Implications of the CMS search for WR on grand unification

The CMS experiment at the Large Hadron Collider has reported a 2.8σ excess in the (2e)(2jets) channel around 2.1 TeV. Interpretation of this data is reconsidered in terms of the production of a right-handed weak gauge boson, WR, of the left-right symmetric model and in an SO(10) grand unified theory abiding by the Extended Survival Hypothesis. The left-right symmetric model can be consistent with this excess if (a) the heavy right-handed neutrino has a mass near WR, or (b) if gL ≠ gR, or (c) the right-handed CKM matrix is nontrivial. Combinations of the above possibilities are also viable. A WR with a mass in the TeV region if embedded in SO(10) is not compatible with gL = gR. Rather, it implies 0.64 ≤ gR/gL ≤ 0.78. Further, a unique symmetry-breaking route — the order being left-right discrete symmetry breaking first, followed by SU(4)C and finally SU(2)R — to the standard model is picked out. The L ↔ R discrete symmetry has to be broken at around 1016 GeV. The grand unification scale is pushed to 1018 GeV making the detection of proton decay in ongoing searches rather unlikely. The SU(4)C breaking scale can be at its allowed lower limit of 106 GeV so that n−n¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ n-\overline{n} $$\end{document} oscillation or flavour changing processes such as KL → μe and Bd,s → μe may be detectable. The Higgs scalar multiplets responsible for SO(10) symmetry breaking at various stages are uniquely identified so long as one adheres to a minimalist principle. We also remark, en passant, about a partially unified Pati-Salam model.


Introduction
The discovery of the Higgs boson at the Large Hadron Collider (LHC) at CERN is a major milestone of the successes of the standard model (SM) of particle physics. Indeed, with all the quarks and leptons and force carriers of the SM now detected and the source of spontaneous symmetry breaking identified there is a well-deserved sense of satisfaction. Nonetheless, there is a widely shared expectation that there is new physics which may be around the corner and within striking range of the LHC. The shortcomings of the standard model are well-known. There is no candidate for dark matter in the SM. The neutrino is massless in the model but experiments indicate otherwise. At the same time the utter JHEP02(2016)023 smallness of this mass is itself a mystery. Neither is there any explanation of the matterantimatter asymmetry seen in the Universe. Besides, the lightness of the Higgs boson remains an enigma if there is no physics between the electroweak and Planck scales.
Of the several alternatives of beyond the standard model extensions, the one on which we focus in this work is the left-right symmetric (LRS) model [1][2][3][4][5] and its embedding within a grand unified theory (GUT). Here parity is a symmetry of the theory which is spontaneously broken resulting in the observed left-handed weak interactions. The leftright symmetric model is based on the gauge group SU(2) L × SU(2) R × U(1) B−L and has a natural embedding in the SU(4) C × SU(2) L × SU(2) R Pati-Salam model [2] which unifies quarks and leptons in an SU(4) C symmetry. The Pati-Salam symmetry is a subgroup of SO (10) [6,7]. These extensions of the standard model provide avenues for the amelioration of several of its shortcomings alluded to earlier.
The tell-tale signature of the LRS model would be observation of the W R . At the LHC the CMS collaboration has searched for the on-shell production of a right-handed charged gauge boson [8] using the process: 1 pp → W R → 2j + ll . (1.1) In the above l stands for a charged lepton, and j represents a hadronic jet.
The CMS collaboration has examined the implication of its findings in the context of a left-right symmetric model where the left and right gauge couplings are equal (g L = g R ) and also the W R coupling to a charged lepton, l, and its associated right-handed neutrino, N l , is diagonal with no leptonic mixing, 2 (i.e., V N l l = 1). In the l = e channel the data shows a 2.8σ excess near 2.1 TeV. Also, regions in the M N l − M W R plane disfavoured by the data, within an LRS theory with g L = g R , have been exhibited. After production, the W R decays through W R → lN l in the first stage. An associated signal of this process will be a peak at M N l in one of the ljj invariant mass combinations. CMS has not observed the latter. One possibility may be that the produced N l has a substantial coupling to the τ -lepton [13,14] -V N l τ is not small. Here we keep M N l as a parameter of the model. Within the LRS model there is room to admit the possibility of g L = g R . Interpretation of the CMS result in the presence of such a coupling asymmetry has also been taken up [15][16][17] keeping M N l = M W R /2 and the implications for grand unification and baryogenesis explored. In [15,16] the coupling parameter V N l l is also allowed to deviate from unity. Other interpretations of the excess have also appeared, for example, in [13][14][15][16][17][18][19][20].
In a left-right symmetric model emerging from a grand unified theory, such as SO(10), one has a discrete symmetry SU(2) L ↔ SU(2) R -referred to as D-parity [21,22] -which sets g L = g R . Both D-parity and SU(2) R are broken during the descent of the GUT to the standard model, the first making the coupling constants unequal and the second resulting in a massive W R . The possibility that the energy scale of breaking of D-parity is different from that of SU(2) R breaking is admissible and well-examined [23]. The difference between these scales and the particle content of the theory controls the extent to which g L = g R .

JHEP02(2016)023
In this work we consider the different options of SO(10) symmetry breaking. It is shown that a light W R goes hand-in-hand with the breaking of D-parity at a scale around 10 16 GeV, immediately excluding the possibility of g L = g R . D-parity breaking at such an energy is usually considered a desirable feature for getting rid of unwanted topological defects such as domain walls [24] and accounting for the baryon asymmetry of the Universe [25]. The other symmetries that are broken in the passage to the standard model are the SU(4) C and SU(2) R of the Pati-Salam (PS) model. The stepwise breaking of these symmetries and the order of their energy scales have many variants. There are also a variety of options for the scalar multiplets which are used to trigger the spontaneous symmetry breaking at the different stages. We take a minimalist position of (a) not including any scalar fields beyond the ones that are essential for symmetry breaking, and also (b) impose the Extended Survival Hypothesis (ESH) corresponding to minimal fine-tuning to keep no light extra scalars. With these twin requirements we observe that only a single symmetry-breaking route -the one in which the order of symmetry breaking is first D-parity, then SU(4) C , and finally SU(2) R -can accommodate a light M W R . We find that one must have 0.64 ≤ g R /g L ≤ 0.78.
The paper is divided as follows. In the following section we give details of the CMS result [8] which are relevant for our discussion within the context of the left-right symmetric model. In the next section we elaborate on the GUT symmetry-breaking chains, the extended survival hypothesis for light scalars, and coupling constant evolution relations. Next we briefly note the implications of coupling constant unification within the Pati-Salam and SO(10) models. The results which emerge for the different routes of descent of SO (10) to the SM are presented in the next two sections. We end with our conclusions.

CMS W R search result and the left-right symmetric model
The results of the CMS collaboration for the search for a W R -boson that we use [8] are based on the LHC run at √ s = 8 TeV with an integrated luminosity of 19.7 fb −1 . The focus is on the production of a W R which then decays to a charged lepton (l) and a right-handed heavy neutrino (N l ), both of which are on-shell. The N l undergoes a threebody decay to a charged lepton (l) and a pair of quarks which manifest as hadronic jets (2j), the process being mediated by a W R . CMS examines the (2l)(2j) data within the framework of an LRS model with g L = g R and presents exclusion regions in the M W R −M N l plane. 3 Interpreting the four-object final state mass as that of a W R CMS presents, in the supplementary material of [8], the 95% CL exclusion limits for the observed and expected that this excess is not consistent with the LRS model with g L = g R , r = 0.5 and no leptonic 3 An alternate explanation of the excess in the data could be in terms of a charged Higgs boson of the LRS model. Figure 1. Feynman diagram of the process under discussion with the right-handed CKM-like mixing matrix taken in a general, non-diagonal, form. The CMS excess corresponds to l α = l β = e.

JHEP02(2016)023
mixing. As we stress later, relaxing these conditions -e.g., r = 0.5 -can make the results agree with the left-right symmetric model. No such excess is seen in the (2µ)(2j) mode.
2.1 A W R signal? Figure 1 shows the Feynman diagram for W R production and its decay in the channels under consideration [26]. Note that the production of the W R will be suppressed compared to that of a left-handed W boson of the same mass by a factor η 2 , where η = (g R /g L ).
The contribution from this diagram is determined by S 2 where S ≡ η|V Nee | 2 . Neglecting the masses of the final state quarks and the charged lepton, the branching ratio of the three-body decay of N e , which we have calculated, is proportional to (1 − r 2 ) 2 (2 + r 2 ), where, as noted earlier, r = M Ne /M W R . A clinching evidence of this process would then be a peak in the (2e)(2j) invariant mass at M W R -for which there is already a hintalong with another around M Ne in the invariant mass of one of the two e(2j) combinations in every event. The absence of the latter in the data could be indicative of more than one N R state being involved and further the coupling of these neutrinos to a τ -lepton with non-negligible strength [13,14,27]. Subsequent leptonic decays of the τ would mimic a dilepton signal but with two missed neutrinos washing out the expected e(2j) peak. It has to be borne in mind that the excess seen in the (2e)(2j) mode is not matched in the (2µ)(2j) data. This would have to be interpreted as an indication that the righthanded neutrino associated with the muon, N µ , is significantly heavier than N e and so its production in W R -decay suffers a large kinematic suppression. Further, the coupling of N e to µ has to be small, i.e., |V Neµ | 1. Interpretation of the excess in terms of N e , a Majorana neutrino, would lead to the expectation of roughly an equal number of like-sign and unlike-sign dilepton events. There are fourteen events in the excess region in the CMS data in the (2e)(2j) channel, of which only one has charged leptons of the same sign. In a similar analysis by the ATLAS collaboration no like-sign events are found [28]. One way around this is to assume that two degenerate right-handed neutrinos together form a pseudo-Dirac state in which case the like-sign events are suppressed [18].
In pectations equal σBR O . To put the plot in context note that CMS has stressed [8] that with η = 1 and V N e = 1 -i.e., S = 1 -the LRS model signal for r = 0.5 is inconsistent with the excess. This is borne out from figure 2 which indicates that for the S = 1 contour, the M W R corresponding to r = 0.5 lies outside the excess region. Consistency of the excess in the data with the LRS model can be accomplished in three ways. Firstly, if r = M Ne /M W R is larger than 0.5 the LRS model signal will be reduced. Indeed, with r > 0.75 the LRS model is consistent with the excess even with S = 1. Alternatively, if η or V N e is less than unity, then too the signal will be less, the suppression being determined by S 2 . In figure 2 it can be seen that for r = 0.5 the excess is consistent with the model for 0.3 S 0.6. The upper limit has been pointed out in [15][16][17]. What we essentially find is that there are large sets of values of r, η and V N e for which the LRS expectation is consistent with the excess. Figure 2 contains information in a somewhat condensed form. In the spirit of the path chosen by the CMS collaboration, we use the exclusion data and plot in the left panel of figure 3 σBR E (red dotted curve) and σBR O (blue dashed curve) as functions of M W R for the fixed value of r = 0.8. The prediction of the LRS model with η = g R /g L = 1 and V Nee = 1 is the black solid straight line. Also shown are the ±50% (green, dark) and ±100% (yellow, light) bands of the expected cross section. In the inset the same results are presented but for r = 0.5. Notice that for r = 0.8 the LRS model expectation passes right through the maximum of the excess while for r = 0.5 it entirely misses the excess region.
The right panel of figure 3 utilises a complementary way of displaying the region in the LRS model parameter space consistent with the result. Here the area in the η − V Nee plane that fits the CMS excess region is shown shaded for two values of r = 0.8 (violet, dark) and 0.5 (green, light). Note that there is an overlap region. It is worth stressing that, as JHEP02(2016)023  in the left panel, for r = 0.8 the model is consistent with the data even for η = 1 and V Nee = 1. For r = 0.5 a suppression through the factor S = η|V N e | 2 is required to bring the model in harmony with the data. If the W R with a mass O(TeV) arises from the SO(10) GUT model we discuss below then η must lie within the two vertical lines.

SO(10) grand unification
SO (10) is an attractive candidate for a unified theory [6,7] as it is the simplest Lie group which includes all the SM fermions and a right-handed neutrino of one generation in a single irreducible representation. We do not include any exotic fermions in the model and deal with three generations. There are a vast number of models characterised by different intermediate symmetries which have SO(10) as the unifying group. In that respect SO(10) is more of an umbrella term, incorporating these different models with alternate symmetry-breaking routes, scalar structures, and physics consequences. What is important for this work is that SO(10) has the Pati-Salam symmetry (G P S ) as a subgroup 4 and includes the discrete D-parity [21,22] which enforces left-right parity, g L = g R . The Left-Right Symmetric group is embedded in G P S . Thus, having reviewed the CMS result in terms of the LRS model, both with and without left-right parity, the obvious next step is to look at it through the lenses of the Pati-Salam partial unified and SO(10) grand unified theories.
In this section we summarize the features of SO(10) GUTs which are relevant for our subsequent discussions. We consider the non-supersymmetric version of this theory.

Symmetry breaking
The different ways in which SO(10) GUT can step-wise break to the SM are graphically represented in figure 4. The intermediate energy scales of various stages of symmetry JHEP02(2016)023 , are common to all routes. The other possible intermediate symmetries, in this notation, are G 422 , G 421 , G 3122D , and G 3122 (see figure 4). All models of SO(10) symmetry breaking (symmetrybreaking chains) are thus defined by the route it belongs to and the Higgs multiplets that it includes. Figure 4 shows the maximum-step chains (chains with maximum number of intermediate symmetries) of each route. Other chains are essentially subcases of these with multiple symmetries breaking at the same scale. This can be achieved if multiple Higgs sub-multiplets gain vacuum expectation value (vev ) at the same scale or if a single sub-multiplet breaks more than one symmetry.

Scalar structure and the Extended Survival Hypothesis (ESH)
The gauge bosons in the model and their masses are determined by the symmetry group and its sequential breaking to the SM. The fermions come in three generations in each of which there are the SM quarks and leptons and a right-handed neutrino. Thus it is only the scalar sector which retains a degree of flexibility.
The generation of quark and lepton masses requires a 10 of SO(10) while the see-saw mechanism for neutrino masses relies on a 126.  In a grand unified theory masses of fermions in the same multiplet are related. In particular, the 10 of SO (10) where M d is the mass matrix of d-type quarks and M l that of the charged leptons. Though these relations are valid only at the scale of unification and at lower energies corrections have to be included, even then they are not in consistency with the measured masses. One way to address this issue is to invoke a [15,2,2] submultiplet which is present in the 126 and the 120 of SO(10) to bring the masses closer to their actual values [29,30].
Two other SO (10) representations which turn out to be useful for symmetry breaking and whose submultiplet structure will be important are the 45  Even using these limited SO(10) multiplets, 6 there remains a variety of options for the scalar submultiplets that can be used for the different stages of symmetry breaking indicated in figure 4. They affect the unification and intermediate energy scales through their role in the evolution of gauge couplings. We make two restrictions: (a) Only renormalisable terms will be kept in the SO(10)-symmetric lagrangian, 7 and (b) The Extended Survival Hypothesis, which is a consequence of minimal fine-tuning, is taken to be valid.
According to ESH [31,32], at any intermediate energy scale only those scalar submultiplets (under the unbroken symmetry at that stage) which are required to spontaneously break a symmetry at that or any lower energy remain massless. All other submultiplets become massive. Because the normal expectation of scalar masses is to be at the highest energy scale the extended survival hypothesis posits the minimal number of fine-tunings in the scalar sector.
With these guiding principles we now turn to the scalar multiplets that are employed for the descent of SO(10) to the SM. The first (at M U ) and last (at M 0 ) stages of the symmetry breaking in figure 4, which are common to all alternate channels, utilise a 54plet and a 126-plet of scalar fields, respectively. D-parity is broken through the vev of 5 We will use the notation [φ4, φL, φR] to specify the behaviour of SO (10)

Renormalisation group equations
The one-loop RG evolution for the coupling α g (µ) corresponding to a gauge symmetry g can be written as: b g ji is the coefficient of the β-function between the scales µ i and µ j : where the three terms are contributions from gauge bosons, chiral fermions, and scalars respectively. N g is the quadratic Casimir corresponding to the particular symmetry group g, N g is 0 for U(1) and N for SU(N ). T (F g ) and d(F g ) are the index and the dimension of the representation of the chiral fermion multiplet F under the group g and the sum is over all fermion multiplets of one generation. n G is the number of fermion generations, 3 in our case. Similarly T (S g ) and d(S g ) are the index and the dimension of the representation S g of the scalar S under g. δ S takes the value 1 or 2 depending on whether the scalar representation is real or complex. It is worth noting that b g is positive for U(1) subgroups and negative 8 for SU(n). Therefore, U(1) couplings grow with increasing energy while SU(n) couplings decrease.
For ease of use, we will rewrite eq. (3.5) as: where w g i ≡ 1 α g (µ i ) and ∆ ji ≡ ln µ j µ i . If the symmetry g is broken to g at the scale µ i then the coupling constant matching condition is simply w g i = w g i unless two groups combine to yield a residual symmetry. As an example of the latter, for U(1) Y of the standard model, resulting from a linear combination of U(1) R and U(1) B−L at the scale M 0 , one has: Matching all the couplings at the boundaries and imposing the unification condition one arrives at three equations: where w U is the reciprocal of the coupling strength at unification. i runs from the unification scale to M 0 . C stands for SU(3) C or SU (4)

Low energy expectations from unification
In the LRS model the energy scales of symmetry breaking can be freely chosen to be consistent with the low energy data. Once embedded in GUTs one must also verify that such choices of intermediate scales are consistent with perturbative unification of the couplings at sub-Planck energies and check their implications for other symmetry-breaking scales. In this section we look at the restrictions imposed by coupling unification on η and the other left-right symmetric model parameters.
SO (10) can descend to the SM through a maximum of four intermediate stages (figure 4). Such four-step symmetry breakings are underdetermined. Accordingly, one is permitted to choose the scale M R in the TeV range, as required by the CMS data, and to check the consistency of the equations. M 0 is always below M R and thus keeping the latter at a few TeV sets the former to an even lower value.

Pati-Salam partial unification
The PS symmetry with D-parity, G 422D , is a common intermediate stage for all the SO(10) symmetry-breaking options. When D-parity is intact, this model has two-independent couplings, namely, g 4C and g 2L = g 2R = g 2 , which achieve equality at the grand unification scale M U . In the DCR route the Pati-Salam G 422 survives at the next step but D-parity no longer holds. In contrast, for the CDR chain the PS symmetry is broken before D-parity. Needless to say, so long as D-parity remains unbroken η = 1.
In Pati-Salam partial unification one has a set of three equations similar to eq. (3.9) sans the constraint of grand unification. In place of an inverse GUT coupling w U one gets two separate couplings -w 4 Thus, the two variables -the GUT coupling and the GUT scale -are replaced by the SU(4) C unification coupling and a D-parity symmetric SU(2) coupling. In the following sections we will look at the results that arise from RG evolution for both PS partial unification and SO(10) grand unification.

Left-right symmetry and unification
The scalar field contributions to gauge coupling evolution play a significant role in achieving coupling unification while keeping a low M R . This has led to a plethora of models where scalar fields have been incorporated in the theory solely for this purpose. This is not the path that we choose. Indeed, the scalar fields which we do include become indispensible in some cases. For example, a subcase which one might imagine from figure 4 will have M R = M 0 . The one-step symmetry breaking of G 3122 → G 321 can be realized through the vev of just a [10, 1, 3] ⊂ 126, dispensing off the submultiplet which breaks SU(2) R → U(1) R . However, without this latter contribution the coupling constants no longer unify. So, M R = M 0 ∼ O(TeV) cannot be accommodated without at least the scalar multiplets that we keep.
As mentioned earlier, the three key ingredients in interpreting the CMS result are the ratio between the left-and right-handed gauge couplings, η, the Majorana mass of the right-handed electron neutrino, M Ne , and the right-handed leptonic mixing V N l l . The The mixing in the right-handed lepton sector -V N l l -is the second relevant quantity in this analysis. It is determined by the generation structure of the Yukawa matrix. Since this does not affect the evolution of couplings, which is the focus, our analysis does not impose any restriction on the choice of this mixing.
The relative strength of the right-handed coupling vis-à-vis the left-handed one at the SU(2) R -breaking scale -η = g R g L -is, however, intimately related to the RG running of the gauge couplings.
The magnitude of η will vary for symmetry-breaking chains depending on the scalar content of the theory and the energy scales at which different symmetries break. Nonetheless, the minimum value that can be attained by η is almost independent of the way in which SO (10) or G P S descends to the standard model, as we now discuss. Firstly the requirement that M R is O(TeV) and M 0 even lower, keeps them close to each other and the two are not too far from M Z either. The other feature, noted earlier, is that U(1) couplings increase as the energy scale µ increases while SU(n) couplings do the opposite. One starts from eq. (3.8) which relates the U(1) couplings when the symmetry breaking G 3121 → G 321 occurs at M 0 . Obviously, and from w B−L The inequality in the first step in eq. (4.2) is due to the evolution of w 1R from M 0 to M R . Since these two energy scales are both in the TeV range this effect is not large. A similar reasoning is also valid for the first inequality in eq. (4.3) but the second could be much more substantial. Using the current values of the low energy couplings 9 and extrapolating them to µ = M 0 one gets η min ∼ 0.59 . (4.5) We stress that eq. (4.5) is an artefact of the LRS model so long as there is a merging of the U(1) B−L with SU(3) C , and so is valid for both PS (partial) and SO(10) (grand) unification. However, this is a limit in principle, accomplishing it will depend on the details of symmetry breaking and the scalar content of the theory. We have previously seen that if r = 0.5 the CMS result is compatible with the LRS model for S as low as ∼ 0.25. From the preceding discussion we see that S lower than ∼ 0.59 cannot be attained by η alone.

The three routes of SO(10) symmetry breaking
In this section we consider one by one the three routes depicted in figure 4 by which SO (10) can descend to the SM. We focus on the scalar fields that are required and the intermediate energy scales involved. We use one-loop renormalisation group equations here but have checked that two-loop effects -on which we comment later on -do not change the results drastically. Since the equations are usually underdetermined, motivated by the CMS data, we will keep 4 TeV ≤ M R ≤ 10 TeV and 1 TeV ≤ M 0 ≤ 4 TeV for the chains of descent.

The DRC route
Restricting M R to the TeV range automatically eliminates the DRC route (Blue dotted in figure 4) -SU(4) C breaking after M R -because then the leptoquark gauge bosons of SU(4) C achieve a mass of the TeV order. Light leptoquarks below 10 6 GeV are forbidden from rare decays of strange mesons, such as K L → µe [2,34,35]. The DRC route of symmetry breaking is thus not compatible with the CMS result.

The CDR route
With all intermediate stages distinct, for this route (Green dashed in figure 4) one has: The scalar submultiplets responsible for the symmetry breaking are shown in   In order to proceed with an elaboration of the consequences associated with this route it is helpful to list the one-loop beta-function coefficients for the stages M R ↔ M D and M D ↔ M C . Including the contributions from the scalars in table 2, fermions, and gauge bosons one finds from eq. (3.5) The SU(2) L and SU(2) R couplings evolve from M R to become equal at M D . This requires Similarly the SU(3) C and U(1) B−L couplings become equal at M C , i.e., The left-hand-sides of eqs.
Using the beta-function coefficients from eq. (5.2), one can reexpress eq. (5.5) as:    (see table 2), the change is in the evolution of the couplings in the M C ↔ M U sector which does not affect this conclusion. The above analysis does not resort to the constraint of grand unification at all. The results hold for PS partial unification as well. So, the CDR route of descent also has to be abandoned for M R ∼ O(TeV).

The DCR route
After having eliminated the other alternatives, the only remaining route of descent has the mass ordering M D ≥ M C ≥ M R (Red solid in figure 4). Keeping all possible intermediate stages separate from each other this corresponds to: In the above we have indicated the SO(10) multiplets which contribute to symmetry breaking at every stage. The scalar submultiplets which contribute to the RG equations as dictated by ESH are shown in table 3. There is, however, an alternative which relies on a 45 of SO(10) whose contents under the Pati-Salam group are given in eq. Denoting by h D , h C , h R the SO(10) scalar multiplets responsible for the breaking of D-Parity, SU(4) C , and SU(2) R respectively, we therefore have the following alternatives: {210,45,45}, {210,45,210}, {210,210,45} and {210,210,210}. Of these, the first employs the lowest dimensional scalar multiplets required to break symmetries at each scale while the last one uses the least number of SO(10) scalar multiplets. Using 45 or 210 for h C makes no difference in the physics since in both cases a [15,1,1] Pati-Salam submultiplet is used. The distinction is relevant only in the choice of h R .
The one-loop beta-function coefficients for the couplings in the M R ↔ M C and M C ↔ M D energy ranges obtained using eq. (3.5) and the scalars in table 3 are: The SU(3) C and U(1) B−L couplings evolve to become equal at M C . Thus Matching of the SU(2) L and SU(2) R couplings at M D implies: As before, we use the approximation w i R w i O w i Z and combine eqs. (5.9) and (5.10) to get: A special limit of the DCR route is when M D = M C , i.e., ∆ DC = ln M D M C = 0. In this limiting case there is no distinction between this route and the CDR one. Indeed, setting M D = M C in eq. (5.11) and substituting the beta-function coefficients from eq. (5.8) one exactly reproduces (5.6) which places the solution in an unacceptable energy regime.
That one should nonetheless expect acceptable solutions can be surmised from the fact that eq. (5.11) implies where in the last step we have used eq. (5.8). This indicates that M C changes faster than M D and so starting from the M D = M C limit solutions in the DCR route with the symmetry breaking scales below M Planck are feasible. Replacing the 210 in h R by a 45 reduces b 2R DC so much that unification of couplings is no longer possible. In the next section we present the allowed soultions in detail.

SO(10) unification with M R ∼ O(TeV)
In the previous section we have seen that of the three routes of symmetry breaking accessible to SO(10), DRC is trivially eliminated when the twin requirements M R ∼ O(TeV) and M R > M C are imposed. We also indicated that for the CDR route with the minimal scalar content and following the extended survival hypothesis the requirement M R ∼ O(TeV) implies M C > M Planck . The only route that can accommodate M R ∼ O(TeV) is DCR.
To simplify the discussion, in eq. (5.11) we have ignored the running of the couplings between M Z and M R . In obtaining the results presented in this section we have not used such an approximation. We have, however, not included the effect of mixing of U(1) R × U(1) B−L under RG evolution from M 0 to M R . As these two scales are close to each other, both in the few TeV range, the impact of the mixing will not be large.

Pati-Salam partial unification for the maximum-step case
The maximum-step symmetry-breaking DCR route has been given in eq. (5.7). Before turning to SO (10) we briefly remark about Pati-Salam partial unification within this route. Because there are four steps of symmetry breaking this is an underdetermined system. For this work, M R is restricted to be in the O(TeV) range. The scale M C is taken as the other input in the analysis. At the one-loop level the results can be analytically calculated using the beta-function coefficients in eq. (5.8). The steps can be identified from eqs. (5.10) and (5.11). The latter determines M D once M C is chosen. η is then fixed using eq. (5.10).
For example, for M C = 10 6 GeV one gets η = 0.63 when M R = 5 TeV. Within the Pati-Salam model the upper limit of M D is set by M Planck . We find that in such a limit one has M C = 10 17.6 GeV and η = 0.87 for M R = 5 TeV.

Coupling unification for the maximum-step case
For SO(10) grand unification one must find the energy scale at which the common SU(2) L,R coupling beyond M D equals the SU(4) C coupling, i.e., g 2 = g 4C . This limits the upper bound of M C compared to the Pati-Salam partial unification.
In the left panel of figure 5 we  then it is unlikely that ongoing proton decay experiments [36] will observe a signal. This is a consequence of our adhering to the principle of minimality of Higgs scalars. One can lower M U by including scalars redundant to symmetry breaking.
We have set the lower limit of M C at 10 6 GeV from the limits on rare meson decays such as K L → µe or B d,s → µe. The current limit on the branching ratio for the former process is Br(K L → µ ± e ∓ ) < 4.7 × 10 −12 at 90% CL [33] which translates to M C 10 6 GeV. LHCb has set the tightest bounds on the latter processes. They find (again at 90% CL) [37] Br(B 0 d → µ ± e ∓ ) < 2.8 × 10 −9 and Br(B 0 s → µ ± e ∓ ) < 1.1 × 10 −8 which yield a weaker limit on M C . It can be expected that these bounds will be strengthened when the results from the newer runs of LHC appear. In addition, n−n oscillations can be mediated through coloured scalars belonging to the [10, 1, 3] ⊂ 126 which also acquire mass at the scale of M C . The current experimental limit, τ n−n ≥ 2.7 × 10 8 s [38] at 90% CL, also translates to M C 10 6 GeV. Therefore, improvements in the measurement of the above-noted rare meson decays and n −n oscillations open the possibility of probing, at least in part, the GUT options that can accommodate a TeV-scale W R . From the inset in the left panel of figure 5 it is seen that for M C ∼ 2 × 10 14 GeV one has M D = M U . As this chain has three intermediate steps, there are no free parameters after setting M 0 and M R . The coupling at unification, w U , comes to be around 47.6, and η, as can be seen from the left panel of figure 5, is near 0.78. An interesting aspect of this chain is that it is minimal in the number of scalar multiplets used.

Two-loop comparison
The discussion till now, based on RG evolution using one-loop β-functions, was amenable to an analytical examination. Our aim was to reason our way through different SO(10) symmetry-breaking options in search of chains which can accommodate a TeV range M R . Now, after finding a specific pattern for which M R ∼ O(TeV) is tenable, we indicate the size of two-loop effects for this chain. In the right panel of figure 5 the evolution of the couplings for a typical choice of M C = 10 10 GeV, M R = 10 TeV and M 0 = 1 TeV are indicated by the dashed lines. It is seen that the essential physics is largely unaltered though there is some change in the various energy scales.
The chain in (5.7) contains many scalar fields until SU(4) C breaking, contributing heavily to the two-loop β-coefficients [39][40][41][42]. These are responsible for some departures from the one-loop results. We have presented the one-loop results for the DCR route JHEP02(2016)023 keeping M C as an input parameter. In figure 6 we compare the one-loop (solid lines) and two-loop (dotted lines) results for M U and M D as a function of M C . The deviation for both scales increases with decreasing M C . This is intuitive because with lower M C SU(4) C remains a good symmetry for a larger energy range over which the two-loop contributions are effective. In the inset a similar comparison is made for η. It is noteworthy that η remains largely unaffected.

Additional scalars: two examples
If fermion masses are generated through scalars which belong to the minimal set required for symmetry breaking then mass relationships do not reflect observed values. One needs to add at least one extra scalar multiplet -which is light, couples to fermions, and develops a vev at the electroweak scale -to get realistic mass ratios. In SO (10) We have examined the behaviour of gauge coupling evolutions for the DCR case including the additional [15,2,2] submultiplet to check if the TeV range M R still remains viable. In figure 6 the variation of M D and M U with M C when an extra [15,2,2] is included are shown (dashed lines). The effect on η (shown in the inset) is negligible. The important change is that the permitted lowest M C is more restricted as M U tends rapidly towards M Planck .
It is clear that the scale M D is governed by the difference in the β-coefficients of SU(2) R and SU(2) L . Submultiplets such as [15,2,2], which contribute symmetrically to the β-coefficients of the left-and right-handed SU(2) groups, will not affect the difference and so do not change the D-parity breaking scale. For the reasoning η is not affected as well.
Another alternative we examined pertains to the impact of additional scalars necessary to produce two right-handed neutrinos nearly degenerate in mass but with opposite CPproperties. As noted in section 2.1 this can explain the lack of like-sign dilepton events in the CMS data [18]. A right-handed neutrino mass matrix of the appropriate nature can be generated [27] by a scalar multiplet χ ≡ (1,2,1) under the SU(2) L × SU(2) R × U(1) B−L symmetry. In an SO(10) GUT χ is a member of the 16 representation and must develop a vev at the scale M 0 . We have examined the impact of adding a 16-plet on η = g R /g L , the symmetry breaking scales, and the coupling at the unification point. We find no serious effect on any of these, the allowed range of M C is about one order of magnitude larger, and for any M C , the unification scale, M U , and the D-parity-breaking scale, M D , are both somewhat lowered. η is essentially unaffected.

Summary and conclusions
The observation by the CMS collaboration of a 2.8σ excess in the (2e)(2j) channel around 2.1 TeV can be interpreted as a preliminary indication of the production of a right-handed gauge boson W R . Within the left-right symmetric model the excess identifies specific values of η = g R /g L , r = M N /M W R , and V Nee . We stress that even with g R = g L and V Nee = 1 the data can be accommodated by an appropriate choice of r.
We explore what the CMS result implies if the left-right symmetric model is embedded in an SO(10) GUT. η = 1 is a consequence of the breaking of left-right D-parity. We find that a W R in the few TeV range very tightly restricts the possible routes of descent of the GUT to the standard model. The only sequence of symmetry breaking which is permitted is M D > M C > M R > M 0 with a D-parity breaking scale ≥ 10 16 GeV. All other orderings of symmetry breaking are excluded. Breaking of left-right discrete parity at such a high scale pushes g L and g R apart and one finds 0.64 ≤ η ≤ 0.78. The unification scale, M U , has to be as high as ∼ 10 18 GeV so that it is very unlikely that proton decay will be seen in the ongoing experiments. The SU(4) C -breaking scale, M C , can be as low as 10 6 GeV, which may be probed by rare decays such as K L → µe and B d,s → µe or n −n oscillations. In table 4 we summarise the essence of the allowed GUT solutions. We have assumed that no extra scalar multiplets are included beyond those needed for symmetry breaking and invoked the Extended Survival Hypothesis to identify scalar submultiplet masses.
The ATLAS collaboration has indicated [43] an enhancement around 2.1 TeV in the di-boson -ZZ and W Z -channels in their 8 TeV data. Our interpretation of the excess in the (ee)(jj) channel can be extended to include the branching ratio of W R to di-boson states [44,45]. One can arrange to accommodate both these findings if r 1 and V Nee < 1. Such a combined analysis is beyond the scope of this work and will be reported elsewhere. Results along somewhat similar directions can be found in the literature [46]. It has also been shown that interpretations of the di-boson observations are possible if the LRS model is embellished with the addition of some extra fermionic states [14,27].