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JHEP09(2016)076

Published for SISSA by Springer

Received: July 22, 2016

Accepted: September 10, 2016

Published: September 13, 2016

A new B − L model without right-handed neutrinos

Sudhanwa Patra,

Werner Rodejohann

and Carlos E. Yaguna

Center of Excellence in Theoretical and Mathematical Sciences,

Siksha ’O’ Anusandhan University, Bhubaneswar-751030, India

Max-Planck-Institut f¨ur Kernphysik,

Saupfercheckweg 1, 69117 Heidelberg, Germany

E-mail: sudha.astro@gmail.com, werner.rodejohann@mpi-hd.mpg.de,

carlos.yaguna@mpi-hd.mpg.de

Abstract: We propose and study a novel extension of the Standard Model based on the

B − L gauge symmetry that can account for dark matter and neutrino masses. In this

model, right-handed neutrinos are absent and the gauge anomalies are canceled instead by

four chiral fermions with fractional B − L charges. After the breaking of U(1)

B−L

, these

fermions arrange themselves into two Dirac particles, the lightest of which is automatically

stable and plays the role of the dark matter. We determine the regions of the parameter

space consistent with the observed dark matter density and show that they can be partially

probed via direct and indirect dark matter detection or collider searches at the LHC.

Neutrino masses, on the other hand, can be explained by a variant of the type-II seesaw

mechanism involving one of the two scalar ﬁelds responsible for the dark matter mass.

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

Symmetry, Neutrino Physics

ArXiv ePrint: 1607.04029

Open Access,

 The Authors.

Article funded by SCOAP

doi:10.1007/JHEP09(2016)076

JHEP09(2016)076

have fractional charges under U(1)

B−L

. These charges forbid any tree level interactions

between the Standard Model particles and the new fermions, rendering the lightest of them

automatically stable and therefore a viable dark matter candidate. Two important features

of this model are thus that the ﬁelds responsible for anomaly cancellation also explain the

dark matter and that the stability of the dark matter particle is automatic — there is no

need to impose any extra discrete symmetries to ensure it.

Besides these four chiral fermions, the model includes two scalar ﬁelds, also singlets of

the SM, with B−L charges 1 and 2, which spontaneously break the B−L symmetry and give

Dirac-type masses to the new fermions. Another scalar ﬁeld, a triplet of SU(2), is further

required to explain neutrino masses via a variant of the type-II seesaw mechanism [33–36].

Interestingly, the necessary induced vacuum expectation value is here generated by one

of the scalar particles responsible for the dark matter mass, thus indirectly connecting

neutrino masses and dark matter.

The plan of the paper is as follows. In the next section the model is introduced and

described in detail. We write down the full Lagrangian, implement symmetry breaking, and

ﬁnd the fermion and scalar mass matrices. The dark matter phenomenology is presented in

section 3. Speciﬁcally, we determine the regions of the parameter space consistent with the

observed value of the dark matter density and discuss the role of current and planned dark

matter experiments in probing them. In section 4 the LHC bounds are examined while

in section 5 we explain how neutrino masses are generated within this model. Finally, we

summarize our results in section 6.

2 A new U(1)

B−L

gauged model

The B −L gauge extension of Standard Model (SM), where the diﬀerence between baryon

and lepton number is deﬁned as a local gauge symmetry, is one of the simplest extensions

from the point of view of a self-consistent gauge theory. It naturally appears in well-

motivated scenarios for physics beyond the SM, such as left-right theories and uniﬁcation

models. Here, we will focus on a model based on the SU(3)

×SU(2)

×U(1)

B−L

gauge symmetry. With just the SM fermions, this model is not anomaly-free as both



U(1)

B−L



= A

(gravity)

× U(1)

B−L

, (2.1)

are non-zero. The usual way of overcoming this problem is to add right-handed neutrinos

, (i = 1, 2, 3), each of which has a B − L charge of −1. In addition, these right-handed

neutrinos may also explain neutrino masses via a type-I seesaw mechanism.

In this paper, we would like to propose an alternative way of canceling the gauge

anomalies that does not invoke right-handed neutrinos. As we will see, this novel sce-

nario provides a direct connection to dark matter and oﬀers also an interesting link to

neutrino masses.

2.1 Particle content

In a model without right-handed neutrinos, the B − L gauge anomalies can be canceled

instead by the following four chiral fermions

(4/3), η

(1/3), χ

(−2/3), χ

(−2/3) , (2.2)

– 2 –

JHEP09(2016)076

Field SU(2)

× U(1)

U(1)

B−L

Fermions Q

≡ (u, d)

(2, 1/6) 1/3

(1, 2/3) 1/3

(1, −1/3) 1/3

≡ (ν, e)

(2, −1/2) −1

(1, −1) −1

(1, 0) 4/3

(1, 0) 1/3

(1, 0) −2/3

Scalars H (2, 1/2) 0

(1, 0) 1

(1, 0) 2

∆ (3, 1) −2

Table 1. Particle content of the U(1)

B−L

model.

which are singlets under the SM gauge group but have fractional charges under B −L (the

number in parenthesis). Here the ﬁelds ξ

and η

are left-handed, while χ

(i = 1, 2) are

right-handed. First of all, let us check that the gauge anomalies indeed vanish



U(1)

B−L



= A



U(1)

B−L



+ A

New



U(1)

B−L



= −3 +



(4/3)

+ (1/3)

− (−2/3)



= 0

(gravity)

× U(1)

B−L

= A

+ A

New

= −3 + [(4/3) + (1/3) − (−2/3) − (−2/3)] = 0 .

In addition to these fermions, the model includes two new scalars, φ

, φ

, also singlets

under the SM, with B −L charges 1, 2 respectively, which break the B −L symmetry and

give masses, via their vevs, to the new fermions. These fermions arrange themselves into

two Dirac particles, the lightest of which is automatically stable — without the need of

ad hoc discrete symmetries — and constitutes a viable dark matter candidate. Thus, the

dark matter is explained in this model by the same ﬁelds that are required to cancel the

gauge anomalies. Moreover, since the correct relic density is obtained, within the thermal

scenario, for dark matter masses around the TeV scale, the B −L breaking scale should also

lie close to TeV and, therefore, not far from the LHC reach. Hence, this scenario predicts

a low B − L breaking scale and could be tested not only via dark matter experiments but

also at colliders.

Finally, one more scalar, ∆, triplet of SU(2) and with B −L = −2, helps neutrinos to

acquire non-zero Majorana masses via a variant of the type-II seesaw mechanism involving

also φ

. Indeed, as explained in section 5, the vacuum expectation value of ∆ is induced by

the SM Higgs H and the scalar φ

, thus linking neutrino masses and dark matter within this

model. The complete particle content, with the respective quantum numbers, is presented

in table 1.

– 3 –

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