Dark Matter from the vector of SO(10)

SO(10) grand unified theories can ensure the stability of new particles in terms of the gauge group structure itself, and in this respect are well suited to accommodate dark matter (DM) candidates in the form of new stable massive particles. We introduce new fermions in two vector 10 representations. When SO(10) is broken to the standard model by a minimal 45 + 126 + 10 scalar sector with $SU(3)_C \otimes SU(2)_L \otimes SU(2)_R\otimes U(1)_{B-L}$ as intermediate symmetry group, the resulting lightest new states are two Dirac fermions corresponding to combinations of the neutral members of the $SU(2)_L$ doublets in the 10's, which get splitted in mass by loop corrections involving $W_R$. The resulting lighter mass eigenstate is stable, and has only non-diagonal $Z_{L,R}$ neutral current couplings to the heavier neutral state. Direct detection searches are evaded if the mass splitting is sufficiently large to suppress kinematically inelastic light-to-heavy scatterings. By requiring that this condition is satisfied, we obtain the upper limit $M_{W_R} \lesssim 25$ TeV.


Introduction
A plethora of astrophysical and cosmological observations have firmly established that non-baryonic dark matter (DM) must exist in our Universe, and contribute to the overall cosmological energy density about five times more than ordinary matter. However, none of the particles of the standard model (SM) can account for the DM, which therefore constitutes a clear hint of new physics. Colorless, electrically neutral and weakly interacting massive particles with mass in the GeV-TeV range are ubiquitous in new physics models, and appear to be well suited to reproduce quantitatively the measured DM energy density if their stability on cosmological time scales can be ensured.
From the model-building point of view, DM stability is most commonly enforced by assuming some suitable symmetry that forbids its decay into lighter SM particles. For example, in supersymmetric models this role is played by R-parity that stabilizes the lightest supersymmetric state, in universal extra dimensional models conservation of Kaluza-Klein parity ensures that the lightest Kaluza-Klein state remains stable [1], T -parity stabilizes the lightest T -odd particle in the littlest Higgs model [2], suitable Z 2 parities play the same role e.g. in the scotogenic model [3,4], in the inert doublet model [5][6][7], and in several other cases. Often these stabilizing symmetries are just imposed by hand on the low energy Lagrangian, and it is certainly more satisfactory when their origin can be traced back to some high energy completion of the model in question. A plausible way to generate unbroken discrete Z N symmetries relies on assuming extra gauged U (1) Abelian factors which are only broken by order parameters carrying N units of the U (1) charge [8] (see also [9][10][11][12]). Such a mechanism renders grand unified theories (GUTs) based on gauge groups of rank larger than four particularly interesting, since they contain extra Cartan generators besides the 2 + 1 + 1 of the SU (3) C × SU (2) L × U (1) Y SM gauge group that, when broken by vacuum expectation values (vevs) of scalars in appropriate representations, yield discrete Z N symmetries which inherit all the good properties of the parent local gauge symmetry. In particular, this type of symmetries remain protected from gravity induced symmetry breaking effects [13][14][15][16] which, although suppressed by the Planck scale, could jeopardize DM stability [17,18].
One of the most interesting GUT groups that allows to preserve at low energies an unbroken discrete gauge parity is the rank five group SO (10). As is well known, SO(10) has many theoretically appealing properties: it unifies all SM fermions in a single 16 dimensional irreducible representation including one right handed (RH) neutrino, it can explain the suppression of neutrino masses via the seesaw mechanism [19][20][21][22][23], it allows for gauge coupling unification at a sufficiently high scale to account for proton stability, and is automatically free from gauge anomalies. In this paper we focus on a SO(10) GUT model in which the breaking to the SM gauge group is driven by vevs of scalars in the 45 H ⊕126 H ⊕10 H representation. It has been recently shown that this model is compatible with unification [24], and that it can fit all charged fermion masses and mixings as well as the low energy neutrino data [25,26], while simultaneously explaining the cosmological baryon asymmetry via leptogenesis [27]. It is therefore interesting to see if this framework can also accommodate automatically stable DM candidates in the fundamental 10 dimensional representation of the group.

Motivations and general considerations
SO(10) is a rank five group and thus with respect to the SM model it contains one additional Cartan generator that, upon breaking of the unified group, can give rise to a new U (1) gauge group factor. The U (1) charges of the component fields are conventionally normalized by setting the smallest charge equal to one. Then, if U (1) is further broken by vevs of scalars carrying n 1 , n 2 , . . . units of charge with N > 1 as their greatest common divisor, a discrete center Z N ∈ U (1) remains unbroken [28][29][30]. 1 In our setup SO(10) is broken to U (1) Q × SU (3) C by vevs of scalars in 45 H ⊕ 126 H ⊕ 10 H , which are all SO(10) tensor representations. With respect to the non-SM U (1) factor singled out in the maximal subgroup SU (5)×U (1), which is the one whose breaking gives rise to the gauge discrete symmetry, all SO(10) tensor representations branch to SU (5) × U (1) fragments which have even values of the U (1) charge. The lowest charge value for the fragments acquiring vevs is 2 [e.g.: 10 → 5(2) ⊕ 5(−2)] and therefore a Z 2 parity survives, which can guarantee the stability of the lightest particles belonging to appropriately chosen representations.
For example, fermions in the vector 10 cannot decay into SM fermions in the 16 since this is a spinorial representation for which all fragments under SU (5) × U (1) carry odd U (1) charges and upon U (1) breaking then acquire odd Z 2 parity. Various proposals for SO(10) DM candidates that are stabilized by the Z 2 parity of gauge origin have been put forth in the recent literature: a dedicated analysis of scalar DM in the 16 was carried out in [32,33], while the possibility of fermionic DM in the 45 was addressed in [34] (see also [35] where the 45 is allowed to mix with a 10). Other more general studies regarding possible embeddings of DM in SO(10) can be found in [36,37]. Indeed, so far a special attention has been devoted to DM in the scalar 16 and in the fermionic 45, and a possible reason for this might be the fact that both these representations contain SM singlets. Needless to say, identifying DM candidates with SM singlets can naturally explain why all experimental direct detection (DD) searches have been eluded so far. In contrast, the 10 dimensional vector representation of SO(10) has not attracted much attention, although this could well be considered as the minimal choice. Perhaps this is due to the fact that the 10 does not contain SM singlets, and in particular all its states carry hypercharge, and thus couple to the Z boson, which might led to the conclusion that this possibility is excluded by DD limits.
In this letter we argue that fermionic DM in the 10 of SO(10) is instead a viable possibility. Our main observation is that in a scenario in which fermions in a vectorlike 10 L ⊕ 10 R acquire tree level masses via a Yukawa coupling with the (antisymmetric) 45 H , loop diagrams involving an insertion of W L -W R mixing produce a mass splitting between the two lightest mass eigenstates, which (in our minimal realization) are two neutral Dirac fermions. We show that the neutral Z L,R gauge bosons couple non-diagonally the light eigenstate to the heavier one and, as a result, at the leading order only inelastic neutral current scatterings of DM off target nuclei is allowed. If the mass splitting between the light and heavy mass eigenstates is larger than the typical DM kinetic energy E K ∼ 200 keV, then the scattering is kinematically forbidden and DD bounds are automatically evaded. Since the loop-induced splitting is suppressed by the RH gauge boson mass, the previous requirement can be translated into an upper-bound on M 2 W R which, combined with the lower bounds from flavour and CP violating processes in the K and B meson systems [38] and from direct searches at the LHC [39][40][41], results in 2.9 TeV < ∼ M W R < ∼ 25 TeV. The fact that the null result of DD DM searches constrains M W R to lie at a relatively low scale, can reinforce the hope that a rich phenomenology could be within the reach of the LHC.
In eq. (3) we have introduced for the gauge groups the short-hand notation e.g. is needed to reproduce realistic fermion masses [44,45] and in particular to accommodate the m t /m b mass ratio. Moreover, to reproduce the complete charged fermion mass spectrum accounting also for Yukawa non-unification of the lepton and down-type quarks of the first two generations, a contribution from the vevs of the electroweak doublets appearing in the 126 H is also necessary [44]. These vevs are unavoidably induced when SU (2) L × U (1) Y is broken by the 10 H [44,45]. All in all, the masses of the SM fermions are generated from the following Yukawa terms: where h, g and f are 3 × 3 symmetric matrices in flavour space and i, j are family indices. The fermion couplings to 10 * H can be forbidden by assigning to the fields a global U (1) Peccei-Quinn charge [44][45][46]. In practice this sets g ij → 0, simplifying the Yukawa structure of the model, and providing a DM candidate for non-supersymmetric SO(10) models in the form of axions. In the spirit of avoiding the introduction of additional symmetries, and given that we are interested in a weakly interacting DM candidate, we will not follow this route, and we allow for g ij = 0. Possible FCNC arising from coupling quarks of the same type to two different Higgs doublets, as it would happen in this situation, can be kept under control in various ways e.g. by assuming a hierarchy g f .

Adding fermions in the vector representation
Let us now add to the SO(10) model outlined above a pair of fundamentals 10 L ⊕ 10 R containing new fermions. 2 The tensor product of two vectors of SO(10) is: where the superscripts denote symmetric and antisymmetric representations. Although our model does not contain a 54 of fundamental scalars, loop corrections can generate mass contributions that mimic the coupling to (effective) representations, as long as these couplings are allowed by the symmetries of the model.
To keep as general as possible it is then convenient to write down all the allowed gauge invariant Yukawa couplings, which are: where, in order not to over-clutter the expressions, we have left understood the usual spinor notations. It is instructive to analyze these couplings in terms of representations of the SU (5) ⊂ SO (10). The branching rule for the SO(10) vector is 10 = 5 +5 so that in the first line the invariant mass term M a multiplies the 5 ·5 singlet from the product of the same 10 a . The second term involves the symmetric 54 = 15 + 15 + 24 and, besides containing a 5 · 24 ·5 coupling involving the SU (5) adjoint, it also includes couplings of the symmetric 15 to a pair of fundamentals: 5 · 15 · 5 + c.c. Let us note at this point that if the colorless SU (2) triplet contained in the 15 (15) acquires a small vev, these terms would generate a Majorana mass for the neutral components of the fermion doublets in the 5 (5). However, the same is not true for the analogous term in the second line since it contains only terms that couple two different 10's. Finally, since the 45 H is antisymmetric, it must couple different representations, and thus it appears only in the second line.
The model we will now study is specified by the following ingredients: (i) the 54 H is absent; (ii) the adjoint vev 45 H which can be written as: acquires a Dimopoulos-Wilczek structure [47,48] with a ∼ Λ GU T and b/a ≈ 0 (since b = 0 breaks SU (2) L × SU (2) R we require b < ∼ Λ I in order to respect the symmetry breaking pattern eq. (3)); (iii) we set M a → 0.
This can be viewed as technically natural since in this limit a global U (1) symmetry 10 L,R → e iα L,R 10 L,R arises (we will briefly comment below on the consequences of relaxing this assumption); (iv) finally, we will also work with M → 0. This is just a simplification: a term proportional to M preserves the global U (1) symmetry obtained for α L = −α R which eventually ensures the Dirac nature of the DM states (see next section), and as long as M ∼ y b a a non-vanishing M would not change the analysis.

Fermion spectrum and neutral current couplings
In the following we label SU Each one of the two 10's contains one (2 L 2 R 1 C ) 0 bi-doublet. We denote the bi-doublet contained in 10 L,R as ξ L,R , with components: where the superscripts carried by the component fields denote the SU (2) L ⊗ SU (2) R isospin eigenvalues of T 3L,3R in units of ± 1 2 . Given that the 10 has vanishing U (1) B−L quantum number, the electromagnetic charge for this representation is simply The mass term for the neutral states arising from 45 H in Eq. (7) reads: with m b = y b. A similar mass term can be written also for the charged components ξ −− L,R , ξ ++ L,R which at lowest order are degenerate in mass with the neutral states. However, electromagnetic corrections from loops involving SM gauge bosons lift this degeneracy inducing a charged-neutral mass difference m ± − m 0 340 MeV [49], thus ensuring that the lightest states are the neutral ones. 3 Eq. (9) describes a pair of neutral Dirac fermions ξ degenerate masses equal to m b . However, these states get mixed via loop diagrams (depicted in figure 2) involving two external 10 H vevs, which generate a mass term: The tree level eq. (9) and the loop induced mass term eq. (10) are sketched in figure 1. Upon diagonalization of the mass matrix (see the Appendix) we end up with one heavier (χ h ) and and one lighter (χ l ) Dirac fermions, respectively with masses The important point is that the couplings of χ h,l to the neutral gauge bosons Z L,R are off-diagonal: both Z L,R couple (with an opposite overall sign) to the vectorlike neutral current Two ingredients are crucial for this result: (i) a complex 10 H : this is in any case needed to reproduce the SM fermion mass pattern. The tensor product of two 10's contains in its symmetric part the 54, which can be regarded here as an effective 54 H coupled to the fermions via the loop diagram. Moreover, since 10 H develops vevs in the L-R doublet components, the effective 54 H ⊂ 10 H ⊗ 10 H contains a non-vanishing mentioned at the end of section 3 remains unbroken also at the loop level. 4 3 We have checked that loops of RH gauge bosons do not contribute. This is because the 10 has vanishing B − L charge, and |T 3R | = 1 2 for both charged and neutral states, so that the corresponding loop contributions cancel in the mass difference. Therefore the SM result [49] for the charged-neutral mass difference for fermion doublets holds also in this case. 4 Allowing for ML,R = 0 would instead break this U (1). As a result, we can expect that the two Dirac states will split into four Majorana fermions with a spectrum determined by the relative sizes of m b , δm and ML,R.

Constraints from direct detection
The mass splitting m h − m l = 2δ m is an important quantity, since for m DM ∼ 1 TeV, the Z L,R mediated inelastic DM scattering off target nuclei χ l +N → χ h +N is kinematically forbidden only if 2δ m > ∼ 200 keV [50], and only in this case χ l could have escaped DD DM searches. Let us then proceed to estimate its value.
We denote the bi-doublet scalar contained in 10 H as φ, with components: where the superscripts have the same meaning as for the fermions eq. (8). Assuming for simplicity that there is only one scalar bidoublet and that its vevs are real, we can write where v 2 u + v 2 d = v 2 is the electroweak breaking vev. One diagram contributing to δ m is depicted in figure 2. The crossed diagrams should also be added, and a similar pair of diagrams can be drawn for external fermions with exchanged LR isospin labels (+−) ↔ (−+). Taking M W R as the largest mass scale in the loop, the diagram can be estimated as where we have used Weinberg angle, and m b δ m is to a very good approximation the DM mass. In the last expression we have which is experimentally bounded in various ways.
Electroweak precision data set the upper limit ϑ LR < 0.013 [51,52]. On the other hand, ϑ LR is bounded from above also by the ratio of gauge boson masses squared: where in the first equality we have approximated g R ≈ g L . Flavour and CP violating processes in the K and B meson systems provide an absolute lower bound on the SU (2) R gauge bosons mass M W R > 2.9 TeV [38]. Similar bounds on M W R have been also obtained from direct searches at the LHC [39][40][41]. Using these figures we obtain the conservative upper bound ϑ LR < 7.7 × 10 −4 . The mass splitting 2δ m between the light and heavy neutral states can then be bounded from above as: The constraint from DM non-observations in DD experiments [50] 2δ m > ∼ 200 keV can then be translated in the following upper limit on the SU (2) R gauge boson masses: Indeed this result suggests the possibility of a non trivial interplay between DM searches in DD experiments, and searches for new physics at the LHC or at a future 100 TeV hadron collider. 5 Before concluding this section, let us note that non-vanishing DD cross sections will appear at the loop level, with leading contributions involving the exchange of pairs of SU (2) L gauge bosons (W L W L or Z L Z L ).
The quantitative effects of the corresponding diagrams have been studied for example in refs. [65][66][67][68], and it was found that the resulting cross sections do not exceed ∼ O(10 −47 ) cm 2 , which is far below the current experimental bounds [69,70]. In the relevant mass range (m χ > ∼ TeV) this remains also below the reach of next generation DD experiments [71] and close to the neutrino scattering background.

Neutrino masses
The relatively low value of the intermediate symmetry breaking scale implied by eq. (18) could be of some concern for what regards the light neutrino masses. In general, the fact that the seesaw mechanism [19][20][21][22][23] can be automatically embedded within SO(10) provides an elegant way to explain why the neutrino masses are so suppressed. SO(10) unification implies relations between the light neutrino, RH neutrinos, and uptype quark masses, which generically require rather heavy RH neutrinos (M N R ∼ m 2 u /m ν ). The natural 5 Let us recall at this point that a certain number of anomalies have been recently reported by both ATLAS [53] and CMS [54,55] which could be explained by a low-scale L-R model with M W R ∼ 2 TeV, see e.g., [56][57][58][59][60][61][62][63][64].
range for M N R is then loosely determined by the up and top quark masses as 10 4 GeV < ∼ M N R < ∼ 10 14 GeV.
On the other hand N R 's acquire their masses from the same 126 H vev that breaks G I and concurs to determine the value of M W R , and thus we would expect their masses to be of the order of M W R , which remains bounded from above by eq. (18). Such a relatively low mass scale for the RH neutrinos does not provide enough suppression. One might then be tempted to appeal to a different suppression mechanism.
For example the inverse seesaw [72] is an elegant option that allows to suppress neutrino masses even when M N R ∼ TeV. However, implementing the inverse seesaw requires the addition of a SO(10) singlet with a (small) Majorana mass term, which couples to N R and to some scalar representation with non-vanishing vev. Given that N R ∈ 16 the only option for writing down a renormalizable coupling is a scalar multiplet 16 H . However, 16 H = 0 would break the Z 2 parity thus allowing DM decays. A similar conclusion can be reached also in the case the new fermion is not a SO(10) singlet but is assigned to some suitable SO (10) representation. We must then conclude that in our framework the inverse seesaw does not provide a viable alternative to explain the neutrino mass suppression.
A straightforward, although not so elegant, way out, is to appeal to cancellations in the neutrino Dirac mass matrix. This relies on the fact that, when projected onto SM multiplets, the fermion mass matrices originating from eq. (4) acquire non trivial Clebsch-Gordan coefficients that weight the various vev contributions and that are different for different fermion species. For the up-type quark and Dirac neutrino masses we have [27,46,73]: where h, g, f are the symmetric Yukawa matrices introduced in eq. (4) and κ u is the up-type induced doublet vev from 126 H . If we take e.g. M N R ∼ 10 TeV, no particular cancellation is needed for the mass entries related to the up quark mass, while for those related to the heavy third generation, a tuning in the cancellation of up to one part in 10 5 is required. While this is certainly unpleasant, we should not forget that non-supersymmetric SO(10) suffers a naturalness problem from the start, which already requires a tuning in the theory at a much higher level than 10 −5 .

Relic density
We now turn to the calculation of the relic density of the DM candidate χ l . Right above the interme-  'cohannihilation' channels can be simply taken into account by including appropriate factors of gauge multiplicities both in the annihilation process and in the counting of particle degrees of freedom. However, in our numerical study we have kept track of the small mass splittings induced by symmetry breaking thus differentiating between annihilation and cohannihilation, but the effects of this more refined treatment remain irrelevant. Of course, eventually the charged states and the heavier neutral state χ h will all decay to χ l , thus adding their contribution to the DM relic abundance.
Before tackling the calculation of the DM relic density in our model, let us first recall the generic features of the analogous computation in the case of a DM SM doublet. The relic density of the neutral component of an SU (2) L doublet of mass m DM > M Z L can be cast as [49]: where σv is the thermally averaged annihilation cross-section times the relative velocity, and includes SM In order to explore quantitatively the mass parameter space for DM, including also the effects of resonant annihilation and co-annihilation via Z R , W R , we have carried out a numerical analysis by implementing our model in Micromegas [79]. 6 The results are summarized in figure 3 where the dependence of the relic den- The velocity-averaged cross-section for χ lχl → W L W L can be estimated as σ W |v| ∼ πα 2 g /(32m 2 l ) ∼ 3×10 −28 (2 TeV/m l ) 2 cm 3 /s with α g the SU (2) fine structure constant. A more accurate estimate, including non-relativistic Sommerfeld corrections, gives for the same mass range an enhancement up to one order of magnitude [82]. In spite of this the signal remains well below the present limits σ W |v| < ∼ (10 −25 − 10 −24 ) cm 3 /s for the mass range 1 TeV < m l < 4 TeV [83].
As regards collider limits, at the LHC the most sensitive searches in our scenario, in which χ a = χ l,h , χ ± are quasi-degenerate, are monojet signatures from processes like pp → χ a χ b j, where the pair of χ's are produced via the s-channel exchange of a SM gauge boson. However, this signal is accompanied by large backgrounds from Z, W +jets, which render the experimental search particularly difficult. A dedicated analysis of signatures of quasi-degenerate Higgsino like DM, which closely resembles our scenario, indicates a surprisingly low reach m l ∼ 250 GeV even for LHC-13 [84]. We have no reason to expect that this limit could be largely exceeded in our case, so that we can conclude that even the non-resonant (lowest mass) DM solution with m l ∼ 0.77 TeV remains unconstrained by collider searches.
Before concluding this section, one additional remark is in order. Our Dirac DM candidates carry hypercharge Y = T 3R , which can be used to distinguish particles (e.g. χ ++ , χ −+ ) from antiparticles (respectively χ −− , χ +− ). During their thermal history, the χ's will unavoidably enter in chemical equilibrium with the thermal bath, inheriting an asymmetry similar to that of all SM hypercharged states. For example, in the temperature range T R > T > T L (T R,L denote the temperatures at which SU (2) R,L get broken) that is the relevant range in which DM annihilates efficiently when the DM mass is above a few TeV, processes like 6 The model file was generated with Feynrules [80] by modifying the model of Ref. [81].
R which occur as long as T > ∼ M W R , or 2 ↔ 2 scatterings mediated by D = 6 effective operators like which occur when T < M W R , will enforce chemical equilibrium between the χ system and the SM thermal bath, and thus an asymmetry will develop in the DM sector as well. The issue whether such an asymmetry could play any relevant role in determining the final DM relic density by quenching the annihilations when the DM density becomes of the order of the density asymmetry was recently studied in [85] for the general case of stable relics belonging to scalar and fermion hypercharged multiplets of dimension D ≥ 2. It was found that for fermion doublets, as for most of the other cases, the effects of the DM asymmetry on the surviving relic density are generally negligible. However, the results of the analysis in [85] rely on certain assumptions, among which: (i) there are no new hypercharged particles besides the multiplet in question; (ii) the same operator responsible for the transfer of the asymmetry is also responsible for the mass splitting between the neutral hypercharged states.
In the present case however, these two conditions are not satisfied: (i) besides the new fermions in the bi-doublets, also the charged gauge bosons W R carry hypercharge Y = T 3R ; (ii) the loop operator that induces the mass splitting (figure 2) is generated only after SU (2) L breaking, while the asymmetry transfer is mediated by tree level interactions with real or virtual W R bosons, and is most efficient well above T L . Moreover, resonantly enhanced annihilation and co-annihilation, that were not present in the scenario analyzed in [85], here play a very important role, and many of the solutions for the correct DM relic density are found in DM mass regions close to the gauge boson resonances. Nevertheless, in spite of all these differences, given that co-annihilation via W R exchange, that is one of the dominant annihilation processes, does not suffer from any asymmetry-related quenching, it is reasonable to expect that in most of the parameter space the DM asymmetry will be largely uninfluential in determining the final value of Ω DM . 7

Conclusions
Breaking the SO (  Before concluding, we should also point out some issues that within our minimal scenario are left open, and that might deserve further studies. In section 6 we have pointed out that, as in all SO(10) derived low scale LR models, there is not enough suppression for the light neutrino masses from the seesaw mechanism, and to accommodate the neutrino mass scale we had to invoke some amount of tuning in the Yukawa sector. A related issue is the fact that the scale of the RH neutrino masses is too low to allow for an explanation of the cosmological baryon asymmetry via the standard leptogenesis scenario [86] (see [87] for a review). This is because the RH neutrinos are too light to provide sufficiently large CP violating asymmetries [88]. Finally, we have verified that with the minimal particle content that we have assumed, gauge coupling unification does not occur; however, we expect that it can be recovered by adding new particles in suitable incomplete SO(10) representations (see for example [89] for different ways to recover gauge coupling unification in low scale LR models).
ξ L,R are multiplets of Weyl fermions coupled to SU (2) L × SU (2) R gauge fields through the gauge-invariant kinetic Lagrangian: where σ µ = (I, − σ) and σ µ = (I, σ) with σ the Pauli matrices acting in Lorentz space, and where τ L are Pauli matrices acting in SU (2) L group space, i.e. on the first superscript labels of the multiplet components in eq. (A.1), while τ R of SU (2) R act on the second labels, and the reversed vector sign reminds that the action is from the right: ξ → ξ = τ L ξ τ R . Let us now definẽ From the free Weyl equation for R-chirality spinors iσ µ ∂ µ ξ R = 0 and using the relation σ 2 σ * σ 2 = − σ it is easily seen thatζ L satisfies iσ µ ∂ µζ L = 0 and thus it contains L-chirality spinors. In terms ofζ L the second term in eq. (A.2) can be rewritten as: where we have integrated by parts the derivative term (neglecting a 4-divergence) and an overall change of sign is due to anticommutation of the fermion fields. Eq. (A.5) shows explicitly that ζ L transforms in the SU (2) L ⊗ SU (2) R conjugate representation (2, 2) with generators τ = − τ * = τ 2 τ τ 2 , where the last relation expresses the pseudoreality of SU (2) representations. It is then convenient to define new ζ L multiplets transforming similarly to ξ L in (2, 2): Focusing now on the neutral fermions, let us define: The neutral current interactions for Ψ i (i = 1, 2) read: with T 3 = diag + 1 2 , − 1 2 . In order to write J nc µ in terms of the mass eigenstates, let us study how the mass terms are rewritten from the original basis of LH and RH Weyl spinors ξ L,R to our basis of LH spinors ξ L and ζ L . The fermion bilinears multiplying the tree level mass term for charged and neutral states eq. (9) are rewritten as: By redefining now χ 1 = χ L and χ R = −σ 2 χ * 2 , the mass term can be written as: where we have introduced the four spinor χ l = [(χ l ) L , (χ l ) R ] T and a similar one for χ h , and we have adopted the usual convention χ l = χ † l γ 0 with γ 0 = ( 0 I I 0 ) in the chiral basis. Thus, upon diagonalization of the mass matrix the new fermions organize into two Dirac mass eigenstates splitted in mass by 2δ.
Coming back to the neutral current gauge interactions, after rotating the interaction eigenstates Ψ i in eq. (A.8) onto mass eigenstates, the neutral current reads: χ h γ µ χ l + H.c. , (A. 16) with γ 0 γ µ = σµ 0 0 σµ . We see that the neutral gauge bosons couple the light mass eigenstates to the heavy ones, a result that follows from the fact that V T 3 V † = − 1 2 τ 2 is anti-diagonal (τ 2 denotes the second Pauli matrix, but with no relation here with gauge group factors or spinors). Obviously, since V ∂ µ V † = ∂ µ the purely kinetic term for the mass eigenstates remains diagonal.