We study a gauge $B-L$ extension of the standard model of quarks and leptons
with unconventional charges for the singlet right-handed neutrinos, and extra
singlet scalars, such that a residual $Z_3$ symmetry remains after the
spontaneous breaking of $B-L$. We discuss the phenomenological consequences of
this scenario, including the possibility of long-lived self-interacting dark
matter and $Z'$ collider signatures.Comment: 11 pages, 2 figures, references added, fig.1 correcte

Ma, Ernest

Pollard, Nicholas

Srivastava, Rahul

Zakeri, Mohammadreza

English

arXiv

We study a gauge B–L extension of the standard model of quarks and leptons with unconventional charges for the singlet right-handed neutrinos, and extra singlet scalars, such that a residual Z3 symmetry remains after the spontaneous breaking of B–L. We discuss the phenomenological consequences of this scenario, including the possibility of long-lived self-interacting dark matter and Z′ collider signatures

Ernest Ma

Nicholas Pollard

Rahul Srivastava

Mohammadreza Zakeri

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Physics Letters B

Gauge B–L model with residual Z3 symmetry

AbstractWe study a gauge B–L extension of the standard model of quarks and leptons with unconventional charges for the singlet right-handed neutrinos, and extra singlet scalars, such that a residual Z3 symmetry remains after the spontaneous breaking of B–L. We discuss the phenomenological consequences of this scenario, including the possibility of long-lived self-interacting dark matter and Z′ collider signatures

Elsevier - Publisher Connector 

Gauge B–L model with residual Z3 symmetry 

Physics Letters B 750 (2015) 135–138Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Gauge B–L model with residual Z3 symmetry
Ernest Ma a, Nicholas Pollard a, Rahul Srivastava b, Mohammadreza Zakeri a,∗
a Department of Physics and Astronomy, University of California, Riverside, CA 92521, USA
b The Institute of Mathematical Sciences, Chennai 600 113, India
a r t i c l e i n f o a b s t r a c t
Article history:
Received 22 July 2015
Received in revised form 28 August 2015
Accepted 3 September 2015
Available online 7 September 2015
Editor: J. Hisano
We study a gauge B–L extension of the standard model of quarks and leptons with unconventional 
charges for the singlet right-handed neutrinos, and extra singlet scalars, such that a residual Z3 symmetry 
remains after the spontaneous breaking of B–L. We discuss the phenomenological consequences of this 
scenario, including the possibility of long-lived self-interacting dark matter and Z ′ collider signatures.
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license 
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.Lepton number L is a familiar concept. It is usually deﬁned as a 
global U (1) symmetry, under which the leptons of the standard 
model (SM), i.e. e, μ, τ together with their neutrinos νe, νμ, ντ
have L = 1, and all other SM particles have L = 0. In the case of 
nonzero Majorana neutrino masses, this continuous symmetry is 
broken to a discrete Z2 symmetry, i.e. (−1)L or lepton parity. In 
this paper, we consider a gauge B–L extension of the SM, such 
that a residual Z3 symmetry remains after the spontaneous break-
ing of B–L. This is then a realization of the unusual notion of Z3
lepton symmetry. It has speciﬁc phenomenological consequences, 
including the possibility of a long-lived particle as a dark-matter 
candidate.
The conventional treatment of gauge B–L has three right-
handed singlet neutrinos νR1, νR2, νR3 transforming as −1, −1, −1
under B–L. It is well known that this assignment satisﬁes all 
the anomaly-free conditions for U (1)B–L . However, another assign-
ment [1]
νR1, νR2, νR3 ∼ 5,−4,−4 (1)
works as well, because
5− 4− 4 = −3, (5)3 − (4)3 − (4)3 = −3. (2)
To obtain a realistic model with this assignment, it was recently 
proposed [2] that three additional neutral singlet Dirac fermions 
N1,2,3 be added with B–L = −1, together with a singlet scalar 
χ3 with B–L = 3. Consequently, the tree-level Yukawa couplings 
ν¯LNR φ¯0 and N¯LνR2χ3, N¯LνR3χ3 are allowed, where  = (φ+, φ0)
is the one Higgs doublet of the SM. Together with the invariant 
* Corresponding author.
E-mail address: zaki1905@gmail.com (M. Zakeri).http://dx.doi.org/10.1016/j.physletb.2015.09.010
0370-2693/© 2015 The Authors. Published by Elsevier B.V. This is an open access article
SCOAP3.N¯LNR mass terms, the 6 × 5 neutrino mass matrix linking (ν¯L, N¯L)
to (νR , NR) is of the form
MνN =
(
0 M0
M3 MN
)
, (3)
where M0 and MN are 3 × 3 mass matrices and M3 is 3 × 2 be-
cause νR1 has no tree-level Yukawa coupling. This means that one 
linear combination of νL is massless. Of course, if the dimension-
ﬁve term ν¯R1NLχ23 also exists, then M3 is 3 ×3 and MνN is 6 ×6.
The form of MνN allows nonzero seesaw Dirac neutrino masses 
for ν [3], i.e.
Mν M0M−1N M3. (4)
Without the implementation of a ﬂavor symmetry, any 3 × 3 Mν
is possible. Although the gauge B–L is broken, a residual global 
L symmetry remains in this model with ν, l, N all having L = 1. 
Because the pairing of any two neutral fermions of the same chi-
rality always results in a nonzero B–L charge not divisible by 3 
units in this model, it is impossible to construct an operator of any 
dimension for a Majorana mass term which violates B–L. Hence 
the neutrinos are indeed exactly Dirac.
We now add two more scalar singlets: χ2 with B–L = 2 and χ6
with B–L = −6. The important new terms in the Lagrangian are
N¯LνR1χ6, χ2NLNL, χ2NRNR , χ
3
2χ6, χ
2
3χ6. (5)
Now B–L is broken by 〈χ3〉 = u3 as well as 〈χ6〉 = u6, and all 
neutrinos become massive. If χ2 is absent, then again a residual 
global L symmetry exists with L = 1 for ν, l, N and L = 0 for χ3,6. 
However, the existence of χ2 shows that the residual symmetry is 
then Z3, such that χ2 and all leptons transform as ω = exp(2π i/3)
under Z3 with χ3,6 ∼ 1. This is thus the ﬁrst example of a lepton  under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by 
136 E. Ma et al. / Physics Letters B 750 (2015) 135–138symmetry which is not Z2 (for Majorana neutrinos), nor U (1) or 
Z4 [4,5] (for Dirac neutrinos). Note that Z3 is also suﬃcient to 
guarantee that all the neutrinos remain Dirac.
Although there is no stabilizing symmetry here for dark matter, 
χ2 has very small couplings to two neutrinos through the Yukawa 
terms of Eq. (5) from the mixing implied by Eq. (3). This means 
that χ2 may have a long enough lifetime to be suitable for dark 
matter, as shown below.
Consider for simplicity the coupling of χ2 to just one N , with 
the interaction
Lint = 12 f Lχ2NLNL +
1
2
f Rχ2NRNR +H.c. (6)
Let the νL–NL mixing be ζ0 =m0/mN and νR–NR mixing be ζ3 =
m3/mN , then the decay rate of χ2 is

(χ2 → ν¯ν¯) = mχ
32π
( f 2L ζ
4
0 + f 2Rζ 43 ). (7)
If we set this equal to the age of the Universe (13.75 × 109 yr), 
and assuming mχ = 100 GeV, f L = f R and ζ0 = ζ3, then f ζ 2 =
8.75 × 10−22. Hence√
f ζ << 3× 10−11 (8)
would guarantee the stability of χ2 to the present day, and allow 
it to be a dark-matter candidate. This sets the scale of mN at about 
1013 GeV, which is also the usual mass scale for the heavy Majo-
rana singlet neutrino in the canonical seesaw mechanism.
In this model, there is of course a gauge boson Z ′ which cou-
ples to B–L. Its production at the Large Hadron Collider (LHC) 
is due to its couplings to quarks. Once produced, it decays into 
quarks and leptons. In the conventional B–L assignment for νR , 
its branching fractions to quarks, charged leptons, and neutrinos 
are 1/4, 3/8, and 3/8 respectively. In this model, the νR charges 
are (5, −4, −4), hence their resulting partial widths are very large. 
Assuming that Z ′ decays also into χ2, the respective branching 
fractions into quarks, charged leptons, neutrinos, and χ2 as dark 
matter are then 1/18, 1/12, 5/6, and 1/36. This means Z ′ has an 
86% invisible width. Using the production of Z ′ via uu¯ and dd¯
initial states at the LHC and its decay into e−e+ or μ−μ+ as sig-
nature, the current bound on mZ ′ assuming g′ = g , i.e. the SU(2)L
gauge coupling of the SM, is about 3 TeV, based on recent LHC 
data [6,7]. However, because the branching fraction into l−l+ is re-
duced by a factor of 2/9 in our B–L model, this bound is reduced 
to about 2.5 TeV, again for g′ = g . There is also a similar bound [8]
from precision e−e+ → e−e+ measurements at the Large Electron 
Positron Collider (LEP), i.e. mZ ′/g′ > a few TeV.
Since χ2 interacts with nuclei through Z ′ , there is also a signif-
icant constraint from dark-matter direct-search experiments. The 
cross section per nucleon is given by
σ0 = 1
π
(
mχmn
mχ + Amn
)2(2g′2
m2Z ′
)2
, (9)
where A is the number of nucleons in the target and mn is the nu-
cleon mass. Consider for example mχ = 100 GeV, then σ0 < 1.25 ×
10−45 cm2 from the recent LUX data [9]. This implies mZ ′/g′ >
16.2 TeV, as shown in Fig. 1. If g′ = g , then mZ ′ > 10.6 TeV. This 
limit is thus much more severe than the LHC bound of 2.5 TeV. 
If g′ < g , then both the LHC and LUX bounds on mZ ′ are relaxed. 
However, it also means that it is unlikely that Z ′ would be discov-
ered at the LHC even with the 14 TeV run.
Consider now the annihilation cross section of χ2χ∗2 for ob-
taining its thermal relic abundance. The process χ2χ∗ → Z ′ → SM 2Fig. 1. Lower bound on mZ ′ /g′ versus mχ from LUX data.
Fig. 2. χ2χ
†
2 annihilation to χ3,6 ﬁnal states.
particles is p-wave suppressed and is unlikely to be strong enough 
for this purpose. We may then consider the well-studied process 
χ2χ
∗
2 → h → SM particles, where h is the SM Higgs boson. If this 
is assumed to account for all of the dark-matter relic abundance 
of the Universe, then it has recently been shown [10] that the 
required strength of this interaction is in conﬂict with LUX data 
except for a small region near mχ =mh/2. On the other hand, an-
other analysis [11] claims that a region with mχ somewhat greater 
than mh is still allowed.
In this paper, we will consider the following alternative sce-
nario. We assume that the hχ2χ∗2 interaction is negligible, so that 
neither Higgs nor Z ′ exchange is important for χ2χ∗2 annihilation. 
Instead we invoke the new interactions of Fig. 2. Since χ3,6 may 
interact freely with h, thermal equilibrium is maintained with the 
other SM particles. This scenario requires of course that mχ to be 
greater than at least one physical mass eigenvalue in the χ3,6 sec-
tor.
To summarize, χ2 ∼ ω under Z3 and decays into two antineu-
trinos, but its lifetime is much longer than the age of the Universe. 
It is thus an example of Z3 dark matter [12–16]. It is also different 
from previous Z2 proposals [17,18] based on Ref. [1]. It has signiﬁ-
cant elastic interactions with nuclei through Z ′ and Higgs exchange 
and may be discovered in direct-search experiments. On the other 
hand, its relic abundance is determined not by Z ′ or Higgs interac-
tions, but by its annihilation to other scalars of this model which 
maintain thermal equilibrium with the SM particles through the 
SM Higgs boson. Note that this is also the mechanism used in a 
recently proposed model of vector dark matter [19].
E. Ma et al. / Physics Letters B 750 (2015) 135–138 137We now discuss the details of the scalar sector of this model. 
Consider the scalar potential
V = −μ20(†) +m22(χ∗2χ2) − μ23(χ∗3χ3) − μ26(χ∗6χ6)
+ 1
2
λ0(
†)2 + 1
2
λ2(χ
∗
2χ2)
2 + 1
2
λ3(χ
∗
3χ3)
2
+ 1
2
λ6(χ
∗
6χ6)
2 + λ02(χ∗2χ2)(†)
+ λ03(χ∗3χ3)(†) + λ06(χ∗6χ6)(†)
+ λ23(χ∗2χ2)(χ∗3χ3) + λ26(χ∗2χ2)(χ∗6χ6)
+ λ36(χ∗3χ3)(χ∗6χ6)
+ [1
2
f36(χ
2
3χ6) +H.c.] + [
1
6
λ′26(χ32χ6) +H.c.]. (10)
Let 〈φ0〉 = v , 〈χ3〉 = u3, 〈χ6〉 = u6, then the minimum of V is de-
termined by
μ20 = λ0v2 + λ03u23 + λ06u26, (11)
μ23 = λ3u23 + λ03v2 + λ36u26 + f36u6, (12)
μ26 = λ6u26 + λ06v2 + λ36u23 +
f36u23
2u6
. (13)
There is one dark-matter scalar boson χ2 with mass given by
m2χ =m22 + λ02v2 + λ23u23 + λ26u26. (14)
There is one physical pseudoscalar boson
A = √2Im(2u6χ3 + u3χ6)/
√
u23 + 4u26 (15)
with mass given by
m2A = − f36(u23 + 4u26)/2u6. (16)
There are three physical scalar bosons spanning the basis [h,√
2Re(χ3),
√
2Re(χ6)], with 3 × 3 mass-squared matrix given by
M2 =
⎛
⎝ 2λ0v2 2λ03u3v 2λ06u6v2λ03u3v 2λ3u23 2λ36u3u6 + f36u3
2λ06u6v 2λ36u3u6 + f36u3 2λ6u26 − f36u23/2u6
⎞
⎠ .
(17)
For illustration, we consider the special case λ03 = λ06 = 0, so that 
h decouples from χ3,6. It then becomes identical to that of the SM, 
and may be identiﬁed with the 125 GeV particle discovered [20,21]
at the LHC. We now look for a solution with
S = √2Re(−u3χ3 + 2u6χ6)/
√
u23 + 4u26, (18)
S ′ = √2Re(2u6χ3 + u3χ6)/
√
u23 + 4u26, (19)
as mass eigenstates. This is easily accomplished for example with
u3 = 2u6, 4λ3 = λ6 − f36/u6. (20)
In this case,
S = −Reχ3 + Reχ6, m2S = 2λ6u26 − 4λ36u26 − 4 f36u6, (21)
S ′ = Reχ3 + Reχ6, m2S ′ = 2λ6u26 + 4λ36u26, (22)
A = Imχ3 + Imχ6, m2A = −4 f36u6, (23)
mZ ′ = 12g′u6. (24)
The couplings of χ2χ∗2 to S and S ′ are given by
χ2χ
∗
2 [u6(λ26 − 2λ23)S + u6(λ26 + 2λ23)S ′]. (25)Since S plays the same role in breaking B–L as the Higgs boson 
h does in breaking SU(2)L × U (1)Y , it is expected to be massive 
of order 
√
u23 + 4u26 = 2
√
2u6. This allows mS ′ to be adjusted to be 
very small, then it may serve as a light scalar mediator for χ2 as 
self-interacting dark matter [22]. This is not a necessary assump-
tion of our model and requires ﬁne tuning of scalar parameters to 
achieve. We merely want to demonstrate that such a possible sce-
nario exists within our model. For mS ′  0, we need λ36 = −λ6/2. 
In that case, using Eq. (20), we ﬁnd
m2S = 16λ3u26, m2A =m2S − 4λ6u26. (26)
We assume that the relic density of χ2 is dominated by the 
χ2χ
∗
2 annihilation to S
′S ′ . This may have to be revised if the semi-
annihilation χ2χ∗2 → χ2S ′ is sizeable. Here we simply assume 
that λ′26 is small. For illustration, we set to zero the χ2χ∗2 S ′S ′
coupling, i.e. λ23 + λ26 = 0, as well as the S S ′S ′ coupling, i.e. 
−12λ3 + 6λ6 + 2λ36 − f36/u6 = 0. This implies λ3 = λ6/2 so that 
the S ′S ′S ′ coupling is also zero and m2A =m2S/2. This choice of pa-
rameters means that only the middle diagram of Fig. 2 contributes 
to the χ2χ∗2 annihilation cross section with
σ × vrel = 164πm2χ
∣∣∣∣∣λ
2
26u
2
6
m2χ
∣∣∣∣∣
2
. (27)
Equating this to the optimal value [23] of 4.4 × 10−26 cm3 s−1 for 
the correct dark-matter relic density of the Universe, we ﬁnd for 
mχ = 100 GeV
λ26 = 0.0295
(
1 TeV
u6
)
. (28)
We assume of course that mA > 2mχ .
For S ′ to be in thermal equilibrium with the SM particles, we 
consider nonzero values of λ03 and λ06. This is possible in our cho-
sen parameter space if 2λ03 + λ06  0, so that the S ′h mixing is 
very small and yet the S ′S ′h coupling λ06v/4
√
2 and S ′S ′hh cou-
pling λ06/16 may be signiﬁcant. Note that the Sh mixing is now 
ﬁxed at (λ06/λ6)(v/2
√
2u6) which may yet be suitably suppressed 
for h to be essentially the one Higgs boson of the SM. Even if λ03,06
are negligible, the gauge interaction S ′AZ ′ may also be suﬃcient 
to maintain thermal equilibrium. This may also affect the magni-
tude of the self-interacting χ2χ∗2 cross section.
The h → S ′S ′ decay width is given by

(h → S ′S ′) = λ
2
06v
2
256πmh
=
(
λ06
0.04
)2
0.5 MeV. (29)
It is invisible at the LHC because S ′ decays slowly to e−e+ only 
through its mixing with h, if mS ′ ∼ 10 MeV for S ′ as a light medi-
ator for the self-interacting dark matter χ2.
In conclusion, we have considered the unusual case of a gauge 
B–L symmetry which is spontaneously broken to Z3 lepton num-
ber. Neutrinos are Dirac fermions transforming as ω = exp(2π i/3)
under Z3. A complex neutral scalar χ2 exists which also transforms 
as ω. It is not absolutely stable, but decays to two antineutrinos 
with a lifetime much greater than that of the Universe. It is thus 
an example of Z3 dark matter. In addition to the one Higgs boson 
h of the SM, there are three neutral scalars S, S ′, A and one heavy 
vector gauge boson Z ′ . From direct-search experiments, mZ ′/g′ is 
constrained to be very large, thus making it impossible to discover 
Z ′ at the LHC even with the current run. The relic abundance of 
χ2 is determined by its annihilation into S ′ which is a candidate 
for the light mediator by which χ2 obtains its long-range self-
interaction.
138 E. Ma et al. / Physics Letters B 750 (2015) 135–138Acknowledgement
This work is supported in part by the U.S. Department of Energy 
under Grant No. DE-SC0008541.
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Gauge $B-L$ Model with Residual $Z_3$ Symmetry

Gauge $B-L$ Model with Residual $Z_3$ Symmetry

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Gauge B−LB-LB−L Model with Residual Z3Z_3Z3​ Symmetry

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Gauge $B-L$ Model with Residual $Z_3$ Symmetry