The Dark Side of Flipped Trinification

We propose a model which unifies the Left-Right symmetry with the $SU(3)_L$ gauge group, called flipped trinification, and based on the $SU(3)_C\otimes SU(3)_L\otimes SU(3)_R\otimes U(1)_X$ gauge group. The model inherits the interesting features of both symmetries while elegantly explaining the origin of the matter parity, $W_P=(-1)^{3(B-L)+2s}$, 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. Viable dark matter candidates can either be a fermion, a scalar or a vector, leading to potentially different dark matter phenomenology.


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 important open questions connected to neutrino masses, the cosmological baryon-number asymmetry, inflation 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 ⊗ SU (2) L ⊗ SU (2) R ⊗ U (1) B−L gauge group, completed by a Z 2 symmetry that interchanges the left and right, is one of the most attractive extensions of the SM [5][6][7][8][9][10]. It gives a manifest understanding for the origin of parity violation in the weak interaction, neutrino mass generation as well as a framework for dark matter [11][12][13][14].
Fully realistic models unifying leftright and 331 electroweak symmetries have, in fact, been recently proposed using a flipped trinification scenario with an extra U (1) X factor [39,40].
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 trinification framework, there is a compelling and minimal solution incorporating dark matter and realistic fermion masses. Such a flipped trinification 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 [41][42][43][44][45][46][47][48][49][50][51][52][53], while generating fermion masses with a minimal scalar sector. Indeed, it suffices to have one triplet (χ L ), one bitriplet (φ), one sextet (σ R ) to generate realistic fermion masses, as opposed to earlier versions where another bitriplet was necessary [38,39]. In order to ensure left-right symmetry further copies of the scalar multiplets are required. Thus, a minimal version of trinification with exact left-right symmetry requires one bitriplet (φ), two sextets (σ L and σ R ) and two triplets (χ L and χ R ).
The rest of this paper is organized as follows: In Sec. 2, we introduce the model with the gauge symmetry and particle content, focusing on the particles with unusual B − L charges. We find the viable patterns of symmetry breaking and show that W -parity is a residual gauge symmetry which protects the dark matter stability. In Sec. 3, we identify the physical fields and the corresponding masses. In Sec. 4, we present detailed calculations of the dark matter observables. Finally, we summarize the results and conclude this work in Sec. 5.

Gauge Symmetry
Trinification is a theory of unified interactions based on the gauge symmetry SU (3) C ⊗ SU (3) L ⊗ SU (3) R , the maximal subgroup of E 6 [24][25][26]. When multiplied by an Abelian group factor, U (1) X , we have the flipped trinification [39,40], This symmetry can be obtained by left-right symmetrizing the 3-3-1 model in order to account for weak parity violation and close both B − L and 3-3-1 algebras (cf. [54]). An 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. [38]).
The electric charge operator is generally given by which reflects the left-right symmetry, where T nL,R (n = 1, 2, 3, ..., 8) and X are the SU (3) L,R and U (1) X generators, respectively. Note that β is an arbitrary coefficient 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 = T 3L + T 3R + 1 2 (B − L), which implies that is a residual gauge symmetry of SU (3) L ⊗ SU (3) R ⊗ U (1) X . Let us note that B − L and SU (3) L neither commute nor close algebraically. Therefore, the present framework, along the 3-3-1-1 gauge theory, constitute a class of models with a fully consistent formulation of gauged B − L symmetry in 3-3-1 extensions of the Standard Model [43,49,[53][54][55].

Fermion Sector
The fermion content in this model results simply from the left-right symmetrization the left-handed fermion sector of the 3-3-1 model, so as to produce the right-handed fermion sector. The fermion sector is given as

Scalar Sector
To break the gauge symmetry and generate the masses properly, we need introduce the scalar multiplets as follows, with the corresponding VEVs, Note that the scalars transform as We emphasize that these three scalar multiplets are sufficient to generate all fermion masses. The scalar multiplets χ L and σ 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 fields 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 σ L , χ L , keeping only the VEVs of σ R , χ R , denoted omitting the subscript "R". We now discuss what types of spontaneous symmetry breaking patterns one may have in our model.

Spontaneous Symmetry Breaking
We now address the issue of which types of symmetry breaking patterns can be achieved within our model.
In this scenario, we assume w, w Λ u, u , leading to the following breaking pattern, Notice that spontaneous symmetry breaking leaves the residual discrete gauge symmetry, W P , conserved along with the electric and color charges. Let us now identify what symmetry is that. The VEV of σ 0 11 , Λ, breaks B−L since [B−L] σ 0 11 = √ 2Λ = 0, where σ 0 11 has B−L = 2. The U (1) B−L transformation that preserves the vacuum is σ 0 11 → e iω(B−L) σ 0 11 = e i2ω σ 0 11 = σ 0 11 , with ω as a transformation parameter. Thus, we obtain e i2ω = 1, or ω = mπ for m = 0, ±1, ±2, ..., and the surviving transformation is M P = e imπ(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 W P = M P × (−1) 2s , which is actually a whole class of symmetries parameterized by m. Among such conserving transformations, we focus on the one with m = 3, which we call the matter parity 2 . We stress that in our model, it emerges as a residual gauge symmetry, and it acts nontrivially on the fields with unusual (wrong) B − L numbers. For details, see Table I. W -parity, W P , is thus named following the "wrong" item as in previous studies.
2. Case 2: Λ w, w u, u For Λ w, w , the gauge symmetry is broken following a different path, The SU (2) R symmetry is generated by {T 6R , T 7R , 1 2 ( √ 3T 8R − T 3R )}, meaning that the left-right symmetry is initially broken in this case. The U (1) X charge is X = √ 3+β The discrete symmetry W P takes a form, W P = (−1) m(βT 8R +X) , which is a residual symmetry of a broken U (1) group, with U (ω) = e iω2(βT 8R +X) transformation. The second stage of the symmetry breaking is driven by φ 0 33 , χ 0 3 fields. The VEV of χ 0 3 breaks the symmetry SU (2) R ⊗ U (1) X , while the VEV of φ 0 33 breaks not only that symmetry but also W P and a U (1) group, with U (ω ) = e iω 2βT 8L transformation, as a SU (3) L subgroup. However, the VEV of φ 0 33 leaves W P unbroken.
which is invariant if ω = π(m + 3k 1+2q ) with k = 0, ±1, ±2.... Choosing k = 0, the residual symmetry coincides with W P after spin parity is included and taking m = 3. Lastly, note that the hypercharge is and the electric charge is Q = T 3L + Y , all of which have the usual form.
3. Case 3: w, w ∼ Λ 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, w ∼ Λ. Therefore, we have only one new physics scale and the gauge symmetry is directly broken down to that of the SM as, Here, W P is the residual discrete gauge symmetry preserved by all VEVs and has the form obtained above.
In summary, regardless of symmetry breaking scheme adopted, they all lead to the residual conserved W -parity, In this way, the matter parity is a direct consequence of the gauge group and as we shall see, 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 I. 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 = (2m − 1)/6, W -parity is nontrivial, with P ± = 1 and (P + ) † = P − . Such new particles, denoted as Wparticles in what follows, have different B − L numbers than those of the standard model.
Since the W -charged and SM particles are unified within the gauge multiplets, W -parity separates them into two classes, • Normal particles with W P = 1: Consist on all SM particles plus extra new fields.
• W -particles with W P = P + or P − : Includes the new leptons and quarks, N a , J a , , 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 + -fields and y P − -fields. The W -parity conservation implies (−1) (6q+1)(x−y) = 1 for arbitrary q which is satisfied only if x = y. Hence, the fields 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 identified as a dark matter candidate.
From Table I, 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 N 0 a , a scalar as the combination of φ 0 13 , φ 0 31 , χ 0 1 , σ 0 13 , and a gauge boson from the mixing of X 0 L,R . The model q = −1 contains two dark matter candidates: a scalar composed of φ 0 23 , φ 0 32 , χ 0 2 and a gauge boson from the lightest mixture of Y 0 L,R . Lastly, the model q = 1 has only one dark matter candidate: the scalar field σ 0 23 .
Before closing this section, it is important to notice that the fundamental field σ 2q 33 , carrying W -parity (P + ) 2 , leads to self-interactions among three W -fields, if it transforms nontrivially under this parity. However, its presence does not alter the results and conclusions given below. See [38] for a proof.

IDENTIFYING PHYSICAL STATES AND MASSES
The Lagrangian of the model takes the form, where M is a new physics scale that defines the effective interactions required to generate a consistent CKM matrix. The scalar potential is We see that φ has trilinear couplings. An SU (2) L doublet contained in φ can be made heavy by taking f at the new physics scale. The remaining Higgs doublet in φ is light and lies in the weak scale, as shown below. If another bi-fundamental field ρ is introduced in this minimal framework, coupling the third quark generation to the first two, there are no such soft-terms for arbitrary values of the β parameter, since its X-charge is nonzero. Thus, both the Higgs doublets contained in ρ would be light as their VEVs are in the weak scale. In order to avoid light scalars, the triplet χ is included in this work instead of ρ in order to generate viable quark masses and mixings.

Fermion Sector
After spontaneous symmetry breaking, the fermions receive their masses via the Yukawa Lagrangian (17). For the up-type quarks and down-type quarks, the corresponding mass matrices are given by The ordinary quarks obtain consistent masses at the weak scale, u, u . The new physics or cut-off scale can be taken as at the largest breaking scale, M ∼ w . The scale M characterizing the non-renormalizable interaction is responsible for generating V ub , V cb , as well as quark CP violation, as required.
The exotic quark, J 3 , is a physical field by itself, with mass, m J 3 = − z 33 w √ 2 , which is heavy, lying at the new physics regime. The two remaining exotic quarks, J α (α = 1, 2), mix via a mass matrix, and are both heavy, at the new physics regime too.
The mass matrix elements for the charged leptons, belong to the weak regime as usual. In contrast, the new leptons, N a , have large masses dictated by the mass matrix Neutrinos have both Dirac and Majorana masses. The mass matrix in the (ν L ν c R ) basis can be written as where M L , M D , M R are 3 × 3 mass matrices, given by As v L u Λ, the mass matrix (27) provides a realization of the full seesaw mechanism, producing small masses for the light neutrinos ∼ ν L , and large masses for the mostly right-handed neutrinos ∼ ν R , of order M R .

Scalar Sector
Since W -parity is conserved, only the neutral fields carrying W P = 1 can develop the VEVs given in (12). We expand the fields around their VEVs as The scalar potential can be written as independent of the fields, and all interactions are grouped into V int . V linear contains all the terms that depend linearly on the fields, and the gauge invariance requires, V mass consists of the terms that quadratically depend on the fields, and can be furhter decom- , which are listed in Appendix A.

The first mass term includes all pseudo-scalars
we see that A 1 , A 5 are massless and can be identified to the Goldstone bosons of the righthanded neutral gauge bosons, Z R , Z R , respectively. The remaining fields A 2 , A 3 , A 4 mix, but their mass matrix produces only one physical pseudo-scalar field with mass which is heavy, at the w, w scale. The remaining fields are massless and orthogonal to A and can be identified with the Goldstone bosons of the neutral boson Z L , analogous to the SM Z boson, and the new neutral gauge boson Z L .
The V S mass term contains all the mass terms of the scalar fields, S 1 , S 2 , S 3 , S 4 , S 5 , as shown in Appendix A. The five scalars mix through a 5 × 5 matrix. In general, it is not easy to find the eigenstates. However, using the fact that u, v w , w, Λ, one can diagonalize the mass matrix perturbatively. At leading order, this matrix yields one massless scalar field, H 1 = 1 √ u 2 +u 2 (uS 2 + u S 3 ), and a massive scalar field, . The H 1 field obtains a mass at next-to-leading order, m H 1 O(u, u ), and is identified with the standard model Higgs boson. The remaining fields, (S 1 , S 4 , S 5 ), are heavy and mixed among themselves via a 3 × 3 matrix. In the limit, Λ w, w , the corresponding physical fields have masses given by where the mixing angle θ H is defined by the relation On the other hand, if one assumes that instead the hierarchy w, w > Λ holds, the masses and mixing of the heavy states, (H 4 , H 5 ) change accordingly to Turning now to the singly-charged Higgs fields, we have three fields plus their conjugates.
The mass matrix extracted from (A3) yields four massless fields, which can be identified to the Goldstone bosons of the W ± L,R gauge bosons, and two singly-charged massive Higgs fields with corresponding masses There is only one doubly-charged Higgs field, σ ±± 22 , and is physical by itself, with mass For q-charged scalars, V q−charged mass contains the fields, φ ±q 13 , φ ±q 31 , σ ±q 13 , χ ±q 1 , as shown in Appendix A. The spectrum in this sector includes four massless Goldstone bosons of the new gauge bosons X ±q L,R , The remaining fields are massive. In the limit, Λ, w, w u, u , their physical states are with masses where , and defined by The other physical fields are massive with corresponding masses, For (q − 1) and 2q−charged scalars, σ ±(q−1) 23 and σ ±2q 33 are already physical fields, with masses m 2

Gauge-boson Sector
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, W ± L,R , X ±q L,R , Y ±(q+1) L,R . The fields W ± L,R , which are defined as W ± L = 1 √ 2 (A 1L ∓ iA 2L ) and W ± R = 1 √ 2 (A 1R ∓ iA 2R ), mix through the mass matrix, Diagonalizing this matrix, the eigenstates and masses are given by where Λ u, u and t 2ξ = −4t R uu 2t 2 R Λ 2 +(t 2 R −1)(u 2 +u 2 ) and t R = g R g L . W 1 is identified as the SM W boson, which implies u 2 + u 2 (246 GeV) 2 . W 2 is a physical heavy state, with mass at the new physics scale.
The mass matrix of the fields X ±q and yields two physical heavy states with masses which provides physical heavy states with masses where the mixing angle ξ 2 satisfies t 2ξ 2 = 4t R u w u 2 +w 2 −t 2 R (u 2 +w 2 +w 2 ) . The neutral gauge bosons, A 3L , A 3R , A 8L , A 8R , B, mix via a 5 × 5 mass matrix. In order to find its eigenstates, we first work with a new basis where .
The gauge boson A is massless and decouples, therefore it is identified with the photon field. The remaining fields, Z L , Z L , Z R , Z R , mix among themselves through a 4 × 4 mass matrix. Given that w, Λ u, u , the mass matrix elements that connect Z L to Z L , Z R , Z R are very suppressed. The mass matrix can be diagonalized using the seesaw formula to separate the light state Z L from the heavy ones Z L , Z R , Z R . Thus, the SM Z boson is identified with Z L whose mass is m 2 Z g 2 L 4c 2 W (u 2 + u 2 ). For the heavy neutral gauge bosons, the mass matrix elements are proportional to the square of the w, w , Λ energy scales. In the general case, it is very difficult to find the physical heavy states. However, if there is a hierarchy between two energy scales w, w and Λ, we can find them. In particular, in the limit Λ w, w , the physical heavy states are where the Z R -Z R mixing angle is With the physical states properly identified, 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.

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.
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 [57].
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, i.e. β = − 1 √ 3 . The neutral W -particles include a fermion N 0 a , a vector gauge boson X 0 1,2 , and a scalar H 0 1,2 4 .

Relic Density
Suppose that H 0 2 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 σv rel 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 H 0 * 2 H 0 2 → H 1 H 1 or mediated by new scalars. In the limit Λ w, w u, u , the interaction between H 0 2 and H 1 is approximated as 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 H 0 * 2 H 0 2 → H 1 H 1 contact interaction. The average cross-section times relative velocity is where the dark matter velocity v satisfies v 2 = 3 2x F , with x F = m H 2 /T F ∼ 20 at the freezeout temperature [58]. Since m 2 Thus, the dark matter candidate H 0 2 reproduces the correct relic density, Ω H 2 h 2 0.11 [56], if σv rel 1 pb, or for scalar couplings of O(1), and using the fact that α 2 /(150 GeV) 2 1 pb. Furthermore, the above condition implies where the upper limit comes from the perturbativity bound λ 1 , λ 2 < 4π. Therefore, the dark matter mass may be in the range few TeVs to 67 TeV, depending on its interaction strength with the SM Higgs boson.
In the scalar dark matter scenario, this scattering takes places through the t-channel exchange of a Z L and a heavy scalar H 1 0 . This scenario is similar to the one studied in [48], 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.

Relic Density
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 Z portal with predictive observables [51,68]. 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 Z L and the other gauge bosons to be small, which can be achived by taking Λ w, w u, u , one finds the relic density to be achieved either by annihilation into fermion pairs, or into Z L Z L . In Fig.1 we show the relic density curve in green.

Direct Detection
The dark matter-nucleon scattering is mostly driven by the t-channel exchange of the Z L gauge boson. This scattering is very efficient 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 [69]. 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 [69]). 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 Z L masses up to 2.5 TeV can reproduce the correct relic density in the Z L resonance regime. Although, LHC results based on heavy dilepton resonance searches with 13.3f b −1 of integrated luminosity exclude Z L masses below few 3.8 TeV [69], for this reason, the Majorana dark matter case has already been ruled out.
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 Z L mass in our model for 36.1f 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 m Z L > 4.2 TeV and m Z L > 5.7 TeV, respectively. These bounds can be seen as vertical lines in Fig.1. We emphasize that other limits stemming from electroweak precision or low energy physics are subdominant thus left out of the discussion [70,71] 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 Z L resonance. FIG. 1. Summary plot for the fermion dark matter. Relic density curve (green), LHC (pink) and current direct detection limit from XENON1T-34 days (red) [64], projected from XENON1T-2 years (blue) [72] and LZ (gray) [73] are overlaid.

Relic Density
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 X 0 1 . It can annihilate into SM particles via following channels, where ν = ν e , ν µ , ν τ , l = e, µ, τ , q = u, d, c, s, t, b. However, the dominant channels are Our predicted result is similar to the one given in [43]. The dark matter relic abundance is approximately given as 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 [41].
One way to circumvent the vector dark matter underabundance is by abdicating thermal production and tie its abundance to inflation, where the inflaton decay or the gravitational mechanism would generate the correct dark matter abundance [53]. Alternatively, we mention that vector DM could be just part of the overall cosmological dark matter within a multicomponent thermal dark matter scenario.

CONCLUSIONS
We have proposed a model of flipped trinification 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. The model offers 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 sufficient to account for realistic fermion masses. In summary, we have presented a viable theory of flipped trinification able to account naturally for the origin of matter parity and dark matter. The scalar fields mix according to the class they belong, and their relevant corresponding mass terms are derived as The gauge interactions of fermions arise from, where Ψ L and Ψ R run on all left-handed and right-handed fermion multiplets, respectively, and P CC L,R = n=1,2,4,5,6,7 T nL,R A nL,R , P N C L,R = T 3L,R A 3L,R + T 8L,R A 8L,R + g X g L,R X Ψ L,R B. The interactions of the physical charged gauge bosons with fermions are where the charged currents take the form, (ν aR γ µ e aR +ū aR γ µ d aR ), The interactions of the physical neutral gauge bosons with fermions are obtained by where f stands for every all the fermion fields, and e = g L s W . The vector and axial-vector Tables II, III, IV, and V. Note that at high energy g L = g R , i.e. t X = t R , 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 t X and t R .