Left-right model with TeV fermionic dark matter and unification

The ingredients for a model with a TeV right-handed scale, gauge coupling unification, and suitable dark matter candidates lie at the heart of left-right symmetry with broken D-parity. After detailing the contents of such a model, with SU(2)R self-conjugate fermions at the right-handed scale aiding in unification of couplings, we explore its dark matter implications and collider signatures.

with (B − L) being normalised by the relation: The scalar sector is given by: Φ ≡ (1 C , 2 L , 2 R , 0 (B−L) ); η ≡ (1 C , 1 L , 1 R , 0 (B−L) ); Under P the multiplets transform as: The scalar sector is modified to accommodate spontaneous breaking of P at a scale M P , where the P odd gauge singlet, η, acquires a vacuum expectation value (vev), v η . Thus, symmetry breaking takes place in three steps. The first being the breaking of P, followed by the breaking of SU (2) R ⊗U (1) (B−L) to SM by the vev, v R , of ∆ R , and finally electroweak symmetry breaking is achieved through the vevs k 1 and k 2 of the bi-doublet Φ, with k 2 1 + k 2 2 = 246 GeV. We show that the other mass scales of the model are M R ∼ O(TeV) and M P at the GUT scale.
The gauge bosons related to SU (2) R ⊗U (1) (B−L) breaking, W ± R and Z , acquire mass at M R , and for (M W /M W R ) 2 1 are given by: with where θ W is the weak mixing angle. For (M W /M W R ) 2 1, W L -W R mixing is negligible. The physical states of the bi-doublet other than the SM Higgs are constrained to be ≥ O(10 TeV) from lepton flavour violation limits [27], and the scalars from ∆ L are all heavy at M P [25,26]. There are no stringent constraints on the masses of the ∆ R scalars and they can be lighter than M W R and even O(100 GeV). Here, for simplicity we take them to be heavier than M W R .
U (1) (B−L) being broken by a scalar with (B−L) = 2, leaves behind a remnant Z 2 symmetry, defined by: Z ≡ (−1) 3(B−L) [9]. LRS fermions (scalars) have odd (even) 3(B − L) and hence are odd (even) under Z. As a result, fermions with even 3(B − L) are forbidden to decay only to SM fermions and/or bosons, and hence the lightest one of them is stable. If this state is neutral, then subject to relic density and direct/indirect detection constraints, it can be taken to be a dark matter candidate.
), typify this scenario with m ∈ N. Each multiplet consists of a Majorana fermion and m pairs of Dirac fermions and their antiparticles with electric charges 1 to m. Thus, these multiplets must be assigned B = L = 0.
The left-right symmetric bare mass and Yukawa terms of these multiplets for a general case of n g such 'generations' is given by: where summation over i= 1, · · · , n g is implicit. The negative signs pertaining to interactions of X i R with η are according to eq. [4]. Because the multiplets can always be rotated into a diagonal basis, we can do away with the cross terms without any loss of generality. From eq. [6], we see that the breaking of P enforces a separation of the masses of the multiplets transforming under SU (2) L and SU (2) R with the corresponding masses given by: With M i ∼ h i v η , X i L multiplets remain heavy at M P and the X i R become light with the exact mass scale dependent on the couplings. We want to underscore that in general for (1 C , (2m + 1) L , 1 R , 0 (B−L) ) ⊕ (1 C , 1 L , (2m + 1) R , 0 (B−L) ) fermion multiplets, the mass scale of either one will be at the larger of M i and h i v η while the other can be tuned to be at lower values. During the evolution of the Universe the superheavy SU (2) L multiplets are Boltzmann suppressed and annihilate and co-annihilate rapidly to lighter states through their couplings to W L and Z.
The framework being discussed can lead to a variety of DM models, which we label as (m, n g ), with the DM particle(s) completely separated from the SM and interacting only with the RH sector. For the rest of the letter we focus on the (1,2) case as this model simultaneously provides a suitable DM sector, gauge coupling unification, and O(TeV) M R .
Gauge Coupling Unification: With the self-conjugate SU (2) R generations of an (m, n g ) ≡ (1, 2) model, i.e., the model with a pair of (1 C , 3 L , 1 R , 0 (B−L) ) + (1 C , 1 L , 3 R , 0 (B−L) ), in the TeV range, the gauge couplings unify with M P = M U . The LR gauge group is a subgroup of SO(10) and with GCU we can embed the model in an SO(10) unified theory. The LRS multiplets of the model belong in the following SO(10) representations: where the subscripts F and H denote whether the multiplets contain fermions or scalars, respectively. There is an element of the SO(10) algebra, 'D' [28], which in the case that all the couplings of the lagrangian are real, plays the role of the parity symmetry P. η ⊂ 210 H is odd under 'D'.
The fermion triplets reside in 45-plets. The SO(10) symmetric mass term for which is:  In Fig. [1], we show the running of the inverses of the fine structure constants (α = g 2 /(4π)), as obtained from 2-loop perturbation theory. As inputs at the Z-pole, M Z = 91.1876 (21), we take α s = 0.1181(11), sin 2 θ W = 0.23129(5), and α EM = 1/128 [31]. We find that when α s (M Z ) and sin 2 θ W are varied over their 1σ allowed ranges the unifcation scale varies between (0.81 − 1.05) × 10 16 GeV and the unification coupling comes out to be, g U = 0.53. The SU (2) R breaking scale lies between 3.78 -9.40 TeV, with the SU (2) R coupling, g R = 0.52. Fig. [1] has been drawn using the central value. The U (1) couplings of the theory are normalised according to GUT (canonical) normalisation, resulting in the matching condition at M R : 1 In between M R and M U the particles flowing in the loops and hence contributing to the β-coefficients are Φ, ∆ R , l L , l R , q L , q R as in traditional D-parity broken models, and the pair of dark sector SU (2) R triplets X 1,2 R . The system of running equations are given by [32,33]: The 1-loop β-coefficients a i and the 2-loop β-coefficients, b ij , for the couplings of the SM are readily available [33], the same for the LRS stage are given in Eq. [12]. Since the only additions on top of the usual LRS particle content are the self-conjugate SU (2) R triplets which transform trivially under the other symmetries, the only change in the β-coefficients are for the SU (2) R coupling for the 1-loop case and the diagonal coefficient corresponding to SU (2) R for the two loop case.
In principle, a complete treatment of 2-loop RGE running should take into account threshold effects [34,35] at all the symmetry breaking scales. However, in this work we do not include threshold effects, as in demanding exact unification of the couplings, we are establishing GCU for a more restricted case. Threshold corrections introduce more parameters to the model and hence such situations are bound to follow suit.
We next estimate the lifetime of the proton in our model. In non-supersymmetric GUTs, scalar induced d = 6 and the d > 6 operators contributing to proton decay are generally highly suppressed in comparison to the gauge induced d = 6 operators [36][37][38], and here we concentrate only on the latter. The decay rate of the proton in the p → e + π 0 channel is expressed as [36,39]: where m p =938.3 MeV [31] is the mass of the proton, f π =130.41(23) MeV [40] is the pion decay constant, α H = −0.0118(0.0021) GeV 3 denotes the relevant hadronic matrix element, D = 0.8(2) and F = 0.47(1) are chiral lagrangian parameters calculated from lattice gauge theory [41][42][43]. g U is the unified coupling constant, M U the unification scale. R L =1.46 is the two-loop long range running effect on the effective proton decay operator, corresponding to running from M Z to m p , while A SL(R) is the short range left-(right-) handed short range renormalisation factor of the proton decay operator corresponding to running from M U to M Z [44]. A SL(R) is a function of the anomalous dimensions and β-coefficients of the running couplings, and also the values of the couplings at the symmetry breaking scales and are taken to be A SL A SR = 2.0 [45][46][47][48][49]. We set the masses of the leptoquark gauge bosons to be degenerate and at M U . Further, the flavor matrices associated with baryon and lepton flavor changing currents have been set to unity [50,51].
With M U = 10 15.97 GeV we get from eq. [13] a proton decay lifetime in this channel, τ p→e + π 0 ∼ 1.5 × 10 35 years, which is larger than the present bound of τ p→e + π 0 = 1.6 × 10 34 years [52], but testable at the Hyper-Kamiokande experiment [53], which is expected to probe lifetimes ∼ 2 × 10 35 yrs. As indicated by eq. [13], this value is extremely sensitive to the unification scale M U . Still, we have checked that for the above chosen values of the parameters, τ p→e + π 0 remains below the Hyper-Kamiokande projection with M U varying between its allowed range, i.e., (0.81 -1.05) ×10 16 GeV. However, extreme choices of the different parameters may make the model not falsifiable even by this experiment.
The mass scales, predicted by unification, are particularly ingratiating for neutrino seesaw masses. In minimal LR models, the left-handed neutrino has both type-I and type-II seesaw [54][55][56][57] contributions. L ↔ R symmetry breaking induces a nonzero SU (2) L triplet vev [58]: Here M η∆ is the dimensionful coefficient of the η∆∆ type term in the potential. The left-and right-handed neutrino masses are given by [59][60][61]: The interaction lagrangian for the constituents of the triplets, X 1,2 R , for the LR stage is given by: where e is the electromagnetic coupling. Presence of charged heavy fermions during big bang nucleosynthesis (BBN) would imply the existence of atom-like bound states, in the present epoch, containing such particles [65,66]. The non-observance of such entities in deep sea water searches [67][68][69][70][71] rules out their existence. This implies positive ∆ 1,2 M , large enough to produce a lifetime for the charged states smaller than the time at BBN ∼ 1 s. In Fig. [2] we plot the decay time of χ ± 1,2 as a function of its mass for different M W R near M R . The intra-multiplet mass splitting, ∆ 1,2 M , calculated using expressions in [13,14], is indicated by the line-styles of the curves, i.e., short-dashed, long-dashed, or solid. Notice that for each curve ∆ 1,2 M changes with M 1,2 . We find that the lifetime of the charged states for all masses near M W R is O(ns), and the mass splitting is O(GeV). Hence the heavy charged states of our model decay well before BBN. As the mass difference is tiny with respect to the masses themselves, χ ± 2 → χ 0 2 χ ± 1 χ 0 1 decay is forbidden from kinematics. Of course the same argument also applies the other way round. Hence, although we have a single stabilising Z 2 , we end up with two component (χ 0 1,2 ) dark matter.
The behaviour of dark matter relic density for this model is illustrated in Fig. [3]. The allowed regions in the M 1 − M 2 plane are those points which fall on the ellipse-like or semi-circle-like plots. We show only the region for which M 1 < M 2 . The allowed values with M 1 > M 2 can be readily obtained by a reflection. In the inset of Fig. [3] we exhibit the relic density as a function of the dark matter mass for M W R = 4 TeV. The observed value of the relic density, Ωh 2 = 0.1198 ± 0.0015 [72], is indicated by the dashed horizontal straight line. As noted, in the model under discussion, there are two dark matter candidates, χ 0 1 and χ 0 2 . In the inset, for simplicity, they have been taken to be degenerate. The dips in the curve reflect resonant χ ± i χ 0 i → W ± R or χ + i χ − i → Z production. Without these dips, the relic density in this model would have been about an order of magnitude larger than the observed limit. The points where the curves agree with the observation are near the two resonant dips. In Fig. [3] the closed ellipse-type curves with an asterisk in the middle correspond to regions where the dark matter candidate χ 0 2 is near the Z resonance (i.e., M 2 M Z /2) while χ 0 1 is close to the W R resonance point (i.e., M 1 M W R /2). The semicircle-like curves with a dot (hexagon) within correspond to the situation where the dark matter particles χ 0 1 and χ 0 2 are near degenerate and also close in mass to M W R /2 (M Z /2). We have kept the lower bound of M 1,2 > 547 GeV, as set by recent searches for heavy singly charged particles [73,74]. For these relic density computations we have utilized the MicrOMEGAS 4.3 [75] package. The model file was written using FeynRules 2.0 [76], modifying the version in [77] to our needs.  At freeze-out temperature, the charged states, χ ± 1,2 , did not have enough time to decay to the neutral ones, and hence annihilation and co-annihilation of all the triplet states contribute to the net annihilation cross section σv . Near the W R resonance, σv is saturated by co-annihilation of χ ± 1,2 with χ 0 1,2 and around the Z resonance by both co-annihilation and annihilation of χ ± 1,2 . As the neutral χ 0 1,2 have no interaction with the Z or Z, it can only annihilate to a W R pair through the t-channel exchange of χ ± 1,2 . This channel, however, opens up only when M 1,2 M W R , and even then it accounts for a minute fraction of the total σv . g R is essentially fixed from the running of gauge couplings and is no more a free parameter while calculating cross sections. Furthermore, the relic density constraint fixes a narrow range for M 1,2 given M W R . The scale of M W R itself is fixed by gauge coupling unification. This makes the model remarkably predictive and free of parameters which can be altered at will. Thus, falsifying the model is quite straightforward.
Present and proposed DM direct detection experiments such as LUX, LZ, XENON1T [78][79][80], are all based on detecting elastic scattering of WIMP DM candidates with nucleons. The dark matter candidates of this model, χ 0 1,2 , do not have any neutral current interactions, neither do they couple to the Higgs boson. Their only possible interaction with nucleons (N ) are through charged current processes, At the direct detection experiments, the N is initially at rest and the DM kinetic energy alone is not large enough to surmount the O(GeV) mass difference between the χ ± 1,2 and χ 0 1,2 . Therefore, an on-shell χ ± 1,2 in the final state is disallowed from kinematic considerations. An off-shell χ ± 1,2 decaying to χ 0 1,2 and (l ν l ) or pions through W * R is in principle possible but highly suppressed due to lack of available phase space and O(TeV) masses in the propagators. Neutral current NLO cross sections for χ 0 i N → χ 0 i N , involving W R and χ ± 1,2 in the loop are naturally negligible. A detailed discussion in case of SU (2) L triplets and a possible way of circumventing the difficulty in detection can be found in [64]. In the absence of any annihilation channels at tree level, the DM parameter region is not constricted by indirect detection [81,82] constraints.
Collider Studies: As noted previously, the dark matter relic density constraint restricts the masses of χ ± 1,2 and χ 0 1,2 to near M W R /2 or M Z /2. The χ ± 1,2 particles, if produced, for example, through W R or Z decay, will be observed as tracks in the CMS and ATLAS pixel detectors and silicon trackers. These particles will typically be at sub-relativistic velocities and can be distinguished from SM charged particles from the higher rate of ionization energy loss (dE/dx). For most of the allowed mass region, the final state particles have 0.3 < βγ(= p/M ) < 1.5 and hence the average energy loss with distance travelled can be modelled by the Bethe-Bloch distribution . Given a lifetime of O(ns) for χ ± 1,2 , as can be seen from Fig. [2], we find their decay lengths to be of the order ∼ 0.1 -1 m. The charged particles will hence decay almost exclusively in the trackers of CMS and ATLAS. The only decay mode of χ ± 1,2 is to χ 0 1,2 , and the mass difference being ∼ O(GeV), the associated jets will be too soft to be reconstructed for a displaced vertex analysis. Hence, the signal of the charged particles will be the observation of disappearing tracks 2 . An energetic initial state radiation jet can be effectively used as a trigger for the event. The neutral states will obviously be missed completely. The charged particle decay length and βγ are also favourable for detection at the MoEDAL detector [84] at LHC. If observed, the masses of the particles can be calculated from information about average energy loss and reconstructed transverse momentum as measured from the curvature of the charged tracks in the magnetic fields [85,86].
The vindication of TeV scale SU (2) R breaking will be the discovery of the W R and the heavy neutrino, in the lljj channel, the event topology being given by: pp → W R → N l l → lljj [87]. If the neutrino is a Majorana particle as predicted by the LRS model, one should observe equal same-sign and opposite-sign final states. We ask to what extent this signal is affected by the presence of the self-conjugate triplets χ 1,2 ? In Fig. [4] we show the cross section times branching fractions of W R production and its subsequent decays in different channels 3 . For this purpose, leading order cross sections were calculated in CalcHep 3.4 [89] using the CTEQ6L1 parton distribution functions [90], and multiplied by the corresponding K-factors, as obtained from [91]. For the sake of comparison, we have chosen the DM multiplets to be mass degenerate and having the smallest mass as allowed by relic density constraints and taken M N l = M χ ± No. of raw events <10 @ 500 fb −1 <10 @ 3000 fb −1 Figure 4: Cross section times branching fraction for production of W R and Z and their subsequent decays into different channels, as labelled, at √ s = 14 TeV. The thick lines represent total cross sections. The deeply (lightly) shaded regions delimit the cross sections for which the total number of raw events drops below 10 at 3000 (500) fb −1 .
The dominant decay mode of W R is obviously to two jets. As can be seen from Fig. [4] the decay W ± R → χ ± i χ 0 i is a few times larger than the subdominant but often searched for leptonic decays (l ≡ e, µ). Nonetheless, the leptonic branching remains substantial and we find that a W R with mass ∼ 6.5 TeV can still be discovered by the ATLAS and CMS collaborations in this channel with √ s = 14 TeV and an integrated luminosity of 3000 fb −1 . Indirect detection of a heavier W R with masses up to ∼ 8 TeV is possible in the studies of K and B meson decays at LHCb [27] where this model has no distinction from the canonical LRS model. Another promising mode for the detection of W R is the di-boson channel (W R → W Z or W R → W H). The branching ratios are almost the same for these two channels. However, due to the suppressed W L -W R mixing, they are small, see Fig. [4], and as M W R approaches 6 TeV this channel becomes unfeasible. Note that as the masses of the triplet fermions are related to the mass of the W R boson from relic density constraints, and since the χ ± 1,2 do not interact with the SM particles, the detection of W R or Z without detection of these will essentially falsify the model.
With M Z R ∼ 1.94 × M W R , the discovery potential of Z is bleak at the LHC. For W R masses above 3.5 TeV, the Z becomes too heavy to be detected at LHC-II as can be seen from Fig. [4]. For HL-LHC luminosities of 3000 fb −1 , the sensitivity increases slightly.

Conclusion:
In this work we have presented a model which rests on left-right symmetry, is amenable to gauge coupling 2 For a recent discussion of the sensitivity of the LHC detectors to such disappearing charged tracks, see for example, [83] 3 The possibility of detecting a virtual heavy W R signal through much lighter RH 'neutrino jets' has recently been examined in [88] unification, and provides suitable dark matter candidates. Aided by two distinct discrete symmetries inherent to the left-right symmetric theory the stability of dark matter and the scales of symmetry breaking are ensured. The model is falsifiable at both the GUT scale and the LRS breaking scale at the Hyper-Kamiokande experiment and the LHC respectively. The model predicts a 'desert' between the LRS and GUT scales. In the absence of multiple symmetry breaking thresholds, the variable parameters of the model viz. the SU (2) R coupling, g R , and scale of the W R mass are essentially fixed from unification. The Dark Matter candidates satisfy the relic density constraint aided by resonant enhancements of the cross section and hence allowed masses are intertwined with M W R /2 and M Z /2. Their direct detection in ongoing and planned experiments is unlikely. Nonetheless, with a very small leeway for the parameters to vary, the model is remarkably predictive, making falsification or vindication more or less straightforward at colliders.