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JHEP10(2019)067

Published for SISSA by Springer

Received: August 9, 2019

Revised: September 20, 2019

Accepted: September 23, 2019

Published: October 7, 2019

Dark Matter from self-dual gauge/Higgs dynamics

Dario Buttazzo,

Luca Di Luzio,

a,b

Giacomo Landini,

a,b

Alessandro Strumia

and Daniele Teresi

a,b

INFN, Sezione di Pisa,

Largo Bruno Pontecorvo 3, I-56127 Pisa, Italy

Dipartimento di Fisica “E. Fermi”, Universit`a di Pisa,

Largo Bruno Pontecorvo 3, I-56127 Pisa, Italy

E-mail: dario.buttazzo@pi.infn.it, luca.diluzio@pi.infn.it,

giacomo.landini@phd.unipi.it, alessandro.strumia@unipi.it,

daniele.teresi@df.unipi.it

Abstract: We show that a new gauge group with one new scalar leads to automatically

stable Dark Matter candidates. We consider theories where the Higgs phase is dual to

the conﬁned phase: it is known that SU(2) gauge theories with a scalar doublet (like the

Standard Model) obey this non-trivial feature. We provide a general criterion, showing that

this self-duality holds for SU(N), SO(N), Sp(N ) and G

gauge dynamics with a scalar ﬁeld

in the fundamental representation. The resulting Dark Matter phenomenology has non-

trivial features that are characteristic of the group, and that we discuss case by case. Just

to mention a few, SU(N) has an accidental conserved dark baryon number, SO(2N + 1)

leads to stable glue-balls thanks to a special parity, G

leads to a Dark Matter system

analogous to neutral kaons. The cosmological Dark Matter abundance is often reproduced

for masses around 100 TeV: all constraints are satisﬁed and lighter dark glue-balls can aﬀect

Higgs physics. These theories acquire additional interest and predictivity assuming that

both the dark and weak scales are dynamically generated.

Keywords: Cosmology of Theories beyond the SM, Gauge Symmetry, Beyond Standard

Model

ArXiv ePrint: 1907.11228

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP10(2019)067

JHEP10(2019)067

1 Introduction

We know that Dark Matter (DM) exists because we observed its collective gravitational

interactions, but we do not know what DM is. Many theories are possible. Since gauge

interactions are maximally predictive in relativistic quantum ﬁeld theory, it makes sense to

explore theories where gauge dynamics leads to DM. We thereby add a new ‘dark’ gauge

group G. Its glue-balls could be DM without any interaction with the Standard Model

sector. In order to thermally reproduce the cosmological DM abundance we minimally

connect the dark sector to the Standard Model by adding one scalar ﬁeld S charged under

G. Depending on G, this leads to non-trivial accidental symmetries that imply DM sta-

bility with non-standard physics. Despite that light elementary scalars are considered as

unnatural by some theorists, interesting DM matter models based on scalars have already

been proposed:

1) The most minimal DM model in terms of new degrees of freedom involves just one

singlet scalar S [1–5]. This is stable imposing an ad-hoc Z

symmetry S → −S and

assuming that the S vacuum expectation value vanishes. Direct detection bounds

excluded a signiﬁcant part of the parameter space of this model [1–5].

2) Next, if the ﬁeld S is complex, describing two scalar degrees of freedom, it can be

charged under a new G = U(1) gauge group. A vacuum expectation value of S breaks

U(1) to nothing and the resulting massive vector A

is a DM candidate, stable thanks

to charge conjugation, S → S

∗

and A

→ −A

, which is a symmetry if the U(1) has

vanishing kinetic mixing with hypercharge [6].

3) A more interesting model where DM stability is automatically implied by the particle

content has been proposed in [7–9], assuming that the scalar S ﬁlls the fundamental

representation 2 of a new SU(2) gauge group. A vacuum expectation value of S

breaks SU(2) to nothing and the DM candidates are the three SU(2) vectors, which

acquire a common mass because of an accidental custodial symmetry.

The SU(2) model admits two apparently diﬀerent phases: Higgs and conﬁned. A non-

trivial feature of the SU(2) model — interesting even from a purely theoretical point of

view — is that the two phases give the same spectrum of asymptotic particles. The lack of

a sharp distinction between the Higgs and conﬁned phases in SU(2) theories with a scalar

in the fundamental has been proved by Fradkin, Shenker et al. [10–13]. A detailed analysis

of how this surprising duality applies to the Standard Model can be found in [14, 15] (we

now know that the SU(2)

gauge group is weakly coupled, so that in the SM this duality

has no physical interest).

We will ﬁnd extra examples of Higgs/conﬁnement dualities, and propose a general

criterion: such a duality holds when a scalar S in a representation R can break the gauge

group G to a unique sub-group H (and thereby with a Higgs phase that is unique). In these

cases S admits a single quartic coupling, and the broken theory contains a single Higgs

scalar, that we call s. This happens when S ﬁlls a fundamental of the SU(N), SO(N),

Sp(N), G

groups (up to equivalences). While in the original model [7–9] G = SU(2) gets

– 1 –

JHEP10(2019)067

fully broken, in our examples H has a non-trivial gauge dynamics — its own conﬁnement

— that must be taken into account. On the other hand, a scalar in the fundamental of

, E

, or in a higher representation of any group, such as a spinorial of SO(10),

instead has multiple quartic couplings and gives inequivalent breaking patterns, leaving

extra scalars in the broken theory.

We will here study theories that satisfy the Higgs/conﬁnement duality, and their ap-

plication to DM. Such theories can be seen as extensions of those previously listed in 1),

2), 3), and give qualitatively new physics. We consider one elementary scalar S in the fun-

damental representation of a gauge group G with vectors G

µν

in the adjoint. We consider

the most generic renormalizable Lagrangian

L = L

−

µν

a µν

− V

(

if S is complex,

/2 if S is real,

(1.1)

with scalar potential

(

−M

|S|

+ λ

|S|

− λ

|H|

|S|

if S is complex,

−M

/2 + λ

/4 − λ

|H|

/2 if S is real.

(1.2)

S is complex when G = SU(N) or Sp(N): in such cases the theory is invariant under an

accidental U(1) global symmetry, dark baryon number, that rotates the phase of S. S is

real when G = SO(N) or G

: we will discuss the accidental symmetries of these theories.

These minimal theories give non-trivial DM physics.

If G conﬁnes, baryons made of scalars S are stable DM candidates. As we will see,

their nature qualitatively depends on the group G. If S gets a vacuum expectation value,

G gets broken to a subgroup H,

SU(N) → SU(N − 1), SO(N) → SO(N − 1), Sp(N) → Sp(N − 2), G

→ SU(3),

(1.3)

and some massive vectors are accidentally stable DM candidates. At lower energy H

conﬁnes, giving rise to various states (dark glue-balls, dark mesons, . . .) and to baryonic

DM, in such a way that the Higgs/conﬁned and G-conﬁned phases are equivalent.

Models with a new conﬁning gauge group G and new fermions F have been explored

in [19–21]: in such models communication with the SM arises if F is charged also under

the SM gauge group: models need to be selected such that the composite DM is neutral.

The dark gauge group can have an extra topological term. In the absence of fermions, it cannot be

rotated away. Such term would violate CP at non-perturbative level. The SU(N ), SO(2N ) and E

groups

with symmetric Dynkin diagrams admit a Z

outer automorphism (complex conjugation) [16] that acts on

vectors by ﬂipping the sign of some vectors, as determined by the vanishing of some f

abc

group structure

constants. More simply, the CP-even vectors are those associated to purely imaginary generators T

some complex representation (e.g. fundamental or spinorial).

In order to avoid conﬁnement, [17, 18] considered non-minimal models with enough multiple scalars that

SU(N ) gets broken to nothing. We accept condensation and focus on the minimal scalar content. Given

that DM is the lightest stable particle, this also approximates the DM phenomenology of more general

theories provided that the extra particles are heavier at least by ∆M & Λ

, where Λ

is the scale at

which g

becomes strongly coupled, if unbroken.

– 2 –

JHEP10(2019)067

In scalar models, instead, we can assume that S is neutral under the SM (resulting into

a neutral DM candidate) because S interacts with the Higgs through the mixed scalar

quartic λ

The paper is structured as follows. In section 2 we consider the group G = SU(N)

and study both the Higgs and condensed phases, focusing on their equivalence, on the

accidental symmetry that protects the stability of DM, as seen by both the dual phases,

and on DM phenomenology. We then extend the analysis to the other groups for which

we ﬁnd that the duality holds: SO(N) (section 3), Sp(N) (section 4) and G

(section 5).

Conclusions are ﬁnally given in section 6, where we summarize our main results.

2 A fundamental of SU(N )

2.1 SU: Higgs phase

Independently of whether symmetry breaking happens dynamically, in the Higgs phase S

can always be written as

S(x) =

√







w + s(x)







(2.1)

such that the gauge group SU(N) gets broken to SU(N −1), leaving one degree of freedom

s in S. While hSi breaks dark baryon number U(1)

(under which S has charge 1), a

stable DM candidate remains thanks to an accidental global U(1) symmetry. Its generator

N(1, . . . , 1, 0)/(N − 1) is the unbroken linear combination of U(1)

and the broken U(1)

gauge symmetry in SU(N) corresponding to the generator (cf. appendix A)

−1

= diag(1, . . . , 1, 1 −N)/

2N(N − 1). (2.2)

Here and in the following, we normalize SU(N) generators in the fundamental represen-

tation as Tr(T

) =

. It is especially interesting to consider dynamical symmetry

breaking through the Coleman-Weinberg mechanism, obtained by setting M

= 0. Assum-

ing that λ

is negligibly small, the scalar S dynamically acquires a vacuum expectation

value w = s

∗

−1/4

where s

∗

is the Renormalization Group Equation (RGE) scale µ at

which the running quartic coupling λ

(µ) crosses 0, becoming negative at low energy, in

view of its RGE at one loop

(4π)

dlnµ

= −

22N−1

, (2.3a)

(4π)

dλ

dlnµ

(N−1)



−



−6Ng



1−



+4(4+N)λ

. (2.3b)

In such a case the scalar s is known as ‘scalon’ [22] and its mass squared is one-loop

suppressed, M

= w

, with β

≡ dλ

/d ln µ [9]. If the Higgs mass term is absent too,

this model can also generate the weak scale v, where v ≈ 246 GeV is the needed Higgs

– 3 –

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