ participates in the electroweak interactions, on the other hand, we may detect the existence of such CP violation in electric dipole moment (EDM) experiments.
In EW-interacting DM (EWIMP) scenarios [11,12], the interactions between the DM and the gauge bosons are fixed by the ordinary gauge couplings, leaving the DM mass the only unknown parameter. However, the DM mass is also completely determined once the thermal relic scenario is assumed. For example, the DM mass should be around 3 TeV in the Wino case [11,12]. In the nonthermal relic scenario, on the other hand, the relic density could be explained by nonthermal production of the DM from heavier particles. In this case, it is conceivable that the DM mass can be as light as O(100) GeV.
In this Letter, we consider a model in which the DM resides in an SU(2) L triplet fermion with hypercharge Y = 0 (Wino-like DM) 1 and the interaction given in Eq. (2). Here we do not confine ourself to the thermal relic scenario and, therefore, the DM mass is taken as a free parameter. In this framework, we study the CP-violating effects coming from the dark sector on the electron EDM in connection with Higgs physics. Throughout the analysis, the singlet scalar S is assumed to be lighter than 1 TeV. For the heavy S case, the interaction between χ and Higgs doublet (H) would be described by the dimension-5 operator H † Hχ c (g ′S + iγ 5 g ′P )χ/Λ after integrating out the S field. Recent studies on the connections between CP violation and the EWIMP using the effective Lagrangian can be found in Refs. [14,15].
The structure of this paper is as follows. In Section II, we describe the DM model, with particular emphasis on the Higgs and dark sectors. Stability and global minimum conditions for the Higgs potential are discussed. We also provide the Higgs couplings with the SM particles and the triplet fermions. Section III discusses observables that can be used to constrain or test the model. Numerical results of these observables are presented in Section IV. Our findings are summarized in Section V.
We consider a model in which the DM candidate arises from an SU(2) L triplet (Winolike) fermion field χ and couples to the SM Higgs sector via an SU(2) L singlet scalar field S.
Both χ and S are assumed to carry no hypercharge. The relevant interactions are described by the Lagrangian where χ a denote 2-component spinors,H = iσ 2 H * andσ µ = (1, −σ i ) with σ i being the Pauli matrices, and the covariant derivative acting on the field χ a is with g 2 being the SU(2) L gauge coupling. We impose the Z 2 symmetry, χ → −χ, so that the third term involving the lepton doublet ℓ L in the square bracket of Eq. (3) drops out, and the neutral component of χ becomes a DM candidate. Phenomenology of DM without the singlet Higgs boson is well studied (see, for example, Refs. [11,12]).
We parameterize the Higgs fields as follows: where v = 246 GeV, and G + and G 0 are the Nambu-Goldstone bosons. The Higgs sector of this model is the same as the real singlet-extended SM (rSM). Here we give a quick review of rSM to make the paper self-contained. The tadpole conditions are where · · · means that the quantity in the bracket is evaluated in the vacuum. These two tadpole conditions can be used to solve for µ 2 H and m 2 S in terms of the other parameters. Assuming v, v S = 0, the squared-mass matrix of the Higgs bosons in the vacuum is cast into the form which can be diagonalized by an orthogonal matrix as where −π/4 ≤ α ≤ π/4. Here we assume that the mass eigenvalues satisfy m H 1 < m H 2 , and m H 1 = 125 GeV. The scenario of no mixing between the H and S fields (α → 0) occurs in both the alignment limit µ HS = −λ HS v S and the decoupling limit − The tree-level effective potential is given by where ϕ and ϕ S are respectively the classical background fields of h and h S , and µ 2 H and m 2 S are given by Eqs. (6) and (7). In order for the potential to be bounded from below, we impose the following conditions on the quartic couplings: where the last condition is needed in particular when λ HS takes negative values. Since to have another vacuum that is lower than the electroweak vacuum specified by (v, v S ). In Ref. [8], the conditions for the electroweak vacuum to be the global minimum are investigated and, as a result, it is found that under the conditions α = µ S = 0 and The left inequality is derived by requiring that the electroweak vacuum has a lower energy than the symmetry vacuum, while the right inequality is obtained by demanding that the electroweak vacuum be lower than another local minimum on the v S axis. 2 It is noted that numerically Eq. (12) is still a good approximation even when α ∼ 0.2 [rad]. Moreover, one can turn the inequalities into Therefore, if another neutral Higgs boson is found experimentally, one can use its mass to bound √ λ S |v S | in the above-mentioned limit of the model.
Note that the constraint in Eq. (12) is derived from the tree-level potential given in Eq. (10). Thus, it may change after including one-loop corrections, especially from the χloops. However, as long as the magnitudes of λ's and α are moderate, which we assume throughout this paper, the tree-level result still remains intact. For the explicit one-loop demonstration in the singlet fermionic DM model, see Ref. [8].
The Higgs coupling constants relevant for our analysis are , where χ +(0) are the 4-component Dirac (Majorana) fermions and with λ = |λ|e iφ λ , M χ = M + λv S = |M χ |e iφ Mχ , and δ φ ≡ φ λ − φ Mχ being the only physical CP-violating phase in the new sector. Here we have also used the shorthand notations s α = sin α and c α = cos α. Naïvely, we expect that the phase δ φ ∼ O(1) and will discuss its effects in various observables. At tree level, χ ± and χ 0 are degenerate in mass, given by |M χ | above. As will be discussed in the next section, such a degeneracy is lifted by radiative corrections. We will thus use m χ ± and m χ 0 to denote the physical masses of χ ± and χ 0 , respectively. 2 The existence of such a nontrivial vacuum commonly happens in the context of strong first-order electroweak phase transition, as needed for successful electroweak baryogenesis [16,17]. However, the condition √ λ HS v ≪ √ λ S |v S | usually does not hold in such cases so that the mass bound (12) is not valid.
Although the Higgs sector of the current DM model is virtually the same as that proposed in Refs. [6][7][8][9], there are significant differences in certain phenomena due to the triplet fermion field χ a . Therefore, we will focus exclusively on the observables with distinctive features in this analysis, especially those being well constrained by experiments and likely to have improvements in the near future.
In this model, the dark sector participates in electroweak interactions. Therefore, under the assumption of a nonzero CP-violating phase, it will contribute to the EDM's of electron, neutron and atoms. The most stringent bound of all comes from the recent experimental measurement of the thorium-monoxide EDM, which places an upper bound on the electron EDM [18]: where e denotes the electric charge of the positron. As is well known, the two-loop Barr-Zee diagrams can have significant contributions [19]. For the electron EDM, the preponderant diagram involves the Higgs boson and photon in the loop and gives where α em = e 2 /(4π), Eq. (15) is used to obtain the second line, and the loop function g(τ ) is defined as In the approximation of m χ ± ≫ m H 1,2 , we have As expected, the EDM is proportional to the sine of the CP-violating phase δ φ . Besides, it would be vanishing if the triplet fermion does not couple with the real scalar or in the limit of α → 0. Finally, the EDM would also be suppressed if the two Higgs bosons are almost degenerate in mass, a consequence of the orthogonality of the mixing matrix O(α).
The spin-independent cross section of the DM with a nucleon at leading order is given by where m N denotes the nucleon mass, µ χ 0 N is the reduced mass of the DM Refs. [8,25]). To have an observable cross section, we also need sufficiently large couplings between S and χ and mixing between the two Higgs bosons.
In the case that the above leading-order contribution is highly suppressed, higher order effects should be taken into account. Ref. [26] has evaluated the dominant electroweak loop corrections induced by the scatterings of the EWMIP with the light quarks and gluon, assuming only one Higgs doublet of the SM. To our knowledge, there is no such a calculation with multiple Higgs bosons, and thus more precise estimates are still unknown. Nevertheless, as we will see in the next section, since the experimentally favored region is cos α > ∼ 0.95, the singlet Higgs boson effect in our model has a suppression factor of (1 − cos 2 α) < ∼ 0.1 and is expected to be subleading. In our numerical study, the higher-order corrections are estimated using the results of Ref. [26] as a first step toward the complete analysis.
Recently, QCD corrections up to next-to-leading order in α s to σ SI in the EWIMP without the singlet scalar have also been finished [27] (see also Ref. [28]). It is found that the Winoproton cross section σ p SI = 2.3 × 10 −47 cm 2 for a wide mass range around 1 TeV. It is noted, however, that if the suppression at the leading order is due to the proximity of the two Higgs mass eigenstates, the cancellation is to all orders in strong interactions.
The Higgs signal strengths are useful observables to probe the structure of the Higgs sector. Without the dark sector, the signal strengths of H 1 are universally scaled by c 2 α , provided Br(H 1 → H 2 H 2 ) = 0, as assumed throughout this paper. Once the dark sector is taken into account, however, the signal strengths are modified mainly due to the contributions of charged χ to the diphoton mode: In what follows, we assume that χ ± , χ 0 are sufficiently heavy so that the last two decays in Eq. (22) are kinematically forbidden. Since the diphoton mode has a relatively small partial width, we have Γ tot ≃ c 2 α Γ tot SM and µ X ≃ c 2 α for the ZZ * , W W * ,f f channels. On the other hand, the signal strength of H 1 → γγ takes the form where A SM = −6.49 [29], Γ tot SM ≃ 4.1 MeV [30], and with τ χ = 4m 2 χ ± /m 2 H 1 and the loop function f (τ χ ) defined in Ref. [31]. In the limits of small α and large m χ ± , µ γγ reduces to where terms of higher order in t α and v/m χ ± have been neglected. Therefore, µ γγ (∼ c 2 α ) would be suppressed in this limit. However, it should be stressed that the reduction factor differs from both the Higgs portal DM models, such as those in Refs. [7,8,25], and the Wino DM case in the EWIMP scenarios [11,12]. Since the CP-violating part does not interfere with the SM contribution, as seen in Eq. (23), its effect is higher order in powers of t α and v/m χ ± .
With a mild dependence on M χ , the mass difference ∆M ∼ O(100 MeV) [38,39]. As a consequence, χ ± have a relatively long lifetime of O(0.1) ns, with the dominant decay mode of χ ± → π ± χ 0 . We can probe such a meta-stable particle at colliders by identifying the disappearance of a charged track. The ATLAS Collaboration has put a constraint on such a long-lived charged particle. With the LHC Run-1 data, the lower bound of m χ 0 is found to be [40] m χ 0 > 270 GeV (95% CL).
The constraints coming from the cosmic rays are also important. Ref. [41] analyzed the observations of gamma-rays from classical dwarf spheroidal galaxies, and found that and m χ 0 < ∼ 2900 GeV from the DM relic abundance constraint. In the following analysis, we also regard the case of m χ 0 = 2900 GeV as the thermal relic scenario inferred by the Wino DM case. This holds as long as the coupling between χ and S is smaller than the gauge couplings. If this is not the case, the DM mass might be changed due to the additional Sommerfeld effect induced by S. Although it is interesting to investigate such a case, the detailed analysis leaves the main scope of this Letter. Throughout our analysis, we take |λ| = 0.1 and focus on sin δ φ ≥ 1/ √ 2, which yields g S H 2χ χ < ∼ 0.07. We first present the results for m χ 0 = 2900 GeV. In Fig. 1, µ  value for B new , the curve will shift downwards. The red dotted lines represent µ γγ = 0.95 (top) and 0.9 (bottom). Since the effects of χ ± are substantially decoupled, the deviation of µ γγ is virtually due to c 2 α (= κ 2 V ), as seen in Eq. (25). The contours of electron EDM are displayed by the black solid lines: |d e | = 10 −29 e cm and 10 −30 e cm from bottom to top. The current bound is outside the region. As discussed above, the cancellation between H 1 and H 2 corrections gets more prominent as m H 2 approaches 125 GeV. Therefore, the maximal CP violation case is still allowed even if the electron EDM is improved to 10 −30 e cm. We emphasize that this possibility cannot be encoded in the effective field theory approach as mentioned in Introduction.
For the δ φ = 45 • case, the contour of the DM direct detection cross section σ p SI = 10 −46 cm 2 is also shown by the blue dot-dashed curve. Similar to the election EDM, the cancellation mechanism is at work when m H 1 ≃ m H 2 . Note that even if the leading contribution in σ p SI vanishes, the NLO contribution (σ p SI ≃ 1.5 × 10 −47 cm 2 ) still remains, which yields the minimum value in the region we are considering here. For the δ φ = 90 • case, on the other hand, there is no leading-order correction since σ p SI ∝ cos 2 δ φ , as shown in Eq. (20). In this For the DM direct detection, the contour of σ p SI = 10 −46 cm 2 is given in the case of δ φ = 45 • . As seen, σ p SI is not sensitive to the DM mass since the leading contribution in σ p SI is mostly controlled by α, g S H 1χ χ , g S H 2χ χ and m H 2 , and the m χ 0 dependence enters only via µ χ 0 N , as mentioned above. Linear Collider [43] and TLEP [44]. For instance, the sensitivity of κ V is expected to be improved up to O(0.1)% at the latter two lepton colliders.
The projected sensitivity of the electron EDM in future experiments is around 10 −30 e cm [45].
In addition to this, the EDMs of nucleons and atoms may also be important (for a recent review, see, e.g., Ref. [46]).
Several DM direct detection experiments are also planned. The XENON1T experiment [47] has a better sensitivity than the current LUX bounds by more than an order of magnitude, i.e., σ p SI = (1.2−50)×10 −47 cm 2 for the DM mass in the range of 100−3000 GeV, which may be further improved to σ p SI = (1.8 − 48) × 10 −48 cm 2 by the LZ experiment [48].
In summary, the entire region for our benchmark points will be fully testable in these future experiments.
We have studied the phenomenology in the electroweak-interacting fermionic dark matter We have also considered the direct detection bounds on the DM and found that if m H 2 > ∼ 150 GeV and κ V < ∼ 0.99, the spin-independent DM-nucleon cross section σ p SI ≃ O(10 −46 ) cm 2 . Therefore, the upcoming XENON1T experiment can readily probe such a region. For m H 1 ≃ m H 2 , on the other hand, the cancellation mechanism is effective so that the leading-order contribution vanishes, as observed in the ordinary Higgs portal DM scenarios. Nevertheless, since the DM participates in electroweak interactions in our model, the residual higher-order corrections still remain and amount to σ p SI ≃ 1.5 × 10 −47 cm 2 . The current analysis have shown that our benchmark scenario will be entirely tested by the future experiments of the electron EDM, DM direct detection and Higgs physics.
Finally, we summarize by pointing out distinctive features of our model in comparison with two existing ones. In the model studied in Ref. [14], the Wino DM couples with the SM Higgs boson via the dimension-5 operator H † Hχ c (g S +iγ 5 g P )χ/Λ, with Λ being a heavy mass scale. Within the effective field theory framework, the regime with accidental cancellation, as explicitly shown in this Letter, is not properly treated. Therefore, the two models will have different signals in CP violation associated with Higgs physics. In the SU(2) L singlet fermionic DM model [9], the Higgs signal strengths are almost the same as those in our model. Even though its DM sector can also accommodate CP violation, the manifestation is so dim that the electron EDM is far below the detectable level, a clear difference from our model.