Sub-dominant Type II Seesaw as an Origin of Non-zero $\theta_{13}$ in SO(10) model with TeV scale $Z^\prime$ Gauge Boson

We discuss a class of left-right symmetric models where the light neutrino masses originate dominantly from type I seesaw mechanism along with a sub-dominant type II seesaw contribution. The dominant type I seesaw gives rise to tri-bimaximal type neutrino mixing whereas sub-dominant type II seesaw acts as a small perturbation giving rise to non-zero $\theta_{13}$ in our model which also has TeV scale right-handed neutrinos and $Z^\prime$ gauge boson thereby making the model verifiable at current accelerator experiments. Sub-dominant type II and dominant type I seesaw can be naturally accommodated by allowing spontaneous breaking of D-parity and $SU(2)_R$ gauge symmetry at high scale and allowing TeV scale breaking of $U(1)_{R} \times U(1)_{B-L}$ into $U(1)_Y$. We also embed the left-right model in a non-supersymmetric $SO(10)$ grand unified theory (GUT) with verifiable TeV scale $Z^\prime$ gauge boson. Drawing it to an end, we scrutinize in detail the evaluation of one-loop renormalization group evolution for relevant gauge couplings and estimation of the proton life time which can be accessible to the foreseeable experiments. And in the aftermost part we make an estimation of branching ratio for lepton flavor violating process $\mu \rightarrow e + \gamma $ as a function of type II seesaw strength due to doubly charged component of the right handed Higgs triplet with mass at the TeV scale, which can be accessible at ongoing experiments.


I. INTRODUCTION
The fact that the most successful phenomenological theory, the standard model (SM) of particle physics, suffers from the inability to address several observed phenomena as well as theoretical questions, has always been a source of excitement for particle physicists. The tiny but non-zero neutrino masses that have been confirmed by the phenomenon of neutrino oscillations detected in solar, atmospheric and reactor experiments [1] is certainly one such phenomena which the SM fails to address. These observations, among others have intensified the urge to ponder beyond the SM which has led to several well motivated beyond SM frameworks. The canonical seesaw mechanism (commonly referred to as type-I seesaw [2]), being the most elegant mechanism for generating small neutrino masses relies on the existence of right-handed (RH) neutrinos. In addition to the canonical seesaw, other seesaw mechanisms have been worked upon as well to explain the tiny masses of the active neutrinos. Picking the type-II seesaw mechanism [3] from them which requires the existence of SU(2) L triplet Higgs fields in addition to the minimal SM particle content, the neutrino mass gets an extra contribution given by m II ν ≃ f v L , where v L is the VEV of the neutral component of the triplet and f is the corresponding Yukawa coupling.
Minimizing the scalar potential of such a model, the VEV of the Higgs triplet is found to be v L = µv 2 /M 2 ∆ , where M ∆ is the mass of the Higgs triplet and µ defines the mixing between SM Higgs and triplet. An obvious setting would be f ≃ O (1) and µ ∼ M ∆ ≃ 10 14−15 GeV in order to explain sub-eV scale of light neutrino masses. Although these seesaw mechanisms look promising while explaining neutrino oscillation data, they lack the direct experimental testability in the ongoing experiments like Large Hadron Collider (LHC) or any future experiment like International Linear Collider.
Retreating not here, the particle phenomenology community has explored beyond standard model physics operative at few TeV scale, the results being repetitive attempts to corroborate neutrino masses and mixing. One such highly motivated and one of the most widely discussed beyond standard model framework is the left-right symmetric model (LRSM) [4] which not only gives a clear description of the origin of parity violation at electroweak scale but also leads a way to the generation of neutrino masses naturally. A thorough study of these models strengthens us with the knowledge that in conventional left-right symmetric model (LRSM), the light neutrino masses arise from two sources: the type-I [2] plus type-II [3] seesaw mechanisms where the parity and SU(2) R gauge symmetry are spontaneously broken at the same scale.
Earlier explorations of the field imply that a deep relation between high energy collider physics and low energy phenomena like neutrino-less double beta decay as well as other lepton flavor violating processes is enrooted by minimal left-right symmetric model (LRSM) valid at TeV scale [5,6]. If it happens that the parity and SU(2) R break at the same scale, then according to the seesaw relation v L v R = γv 2 (with γ being a dimensionless parameter) the microscopic value of v L as required for type-II seesaw depends on large value of v R , which further implies v R ≈ (10 13 ∼ 10 14 ) GeV making itself incapable of direct detection in near future. In a contrast way, if the right-handed scale is assigned with more moderate values, say in the range of few TeV, one can expect to have observable consequences at experiments in the near future. More willingly, if we assume both parity and SU(2) R to be broken at TeV scale i.e, v R ≃ TeV that is the scale of RH heavy neutrino mass, then we strictly need to calibrate the Higgs couplings up to the order of γ ≤ O (10 −10 ) in order to fit neutrino data from the seesaw relation. To refine this, studies have been done on left-right symmetric models to come upon with spontaneous D-parity breaking [7][8][9][10][11] where parity gets broken much earlier than SU(2) R gauge symmetry. In this work, we shall be discussing such a class of left-right symmetric models in which the spontaneous breaking of Dparity occurs at reasonably high scale along with SU(2) R × U(1) B−L gauge symmetry breaking down to U(1) R × U(1) B−L . We then check numerically whether U(1) R × U(1) B−L breaking occurs at TeV scale (provided parity breaks at much higher scale) and tiny neutrino masses can be obtained without too much fine tuning. In this class of models, the TeV scale breaking of U(1) R × U(1) B−L results in the TeV scale masses of the right-handed neutrinos as well as Z ′ boson while D-parity breaks at a high energy scale (≃ 10 9−11 GeV). As will be discussed later, this allows the possibility of dominant type I seesaw contribution to neutrino mass whereas type II seesaw contribution can naturally remain sub-dominant. We use such a sub-dominant type II seesaw contribution as the origin of non-zero θ 13 , the reactor mixing angle. It should be noted that, most of the earlier attempts to explain the non-zero θ 13 incorporate different corrections to the µ − τ symmetric tri-bimaximal (TBM) neutrino mass matrix which can naturally originate in generic flavor symmetry models like A 4 . Motivated by this, we consider the dominant type I seesaw contribution giving rise to TBM type neutrino mass matrix whereas the sub-dominant type II term giving rise to non-zero θ 13 . We also constrain the D-parity breaking scale from the demand of generating the experimentally allowed range of θ 13 . Apart from this, we also investigate whether such a choice of intermediate symmetry breaking scales allows the possibility to unify all the gauge couplings while being embedded in a non-supersymmetric SO(10) grand unified theory.
With all these motivations, we present a SO(10) model with a novel chain of symmetry breaking having left-right symmetry as an intermediate step giving neutrino masses through type-I plus type-II seesaw mechanisms, unification of three fundamental forces, prediction of proton life time accessible to the ongoing search experiments and most importantly, a low mass Z ′ gauge boson which can be probed at LHC. While preparing this manuscript, an interesting work appeared online [12] with similar symmetry breaking chains and scales as the one we are discussing here.
However, the neutrino mass phenomenology in that work is completely different from the one we pursue here. The plan of the paper can be sketched as follows. In section II we briefly discuss the left-right symmetric models, elucidating the spontaneous breaking of D-parity. In section III we discuss neutrino masses and mixing via dominant type-I seesaw giving rise to TBM type neutrino mixing and sub-dominate type-II seesaw giving rise to deviations from TBM mixing and hence non-zero θ 13 . In Sections IV and V, we give a possible path for embedding the present left-right symmetric models in the non-SUSY SO(10) GUT with its symmetry breaking pattern and oneloop gauge coupling unification. In Section VI, the proton lifetime is estimated using the value gauge coupling at GUT scale. In section VII, we estimate the branching ratio for lepton flavor violating decay µ → e + γ as a function of type II seesaw strength and finally conclude in section VIII.

II. LEFT-RIGHT SYMMETRIC MODEL WITH SPONTANEOUS D-PARITY BREAKING
In left-right symmetric models with spontaneous D-parity breaking, the discrete symmetry called D-parity gets broken earlier compared to the SU(2) R gauge symmetry. Here the gauge group can be written effectively as SU (2) is the discrete left-right symmetry or D-parity. In matter sector, the left and right handed fermions are doublets under SU(2) L and SU(2) R gauge groups, respectively. The transformation of quarks and leptons under the left-right symmetric group can be summarized as Notably the difference between Lorentz parity and D-parity is that Lorentz parity acts on the Lorentz group and interchanges left-handed fermions with the right-handed ones but the bosonic fields remain the same whereas D-parity acts on the gauge groups SU(2) L ×SU(2) R interchanging the SU(2) L Higgs fields with the SU(2) R Higgs fields in addition to the interchange of fermions.
The spontaneous breaking of D-parity creates an asymmetry between left and right handed Higgs fields making the coupling constants of SU(2) R and SU(2) L evolve separately under the renormalization group.
The Higgs sector of the left-right model with spontaneous D-parity breaking mechanism consists of a SU(2) singlet scalar field σ which is odd under discrete D-parity, two SU ( By assigning a non-zero VEV to D-parity odd singlet σ ≃ M P , the left-right symmetry is spontaneously broken but the gauge symmetry G 2213 remains unbroken resulting in where M ∆ is the mass term for triplets i.e, M 2 ∆ Tr ∆ † L ∆ L + ∆ † R ∆ R , and λ ∆ is the trilinear coupling in the term MσTr ∆ † L ∆ L − ∆ † R ∆ R . In this scenario M ∆ , M, σ all are of order of M P which is the scale of D-parity breaking thereby resulting TeV scale masses for right-handed Higgs triplets and D-parity breaking scale for their left-handed counterparts by suitable adjsutment of trilinear coupling λ ∆ . In order to have W R and Z R mass predictions at nearly the same scale along with the generation of Majorana neutrino masses, it is customary to break SU (2) Instead of pursuing the aforementioned left-right symmetric model with D-parity breaking mechanism, we consider a more appealing phenomenological scenario: with M W R >> M Z R via two step breaking of the left-right symmetric gauge theory to the SM.
The Higgs sector of the present model with spontaneous D-parity breaking mechanism consists of two SU(2) L triplets ∆ L and Ω L , two SU(2) R triplets ∆ R , Ω R and a bidoublet Φ which contains two copies of SM Higgs transforming under the LR gauge groups is shown in Table. I.

Higgs Fields Under G 2213
Under G 2113 We have chosen those fields in the third column under G 2113 which acquire a non-zero vacuum expectation value and in particular, the U (1) R values corresponds to the z-components of Isospin i.e, T 3R of SU (2) R satisfying Q = T 3L + T 3R + (B − L)/2 valid both for G 2113 as well as G 2213 gauge groups.
The first step of symmetry breaking i.e, SU(2) L × SU ( TeV. This unique scenario gives us the knowledge that W R scale completely decouples from Z R scale and hence, the LHC signatures of these gauge bosons and corresponding bounds on their mass scales should be revived again. The right handed neutral gauge boson Z R gets mass around few TeV staying very close to the experimental lower bound M Z ′ ≥ 1.162 TeV allowing its visibility at high energy accelerators in near future.
Apart from the right handed triplets whose VEV give masses to the right handed gauge bosons, the left handed triplets can also acquire non-zero VEV due to several scalar mixing terms in the Lagrangian. The analytic expression for VEV of the neutral component of ∆ L can be expressed where we have used v = 246 GeV and β is a coupling constant of O(1). Noticeably in the above eq.(3), the smallness of the VEV of ∆ L is decided by the parity breaking scale and not by the SU(2) R breaking scale thereby putting no constraints on v R from the type-II seesaw point of view. Therefore, the type-II seesaw relation is modified for left-right models accompanied by spontaneous D-parity breaking scenario instead of its usual expression valid for conventional left-right symmetric model. As a result, the type-I [2] seesaw term decouples completely from Dparity breaking scale and become sensitive to the U(1) R × U(1) B−L breaking scale M 0 R while the type-II [3] seesaw contribution becomes sensitive to the D-parity breaking scale. In the following section we shall briefly discuss how a particular value of D-parity breaking scale M P = 10 9 −10 10 GeV leads to sub-dominant type-II seesaw giving rise to correct deviations from TBM neutrino mixing in order to generate non-zero θ 13 . As we show later, the D-parity breaking scale M P ∼ M is constrained to be greater than around 3 × 10 9 GeV. Hence, for v R ∼ 1 TeV and order one dimensionless couplings, the type II contribution comes out to be 0.001 eV or less. The leading order TBM type neutrino mass matrix can originate from usual type I seesaw term due to the TeV scale right handed neutrinos originating from the TeV scale breaking of U(1) B−L .

III. NEUTRINO MASS
The renormalizable invariant Yukawa Lagrangian that gives rise to the G 2113 invariant interactions, near the TeV scale for the model considered in our present analysis, is resulting in 6 × 6 neutral fermion mass matrix after electroweak symmetry breaking One should note here that all the mass scales used in above mass matrix M ν have their dynamical can be written as a seesaw formula given by where the usual type I seesaw formula is given by the expression, Here M LR is the Dirac neutrino mass matrix. Thus, for type I seesaw dominance with TeV scale TeV, the Dirac Yukawa copulings should be fine tuned to y ν ∼ 10 −5 for f R ∼ 1. The type II seesaw term (m II LL = f L v L ) however, is directly proportional to the Majorana Yukawa couplings f L which have to be large in order to have sizeable contribution to neutrino masses.
The induced VEV for the left handed triplet v L can be shown for generic LRSM to be . This expression for type II seesaw term is valid for those class of minimal models where Dparity and SU (2) [13] high scale whereas U(1) R × U(1) B−L gets broken down to U(1) Y of standard model at TeV scale.
The VEV of the left handed triplet is given by equation (3) in such a case.
Before doing a numerical analysis of neutrino mass and mixing in our model, we note that prior to the discovery of non-zero θ 13 , the neutrino oscillation data were compatible with the well motivated TBM form of the neutrino mixing matrix discussed extensively in the literature [14].
However, since the latest data (last five references in [1]) have ruled out sin 2 θ 13 = 0, one needs to go beyond the TBM framework to incorporate non-zero θ 13 . Since the experimental value of θ 13 is much smaller than atmospheric and solar neutrino mixing angles, TBM type mixing can still be a valid approximation and the non-zero θ 13 can be accounted for by incorporating small perturbations to TBM mixing coming from different mechanisms like charged lepton mass diagonalization, for example. There have already been a great deal of activities in this context [15,16] which can successfully explain the latest data within the framework of several interesting models.
Since non-zero θ 13 can be very naturally explained by incorporating corrections to TBM mixing and our model naturally provides such small correction in the form of type II seesaw term, we find it interesting to explore the possibility of TBM type mixing coming from type I seesaw term and the origin of non-zero θ 13 through the type II seesaw term. Similar attempts to study the deviations from TBM mixing by using the interplay of two different seesaw mechanisms were done in [17,18]. Our analysis here differs from these in the sense that we implement our model within a grand unified theory where the strength of seesaw terms can be naturally explained from gauge coupling unification point of view. We also extend our earlier discussion [18] to include two different cases: one where the light neutrinos are almost degenerate, and the other in which there exists a moderate hierarchy between them, both obeying the cosmological upper limit on the sum of absolute neutrino masses.
Type I seesaw giving rise to µ − τ symmetric TBM mixing pattern for neutrinos have been discussed extensively in the literature. The neutrino mass matrix in these scenarios can be written in a parametric form as which is clearly µ −τ symmetric with eigenvalues m 1 = x−y, m 2 = x+ 2y, m 3 = x−y + 2z. It predicts the mixing angles as θ 12 ≃ 35.3 o , θ 23 = 45 o and θ 13 = 0. Although the prediction for first two mixing angles are still allowed from oscillation data, θ 13 = 0 has been ruled out experimentally at more than 9σ confidence level. This has led to a significant number of interesting works trying to explain the origin of non-zero θ 13 . Here we study the possibility of explaining the deviations from TBM mixing and hence from θ 13 = 0 by allowing the type II seesaw term as a perturbation.
It should be noted that the structure of the type I seesaw mass matrix (7) does not constrain the Dirac neutrino mass matrix M LR or the right handed neutrino mass matrix M RR to have some specific form. However, choosing one to have some particular form restricts the other so as to get the desired type I seesaw structure (7). For example, if we choose the Dirac neutrino mass matrix to have a diagonal structure then the M RR is restricted to have the following form Before choosing the minimal structure of the type II seesaw term, we note that the parametrization of the TBM plus corrected neutrino mass matrix can be done as [16].  where w denotes the deviation of m LL from that within TBM frameworks and setting it to zero, the above matrix boils down to the familiar µ − τ symmetric matrix (7). Thus, the minimal structure of the perturbation term to the leading order µ − τ symmetric TBM neutrino mass matrix can be taken as Such a minimal form of the type II seesaw term can be explained by incorporating additional flavor symmetries as outlined in [18].
We first numerically fit the leading order µ − τ symmetric neutrino mass matrix (7) by taking the central values of the global fit neutrino oscillation data [13] as presented in table II. We also incorporate the cosmological upper bound on the sum of absolute neutrino masses i m i < 0.23 eV [19] reported by the Planck collaboration recently. For normal hierarchy, the diagonal mass matrix of the light neutrinos can be written as m diag = diag(m 1 , m 2 1 + ∆m 2 21 , m 2 1 + ∆m 2 31 ) whereas for inverted hierarchy it can be written as We then incorporate the type II seesaw contribution which breaks µ − τ symmetry and hence gives rise to non-zero θ 13 . We show the variation of other neutrino parameters with respect to sin 2 θ 13 in figure 1, 2 for normal and inverted hierarchies respectively. It can be seen that the differences in the lightest active neutrino mass show up only in the variation of ∆m 2 21 . In case of normal hierarchy, all the parameters lie in the 3σ range for m lightest = 0.07 eV whereas for inverted hierarchy we see a preference for lighter m lightest namely, 0.001 eV. We then show the variation of sum of absolute neutrino masses in figure 3 and for all the cases considered, the sum is found to be within the cosmological limit. We also show the variation of sin 2 θ 13 as a function of type II seesaw strength w in figure 4. It is seen that for higher values of m lightest , we require a lower strength of the type II seesaw term to give rise to the desired θ 13 . For m lightest = 0.065, 0.07 eV, one can see Taking the dimensionless couplings to be of order unity and v = 10 2 GeV, v R = 10 4 GeV, one gets a constraint MM P ∼ 5 × 10 19 GeV 2 . Similarly, for m lightest = 0.001 eV, one can estimate this bound to be around 2 × 10 19 GeV 2 . Thus, from the constraint of neutrino mass, we get a bound on the SU(2) R × D breaking scale to be of the order of 10 9 − 10 10 GeV which is consistent with the gauge coupling unification as will be discussed below.
The variation of the neutrino parameters with the perturbation strength can be understood simply by calculating the diagonalizing matrix of the neutrino mass matrix considered in the study.
which has eigenvalues m 1 = x − y + z − √ 3w 2 + z 2 , m 2 = x + 2y and m 3 = x − y + z + √ 3w 2 + z 2 . Assuming m 1 < m 2 < m 3 we calculate the neutrino parameters by first identifying the diagonalizing matrix. Assuming w to be small such that higher order terms beyond w 2 can be neglected, we arrive at the following approximate variations of neutrino parameters where h.o. refers to higher order terms in w. It can be easily seen that for w = 0, the mixing angles correspond to the values predicted by TBM mixing.
With the above choice of symmetry breaking, the SO (10) The remaining symmetry breaking SU (2) (

V. GAUGE COUPLING EVOLUTION WITH ONE-LOOP ANALYSIS
In this section we study the one loop renormalization group evolution (RGE) equations for gauge couplings relevant for our model. The one loop RGE equations for the gauge couplings can be written as where t = ln(µ), α i = g 2 i /(4π) are the fine structure constants and a i a i a i are the one-loop beta coefficients derived for the corresponding i th gauge group for which coupling evolution has to be determined. The analytic formula for a i a i a i is with no summation over i. We denote C 2 and T 2 as quadratic Casimir of a given representation, d S i as the multiplicity factor for a particular gauge group G i due to other SU(N) j group present in the model, N G as the number of fermion generation (which is 3 in our model). We take κ = 1, 1 2 for Dirac and Weyl fermions, η = 1, 1 2 for complex and real scalar fields, respectively.
. (22) Similarly, the appropriate gauge coupling matching conditions at the scale M P valid for the gauge . (23) Also, one can write down the gauge coupling matching conditions at the unification scale M U as With the above gauge coupling matching conditions, one can express the RGE equations for where the one-loop beta coefficients for our model determined by the particle spectrum in the , a a a Y , a a a 3C }, {a a a ′ 2L , a a a ′ 1R , a a a ′ B−L , a a a ′ 3C }, and {a a a ′′ 2L , a a a ′′ 2R , a a a ′′ B−L , a a a ′′ 3C }, for gauge groups G 213 , G 2113 and G 2213D , respectively. Fixing M 0 R around few TeV, and using particle data group values [20] sin 2 θ W = 0.23166 ± 0.00005, α S = 0.1184 ± 0.003, and α em = 1/127.94, a simple one-loop analytical survey of the gauge coupling running equations yields two important relations for M P and M U as [11] ln ln with In the following subsection, the value of the D-parity breaking scale M P and the unification scale

Range of Masses (GeV)
Higgs content a a a     Similarly, for the evaluation of a i a i a i , a a a ′ i , and a a a ′′ i presented under column C2 of table V, the Higgs field ξ (2,2,0,8) is introduced at or above the scale M P .

) Y via Higgs triplet (∆) plus Higgs doublet (χ)
It should be noted that, shifting the parity breaking scale M P towards the GUT scale provides us with more possibilities to achieve unification with more minimal set of additional fields than discussed above. However, to keep a sizable contribution of type II seesaw so that it can give rise to the observed θ 13 , we intend to keep M P as low as 10 9 − 10 10 GeV. Here lies the need to include additional field content discussed in previous subsection. Apart from the scenario where U(1) R × U(1) B−L gauge symmetry is broken by Higgs triplets, there can be one more possibility to achieve the same using both triplets and doublets. For the sake of completeness we discuss this case as well and check the gauge coupling unification.
In such a scenario, we allow the breakdown of the intermediate symmetry SU (2) (3) Table.VII along with color octet Higgs scalar (1,1,0,8) where the predicted mass scales are Hence, there will be little modification to the type II seesaw contribution which is m II M M P if one incudes two-loop RG effect. One can fix the neutrino mass arising from type-II seesaw by suitably adjusting the other free parameters like Higgs coupling β and M even if we include the effect of two-loop RG corrections on v R and M P .
It should be noted that the LRSM where the breaking of U(1) R × U(1) B−L → U(1) Y occurs through Higgs triplet (∆) and SU(2) R × D gets broken by Higgs triplet Σ can also be constrained from the cosmologial constraints on the successful disappearance of domain walls. Domain walls generically arise in such models (due to the spontaneous breaking of discrete symmetry called D-parity) which, if stable, can overclose the Universe conflicting with standard cosmology. As discussed in [21], for M 0 R = 10 TeV, domain wall disappearance requires M P < 10 9 GeV, which are very close to the symmetry breaking scales in our present model. Similar constraint on the second model (the one with both Higgs triplet and doublet) have not been studied yet and left for future investigations.
where A L = 1.25 is the renormalization factor from the electroweak scale to the proton mass, D = 0.81, F = 0.44, α H = −0.011 GeV 3 , and f π = 139 MeV are extracted as phenomenological parameters by chiral perturbation theory and lattice gauge theory. Also, m p = 938.3 MeV is the proton mass, and α U ≡ α G is the gauge fine structure constant derived at the GUT scale.
The renormalization factor R = ( for SO (10), the (1, 1) element of V CKM is V ud = 0.974 = with A SL (A SR ) being the short-distance renormalization factor in the left (right) sectors.
Redefining α H = α H (1 + F + D) = 0.012 GeV 3 , and A R ≃ A L A SL ≃ A L A SL , the proton life time can be expressed as where F q ≃ 7.6 In order to estimate the proton life time, we should have knowledge about the short distance enhancement renormalization factors which are fully model dependent, a few of which are known while a few others have been already determined in the present model. For the particular choice of symmetry breaking considered in present non-SUSY SO(10) model and assuming no threshold corrections at or below the GUT scale, the short distance renormalization factors evaluated at one loop level are given as where, We have used the anomalous dimensions taken from [22,23]  e + π 0 channel is τ (p → e + π 0 ) SK,2011 > 8.2 × 10 33 yrs [24] while it can be accessible to future planned experiment such as τ (p → e + π 0 ) HK,2025 > 9.0 × 10 34 yrs and τ (p → e + π 0 ) HK,2040 > 2.0 × 10 35 yrs [25].

VII. LEPTON FLAVOR VIOLATING DECAYS
In FIG. 7: One loop Feynman diagrams for lepton number violating decays ℓ i → ℓ j + γ(i = j). Contribution from the W L exchanges involving mixing between left-and right-handed neutrinos is presented in (a) whereas contribution from the W R exchanges with heavy RH Majorana neutrinos is presented in (b).
The dominant contribution to lepton flavor violation (LFV) decays via doubly charged RH Higgs triplet exchanges is presented in (c).
these contributions are given below where θ W is the weak mixing angle, θ R is the mixing angle between left and right handed neutrino sector, Γ µ = 2.996 × 10 −19 GeV, G µe γ contains left-right neutrino mixing plus the loop factor and α W = g 2 2L /(4π) is the fine structure constant for SU(2) L valid at M Z scale and is found to be 0.18389. There have been several attempts to calculate the enhanced LFV signal in µ → eγ process for example, in [26] and recently, it has been pointed out in refs. [27] that the LFV branching ratios can be significant if the heavy-light neutrino mixing is large.
Assuming the left-right mixing to be small, one can neglect the contribution Br (µ → e + γ) (a) W L in comparison to other contributions. Also, in our model the W R gauge boson mass is found to be ≥ 10 8 GeV making the Br (µ → e + γ) (b) W R contribution suppressed. The remaining dominant contribution due to TeV scale right-handed Higgs triplet contribution is We have numerically estimated this contribution represented by a plot as shown in figure  0. ≤ 5.7 × 10 −13 [28,29] for type II seesaw strength f v L = 0.013 eV.
This is consistent with our model where the required type II seesaw strength is of the order of 0.001 eV.

VIII. CONCLUSIONS
We have studied a left-right symmetric gauge theory SU(2) L ×SU (2)  while keeping type II seesaw term as sub-dominant but sizeable enough to give rise to the required deviation from TBM mixing in order to explain non-zero θ 13 . First we have performed a numerical analysis taking type I seesaw term as TBM type and type II seesaw term as a perturbation which breaks µ − τ symmetry. We have done this exercise for both normal and inverted hierarchical neutrino mass spectra as well as two possible values of lightest neutrino mass (one being close to the maximum allowed by cosmological upper bound and one slightly lower). We have constrained the type II seesaw strength by demanding the required deviation from TBM to produce non-zero θ 13 . For dimensionless couplings to be of order one and U(1) R × U(1) B−L breaking scale of around 10 TeV, the parity breaking scale has been restricted to be 10 9 − 10 10 GeV.
We have also made an attempt to embed this model within SO(10) GUT and check whether the above mentioned symmetry breaking steps can be naturally realized along with successful gauge coupling unification at a scale which lies close to the bound coming from proton lifetime constraint. We have shown that in the framework of non-SUSY SO(10) GUT invoking spontaneous D-parity breaking, one-loop RGE analysis of gauge couplings allow mass ranges M 0 R = 3 − 6 TeV, M P = 10 9 − 10 11 GeV and M U = 10 14.5 − 10 16.5 GeV for several possible additional Higgs structures. We have also calculated the proton lifetime from the unification scale and find it to be within future experimental reach. At the end, we have made an estimate of branching ratio for the LFV decays of µ → e + γ due to the presence of TeV scale doubly charged component of right handed triplet Higgs and found it to be lying close to the experimental limit.

IX. ACKNOWLEDGMENT
Two of the authors, Debasish Borah and Sudhanwa Patra would like to thank the organizers of the workshop entitled "Majorana to LHC: Origin of neutrino Mass "at ICTP, Trieste, Italy during 2-5 October, 2013 where part of this work was completed.