Ark matter in minimal 3-3-1-1 model

TẠP CHÍ KHOA HỌC – ĐẠI HỌC TÂY BẮC Phạm Ngọc Thư, Lê Thu Lam (2020) 
Khoa học Tự nhiên và Công nghệ (20): 82 - 88
 DARK MATTER IN MINIMAL 3-3-1-1 MODEL
 Pham Ngoc Thu, Le Thu Lam 
 Tay Bac University
 Abstract: In this work, we investigate a minimal SU(3)C⊗ SU (3) LXN ⊗⊗ U (1) U (1) (3-3-1-1) model 
and argue how it improves the phenomenological and theoretical aspects of the known minimal 3-3-1 model. The 
lepton content includes only the standard model leptons, which form SU (3) L (anti)triplets, and the right-handed 
neutrinos, which transform as SU (3) L singlets, in order to cancel the BL− anomalies. We show that the 
electroweak and BL− interactions are unified, similarly to the electroweak theory, at the TeV scale. There is a 
residual symmetry of the gauge symmetry, called W -parity, which makes extra wrong BL− particles stabilized, 
provides dark matter candidates.
 Keywords: dark matter, standard model, 3-3-1-1 model, BL− anomalies, W - parity .
 I. INTRODUCTION our work and make conclusions in Sec. V.
 The greatest successes of the 20th century II. EXTENDING SU (2)L TO SU (3) L
physics must include the general theory of 
 In the standard model, the fermions come in 
relativity of gravitational interaction that acts at 
 generations. On the theoretical side, the number 
macroscopic scales and the standard model of 
 of fermion generations is arbitrary. However, 
electromagnetic, weak, and strong interactions 
 the experiments have proved that there are 
that governs the microscopic world. In spite of 
 only three generations of fermions in which 
those successes, the standard model and even 
 each generation includes two flavors of leptons 
the general relativity have limits, leaving many 
 ν , e and two flavors of quarks ud, , where 
open questions of the nature unsolved [1]. aa aa
 a = 1, 2, 3 is the generation index.
 The standard model explains only about 
 Under the standard model symmetry, 
5% mass-energy density of the universe. 
 SU(3)⊗⊗ SU (2) U (1) , the left-
What remains unknown is roundly 25% dark C LY
 handed fermions in each generation are 
matter and 70% dark energy. What causes the 
 arranged in doublets of SU (2) while 
inflationary expansion of the early universe? L
 the right-handed fermions are in singlets 
Those issues are the basic barriers, preventing T
 of this group: (ν aLe aL ) ~ (1, 2,− 1 / 2)
our next understanding of the macroscopic T
 , (ud ) ~ (3, 2,1 / 6) , e ~ (1,1,− 1) , 
and microscopic worlds, which have been aL aL aR
 u ~ (3,1, 2 / 3) , and d ~ (3,1,− 1 / 3) . The 
extensively studied. aR aL
 quantum numbers in parentheses are given upon 
 In this work, we pursue a new extension of the above gauge groups, respectively. There are 
the standard model, named the 3-3-1-1 model no right-handed neutrinos. The electric charge 
[2, 3], in hoping that it can provide some operator is related to the weak hypercharge as 
insights of the questions. QT=3 + Y, where T3 is the diagonal generator 
 of SU (2)L .
 In Sec. II, we show why SU (3) L is needed. 
In Sec. III, we show why 3-3-1-1 symmetry It is easily checked that each generation of 
is required. The minimal 3-3-1-1 model is the standard model is free from all the anomalies 
constructed in in Sec. IV, where dark matter associated with the gauge symmetry; therefore, 
candidates are identified. Finally, we summarize the anomalies are independent of generation 
*Tel: 0918201669, Email: phamngocthutb@gmail.com
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number which the model presents. In addition, 
 3 transforms as 3 under SU (3) L . The item 
the chiral anomaly [SU (2)L ] always vanishes 
without depending on any chiral fermion content “minimal” means that the lepton triplets are 
of the model since the anomaly contribution constructed from just lepton doublets and 
 singlets of the standard model (no extra particle 
 Aijk = Tr[{ TTi , j } T k ]= 0 for any chiral fermion 
representation T (i = 1, 2, 3) of SU (2) . The is required for the lepton triplets). Here, we 
 i L c cc
 have denoted eeRR≡=() () e L ( c indicates 
SU (2)L is a safe group because its anomaly is 
trivial. to charge conjugation) which implies that all 
 the components of triplets are left-handed as a 
 However, for an extension with higher chiral consequence of the commutation between the 
 =
symmetry such as SU() N ( N 3, 4, 5 ), the space-time and gauge symmetries.
corresponding Aijk generally does not vanish 
[4]. This subsequently gives a constraint on The SU (3) L self-anomaly is cancelled if 
the fermion content of the new model to cancel the number of fermion triplets equals to that 
the anomaly. Indeed, the minimal extension, of fermion antitriplets. Therefore, the first 
 two generations of quarks can be arranged 
SU (2)L to SU (3) L , provide a solution to the 
issue of three fermion generations observed in in antitriplets while the third generation of 
 − T *
the nature (because the generation nu ... (3) L L
generators). It is easily checked that model case). We can show this explicitly by 
 considering for a lepton triplet, therefore the 
 [Q, T± iT] =±±≠ ( T iT ) 0,
 12 12 BL− charge is given by BL− = diag(−1, −1, 
 [Q, T45±= iT]  ( T 45 ±≠ iT ) 0, (1) 1). We have nonzero commutations:
 [Q, T67±= iT]  2( T 67 ±≠ iT ) 0.
 [B− L, T45 ±= iT]  2( T 45 ±≠ iT ) 0,
 The non-closed algebras can be deduced 
 [B− L, T67 ±= iT]  2( T 67 ±≠ iT ) 0. (3)
from the fact that in order for Q to be some 
 Also, if BL− is algebraically 
generation of SU (3) L , we have a linear 
 closed with , it is a linear combination 
combination Q= xT and thus TrQ = 0 , which SU (3) L
 ii −=
is invalid for right-handed quark singlets such of Ti generators, B L aTii and thus Tr
 (BL−= )0. It is incorrect for a lepton 
as uR and dR as well as left-handed quark 
triplets and antitriplets. triplet, Tr(BL−=− ) 1, thus the algebras are 
 non-closed. Let us introduce a new Abelian 
 In other words, the Q and T by themselves 
 i charge N so that BL− is a residual charge 
do not make a symmetry under which our 
 of SU(3)⊗ U (1) , i.e. B−= L a T + bN . 
theory based on is manifest. Therefore, to have LN ii
 Acting BL− on a lepton triplet, we have
a closed algebra, we must introduce at least a 4
new Abelian charge X so that Q is a residual BL−=− T8 + N. (4)
 3
 ⊗
symmetry of closed group SU(3)LX U (1) , Let us note that the N cannot be identified as 
 = +
i.e. Q xii T yX . The coefficients xyi , can X since they are generally different for right-
be obtained when Q acts on a lepton triplet, for handed fermion singlets, even for left-handed 
example, to yield fermion triplets. They are independent charges. 
 −
 QT=−+383. T X (2) Similarly to the case of Q , BL and N are 
 gauged charges since T is a gauged charge. This 
 The weak rhyperchage can be identified as 8
 is a consequence of non-commuting BL− with 
Y=−+3 TX8 . If Q is an exact symmetry of 
 SU (3) L which is unlike the standard model. In 
the SU (3) L theory, the introduction of X is 
necessary since in the theory’s Lagrangian we other words, the theory is only consistent if it 
 contains at least the SU(3)LXN⊗⊗ U (1) U (1) 
can always find a conserved symmetry U (1) X 
so that Q is obtained by the above relation is gauge symmetry. Let us remind the reader that 
also conserved. By contrast, a violation in X the minimal 3-3-1 model always conserves 
 BL− . Therefore, in its Lagrangian we can find 
leads to the violation in Q . Since Ti are gauged 
charges, the Q and X must be gauged charges. a conserved Abelian charge N so that BL− 
The gauging of Q is a consequence of non- is related to as given above. The gauging of 
 BL− and N as a necessary requirement for 
commuting between Q and SU (3) L . In other 
words, the theory is only consistent if it contains a consistent theory were neglected.
at least the SU(3)LX⊗ U (1) gauge symmetry. In summary, the chiral symmetry theory 
This is like a similar one in the standard model.
 of weak interaction SU (3) L contains in it 
 It is evident that BL− is a good two conserved non-commuting charges, Q 
symmetry. In the standard model and minimal similarly to the standard model case while 
3-3-1 model, it is an exact symmetry. We have BL− characteristically for the new extension. 
not observed the violation of BL− on the The theory is only manifest or self-consistent 
experimental side. Therefore, the conservation if it works with a closed algebra based on the 
of B−L is naturally supposed in this theory [10]. following gauge symmetry: 
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 handed counterparts, which are unlike ordinary 
 SU(3)C⊗ SU (3) LXN ⊗⊗ U (1) U (1) , (5)
 quarks. The ν R fields are necessarily included 
 which is named 3-3-1-1 symmetry as just to cancel the BL− anomalies. We see that 
the component groups mean. A model based on ν R is not a sterile particle since it has BL− 
it is called 3-3-1-1 model, a self-contained and interaction, which is unlike that in the ordinary 
unified theory of strong, electroweak and BL− seesaw extension of the standard model. For the 
interactions. It is noteworthy that even for the phenomenological purpose, we also introduce a 
typical 3-3-1 models [6–8], we can always find variant of it, called ν ′ , which has only BL− 
 ′
a conserved symmetry U (1) N so that BL− (abnormal) charge, e.g. ν ~ (1,1, 0, 2) . This 
is the residual charge of SU(3)LN⊗ U (1) . particle is a standard model singlet, vectorlike, 
And, this charge has to be gauged since T8 is and often neglected. However, its presence may 
gauged. The 3-3-1 model is only self-consistent imply to the simplest case of a fermionic dark 
if we additionally include the gauge symmetry matter.To break the 3-3-1-1 gauge symmetry 
U (1) N [2, 3]. Otherwise, the 3-3-1 models must and generate the masses in a correct way, we 
contain the interactions which explicitly violate introduce the following scalars,
the lepton number. In fact, the 3-3-1 models η 0
are only survival if the lepton symmetry is 1
 ηη= −
an approximate symmetry, which have been 2 ~ (1, 3, 0, 2 / 3),
 η +
studied in [9, 11]. 3
 IV. PARTICLE CONTENT, R-PARITY +
 ρ1
AND TOTAL LAGRANGIAN 0
 ρρ= 2 ~ (1, 3,1, 2 / 3), 
 Under the 3-3-1-1 gauge symmetry, the ρ ++
 3
fermion representations of the model are given 
 −
as χ
 1
 ν aL −−
  χχ=2 ~ (1, 3, −− 1, 4 / 3), (10)
 ψ ≡−e ~ (1, 3, 0, 1/ 3) ,
 aL aL 0
 c χ
 e ν aR ~ (1,1, 0,− 1), 3
  aR (6)  
 φ ~ (1,1, 0, 2). (11)
 dα L
 
 ≡−* − − Here, the scalar triplets are given similarly 
 QuααLL ~ (3,3 , 1/3, 1/3),
  to the minimal 3-3-1 model, while φ is 
 Jα L
 necessarily included to break U (1) N totally 
 u3L as well as providing the masses for the right-
 
 ≡ handed neutrinos. The electrically-neutral 
 Q33LL d ~ (3, 3, 2 / 3,1) , (7)
  scalar components can develop VEVs as
 J3L
 u 0
ud~ (3,1, 2 / 3,1 / 3), ~ (3,1,− 1 / 3,1 / 3) (8) 1  1 
 aR aR 〈η 〉= 0,〈ρ 〉= v ,
 2  2
 0 0
Jα R ~ (3,1,−− 4 / 3, 5 / 3), J3R ~ (3,1, 5 / 3, 7 / 3). (9)  
 0
 The quantum numbers in parentheses are 1
 1  〈φ〉= Λ , (12)
defined upon the 3-3-1-1 groups, respectively. 〈χ 〉= 0,
 2  2
The X and N charges have been obtained, based w
on the relations (2), (4) and the known BL− 
 which can be given from the scalar potential 
and Q values of ordinary particles. Let us note 
 minimization. For the phenomenological 
that the exotic quarks J have anomalous Q and 
 a purpose, we also include a standard-
BL− charges as given by the last two numbers 
 model singlet scalar as a variant of φ
in the corresponding parentheses for their right-
 , e.g. φ′ ~ (1,1, 0,1) , which has only BL− 
 85
(abnormal) charge and cannot develop a VEV 3-3-1-1 gauge symmetry, when including the 
due to a reason discussed below. Furthermore, spin one. Remark: (i) all the particles that are 
this scalar singlet can couple the fermion singlet those of the minimal 3-3-1 model, U (1) N gauge 
ν ′ to the right-handed neutrinos, and all them boson, φ , and ν R acting as the essential blocks 
 ′
have U (1) N interactions too. The φ field may for a self-consistent theory are R -even (they 
be the simplest case of a scalar dark matter. have normal BL− number or only differ from 
 The 3-3-1-1 gauge symmetry is broken that value by even units), whereas (ii) the most 
 trivial particles, the singlet fermion (ν ′ ) and the 
down to SU(3)⊗⊗ U (1) U (1) 
 C Q BL− φ′
due to the VEVs of η , ρ and χ . This singlet scalar ( ), are R -odd, assumed that 
 −
breaking process undergoes two stages. they have wrong (abnormal) BL number. 
The first stage is the 3-3-1-1 symmetry to This stuff of particles can contribute dark 
 matter, and the candidate is either ν ′ or φ′ if it 
SU(3)C⊗ SU (2)L ⊗⊗ U (1) Y U (1) BL− 
due to w . The second stage is is lightest between the two. 
SU(3)C⊗ SU (2)L ⊗⊗ U (1) Y U (1) BL− to The total Lagrangian, up to the gauge mixing 
SU(3)C⊗⊗ U (1) Q U (1) BL− due to uv, . Note and ghost terms, is given by
that uv, also break the N charge to BL− , like µµ†
 =∑∑Fiγ Dµµ F + ( D S )( D S )−+ V Y 
w . Simultaneously, the VEV of φ breaks the FS
N or BL− symmetry to a discrete symmetry, 
denoted by P , UP(1) → . Above, Q 1 µν µν µν µν
 BL− −(Giiµν G + A iiµν A ++ BBµν CCµν ), (15)
is always conserved, responsibly for the 4
 where F and S indicate to the fermion 
electromagnetic interaction, while BL− must 
 multiplets and scalar multiplets, respectively. 
be broken (down to P ) because this kind of 
 The covariant derivative takes the form,
interaction should be short range, and its gauge 
 D=∂+ ig t G + igT A
boson get a large enough mass to escape from µµsi i µ i i µ (16)
 ++
the detection. In summary, the gauge symmetry igXN XBµµ ig NC ,
of the theory is broken as
 uvw,, ,Λ where (,tTXNii , , ), (gs ,, gg XN , g ), and 
 ⊗ ⊗⊗ → (13)
SU(3)C SU (3) LXN U (1) U (1) (,Gii A ,,) BC denote the generators, gauge 
SU(3)CQ⊗⊗ U (1) P . coupling-constants, and gauge bosons of the 
 3-3-1-1 groups, respectively. The field strength 
 Here, P is obtained by the reduction process: 
 tensors are
 SU(3)LN⊗→ U (1) U (1) BL− → P .
 Giµν=∂−∂− µ Gi ν ν G i µ gf s ijk G jµ G k ν , (17)
 First, BL− transforms component fields by 
 iα () BL− Aiµν=∂−∂− µ Ai ν ν A i µ gf ijk A jµ A k ν , (18)
Ue()α = , where BL−=−(4 / 3) T8 + N 
and α is a transforming parameter. Next, P Bµν=∂ µ B ν −∂ ν BC µ,, ∂ µν =∂ µ C ν −∂ ν C µ (19)
is a BL− subgroup that conserves the φ where f is the fine structure constant of
 i2α ijk
vacuum, i.e. U ()α Λ=Λ, with Ue()α = SU (3) group. The Yukawa Lagrangian and 
 i2α
for φ . We have e =1, thus απ= k , for scalar potential are obtained, respectively, by
k =0, ±± 1, 2, The discrete symmetry is ννc
 kBL()−  =hhψ ην + ′ ν ν φ
 P= Uk(π ) = ( − 1) . Considering only Y ab aL bR ab aR bR
 3(BL− ) e mnp c
the solution, k = 3, it becomes P =( − 1) +hab (ψψη aL )( m bL )( n ) p
. The angular momentum conservation implies JJ*
 ++hQ33 3LRχχ J 3 hαβ Q α L J β R (20)
that the spin parity (− 1) 2s is always conserved 
 ++udηρ
and unbroken. Therefore, P can be rewritten in hQ33a L u aR hQ 33 a L d aR
 = − 3(BL−+ ) 2 s du**
a convenient form, P ( 1) , which is ++hQααa Lηρ d aR hQ αα a L u aR + Hc.,
commonly called R -parity. Hence, the R -parity 
as studied in SUSY is interpreted as a residual 
86
 2† 2 † 2 † Kim, and N. T. Thuy, (2015) Phys. Rev. 
 V =++µηη µρρ µχχ
 12 3 D 91, 055023 
 2† † 2 † 2
 ++µφφ41 λ() ηη + λ 2 ( ρ ρ )
 3. P. V. Dong, Phys. Rev. D (2015), 92, 
 +λ( χ† χ ) 2 ++ λ ( φφ †2 ) λ ( ηη † )( ρ † ρ )
 3 45 055026 (2015); P. V. Dong and D. T. Si, 
 †† † †
 ++λ67( ηη )( χχ ) λ ( ρρ )( χχ ) “Kinetic mixing effect in the 3-3-1-1 
 †† † † model”, arXiv:1510.06815 [hep-ph]. 
 ++λ89( φφ )( ηη ) λ ( φφ )( ρρ )
 †† † † (21)
 ++λ10 ( φφ )( χ χ ) λ11 ( η ρ )( ρη ) 4. D. J. Gross and R. Jackiw, (1972) Phys. 
 ++ληχχηλ(†† )( ) ( ρχχρ †† )( ) Rev. D 6, 477; H. Georgi and S. L. 
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 Phạm Ngọc Thư*, Lê Thu Lam
 Trường Đại học Tây Bắc
 Tóm tắt: Trong bài báo này, chúng tôi khảo sát mô hình 3-3-1-1 tối thiểu và thảo luận về 
cách phát triển các hiện tượng luận và lý thuyết dựa trên mô hình 3-3-1 tối thiểu đã biết. Thành 
phần lepton chỉ bao gồm các lepton mô hình chuẩn được sắp xếp trong tam tuyến (phản tam tuyến) 
SU (3) L , và các neutrino phân cựa phải biến đổi như một đơn tuyến của SU (3) L , nhằm mục đích 
khử dị thường B - L. Chúng tôi cũng chỉ ra rằng các tương tác yếu và B – L là không đồng nhất 
tương tự như trong lý thuyết điện yếu ở thang đo TeV. Tồn tại một đối xứng gián đoạn của đối xứng 
chuẩn là W - parity. W - parity làm các hạt lepton-sai ổn định và cung cấp các ứng viên cho vật 
chất tối.
 Từ khóa: vật chất tối, mô hình chuẩn, mô hình 3-3-1-1, dị thường BL− , W - parity.
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Ngày nhận bài: 9/4/2020. Ngày nhận đăng: 2/5/2020
Liên lạc: thupn@utb.edu.vn
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