Multicomponent dark matter in noncommutative $B-L$ gauge theory

It is shown that for a higher weak isospin symmetry, $SU(P)_L$ with $P\geq 3$, the baryon minus lepton charge $B-L$ neither commutes nor closes algebraically with $SU(P)_L$ similar to the electric charge $Q$, which all lead to a $SU(3)_C\otimes SU(P)_L\otimes U(1)_X\otimes U(1)_N$ gauge completion, where $X$ and $N$ determine $Q$ and $B-L$, respectively. As a direct result, the neutrinos obtain appropriate masses via a canonical seesaw. While the version with $P=3$ supplies the schemes of single-component dark matter well established in the literature, we prove in this work that the models with $P\geq 4$ provide the novel scenarios of multicomponent dark matter, which contain simultaneously at least $P-2$ stable candidates, respectively. In this setup, the multicomponet dark matter is nontrivially unified with normal matter by gauge multiplets, and their stability is ensured by a residual gauge symmetry which is a remnant of the gauge symmetry after spontaneous symmetry breaking. The three versions with $P=4$ according to the new lepton electric charges are detailedly investigated. The mass spectrum of the scalar sector is diagonalized when the scale of the $U(1)_N$ breaking is much higher than that of the usual 3-4-1 symmetry breaking. All the interactions of gauge bosons with fermions and scalars are obtained. We figure out viable parameter regimes given that the multicomponent dark matter satisfies the Planck and (in)direct detection experiments.


Contents
1 Introduction The standard model of fundamental particles and interactions has been extremely successful in describing observed phenomena, especially predicting the existence of the Higgs boson. However, it leaves a number of striking physics features of our world unexplained. The experimental evidences of neutrino oscillation have indicated that the neutrinos have nonzero small masses and flavor mixing, which cannot be solved within the framework of the standard model [1,2]. Additionally, the standard model fails to account for the cosmological issues relevant to particle physics, such as the matter-antimatter asymmetry of the universe and the fact that the standard model addresses only about 5% matter content of the universe [3]. The rest includes 26.5% dark matter and 68.5% dark energy, which all lie beyond the standard model [4,5]. In this work, we concentrate on the dark matter issue, finding its implication to the other puzzles.
The most widely studied dark matter candidate in particle physics and cosmology is a new kind of colorless, electrically-neutral, and weakly-interacting massive particles, called WIMPs [6,7]. Such particles arise naturally in many extensions of the standard model from the supersymmetric models [8][9][10][11] to models with universal extra dimensions [12][13][14][15], little Higgs models [16][17][18][19], and other interesting scenarios . Despite severe constraints from stability condition on cosmological timescales, relic density [4,5], direct [49][50][51][52][53] and indirect [54][55][56][57][58] searches, and particle colliders [59,60], the dark matter candidate can be viable to be a fermion, a vector, or a scalar, with mass scales ranging from a few GeV to several TeV. 1 The stability of the dark matter candidate is usually ensured by an unbroken discrete symmetry, such as R-parity in supersymmetry, KK-parity in universal extra dimension, T -parity in the little Higgs theory, matter parity in B − L extensions [30,63,64], or lepton parity in neutrino mass generation schemes [65]. Generally, all of the standard model particles are even, whereas the relevant new particles are odd, such that the lightest odd particle is stabilized, contributing to dark matter. A lot of such discrete symmetries must be imposed by hand, assumed to be exact or appropriately violated. The possibility of discrete symmetry that arises as a residual gauge symmetry is compelling, because the gauge symmetry not only determines and stabilizes dark matter candidates but also sets dark matter interactions and observables.
The mentioned experiments on relic density, direct and indirect detections, and particle colliders have not yet provided the particle picture of dark matter. Obviously, the mentioned theories often assume dark matter to be composed of a kind of a single particle-the lightest particle that is odd under the discrete symmetry. Since the constituent of dark matter is still an open question, there is no reason why dark matter comes from such a single particle kind. The scenario of multicomponent dark matter seems to be naturally in view of the rich structure of stable normal matter-the atoms. Furthermore, they have been phenomenologically and/or theoretically motivated [66][67][68][69][70][71][72][73][74][75][76][77][78][79][80][81][82][83][84][85] and revealed interesting consequences for galaxy structure [86,87]. The schemes of multicomponent dark matter have been especially to accommodate the multiple gamma-ray line and boosted dark matter signals [88][89][90][91][92][93][94] as well as dark matter self-interactions [95,96]. Theoretically, to have a multiple dark matter scenario, the simplest way adds to the standard model an exact discrete symmetry Z 2 ⊗ Z 2 . One can also add an exact Z 2 symmetry to supersymmetric models, or to universal extra dimension models, or to U (1) B−L models. Besides, there are other interesting approaches [97][98][99][100][101][102][103][104][105][106][107][108][109][110][111][112]. Since the dark matter structure is enriched, such scenarios possess interesting phenomenological consequences, attracting current research. Among the standard model extensions, the models that include B −L as a gauge charge have intriguing features. They can explain small neutrino masses through the exchange of heavy right-handed neutrinos, which arise as a result of B − L anomaly cancelation, while the right-handed mass scale is induced by B − L breaking [113][114][115][116][117][118][119][120][121]. The theories that contain noncommutative B − L charge define the dark sector to be nontrivially unified with the normal sector in gauge multiplets, while the residual B −L charge stabilizes dark matter candidates [30, 32-38, 42, 46]. Since dark matter takes part in gauge multiplets, the gauge symmetry would govern the dark matter observables, providing very predictive signals. The inflation and leptogenesis can be derived by the B − L symmetry when its breaking scale is high enough [36,43,47,122]. In this article, we develop a gauge theory that contains noncommutative B − L charge. Starting from a higher weak isospin symmetry SU (P ) L , we prove that the complete gauge symmetry must be SU (3) C ⊗SU (P ) L ⊗U (1) X ⊗U (1) N , called 3-P -1-1, where the last two factors determine the electric charge and B−L, respectively. We show that this theory provides multicomponent dark matter naturally for P ≥ 4. Whereas, the model with P = 3 yield single component dark matter, which has been well established in the literature [30,32,33,36,37,43,46].
The rest of this work is organized as follows: In Sec. 2, we construct a general noncommutative B − L gauge theory for multicomponent dark matter, discussing the spontaneous symmetry breaking, residual symmetries, dark matter stability, and fermion masses. In Sec. 3, we study the mass spectra of the scalar and gauge boson sectors according to the minimal model of multicomponent dark matter, i.e. the 3-4-1-1 model. All the interactions of gauge bosons with fermions and scalars are obtained in Sec. 4. In Sec. 5, we consider the three different scenarios of multicomponent dark matter and examine the dark matter observables. We summarize the results and make conclusions in Sec. 6.

Noncommutative B − L gauge theory
The purpose of this section proposes a general B − L gauge model responsible for multicomponent dark matter. The minimal realization of the model is presented.

General setup
Apart from the QCD group, let the SU (2) L symmetry of weak isospin be enlarged to SU (P ) L for P = 3, 4, 5, · · · , a higher weak isospin symmetry. Correspondingly, let each standard model fermion doublet be enlarged to either the defining representation (P -plet) or the complex conjugate of defining representation (anti-P -plet) of SU (P ) L . The [SU (P ) L ] 3 anomaly cancellation demands that the number of fermion P -plets equals that of fermion anti-P -plets, since a representation and its conjugate have opposite anomaly contributions. It follows that the number of generations is a multiple of fundamental color number, 3. Further, the QCD asymptotic freedom condition implies that the number of generations is not larger than [33/(2P )] = 5, 4, 3 for P = 3, 4, 5, respectively. Hence the generation number is just three, matching that of colors, and P ≤ 5. That property disappears in the standard model due to vanishing [SU (2) L ] 3 anomaly for every representation, unlike that of SU (P ) L . The higher weak isospin extension is thus motivated. Therefore, the fermion content under SU (P ) L is arranged as plus the corresponding right-handed components transforming as SU (P ) L singlets. The generation indices are a = 1, 2, 3 and α = 1, 2. The new fields E's and J's are necessarily included to complete the representations. Their subscripts are SU (P ) L indices, while the superscripts of all fields are clarified below. Without loss of generality, take a lepton P -plet into account. Since the new fields E's are unknown, let their electric charge Q and baryon minus lepton charge B−L be q's and n's, respectively. Thus, each lepton field (ν, e, E's) possesses a pair of the characteristic charges (Q, B−L) as superscripted, respectively. Suppose that Q and B−L are conserved, which are all approved by the standard model and the experiment. Both Q and B−L neither commute nor close algebraically with SU (P ) L . Indeed, we have Q = diag(0, −1, q 1 , q 2 , · · · , q P −2 ) and B − L = diag(1/3, 1/3, n 1 , n 2 , · · · , n P −2 ) for ψ aL , which are generally not commuted with the SU (P ) L weight raising/lowering generators:
To close the symmetries, two Abelian charges must be imposed, yielding a complete gauge symmetry, called 3-P -1-1, where the color group is also included, and X, N determines Q and B − L, 15 , · · · , T P 2 −1 according to k = 0, 1, 2, · · · , P − 2 are the SU (P ) L Cartan generators. Note that X and N are linearly independent as Q and B − L are. All the charges Q, X, B − L, and N must be gauged, since H k are gauged charges, a consequence of the noncommutation. The coefficients β's and b's can be obtained by acting Q and B − L on ψ aL , which obey where the initial conditions are (β 0 = 1, q 0 = −1) and (b 0 = 0, n 0 = −1), respectively. For P ≤ 5, we find Last, but not least, acting Q and B − L on the quark multiplets, Q αL and Q 3L , we obtain the pairs of the corresponding (Q, B − L) charges for component quark fields, as already superscripted in (2.2) and (2.3).
The important remark is that the higher weak isospin symmetry SU (P ) L contains two noncommutative charges Q and B − L, and the algebraic closure among them yields the gauge model 3-P -1-1, where B − L is nontrivially unified with the weak interaction in the same manner the electroweak theory does so for the electric charge. We will shortly prove that the new fermions E's and J's including new scalar and gauge bosons transform nontrivially under a residual gauge symmetry survived after spontaneous breaking, which contribute to dark matter. This theory determines dark matter nontrivially unified with normal matter in gauge multiplets by the gauge principle, a consequence of the noncommutative B − L charge. The multicomponent dark matter arises due to a nontrivial structure of the residual gauge symmetry relevant to the existence of multi B − L charges n 1 , n 2 , · · · , n P −2 , or in other words, the 3-P -1-1 extension for P ≥ 4.
To summarize the full fermion content transforms under the 3-P -1-1 symmetry as where we denote k = 1, 2, 3, · · · , P − 2, q ≡ q 1 + q 2 + · · · + q P −2 , and n ≡ n 1 + n 2 + · · · + n P −2 . Of course, the right-handed fermions have the same (Q, B − L) charges as those of the left-handed fermions in (2.1), (2.2), and (2.3), which were not supperscripted (i.e. omitted) without confusion. It is easily verified that all the anomalies vanish, as indicated in Appendix A. Especially, ν aR must be presented to cancel the U (1) N anomalies which are relevant to B − L charge. To break the gauge symmetry and generate the particle masses properly, we introduce P scalar P -plets plus a scalar singlet, . . .
Here φ owning such quantum numbers is given in order to break U (1) N and produce righthanded neutrino masses through the couplings ν aR ν bR φ, when it develops a vacuum expectation value (vev), φ ∼ Λ. The scalar P -plets correspondingly possess the components ϕ 11 , ϕ 22 , ϕ 33 , · · · , ϕ P P which have both Q = 0 and B − L = 0, hence possibly developing vevs, such as v 1 , v 2 , v 3 , · · · , v P , respectively. The remaining scalar fields have vanishing vev due to the electric charge conservation. The first two vevs v 1,2 break the standard model symmetry and give mass for ordinary particles, while v 3,4,··· ,P including Λ break the extended symmetry and provide mass for new particles. To be consistent with the standard model, we impose v 1,2 v 3,4,5,··· ,P , Λ. The scheme of the gauge symmetry breaking is therefore summarized as k=1 β k H k + X is the hypercharge, while P is a residual symmetry of B − L obtained below. To achieve the appropriate scenario of multicomponent dark matter, one must take v 3,4,··· ,P at TeV scale, while Λ that would define both the seesaw scale and the multiple matter parity P can take values from TeV to the inflation scale ∼ 10 15 GeV.
As indicated before, , which is actually larger than a Z 2 (at least homomorphic to Z 6 ) for z = 0, ±1, ±2, · · · . We consider an invariant (or normal) subgroup of it that is generated by the survival transformation P ≡ U (3π) = (−1) 3(B−L) according to z = 3. We further redefine by multiplying the spin parity (−1) 2s , which is always conserved by the Lorentz symmetry. The transformation P is similar to R-parity in supersymmetry, but in our model it arises from the gauge symmetry and means a multiple matter parity. Indeed, let us calculate all P values for fields, as collected in Table 1. First note that (P ± k ) † = P ∓ k and (P ± k ) 2 = 1 which generally differ from a parity. 3 It is clear that the fields that have a B − L charge dependent on n k or n k − n l transform as P ± k or P ± k P ∓ l , respectively, while the other fields including the standard model ones have P = 1, called normal fields. P ± k and P ± k P ∓ l are nontrivial ( = 1) if n k = (2z − 1)/3 = ±1/3, ±1, ±5/3, · · · and n k − n l = 2z/3 = 0, ±2/3, ∓4/3, · · · for every z integer, respectively. Such conditions generally apply since n k , n l can in principle be arbitrary. Hence, the fields that have a nontrivial P value are called wrong fields, since they possess an abnormal B − L charge, opposite to the standard model definition of B − L for normal fields (B − L = −1, 1/3, and 0 for leptons, quarks, and bosons, respectively). Table 1. The multiple matter parity P value of the model particles, where P ± k,l ≡ (−1) ±(3n k,l +1) and k, l = 1, 2, 3, · · · , P − 2. Additionally, G, B, C, A and W 's denote the gauge bosons associated with the color, X, N , the Cartan and weight raising/lowering generators.
Because our theory conserves the P -transformation, there is no single wrong field in interactions. Hence, the wrong fields are only coupled in pairs or self-interacted in interactions. Further, if an interaction has r k of P + k fields and s l of P − l fields, where r k , s l are integer, the P conservation implies k r k (3n k + 1) − l s l (3n l + 1) = 2z for z integer. This is valid for arbitrary n k , n l charge parameters if and only if k = l and r k = s l , i.e. P + k and P − k always appear in pairs. If an interaction has t kl of P + k P − l fields for t kl integer, the P conservation leads to kl t kl (3n k − 3n l ) = 2z for z integer. This happens for arbitrary n k , n l charge parameters if t kl = t lk , i.e. P + k P − l and its conjugate always appear in pairs. Last, but not least, if an interaction contains both kinds of the above wrong fields, then the P + k P − l field can self-interact with two P − k and P + l fields, such that P is conserved. The above analysis yields that P ± k is separately conserved, for each k. Thus the invariant subgroup that includes P must span where the operator P k has values P ± k or 1 when acting on a field. Each field would possess a multiple P value such as P = (P 1 , P 2 , · · · , P P −2 ). The normal fields have P = (+, +, · · · , +), while the wrong fields have at least a P k = P ± k = 1 in P . For the latter, we call singly-, doubly-, triply-, etc. wrong fields if they contain one, two, three, etc. nontrivial P k 's in P , respectively. Considering the special case, n k = 2z/3 for each k and any z integer, we have P ± k = −1 for every k. In this case, each P k is a Z 2 , and the invariant subgroup that contains closes by itself, called multiple matter parity, as stated before. This structure will be used for investigating realistic models, where P always possesses odd/even values when acting on fields, likely P = (· · · ±, · · · , ±, · · · , ± · · · ). It is noteworthy that even if the number of odd values are even, the corresponding field always belongs to the class of wrong fields, despite having a product of partial parities P = 1. Three remarks are in order 1. It is sufficiently to consider P , since the theory automatically conserves the quotient group of the residual symmetry by P . 4 2. The wrong scalar fields always have vanishing vev, even if electrically neutral, due to the P conservation. This validates the choice of vevs from the outset.
3. The lightest P ± k field is stabilized responsible for dark matter due to the P k conservation, for each k. Hence, there are simultaneously P − 2 stable dark matter candidates corresponding to the P 1 , P 2 , · · · , P P −2 transformations, called multicomponent dark matter, as stated before. The 3-4-1-1 and 3-5-1-1 models contain two-component dark matter and three-component dark matter, respectively, whereas the well-established 3-3-1-1 model has only single-component dark matter.
Furthermore, dark matter must be colorless and electrically neutral. We have various schemes for dark matter candidates, 1. q k = 0: The candidate is either a lepton E k , a non-Hermitian gauge boson that couples νE k , or a scalar combination of ϕ k+2,1 and ϕ 1,k+2 .
3. q r = q s : The candidate is either a non-Hermitian gauge boson that couples E r E s or a scalar combination of ϕ r+2,s+2 and ϕ s+2,r+2 .
The multicomponent dark matter scenarios are given by composing the above P − 2 conditions. For instance, q k = 0 for all k, the multicomponent dark matter may contain either only lepton candidates E 1 , E 2 , · · · , E P −2 , only scalar candidates as combinations of ϕ k+2,1 and ϕ 1,k+2 for k = 1, 2, 3, · · · , P − 2, or composition of those N lepton plus P − 2 − N scalar candidates. Alternatively, q l = −1 for all l, the multicomponent dark matter may include only scalar candidates as combinations of ϕ l+2,2 and ϕ 2,l+2 for l = 1, 2, 3, · · · , P − 2. Similarly, q k = 0 for k = 1, 2, · · · , N and q l = −1 for l = N + 1, N + 2, · · · , P − 2, this case may consist of N lepton and P − 2 − N scalar candidates. Above, we do not interpret the vector candidates, since they annihilate entirely before freeze-out due to the gauge interactions with the standard model W, Z, which do not contribute to the present dark matter relic.
In Appendix B we present the fermion mass generation. There, the neutrino masses are given along with the determination of the seesaw scale Λ ∼ [(h ν ) 2 /f ν ] × 10 14 GeV, which is proportional to the inflation scale. Therefore, it is naturally to impose where the intermediate physical regime v 3,4,...,P −2 ∼ 5-10 TeV sets the multicomponent dark matter observables.

Minimal realization
The minimal realization of multicomponent dark matter corresponds to P = 4, i.e. the 3-4-1-1 model. We now provides the necessary features of the model as well as obtaining the two-component dark matter scenarios that the model contains.
The gauge symmetry is given by The Q and B − L charges are embedded as where we redefine the coefficients β ≡ β 1 , γ ≡ β 2 , b ≡ b 1 , and c ≡ b 2 for brevity, and note that the SU (4) L charges are T i for i = 1, 2, 3, . . . , 15.
The fermion and scalar contents under the gauge symmetry are expressed in Table 2, which are identical to the X, N charges of the corresponding right-handed fermions, and the scalar multiplets η ≡ ϕ 1 , ρ ≡ ϕ 2 , χ ≡ ϕ 3 , Ξ ≡ ϕ 4 , for clarity. The charge parameters q, p, n, m are related to the embedding coefficients as The vevs of the scalar multiplets are written as where the partial parities P n and P m have values to be either 1 or P ± n = (−1) ±(3n+1) = −1 and P ± m = (−1) ±(3m+1) = −1, provided that n, m = 2z/3 = 0, ±2/3, ±4/3, ±2, · · · , respectively. P classifies the particles, such as They are all collected in Table 3, along with the corresponding Q, B − L charges and P multiple matter parity.
The two-component dark matter scenarios can be extracted when imposing color and electric neutrality conditions for both kinds of the candidates (i.e., P n and P m odd fields). With the aid of the physical scalar and gauge boson fields identified in the next section, we derive such dark matter schemes as given in Table 4. As mentioned, since the wrong gauge bosons do not contribute to dark matter, the model in the last raw is not appropriate for multicomponet dark matter, whereas the first four models do. The model with q = p = 0 has a rich two-component dark matter structure, to be further investigated in this work.
The total Lagrangian is given, up to the gauge fixing and ghost terms, by where F , S, and A run over the fermion, scalar, and gauge-boson multiplets, respectively. The covariant derivative and field strength tensors are Table 3. Q, B − L charges and P multiple matter parity of the model particles. Table 4. Schemes of two-component dark matter.
where (g s , g, g X , g N ), (t r , T i , X, N ), and (G r , A i , B, C) are the coupling constants, generators, and gauge bosons according to the 3-4-1-1 subgroups, respectively. f rst and f ijk are the SU (3) and SU (4) structure constants, respectively. The Yukawa interactions can be extracted from Appendix B, such as The scalar potential is given by where the last term is the potential of φ plus the interactions of φ with η, ρ, χ, and Ξ, We will identify the scalar mass spectrum and calculate the gauge interactions of scalars and fermions, which were all skipped in [44].

Scalar and gauge sectors
The necessary conditions for the scalar potential to be bounded from below and to induce the gauge symmetry breaking properly are The hierarchies u, v w, V Λ can be obtained by requiring We consider the large hierarchy case |µ| |µ 1,2,3,4 |, such that φ is decoupled. The φ field obtains a large vev, where H N and G Z N are heavy Higgs and massless Goldstone bosons associated with the U (1) N breaking, with the gauge boson Z N ≡ C. The U (1) N gauge and Higgs bosons have masses, m Z N 2g N Λ and m H N √ 2Λ, which are proportional to Λ scale and decoupled from the particle spectrum.
Below Λ, integrating φ out, we find that the effective potential at the leading order coincides with V Higgs when omitting V (φ), We expand the scalar quadruplets, Substituting them to the effective potential, we derive which includes vacuum energy, linear, mass, interaction terms, respectively. Because of the gauge invariance, the coefficients of V linear vanish, which provide a solution for u, v, w, V related to the potential parameters. The mass terms can be separated into, where the first two terms contain the mass terms of the CP-even and CP-odd scalar fields, respectively, while the last one consists of the mass terms of the charged scalar fields. Using the potential minimum conditions,(3.10), (3.11), (3.12), and (3.13), V S mass is given by Due to the condition u, v w, V , the above mass matrix would provide a small eigenvalue identical to the standard model Higgs boson (H 1 ) mass and three large eigenvalues corresponding to new neutral Higgs bosons (H 2,3,4 ). Indeed, at the leading order, we obtain

17)
is finite, given that w ∼ V .
In the new basis, (H 1 , H 2 , H 3 , H 4 ), the mass matrix M 2 S has a type I and II seesaw form, which includes m 2 H 1 ∼ (u, v) 2 plus the mixing terms between the Higgs bosons H 1,2,3,4 proportional to (u, v)(w, V ), while the mass terms of H 2,3,4 depend on (w, V ) 2 . Hence, at the next-to-leading order, the standard model Higgs boson H 1 gains a mass via the seesaw formula, approximated to be while the heavy Higgs bosons H 2,3,4 have the masses as retained. The mixings between the Higgs bosons are suppressed by (u, v)/(w, V ) 1, as neglected. The second term V A mass is given by This mass matrix provides a physical pseudoscalar with a corresponding mass, The remaining eigenstates are three massless pseudoscalars identical to the Goldstone bosons of the corresponding neutral gauge bosons, Z 1,2,3 , such that where Z 1 and G Z 1 are identical to those of the standard model, while the others Z 2,3 and G Z 2,3 are new physical states.
Concerning the charged scalar term, we obtain Each of the mass matrices M 2 's of the charged scalars yields a massless eigenstate identical to the Goldstone boson of a corresponding non-Hermitian gauge boson and a massive eigenstate with a mass at w, V scale. They are summarized as In summary, at the effective limit u, v w, V , the physical scalar states are related to the gauge states as follows: where the α 2,3 angles are defined by tan(α 2 ) = v/u and tan(α 3 ) = w/V , respectively. Let us investigate the mass spectrum of the gauge bosons, given by where the U (1) N gauge and Higgs sectors are decoupled as presented above. Note that in this case, the kinetic mixing between the two U (1) gauge bosons does not contribute. Substituting the vevs of the scalar quadruplets, we get where we have denoted the non-Hermitian gauge bosons as The non-Hermitian gauge bosons are physical eigenstates by themselves with corresponding masses The W boson has a mass at the weak scale identified to the standard model W boson, thus u 2 + v 2 = (246 GeV) 2 , as mentioned. Whereas, the remainders are new charged gauge bosons with large masses at w, V scale. For the neutral gauge bosons, the mass matrix is given by where we define t X = g X /g and the scalar X-charges are given above. The diagonalization of the mass matrix M 2 0 can be read off from [44], which yields the neutral gauge bosons, with the corresponding masses, where s W = e/g = t X / 1 + (1 + β 2 + γ 2 )t 2 X is the sine of the WeinbergâĂŹs angle. The photon field A is an exact massless eigenstate, decoupled from the other states. The field Z 1 slightly mix with the two heavy states Z 2,3 , which at the effective limit, (u, v) 2 /(w, V ) 2 1, the Z 1 is identified to the standard model Z boson. There is a finite mixing between Z 2 and Z 3 , which produces two new eigenstates, with corresponding masses, at w, V scale. The mixing angle is given by In summary, the physical neutral gauge bosons are related to the beginning states by (3.74)

Gauge interactions for fermions
The interactions of fermions with gauge bosons are derived from the Lagrangian, where the charged and neutral currents couple to the gauge bosons by 2,4,5,6,7,9,10,11,12,13,14 T i A iµ , (4.2) Substituting the charged gauge bosons from (3.54), (3.55), and (3.56) into (4.2), we get The interactions of the charged gauge bosons with fermions are where the corresponding charged currents are determined by Substituting the neutral gauge bosons from (3.74) into (4.3), we obtain Hence, the interactions of the neutral gauge bosons with fermions are given by where e = gs W and f indicates every charged fermion of the model. The first term in (4.13) yields electromagnetic interactions, as usual. Concerning the remaining interactions, for the neutrinos we have For the other fermions, the vector and axial-vector couplings can be obtained as In Appendix C, we compute the couplings of Z 1 with fermions as given in Table 5, which are consistent with the standard model. Additionally, the couplings of Z 2 with fermions are derived as collected in Table 6. Here, it is noted that the couplings of Z 3 with fermions can be obtained from those of Z 2 by replacing, c ϕ → s ϕ and s ϕ → −c ϕ , which need not necessarily be determined.

Gauge interactions for scalars
The relevant interactions arise from where S = S + S takes the forms, with the vevs and physical states explicitly shown. The Lagrangian is correspondingly expanded by In Appendix D, we calculate all the gauge boson and scalar interactions and express the corresponding couplings from Table 7 to Table 18. There, the new labels are, which differ from those in the electric charge operator, without confusion.

Multicomponent dark matter phenomenology
We consider the model with q = p = 0. In this case, the neutral particles that transform nontrivially under the multiple matter parity P = P n ⊗ P m are E 0 a , F 0 a , H 0 2 , H 0 3 , H 0 6 , W 0 13 , W 0 14 , and W 0 34 , as explicitly shown in Table 4. We divide into three possibilities of two-component dark matter existence.

Scenario with two-fermion dark matter
We assume that E (one of three particles E 0 a ) and F (one of three particles F 0 a ), which are singly-wrong particles according to the separately conserved single parities P n and P m , are the lightest particles within the classes of singly-wrong particles of the same kind (E a , H 2 , W 13 ) and (F a , H 3 , W 14 ), respectively. Note that E and F are only coupled via the new gauge boson W 34 and new Higgs scalar H 6 , due to the gauge and P n,m invariances. We further assume that the net mass of E and F is smaller than each mass of W 34 and H 6 . Hence, they are stable and can play the role of two-component dark matter candidates.
The dominant channels of the dark matter pair annihilation into the standard model particles are given by In addition, there is the conversion between dark matter, which plays the key role in the multicomponent dark matter scenario, in which the heavier dark matter component would annihilate into the lighter one. In this sense, there adds the annihilation process either The relevant Feynman diagrams which describe the dark matter pair annihilation into the standard model particles and the conversion between dark matter components are given in Figures 1 and 2, respectively. Hereafter, note that Z N is superheavy, hence not contributing to the dark matter observables.

Figure 2. Conversion between fermion dark matter components.
We compute the dark matter relic abundance due to the thermal freeze-out of two dark matter components E and F . The dark matter relic abundance is obtained by solving the coupled Boltzmann equations (BEQs), which govern the evolution of Y E(F ) ≡ n E(F ) s with n E(F ) referring to the number density of the dark matter component E(F ) and s to be the entropy density, given by where M Pl = 1.22 × 10 19 GeV, g * = 106.75 is the effective total number of degrees of freedom, µ = m E m F m E +m F , and with g = 2 being the number of degrees of freedom for dark matter components.
In equations (5.5) and (5.6), the thermal average annihilation cross-section times the relative velocity for the dark matter components is given in the non-relativistic approximation at the leading order as where i, j = 2, 3, and f refers to every fermion of the standard model. Above, we have assumed that the masses of the new gauge bosons are much larger than the masses of the standard model fermions. By solving numerically these equations with the following boundary condition (5.13) corresponding to that the dark matter species are in equilibrium with the thermal bath at start, one can obtain the individual relic abundance of each dark matter component as 14) where x ∞ refers to a very large value of x after the thermal freeze-out. In particular, when the production of the lighter dark matter component from the heavier dark matter component is less significant compared to its annihilation to the standard model particles, an approximate analytic solution of BEQs is given by [72] ) for the opposite case m E < m F . The dark matter relic abundance is a sum of the individual contributions as   Let us investigate the case that the total relic density (5.22) varies as a function of dark matter masses (m E , m F ) for m E + m F < m W 34 , satisfying the experimental bound Ω DM h 2 < 0.12 [5]. In Figure 3, we show the viable dark matter mass regime as the overlap of the two colored regions according to the relic density and the stability condition, respectively. Note that the condition for m E + m F < m H 6 is easily evaded by imposing an appropriate λ 16 value, which is not taken into account. Moreover, the selections of the new physics scales w, V always satisfy the constraints from the ρ-parameter, Zf f couplings, FCNCs, and collider bounds, as studied in [44].  To see the physical effect of each dark matter component that contributes to the relic abundance, in Figure 4 we depict the total relic density as a function of m F for several choices of w, V and m E as related to m F , which are viable from the above contours. Typically, we determine four resonances in each density curve, corresponding to m F = 2 , respectively. It is noted that we always have m Z 3 > m Z 2 for the mediators and the details of the resonances can be seen in the next figure. Further, due to the contributions of both E and F , the total density does not vanish at the resonances. In this case, the viable dark matter mass regime is given below the correct abundance Ωh 2 < 0.12 and before the dark matter unstable regime (red) according to m E + m F > m W 34 .  Correspondingly, in Figure 5 we make a comparison between the partial relic densities of dark matter components with the choices of the new physics scales w, V and m E via m F , as mentioned. It is noteworthy that the region above the line Ω F /Ω = 0.5, the candidate F dominantly contributes to the density, and the two peaks at which are due to the m E resonances. Whereas, below the line Ω F /Ω = 0.5, E dominates the density, where the two resonances correspond to the m F ones. The dark matter unstable region (red) according to m E + m F > m W 34 is also included for completion.
The above analysis is relevant to m F > m E . It is noted that the dark matter phenomenologies happen similarly to the case with m F < m E , thus to the whole dark matter mass regime which is viable from Figure 3.
We study the direct detection for the dark matter components in our model through their spin-independent (SI) scattering on nuclei. First, let us write the effective Lagrangian describing dark matter-nucleon interaction at the fundamental level through the exchange of the new neutral gauge bosons Z 2,3 as From this effective Lagrangian, one can obtain the SI cross-section for the scattering of the dark matter components on a target nucleus N as m N is the dark matter-nucleon reduced mass, Z and A are the atomic number and atomic mass of the nucleus N , respectively. However, in the twocomponent dark matter scenario, the SI cross-section for each dark matter component is calculated as follows Supposing that the two-component dark matter obtains the correct total density, in Figure 6 we plot the SI cross-sections of dark matter components corresponding to the above choices of (w, V ) parameters, respectively. Here in each limit curve, we explicitly show which part the case m F > m E (blue) and vice versa the opposite case m F < m E (red) take place. The experimental bounds [52,53] are also included. That said, the dark matter masses that have been obtained from the relic density and the stable condition also satisfy the direct detection if (w, V ) = (8,9) TeV and higher. Correspondingly, both m E and m F should be above 1 TeV.

Scenario with two-scalar dark matter
We consider the second case where two-component dark matter contains the scalar particles H 2 and H 3 . Here we assume that they are the lightest particles within the classes of singlywrong particles of the same kind as mentioned, respectively, and they have a net mass  Figure 6. The spin-independent dark matter-nucleon scattering cross-section limits as a function of dark matter masses according to (w, V ) = (5, 6), (8,9), and (11,12) TeV, assuming the correct abundance Ω DM h 2 = 0.12.
smaller than those of W 34 and H 6 . The dark matter pair annihilation into the standard model particles are given by the following dominant channels as presented in Figure 7, while the dark matter conversions are given in Figure 8. The thermal average annihilation cross-section times the relative velocity for the scalar dark matter components is approximately given by Figure 7. Dominant contributions to annihilation of the two-component scalar dark matter into standard model particles. σv where the remaining annihilation cross-sections are Above, the annihilation (5.32) happens for m H 2 > m H 3 , whereas the annihilation (5.34) exists for m H 3 > m H 2 .  Corresponding to each choice of (w, V ), we make a contour of the total relic density Ω DM h 2 < 0.12 due to the contributions of both scalar dark matter candidates to be a function of their masses as in Figure 9, where the regions constrained by the dark matter stable conditions m H 2 + m H 3 < m H 6 and m H 2 + m H 3 < m W 34 are also indicated. The viable dark matter mass regime is the overlap of the three regions corresponding to the relic density and the stable conditions.
To examine the contribution effects of each scalar dark matter component, we consider the case m H 3 > m H 2 . The total relic density is depicted in Figure 10  the electroweak scale, the standard model Higgs portal H 1 negligibly contributes to the relic density. Furthermore, the new Higgs portal H 2 gives negligible contributions because it weakly couples to the dark matter components.
One can consider the total relic density for the case m H 2 > m H 3 as a function of m H 2 with the several values of w, V and H 2,3 mass relation. The process happens analogous to the case m H 2 > m H 3 . Hence, the common remark for both cases is that the correct density and stability condition require the scalar dark matter masses to be not too large, limited below several TeVs. Additionally, some H 3,4 resonances at the high mass region are already excluded by the stability condition.
The effective Lagrangian describing dark matter-nucleon interaction in the limit of zero-momentum transfer through the exchange of the Higgs boson H 1 is given as where Note that H 2,3,4 give smaller contributions, as neglected. Then, the SI cross-section for the scattering of each dark matter component on a target nucleus N is expressed as where σ SI H 2(3) N is given by m N to be the dark matter-nucleon reduced mass, and the nucleus factor C N is given by  Figure 11. The spin-independent dark matter-nucleon scattering cross-section limits as a function of dark matter masses according to each choice of w, V and m H2,3 relation, where the dark matter unstable regime is input as red.
According to the above choices of w, V and the dark matter mass relations, we plot the SI dark matter nucleon scattering cross-section as the function of the corresponding dark matter mass, where the experimental bounds [52,53] are included. The general remark is that the scalar dark matter masses below around 700 GeV are excluded by the direct detection experiment. Additionally, they with a low mass give small contributions to the abundance as seen from Figure 10.

Scenario with a fermion and a scalar dark matter
In this case we consider E and H 3 to be two-component dark matter candidates, without loss of generality. Their annihilation cross-sections to the standard model particles have been obtained above. Let us examine the fermion and scalar dark matter conversion, as given by the diagram in Figure 12  The thermal average annihilation cross-section times the relative velocity for the dark matter-dark matter conservation is obtained in the non-relativistic approximation at the leading order as  For numerical calculation, we use the following parameter values, λ 1 = 0.1, λ 3,4,6,7,9,10 = 0.3, λ 5 = −0. 19, (5.49) throughout this section. Note that some of them differ from the two-scalar dark matter section since the three cases of two-component dark matter are alternative. We contour the total relic density as the function of dark matter masses as well as imposing the dark matter stable conditions as displayed in Figure 13, corresponding to the fixed values of the new physics scale w, V . From the density contours according to each pair value of w, V , we select the viable dark matter mass relations and plot the total relic density to see the contribution effect of each dark matter component and the direct detection cross-section, which are all given as the functions of a dark matter mass, presented in Figures 14 and 15, respectively. Here, the dark matter unstable regimes are shown and note that the resonances that are presented in the excluded regimes are omitted. Typically, we obtain the resonance phenomena similar to the two cases above, two resonances for the fermion candidate set by the new gauge portal and other two for the scalar candidate set by the new Higgs portal. With the selection of parameters, in these cases, the resonances are important to govern the mark matter observables. The viable dark matter masses are around one to a few TeV.

Conclusion
We have shown that a gauge theory that includes a higher weak isospin symmetry SU (P ) L must possess a complete gauge symmetry of the form where the last two Abelian groups define the electric charge and baryon-minus-lepton charge, respectively. The last charges are unified with the weak charge in the same manner as the electroweak theory. Additionally, the neutrino masses are appropriately induced by the gauge symmetry breaking, supplied in terms of a canonical seesaw mechanism.
The multiple matter parity P = P −2 k=1 P k , where each P k is a Z 2 , is obtained as a residual gauge symmetry. This parity makes P − 2 wrong particles stable, providing multicomponent dark matter candidates. The noncommutation of B − L with SU (P ) L yields that the dark matter candidates are nontrivially unified with normal matter in gauge multiplets; in other words, multicomponent dark matter is required to complete the SU (P ) L representations enlarged from the standard model. Therefore, the gauge interactions would govern the dark matter observables.
The minimal multicomponent dark matter model corresponds to P = 4, the so-called 3-4-1-1 model. In this case, we have fully diagonalized the scalar and gauge sectors and identified two-component dark matter schemes according to the multiple matter parity P = P n ⊗ P m . All the interactions of fermions and scalars with gauge bosons have been obtained. The 3-4-1-1 model with q = p = 0 obeys three possibilities of two-component dark matter, including two fermions (E, F ), two scalars (H 2,3 ), and a fermion and a scalar e.g. (E, H 3 )

A Anomaly cancelation
The nontrivial anomalies include Let us compute each of them, Hence, all the anomalies are cancelled, independent of P and the U (1)'s charge parameters. Additionally, the anomalies (A.7), (A.8), (A.9), and (A.11) relevant to U (1) N vanish with the presence of the right-handed neutrinos.

C Vector and axial-vector couplings
This appendix is devoted to the neutral gauge boson couplings with fermions.
No data No data No data Table 5. The couplings of Z 1 with fermions. Table 6. The couplings of Z 2 with fermions.

D The gauge couplings of scalars
This appendix is devoted to all the gauge boson and scalar couplings.

Vertex
Coupling Vertex Coupling