Triplet-quadruplet dark matter

We explore a dark matter model extending the standard model particle content by one fermionic SU(2)L triplet and two fermionic SU(2)L quadruplets, leading to a minimal realistic UV-complete model of electroweakly interacting dark matter which interacts with the Higgs doublet at tree level via two kinds of Yukawa couplings. After electroweak symmetry-breaking, the physical spectrum of the dark sector consists of three Majorana fermions, three singly charged fermions, and one doubly charged fermion, with the lightest neutral fermion χ10 serving as a dark matter candidate. A typical spectrum exhibits a large degree of degeneracy in mass between the neutral and charged fermions, and we examine the one-loop corrections to the mass differences to ensure that the lightest particle is neutral. We identify regions of parameter space for which the dark matter abundance is saturated for a standard cosmology, including coannihilation channels, and find that this is typically achieved for mχ10~2.4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {m}_{\chi_1^0}\sim 2.4 $$\end{document} TeV. Constraints from precision electroweak measurements, searches for dark matter scattering with nuclei, and dark matter annihilation are important, but leave open a viable range for a thermal relic.


Introduction
With the discovery of the ∼ 125 GeV Higgs boson at the LHC [1,2], the Standard Model (SM) of particle physics has been proven to be a self-consistent SU(3) C × SU(2) L × U(1) Y gauge theory describing the strong and electroweak interactions of three generation quarks and leptons. However, the SM fails to describe astrophysical and cosmological observations, which are best explained by the existence of a massive neutral species of particle -dark matter (DM) [3][4][5]. While a variety of DM candidates are provided by extensions of the SM, among the most attractive are weakly interacting massive particles (WIMPs), which have roughly weak interaction strength and masses of O(GeV)-O(TeV). If WIMPs were thermally produced in the early Universe, they could give a desired relic abundance consistent with observation.
WIMPs typically appear in popular extensions of the SM aimed at addressing its deficiencies, such as e.g. supersymmetric [6,7] and extra dimensional models [8,9]. However, the need for their existence is independent of deep theoretical questions and it behooves us to leave no stone unturned in exploring the full range of possibilities. It is further natural to explore dark sectors containing SU(2) L multiplets, whose neutral components are JHEP03(2016)204 natural DM candidates and whose interactions suggest the correct relic density for weak scale masses. Within the broad class of such models, both theoretical considerations and experimental results (most importantly, the null results of searches for WIMP scattering with heavy nuclei) provide important constraints on the viable constructions.
In minimal dark matter [10], the dark sector consists of a single scalar or fermion in a non-trivial SU(2) L representation. For even-dimensional SU(2) L representations, nonzero hypercharge is required to engineer an electrically neutral component, and typically results in a large coupling to the Z boson, which is excluded by direct searches for dark matter [11]. Odd-dimensional SU(2) L representations have much weaker constraints, and lead to thermal relics for masses in the range of a few TeV.
If the dark sector consists of more than one SU(2) L representation, electroweak symmetry-breaking allows for mixing between them, resulting in a much richer theoretical landscape. If the dark matter is a fermion, tree level renormalizable couplings to the Standard Model Higgs are permitted provided there are SU(2) L representations differing in dimensionality by one. Such theories provide a theoretical laboratory to explore the possibility that the dark matter communicates to the SM predominantly via exchange of the electroweak and Higgs bosons. 1 The minimal module consists of a single odd-dimensional SU(2) L representation Weyl fermion together with a vector-like pair (such that anomalies cancel) of even-dimensional representations with an appropriate hypercharge. Two such constructions which have been previously considered are singlet-doublet dark matter [12][13][14][15][16][17] and doublet-triplet dark matter [17,18]. Both of these sets look (in the appropriate limit) like subsets of the neutralino sector of the minimal supersymmetric standard model (MSSM), and share some of its phenomenology.
In this work we investigate a case which does not emerge simply as a limit of the MSSM, triplet-quadruplet dark matter, consisting of one Weyl SU(2) L triplet with Y = 0 and two Weyl quadruplets with Y = ±1/2. After electroweak symmetry-breaking, the mass eigenstates include three neutral Majorana fermions χ 0 i , three singly charged fermions χ ± i , and one doubly charged fermion χ ±± , leading to unique features in the phenomenology. After imposing a discrete Z 2 symmetry, and choosing the lightest neutral fermion χ 0 1 to be lighter than its charged siblings, we arrive at an exotic theory of dark matter whose interactions are mediated by the electroweak and Higgs bosons.
As with the singlet-doublet and doublet-triplet constructions, this theory is described by four parameters encapsulating two gauge-invariant mass terms (m T and m Q ) and two different Yukawa interactions coupling them to the SM Higgs doublet (y 1 and y 2 ). The limit y 1 = y 2 realizes an enhanced custodial global symmetry resulting in χ 0 1 decoupling (at tree level) from the Z and Higgs bosons (provided m Q < m T ), greatly weakening the bounds from direct searches. It further implies that χ 0 1 is degenerate in mass with one of the charged states (and sometimes χ ±± ) at tree level. For small deviations from this limit, the degeneracy is mildly lifted, requiring inclusion of the one-loop corrections to reliably establish that the lightest dark sector fermion is neutral.

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This paper is outlined as follows. In section 2 we describe triplet-quadruplet dark matter in detail and establish notation. In section 3 we discuss the interesting features in the custodial symmetry limit. In section 4 we compute the corrections to the mass splittings at the one-loop level. In section 5 we identify the regions of parameter space resulting in the correct thermal relic abundance for a standard cosmology (including coannihilation channels) as well as the constraints from the electroweak oblique parameters and from direct and indirect searches. Section 6 contains our conclusions and further discussions. Appendix A gives the explicit expressions for the interaction terms, while appendix B lists the self-energy expressions which are used in the calculations of the mass corrections and electroweak oblique parameters.

Triplet-quadruplet dark matter
The triplet-quadruplet dark sector consists of colorless Weyl fermions T , Q 1 , and Q 2 transforming under (SU(2) L , U(1) Y ) as (3, 0), (4, −1/2), and (4, +1/2). We denote their components as: The two quadruplets are assigned opposite hypercharges in order to cancel gauge anomalies. Gauge-invariant kinetic and mass terms for the triplet and the quadruplets are given by and which specify their interactions with electroweak gauge bosons. They also couple to the SM Higgs doublet H through Yukawa interactions where we use the tensor notation (see e.g. ref. [19]) to write down the triplet and quadruplets with SU(2) L 2 (upper) and2 (lower) indices explicitly indicated. We further assume there is a Z 2 symmetry under which dark sector fermions are odd while SM particles are even to forbid renormalizable operators T LH and nonrenormalizable operators such as T eHH, Q 1 L † HH † , and Q 2 LHH † (where L is a lepton doublet and e is a charged lepton singlet), which would lead the lightest dark sector fermion to decay. In decomposing the SU(2) components, a traceless tensor T i j in the 3 representation is constructed from a 2, u i , and a2, v i , as which is symmetric in the upper indices i and j, and satisfies k Q kj k = k Q ik k = 0. Taking into account the normalization of the Lagrangians (2.2) and (2.3), we can identify the components of T , Q 1 , and Q 2 in the vector notation (2.1) with those in the tensor notation via: They are diagonalized by three unitary matrices, N , C L , and C R : with the gauge eigenstates related to the mass eigenstates by Therefore, the dark sector fermions consist of three Majorana fermions χ 0 i , three singly charged fermions χ ± i , and one doubly charged fermion χ ±± . Here we denote the particles in order of mass, i.e., m χ 0 . The lightest new particle is stable as a result of the imposed Z 2 symmetry. Consequently, we identify parameters such that χ 0 1 is lighter than χ ± 1 and χ ±± , in order for χ 0 1 to effectively play the role of dark matter.
We can construct 4-component fermionic fields from the Weyl fields: where

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And the mass basis is defined such that they have diagonal mass terms:

Custodial symmetry
If y 1 is equal to y 2 , there exists a global custodial SU(2) R global symmetry, as is well known in the SM Higgs sector. Under this symmetry the triplet is an SU(2) R singlet, while the quadruplets and the Higgs field are both SU(2) R doublets: where H i ≡ ε ij H j and A is an SU(2) R index. L Q and L HTQ can be expressed in an SU(2) L × SU(2) R invariant form: where y = y 1 = y 2 . This symmetry is also found in the singlet-doublet model [13][14][15] and the doublet-triplet model [18]. Though broken by the U(1) Y gauge symmetry, nonetheless it dictates some tree level relations with important implications. We describe the cases m Q < m T and m Q > m T separately below.

m Q < m T
If m Q < m T , the leading order (LO) dark sector fermion masses can be derived to be: while the mixing matrices take the form Thus each of the neutral fermions is degenerate in mass with a singly charged fermion, and the lightest one is also degenerate with the doubly charged fermion, which always  has a mass of m Q . In figure 1(a), we show the mass spectrum for m Q = 200 GeV and m T = 400 GeV. If y = 0, the quadruplets would not mix with the triplet, and we would have m LO , and χ ± 3 become heavier. At loop level the custodial symmetry realizes that it is broken by U(1) Y , and corrections from the loops of electroweak bosons lift the degeneracies [10,20,21]. We examine the next-to-leading (NLO) corrections to the masses in detail in section 4.
In general, the χ 0 1 couplings to the Higgs boson and to the Z boson are proportional to (y 1 N 21 − y 2 N 31 )N 11 and (|N 31 | 2 − |N 21 | 2 ), respectively. In the custodial symmetry limit, the interaction properties of χ 0 1 are quite special. From the explicit expression of N in eq. (3.6), we can find that there is no triplet component in χ 0 Therefore, the χ 0 1 coupling to the Higgs boson vanishes because this coupling exists only when the T 0 component is involved. Moreover, there is no χ 0 1 coupling to the Z boson, since Q 0 1 and Q 0 2 have opposite hypercharges and opposite eigenvalues of the third SU(2) L generator. As a result, χ 0 1 cannot interact with nuclei at tree level and generically escapes from direct detection bounds.

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and χ 0 1 is a mixture of T 0 , Q 0 1 , and Q 0 2 : In this case, the coupling to the Higgs boson does not vanish, that with the Z boson still vanishes because |N 21 | 2 = |N 31 | 2 = 1/b 2 . Consequently, χ 0 1 can interact with nuclei through the Higgs exchange at tree level. Figure 1(b) shows the mass spectrum for m Q = 400 GeV and m T = 200 GeV. If and hence m χ 0 1 = m Q and χ 0 1 = (Q 0 1 + Q 0 2 )/ √ 2, whose interactions are similar to the case of m Q < m T described above.

One loop mass corrections
In this section, we calculate the dark fermion mass corrections at NLO, determining the parameter space for which χ 0 1 is lighter than χ ± 1 and χ ±± . For mixed fermionic fields X i (either X 0 i or X + i ), renormalized one-particle irreducible two-point functions can be written down as [22,23] where P L ≡ 1 2 (1 − γ 5 ) and P R ≡ 1 2 (1 + γ 5 ) are chiral projectors and δM ij are mass renormalization constants defined byM ij,0 =M ij + δM ij , where the subscript 0 denotes a bare quantity and the diagonalized mass matrixM stands for eitherM N orM C . The wave function renormalization constants δZ L ij and δZ R ij are defined as X i,0 = X i + 1 2 (δZ L ij P L + δZ R * ij P R )X j . The self-energy Σ X i X j (q) can be decomposed into Lorentz structures: and Hermiticity relates these functions: There are additional constraints for Majorana fields X 0 i : which we utilize as a cross-check on our calculations.
On-shell, there should be no mixing between states in the mass basis. Using the definition of the pole mass in the on-shell scheme leads to the renormalization condition:

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where Re takes the real parts of the loop integrals in self-energies but leaves the couplings intact. This condition fixes the mass renormalization constants to As in refs. [22,24] for the renormalization of neutralinos and charginos, we introduce renormalization constants δM N and δM C to shift the mass matrices M N and M C , but the mixing matrices N , C L , and C R remain the same at NLO as at LO. Therefore, we have and Furthermore, we choose to renormalize the Majorana fermion masses on-shell, i.e., and compute the relative shifts in the masses of χ ± i and χ ±± . In this scheme eq. (4.7) provides the NLO shifts in the parameters m T , m Q , y 1 , and y 2 : where δM N is given by eq. (4.6): (4.12) The shifts on these parameters shiftM C through eq. (4.8). As a result, the physical masses of χ ± i at NLO are given by The physical mass of χ ±± is affected by the shift in m Q :    Explicit expressions for the self-energies of dark sector fermions at NLO can be found in appendix B. We evaluate the mass corrections numerically with LoopTools [25]. In the custodial symmetry limit y = y 1 = y 2 , the mass differences between charged and neutral fermions at NLO are presented in figure 2 where the triplet has no mixing with the quadruplets, and the mass splitting is solely induced by the irreducible O(100) MeV contribution from the electroweak gauge interaction at one loop [10]. This degeneracy lifts for y = 0. When m Q = 200 GeV and m T = 400 GeV, the charged fermions are always heavier than their corresponding neutral fermions for |y| ≤ 1. When m Q = 400 GeV and m T = 200 GeV, χ ± 3 becomes lighter than χ 0 3 for 0.25 |y| ≤ 1. In both cases, χ 0 1 is always the lightest dark sector fermion as required for a DM candidate.
Moving beyond the custodial symmetry limit, in figure 3, we fix m Q , m T , and y 1 = 1, and plot the fermion masses as functions of y 2 . We find that a value of y 2 unequal to y 1 tends to drive χ 0 1 lighter, especially when the sign of y 2 is opposite to y 1 . The charged fermions remain rather degenerate with the corresponding neutral fermions. In figure 4, we present the corresponding mass differences, which change sign frequently as y 2 varies. For −1.95 y 2 −0.5 (−1.95 y 2 −0.85) in the m Q < m T (m T < m Q ) case, χ 0 1 becomes lighter than χ ± 1 and fails to describe viable DM.

Constraints and relic density
In this section, we investigate the constraints on the parameter space from electroweak precision measurements, direct and indirect searches, and identify regions where the observed DM relic abundance is obtained for a standard cosmology. We discuss each of these regions in greater detail below, but begin with a summary presented in figure [26], while the light blue bands denote its 2σ range and the dark blue regions indicate DM overproduction in the early Universe. The violet, orange, green, and red regions are excluded by the condition m χ ± 1 < m χ 0 1 , electroweak oblique parameters [27], the LUX direct detection experiment [28], and the Fermi-LAT gammaray observations on dwarf galaxies [29], respectively. The gray dashed lines indicate contours of fixed m χ 0 1 .

Relic abundance
To begin with, we identify the regions in which the dark matter abundance saturates observations for a standard cosmology. As we have seen, χ 0 1 is always nearly degenerate in mass with χ ± 1 . Furthermore, for m Q < m T , we may have m χ ±± m χ 0 1 , as well as m χ 0 2 m χ ± 2 m χ 0 1 when m Q |y 1,2 v|. These fermions, with close masses and comparable interaction strengths, tend to decouple at the same time, with coannihilation processes playing a significant role in their final abundances. Since after freeze-out they decay into χ 0 1 , we compute their combined relic abundance using the technology of ref. [30]. We implement the triplet-quadruplet model in Feynrules 2 [31], and compute the relic density with MadDM [32] (based on MadGraph 5 [33]).

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In figure 5, the parameter space consistent with the DM abundance measured by the Planck experiment, Ωh 2 = 0.1186 ± 0.020 [26], is plotted as the dot-dashed blue lines, with the 2σ region around it denoted by the light blue shading. As is typical for an electroweaklyinteracting WIMP, the observed DM abundance is realized for m χ 0 1 ∼ 2.4 TeV. When χ 0 1 is heavier, there is effectively overproduction of DM in the early Universe, as shown by darker blue shaded regions in figure 5. Regions with lighter masses and underproduction of dark matter are left unshaded.

Precision electroweak constraints
The dark fermions contribute at the one loop level to precision electroweak processes. Since there are no direct coupling to the SM fermions, these take the form of corrections to the electroweak boson propagators, and are encapsulated in the oblique parameters S, T , and U [34,35], where s W ≡ sin θ W , c W ≡ cos θ W with θ W denoting the Weinberg angle. Π IJ (p 2 ) is the g µν coefficient for the vacuum polarization amplitude of gauge bosons I and J, which can be divided as iΠ µν IJ (p 2 ) = ig µν Π IJ (p 2 ) + (p µ p ν terms), and Π IJ (0) ≡ ∂Π IJ (p 2 )/∂(p 2 )| p 2 =0 . The contributions to Π ZZ (p 2 ), Π W W (p 2 ), Π AA (p 2 ), and Π ZA (p 2 ) from dark sector fermions are given in appendix B. In the custodial symmetry limit, T and U remain zero, while S is positive and increases as |y| increases. Outside of the custodial limit, all are typically nonzero, with U typically much smaller than S and T , as is expected given the fact that it corresponds to a higher dimensional operator.
A global fit to current measurements of precision data by the Gfitter Group yields [27] S = 0.05 ± 0.11, T = 0.09 ± 0.13, U = 0.01 ± 0.11, These results exclude the orange regions in figures 5(a) and 5(b) at the 95% CL. In the custodial symmetry limit y 1 = y 2 = 0.5, a region limited by m Q 300 GeV and m T 1.8 TeV is excluded. For y 1 = 0.5 and y 2 = 1, a region limited by m Q 400 GeV and m T 4.1 TeV is excluded.

Scattering with heavy nuclei
Spin-independent scattering with heavy nuclei is mediated at tree level by the exchange of a Higgs or Z boson. The coupling strength of χ 0 1 to Higgs (see appendix A) is:

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In the zero momentum transfer limit, this induces a scalar interaction with nucleons N : (5.8) and the nucleon form factors f N i are determined to be roughly [36]: The tiny up and down Yukawa couplings imply approximately iso-symmetric couplings, G S,n G S,p , yielding a spin-independent (SI) scattering cross section of is the χ 0 1 -N reduced mass. As a Majorana fermion, χ 0 1 couples to Z with an axial vector coupling of strength (5.11) leading to axial vector interactions with nucleons: The form factors are ∆ p u = ∆ n d = 0.842 ± 0.012, ∆ p d = ∆ n u = −0.427 ± 0.013, ∆ p s = ∆ n s = −0.085 ± 0.018 [37]. These interactions lead to a spin-dependent (SD) scattering cross section Current limits on σ SI χN are lower than those on σ SD χN by several orders of magnitude (owing to the coherent enhancement of the SI rate for heavy nuclear targets such as Xenon). The green regions in figure 5 are excluded by the 90% CL exclusion limit on the SI DMnucleon scattering cross section from LUX [28]. The profiles of these regions depend on the relation between y 1 and y 2 . As mentioned in section 3, in the custodial symmetry limit g hX 0 1 X 0 1 and g ZX 0

Dark matter annihilation
Finally, we consider bounds on the annihilation cross section σ ann v rel (where v rel is the relative velocity between two incoming DM particles) based on the non-observation of anomalous sources of high energy gamma rays. We adapt MadGraph 5 to calculate the annihilation cross sections in the non-relativistic limit for all open two-body SM final states. The dominant channels 2 are W + W − , ZZ, and Zh. The W + W − channel is typically dominant over ZZ and Zh by one to two orders of magnitude. Thus, we compare predictions for annihilation into W + W − with the null results for evidence of DM annihilation into gamma rays in dwarf spheroidal galaxies based on 6 years of data collected by the Fermi -LAT experiment [29]. Fermi provides 95% CL upper limits on σ ann v rel for annihilation into W + W − as a function of the DM particle mass, which we translate into the exclusion regions shown as the red shaded regions in figure 5. The Fermi data basically excludes m χ 0 1 1 TeV for m Q < m T and m χ 0 1 700 GeV for m T < m Q .

Constraints on the y 1 -y 2 plane
By fixing the mass parameters m T and m Q , we can see how the constraints vary in the y 1 -y 2 plane, as shown in figure 6. The plots are symmetric under the simultaneous transformations of y 1 → −y 1 and y 2 → −y 2 . In figures 6(a) and 6(c), we have m Q < m T , and the condition m χ ± 1 < m χ 0 1 excludes some regions where y 1 and y 2 are sufficiently large and their signs are opposite to each other. The contours of m χ 0 1 are parallel to the diagonals, which correspond to the custodial symmetry limit and have the largest values of m χ 0 1 . In figures 6(b) and 6(d), we have m T < m Q , and m χ 0 1 is larger at the corners of y 1 = y 2 = 1.5 and y 1 = y 2 = −1.5 than at the corners of y 1 = −y 2 = 1.5 and y 1 = −y 2 = −1.5.
In figures 6(a) and 6(b), the fixed values of m T and m Q are suitable for obtaining an observed DM abundance. The contours corresponding to the mean value of the measured Ωh 2 appear as ellipses, inside which Ωh 2 is larger. The mass parameters are chosen to show comparable sensitivities of LUX and Fermi -LAT in figures 6(c) and 6(d), where both the LUX and Fermi exclusion regions enclose the point y 1 = −y 2 = 1.5 as well as the point y 1 = −y 2 = −1.5. In figure 6(d), the LUX bound also excludes the regions around y 1 = y 2 = 1.5 and y 1 = y 2 = −1.5. Both the LUX and Fermi limits roughly coincide with the contours of m χ 0 1 .

Conclusions and outlook
In this work, we explore a dark sector consisting of a fermionic SU(2) L triplet and two fermionic SU(2) L quadruplets. This set-up is a minimal UV complete realistic model of electroweakly interacting dark matter with tree level coupling to the SM Higgs boson, the simplest such construction which is distinct from any limit of the MSSM. After electroweak symmetry-breaking, the dark sector consists of three Majorana fermions χ 0 i , three singly charged fermions χ ± i , and one doubly charged fermion χ ±± . The lightest neutral fermion χ 0 1 is a wonderful DM candidate provided it is the lightest of the dark sector fermions.     When two Yukawa couplings are equal, i.e., y 1 = y 2 , there is an approximate global custodial symmetry, implying that χ 0 i is mass-degenerate with χ ± i at tree level. We compute the one-loop mass corrections to determine the precise spectrum. Fortunately, in the custodial limit these corrections always increase the masses of charged fermions. Another gift from this symmetry is the tree-level vanishing of the χ 0 1 couplings to Z and h, rendering the current DM direct searches impotent as far as constraining it. Beyond the custodial symmetry limit, at tree level m χ 0 i and m χ ± i are slightly different, but nonetheless still quite degenerate. At the one-loop level, mass corrections suggest that we may have m χ ± 1 < m χ 0 1 when y 1 and y 2 have opposite signs. When that happens, χ 0 1 is no longer a viable DM candidate, decaying into the lightest charged state.
Due to the mass degeneracy, coannihilation processes among dark sector fermions strongly affect the abundance evolution of χ 0 1 in the early Universe, and must be included. The calculation suggests that m χ 0 1 ∼ 2.4 TeV to saturate the observed relic density for a JHEP03(2016)204 standard cosmology. We also investigate the constraints from the electroweak oblique parameters and direct and indirect searches. The global fit result of S, T , and U parameters excludes a region up to m Q 300 (400) GeV and m T 1.8 (4.1) TeV for y 1 = y 2 = 0.5 (y 1 = 0.5 and y 2 = 1). The LUX exclusion region significantly depends on the relation between y 1 and y 2 . When y 2 has a sign opposite to y 1 , the LUX result excludes m χ 0 1 up to several TeV for m Q m T , cutting in to some regions favored by the relic abundance. Annihilation into W + W − is the dominant channel in the non-relativistic limit, and Fermi -LAT dwarf galaxy limits exclude m χ 0 1 1 TeV and 700 GeV for m Q < m T and m T < m Q , respectively. Nonetheless, there is still plenty of room in the parameter space that is consistent with the observed DM abundance and escaping from phenomenological constraints.
As the charged fermions in the dark sector couple to the Higgs boson, the h → γγ decay is a possible indirect probe of their presence. However, the current LHC data are not sufficiently precise to give a meaningful limit, though LHC high luminosity running may reach the correct ballpark [39]. LHC direct searches for exotic charged particles decaying into missing momentum may also be able to explore the model, but the electroweak production rates of the dark sector charged fermions are quite low for multi-TeV fermions, and it may ultimately fall to future higher energy colliders to have the last word [40,41].

Acknowledgments
TMPT acknowledges Randy Cotta, JoAnne Hewett, and Devin Walker for earlier collaboration on related topics. ZHY would like to acknowledge helpful discussions with Arvind Rajaraman, Philip Tanedo, Alexander Wijangco, Mohammad Abdullah, and Ye-Ling Zhou and also thanks the Particle Theory Group at UC Irvine for hospitality during his visit. The work of TMPT is supported in part by NSF grant PHY-1316792 and by the University of California, Irvine through a Chancellor's Fellowship, and that of ZHY was supported by the China Scholarship Council (Grant No. 201404910374).

A Detailed expressions for interaction terms
In this appendix, we derive explicit expressions for the interaction terms in the tripletquadruplet model. The covariant derivatives for the triplet and quadruplets are where Y Q 1 = −1/2, Y Q 2 = +1/2, and the generators of SU(2) L are:

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and We can express the gauge interaction terms in eqs. (2.2) and (2.3) as and Including the would-be Goldstone bosons, eq. (2.4) becomes

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where the Goldstone bosons G 0 and G ± are defined as For convenience, we would like to express the interaction terms with 4-component fermionic fields. Here we define Now eq. (2.22) is equivalent to We can use chiral projection operators to divide every fermionic field into two parts: Thus we have The interaction terms in eqs.
where the coupling coefficients read The coupling coefficients that have not mentioned above are zero. Converting the gauge bases into the physical bases, we have + (a G ± X ++ X + i G +X ++ R X + iL + h.c.) + (b G ± X ++ X + i G +X ++ L X + iR + h.c.), (A. 16) JHEP03 (2016)204 where the coupling coefficients are related to those in (A.15) through the mixing matrices: (A.17)

B Self energies
In this appendix, we give useful expressions for the self-energies of χ 0 i , χ ± i , χ ±± , γ, Z, and W , used for both the calculation of the mass corrections for dark sector fermions and the electroweak oblique parameters. In these calculations, we use the one-loop integrals whose definitions are consistent with ref. [23]: .

(B.4)
We calculate the self-energies of χ 0 i , χ ± i , and χ ±± in the DR scheme with the 't Hooft-Feynman gauge, as in ref. [42]. At NLO, the χ 0 i -χ 0 k self-energy has contributions from loops