The dark side of flipped trinification

HJE The dark side of ipped trini cation P.V. Dong 0 1 2 5 6 7 8 9 10 11 D.T. Huong 0 1 2 5 6 7 8 9 10 11 Farinaldo S. Queiroz 0 1 2 4 6 7 8 9 10 11 Jose W. F. Valle 0 1 2 6 7 8 9 10 11 C.A. Vaquera-Araujo 0 1 2 3 6 7 8 9 10 11 0 Campus Universitario , Lagoa Nova, Natal-RN 59078-970 , Brazil 1 Saupfercheckweg 1 , 69117 Heidelberg , Germany 2 10 Dao Tan , Ba Dinh, Hanoi , Vietnam 3 Departamento de F sica, DCI, Campus Leon, Universidad de Guanajuato 4 International Institute of Physics, Federal University of Rio Grande do Norte 5 Institute of Physics, Vietnam Academy of Science and Technology 6 Cosmology of Theories beyond the SM , Discrete Symmetries, Gauge 7 Del. Benito Juarez , C.P. 03940, Ciudad de Mexico , Mexico 8 Av. Insurgentes Sur 1582. Colonia Credito Constructor 9 Loma del Bosque 103, Lomas del Campestre C. P. 37150, Leon, Guanajuato , Mexico 10 2 E-46980 Paterna (Valencia) , Spain 11 Edi cio de Institutos de Paterna, C/Catedratico Jose Beltran We propose a model which uni es the Left-Right symmetry with the SU(3)L gauge group, called ipped trini cation, and based on the SU(3)C Symmetry ArXiv ePrint: 1710.06951 SU( 3 )L SU( 3 )R U( 1 )X gauge group. The model inherits the interesting features of both symmetries while elegantly explaining the origin of the matter parity, WP = ( 1 )3(B L)+2s, and dark matter stability. We develop the details of the spontaneous symmetry breaking mechanism in the model, determining the relevant mass eigenstates, and showing how neutrino masses are easily generated via the seesaw mechanism. Moreover, we introduce viable dark matter candidates, encompassing a fermion, scalar and possibly vector elds, leading to a potentially novel dark matter phenomenology. 1 2 3 4 5 A B 1 A 2.1 2.2 2.3 2.4 3.1 3.2 3.3 ipped trini cation setup Gauge symmetry Fermion sector Scalar sector Spontaneous symmetry breaking 2.4.1 2.4.2 2.4.3 Identifying physical states and masses Dark matter 4.1 Scalar dark matter 4.2 Fermion dark matter 4.1.1 4.1.2 4.2.1 4.2.2 4.2.3 Relic density Direct detection Relic density Direct detection Collider 4.3 Gauge-boson dark matter 4.3.1 Relic density Conclusions Relevant scalar mass terms Fermion gauge-boson interactions Introduction Contents Introduction The mystery of Dark Matter (DM) is one of the biggest open questions in science [1{4]. Despite the fact that its existence has been ascertained at several distance scales of our universe, its nature has not yet been resolved and the Standard Model (SM) fails to account for it. The need to extend the SM goes beyond the DM problem, due to the existence of { 1 { important open questions connected to neutrino masses, the cosmological baryon-number asymmetry, in ation and reheating. Besides, from the theoretical side, the SM fails to explain the existence of (just) three fermion families as well as the origin of the observed parity violation of the weak interaction. The purpose of this paper is to study how an extension of the SM addressing these two issues, while hosting a viable DM candidate. The minimal left-right symmetric model based on the SU(3)C for the origin of parity violation in the weak interaction, neutrino mass generation as well as a framework for dark matter [11{15]. By the same token, models based on the SU(3)C SU(3)L U(1)N gauge group, for short 3-3-1, o er plausible explanations for the number of generations and a hospitable scenario for neutrino mass generation as well as implementing a viable dark sector [16{ 20, 20{24]. Hence it is theoretically well motivated to build a model where both groups are described in a uni ed way. Indeed, models have been proposed in the context of the SU( 3 )C SU( 3 )L SU( 3 )R gauge group [25{42]. Since they are based on a three copies of the SU( 3 ) non-Abelian group, it has been coined the term trini cation. The motivation for trini cation lies in the uni ed description of both strong and electroweak interactions using the same non-Abelian gauge group, while incorporating nice features of both left-right and 3-3-1 gauge groups. Fully realistic models unifying left-right and 331 electroweak symmetries have, in fact, been recently proposed using a ipped trini cation scenario with an extra U( 1 )X factor [40, 41]. In this paper we focus on an interesting question, namely, can we build a model preserving the nice features of the left-right and 3-3-1 symmetries while naturally explaining the origin of the matter parity and dark matter? We argue that, using the gauge principle to extend the trini cation framework, there is a compelling and minimal solution incorporating dark matter and realistic fermion masses. Such a ipped trini cation setup is better motivated because inherits the good features of both left-right and SU( 3 )L U( 1 )N symmetries and, in addition, elegantly addresses the origin of matter parity and dark matter stability in the context of 3-3-1 type models [43{55], while generating fermion masses with a minimal scalar sector. Indeed, it su ces to have one triplet ( L), one bitriplet ( ), one sextet ( R) to generate realistic fermion masses, as opposed to earlier versions where another bitriplet was necessary [39, 40]. In order to ensure left-right symmetry further copies of the scalar multiplets are required. Thus, a minimal version of trini cation with exact left-right symmetry requires one bitriplet ( ), two sextets ( L and R) and two triplets ( L and R). The rest of this paper is organized as follows: in section 2, we introduce the model with the gauge symmetry and particle content, focusing on the particles with unusual B L charges. We nd the viable patterns of symmetry breaking and show that W -parity is a residual gauge symmetry which protects the dark matter stability. In section 3, we identify the physical elds and the corresponding masses. In section 4, we discuss the dark matter phenomenology. Finally, we summarize the results and conclude this work in section 5. { 2 { HJEP04(218)3 2.1 Gauge symmetry Q = T3L + T3R + (T8L + T8R) + X; Trini cation is a theory of uni ed interactions based on the gauge symmetry SU( 3 )C SU( 3 )R, the maximal subgroup of E6 [25{27]. When multiplied by an Abelian group factor, U( 1 )X , we have the ipped trini cation [40, 41], alternative motivation is that it can be achieved from the minimal left-right symmetric model by enlarging the left and right weak isospin groups in order to resolve the number of fermion generations and accommodate dark matter (cf. [39]). The electric charge operator is generally given by which re ects the left-right symmetry, where TnL;R (n = 1; 2; 3; : : : ; 8) and X are the SU(3)L;R and U(1)X generators, respectively. Note that is an arbitrary coe cient whose values dictate the electric charge of the new fermions present in the model. As usual, the baryon minus lepton number is embedded as Q = T3L + T3R + 12 (B L), which implies that B L = 2[ (T8L + T8R) + X] is a residual gauge symmetry of SU( 3 )L aL NaL d L 1 J uLq L13 CA 0 u3L 1 J3qL+ 23 A aR NaR d R J uRq R13 CA 0 u3R 1 1 J3qR+ 23 A { 3 { can be easily shown that all triangle anomalies vanish, since both SU( 3 )L or SU( 3 )R groups match the number of fermion generations to be that of fundamental colors, in agreement with the current observations [58]. This choice of fermion representations is the minimal for a ipped trini cation [25{27]. 2.3 Scalar sector scalar multiplets as follows, To break the gauge symmetry and generate the masses properly, we need introduce the L = B = B 0 0 0 0 + 12 0 22 1+q L = BBB p 12 2 13 2 R = BBB p 12 2 13 2 22 q 1 23 p 2 12 2 22 q 1 23 p 2 11 p 23 A 1 B 0 0 0 C : generate all fermion masses. The scalar multiplets L and L have been added to ensure the left-right symmetry, but they do not play any role in our phenomenology because the VEV of these elds are neglible hence contributing neither to gauge boson masses nor to the spontaneous symmetry breaking pattern.1 Therefore, for simplicity hereafter we ignore the VEVs of L; L, keeping only the VEVs of R; R, denoted omitting the subscript \R". We now discuss what types of spontaneous symmetry breaking patterns one may have in our model. 1They only contribute to the tiny neutrino masses. { 4 { (2.7) (2.8) (2.9) (2.10) (2.11) (2.12) We now address the issue of which types of symmetry breaking patterns can be achieved within our model. 2.4.1 u; u0, leading to the following breaking pattern, for m = 0; 1; 2; : : :, and the surviving transformation is MP = eim (B L) = ( 1 )m(B L). Since the spin parity ( 1 )2s is always conserved due to Lorentz symmetry, the residual discrete symmetry preserved after spontaneous symmetry breaking is WP = MP ( 1 )2s, which is actually a whole class of symmetries parameterized by m. Among such conserving transformations, we focus on the SU( 3 )C 2We note that the matter parity present in our model coincides with R-parity in supersymmetry. one with m = 3, gauge symmetry, 2.4.2 For which we call the matter parity.2 We stress that in our model, it emerges as a residual WP = ( 1 )3(B L)+2s; WP = ( 1 )6[ (T8L+T8R)+X]+2s; and it acts nontrivially on the elds with unusual (wrong) B L numbers. For details, see the left-right symmetry is initially broken in this case. 033 breaks not only that symmetry but also W P0 and a U( 1 ) group, with U(!0) = ei!02 T8L transformation, as a SU( 3 )L subgroup. However, the VEV of 033 leaves WP unbroken. Indeed, 033 transforms under U( 1 )2 T8L W P0 as, HJEP04(218)3 0 33 ! 0330 = ei 32 (!0 m )(1+2q) 033 ; which is invariant if !0 = (m + 1+3k2q ) with k = 0; 1; 2 : : :. Choosing k = 0, the residual symmetry coincides with WP after spin parity is included and taking m = 3. Lastly, note that the hypercharge is Y = T8L + p 3 4 1 p ( 3T8R T3R) + X0; and the electric charge is Q = T3L + Y , all of which have the usual form. 2.4.3 Another possible breaking pattern takes place when assuming that the symmetry breaking of the left-right and SU( 3 )L symmetry occurs at the same scale, i.e. w; w0 . Therefore, we have only one new physics scale and the gauge symmetry is directly broken down to that of the SM as, (2.15) (2.16) SU( 3 )C SU( 3 )L SU( 3 )R U( 1 )X SU( 3 )C it naturally leads to the existence of stable dark matter particles. The transformation properties of the particles of the model under B L number and W -parity are collected in table 1. Notice that the B L charge for the new particles depends on their electric charge, i.e. on the basic electric charge parameter q, with W parity values P ( 1 ) (6q+1). When the new particles have ordinary electric charges q = m=3 for m integer, they are W -odd, P = 1, analogously to superparticles in supersymmetry. Generally, assuming that q 6= (2m 6= 1 and (P +)y = P . Such new particles, denoted as W -particles in what follows, 1)=6, W -parity is nontrivial, with { 6 { 2q P + 1 0 P P 3 ( q) 1 ZL0;R 0 1 3 P + (q+1) 2 P WL;R 0 1 XL;R 1+2q P + + 0 1 0 11 2 1 XL;qR (1+2q) P (1+2q) P q 12 2 1 YLq;+R1 1+2q P + 0 1 q 13 1+2q P + YL;R(q+1) (1+2q) P work as they require fractional charges for the new leptons. Since the W -charged and SM particles are uni ed within the gauge multiplets, W parity separates them into two classes, Normal particles with WP = 1: consist on all SM particles plus extra new elds. Explicitly, the particles belonging to this class are the fermions, a, ea, ua, da, the scalars, 011; 12; 21; 202; 303; 03; 101; 12; 22 ; 332q, the gauge bosons, A, ZL;R, ZL0;R, and the gluon. new scalars, Hermitian gauge bosons, XL;R; YL;R q W -particles with WP = P + or P : includes the new leptons and quarks, Na; Ja, the (q+1) q (q 1), and the new non It can be easily shown that W -particles always appear in pairs in interactions, similarly to superparticles in supersymmetry. Indeed, consider an interaction that includes x P +elds and y P - elds. The W -parity conservation implies ( 1 )(6q+1)(x y) = 1 for arbitrary q which is satis ed only if x = y. Hence, the elds P + and P are always coupled in pairs. The lightest W -particle (often called LWP) cannot decay due to the W -parity conservation. Thus, if the lightest W -particle carries no electrical and color charges, it can be identi ed as a dark matter candidate. From table 1, the colorless W -particles have electrical charges q; (1 + q); (q 1), and therefore three dark matter models can be built, corresponding to q = 0; 1.3 The model q = 0 includes three dark matter candidates, namely, a lepton as the lightest mixture of Na0, a scalar as the combination of 013; 031; 01 ; 103, and a gauge boson from the mixing of XL0;R. The model q = 1 contains two dark matter candidates: a scalar composed of has only one dark matter candidate: the scalar eld 203. 023; 032; 02 and a gauge boson from the lightest mixture of YL0;R. Lastly, the model q = 1 3The q = 1 case might be ruled out in the manifest left-right model [38]. { 7 { Before closing this section, it is important to notice that the fundamental eld carrying W -parity (P +)2, leads to self-interactions among three W - elds, if it transforms nontrivially under this parity. However, its presence does not alter the results and conclusions given below. See [39] for a proof. 3 Identifying physical states and masses The Lagrangian of the model takes the form, L = Lgauge + LYukawa V , where the rst term contains all kinetic terms plus gauge interactions. The second term includes Yukawa interactions, obtained by where M is a new physics scale that de nes the e ective interactions required to generate a consistent CKM matrix. The scalar potential is V = V + V + V + Vmix, where + t3 Q3L M and lies in the weak scale, as shown below. If another bi-fundamental eld is introduced in this minimal framework, coupling the third quark generation to the rst two, there are no such soft-terms for arbitrary values of the parameter, since its X-charge is nonzero. Thus, both the Higgs doublets contained in would be light as their VEVs are in the weak scale. In order to avoid light scalars, the triplet is included in this work instead of in order to generate viable quark masses and mixings. 3.1 Fermion sector matrices are given by After spontaneous symmetry breaking, the fermions receive their masses via the Yukawa Lagrangian (3.1). For the up-type quarks and down-type quarks, the corresponding mass Mu = 1 p 2M t31uw0 t32uw0 p 2M t13u0w0 1 p 2M t23u0w0 C ; p 2M A z33u 1 p 2M t31u0w0 t32u0w0 z12u z22u p 2M z33u0 t13uw0 1 p 2M 2M A t23uw0 C : (3.6) p The ordinary quarks obtain consistent masses at the weak scale, u; u0. The new physics or cut-o scale can be taken as at the largest breaking scale, M w0. The scale M Md = { 8 { and are both heavy, at the new physics regime too. The mass matrix elements for the charged leptons, HJEP04(218)3 belong to the weak regime as usual. In contrast, the new leptons, Na, have large masses dictated by the mass matrix Neutrinos have both Dirac and Majorana masses. The mass matrix in the ( L basis can be written as characterizing the non-renormalizable interaction is responsible for generating Vub, Vcb, as well as quark CP violation, as required. The exotic quark, J3, is a physical eld by itself, with mass, mJ3 = 2 zp33w , which is heavy, lying at the new physics regime. The two remaining exotic quarks, J ( = 1; 2), mix via a mass matrix, (3.7) (3.8) (3.9) c ) R (3.10) (3.11) (3.12) (3.13) MJ = 1 p 2 = BBB = BB +Sp1+iA1 2 12 2 q 13 2 p p 2 21 q 31 1 q (q+1) { 9 { where ML; MD; MR are 3 3 mass matrices, given by [MD]ab = 1 p2 yabu; with h L11i = vL=p2. As vL 0 the full seesaw mechanism, producing small masses for the light neutrinos L, , the mass matrix (3.11) provides a realization of m = ML MDMR 1MDT u2= vL; and large masses for the mostly right-handed neutrinos R, of order MR. 3.2 Scalar sector Since W -parity is conserved, only the neutral elds carrying WP = 1 can develop the VEVs given in (2.12). We expand the elds around their VEVs as The scalar potential can be written as V = Vmin + Vlinear + Vmass + Vint, where Vmin is independent of the elds, and all interactions are grouped into Vint. Vlinear contains all the terms that depend linearly on the elds, and the gauge invariance requires, 1)-charged + Vm2qa-scsharged, which are listed in appendix A. Vmass consists of the terms that quadratically depend on the elds, and can be furhter decomposed as Vmass = VmAass+VmSass+Vmsiansgsly-charged+Vmdaosusbly-charged+Vmq-acshsarged+Vm(qa+ss1)-charged The rst mass term includes all pseudo-scalars A1; A2; A3; A4; A5. From appendix A, we see that A1; A5 are massless and can be identi ed to the Goldstone bosons of the righthanded neutral gauge bosons, ZR; ZR0, respectively. The remaining elds A2; A3; A4 mix, but their mass matrix produces only one physical pseudo-scalar eld with mass A = pu2w2 + u02w2 + u2u02 1 u0wA2 + uwA3 + u0uA4 ; m2A = [u02w2 + u2(u02 + w2)][2 2(u02 w2) 2u2(w2 u02) ; which is heavy, at the w; w0 scale. The remaining elds are massless and orthogonal to A GZL = s u2(w2 + u02) w2u02 + u2(w2 + u02) u0 GZL0 = pw2 + u02 A3 w pw2 + u02 A4; u0w2 u02w A2 + u(w2 + u02) A3 + u(w2 + u02) A4 ; and can be identi ed with the Goldstone bosons of the neutral boson ZL, analogous to the SM Z boson, and the new neutral gauge boson ZL0. The VmSass term contains all the mass terms of the scalar elds, S1; S2; S3; S4; S5, as shown in appendix A. The ve scalars mix through a 5 5 matrix. In general, it is not easy to nd the eigenstates. However, using the fact that u; v w0; w; , one can diagonalize the mass matrix perturbatively. At leading order, this matrix yields one massless scalar eld, H1 = pu21+u02 (uS2 + u0S3), and a massive scalar eld, H2 = pu21+u02 (u0S2 uS3), with 6w02 + 2 2w2 . The H1 eld obtains a mass at next-to-leading order, mH1 ' O(u; u0), and is identi ed with the standard model Higgs boson. The remaining elds, (S1; S4; S5), are heavy and mixed among themselves via a 3 3 matrix. In the limit, (3.15) (3.16) m2H4 = m2H5 = 1 2 + 1 2 s s H5 = sH S4 +cH S5; w; w0, the corresponding physical elds have masses given by and two singly-charged massive Higgs elds with corresponding masses where the mixing angle H is de ned by the relation t2 H = ww0 n 1+ 2 2 4 + 2( 1 + 6) 2 n ( 1 + 2)w2 + w02 + 224w(21+ 2) o On the other hand, if one assumes that instead the hierarchy w; w0 > holds, the masses and mixing of the heavy states, (H4; H5) change accordingly to t2 H = m2H4 = m2H5 = 1 2 1 2 ww0( 1 + 6) w02 ( 1 + 2)w2 ; q q ( 1 + 2)w2 + w02 + [( 1 + 2)w2 Turning now to the singly-charged Higgs elds, we have three elds plus their conjugates. The mass matrix extracted from (A.3) yields four massless elds, which can be identi ed to the Goldstone bosons of the WL;R gauge bosons, GWL = pu2 + u02 1 GWR = q 1 1 + uu022 + 2u(2u(2u+2u02)2 2 u02)2 12 + 12 + ) u0 u 21 ; (3.22) u02 p2u u 2 12 + 12 + u0 u 21 ; w2) + 6w2w02gf(u2 There is only one doubly-charged Higgs eld, 22 , and is physical by itself, with mass appendix A. The spectrum in this sector includes four massless Goldstone bosons of the new gauge bosons XL;qR, GXqL = pu4 +w4 +u2(w02 +2 2 2w2) 1 The remaining elds are massive. In the limit, ; w; w0 u; u0, their physical states are where t q = H q = (w02 + 2 2) 2(w02 + 2 2) w02( 5 + 2t2u 6) + 2 5 2 + 6w2(w04(1 tu) + 2t2uw02 2 2 4 2 w02( 5 2t2u 6) 2 +2 5 4 + 6w2(w02 t2uw02 2 2) 2 2(t2u 1)w2(w2 +2 2) 2w2w02(w2 +w02 +2 2) [(t2u 1)(2 2w2 + 6w02) 2 6 2]2 + H q; n(w02 +2 2) 31q +p2w 13 q ww0 1 qo ; n(w02 +2 2) 31q +p2w 13 q ww0 1 qo ; H q (3.27) (3.28) (3.29) (3.30) ; (3.31) Vm(qa+ss1)-charged contains the mixing terms of 23 extracted from (A.5) yields four massless elds, identi ed with the Goldstone bosons of the new gauge bosons YL;R(q+1), and de ned by 1 The other physical elds are massive with corresponding masses, HY (q+1) = pw02(w2 +u02)+(u02 1 w2)2 2q are already physical elds, with w2) u2(u02 mix through the mass matrix, Let us now study the physical gauge boson states and their masses. In the non-Hermitian gauge boson sector, there are three kinds of left-right gauge bosons, WL;R; XL;qR; YL;R The elds WL;R, which are de ned as WL = p12 (A1L iA2L) and WR = p12 (A1R (q+1). iA2R), g 2 L 4 Diagonalizing this matrix, the eigenstates and masses are given by W1 = c WL s WR; W2 = s WL +c WR; m2W1 ' 4 m2W2 ' 4 g 2 L g 2 R u2 +u02 u2 +u02 +2 2 + 4t2Ru2u02 (t2R 1)(u2 +u02)+2t2R 2 ; 4t2Ru2u02 (t2R 1)(u2 +u02)+2t2R 2 1)(u2+u02) and tR = ggRL . W1 is identi ed as the SM at the new physics scale. The mass matrix of the elds XL q = p12 (A4L iA5L) and XR q = p12 (A4R g 2 L 4 and yields two physical heavy states with masses m2X1 ' 4 m2X2 ' 4 g 2 L g 2 R X1 q ' c 1 XL s 1 XR q; X2 q ' s 1 XL + c 1 XR q; with t2 1 = u2+w2 t2R(u2+w02+w2+2 2) . following mass matrix u2 + w2 + 4t2Ru2w2 u2 + w2 which provides physical heavy states with masses m2Y1 ' 4 m2Y2 ' 4 g 2 L g 2 R Y1 (1+q) = c 2 YL (1+q) s 2 YR (1+q); Y2 (1+q) = s 2 YL (1+q) + c 2 YR (1+q); u02 + w2 + 4t2Ru02w2 u02 + w2 t2R(u02 + w02 + w2) ; u02 + w02 + w 2 4u02w2 u02 + w2 t2R(u02 + w02 + w2) ; The neutral gauge bosons, A3L; A3R; A8L; A8R; B, mix via a 5 order to nd its eigenstates, we rst work with a new basis (3.37) (3.38) (3.39) (3.40) (3.41) (3.42) (3.43) (3.44) (3.45) (3.46) (3.47) A = sW A3L + cW ZL = cW A3L sW ZL0 = &1tX tW A3R &1 & ZR = A3R + &&1t2X A8R + &&1tX tRB; ZR0 = &(tRA8R tX B); tR tR tW &1tX tR tW A3R + tW A8L + tW A3R + tW A8L + tR tR tW A8R + tW A8R + tX tW B ; tX tW B ; A8L + &1tX tW 2A8R + &1tRtW B; where tX = ggXL , & = pt2R+ 2t2 1 X 1 , &1 = pt2R+(1+ 2)t2X , and sW = pt2X (1+ 2)+t2R(1+t2X (1+ 2)) . tX tR identi ed with ZL whose mass is m2Z ' 4c2W g2 L The gauge boson A is massless and decouples, therefore it is identi ed with the photon eld. The remaining elds, ZL; ZL0; ZR; ZR0, mix among themselves through a 4 4 mass matrix. Given that w; u; u0, the mass matrix elements that connect ZL to ZL0; ZR; ZR0 are very suppressed. The mass matrix can be diagonalized using the seesaw formula to separate the light state ZL from the heavy ones ZL0; ZR; ZR0. Thus, the SM Z boson is u2 + u02 . For the heavy neutral gauge bosons, the mass matrix elements are proportional to the square of the w; w0; energy scales. In the general case, it is very di cult to nd the physical heavy states. However, if there is a hierarchy between two energy scales w; w0 and , we can nd them. In particular, in the limit w; w0, the physical heavy states are HJEP04(218)3 ZL0 ' ZL0; ZR ' c 3 ZR m2ZL0 ' 3 s 3 ZR0; gL2 (1 + &12t2Rt2X 2)2t2W w2 ) p where the ZR-ZR0 mixing angle is With the physical states properly identi ed, we list in appendix B the most important interactions between the gauge bosons and fermions in the model. Now we turn to thed to discussion of the dark matter phenomenology. 4 Dark matter Despite the multitude of evidence for the existence of dark matter in our universe, its nature remains a mystery and it is one of the most exciting and important open questions in basic science [2]. In this work, we will investigate the possible dark matter candidates in our model and discuss the relevant observables, namely relic density and direct detection cross section. Indirect detection is not very relevant in our model because we will be discussing multi-TeV scale dark matter, a regime for which indirect dark matter detection cannot probe the thermal annihilation cross section [59]. In this section we will brie y address the possibility of scalar and fermion WIMP dark matter. First we note that, in some limiting cases, it is very similar to the one present in 3-3-1 models [43, 44, 47, 51, 60, 61]. As a result we opt for a more sketcky presentation here, primarily aimed at establishing the viabiblity of the present scenario, rather than covering it in full generality. Our main goal in this section is to illustrate our reasoning concerning the origin of the dark matter stability in a uni ed framework involving 3-3-1 and left-right symmetries, by simply showing that we have do viable dark matter candidates. i.e. = and a scalar H10;2. 4 That said, we have seen that the W -parity symmetry is exact and unbroken by the VEVs. Thus, the lightest neutral W -particle is stable and can be potentially responsible for the observed DM relic density. For concreteness we will study the model with q = 0, p13 . The neutral W -particles include a fermion Na0, a vector gauge boson X10;2, Suppose that H20 is the lightest W -particle (LWP). It cannot decay and can only be produced in pairs. The scalar dark matter has only s-wave contribution to the annihilation cross-section. Hence, the dark matter abundance can be approximated as where h vreli is the thermally averaged cross-section times relative velocity. As our candidates are naturally heavy at the new physics scale, the SM Higgs portal is inaccessible. The main contribution to the cross-section times relative velocity is determined by the direct annihilation channel H20 H2 ! 0 H1H1 or mediated by new scalars. In the limit w; w0 u; u0, the interaction between H20 and H1 is approximated as LH20 H1 = 2u2 + 1(u2 + u02) c2 It can be shown that the new Higgs portal gives a contribution of the same magnitude as the one above. Therefore in our estimate it is enough to consider only the H20 H2 ! H1H1 0 contact interaction. The average cross-section times relative velocity is h vreli = 1 ; freezeout temperature [1]. Since m2 where the dark matter velocity v satis es hv2i = 2x3F , with xF = mH2 =TF 20 at the h vreli = n 150 GeV H2 m2H1 , we approximate o2 ( 2u2 + 1(u2 + u02) c2q )2 Thus, the dark matter candidate H20 reproduces the correct relic density, for scalar couplings of O( 1 ), and using the fact that 2=(150 GeV)2 ' 1 pb. Furthermore, the above condition implies mH2 < 2:66( 1 + 2) TeV < 67 TeV; 4The other two dark matter models with q = 1 can be examined in a similar way. (4.1) (4.2) (4.3) (4.4) (4.5) (4.6) where the upper limit comes from the perturbativity bound 1 dark matter mass may be in the range few TeVs to 67 TeV, depending on its interaction strength with the SM Higgs boson. The detection through low energy nuclear recoils constitute a clear signature for dark matter particles. Since no signal has been observed thus far, stringent limits have been derived on the dark matter-nucleon scattering cross section [62{70]. In the scalar dark matter scenario, this scattering takes place through the t-channel exchange of a ZL0 and a heavy scalar H01. This scenario is similar to the one studied in [50], where it has been shown that one can obey direct detection limits from the XENON1T experiment with 2 years of data for the dark matter masses above 3 TeV, while reproducing the correct relic density. We emphasize that our goal here was simply to show that we do have viable dark matter candidates, rather than investigating in detail their phenomenology. In fact, the latter is substantially more complex than found in previous 3-3-1 models. However it su ces to exemplify the viability of our dark matter candidates under certain assumptions. Let us now assume that the LWP is one of the neutral fermions denoted by N . The model predicts that N is a Dirac fermion. The covariant derivative (i.e., gauge interactions) dictates the dark matter phenomenology. The dark matter might annihilate into SM particles via the well known Z0 portal with predictive observables [53, 71]. The relic density is governed by s-channel annihilations into SM fermions, whose interactions are presented in appendix B. Assuming that the mixing between the gauge boson ZL0 and the other gauge bosons to be small, which can be achived by taking w; w0 u; u0, one nds the relic density to be achieved either by annihilation into fermion pairs, or into ZL0ZL0. The role of ZR is analogous to ZN in the 3-3-1-1 model [51], which is not discussed further. In gure 1 we show the relic density curve in green. 4.2.2 Direct detection The dark matter-nucleon scattering is mostly driven by the t-channel exchange of the ZL0 gauge boson. This scattering is very e cient since it is governed simply the couplings with up and down quarks without much freedom. Taking into account the current and projected sensitivities on the dark matter-nucleon scattering cross-section, one can conclude that the dark matter mass must lie in the few TeV scale, as already investigated in [61]. Notice that this conclusion holds for a Dirac fermion (the possibility of having a Majorana fermion has already been ruled out by direct detection data [61]). The Majorana dark matter case leads to an annihilation rate which is helicity suppressed and therefore the range of parameter space that yields the correct relic density is smaller compared to the Dirac fermion scenario, only ZL0 masses up to 2:5 TeV can reproduce the correct relic density in the ZL0 resonance regime. HJEP04(218)3 5,000 4,000 ss3,000 t M k r a ta2,000 D1,000 0 0 X E N O N 1 T 3 4 d X E N O N 1 T 2 y L H C 3 6 . 1 f b 1 L H C 1 a b 1 L Z Ω h and current direct detection limit from XENON1T-34 days (red) [67], projected from XENON1T-2 years (blue) [74] and LZ (gray) [75] are overlaid. See text for detail. 4.2.3 As for collider bounds, LHC results based on heavy dilepton resonance searches with 13:3 f b 1 of integrated luminosity exclude ZL0 masses below few 3:8 TeV [61]. This is very important because in light of this bound, the Majorana dark matter case has already been ruled out, since the entire parameter space which yields the correct relic density in within the LHC exclusion region. In light of the importance of this collider bound we took the opportunity to do a rescalling with the luminosity to obtain current and projected limits on the ZL0 mass in our model for 36:1 f b 1 and 1000f b 1 keeping the center-of-energy of 13 TeV, using the collider reach tool introduced in.5 The limits read mZL0 > 4:2 TeV and mZL0 > 5:7 TeV, respectively. These bounds can be seen as vertical lines in gure 1. We emphasize that other limits stemming from electroweak precision or low energy physics are subdominant thus left out of the discussion [72, 73]. In summary, one can conclude that our model can successfully accommodate a Dirac fermion dark matter in agreement with existing and projected limits near the ZL0 resonance. Once again, our discussion clearly shows the present of a viable dark matter candidate. We stress that, in full generality, the phenomenology of our model is substantially more complex than presented in previous literature. Here we only consider simple benchmarks, for which a detailed study of the dark matter phenomenology has nnlo68cl.LHgrid. already been performed earlier, hence we have not repeated it here. A complete study of the phenomenological implications of our model lies beyond the scope of this paper and will be taken up elsewhere. Nevertheless, our results su ce to establish the consistency of our dark matter candidates. Finally, let us give a comment on the possibility of vector gauge boson dark matter. In this case one assumes that the LWP is the gauge boson X10. It can annihilate into SM particles via following channels, X10X10 ! W1+W1 ; ZLZL; H1H1; c; llc; qqc; where X10X10 e ; ; , l = e; ; , q = u; d; c; s; t; b. However, the dominant channels are ! W1+W1 ; ZLZL. Our predicted result is similar to the one given in [45]. The dark matter relic abundance is approximately given as X1 h2 ' 10 3 mm2X2W1 : Since the annihilation cross section is large and it is dictated by gauge interactions, the abundance of this vector dark matter is too small. In the context of thermal dark matter production, the vector dark matter in our model can contribute to only a tiny fraction of the dark matter abundance in our universe. A similar conclusion has been found in [43]. One way to circumvent the vector dark matter underabundance is by abdicating thermal production and tie its abundance to in ation, where the in aton decay or the gravitational mechanism would generate the correct dark matter abundance [55]. Alternatively, we mention that vector DM could be just part of the overall cosmological dark matter within a multicomponent thermal dark matter scenario. 5 Conclusions We have proposed a model of ipped trini cation that encompasses the nice features of left-right and 3-3-1 models, while providing an elegant explanation for the origin of matter parity and dark matter stability, which follows as a remnant of the gauge symmetry. The model o ers a natural framework for three types of dark matter particles, which is an uncommon feature in UV complete models. One can have a Dirac fermion, as well as a scalar dark matter particle, with masses at the few TeV scale. Both scenarios reproduce the correct relic density, while satisfying existing limits, in the context of thermal freeze-out. As for the vector case, thermal production leads to an under-abundant dark matter. We have also discussed other features of the model such as the symmetry breaking, driven by a minimal scalar content, but su cient to account for realistic fermion masses. In summary, we have presented a viable theory of ipped trini cation able to account naturally for the origin of matter parity and dark matter. (4.7) (4.8) Acknowledgments The authors thank Carlos Yaguna, Carlos Pires, Alex Dias. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant numbers 103.01-2016.77 and 103.01-2017.05, and is supported by the Spanish grants FPA2014-58183-P and SEV-2014-0398 (MINECO), and PROMETEOII/2014/084 (Generalitat Valenciana). FSQ acknowledges support from MEC, UFRN and ICTP-SAIFR FAPESP grant 2016/01343-7. C.A.V-A. acknowledges nancial support from Catedras CONACYT, proyect 749. HJEP04(218)3 A Relevant scalar mass terms The scalar elds mix according to the class they belong, and their relevant corresponding mass terms are derived as VmAass = A22 w2u02(2 2(w2 u02)+ 6w02) 4u2(u02 w2) +A2A3 w2u0(2 2(w2 u02)+ 6w02) wu0(2 2(w2 u02)+ 6w02) +A32 w2(2 2(w2 u02)+ 6w02) 4(u02 w2) +A42 u02(2 2(w2 u02)+ 6w02) : 4(u02 w2) VmSass = ( 1 + 2) 2S12 + (u02 u2)[2 2(u2 w2)(u02 w2)+ 6w2w02] u(u02 w2) + 2u S1S2 + 2u0 S1S3 + 2w S1S4 + 4w0 S1S5 + 2(u02 w2)[2( 1 + 2)u4 2u02w2]+ 6u02w2w02 2 S 2 4u2(u02 w2) u0 h4 1u2 +w2 2 2 + w26wu0202 i w 4 1u2 +u02 h2 2 + w26wu0202 i + + + 1u02 + 2(u02 w2 ) 1w2 + 2 w 2u 6w2w02 (u02 u2)(2 2(u2 w2)(u02 w2)+ 6w2w02 u0 12 1+2 + u 12 21 + 2u02(u2 w2) u2 + 6u02w2w02 12 2+1 +H:c : (A.1) (A.2) + 6u02w02w2 ( (u2 u02)(u2 w2) 2 (u2 w2)(u02 w2)+ 6w2w0 + 5(u02 w2)u2w02 2 ) 4u2 2(u02 w2) 2 32 +q q +q q ) (A.4) (A.5) (B.1) 2 2 2 2 2 2 2 B Fermion gauge-boson interactions The gauge interactions of fermions arise from, L i (PLCC + PLNC ) L gR R (PRCC + PRNC ) R; where L and R run on all left-handed and right-handed fermion multiplets, respectively, and PLC;RC = P n=1;2;4;5;6;7 TnL;RAnL;R; PLN;RC = T3L;RA3L;R + T8L;RA8L;R + ggLX;R X L;R B: The interactions of the physical charged gauge bosons with fermions are LCC = J1W W1+ + J2W W2+ + J1Xq X1q + J2Xq X2q + J1Y(q+1) Y q+1 + J2Y(q+1) Y q+1 + H:c:; 1 2 where the charged currents take the form, gpLc ( aL eaL +uaL daL)+ gpRs ( aR eaR +uaR daR); p ( aL eaL +uaL daL) p ( aR eaR +uaR daR); gRc 2 2 aL d L J L +J3L u3L)+ gRps 1 (NaR aL d L J L +J3L u3L) gRpc 1 (NaR aR d R J R +J3R u3R); aR d R J R +J3R u3R); gLpc 2 (NaL eaL +u L J L +J3L d3L)+ gRps 2 (NaR eaR +u R J R +J3R d3R); gLps 2 (NaL eaL +u L J L +J3L d3L) gRpc 2 (NaR eaR +u R J R +J3R d3R): The interactions of the physical neutral gauge bosons with fermions are obtained by LNC = gL L gL f [gVZR (f ) gAZR (f ) 5]f ZR 2cW gL f [gVZR0 (f ) gAZR0 (f ) 5]f ZR0 ; (B.2) 2cW gL f [gVZL (f ) gAZL (f ) 5]f ZL 2cW gL f [gVZ0L (f ) gAZ0L (f ) 5]f Z0L gVZL( a) gAZL( a) gVZL(ea) gAZL(ea) gVZL(Na) gAZL(Na) gZL(u ) V gZL(u ) A gVZL(u3) gAZL(u3) gZL(d ) V gZL(d ) A gAZL(d3) gZL(J ) V gZL(J ) A gVZL(J3) gAZL(J3) gZL(f) V ft2R[3 (tR 1)tX(3+ 3 )]+3(1+t2R)t2X(1+ 2)gc2W p p tXt2Rf4 3 tX+(1+tR)(3+ 3 )gc2W 6[t2R+t2X+(1+t2R)t2X 2] (tR 1)t2RtX(3+ p 3 )c2w f3t2R+(1+tR)tXt2R(1+ 3 )+t2X[3+3 2+t2R(3 2+2p3 3)]gc2W p p f3t2R (tR 1)t2RtX(1+ 3 )+3(1+t2R)t2X(1+ 2)gc2w p f3t2R+t2R(1+tR)tX( 3 p f3t2R (tR 1)t2RtX( 3 1)+3(1+t2R)t2X(1+ 2)gc2W p 6[t2R+t2X+(1+t2R)t2X 2] 2] 2] 2] 2] 2] p 3tR + 3(1 4tX) ]c2W 6[t2R+t2X+(1+t2R)t2X 2] (tR 1)t2RtX(1+ 3 )c2W 6[t2R+t2X+(1+t2R)t2X 2] 6[t2R+t2X+(1+t2R)t2X 2] (tR p 1)t2RtX( 3 6[t2R+t2X+(1+t2R)t2X 1)c2W 2] where f stands for every all the fermion elds, and e = gLsW. The vector and axial-vector couplings g ZL;ZL0;ZR;ZR0(f) are collected in tables 2, 3, 4, and 5. Note that at high energy V;A gL = gR, i.e. tX = tR, due to the left-right symmetry. However, at the low energy, such relation does not hold anymore. Therefore, the couplings we provide are general, depending on both tX and tR. Z0 gAL( a) Z0 g L(ea) V Z0 g L(ea) A Z0 g L(Na) V Z0 g L(Na) A Z0 g L(u ) V Z0 g L(u ) A Z0 g L(u3) V Z0 g L(u3) A Z0 g L(d ) V Z0 g L(d ) A Z0 g L(d3) V Z0 g L(d3) A Z0 g L(J ) V Z0 g L(J ) A Z0 g L(J3) V Z0 g L(J3) A f f f f f f f f p3(t2R+t2X) 3 tXt2R(1+tR tX) p3 2tX(1+tR)(t2R+tX tRtX)g 1 tR1tX1c 1 1 W tW6[t2R+t2X+(1+t2R)t2X 2] 3t2X(1+ 2) t2 [p3+tX(1+tX) (3+ 3 )]g p R 1 tR1tX1c 1 1 W tW6[t2R+t2X+(1+t2R)t2X 2] 3 2tX[tX+t2R(1 tR+tX)]g f2p3t2R 1 tR1tX1c 1 1 p p 3tX[tX+t2R(1+tX tR)] 2g 1 t 1 1 R tX1c 1 W tW6[t2R+t2X+(1+t2R)t2X 2] p 3 t2X +tX (1+tR)(3+p3 )]+s 2 1 2[p3t2R+tX (3+p3 )(1+tR)]gcW fc 2 tR 1 1[ 3 2 p 3 t2X +tX (1+tR)(3+p3 )]+s 2 1 2[p3t2R+tX (3+p3 )(1+tR)]gcW p3tR 3 (1+2tX )]+s 2 1 2[2p3t2R+tX (1 p a ea Na u3 d d3 J J3 ea Na u u3 d d3 J 6 1 1 2 f c 2 tR 1 1[3 2+tX (tR 1)+p3 tX (tX tR+1)]+s 2 1 2[p3t2R+tX (1 p fc 2 tRtX 1 1[2p3tX +(tR 1)(p3 1)]+s 2 1 2[ 2p3t2R+tX (p3 1)(tR 1)]gcW ea Na u3 d3 J J3 a ea Na u3 d d3 J p3tR p 3 (1+2tX )]+c 2 1 2[2p3t2R+tX (1 p 6 1 1 2 6 1 1 2 6 1 1 2 1 1 1 1 2 2 2 2 6 1 6 1 1 2 6 1 6 1 6 1 6 1 1 1 1 1 2 2 2 2 6 1 6 1 6 1 6 1 1 2 6 1 1 2 6 1 1 2 fs 2 tR 1 1[3 2+tX (tR 1)+p3 tX (tX tR+1)]+c 2 1 2[p3t2R+tX (1 p 3 )(1+tR)]gcW fs 2 tRtx 1 1[(1 tR)+p3 (1 tR+2tX )]+c 2 1 2[2p3t2R+tX (p3 +1)(tR 1)]gcW fs 2 tRtX 1 1[2p3tX +(tR 1)(p3 1)] c 2 1 2[ 2p3t2R+tX (p3 1)(tR 1)]gcW fs 2 tRtX 1 1[2p3tX +(tR 1)(3+p3 )]+c 2 1 2[2p3t2R+tX (3+p3 )(1 tR)]gcW Open Access. 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