Planck-scale induced left-right gauge theory at LHC and experimental tests

Recent measurements at LHC has inspired searches for TeV scale left-right gauge theory originating from grand unified theories. We show that inclusion of Planck-scale induced effects due to ${\rm dim.}5$ operator not only does away with all the additional intermediate symmetries, but also it predicts the minimal set of light Higgs scalars tailored after neutrino masses and dilepton, or trilepton signals. The heavy-light neutrino mixings are predicted from charged fermion mass fits in $SO(10)$ and LFV constraints which lead to new predictions for dilepton or trilepton production signals. Including fine-structure constant matching and two-loop, and threshold effects predicts $M_{W_R}= g_{2R}\times 10^{4.3\pm 1.5 \pm 0.2}$ GeV and proton lifetime $\tau_p=10^{36.15\pm 5.8\pm 0.2}$ yrs with $W_R$ gauge boson coupling $g_{2R}=0.56-0.57$. Predictions on lepton flavour and lepton number violations are accessible to ongoing experiments. Current CMS data on di-electron excess at $\sqrt s= 8$ TeV are found to be consistent with $W_R$ gauge boson mass $M_{W_R}\ge 1.9-2.2$ TeV which also agrees with the values obtained from dijet resonance production data. We also discuss plausible explanations for diboson production excesses observed at LHC and make predictions expected at $\sqrt s =14$ TeV


Introduction
The standard model SU (2) L × U (1) Y × SU (3) C (≡ G 213 ) partially unifies electromagnetic and weak interactions but fails to explain neutrino masses and why parity violation occurs only in weak interaction. Left-right symmetric (LRS) gauge theory [1][2][3] SU (2) L × SU (2) R × U (1) B−L × SU (3) C (g 2L = g 2R )(≡ G 2213D ) predicts a number of phenomena beyond the standard model including neutrino masses and parity violation. It also goes further to suggest that the right-handed (RH) neutrino (N ), a member of its fundamental representation, could be a heavy Majorana fermion driving seesaw mechanism for light neutrino masses and acting as a seed for baryogenesis. As possible experimental evidence of LRS theory, it would be quite pertinent to associate these RH neutrinos to be mediating dilepton production events recently observed at the Large Hadron Collider (LHC) [4,5] which can discriminate even if W R gauge coupling is different from the standard W L boson [6]. The minimal left-right symmetric GUT that unifies strong, weak, and electromagnetic interactions is SO(10) that leaves out gravity 1 . It would be quite interesting if spontaneous symmetry breaking of non-SUSY SO(10) through any one of the following two minimal symmetry breaking chains gives the LHC accessible W R , Z R bosons In eq.(1) G 2213 represents the same left-right gauge theory G 2213D but without the D-parity for which g 2L = g 2R [6]. When the symmetry breaking proceeds through G 2213D , consistent with extended survival hypothesis (ESH) [9], it needs large values of sin 2 θ W (M Z ) ∼0.27-0.28 in direct conflict with electroweak precision data (sin 2 θ W ) expt = 0.2311 ± 0.00013. On the other hand, solutions to renormalisation group (RG) equations consistent with this value needs an unusually larger number of nonstandard Higgs scalars and/or exotic fermions in direct violation of the ESH [12]. When the symmetry breaks through G 2213 , the allowed solutions also require a number of additional light particle degrees of freedom [13], although less than the G 2213D case. In this case also the ESH has to be abandoned. Although several possibilities have been discussed earlier [6,7,11], allowed solutions for TeV scale W R , Z R in the best identified chain of ref. [7] have been noted recently to be in concordance with neutrino oscillation data [10]. This model has four intermediate symmetries In eq.
with left-right discrete symmetry and G 224 denotes the same gauge symmetry without D-Parity. Of course a theory is finally accepted or falsified on the basis of its experimental tests and until those are carried out both the approaches to achieve LHC scale LR theory may be equally likely. 2 Also in view of the LHC capability to discriminate among different models, alternative theoretical explorations for LR models with parity restoration at low scales (g 2L = g 2R ) or high scales (g 2L = g 2R ) with additional signatures would be interesting. The purpose of this letter is to show that when effect of gravity is included through a dim.5 operator, the SO(10) model gives LHC scale LR gauge theory G 2213 in the minimal symmetry breaking chain with drastically reduced size of the light Higgs spectrum that depends upon neutrino masses and the manifestation of W R through dilepton or trilepton signals. The model predicts heavy-light neutrino mixings which forms an important ingredient for multi-lepton signals at LHC especially in the W L production channel. The heavy sterile Majorana neutrinos act as additional source of dilepton signals. Other predictions of the model can be verified by charged lepton flavor violating (LFV) decay branching ratios or dominant contributions to 0νββ decay rates, or both in near future.
This letter is organized as follows. In Sec.2 we discuss the predictions of left-right gauge theory breaking scale and the grand unification scale in relation to the scale of dim.5 operator. In Sec.3 we give a short description on neutrino masses and LFV decay and in Sec.4 we discuss lepton number violation. In Sec.5 we discuss LHC signals for dilepton and trilepton production. Finally we give a brief summary of our results.

LHC scale LR theory
We attempt to predict the scale of LR gauge theory G 2213 in the minimal symmetry breaking chain of eq.(1).
We include the standard two-loop RG equations for gauge couplings We also include the effect of dim.5 operator [14,15] where φ 210 ≡ 210 H Higgs representation that breaks SO(10) → G 2213 at the GUT scale by acquiring vacuum expectation value (VEV) along its G 2213 singlet direction and M C is the scale 2 Compatibility of such LR theories [12,13] with neutrino oscillation data is yet to be investigated. of the dim.5 operator 3 . For the gauge kinetic field tensor we have where 1 2 σ ij (W ij µ ) ,and i,j=1,2,3......10 denote the 45 generators (gauge bosons) of SO (10). The GUT-scale boundary conditions are modified by the dim.5 operator where the ǫ i terms arise due to the dim.5 operator and α G is the effective GUT fine structure constant. The resulting analytic formulas for the unification mass M U and the LR scale M R are [16] log log The terms on the right-hand side reduce to the usual renormalisable ones in the limit ǫ ′ = ǫ ′′ = 0. Various notations occurring in eq.(7) and eq.(8) are where we have used Heavy pseudo Dirac neutrinos In this case 210 H breaks SO(10) and D − Parity to G 2213 [15] which further breaks to SM by the doublet χ R (1, 2, −1, 1) ⊂ 16 H . The SM theory breaks to the low-energy symmetry by the standard Higgs doublet h(2, 1, 1) ⊂ φ(2, 2, 0, 1) ⊂ 10 H . With such minimal Higgs content, respective beta function coefficients are presented in Table 1. Three additional singlet fermions (S i , i = 1, 2, 3), one for each generation are added in case of SO(10) theory 4 . The VEV of the RH doublet χ R also generates the N − S mixing mass term M = Y χ < χ R > which, along with the Dirac neutrino mass M D , is known to contribute to leptonic non-unitarity , leptonic CP violation, and lepton flavor violation. After block diagonalisation of the 9 × 9 neutral fermion mass matrix the RH neutrinos form three pairs of heavy pseudo-Dirac neutrinos with |M Ni | ≃ |M i |(i = 1 − 3). In combination with M D , M and the global lepton number violating fermion singlet mass term µ S , the model drives the inverse seesaw mechanism for neutrino masses [18,23] where, for simplicity, we use Using the numerical values of the available data sin 2 θ W (M Z ) = 0.23116 ± 0.00013, α(M Z ) = 1/127.9 and α 3C (M Z ) = 0.1184 ± 0.0007 in eq.(7), eq.(8) , we obtain solutions to the values of M R , M U , σ and M C which are presented in Table 2. It is clear that the gravity induced solutions as low as V χR ∼ M R ≃ 9 − 25 TeV are allowed and the model predicts the W R mass M WR ≃ g 2R V χR ≃ 3 − 10 TeV in the case of minimal combination of the light Higgs sector with only two doublets, D φ = D χ = 1. The scale of dim.5 operator turns out to be almost at the reduced Planck scale signifying that the induced nonrenormalisable corrections could be genuinely due to gravitational effects of four dimensional space time. The estimated proton lifetime for the p → e * π o mode is found to be beyond the range accessible to Super-Kamiokande or Hyper-Kamiokande collaborations. It is interesting to note that σ = g 2 2L g 2 2R ≃ 1.27 is only 27% larger compared to the value σ = 1 in the manifest LRS model [2].

Heavy RH Majorana neutrinos
For this purpose, in addition to the Higgs representations of Model-I, we require the SO(10) representation 126 H ⊃ ∆ R (1, 3, −2, 1) under G 2213 that carries B − L = −2 with corresponding coefficients given in Table 1. When the RH triplet acquires VEV ∆ 0 R = V ∆R , G 2213 symmetry    Model  (12). The RH doublet χ R (1, 2, −1, 1) ⊂ 16 H , apart from taking part in symmetry breaking process rather weakly, generates the N-S mixing mass term M as noted in the case of Model-I leading to gauged inverse seesaw formula for neutrino masses provided M N >> M > M D , µ S , a condition well known in extended seesaw mechanism [22]. The would-be dominant type-I seesaw term in this model cancels out in such decoupling limit [10,17,24] leading to gauged inverse seesaw formula of eq.(13) to explain the neutrino oscillation data. There are two heavy Majorana neutrino mass matrices: m N for RH neutrino and m s for sterile neutrino under the constraint m N >> m s The heavy RH Majorana neutrino mass matrix is very close to its gauged value, These two types of heavy Majorana neutrinos emerging as outcome of the extended seesaw mechanism mediate neutrino-less double beta decay in the W L − W L , W L − W R , and the W R − W R channels. Further both of them are capable of mediating the dilepton production process.
Using numerical values of a i and b ij in eq.(7), eq.(8), and eq.(9) and following the same procedure as outlined for Model-I, the solutions for mass scales M R , M U , M C , and σ are also presented for this Model-II in Table 2. It is clear that in this case low-mass RH gauge bosons are also permitted at the LHC energy scale for σ ≃ 1.2 and M C values almost at the reduced Planck scale which leads to the interpretation that the additional corrections could be due to gravity effects.

Neutrino masses and lepton flavor violation
The Dirac neutrino mass matrix M D occurring in eq.(13) is determined by the GUT-scale fitting of the extrapolated values of all charged fermion masses obtained by following the bottom-up approach [19] and running it down to the TeV scale following top-down approach as explained in the corresponding cases [10,23,24]. While the procedure followed in ref. [23] is used for Model-I, the procedures followed in ref. [10,24] is utilized for Model-II. 5 An additional bidoublet φ ′ ⊂ 10 H ′ is needed to fit fermion masses without affecting coupling unification substantially. The Higgs bidoublet ξ(2, 2, 15) ⊂ 126 H acquires the induced VEV v ξ ≃ 10 − 50 MeV [20] which ,along with the direct VEVs of the two bidoublets, enables fitting all charged fermion masses in Model-II. A byproduct of this fitting is the diagonalised version of the heavy RH neutrino mass matrix , where, in our Model-II,M In The dominant source of LFV is through the W L -loop in both the models and there are two types of heavy Majorana Fermion exchange contributions in case of Model-II. The RH neutrino exchange contribution can be considered subdominant since M Ni >> M i . Using the relevant analytic formulas [21] we estimate LFV decay branching ratios µ → e + γ, τ → e + γ and τ → µ + γ as shown in Table.3 where the allowed values of M i , (i = 1, 2, 3) satisfying the nonunitarity constraints have been also given [10,23]. As the predicted values are 3 − 5 orders smaller than the current experimental limits they may be accessible to ongoing or planned searches with improved accuracy. Using the set of values on M from Table. 3 and the Dirac neutrino mass matrix from eq.(18), we fit the available data on neutrino masses and mixings through inverse seesaw formula of eq.(13) for all the three types of mass hierarchies: NH, IH, and QD. In each case the fit gives a set of elements for µ S . As an example by using M = diag.(500, 500, 1140) GeV and the QD light neutrino hierarchy of common mass m 0 = 0.2 eV [23] gives Different aspects of LFV in non-SUSY SO(10) have been discussed in ref. [10,23,24] and our predictions in the corresponding cases are similar.

Lepton number violation
The standard contribution in the W L − W L channel is due to light neutrino exchanges. But, one important aspect of this Model-II is that even in the W L −W L channel, the singlet fermion exchange allowed within the extended seesaw mechanism, can yield much more dominant contribution to 0νββ decay rate than the standard one. The contributions due to the exchanges of heavy W R ,∆ ++ R and the RH neutrino in the extended seesaw frame work are negligible compared to the light neutrino and singlet fermion exchange contributions in the W L − W L channel. The leading order contributions have been summarized in in the Table.4.   Table 3, and the computed values of m Si through eq. (14) and eq. (17).
Out of various contributions in the W L −W L channel analysed in detail [24], we give an example of scattered plot of predicted half life as a function of the lightest sterile neutrino mass eigen value in Fig. 4 in the normally hierarchical (NH) case of the light neutrino masses.
Saturation of current experimental bound on 0νββ decay half life gives the lower bound on the lightest sterile neutrino mass m S1 ≥ 17 ± 3 GeV. In a similar manner bounds evaluated in the IH and QD cases have been found to be consistent with this value.

LHC signals of heavy pseudo Dirac and Majorana neutrinos
At the LHC, the parton-level generation of a heavy neutrino can be realized in the following way provided this process is kinematically feasible. This has lepton-number conserving (LNC) or lepton number violating (LNV) decay modes depending on whether N is pseudo Dirac as in Model-I or Majorana as in Model-II. We use the parton level differential cross section [28] dσ LHC d cos θ = kρ 32πŝŝ where k = 3.89 × 10 8 pb,ŝ is the square of centre-of-mass energy of the colliding partons, M is mass of N , and ρ = (ŝ − M 2 )/(ŝ + M 2 ). The total production cross section at the LHC is where τ =ŝ/E 2 CM and E CM is centre-of-mass energy of the LHC. The Feynman diagrams for trilepton(dilepton) production mechanism is shown in the left-panel (right-panel) of Fig.2 Trilepton signals The RH neutrino in Model-I being pseudo Dirac can not mediate like-sign dilepton production and the best channel for the trilepton mode is where W L /W R decays to leptonic final states: The inclusive cross-section for the trilepton state in a generic seesaw model is given by [28] σ(pp → l 1 l 2 l 3 + T me ) = σ prod (pp → W ⋆ → N l 1 )Br(N → l 2 W )Br(W → l 3 ν).
Here, T me stands for the missing transverse energy and the W L branching ratio Br(W → lν)=0.21 [29]. We have assumed m N > m W . Although this condition is needed for kinematic feasibility of the decay N → l 2 W → l 2 l 3 ν when the W is the real intermediate boson of the SM, this is not required for virtual W * exchange to give N → l 2 W * → l 2 l 3 ν. One important aspect of this model is that the fermion mass fitting and LFV constraint predicts all the elements of the heavy-light neutrino mixing matrix V νS = MD M . For example using eq.(18) and M N2 ≃ M 2 = 50 GeV, the light-heavy neutrino mixing parameter is |V µN2 | 2 = 9.8 × 10 −5 . Thus the heavy-light neutrino mixing is determined and varies inversely as the corresponding heavy pseudo Dirac neutrino mass exchanged.
For computation of the production cross section we have utilized the MRST parton distribution functions [31] in eq.(23). Using our ansatz for heavy light neutrino mixing matrix M D /M in the pseudo Dirac case, eq.(22), eq.(23), and eq.(24), our predicted results on trilepton signals in the LL channel are shown for LHC energy √ s = 14 TeV in Fig.3 where l (1) l (2) = e ± e ∓ and l 3 = e ± or µ ± in the upper blue curve and the mediating heavy fermion is the pseudo Dirac N 2 . The corresponding trilepton signal as a function of the pseudo Dirac mass M N1 is shown as lower red curve in the same figure for which l (1) l (2) = µ ± µ ∓ but l 3 = e ± or µ ± . At √ s = 14 TeV, the predicted trilepton signal cross sections in the W R − W R channel are shown in Fig.4(a) when l (1) l (2) = e ± e ∓ and l 3 = e ± or µ ± . In Fig. 4(b) the predicted signal cross sections are for l (1) l (2) = µ ± µ ∓ , and l 3 = e ± or µ ± also in the same channel. In this case

Dilepton signals
The Model-II has two types of heavy Majorana neutrinos. In this case, the collider signatures of heavy RH neutrino and the sterile neutrino can manifest at LHC energies through dilepton production in various channels of which we have examined the W L − W L and W R − W R , channels. Thus compared to other models [26,27] this model gives an additional source of dilepton production especially in LL, LR and RL channels due to heavy sterile neutrino exchanges. The signal cross-section for the production of the RH neutrino or sterile neutrino including the real or virtual W L , W R exchanged at the second stage is given by where the branching ratio here Br(W → jj)=0.676 [29]. For heavy Majorana neutrino exchange, our results are shown for √ s = 14 TeV with MRST parton distribution functions in Fig. 5 and Fig. 6(a), respectively, in the LL and RR channels. For sterile neutrino exchanges, our computed results are shown in Fig.  7 and Fig. 8, respectively, in the LL and RR channels. At 30 f b −1 luminosity, the number of events for heavy neutrino mass M N =100 GeV are 22.17 for W L − W L channel and 1380 for W R − W R channel respectively. Here, it is noteworthy that the dilepton signal is more dominant over trilepton signal in the W R − W R channel. It is also found that our estimated signal cross section for heavy neutrino in W L − W L channel lies in between those of two bench mark scenarios as discussed in ref. [27]. Hence this model predicts a signal cross section having a better probability of observing heavy neutrinos at the LHC or the other ongoing experiments.
Using √ s = 8 TeV we have shown our model predictions of the Di electron and dimuon signal cross sections in the left-panel and the right panel, respectively in Fig. 9 for W R production, and compared them with the CMS data [5]. The line I with uncertainty band is the prediction of the manifest LR model ( σ = 1). The line II is our Model-II prediction for (g 2L /g 2R ) 2 = σ = 1.3 and V 2 e1 = 1 in both the panels whereas the line III represents our prediction in the same model for the same value of σ but V 2 e1 = 0.8 (left-panel) and V 2 µ2 = 0.7 (right-panel). These mixings among the RH neutrinos are similar to the PMNS mixings for light neutrinos. The possibility of trilepton signals by heavy Majorana fermion exchanges [30] would be presented elsewhere.  Figure 9: Predictions of dielecton (left-panel) and dimuon (right-panel) signal cross section (lines II and III) for W R production at √ s = 8 TeV and their comparison with the LHC data for which the green (red) band is the 1σ(2σ) limit. The zig-zag dotted (solid) curve represents expected (observed) results of measurements. The line III with spreaded uncertainty is the prediction of manifest LRS model g 2L = g 2R [3].
Summary : In summary we have shown the realization of LHC scale LR gauge theory in the minimal chain with minimal light Higgs spectrum consistent with neutrino oscillation data and all charged fermion mass fit in non-SUSY SO(10) model. The content of the light Higgs spectrum depends upon the pseudo-Dirac (Majorana) nature of the heavy RH neutrino that predicts trilepon (like-sign dilepton) signal for W R production at the LHC. Only for gauge coupling unification in the pseudo Dirac case the Model-I has just one bidoublet and one RH doublet carrying B −L = −1 and one more bidoublet-doublet when all fermion mass fit is desired. In the heavy RH Majorana neutrino case, in addition to the light Higgs of Model-I in the corresponding cases, the Model-II needs just one more RH triplet Higgs scalar carrying B − L = −2. We have reported predictions of trilepton (dilepton) signal cross sections in the Model-I (Model-II) in the LL and RR channels at √ s = 14 TeV and also predicted new signal cross sections due to the mediation of heavy sterile neutrinos in Model-II. The heavy-light neutrino mixings are predicted because of the determined value of the Dirac neutrino mass matrix and LFV constraints. The model predicts g 2 2L /g 2 2R = 1.2 − 1.3 leading to the lower bound M WR ≥ 2.5 − 2.8 TeV from the CMS data. In addition, the model also predicts LFV branching ratios and dominant 0νββ decay rate accessible to ongoing experiments even when light neutrino masses are normally hierarchical.