Dark matter in minimal 3-3-1-1 model

Nội dung text: Dark matter in minimal 3-3-1-1 model

1. 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 82
2. 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 number must quarks is in triplets: (dαL uJ αα LL) ~3 and (udJ333LLL )~3, where α =1, 2 is a generation be an integer multiple of three due to Aijk = 0 TT , simultaneously the QCD asymptotic freedom index. Note that (d−= u ) iσ 2 () ud T condition bounds it not larger than five) [5]. transforms as a SU (2)L antidoublet if ()ud is a doublet, where iσ is the SU (2) In this work, let us reexamine the extension 2 L asymmetric tensor. There are 6 fermion triplets SU (2) to SU (3) and derive a new consistent L L (3 for three lepton generations plus 3 for third gauge-symmetry theory which arises as a generation of quarks since each quark having 3 natural result. For this aim, we write the new fundamental colors) and 6 fermion antitriplets theory that firstly contains two gauge groups, (for first two generations of quarks with each SU (3) for QCD and SU (3) for electroweak. C L quark possessing 3 fundamental colors) which From the necessary algebraic-closure condition match as expected. The right-handed quarks of the theory with other conserved non- transform as SU (3) singlets: u ~1, d ~1 commuting charges, e.g. electric charge and L aR aR and J ~1. The third components of quark BL− , we finally complete the gauge symmetry aR triplets and antitriplets have to be the new (which is needed for a manifest theory). The particles J since the ordinary right-handed resulting theory (see, for example, [2, 3]) will a quarks (u and d ) cannot take place (unlike significantly differs from the traditionally- R R the lepton triplets). It is due to the fact that the studied 3-3-1 model [6–8]. Under this view, the SU (3) , SU (3) and space-time symmetry in 3-3-1 model is not self-consistent; however, for C L each pair commute as such we cannot reverse an alternative in trying to keep the 3-3-1 model the chirality of right-handed quarks that remains surviving, see [9]. the same color index as the doublet components While the SU (3)C representations retain, (furthermore, it is known that their electric the electroweak doublets will be enlarged to charges are not satisfied too). become respective representations of SU (3) L III.ALGEBRAIC CLOSURE CONDITION . Notice that a fundamental representation of * AND 3-3-1-1 GAUGE SYMMETRY SU (3) L , 3 or 3 , is composed from a doublet or antidoublet and a singlet of SU (2)L Similarly to the standard model, the electric , i.e. 321= ⊕ or 321 = ⊕ , respectively. charge operator does not commute and non- Therefore, for a minimal extension, the chiral closes algebraically with the gauge symmetry cT leptons are arranged as ()ν aLee aL aR which SU (3) L . To be concrete, for a lepton triplet, 83
3. we have Q =diag(0,1,1) −+, which is not Similarly to Q, we can show that BL− does 1 not commute and non-closes algebraically with commuted with Ti = for i =1,2, 4,5,6,7 2 SU (3) too (thus, it differs from the standard (where Ti , i =1,2,3, ,8 , are SU (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: 84
4. 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
5. (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
6. 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. 12 13 Glashow, (1972), Phys. Rev. D 6, 429; J. ++µmnp ηρχ ( mn p Hc. ), Banks and H. Georgi, (1976), Phys. Rev. where the Yukawa couplings h ’s and the D 14, 1159; S. Okubo, (1977), Phys. Rev. scalar couplings λ ’s are dimensionless, whereas D 16, 3528. µ and µ have the mass dimension. 1,2,3,4 5. P. H. Frampton in [6]; A. Doff and F. V. CONCLUSION Pisano, (2000), Mod. Phys. Lett. A 15, Let us note that the mass spectra of the 1471 ; P. V. Dong and H. N. Long, (2006), normal scalars, normal fermions, and gauge Int. J. Mod. Phys. A 21, 6677. bosons as given by the above Lagrangians are 6. F. Pisano and V. Pleitez, (1992), Phys. independent of the dark sector (ν ′ , φ′ ), because Rev. D 46, 410; P. H. Frampton, (1992), of the R -parity conservation. There may be Phys. Rev. Lett. 69, 2889; R. Foot, O. a kinetic mixing effect between the U (1) X F. Hernandez, F. Pisano and V. Pleitez, and U (1) N gauge bosons. The dark matter (1993) Phys. Rev. D 47, 4158. candidate (ν ′ or φ′ ) is stabilized by the parity 7. M. Singer, J. W. F. Valle and J. Schechter, conservation. It can obtain a correct abundance (1980), Phys. Rev. D 22, 738; J. C. via the annihilation to standard model particles Montero, F. Pisano and V. Pleitez, (1993) via new gauge and Higgs portals. Phys. Rev. D 47, 2918; R. Foot, H. N. Long and Tuan A. Tran, (1994), Phys. REFERENCE Rev. D 50, 34. 8. P. V. Dong, L. T. Hue, H. N. Long, and D. 1. B.C.H. Steele (1994), “Oxygen transport V. Soa, (2010), Phys. Rev. D 81, 053004; and exchange in oxide ceramics”, J. P. V. Dong, H. N. Long, D. V. Soa, and Power Sources, 49, tr.1. V. V. Vien, (2011), Eur. Phys. J. C 71, 2. C.C. Wei, K. Li (2008), “Yttria-Stabilized 1544; P. V. Dong, H. N. Long, C. H. Nam, Zirconia (YSZ)-Based Hollow Fiber and V. V. Vien, (2012), Phys. Rev. D 85, Solid Oxide Fuel Cells”, Ind. Eng. Chem. 053001. Res., 47, tr.1506. 9. P. V. Dong, T. Phong Nguyen, and D. V. 3. Jagadish C. Ray, Soa, (2013), Phys. Rev. D 88, 095014; P. 1. K. A. Olive et al (Particle Data Group), V. Dong, N. T. K. Ngan, and D. V. Soa, (2014), Chin. Phys. C 38, 090001 and (2014), Phys. Rev. D 90, 075019; P. V. partial updates at Dong, C. S. Kim, N. T. Thuy, and D. V. Soa, (2015), Phys. Rev. D 91, 115019. 2. P. V. Dong, T. D. Tham, and H. T. Hung, (2013), Phys. Rev. D 87, 115003; P. V. 10. M. B. Tully and G. C. Joshi, (2001) Phys. Dong, D. T. Huong, F. S. Queiroz, and Rev. D 64, 011301. N. T. Thuy, (2014), Phys. Rev. D 90, 11. W. A. Ponce, Y. Giraldo and L. A. Sanchez, 075021; D. T. Huong, P. V. Dong, C. S. (2003) Phys. Rev. D 67, 075001; P. V. 87
7. Dong, H. N. Long, D. T. Nhung and D. de S. Pires, and P. S. Rodrigues da Silva, V. Soa, (2006), Phys. Rev. D 73, 035004; (2011), Phys. Rev. D 84, 095019. J. G. Ferreira, Jr., P. R. D. Pinheiro, C. A. 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. ___ Ngày nhận bài: 9/4/2020. Ngày nhận đăng: 2/5/2020 Liên lạc: thupn@utb.edu.vn 88