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JHEP08(2019)058

Published for SISSA by Springer

Received: February 25, 2019

Revised: June 24, 2019

Accepted: July 7, 2019

Published: August 9, 2019

Scalar electroweak multiplet dark matter

Wei Chao,

Gui-Jun Ding,

Xiao-Gang He

c,d,e

and Michael Ramsey-Musolf

f,g

Center for advanced quantum studies, Department of Physics, Beijing Normal University,

100875, Beijing, China

Interdisciplinary Center for Theoretical Study and Department of Modern Physics,

University of Science and Technology of China,

Hefei, Anhui 230026, China

Tsung-Dao Lee Institute, and School of Physics and Astronomy,

Shanghai Jiao Tong University,

Shanghai 200240, China

Department of Physics, National Taiwan University,

Taipei 106, Taiwan

National Center for Theoretical Sciences,

Hsinchu 300, Taiwan

Amherst Center for Fundamental Interactions,

University of Massachusetts-Amherst, Department of Physics,

Amherst, MA 01003, U.S.A.

Kellogg Radiation Laboratory, California Institute of Technology,

Pasadena, CA 91125, U.S.A.

E-mail: chaowei@bnu.edu.cn, dinggj@ustc.edu.cn, hexg@phys.ntu.edu.tw,

mjrm@physics.umass.edu

Abstract: We revisit the theory and phenomenology of scalar electroweak multiplet ther-

mal dark matter. We derive the most general, renormalizable scalar potential, assuming

the presence of the Standard Model Higgs doublet, H, and an electroweak multiplet Φ of

arbitrary SU(2)

rank and hypercharge, Y . We show that, in general, the Φ-H Higgs portal

interactions depend on three, rather than two independent couplings as has been previously

considered in the literature. For the phenomenologically viable case of Y = 0 multiplets,

we focus on the septuplet and quintuplet cases, and consider the interplay of relic density

and spin-independent direct detection cross section. We show that both the relic density

and direct detection cross sections depend on a single linear combination of Higgs por-

tal couplings, λ

eﬀ

. For λ

eﬀ

∼ O(1), present direct detection exclusion limits imply that

the neutral component of a scalar electroweak multiplet would comprise a subdominant

fraction of the observed DM relic density.

Keywords: Beyond Standard Model, Cosmology of Theories beyond the SM, Higgs

Physics

ArXiv ePrint: 1812.07829

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP08(2019)058

JHEP08(2019)058

detection (DD) experiments and LHC searches place severe constraints on this possibility,

though some room remains depending on the speciﬁc model realization.

An alternative possibility is that the dark matter consists of the neutral component of

an electroweak multiplet, χ

. A classiﬁcation of these possibilities is given in [1]. Those

favored by the absence of DD signals carry zero hypercharge (Y ), thereby preventing overly-

large WIMP-nucleus cross sections mediated by Z

exchange. Tree-level stability of the χ

requires imposition of a discrete Z

symmetry unless the representation of the electroweak

multiplet is of suﬃciently high dimension: d=5 for fermions and d=7 for scalars. These

scenarios with suﬃciently high dimension representation go under the heading “minimal

dark matter”.

In this work, we consider features of scalar electroweak multiplet dark matter Φ, in-

cluding but not restricting our attention to minimal dark matter (as conventionally de-

ﬁned). The phenomenology of scalar triplet dark matter, involving a multiplet transform-

ing as (1, 3, 0) under SU(3)

and electroweak symmetries, has been considered previously

in refs. [2–9]. Extensive studies for other electroweak multiplets of dimension n have been

reported in refs. [10–35]. The authors of ref. [34] considered the Inert Doublet model and

the n = 3, 5, 7 scalar electroweak multiplets and discussed the impact of non-vanishing

Higgs portal interactions on the relic density, spin-independent dark matter-nucleus cross

section, σ

, and indirect detection (ID) signals. Ref. [35] also considered the impact of

Higgs portal interactions on the relic density and σ

but did not analyze the implications

for indirect detection. The latter study also focused on a relatively light mass for the dark

matter candidate, for which it would appear to undersaturate the relic density.

In what follows, we revisit the topic of these earlier studies, taking into account several

new features that may require modifying some of the conclusions in refs. [34, 35]:

• We ﬁnd that the scalar potentials V (H, Φ) given in refs. [34, 35] are not the

most general renormalizable potentials and that, depending on the representation

of SU(2)

×U(1)

there exist one or more additional interactions that should be in-

cluded. For the Y = 0 representations, the Φ-H interaction relevant for both the

relic density and DD cross section involves an eﬀective coupling λ

eﬀ

that is linear

combination of two of the three possible Higgs portal couplings. The speciﬁc linear

combination is representation dependent.

• We update the computation of σ

taking into account the nucleon matrix elements

of twist-two operators generated by gauge boson-mediated box graph contributions

as outlined in refs. [36–39]. We note that ref. [35] considered only the Higgs portal

contribution to σ

and did not include the eﬀect of electroweak gauge bosons. We

ﬁnd that the gauge boson-mediated box graph contributions are smaller in magnitude

that given in ref. [34], which used the expressions given in ref. [1]. In general, the

Higgs portal contribution dominates the DD detection cross section except for very

small values of λ

eﬀ

• The presence of a non-vanishing λ

eﬀ

can allow for a larger maximum dark matter

mass, M , to be consistent with the observed relic density than one would infer when

– 2 –

JHEP08(2019)058

considering only gauge interactions. For the cases we consider below, this maximum

mass be as larger as O(20) TeV for perturbative values of λ

eﬀ

• For moderate values of the Higgs portal couplings, the spin-independent cross section,

scaled by the fraction of the relic density comprised by Φ

, is a function λ

eﬀ

and M .

The present DD bounds on σ

generally require M . 5 TeV for perturbative values

of λ

eﬀ

— well below the maximum mass consistent with the observed relic density.

In what follows, we provide the detailed analysis leading to these conclusions. For

the structure of V (H, Φ) we consider Φ to be a general representation of SU(2)

×U(1)

Previous studies have considered in detail electroweak singlets (n = 1), doublets (n = 2),

and triplets (n = 3). In all three cases, stability of the DM particle requires that one impose

a discrete symmetry on the Lagrangian. Going to higher dimension representations, it has

been shown in ref. [1] that for n = 4, stability of the neutral component also requires

imposition of a discrete symmetry, while for n = 5, the neutral component can only decay

through a non-renormalizable dimension ﬁve operator with coeﬃcient suppressed by one

power of a heavy mass scale Λ. In the latter case, it is possible to ensure DM stability

on cosmological time scales by either imposing a discrete symmetry or by choosing Λ to

be well above the Planck scale. At n = 7, the ﬁrst non-renormalizable, decay-inducing

operator appears at higher dimension, and DM stability at tree-level may be ensured even

without imposition of a discrete symmetry by choosing Λ below the Planck scale. It was

subsequently noted in refs. [40, 41], however, that there exists a dimension ﬁve operator

leading decay of the neutral component of the septuplet at one-loop level. For a Λ at the

Planck scale and septuplet mass of O(10) TeV, the septuplet would not be suﬃciently stable

on cosmological time scales to provide for a viable DM candidate. Consequently, one must

again either choose a trans Planckian cutoﬀ or impose a stabilizing discrete symmetry.

With the foregoing considerations in mind, we focus on the n = 5 and 7 cases for

purposes of illustrating the dark matter phenomenology. Since the group theory relevant to

construction of V (H, Φ) is rather involved, we provide a detailed discussion in appendices A

and B. In section 2, we start with a general formulation, followed by treatment of speciﬁc

model cases. Section 3 gives the calculation of the relic density, including the eﬀects of

co-annihilation and the Sommerfeld enhancement. We compute σ

in section 4. We

summarize in section 5. Along the way, we point out where we ﬁnd diﬀerences with

earlier studies.

2 Models

We consider the renormalizable Higgs portal interactions involving H and Φ for two illus-

trative cases. We restrict our attention to Φ being a complex scalar with Y = 0. The form

of the potential for Φ being a real representation of SU(2)

with Y = 0 is relatively simple.

The corresponding features have been illustrated in previous studies wherein Φ is either an

SU(2)

singlet or real triplet. Consequently, we focus on complex representations, using

the n = 5 and n = 7 examples, to illustrate the new features not considered in earlier work.

– 3 –

JHEP08(2019)058

To proceed, we ﬁrst introduce some notation. It is convenient to consider both Φ and

the associated conjugate Φ, whose components are related to those of Φ as

j,m

= (−1)

j−m

∗

j,−m

, (2.1)

where j refers to the isospin of the scalar multiplet Φ. As we discuss in appendix A, Φ

and Φ transform in the same way under SU(2)

. The scalar multiplet Φ of integer isospin

can be either real or complex. If Φ is a real multiplet, there is a redundancy Φ = Φ such

that the constraint φ

j,m

= (−1)

j−m

∗

j,−m

should be fulﬁlled. For complex multiplet, each

component represents a unique ﬁeld, and it can be decomposed into two real multiplets

as follows

A =

√



Φ + Φ



, B =

√



Φ − Φ



. (2.2)

It is easy to verify that both A and B fulﬁll the realness condition A = A and B = B.

Therefore a general model with a complex multiplet Φ is equivalent to a model of two

interacting real multiplets A and B. Notice that a scalar multiplet Φ of half integer isospin

is always complex since the realness condition Φ = Φ can not be fulﬁlled anymore. As we

note below, under certain assumptions about the model parameters, the complex scalar

multiplets may reduce to a pair of degenerate real multiplets, allowing for a two-component

DM scenario. Since the case of the real triplet and singlet DM as singlet component DM

have been analyzed elsewhere, we do not consider higher dimensional real representations

here. Instead, we focus on the complex Y = 0 examples that, in principle, can embody

two-component real multiplet DM scenarios.

One may then proceed to build SU(2)

invariants by ﬁrst coupling Φ, Φ, H, and H

pairwise into irreducible representations and ﬁnally into SU(2)

invariants. For example,



ΦΦ



(−1)

√

2j + 1

†

Φ, (2.3)

which in general is a distinct invariant from (ΦΦ)

except in special cases when Φ is a

real scalar multiplet satisfying Φ =

Φ. We shall denote with (. . .)

a contraction into the

irreducible representation with isospin J throughout this paper. Note that for j = 1/2,

(ΦΦ)

vanishes, so that there is only one quadratic invariant in this case as well. Quartic

interactions can be constructed in a variety of ways, such as



(ΦΦ)



Φ Φ



, J = 0, 1, . . . , 2j (2.4)

for Φ self-interactions or







ΦΦ



(2.5)

with L = 0, 1 for the Higgs portal interactions. Note that there exists a third such inter-

action





(ΦΦ)

(2.6)

that is distinct from the L = 0 operator in eq. (2.5) for Φ being a complex integer represen-

tation. We note that previous studies have not in generally included all three of the possible

Higgs portal interactions. The classiﬁcation of the Φ self-interactions is more involved, and

it is most illuminating to consider them on a case-by-case basis.

– 4 –

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