Dirac neutrinos in a $SU(2)$ left-right symmetric model

In a left-right symmetric model, with the scalar sector consisting of several bi-doublets and two doublets, neutrinos remain as Dirac fermions in all order in perturbation theory. Although with only two bi-doublets the neutrino masses need still a fine tuning, this is not the case when a third bi-doublet is added. One of the scalar doublet may be of the inert type since it is protected by the left-right symmetry.


I. INTRODUCTION
At present it is very well established that neutrinos are massive particles and that there is mixing in the leptonic charged currents [1]. However, the nature of neutrinos is still unknown. They may be purely Majorana (equal to their charge conjugated fields) at all order in perturbation theory, purely Dirac (different from their charge conjugated fields) at any order of perturbation theory, quasi-Dirac when two active (left-handed) Majorana neutrinos are mass degenerate, or pseudo-Dirac when the mass degeneration occurs with an active (lefthanded) and a sterile (right-handed) neutrino, see [2,3] and references therein. In the last two cases, the mass degeneracy occurs at tree level but quantum corrections usually imply an additional small Majorana mass and, eventually neutrinos become Majorana particles. In fact, it is difficult to keep the lepton number L automatically conserved in most extensions of the standard model (SM) and neutrinos are in these models Majorana particles. With Majorana neutrinos it is possible to explain the smallness of their masses, even at the tree level, using the so-called type I and II seesaw mechanism in SU(2) L ⊗ U(1) Y models, if complex (Y = 2) scalar triplets and right-handed neutrinos ν R are added [4,5]. These mechanism can be implemented, for instance if new physics does exist at the TeV scale, in the context of models with SU(2) L ⊗ SU(2) R ⊗ U(1) B−L [6], in unified theories [7], and in models with SU(n) F ⊗ SU(2) ⊗ U(1) Y symmetries [8]. The type III seesaw mechanism requires the introduction of a self conjugate (Y = 0) triplet of fermions of SU(2) and can be implemented in SU(2) ⊗ U(1) y or SU(2) L ⊗ SU(2) R ⊗ U(1) B−L models [9].
One of the most motivated extensions of the electroweak standard model (ESM) are those with SU(2) L ⊗ SU(2) R ⊗ U ′ (1) gauge symmetry with left-right parity symmetry [10][11][12]. We will call them LR symmetric models for short. Although it is possible to introduce instead of a parity a generalized charge conjugation symmetry [13] here we will consider only the case of parity. In particular, in these models the parity may be spontaneously broken [14] and, moreover the U ′ (1) factor can be identify with B − L allowing to implement quarks and leptons correspondence since they are only distinguished by the B − L quantum number [6,15]. We must bear in mind that this correspondence would be stronger if neutrinos were Dirac particles. However, in the minimal LR model with one bi-doublet and two doublets the smallness of the neutrino masses is not easily explained [16]. In the context of this model quarks and leptons are those of the SM plus three right-handed neutrinos which are incorporated naturally in a doublet together the right-handed charged leptons.
In LR symmetric models Majorana neutrinos and the seesaw mechanism are obtained if, instead of scalar doublets χ L,R [16], scalar triplets ∆ L,R are introduced [6,17]. However, as we said above, we already do not known the nature of neutrinos purely Dirac or Majorana (with or without seesaw mechanism). Hence, we may wonder ourselves, what would happen if the neutrinos are, in reality, pure Dirac fermions? In addition, can we have Dirac neutrinos if the only additional neutral fermions were right-handed neutrinos? After all it would be interesting if the lepton-quark correspondence is maintained when all particle gain masses but this implies that neutrinos have to be Dirac fermions. In LR models with the scalar sector consisting of only one bi-doublet and two doublets [16] it is possible at least to accommodated Dirac neutrino masses.
Models have been proposed with Dirac neutrinos in which the smallness of theirs masses can be explained. For instance, calculable Dirac neutrino masses in the context of LR symmetric models were obtained but only by introducing extra heavy singlet leptons and/or charged and neutral scalars [18][19][20], or even doubly charged scalars [21]. Recently, anomaly free models that allow Dirac or inverse seesaw neutrino masses, which include sterile neutrinos with exotic lepton number assignment [22] was proposed in Ref. [23]. In scotogenic models in order to obtain calculable Dirac masses for neutrino we have for instance, i) to add, besides the right-handed neutrinos two neutral leptons N, N c per family, or ii) two new fermion singlets and one fermion doublet [24]. It is also possible to implement the inverse sewsaw mechanism, without the introduction of triplets but we must add more neutral singlet leptons [25]. An alternative formulation of the LR symmetric models in which B − L is a global unbroken symmetry and in which neutrinos are Dirac particles has been formulated recently [26]. However the price to be paid is the introduction of extra quarks and charged leptons. It means that, with only the known leptons plus right-handed neutrinos and renormalizable interactions, purely Dirac neutrinos do not arise easily in any model. Hence, it is interesting to search mechanisms that at least allow to accommodate light Dirac neutrinos in the context of a renormalizable electroweak model with a representation content in a completely analogy with the SM regarding the charged fermions, being the only extra neutral fermions three right-handed neutrinos.
On the other hand, although the resonance discovery at LHC [27,28] is consistent with the neutral scalar of the SM, it does not discard the existence of more neutral scalars (and their charged partners if they are not singlets of the gauge symmetry). Since the scalar content in any model is not fixed by the gauge symmetry, and also we do not known yet the complete spectra in the scalar sector we can add, in any model, more scalar multiplets.
Hence, the issue of the number of scalars is added to the generation problem: how many scalars?. The interesting possibility is that this number is equal to the number of fermion generations, i.e., three [29]. Although in the context of the SM, the introduction of three doublets is well motivated, say, for implementing CP violation [30] and/or dark matter candidates [31], in models with larger gauge symmetries a given number of scalar multiplets is not, necessarily, well motivated. It will depend on the phenomenological results. This is the case in the LR symmetric electroweak models in which there are several way to introduce scalar multiplets. Here, we will consider an extension of the minimal LR model by adding more bi-doublets and no triplets. In particular we show that the case of three bi-doublets it is possible to avoid a fine tuning in the neutrino Yukawa couplings. However, the details of the scalar potential are given only for the case of two bi-doublets.
The outline of this work is as follows. In the next section we consider the model and the symmetries that make it invariant under a generalized parity and other discrete symmetries in such a way that one bi-doublet is coupled only with leptons and the other only with quarks.
In Sec. III we consider the most general scalar potential invariant under the symmetries of the model. We show that for the two bi-doublets case an approximate Z 5 symmetry allows to consider a more simplified potential. The gauge vector boson sector in analyzed in Sec. IV, while Yukawa interactions and fermion masses are considered in Sec. V. Next, in Sec. VI the fermion-vector boson interactions are given, while in Sec. VII we analyze the case when we add a third bi-doublet. Some phenomenological consequences appear in Sec. VIII and in Secs. IX the case when parity is breakdown first is consider. Finally our conclusions appear in the last section.

II. THE MODEL
The model to be consider has the following electroweak symmetry: We omit the SU(3) C factor because is as in the standard model (SM). The electric charge operator defined as usual Q = T 3L + T 3R + (B − L)/2.
The left-and right-handed fermions transform non trivially under different SU(2) transformation. In the lepton sector L ′T , with l = e, µ, τ and the primed states denote symmetry eigenstates. Similarly in the quark sector, Q L ∼ (2 L , 1 R , −1/3) and Q R ∼ (1 L , 2 R , −1/3). The scalar sector consists of at least two and three bi-doublets transforming as (2 L , 2 * R , 0) and two doublets to break the parity and the gauge symmetry down to U(1) Q [14,16].
We also impose a generalized parity under which whereΦ i = τ 2 Φ * i τ 2 ; W µL,R are the gauge bosons of the factors SU(2) L,R , respectively, f denotes a quark or a lepton doublet, and Φ i and χ L,R are the scalar multiplets introduced above. The coupling constants g L,R , g ′ correspond the the groups SU(2) L,R and U(1) B−L , respectively. However, the invariance under P implies equality of gauge couplings g L = g R ≡ g at the energy at which these symmetries are realized. Under this condition the model has only two gauge couplings, g and g ′ . Although as a result of running couplings we have g L = g R [32] we will consider in this paper the case when these two coulings are equal at any energy scale but this has to be seen just as an approximation.
For the case of two bi-doublets we will impose also the discrete symmetries Z 2 × Z ′ 2 in such a way that under the first factor L R , Φ 1 → −L R , −Φ 1 , and under the second one

III. THE SCALAR POTENTIAL
Firstly, we consider the most general scalar potential invariant under the gauge symmetries and parity and then we see the effect of imposing discrete symmetries. Since some of our results are valid for an arbitrary number of bi-doublets we consider the scalar po-tential involving n bi-doublets and two doublets. In general a bi-doublet transforms un- Under these conditions, the scalar potential is given by: where We have omitted the redundant terms, for instance Tr(Φ † i Φ j ) = Tr(Φ † iΦ j ), and so on. Let us consider explicitly the case of two bi-doublets, n = 1, 2 in (4) with The vacuum expectation values (V EV s) are In general we will write the neutral components of the scalars as where v i , θ i may be complex numbers and R i , I i , Hermitian fields. However, here we will consider all VEVs real, i.e., θ i = 0 for all i running over the bi-doublets and doublets.
In this case, the invariance under the parity transformations defined in (2) im- However, we will allow for the moment a soft broken of these symmetries and use µ 2 = 0.
The constraint equations t X = ∂V /∂X, where A = Λ 11 + Λ 11 , B = Λ 12 + Λ 21 + Λ 12 + Λ 21 , C =Λ 12 +Λ 21 + Ω ′ 12 + Ω 21 and similarly we obtain t 2 and t ′ 2 for k 2 and k ′ 2 , respectively, but we will not write them explicitly. Finally, we have where Notice that only v L and v R can be zero, however this solution is not accepted for v R . We then we obtain from Eqs. (6) and (7), respectively, the constraint equations become In fact, we further restrict the Higgs potential so that it is invariant under the Z 5 symmetry, (defined as ω i = e −2ın/5 n = 1, · · · , 5) under which also other fields are invariant, the scalar potential in Eq. (4) becomes: and the constraints in Eq. (13) arise from this potential. It means that these conditions are protected by the Z 5 symmetry and may be naturally small. We can consider the potential in Eq. (15), and the respective mass spectra, as a good approximation. Notice that all VEVs may be zero, in particular the solutions k ′ 1,2 = 0 and v L = 0 are allowed. The SM-like Higgs scalar is in the bi-doublet Φ 2 . .
It is important to note that since the doublet χ L was introduced just to implement the invariance of the Lagrangian under parity and it does not couple to fermions, if the respective VEV is zero it is an inert doublet an the left-right symmetry protec its inert character, hence it is a candidate for dark matter. However, notice that v L = 0 is also a solution hence the possibility to have a model without any bi-doublet, with fermion masses arisen from nonrenormalizable interactions [33], in which case A = H = I = J = 0 in Eq. (14), it is posible.
However, in this case the model needs an ultraviolet completion. We stress that although the constraint equations in Eq. (14) were obtained using the potential in Eq. (15) by considering the most general potential (without the Z 2 ⊗ Z ′ 2 symmetry) we still obtain and the solution v L = 0 is still allowed even without a soft breaking of parity symmetry [34].

IV. GAUGE BOSONS MASS EIGENSTATES
The covariant derivative for the bi-doublets Φ i , i = 1, 2 and for the doublets χ L and χ R are given by where we have already did g L = g R = g. (However, see Sec. II.) With the VEVs given in Sec. II we obtain for the charged vector bosons: , and the respective eigenvalues are given by where ∆ = 4z + 1 4 (y − 1) 2 . These expressions can be generalized for an arbitrary number of bi-doublets K 2 = n i k 2 i andK 2 = n i k i k ′ i and the results of this section are valid for n bi-doublets.
Symmetry and mass eigenstates are related by an orthogonal matrix: where Y = 1 − y + 2 √ ∆, and c ξ = cos ξ, etc. Notice that since we will always consider that W 1 , and we identify the W ± of the standard model with W ± 1 . In the limit v R → ∞ (x, z, y ≪ 1), we obtain In the neutral vector bosons we have the massa matrix: where we have defined x and y as before and r ≡ g ′ /g.
The determinant of the matrix in (23) is zero and its eigenvalues are without any approximation M A = 0, we have defined The symmetry eigenstates (W 3L , W 3R , B) are linear combinations of the mass eigentates n n 12 n 13 n n 22 n 23 n ′ n 32 n 33 Although we have all the elements of n ij exactly calculated, here we write for the sake of space only n and n ′ exactly, while the other entries are in the approximation where we have defined φ = (x − r 2 y)/(1 + r 2 ) 3/2 . The angle θ is defined below, see Eq. (46).
In the limit v R → ∞ i.e., x, y → 0 (φ → 0 also), the matrix in Eq. (27) becomes to the usual form in literature: Going back to the masses of vector bosons we note that in the limit v R ≫ v where v is any VEVs, we obtain from Eq. (24) and we see from (22) and (29) that Notice that only in the limit v R → ∞ the angle θ in this model has a relation with the θ W of the SM. However, it is important that v R is keep to be large but finite in order to obtain a lower bound on the right-handed vector bosons, W 2 and Z 2 [35] and the respective coupling with fermions. If χ L is an inert doublet, we simply put v L = 0, or equivalently y = 0, in the expressions above.
Using the exact results in Eqs. (19) and (24), where v SM = 246 GeV, and M W /M Z = 0.88147 ± 0.00013 [1] we obtain (using 2σ value of that ratio) that v R > 24 TeV. With this lower limit on v R we can calculate the lower limit for the masses of W + 2 and Z 2 , using Eqs. (19) and (24), respectively, obtaining (in TeV): and the mixing angle W L − W R defined in Eq. (21) has an upper limit sin ξ < 10 −4 . Recent analysis comparing the experimental limits to the theoretical calculations for the total W 2 resonant production and the decay W 2 → W Z implies that ξ is between 10 −4 − 10 −3 [36].

V. YUKAWA INTERACTIONS AND FERMION MASSES
The Yukawa interactions in the lepton sector are given by where L ′ and R ′ are defined in Sec. II and we have omitted generations indices. Since Φ i ↔ Φ † i under the left-right symmetry implies that G † = G and F † = F . With these interactions and the vacuum alignment the mass matrices in the lepton sector A similar expression arises in the quarks sector but now . We recall that the Z 2 ⊗ Z ′ 2 symmetry forbids the coupling of the bi-doublet Φ 2 with leptons and Φ 1 with quarks.
Primed fields denote symmetry eigenstates and unprimed ones mass eigenstates. In general G, F and VEVs are complex, and the mass matrices are diagonalized by bi-unitary transformations as follows: whereM l = diag(m e , m µ , m τ ) andM ν = diag(m 1 , m 2 , m 3 ) for charged leptons and neutrinos respectively.
For given an appropriate mass to the quarks, we have introduce the bi-doublet Φ 2 , and it is possible to implement the analysis as in Ref. [37]. Notice that this means that the neutral scalar with VEV and mass about 174 and 125 GeV is part of this bi-doublet.
We will assume that k ′ 1 = 0 (see Sec. III) and in this case the lepton mass matrices are given by where G and F are symmetric complex matrices that are diagonalized by the bi-unitary transformation in Eq. (34). Hereafter we will consider, just for the sake of simplicity, all VEVs being real.
From these matrices and the lepton measured masses we found the Yukawa coupling and we use for numerical calculations |k 1 | = 2 GeV since this VEV is the only one for generating the lepton masses. We will work for the sake of simplicity in the basis in which the charged lepton mass matrix is diagonal and consider the matrices G, F and all VEVs real. In this case U ν L = U ν R ≡ U ν , and U ν = V L P M N S = V R P M N S ≡ V l , and we have being the unitary matrix V l parametrized in the same way for Dirac particles. We use the PDG parametrization for Dirac neutrinos, for the interactions with W + L,R : with s l ij = sin θ l ij , · · · and we have considered δ l = 0. In the case the Yuakawa interactions are and V l given in (38). Notice that in this case (charded leptons in the diagonal basis), the Higgs φ 0 1 is the one whose couplings with charged leptons are proportional to respective masses and the couplkings with η 0 1 are suppressed by the neutrino masses in the charged lepton sector. In the neutrino sector the situation reverses the enhanced interactions ar those with η 0 1 since they are proportional to the charged lepton mases. For instance, the vertexν 3R ν 1L η 0 * 1 has the strengh proportional to s l 13 c l 13 c l 23 m τ and η 0 1 can decay through out its mixing with the other neutral scalar, into two of the other particles, bosons or fermions.
For completeness, we write the Yukawa interactions in the quark sector (mass matrices diagonalized by the unitary matrices In the quark sector we shall not consider the solution k ′ 2 = 0 since for the case of generalized parity, P, it has been shown that k ′ 2 ≪ k 2 is ruled out by the CP violating parameters ǫ and ǫ ′ , however this hierarchy is allowed in the case of genelaized C [38].
Notice that there are flavor changing neutral currents mediated by scalars in both, lepton and quark sectors.

VI. FERMION-VECTOR BOSON INTERACTIONS
The covariant derivatives are given by and similarly for quarks. The lepton-gauge boson interactions are obtained from L =L ′ l γ µ D µL L ′ l and similarly for the right-handed doublets.

A. Charged currents
The charged current interactions in the mass eigenstates basis are given by the Lagrangian where J lµ L =ν L γ µ V l l L and J lµ R =ν R γ µ V l l R . In the general case where the Yukawa couplings in Eq. (33) are complex the right-handed CKM matrix is different from the left-handed one.
This case was considered in Ref. [39].
In the quark sector where J qµ L =ū L γ µ V CKM d L and J qµ R =ū R γ µ V CKM d R with V CKM being the same as in the left-handed sector with 3 angles and one physical phase.
The introduction of the phase φ l and φ q in Eqs. (42) and (43), respectively, need an explanation. In the mixing matrix for n Dirac fermions 2n − 1 phases are absorbed in the Dirac fields since one is a global phase. In the SM this is enough because there is only one charged currents and the global phase never appear in amplitudes. However, in this sort of models there are also right-handed charged currents and the there is a relative global phase between both charged currents. This phase is φ l for lepton and φ q for quarks.

B. Electromagnetic interactions
The interaction with the photon arises from the projection of W 3L , W 3R and B over A using the matrix in Eq. (26). Then, it is possible to verified that the electric charge is written in terms of g and g ′ as and we obtain 1 where g Y is the coupling constant of the SM. These relations are valid only at the energy scale at which g L = g R ≡ g. Hence, we have the relations From the relations in Eq. (46) we have r = g ′ /g = sin θ/ √ cos θ. We have also that and the model has a Landau-like pole in g ′ when s 2 θ = 1/2 but it happens at energies larger than the Planck scale. However, this only implies that the energy scale at which g L (µ) = g R (µ) must be below the scale at which s θ (Λ) = 1/2, µ < Λ.

C. Neutral currents
Next, we parametrize the neutral interactions of a fermion i with the Z 1µ and Z 2µ neutral bosons as follows: Let us consider the case when the VEV of the doublet χ L is not zero, v L = 0. Using (26) and r = s θ / √ c 2θ in Eq. (27), we obtain: Defining where a f L and a f R are the couplings of the left-and right-handed compnents of a fermion f .
Notice that when v R → ∞(x, y → 0), we obtain and the same happens with the coefficients of the quark sector in that limit, we obtain It is worth noting that the vector couplings are the same for Z 1 and Z 2 . We see once again that only when v R is strictily infinite we can identify, at tree level, the angle θ with θ W of the SM. Assuming the measured values g l V = 0.03783 ± 0.00041 does not implies a stronger lower bound on v R and the W 2 and Z 2 masses obtained from the M W /M Z ratio in Eq. (31).
Recently, the CMS Collaboration using W 2 → B + t or W 2 → T + b (T, B are vectorlike quarks VLQ) a W 2 with a mass below 1.6 TeV is excluded at 95% CL assuming equal branching fractions for W ′ boson to tB and bT and 50% for each VLQ to qH where H is a neutral escalar [40]. If T, B are the known t, b quarks and assuming W 2 with coupling to the SM particles equal to the SM weak coupling constant, masses below 3.15 TeV are excluded at the 95% confidence level [41].
Furthermore, if right-handed gauge bosons decay into a high-momentum heavy neutrino and a charged lepton at LHC has excluded values of the W R smaller than 3.85 TeV for N R in the mass range 0.11.8 TeV [42]. Of course, if there are no extra quarks like T and B and neither heavy right-handed neutrinos, these restrictions for the mass of W R are not valid anymore.
Only for ilustration, we give the partial widths at tree level and neglecting the fermion masses, For the Z 2 and also neglecting the fermion masses we have Notice that scalar doublets χ L,R do not couple to fermions and we will assume that vacuum alignment is such that v L = 0, and this scalar field does not contribute to the gauge boson masses, hence χ L is an inert doublet [43]. Hence, in this case the inert character is protected by the left-right symmetry.

VII. LEPTON MASSES AND MIXING
Here we will obtain, assuming the measure matrix elements of the PMNS matrix, the Yukawa coupling generating the correct charged lepton and neutrino masses. Firstly, considering the present case, i.e., two bi-doublets and then we briefly discuss the case with three bi-doublets.
A. The two bi-doublet case Recall that in the case we are considering here, the PMNS mixing matrix is given by V l .
Using the numerical values for the neutrinos masses in Eq. (56) and the PDG's angles, In the quasi-degenerate case, in (58), we obtain also up to a factor 10 −11 , in this case all the other G's vanish for all practical purposes.
Although this model with two bi-doublets contains a fine adjustment as in the SM, which is avoided if we introduce the scalar triplets, this would be the price to pay for having Dirac neutrinos and only the known charged leptons plus the right-handed neutrinos. However, we will show that when three bidoublet is considered it seems possible to avoid a fine tuning in the lepton masses.

B. Three bi-doublets case
It is interesting that one of the natural hierarchy in field theories are those in the values of the VEVs which are responsible by the spontanously breaking of symmetries. This is because their values depend on the vacuum alignment and heavy scalars may have small VEVs. Probably this was first noted by Ma [44] and we have seen an example in Sec. III in the case of k ′ 1 . Moreover, as we have strssed before, we already do not known the number and sort of scalars and we can think of an extension of the present model in which three bi-doublets (and the two doublets χ L,R ) are introduced.
In this case, the Yukawa interactions is the sector of the model which is more affected by the existence of a third bi-doublet is the Yukawa. Let us denote Φ ν , Φ l and Φ q the three bi-doublets. We introduce the discrete symmetreis D under which [45] and all the other fields being invariant under D. Under this conditions thw Yukawa interactions are given by Notice that the D symmetry forbid the interactions asL ′Φ ν R ′ andL ′Φ l R ′ , whereΦ = τ 2 Φ * τ 2 . Denoting the respective VEVs k ν , k ′ ν , k l , k ′ l and k q , k ′ q , we see that if the vacuum alignment allows that k ν , k ′ ν , k l ≪ k ′ l ≪ k q , k ′ q , then neutrino masses arise from k ν , the charged lepton masses from k ′ l (these leptons receive a small contributions from k ′ ν ). If with X = S, A, ǫ = m τ /m µ , λ = m µ /M X and Q S (x) = x 2 (1 + ǫ − x) for a scalar S, and Below, we will consider mainly the difference with the case of the model with Majorana neutrinos (with triplets in the scalar sector), in particular when heavy neutrino do exist, with the present model woth Dirac neutrinos.
• Obviously, in the present case there is no heavy neutrinos that can decay into a Higgs boson plus an active neutrino, N R → H + ν.
• Although the processes µ → e+γ occurs in this model, they have not the (logarithmic) enhancement produced by the doubly charge scalar bosons [47,48].
• Flavour lepton number processes as µ → eeē and µ − e conversion cannot occur in this model.
Notice that in the KS process there are no missing energy. However, in both Majorana and Dirac neutrino cases there are processes like . We recall that in the SM the processes madiated by the W ± , for instance pp Hence, at least in principle, it is possible to use these processes to distinguish the Dirac from the Majorana case. It is worth to note that if neutrino are Majorana particles through the type II seesaw mechanism implemented in the SM plus a scalar triplet with Y = 2 the processes are pp → H + → ν L l + L → l + L l − L l + R ν L , if a charged scalar H + is in the second lepton vertex; or pp → H + → ν L l + L → l + L l − R l + R ν L , if a W + R is in the second leptonic vertex. The case of trimuons, l = µ, could be the more interesting in all these processes.
Finally, but not least, we note that in LR or other models with s second charegd current, there is a contribution to the electric dipole moment (EDM) of an elementary particle, say electron or quarks (neutron), at the 1-loop level. The phase in the CKM or PMNS matrix do not contribute at this level since in this the diagram CP violating phase cancel out because on vertex in the complex conjugate of the other, say V CKM V * CKM in the quark secto, and the diagram is real [50]. However, if there is a second charged currents as in LR models, the relative phase φ l and φ q in Eqs. (42) and (43) cannot be absorved. In particular it implies a contribution to the EDM a quark: where M 2 W i = M 2 W 1 ,W 2 , and the larger contributions is that of W + 1 . As an illustration, the EDM of a light quarks with m q ∼ 10 −3 GeV and M W 1 ∼ 80 GeV, s ξ ∼ 10 −3 we obtain µ q E ≈ 3.08 × 10 −26 sin φ q e cm, which is almost the present limit for the EDM of the neutron µ n E < 0.30 × 10 −25 e cm CL 90% [1]. Hence, the phase φ q is not restricted with the present experimental data. In the lepton sector the most restringent EDM is that of the electron µ e E < 0.87 × 10 −28 e cm, CL=10% [1] which implies sin φ l < 10 −2 . In the lepton sector the 1-loop contribution (induced by the W L − W R mixing) to the electron EDM is suppressed by the neutrino masses and the phase φ l is not suppressed by this observable. On the other hand, a neutrinos have magnetic and electric dipole moment (induce by the mixing) which are not suppressed since they are proportional to the heavy charged lepton.

IX. BREAKING PARITY FIRST
Any model beyond the standard model (BSM) must match with that model at a given energy, say the Z-pole. In the SM coupling constants g and g Y have different running with the energy. In the case of LR symmetric models, the same happens with g and g ′ ≡ g B−L as was noted in Ref. [32]. It means that we cannot keep g L = g R for all energies since quantum corrections imply a finite ∆g = g L − g R = 0. This is due to the fact that both constants feel different degrees of freedom. Hence, it is interesting to search for models with gauge symmetries as in Eq. (1) but in which parity is broken spontaneously by non-zero VEVs [32] or softly if quadratic terms in the scalar potential are different µ 2 L = µ 2 R as in Ref [12]. Let us consider as in [32] the possibility that the symmetries in Eq. (1) are broken spontaneously but in the following way: First the parity P is breaking at an energy scale µ P by introducing a neutral pseudoscalar singlet, η ∼ (1, 1, 0) with η → −η under parity.
The relevant terms in the scalar potential involving the doublets χ L , χ R and the isosinglet η are the following: At the energy µ P , µ 2 η < 0 with η = v η ≃ µ P , and all the other VEVs are still zero. We obtain with the singlet VEV v η = −µ 2 η /2λ η . Next, if µ 2 R < 0 and |µ 2 R | ≪ v η we have that χ R = v R = 0. This leads to the interesting case in which the SU(2) R symmetry breaking scale is induced by the parity breaking scale as noted in Refs. [32]. It happens also that g L = g R , for energies in the range v R < µ < v η , and also V L P M N S = V R P M N S , with V L P M N S = V l † L U ν L , V R P M N S = V l † R U ν R . In this case we have to consider the most scalar potential involving two or three bi-doublets, Φ i , two doublets χ L,R , and the singlet η. We stress that, since the early eighties most phenomenology of the left-right symmetric models includes triplets and Majorana neutrinos [17]. Since then, the model with the following scalar multiplets: one bi-doublet and two triplets was considered the minimal left-right symmetric model. There is no doubt that this proposal was, and still is, well motivated [51]. However, if the neutrinos ultimately turn out to be Dirac particles, all that efforts will have been in vain. For this reason we have to pay attention to Dirac neutrinos, even in the context of the left-right symmetric models.