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Study of baryon number and lepton flavour violation in the new minimal supersymmetric SO(10)GUT

We study the so-called new minimal supersymmetric SO(10) GUT(NMSGUT) where explicit spontaneous symmetry breaking allows determination of superheavy spectrum and thus threshold corrections to the effective MSSM couplings. This provides a generic mechanism to resolve the long standing super fast proton decay in Susy GUTs. We estimate lepton flavor violation associated with realistic charged fermion and (Type I seesaw) neutrino fit and show compatibility with baryon number and lepton flavour violation limits. We improve NMSGUT fits by including important loop corrections to sparticle spectra. Our fits use 5 GUT compatible soft supersymmetry breaking parameters of the Supergravity with Non-Universal Higgs Masses(SUGRY-NUHM) type. We calculate the full two loop NMSGUT gauge-Yukawa beta functions to study feasibility of the NUHM parameters via strong renormalization of SO(10) Higgs soft masses. Focus on MSSM Higgs allows formulation of a"Yukawonification"strategy for gauged flavour unification.


Abstract
We study baryon number (B) and lepton flavour violation (LFV) in a supersymmetric model based on SO(10) gauge group called New Minimal Supersymmetric SO(10) Grand Unified Theory (NMSGUT).
We calculated one loop GUT scale threshold corrections to the relation between the NMSGUT and effective minimal supersymmetric standard model (MSSM) Yukawa couplings. Strong renormalization of the Higgs line entering these vertices, due to the large number of GUT fields coupled to them allows lowering of the SO (10) couplings required for fitting MSSM couplings. Since the same SO(10) Yukawas are responsible for B-violation, proton decay lifetimes compatible with experimental limits are generically achievable. We successfully searched the NMSGUT parameter space for values that allowed accurate fits of known MSSM couplings and have acceptable dimension five operator mediated B violation rates.
The spectra of sparticles used in the B violation calculations were improved by including one loop corrections. Searches including these corrections require careful control to avoid instability. We found fits compatible with the MSSM data and B violation limits even after inclusion of loop corrections.
The effective theory of the NMSGUT includes lepton number violating couplings and thus our fits also imply predictions for LFV rates. We computed NMSGUT estimates-based on successful fits-for important observables in the lepton sector such as lepton flavour violating processes (e.g. l i → l j γ, l i → 3l j ), the muon g-2 anomaly (a µ ) and the CP violation parameter ( CP ) relevant for high scale leptogenesis scenarios. For LFV estimation we have included heavy right handed neutrino thresholds.
We computed the two loop renormalization group evolution equations of the    c W (s W ) = cos θ W (sin θ W ), θ W is Weinberg angle, defined as tan θ W = g g (g and g are gauge couplings of U (1) Y and SU (2) L respectively). 8 gluons (G a , a = 1...8) corresponding to unbroken SU (3) C remain massless. Fermionic matter consists of three generations, embedded in the SM gauge group with following quantum numbers : where A= 1, 2, 3 is the generation index. Clearly, all the fermion fields are present in chiral pairs (i.e. both Ψ L and Ψ R ) except the neutral fermion ν L . Dirac mass terms like : m ψψ ψ are not allowed by gauge invariance. However, charged fermion masses are generated via the Yukawa couplings of the fermions with the Higgs doublet since the Higgs acquire a VEV. Visible, stable, matter content of the universe is thought to be made of first generation fermions (except for ν µ , ν τ which persist) since the heavier generations decay to them with lifetimes shorter than 10 −10 s.
Discovery of Higgs boson at Large Hadron Collider (LHC) [1,2] confirmed the existence of all the SM particles. Although the SM seems to be a very successful theory but there seems to be no justification for its basic assumptions like the existence of arbitrary Yukawa couplings. There is no profound explanation for the origin of families and their observed mixing structure. Furthermore, charge quantization remains unexplained. The fourth fundamental interaction-gravity-is not included.
Besides these structural defects other flaws of the SM are the following unexplained experimental observations: 1. Neutrino oscillations indicate non zero neutrino masses [3,4,5,6,7,8]. Although neutrino masses can be included via (non-renormalizable) dimension ≥ 5 operators [9] in the effective Lagrangian, there is no way to generate neutrino masses in the renormalizable SM without any extension. This indicates SM is incomplete.

Supersymmetric Grand Unification
2. SM does not explain observed baryon asymmetry of the universe: n B /n γ ∼ 10 −10 .
3. Only 4% of the universe is visible matter, while the remaining 96% is dark matter (DM) and dark energy. SM has no DM candidate. Dark energy is an even greater mystery.
Answers to these questions require physics beyond SM. Therefore in spite of its experimental successes, the SM suffers from a number of limitations, and can't be an ultimate theory of nature. It is only an effective theory at low energies (≤ 200 GeV) of some more profound and complete theory. For instance one can generate tiny neutrino masses through seesaw mechanism by extending the particle content of SM [10]. Depending upon the nature of additional particle one can have Type I [10], II [11] or III [12] seesaw contribution. Type I seesaw requires addition of a gauge singlet right handed ("sterile") neutrinos. Type II and III need Higgs and fermion which are triplet irreducible representations (irreps) (of SU (2) L ) respectively. Several beyond SM approaches like Supersymmetry (Susy), Grand Unified Theories (GUT), Extra Dimension, String Theory, Technicolor Model and 4th generation model etc. have been considered. String theory is a theoretical tool which replaces point particle by one dimensional string whose excitations are the usual point particle fields. Extra dimensions models are based on the assumption that real world is higher dimensional and its extra spatial dimensions are compact. Susy and GUT together provide an attractive framework which has explanation of most of the above mentioned open questions. From now onwards we will focus on this approach.

Supersymmetry
Supersymmetry is the most appealing extension of the SM. It relates the existence of the fermions with bosons and vice versa. Each SM particle has its partner called superpartner which has the same quantum numbers as the particle except the spin which differs by half. Certain relations among the allowed coupling constants ensure

Supersymmetric Grand Unification
the invariance of the actions under transformations with anti-commuting Lorentz spinor parameter of Susy [13]. This leads to conserved supercurrents and Noether supercurrents that are constants of motion. Higgs mass is sensitive to radiative corrections from new physics which vary quadratically with the scale of heavy particles associated with the new physics. Stabilization of Higgs mass was the main motivation to introduce Susy. It stabilizes Higgs mass against radiative corrections by cancelling the loop contribution of a particle with the contribution from its superpartner [14].   16 GeV which has been a strong motivational hint to study Susy.
Both fermions and bosons are placed into an irreducible representation of Susy algebra called supermultiplet. Supersymmetric gauge theories typically use chiral and vector supermultiplets. Interaction and mass terms for the various superfields are described by analytic function (of chiral superfields (Φ)), called superpotential : where M ij is the mass matrix and y ijk is the Yukawa coupling. Scalar potential is computed from the superpotential : Index a run over the adjoint representation of the group so the corresponding term exhibit gauge interactions.
Clearly, Susy is broken in nature because none of the Susy particles has been observed till date. Even without being exact symmetry, Susy can solve the hierarchy problem in a elegant way provided Susy is softly broken. 'Soft breaking' means that the symmetry breaking terms are super-renormalizable (i.e. mass or scalar trilinear terms). Soft mass terms are introduced by hand in the Lagrangian (L soft ) to distinguish the mass of Susy particles from their SM partners. Supersymmetric Lagrangian is given by Here L Susy (φ, ψ, A µ , λ, F, D) is globally supersymmetric action coupling matter and gauge fields to each other and their superpartners. The generic form is: (1.6) We follow notation of [15,16] (see these Refs. for reviews). L soft (φ, λ) is the Susy violating part having the generic form of gaugino mass, scalar bilinear and trilinear terms: Note that all the parameters (M , m φ , A 0 ) in L soft have mass dimensions. If one consider the different soft masses and trilinear couplings for the MSSM scalars and gaugino masses then one would end up with 105 free parameters [17] and this scenario is called unconstrained MSSM. However it is known that flavour violation in the soft breaking terms must be nearly absent to avoid disastrous levels of flavour changing interactions not observed at low energy [18]. There are various theoretical mechanisms that may explain Susy breaking like gravity [19,20], gauge [21] and anomaly [22,23] [24]. The superpotential of the MSSM is : Higgs Boson H u HiggsinoH u H dHd Here Q, L,ū,d,ē are the chiral superfields introduced in the generate mass for the up and down type fermions. The first three terms are the familiar SM Yukawa interactions while the last term is the Higgs mixing or µ term. Besides these terms in the superpotential the following baryon number (B) and lepton number (L) violating terms are also allowed by gauge invariance (but are excluded from the MSSM on phenomenological grounds) These operators imply fast proton decay rate unless the couplings are suppressed (e.g. |λ λ | ≤ 10 −24 M Susy 100 GeV 2 ). To forbid these d = 4 operators, a discrete symmetry known as R-parity is introduced : Here s is the spin of the particle. SM particles have R=1 and Susy particles have -1 value. Exact conservation of R-parity require each vertex must have even number of R = −1 particle, so W ∆B/∆L=1 is not allowed. Further, the "lightest supersymmetric particle (LSP)" is stable by R-parity conservation and proves a suitable "weakly interacting massive particle (WIMP)" dark matter candidate. This is a crucial phenomenological virtue of R-parity conserving MSSM.

Grand Unified Theories
The quest for unification started back in the 19 th century with the unification of electric and magnetic forces as electromagnetic force by Clerk Maxwell. Unification of weak and electromagnetic interaction is successfully achieved in the SM but leaves many unanswered questions. Unification of all the known interactions except gravity is known as Grand Unified Theory. It offers a framework to solve many of the shortcomings of the SM. The first evidence for physics beyond SM came with discovery of neutrino masses implied by neutrino oscillation data [3,4,5,6,7,8] which require theoretical explanation. The seesaw mechanism [10] is the most appealing mechanism for explaining smallness of neutrino masses although simply tuning of Yukawa couplings is also still viable. It relates smallness of neutrino masses with the existence of heavy particles. It provides some hint of high scale physics. Essentially the d=5 neutrino mass or Weinberg operator [9] and its higher dimension generalization are generated when heavy right handed neutrino (ν(1, 1, 0)) are integrated out (Type I) or a heavy triplet scalar mass suppress a neutrino mass inducing VEV (Type II) and so on. Nearly exact unification of the RG evolved three MSSM gauge couplings at high scale ∼ 10 16 GeV [25,26] is strongest motivation to study GUTs.
All these clues reinforced the proposal [27,28] to search for the larger gauge symmetry of the nature represented by some higher group whose effective theory is the SM.
The basic idea is to unify gauge as well as matter content to reduce arbitrariness of SM. G SM ⊂ G GU T , so GUT group should be of rank ≥ 4.
In 1974, Pati-Salam proposed the first-ever GUT based upon G P S ≡ SU (4) × SU (2) L × SU (2) R gauge group [27]. G P S contains the left-right symmetric gauge (LR) symmetric models offer an appealing understanding of the origin of parity violation in the SM. LR symmetry requires existence of the right handed neutrino.
If the Majorana mass ofν is large a small neutrino mass is generated through seesaw mechanism [10]. SM fermion andν are embedded in the LR models as where L, Q,ē,d,ū are written with their SM quantum numbers and the indices µ, α,α refer to SU (4), SU (2) L and SU (2) R respectively. 4-plet of SU(4) treats lepton as a fourth color. Higgs triplets (∆ L (1, 3, 1) and ∆ R (1, 1, 3)) or doublets (χ L (1, 2, 1)⊕χ R (1, 1, 2)) and bidoublet (φ(1, 2, 2)) can implement symmetry breaking W L and W R are gauge bosons corresponding to SU (2) L and SU (2) R respectively with m W R >> m W L as these are missing experimental signature. Electric charge is given by Although G P S unified matter content but still we have 3 gauge couplings, therefore no reduction in number of gauge parameters.  (5) : shows that E 6 is a "maximal" GUT gauge group [34]. This thesis is based upon a successful supersymmetric SO(10) model so we will elaborate SO(10) properties in detail in the following section.
Antisymmetric generators in the fundamental 10-plet representation are: here square bracket represents antisymmetrization and these generators obey algebra The fundamental (10-plet) representation H i transforms as : Taking tensor product of the fundamental representation one can form higher dimensional symmetric or anti-symmetric representation. For example 45, 120, 210, 54 are (2, 3, 4) index antisymmetric and two index symmetric traceless vector representations respectively. 126 (Σ) is self-dual 5 index anti-symmetric representation which requires special attention because it plays a crucial role in SO(10) model building specially for neutrino masses.
Similarly one can project out the anti self-dual 126 : Apart from tensor representations, orthogonal groups have spinor representations generated using Clifford algebra of the 2 N (N=5 for SO(10)) dimensional Γ i matrices: From these generators one constructs : The generators Σ ij obey the SO(10) commutation algebra (see Eq. (1.17)). The explicit form of Γ and hence Σ matrices can be found in [35,36]. Spinor representation of SO(2N) is 2 N dimensional (32 dim for SO (10)) which transform as Irreducible 16(ψ) dimensional spinor representation is constructed using projectors: By taking direct products with tensors, spinor representation lead to an additional class of ("double valued") representations.
We strictly follow the notations of Ref. [36]  GeV.
• It is a natural home for Type I and Type II seesaw mechanism which generate   neutrino masses in the milli-eV range, via high scale B-L breaking, without any tuning of Yukawa couplings as is required when the B-L breaking scale is small. This follows since minimal Susy SO(10) embeds the minimal Susy LR models [37,38,39,40,41] which have high scale breaking of B-L symmetry.
Type II seesaw dominated models relate atmospheric neutrino mixing angle with b − τ unification [43].
• Another important aspect of SO(10) gauge group is that M-parity ((−1) 3(B−L) ) is effectively part of SO(10) gauge symmetry since U (1) B−L ⊂ SO (10). It can be preserved [39,40,44] till low energy with suitable choice of VEV of Higgs field. Using only B-L even VEVs, R/M-parity preservation ensures stable LSP which can act as a cold dark matter candidate.
• Observed baryon asymmetry of the universe can be understood via leptogenesis which explains baryogenesis through sphaleron processing of a lepton asymmetry created in L and CP violating decays of heavy neutrino.

Model Building
As mentioned spinor representation of an orthogonal group provides special motivation to study SO(10) GUT. 16 These breaking chains proceed via different Higgs sectors. As clear from the G P S and SU (5) (10) models [43,53]. MSGUT, a fully specified theory with only 26 real (hard) parameter failed [54,55,56] to fit realistic fermion mass mixing data because Type I seesaw contribution which dominates over Type II seesaw yields too small neutrino masses.
Faced with this impasse, Aulakh and Garg [55,57] investigated the role of 120 plet (which can couple to matter bilinear) in the context of its direct contribution to fermion masses. Earlier the 120-plet was considered [56,58,59]

Thesis Outline
This thesis is a report on development of a realistic Susy SO (10)  • In Ref. [57] tree level sparticle spectrum were used. One loop corrections to sparticle masses can be large specially for the minisplit Susy spectra with large A 0 , µ and M A parameters found in [57]. We incorporate these corrections in the search program.
• Since the NMSGUT generates neutrino masses from B-L violating VEVs and the GUT scale slepton soft masses and trilinear couplings are renormalized by loop corrections and right handed neutrino thresholds one expects [61,62,63] significant lepton flavour violation (LFV) in the benchmark observable like l i → l j γ rates. We calculate these and other lepton sector predictions of the NMSGUT.
• One expects the soft Susy parameters to obey GUT relations at some high scale (e.g. Planck scale or string scale) which need not coincide with the MSSM coupling unification scale M 0 X = 10 16.33 GeV. Thus the RG equations predicting flow of NMSGUT couplings between these scales should be calculated and used to improve the estimate of plausible soft parameter values at M 0 X . We calculated the complete two loop NMSGUT RG equations for soft and hard parameters and quote the hard parameter equations in the thesis.
The soft parameter RG equations are available in [64,65].
• The successful fitting of the fermion hierarchy in the NMSGUT naturally motivates attempts at flavour unification which generate the successful NMSGUT and ultimately MSSM fermion couplings dynamically. We have proposed a novel dynamical scenario implementing this idea using the experience gained from unifying the MSSM fermion hierarchy in the NMSGUT.
Our aim is to check the model compatibility with experimental data. First of all successful GUT should be able to fit the SM fermion mass mixing data (m q,l , θ CKM 12,23,13 , δ CKM , θ PMNS 12,23,13 and δ PMNS , ∆m 2 ν ). Proton decay is a peculiarity of GUT so it becomes a fundamental test. In the lepton sector the experimental upper bound [66,67,68] for the branching ratio of lepton flavour violating decays µ → eγ, τ → µγ, etc. and the muon anomaly [69] require thorough investigation of how these limits constrain the parameter space of the model. We will discuss all these issues chapterwise.
In Chapter 2 we review the structure of the model, in particular, fermion mass generation, effect of superheavy thresholds on gauge couplings and FORTRAN fitting program algorithm. Tree level Susy spectrum is presented in the Appendix. In Chapter 3 a generic mechanism is introduced to suppress fast d=5, baryon decay in SO(10) GUT with an example solution. Appendix contains detailed formulae for Higgs renormalization factors. Loop corrected Susy spectrum is presented in the Chapter 4 with approximate formulae which clarify the dominant contributions and fits incorporating these loop corrections. Chapter 5 is devoted to the lepton sector phenomenological implications of the model which includes ∆F =1 LFV processes, ∆F =0 (a µ ) and calculation of leptogenesis parameters. In Chapter 6, we present the SO(10) renormalization group equations for the NMSGUT soft and hard parameters. In Chapter 7, dynamical Yukawa generation in SO(10) GUTs extended with the family group O(N g ) is discussed. In Chapter 8, we summarize our work and conclusions. We also indicate avenues for further research.

Chapter 2
New Minimal Supersymmetric SO(10) Grand Unified Theory
All the SM fermions along with the right handed neutrino are accommodated in three copies (ψ A ) of 16-plet. 126, 126, 210 Higgs irreps participate in spontaneous symmetry breaking (SSB) from Susy SO(10) to MSSM in steps or at once and are therefore called adjoint type Higgs multiplets (AM). 10, 126, 120 Higgs irreps couple to matter bilinear to generate fermion masses and are hence called [70] fermion mass (FM) type Higgs multiplets. 120-plet has no MSSM singlet so it does not participate in symmetry breaking. Use of 126 irrep in the GUT scale SSB offers automatic implementation of high scale Type I and Type II seesaw. 126 is introduced to preserve Susy in the GUT scale SSB which exhibits crucial R-parity conservation ( only B-L=2 even fields have VEVs). 10 and 120-plet are mainly responsible for generating charged fermion masses and small Yukawa coupling of 126 is crucial for viable neutrino masses. The heavy right handed neutrino in range 10 8−12 GeV (compatible with leptogenesis) and milli-eV neutrino masses as required by neutrino oscillations, through seesaw mechanism are achievable. Thus this model is capable of fitting known SM fermion mass mixing data.
The structure of the theory includes its mass spectra, RG evolution, effective MSSM, B violation effective superpotential, threshold effects, fermion fitting etc. and has already been elucidated [70,48,36,54,55,57]. Extensive computer codes were developed, incorporating the NMSGUT formulae to search the GUT parameter space for viable parameter sets [71]. In this chapter we review the structural features and predictions of the model. We include the description of how the GUT threshold corrections modify MSSM Yukawas that are incorporated in the NMSGUT code.
The actual threshold corrections are given in the next chapter.

Spontaneous Symmetry Breaking
The superpotential of theory which involves Yukawa couplings for Higgs and matter fermions is given by Here h, f and g are Yukawa couplings of 10, 126, 120 Higgs. These are complex symmetric (h, f ) and anti-symmetric (g) matrices in flavour space. We can diagonalize one out of these by performing U(3) rotations in the flavour space since the kinetic terms of the three 16-plets are invariants under U(3). These Yukawas contribute 21 real parameters (real diagonal h(3)+ complex symmetric f (12) + complex antisymmetric g (6)). In addition to these, superpotential has trilinear couplings (λ, η, γ, γ, k, ρ, ζ,ζ) and masses (M H , M, m Θ , m) which contribute 24 parameters. In total model has 45 real parameters out of which 2 can be fixed by using fine tuning condition for Higgs mass (which we will explain later) and 5 phases can be removed by redefining Higgs fields (therefore we choose real (γ, γ, m Θ , m, M )). M is determined from λ, η, m, ξ (M = ξηm λ ) and further ξ parameter is determined from x (solution of Eq. 2.5). We are left with 37 parameters. Although this seems a lot, it is minimal in comparison to any other SO(10) GUT which provides realistic fermion mass mixing data and experiment compatible B-decay rates. The GUT scale (SM neutral) VEVs that break the gauge symmetry down to the SM symmetry (in the notation of [36]) are As a function of these VEVs the superpotential becomes W = m(p 2 + 3a 2 + 6ω 2 ) + 2λ(a 3 + 3pω 2 + 6aω 2 ) + (M + η(p + 3a − 6ω))σσ (2. 3) It is sufficient to calculate F terms from the above superpotential and investigate the conditions for them to vanish. The VEVs of 210 do not contribute to any D term leaving only D B−L ∼ (|σ| 2 − |σ| 2 ) contribution to the D-term potential.
Note that Arg(σ)+Arg(σ) is B-L invariants while Arg(σ)−Arg(σ) can be set to zero by a B-L transformation. Thus effectively σ =σ. Then F p,a,ω = 0 and vanishing of F σ,σ requires [70,48,72] : where ξ = λM ηm . For each value of ξ, three solutions of x are available. The complex parameter x is used for systematic survey of parameter space of the model [70,54,55] since its variation directly affects the VEVs and thus the masses in the theory and each value of x fixes a unique ξ, whereas solving (Eq. 2.5) for x given ξ requires checking three solutions separately.

Superheavy Spectrum
Hypercharge is given by so that one finds the beautiful LR symmetric electromagnetic charge formula : PS to the MSSM decomposition can be easily achieved using SU(4) decomposition (µ =μ + 4,μ = 1, 2, 3) and hypercharge formulae given above. Under the SM gauge group (SU (3) C × SU (2) L × U (1) Y ) the 10 plet decomposes as :  [70,57]. The decomposition of SO(10) in terms of its "Pati-Salam" labels (i.e. the maximal subgroup SU (4)×SU (2) R ×SU (2) L ) provided a translation manual [36] from SO(10) to unitary group labels. Using this technology all the invariants of the superpotential are decomposed into PS labels. Decomposition of invariants corresponding to 10 + 126 + 126 + 210 are given in [70,36] and the ones involving 120-plet are given in [57]. To illustrate we give decomposition of one invariant ηφΣΣ The supermultiplet masses are determined from this decomposition by using sym-  2 3 ] multiplets that belong to this category. We will give explicit form of h mass matrix when we will discuss the emergence of effective MSSM Higgs.
3. Mixed chiral gauge are the multiplet which mixes among themselves and also with gauge particles. These multiplets are named as E Rows and columns are labelled by ( µ4α1 ,λμ α1 ,Θ 41 σα ). 5 th row and column are gaugino contributions.
As a check, SU(5) irreps mass spectra is generated from above spectra using special direction p = a = ±ω (2.30) of VEVs [57]. The superheavy fields play a crucial role as they provide threshold corrections to the unification scale, gauge couplings and Higgs fields etc. that we will explain in the subsequent sections and chapters. The complete GUT scale spectrum and couplings of NMSGUT have been given in [70,36,57,72,73].

RG Analysis
After symmetry breaking large number of fields get mass of order of GUT scale which give threshold correction to the gauge couplings [75,74]. In [36] effect of superheavy thresholds on α G (M X ), sin 2 θ W and M X is investigated using Weinberg and Hall approach. Alternatively [57] one can predict α 3 (M Z ) instead of sin 2 θ W . Effect of these superheavy thresholds on α 3 (M Z ), α −1 G (M X ) and log 10 M X (∆ G , ∆ 3 , ∆ X ) is calculated in [76] using precisely measured value of sin 2 θ W (M Z ). It was shown that although there are large number of superheavy fields their spectrum spread around M 0 X gives contribution of both signs so that the sums can be reasonable modification of the tree level results. This analysis has disproved the conjecture that large number of fields will give huge corrections to the untameable observables making Susy SO(10) unification meaningless [77]. In [36,57,76] mass of the lightest vector particle mediating proton decay (X[3, 2, ) and GUT (α G ) is given by [75,74] 1 here second and third term represent one-loop and two-loop gauge running 3) are the one-loop and two-loop gauge beta function coefficients : , M X and α G (M X ). The threshold correction [70,54] formulae are : for multiplets with mass M and λ i are the leading contributions of the superheavy thresholds [74,70]. threshold corrections are estimated. Two expressions of M X = M 0 X 10 ∆ X and M X = m λ X = |m/λ|g 4|ã +w| 2 + 2|p +ω| 2 determine m parameter : is the threshold corrected SO(10) gauge coupling. Theory should remain perturbative after including threshold effects and mass of the proton decay mediating gauge boson should not be lowered too much so as not to violate experimental bounds.
These requirements constrain the corrections as : All the superheavy VEVs and hence masses are determined in terms of parameter x. Solution of this variable depend upon the superpotential parameters through ξ = λM ηm parameter. Threshold corrections are not very sensitive to λ, η, γ,γ as shown by scanning the parameter space. Systematic survey of behavior of these unification stability monitoring parameters versus x and ξ is shown in [70,54,55].

Fermion Masses
As mentioned the Yukawa couplings of the pairs of bidoublets contained in the Higgs set 10 ⊕ 126 ⊕ 120 give rise to charged fermion masses. Therefore Yukawa coupling matrices y l , y u , y d and y ν at high scale (M X ) are predicted in terms of the Y 10 AB = h AB , Y 126 AB = f AB and Y 120 AB = g AB which specify the Yukawa couplings of the 10, 126, 120 to 16.16. For explicit relations of the fermion Dirac masses decomposition of 16 · 16 · (10 ⊕ 120 ⊕ 126) is required, which can be found in [70,57]. For illustration we present decomposition of 16 · 16 · 10 in terms of the PS and the SM labelles: From these invariants one obtains the Yukawa couplings just by replacing the MSSM Higgs by corresponding Higgs fractions as By multiplying these Yukawas with electroweak Majorana mass term and Dirac mass term (which mixes the left and right neutrinos) give rise to Type I seesaw contribution by eliminatingν A In addition to this there is another contribution to the neutrino mass known as Type II seesaw. The Type II neutrino mass is where Ō − is SU (2) L triplet VEV (∈ (10, 3, 1)) of 126 whose computation requires inspection of relevant terms in the superpotential : and M O = 2(M + η(3a − p)) is mass of superheavy O multiplet [57].

Viable Parameter Space Search
With appropriate formulae in hand, next task is to check the compatibility of the model with experimental data. FORTRAN and Mathematica codes were developed [71] for fermion fitting along with viable unification, electroweak symmetry breaking, including Susy threshold corrections and for B-decay calculations. χ 2 analysis is performed to fit SM mass-mixing data at two scales-GUT scale (M 0 X ) and electroweak scale (M Z ). GUT scale fitting is based upon the random searches of 37 model parameters (listed in Section 2.2.1 ) and x parameter (which is chosen complex and later the phase of λ is fixed to remove this freedom). M Z scale calculations involve Susy threshold corrections which require estimation of Susy spectra. Soft Susy breaking Lagrangian is :    2. Subroutine AMOEBA1/AMOEBA2-There are various public available packages to find minima of non-linear functions. Downhill simplex method of Nelder and Mead [78] has been used to find the minima of highly non-linear function in a multi-dimensional space because it involves function evaluations only. It is based upon the n-simplex having n+1 vertices in n-dimensional space. The AMOEBA subroutine [78] [70,57]. It contains penalties to get ∆ G , ∆ 3 and ∆ X within the required range (see Eq. 2.39). It provides ∆ G , ∆ 3 , ∆ X and m parameters as output.

Function FUNKFERM-It calculates χ 2
X value by comparing model predicted values with target (which is run up SM experimental mass mixing data). For a given set of values of superpotential parameters and Yukawa couplings, fermion Yukawas are calculated using Eq. (2.44). Then it calculates the eigenvalues and mixing angles in quark and lepton sector. After that fitting of SM mass mixing data (18 parameters) is done. δO i is an estimate of the uncertainty in the GUT scale value based upon extrapolation of uncertainty at the measured scale (see e.g. [79]).Ō and O are experimental and model predicted values respectively. Here the sum (i) run over the Yukawa couplings (9), quark and lepton mixing angle, CKM phase and neutrino mass square differences. This function returns argument funk (χ 2 X ). [80,81] to run the hard (Yukawa and gauge coupling) and soft parameters from GUT scale to electroweak scale and vice versa.

Subroutine PMX2PS uses two-loop MSSM renormalization group equations
where t 1,2 are tadpoles of the effective potential, calculated using a SPheno subroutine based on the formulae of [83]. These can be extrapolated back to M 0 X to find µ and B at M 0 X since the RGEs of the other soft masses are independent of these. Both µ and B are assumed real and positive. M A (pseudo-scalar mass) is calculated using the above one-loop corrected value of B parameter Tree level spectrum is again calculated using updated value of µ and B parameter. Susy threshold corrections to the Yukawa couplings are calculated. Then using off-diagonal run down values of Yukawas, tree level spectra is calculated to verify that ignoring generation mixing does not make much difference. Aim is to find the suitable set of soft parameters which give appropriate corrections to the NMSGUT run down Yukawas to fit them with the SM data. Following penalties are imposed upon soft parameters [57] : Higgs mass measurements are available since December 2011, in [57] SM Higgs was required to be heavier than 114 GeV (LEP limit) and the Bino lighter than the lightest sfermion. Susy threshold corrected run down Yukawas values are compared with SM Yukawa via a χ 2 Z .
Here y MSSM i are threshold corrected Yukawas. This function returns χ 2 Z . AMOEBA2 subroutine uses this function and calculates its value at each vertex of the simplex.
Program starts with the GUT scale searches. It requires target extrapolated SM data. Using two loop RGEs of MSSM [80,81], central fermion experimental data (Yukawa + neutrino mass difference and (quark, lepton sector) mixing data) is extrapolated to one loop unification scale M 0 X =10 16.33 GeV ignoring right handed neutrino thresholds and assuming normal hierarchy for left handed neutrino masses (neutrino mass splitting is calculated from extrapolated coefficient of d=5 operator [9,84]). At M 0 X canonical parameters are extracted which serves as target for model calculations. This target file and two other input files having random set of NMSGUT superpotential couplings and SUGRY-NUHM parameters are provided. To start with, the value of variable bestfunk (whose role will be explained later) is fixed along with the initialization of many other parameters. In a downhill simplex method one needs to provide the initial vertex of simplex around which it starts searching in the parameter space. Random input provided is that initial vertex. With slight random changes from this point all the vertices of the simplex are generated. At each vertex of simplex GUTTHRESH calculations and function FUNKFERM is calculated. AMOEBA1 subroutine also calls GUTTHRESH and function FUNKFERM for viable unification and function evaluations and it compares the value of funk at each vertex and select the one with minimum funk. If calculated funk at that point is less than or equal to initial bestfunk value then it replaces that parameter set in the input file. In the next iteration it start searching around that point. If the lowest funk value is more than the initial bestfunk then in the next iteration program starts searching around the old point, but as the vertices of simplex are randomly generated so simplex will be different from the earlier one.

Large A 0 and µ Parameter
If only 10 and 120-plet fit charged fermion masses with 126 Yukawa coupling (f AB ) chosen tiny the NMSGUT can only generate y d,s (M 0 X ) values which are smaller by a factor of 3-5 than the extrapolated values of SM Yukawas at M 0 X . Therefore SM down and strange quark Yukawa require lowering by a factor of 5 to fit these with run down values of Yukawas. This lowering is achieved by large tan β (preferred by SO(10) GUTs for third generation Yukawa unification ) driven threshold corrections which require specified soft Susy breaking parameters [57]. Gluino contribution is a dominant one loop correction which is proportional to Susy breaking parameter µ, so large µ parameter provides significant Susy threshold corrections to y d and y s to match it with GUT renormalized value down to M Z scale. For third generation, bottom quark, slight raising is required, so Susy threshold corrections should not change it too much. This cancellation can be achieved by large A 0 (soft trilinear couplings) parameter. After the Higgs discovery large A 0 is favoured in Susy-GUTs for Higgs mass of 126 GeV [86]. Heavy third s-generation and large A 0 raise the tree level Higgs mass from 91 GeV to 126 GeV.

Bino LSP and Light Smuon
Solutions found have pure Bino LSP and the light charginos are pure Winos. A striking feature is that there are solutions with next to LSP (NLSP) as light smuon which can generate a significant corrections to the muon g-2 and thus remove the observed anomaly a µ ∼ 10 −9 . Moreover a light smuon provides DM co-annihilation channel to get the acceptable relic density. tions. Solution sets presented in [57] have large proton decay rates of order of 10 −27 yrs −1 . However optimized search with respect to baryon decay, including GUT scale threshold correction will be discussed in the next chapter.

Discussion
Among the superpotential parameters h 33 (largest of all elements of Yukawash, f , g ) is crucial for fitting fermion mass-mixing data, it alone can fit third generation within 5% error. Next relevant parameter is g 23 . f AB is irrelevant for charge fermion masses and its small value is crucial for neutrino masses as it enhances Type I and suppress Type II seesaw contribution. Fits yield right handed neutrino mass in 10 8 -10 13 GeV range which is compatible with leptogenesis. Heavy right handed neutrino and small f AB generates neutrino masses of order of meV. Fermion [87,88,89,90,91] based upon the 10 − 120-plet FM Higgs system. In most of solutions, superheavy thresholds raise unification scale M X closer to M P . Superheavy spectrum varies from 10 15 GeV to 10 19 GeV.
In addition to the superpotential couplings, tan β is also a crucial parameter for realistic fermion mass generation by the GUT Yukawas. For third generation Yukawa unification (y t ≈ y b ≈ y τ ), NMSGUT prefer large tan β (∼ 50). As discussed in [57] large tan β driven Susy threshold corrections provides a route for successful fitting of fermion masses at M Z . Fermion fitting and experiment compatible MSSM Higgs mass require decoupled/mini-split Susy spectrum [86,92] : |M | H,H ∼ 100 TeV, heavy CP-odd Higgs M A , large µ and trilinear coupling A 0 , light gaugino (< 1 TeV) with pure Bino as LSP, normal s-hierarchy with mf 3 ∼ 50 TeV and degenerate first two generations. With this kind of soft spectrum gaugino mass deviate from the Susy-GUT ratio 1:2:7 operative at one loop level. Fits prefer large A 0 and µ parameter ∼ 100 TeV which suppress flavour changing neutral current processes and avoid problem with charge and color breaking/unbounded from below vacua [93].
Besides realistic B-decay rates, other improvements in the fitting programs which are part of the thesis are loop corrected Susy spectrum, inclusion of right handed neutrino thresholds for LFV estimation and to consider the effect of soft parameter RG running from M P to M 0 X . This will be discussed in Chapter 4, 5 and 6 respectively.

Appendix: MSSM Tree Level Spectrum
The MSSM is an effective theory of NMSGUT. Its particle content is given in the

Sparticle Masses
Squark and slepton mass term in the Lagrangian is given by wheref representsũ,d,l andν. Mass matrices are : Here m 2 L , m 2Q , m 2 u , m 2 d and m 2 l are soft mass parameters. A f , Y f and m f are the soft trilinear couplings, fermion Yukawa couplings and masses. Tree level sparticle masses are obtained by diagonalizing above (Hermitian) mass matrices via unitary where Λ 2 u , Λ 2 d , Λ 2 e and Λ 2 ν are positive definite mass square parameter.

Higgs Masses
Physical Higgs particles of the MSSM are : CP-odd neutral A, CP-even h and H, and charged Higgs H ± . Masses of these particles can be computed from M A ( and tan β) which itself is determined from B parameter : B is calculated using EW symmetry breaking conditions.

Gaugino and Higgsino Masses
Gauginos and Higgsino mix to form chargino and neutralino eigenstates.
HereB,W 3 ,H d andH u are Bino, Wino and Higgs components. Neutralino and chargino mass matrices are obtained from L soft , L int (matter-gauge-Higgs) and superpotential : Matrices Mχ± and Mχ0 are diagonalized to get masses : Here U − , U + and N are unitary matrices and Λ C , Λ N are positive definite masses.

Chapter 3
Baryon Decay and GUT Scale Threshold Corrections

Introduction
GUTs place quarks and leptons in common irreducible representations. Quarks can transform into leptons by exchanging gauge and Higgs leptoquarks, thus GUTs predict baryon violating processes such as proton decay e.g. (p → π 0 e + ). However non observation of proton decay has put a stringent lower limit [94] on its life time and this contradicts the simplest models. Hence one must investigate more refined models, among which the most appealing are supersymmetric GUTs. In Susy GUTs B and L are violated by the exchange of superheavy color triplets. In SO(10) GUTs, B-L is preserved by all the vertices since it is part of gauge symmetry. Gauge mediated dimension 6 operator proton decay rate is estimated as Here M G is mass of superheavy gauge boson. Even with M G ∼ 10 16.25 GeV this gives τ p ∼ 10 36 yrs. Threshold corrections can raise the unification scale near to the Planck scale so this contribution can be even more strongly suppressed. R-parity forbids fast d=4 baryon number violating operators. The remaining contribution is d=5 operators (involving two fermion and two scalars exchanging triplet Higgsino).
Scalars are converted into fermions via gaugino or higgsino dressing [95,96,97]. In Susy GUTs dimension 5 operators thus give leading contribution to proton decay [98,99] as these are suppressed only by where M H is the mass of triplet Higgsino. Experimental limits put the stringent constraint on the model parameters [100]. In this chapter we will investigate d=5 operator baryon decay and uncover a generic and natural mechanism to suppress these.

SO(10) Yukawa interaction include many superheavy Higgs-fermion interactions.
By using the superpotential equations of motion for the heavy fields (just as we eliminate W ± , Z to get the Fermi effective theory of weak interactions from the SM), we obtain two types of d=5 operator which lead to proton decay. The effective superpotential has generic form : where the first and second term represent contribution of SU (2) L doublets and singlets, therefore are called LLLL and RRRR operator respectively. In NMSGUT 10, multiplets that couple to fermion and violate B+L. From PS decomposition of the relevant superpotential invariants L ABCD and R ABCD are obtained [36,70,57]: and

] multiplets mass matrix and
where α, β and γ are the colour indices and SU (2) indices are suppressed. Different Susy GUTs will furnish different coefficient arrays L ABCD , R ABCD and the task is to convert this information together with the assumptions for the soft Susy breaking terms (till the superpartners are discovered) into predictions for the baryon decay rate into different channels.

Baryon Decay Rate
We calculate proton decay rates due to d=5 (∆B = ±1) operators using formulas of [95]. As a check we compare the result calculated by using the formalism of [96] separately and verify they are the same. The calculation of baryon decay rates is done in steps as follows.
• Firstly one has to renormalize the 2 × 3 4 = 162 component arrays L ABCD and R ABCD from M X down to M S ∼ M Z using the MSSM RG equations supplemented by the RGEs for the coefficients L ABCD ,R ABCD . The RGE for but also diagonalize the 6 × 6 charged sfermion mass squared matrices given in the Appendix of Chapter 2. These are written in the weak basis (denoted by hat onF ) or the mass diagonal basis where Λ 2F is diagonal positive definite. Similarly one must also diagonalize the • The superpotential will yield 2 fermion, 2 scalar terms in the Lagrangian : So from W ∆B=±1 , one obtain the following d=5 terms: We have mentioned the coefficients corresponding to channel: proton decays into the charged lepton. Contribution for the other decay channels can be found in [95]. Here a, b and c are SU(3) fundamental indices. Subscript L and R represent fermion chirality. The coefficients C(ff f f ) are defined in terms of L ABCD and R ABCD as follows: Here the indices M, N, i, j, k run from 1 to 3 andŨf , U f , V f are the unitary matrices which diagonalize scalars and fermions respectively.
The gluino mass matrix does not need diagonalization so one can evaluate the diagram in Fig 3.1 to give a contribution to L ef f Here the function H(ũ N ,d M ) arises from the standard loop integration, having form : and the arguments of loop function H are Susy particle mass squared ratios: • After calculating one-loop (gluino, neutralino and chargino) dressing diagrams, effective Lagrangian containing four-fermi interaction terms relevant to the proton decay into charged lepton channels is given by: C coefficients can be found in [95].
• Finally matrix elements of the four Fermi operators involving quark and lepton fields must be evaluated between the baryon and meson initial and final states to obtain the amplitude for baryon decay in any channel (e.g. p−→ e + π 0 ). Chiral Lagrangian technique [101] is used to convert the effective quark Lagrangian to the effective hadronic Lagrangian. Then partial decay widths of the nucleon are given as where m i and m j are the masses of baryon and meson respectively. f π is the pion decay constant having value 139 MeV. A factor of A Long ≈ .22 is used to take into account the renormalization from M Z to 1 GeV. The explicit expressions for A ijk L,R (defined in terms ofC coefficients) can be found in [95] As shown in [57] using the tree level Yukawas successful fitting of fermion mass mixing data is obtained in the NMSGUT but proton life time is 6-7 order of magnitude smaller than the experimental limit. In the literature particular textures of Yukawa couplings and discrete symmetries are considered to suppress fast B-decay rates [58].
In the next section we will discuss a generic mechanism to suppress fast B-decay rates in Susy-GUTs.

GUT Scale Threshold Corrections
We have computed one-loop GUT scale threshold correction to a Yukawa coupling of matter field due to heavy fields running in self energy loops on lines leading into the Yukawa vertex when the external light Higgs comes from any of the 6 possible components [57] using technique of [102]. These threshold corrections are very significant due to the large Higgs representation used and also play a crucial role in obtaining parameter sets compatible with constraints on B violation.

Formalism
In supersymmetric theories, non-renormalization theorem [103] implies that superpotential couplings are modified only by wave function renormalization. We have calculated the large number of the NMSGUT Yukawa vertices that couple light fermions and Higgs field. To calculate corrections we need to move into the basis where mass matrices of heavy fields are diagonal. We can redefine heavy field Φ to diagonalize the mass term in the superpotential As shown in Fig.3.2 heavy superfields can circulate on any of the three chiral superfields which give wave function renormalization in the kinetic terms : Here α i andᾱ i are the Higgs fractions which describe the contribution of different Higgs fields to light Higgs and are first columns of the unitary matrices which diagonalize the Higgs mass matrix. One needs to define a new basis to write kinetic terms of light matter and Higgs fields in canonical form as : Here As a result the MSSM Yukawa couplings in terms of the tree level (SO(10) determined) Yukawas change as Generic form of correction factor for any chiral field Φ i is (Z = 1 − K) : Here first term is the contribution of coupling of Φ i to gauge field (A µ ) in (10) gauge coupling) and second term is Yukawa . F is symmetric Passarino-Veltman loop function. When both the fields running in the loop are heavy fields then F (m 1 , m 2 ) has the form which reduces to just when one field is light (M B → 0). One should avoid the sum over light index when both the fields running in the loop are Higgs fields.

Explicit Form of Correction Factors
Here To illustrate the correction factor from the JD channel on Higgs line is given by:

Threshold Effects on
As discussed SO(10) Yukawa couplings (h, f , g) and heavy masses M i determine both fermion masses and coefficient of d=5 baryon decay operator As a result the coefficient L ABCD , R ABCD of d=5, ∆B = ±1 decay operator in terms of the Yukawa eigenstate basis transform as

Fits Including Threshold Corrections
Besides d=5, B decay operator coefficients, wavefunction renormalization also modifies the relation between other GUT and MSSM parameters. MSSM µ and B parameters are larger than the same GUT parameters by the factor of (Z H ZH) −1/2 . Scalar soft masses and soft Higgs masses will be modified by a factor of Z −1 f and Z −1 respectively. Trilinear soft parameters will remain same as wavefunction renormalization factors are absorbed by the Yukawa couplings (A = A 0Ỹf ). These thresholds redefine f AB asf This changes the right handed neutrino Majorana masses ((Mν) AB ∼ f ABσ ). Y ν and Higgs field redefinition modify the Type I seesaw formula.   is used in ∆ Susy αs as given in [105]. (3) in the MSSM respectively. ∆ Susy αs can be significant so it changes the allowed range at GUT scale. We considered the following limits for ∆ Susy αs in the search program.
− .0146 < ∆ Susy αs < −0.0102 (3.45) Typically we find M Susy ≈ 10 TeV. Our constraints on the gauge unification parameters are thus : 3.6.1 Example Fit 1. In Table 3.2 we give the values of the NMSGUT superpotential parameters, changes in gauge unification parameters-∆ x , ∆ α 3 , ∆ α G (from GUT and Susy), x parameter, the superheavy spectrum and the mSUGRY-NUHM parameters preferred by the fitting search program. We also give heavy right handed neutrino masses along with Type I and II contribution to light neutrino masses.
2. Table 3 (10) Yukawa couplings therefore Z f , Zf are close to unity since these are determined by the lowered Yukawas. In Table 3.4 (column 2) we show, run down to M Z , values of the fermion masses generated by GUT.
Notice that y d and y s are smaller by a factor of 3 as compared to their SM values. Fermion masses including large tan β driven Susy threshold corrections are given in 4 th column.
3. In Table 3.5 we have run down values of soft Susy masses (including M 2 H,H ) and trilinear couplings. These parameters determine Susy threshold corrections to fermion Yukawas. µ and B are determined by electroweak symmetry breaking conditions (their values at M X were obtained by running backup to M X ). We used the tree level formulae given in the Appendix (of Chapter 2), to calculate Susy spectra. Tables 3.6 and 3.7 show Susy spectra ignoring and including generation mixing. Off-diagonal running changes the spectra marginally. 4. Here the spectrum has same characteristic as shown in [57] and mentioned in the previous chapter: µ, A 0 , B > 100 TeV and normal s-hierarchy. Sometimes we get light smuon having mass 100-200 GeV along with Bino (LSP).
These special solutions are crucial for model phenomenology as they offer a co-annihilation channel to LSP and can predict appropriate contribution to (g − 2) µ . Unoptimized values of important beyond SM (BSM) observables are presented in Table 3.8.

5.
Using the formalism of [95,96] we have calculated the decay rate of proton and neutron to the different channel as shown in Table 3.9. Clearly, B-decay rates are compatible with the experiment. In Table 3.10 we consider the chargino and gluino contribution separately and it shows chargino dominance.
6. Other solution with large value of Mμ are also obtained. They may be found in [65,104].

Conclusions and Outlook
We have computed the one-loop GUT scale threshold corrections [104] to the tree level Yukawas at an SO (10) Table 3.2: NMSGUT superpotential couplings and SUGRY-NUHM soft parameters at M X which accurately fit fermion mass-mixing data respecting RG constraints. Unification parameters and mass spectrum of superheavy and superlight fields are also given.
We have also implemented the effects of Susy spectra on gauge unification parameters α 3 (M Z ), M X and α(M X ). Susy spectra of the fits including superheavy threshold corrections have same characteristics as the one without GUT scale threshold corrections (exhibited in [57]). Thus they have large A 0 , µ ∼ 100 TeV (for realistic fermion mass and mixing data and M h ∼ 125 GeV) parameters and heavy s-particle spectra which seems to be a likely scenario after Higgs discovery. Fits obtained deviate significantly from y b −yτ ys−yµ ≈ 1 which is characteristic of 10 − 120-plet generated fermion fits [87,88,89,90,91].
This mechanism of suppressing fast dimension 5 proton decay rates including GUT scale threshold to light fields, is generic for all realistic Susy GUTs in which the light MSSM Higgs arise from a mixture of GUT Higgs doublets coupled to a large number of superheavy fields. The effect of d=6 B violation operator with one external Higgs line remains to be checked.

Appendix: Higgs Field Correction Factors
In this Appendix, we give correction factors from all type of GUT multiplets to the H andH (see Eq. 3.37) :

Chapter 4
Loop Corrected Susy Spectra

Introduction
The discovery that the Higgs mass is, at around 126 GeV, almost 28% larger than the tree level upper limit in the MSSM [1,2], has emphasized the crucial role of loop corrections in the MSSM [106]. For the precise estimation of Susy particle masses, it is essential to consider the loop effects. Susy threshold corrections to the SM Yukawas are already incorporated in the NMSGUT [57]. The NMSGUT requires GUT scale [104] and Susy threshold corrections [57] to suppress fast B-decay rates and for fermion fitting respectively. In the previous chapter tree level Susy spectrum is presented (however the Higgs mass is one-loop corrected). The next step regarding inclusion of quantum effects is to calculate one loop corrections to the Susy spectrum.

• Squarks and Sleptons
Complete formulae of these self energy calculations are presented in [83]. In the next section we will discuss dominant corrections with explicit expressions.

Dominant Corrections
FORTRAN subroutines implementing formulae of [83] are also available [82]. We have interfaced these subroutines with our FORTRAN code. We calculated the loop corrections to the tree level spectrum presented in the previous chapter (see Tables   3.2

Gluino Mass
Tree level gluino mass is The physical gluino mass is given by the solution of where Σg(M 2 g ) is gluino self-energy : where Q is the renormalization scale (which we take to be M Z ), g 3 and θ q are SU (3) gauge coupling constant and squark mixing angle respectively. One loop correction to gluino mass comes from gluon/gluino and quark(m q )/squark(mq) loops [83]. The loop-function B 0 has the form and 1/ˆ represents infinite contribution. The function B 1 is defined in terms of loop-functions B 0 and A 0 : First term of Eq.(4.4) represents the gluon/gluino contribution. Quark/squark contribution is given by second and third terms. Out of these the third term has negligible contribution since it is proportional to quark masses. First and second term have opposite contribution. Second term involves the loop function A 0 which is proportional to m 2 q , therefore it provides the dominant contribution. Gluino mass gets approximately 30% corrections (see Tables 4.1

Neutralino and Chargino Masses
MSSM gauginos and Higgsinos form chargino and neutralino eigenstates. Neutralino mass matrix including radiative corrections is given by: Here Mχ0 is the tree-level neutralino mass matrix (Eq. 2.69), and the factors Σ 0 L,R,S (p 2 ) are matrix corrections. Self energy has contributions from (quark/squark, lepton/slepton, neutrino/sneutrino), chargino/W-boson, neutralino/Z-boson and gaugino/Higgs loops. Dominant contribution comes from-quark/squark, lepton/slepton, neutrino/sneutrino loops. Third generation quark/squark provide maximum corrections : Here q represents top/bottom quark andq refer to the corresponding scalar. aχ0 j qq k are neutralino-fermion-sfermion couplings and sum over k includes contribution of scalar partners of both left and right handed quarks. Similarly, one-loop chargino mass matrix is as follows : where Mχ+ is the tree-level chargino mass matrix (Eq 2.68). Quark/squark corrections have form : Here q denotes bottom (top) when q is top (bottom). Σ 0,+ R can be obtained from Σ 0,+ L by replacing the couplings. Neutralino/chargino corrections are approximately 7%.

Higgs Mass
The MSSM has two Higgs doublets H 1 and H 2 whose VEVs generate fermion masses, so to start with we have 8 degrees of freedom. W ± and Z bosons eat 3 degrees of freedom and become massive so we are left with 5 degrees of freedom i.e. the neutral Higgs (h, H), charged Higgs H ± , CP odd Higgs A. Higgs soft mass parameters m 2 H 1,2 at M Z determine µ and B : M A (B(tan β + cot β)) is computed from B and then tree level Higgs masses are calculated. Tadpoles need to be calculated to take into account the one loop radiative corrections to the Higgs masses. Vanishing of tadpoles provides : Here Π ZZ and Π AA are self energies of Z-boson and CP odd pseudo-scalar A.
Charged Higgs mass including loop corrections is given by: Remaining two CP-even Higgs mass are determined from the pole of matrix : HereM 2 Z andM 2 A are DR masses. Explicit form of self-energies (Π ss ) can be found in [83].

Squark and Slepton Masses
Tree level squark and slepton masses are calculated from 6 × 6 mass matrices. It is convenient to breakup self energy corrections as 3 × 3 blocks (Πf LfL , Πf LfR , Πf RfR ): Here H 0 n denotes H, h, G 0 and A 0 , H + 1 (H + 2 ) refer to H + (G + ), Y u /Y d is the Yukawa coupling of up/down quarks, the factors D nu and C n are sine/cosine functions of Higgs mixing angles (α, β) [83]. The parameters I u 3 , g and g f (I f 3 − Q f sin 2 θ W ) represent SU(2) quantum number, SU(2) gauge coupling and weak neutral current couping. Self energy Π Higgs-sector f RfR (p 2 ) can be obtained by replacing g u L by g u R and Chargino/neutralino contribution is appreciable for third generation : Couplings f and g are defined as Loop function G is defined in terms of functions B 0 and A 0 .
Chargino/neutalino and Higgs corrections have opposite contribution. One needs to decrease the magnitude of Higgs corrections to get positive first and second generation sparticle masses. The chargino and neutralino masses are proportional to µ parameter. We have softened this contribution via penalty on µ and M A which we will discuss in the next section. In the tree level example spectra presented in the previous chapter M A ∼ 3µ.

Numerical Procedure
Our fitting criteria are the same as discussed in the previous chapters.    of SPheno [82] with our low scale calculation subroutine FUNKTUNE (see flowchart 2.1). We will discuss only the extra penalties imposed and structure of the additional subroutine. Algorithm of LoopmassesMSSM is represented by a flowchart 4.1. It calculates one-loop radiative [83] corrections to the Susy particles but for the Higgs masses two-loop corrections [113,114,115] are also included. We run the MSSM RGEs from M GU T to M Z to get hard and soft parameters at M Z (considered the renormalization matching scale), which are the inputs to LoopmassesMSSM. First of all the subroutine calculates µ and B parameter using electroweak symmetry breaking conditions (EWSB) (Eq. 4.16). Using calculated µ and B tree-level Susy This VEV is used to calculate one-loop and two-loop tadpoles further in the spectrum calculations. µ and B are calculated using two-loop effective potential. This process is repeated until consistent values are achieved. Then the tree level sparticle spectrum is calculated using updated µ and B value.
As the Higgs sector masses are parameterized in terms of CP-odd pseudoscalar Higgs mass (M A ) and tan β, so first of all two loop corrections to pseudoscalar Higgs are calculated and the process is repeated to get the convergent value. Output kont of LoopmassesMSSM is added to χ 2 Z . In other words penalties are imposed to get positive masses.
Large value of M A gives huge corrections to sleptons and squarks masses and can turn these negative. To overcome these obstacles one way is to decouple each Susy particle at its mass threshold [117]. But implementation of this procedure is not trivial in multiple iteration search code. Other possibility is to limit the size of loop correction  Tables 4.11 and 4.18. Since the two-loop corrections are small the spectrum is slightly modified.
The values of µ and B calculated using one-loop EWSB conditions are also provided.

Discussion
We have found the NMSGUT superpotential couplings and mSUGRY-NUHM parameter sets which provide realistic fermion mass-mixing data, B-decay rates respecting experimental limits and consistent loop corrected Susy spectra as shown in Table 4 Target  Parameter   Besides explaining smallness of neutrino masses, another attractive feature of the seesaw mechanism is that it provides a natural mechanism for generating the observed baryon asymmetry of the universe through leptogenesis. Once the MSSM is extended with three right handed neutrinos, the lepton number asymmetry can arise via a lepton number and CP violating out of equilibrium decay of the right handed neutrino. This can be followed by processing of the net lepton number into baryon asymmetry via sphaleron processes at scales near to but higher than M Z [60]. The E821 experiment [120] at Brookhaven National Laboratory observed a significant deviation of the muon g-2 from its SM prediction. This observation provides hint for physics beyond SM . In Susy GUTs extra contribution to muon g-2 than SM will come from loop diagrams involving Susy particles. In this chapter we discuss rare muon decay in extended SM and lepton flavour violation in Susy-GUTs and in particular NMSGUT predictions for mentioned lepton sector observables.

Standard Model Muon Decay
Muon decay (µ → eν eνµ ) in the SM is mediated by W ± gauge bosons. Due to suppression of heavy gauge boson propagation this decay is represented as point like

4-fermion interaction governed by the interaction Lagrangian
The µ → eν eνµ channel accounts for nearly 100 % of the muon decay width.
However many other channels for muon decay width are conceivable in the SM and beyond. For example the diagram 5.1 accounts for the lepton flavour violating decay µ → eγ in the SM supplemented by Majorana mass term for neutrino mass (coded in the Weinberg operator ((Mν) −1 ij L i HL j H). Decay rate and branching ratio (BR) are given by : Here U is the neutrino mixing matrix, ∆m 2 is neutrino mass square difference param-  Table 5.1). One needs to include the contribution of new physics (beyond SM) to get BR comparable to the experimental limit.

LFV in Supersymmetric GUTs
The SM extensions predict lepton flavour violation. In the unified scenario LFV has been studied by Barbieri   The main consequence of the seesaw mechanism in supersymmetric theories is the violation of lepton number in the scalar sector leading to rare lepton flavour violating decays. Many groups have discussed this issue considering different frameworks [63,121,122,123,124,125,126]. However NMSGUT also predicts neutrino Yukawa coupling like the Yukawa couplings of other SM fermions. LFV ultimately originates from the neutrino Yukawa coupling, because neutrino Yukawa couplings in general are not diagonal in the basis in which charged lepton Yukawa coupling and right-handed neutrino mass matrix are diagonal. The off-diagonal neutrino Yukawa couplings will give rise to LFV in the soft slepton masses through the RGEs [63,121].
Lepton flavour violation will be generated through the RGE even if the supersymmetry breaking mechanism at the high scale conserves flavour as it does in mSUGRY.
Leptonic superpotential of the MSSM + three right handed neutrino is given by :- Here where theL,ẽ R andν R are the leptons Susy partners (sleptons). The parameters (m 2 L , m 2 ν ) and (A e , A ν ) are slepton's soft mass matrices squared and trilinear couplings. Additional contribution by neutrino Yukawa couplings to the one loop RGE of the slepton masses (m 2 L ) is given by [63] 1 The leading contribution to the off-diagonal entries of the slepton mass squared matrix can be estimated as [63] ∆(m 2 L ) ij The flavour violation in the slepton sector contributes through the one loop diagrams to charged flavour violating processes such as rare muon and tau decay (µ → eγ, τ → µγ, τ → e, e, e etc.) and yields BR : Where α is fine structure constant and M S is Susy particles mass scale. Feynman Figure 5.2: Feynman diagrams representing the contribution of neutralino − slepton and chargino − sneutrino loop for the process l i → l j γ. The photon line is to be attached on each of the charged particle lines.
diagrams contributing to these processes are shown in Fig. 5.2. The upper bounds on BR of these processes may constrain very strictly the elements of soft mass matrices. It is important to estimate these LFV rates from NMSGUT on the basis of fits of the fermion data. To estimate LFV, one needs to consider the effect of right handed neutrino evolution upto their mass scale which we will discuss in the following section.

NMSGUT LFV Predictions
In NMSGUT using the formulae of Eq. need to be integrated out below their mass scale. We shift to the right handed neutrino and charged lepton diagonal basis by redefining the fields : HereỸ f (f = ν, e) is threshold corrected Yukawa coupling, redefined in a basis where kinetic terms have canonical form.

Right Handed Neutrino Thresholds
We incorporated the three thresholds associated with the heavy right handed neutrino masses in the theory. For this neutrino Dirac coupling is progressively introduced [127] on passing these thresholds when going higher in energy and right handed neutrino are decoupled one by one going the other way.
To do this one needs to consider different effective theories (EFTs) in each of the energy ranges defined by right handed neutrino masses. We use the RGEs of the MSSM parameters extended with three right handed neutrino superfields [128] upto the energy scale of the lightest right handed neutrino by using the matching condition at the thresholds of right handed neutrino as discussed below. Along with these parameters we also consider the running [129] of dimension five (Weinberg [9]) operators which provide effective neutrino mass matrices below successive heavy mass thresholds. In the full theory there is no κ (coefficient of dimension five operator). Once third generation right handed neutrino is eliminated, a dimension five Weinberg operator will appear in the theory. Leptonic superpotential in the above discussed basis is: here A, B = 1, 2, 3 are generation indices. Split A, B into a, 3 and b, 3 where a, b run from 1 to 2. At scale Mν 3 we diagonalize the mass matrix of right handed neutrinos. Writing only last 2 terms of the superpotential (Eq. 5.11) with indices a, b and 3 in right handed diagonal basis : By solving the superpotential equation of motion for the heavy singletν 3 to leading order in (Mν) (−1) we getν Substituting back in (5.12) the effective superpotential is given by Mν 33 ν B (5.14) From third term we identify the coupling in EFT3 κ (3) : This condition should be imposed at µ = Mν 3 (largest eigenvalue of Mν). Now the Yukawa matrix Y aB is 2 × 3 and M ab is 2 × 2.
In the full theory light neutrino mass matrix is given by Scale Mν 2 superpotential is given by Eq. (5.16). Again using the superpotential equation of motion forν 2 effective superpotential is given by Again κ (2) can be obtained from matching of m EFT3 ν and m EFT2 ν at µ = M 2 . After a further RGE running to µ = Mν 1 , the above steps are repeated forν 1 , so that κ (1) is given by

Soft Sector
In the soft sector we have terms involving right handed sneutrino in addition to the We treat heavy right handed sneutrino in the same way as their fermion partner.
At each threshold we move to diagonal basis of the right handed neutrino. We apply the same rotation to the right handed sneutrino, because sparticle also follow superpotential equation of motion. In order to integrate out right handed sneutrino, we simply remove the last row and column from soft mass matrix in right handed neutrino diagonal basis. Similarly our trilinear coupling A ν is modified at each threshold. These thresholds have very small effect on gauge and Yukawa unification as shown in Table 5.2. But these thresholds are important for lepton flavour violation phenomenology which is our next task.

LFV Decay Rate
The decay rates for these processes are calculated using the amplitudes depicted by One obtains [63] A (n)L = 1 32π 2 , m 2 ν X are ratios of neutralino mass squared to the charged slepton mass square and chargino mass squared to the sneutrino mass square respectively. The neutralino-slepton vertices used in the Eqns. 5. 27-5.29 have form Similarly chargino-sneutralino vertices are :-

Numerical Fit
We used the solution sets presented in the previous chapter to calculate LFV. If we directly use these fits (which are found integrating out all the heavy right handed neutrinos at GUT scale) and run down hard and soft parameter using neutrino thresholds then soft Susy parameters at M Z are slightly changed as shown in Tables   5.3  BR for the LFV processes for these fits is given in Table 5.9. Complete solution with acceptable down type quark masses is presented in Table 5.10-5.15 and LFV BR is given in    The precisely measured [120] magnetic moment of muon has a significant deviation [130] from the theoretical prediction :

Anomalous Magnetic Moment of Muon
The ∆a µ may thus represent the contribution of new physics beyond SM. Supersymmetry is one of the leading candidate for new physics. The deviation a µ may be due to heavy sparticle contributions [69].    The lowest order supersymmetric contribution (a Susy µ ) to a µ from sneutrinochargino and smuon-neutralino loops [63,69] is shown in Figure 5    Fits presented have heavy smuon so we use exact formulae of Eq. (5.37) and (5.38) for Susy contribution to a µ . There are tan β enhanced two-loop diagrams also [69] which can modify the leading order results by 10 % but we calculated only one loop diagrams. NMSGUT fits prediction for ∆a µ is given in Table 5. 17. We see that except for the light smuon solution (presented in Chapter 3), the ∆a µ values are too small to resolve the muon g-2 anomaly.

Leptogenesis
The baryon asymmetry of the universe is defined as : This can be generated via :-B violation, C and CP violation and departure from thermal equlibrium as pointed out by Sakharov [135]. Leptogenesis [60,136] is most promising mechanism to explain observed baryon asymmetry of the universe as CP violating decay of right handed neutrino ( shown in Fig. 5.6) can fulfill these conditions. Leptonic CP asymmetry can be converted to baron asymmetry through sphaleron processes. Non-thermal leptogenesis involves right handed neutrinos generation through inflaton decay whose further decay can generate lepton and ( after sphaleron processes) baryon asymmetry. CP asymmetry parameter relevant for leptogenesis is given by In case of hierarchical spectrum: Model independent upper bound on CP asymmetry [137] :- Here m 1,3 are left handed neutrino masses. The CP parameter for the three discussed fits is given in Table 5.18. The desirable range of values for successful leptogenesis [137] is CP ∼ 10 −7 . Although the generic values we obtain are somewhat small, it should be noted that we have not optimized our fits to improve CP yet we are not too far off. Since it depends sensitively on the off diagonal structure of Y † ν Y ν and linearly on Mν 1 /Mν 2 , optimization could easily yields more satisfactory values. We will return to these questions in future research.

Conclusion
Neutrino Dirac Yukawa coupling is crucial for LFV predictions which require inclusion of heavy right handed neutrino thresholds. We found a reasonable fit implementing these thresholds and calculated BR for l i → l j γ and l i → 3l j LFV processes.
NMSGUT predict BR for the these processes several order of magnitude smaller than the upper bound from experiments because the fits have negative soft Higgs masses (M 2 H ) and heavy Susy spectrum. Negative soft Higgs masses provide cancellation in the off-diagonal entries of slepton mass matrices [138] which estimate the LFV. Solution presented has all the NMSGUT superpotential and soft parameters in the range corresponding to the previous chapter solutions. The smuon is heavy in all the solutions with loop corrected sfermion masses that we have so far found.
Hence Susy contribution to muon g-2 is very small ∼ 10 −12 . ∆a µ is not sensitive to neutrino Yukawa coupling but depend upon the soft Susy spectrum which further depend upon mSUGRY parameters at GUT scale. The value of the leptogenesis CP violation parameter is roughly in the desired ball park even without optimization.
This study shows that NMSGUT fits found are compatible with LFV constraints but further searches are needed to explore whether light smuon and adequate CP are achievable.

Chapter 6 Renormalization Group Evolution
Equations of the NMSGUT

Introduction
Renormalization group equations (RGEs) are used to evolve the gauge couplings, superpotential parameters and soft terms from UV scales into physically meaningful quantities that describe physics near the electroweak scale. The scale of generation of soft Susy breaking parameters in mSUGRY and string motivated mechanisms is above the GUT scale, typically M P . Thus a complete RG study of a Susy GUT requires evolution of GUT parameters from M P to GUT scale and then of the effective MSSM to electroweak/Susy breaking scale. Evolution between M P and M GU T can be (and is) very important for both the hard and soft Susy breaking parameters of the NMSGUT due to the large RG β functions in SO (10). In this chapter we give formulae for NMSGUT β function upto two loops for the first time and examine their effect. The form of the RGEs for supersymmetric theories is governed by the supersymmetric non-renormalization theorem [103]. According to this theorem the logarithmically divergent contributions to a particular process arise only from wave-function renormalisation, without any superpotential coupling renormalization. Variation of parameter X with energy scale is given by where t = log(Q/Q 0 ), Q and Q 0 are the renormalization and reference scale respectively. β

Numerical Analysis
We throw SO(10) gauge and Yukawa couplings and soft parameters randomly in the perturbative range. Along with this we choose soft breaking parameters according to SUGRY soft term form (assuming canonical soft terms). Susy breaking i.e. with all gaugino masses zero, all soft scalar masses equal, A 0 =2m 3/2 , b i =(A 0 -m 3/2 )µ i at the Planck scale. We chose m 3/2 =20 TeV and renormalize them from M P and M 0 X . Large coefficient of the trilinear couplings in the anomalous dimension make these RGEs to evolve fast between M P to M 0 X . The values of hard and soft parameters at two scales (M P and M 0 X ) are given in Tables 6.1 and 6.2 respectively. This shows that the evolution can be very significant and in particular the soft masses change rapidly. The large value of β g makes a UV fixed point impracticable. Soft mass evolution is shown in Fig. 6

Discussion and Outlook
We derived the NMSGUT RG equations to determine the RG evolution of couplings between M P and M 0 X (the matching scale between GUT and effective theory) assuming pure supergravity canonical scenario for the starting parameter ansatz.
Evaluating the effects of the evolution on randomly chosen sets of parameter values we see that the RG evolution has dramatic effects on the soft susy breaking parameters. Firstly most of the soft Susy squared masses of the SO(10) Higgs irreps become negative. It provides a potentially robust justification of the negative values of M 2 H,H the NMSGUT fits already needed. Note that the distinctive normal s-hierarchy at low scale is strongly correlated with the large negative M 2 H,H we use in the fits. Gaugino masses will be generated by two loop RG evolution, however
The D-terms of the family group vanish automatically only for trivial solutions of the F-terms conditions. We are interested in non-trivial solutions because only these can generate generation mixing. One needs to introduce additional fields to cancel GUT sector contribution to the family D-terms. F-terms corresponding to extra fields should not interfere with the GUT F terms so as not to disturb the MSGUT SSB. The best possible choice is to locate these fields in the hidden sector.
In [146] it has been shown that Bajc-Melfo (BM) two field superpotential is an appropriate candidate. In the next section we will discuss how BM superpotential enables YUMGUTs.

Bajc-Melfo Superpotential
Two field (S s , φ s ) BM superpotential reads: It has a Susy preserving global minima at S s = φ s = 0 and S s = 0, φ s = − µ B λ B and Susy breaking local minima at φ s = − µ B 2λ B where S s remains undetermined with a condition | S s | ≥ | φs √ 2 |. S can be fixed either by radiative corrections [147] or by couplings to N=1 supergravity [148,149]. In [146] S is determined by coupling S, φ fields to N=1 supergravity as reviewed below.

Coupling to Supergravity
Supergravity potential [150,151,152,153] for the scalar field Z I is ; E ≡ e κ 2 i |Z I | 2 (7.14) Visible sector VEVs (z i ) and φ s preserve global Susy ( ∂W ∂Z I = 0 = D(Z I )) The solution is The globally undetermined VEV S s ∼ M p is now fixed. In gravity mediated scenario, gravitino mass is given by Typical range of gravitino mass require cancellation among W and W GU T such that BM superpotential parameters determine Susy breaking scale : λ B ∼ 10 10.5 − 10 11.5 GeV (7.23)

Gauged O(N g ) with Hidden Sector Superpotential
Considering BM superpotential as a hidden sector of supergravity, total Superpotential is given by where W H (S s , φ s ) = W 0 +S s (µ B φ s +λ B φ 2 s ), z i represents GUT chiral and symmetric multiplets of O(N g ) whose VEVs determine supersymmetric vacuum from vanishing of F and D terms Supergravity potential representing D-term contribution is Since F terms vanish for all z I except S s (which is gauge singlet) we are left with just global supersymmetric D-terms. The O(N g ) D terms are given as : where D a X is the visible sector contribution, and T a , T a are O(N g ) generators in the fundamental and generic representations. We can consider S, φ as traceful multiplets of O(N g ), traceless part still remains undetermined, to fix family D-terms : here I Ng is the unit matrix of order N g . All O(N g ) non-singlet fields :Ŝ ab ,φ ab and visible sector Higgs fields enter into the family D-terms. As discussed in the previous section flavour singlet VEV (S s ) is detemined by supergravity effects and non-singlet part is fixed by the O(N g ) D-terms and supergravity soft mass terms.

S fixation
A. N g = 2 Using O(2) ≈ U (1) isomorphism, S (similarly φ) is defines as where S ± , S s are properly normalized fields so that The superpotential becomes F s is non zero(θ) and all other F-term vanish forφ s = − µ B 2λ B ,φ ± = 0 and S s,± remain undetermined. Using the values of fields : hereD X = q i |z i | 2 , q i is the family symmetry charge of the visible sector VEV Z i .
The minimum will occur when Detailed mass spectrum of S, φ fields can be found in [146].
where S s,0,±,±2 are properly normalized fields so that where V a =D a X . The potential for the flat directions fromŜ is now Solution found is In [146] it is shown that BM type hidden sector necessarily imply a number of

Analytical and Numerical Analysis
Writing VEVs {p, ω, a, σ,σ, χ} = m λ {P, W, A,σ,σ,χ},χ A =χ AA + ξ in units of m/λ, we can eliminate all the parameters in F-term equations (Eq. 7.5-7.9) except two ratios ξ = λM ηm and λ η : It is convenient to use dimensionless form of equations for SSB analysis because we can get most of the VEVs independent of model parameters. Before analyzing realistic SM case (N g =3) we will study the simplest toy model (N g = 2).

Toy Model (N g = 2)
For N g = 2,Σ = {σ 11 ,σ 22 ,σ 12 }, the matrix Ξ involves the combinationsχ A = χ AA + ξ: Then Yukawas corresponding to all allowed value of M H are determined. Yukawa eigenvalues, mixing angles and neutrino masses are presented in Table 7.1 for f ∼ h and in Table 7.2 when f is smaller by a factor of 10 −3 . In f ∼ h case we have acceptable fermion hierarchy and mixing but too small neutrino masses which is the main failure of MSGUT. One can boost Type I seesaw contribution by suppressing  f which implies Y u = Y ν and Y d = Y l (see Table 7.2). Type II seesaw contribution is generated by VEV of O[1,3,-2] multiplet [36,70,55,57] which is now family group triplet. In the MSGUT(NMSGUT) Type I dominates over the Type II. Type II contribution needs to be re-examined in the YUMGUT. Complete superheavy spectra (in units of m/λ) for the solution found is presented in Table 7 Matter Yukawas can be written by the same procedure as in the N g =2 case :  Using superpotential equation of motion : In a right handed neutrino diagonal basis, the effective superpotential reads : Mν 33 ν B (7.60) We can write a dimension five (κ ) operator for three left handed neutrinos: The light sterile neutrino will get Dirac mass only, so the mass matrix is given by : Using the above solution and random superpotential parameters, we have calculated the Yukawa structure and neutrino masses for all the M H values as shown in  Table 7.5: Yukawa eigenvalues and mixing angles for N g =3 (Rank[Ξ] = 4) and the same parameter values as in Table 7.4 but with much smaller f = -0.00011 + .00002 i. M ν c ≡ λM ν c /m. m/λ is taken to be 10 16 GeV to estimate ∆m 2 ν .

Discussion and Outlook
We proposed [145] dynamical generation of flavour based upon the Susy SO (10)  This motivates reconsideration of the no-go [55] in the MSGUT using light sterile neutrinos. Rank reduction of homogeneous system provides a possible route to find non-zero eigenvalues of σ(σ) VEV. We have also studied the case using tracelessdimensional representation. We can't use traceless representation for N g =2 because cubic invariants in the superpotential do not contribute. We have calculated the complete superheavy spectrum for all the cases considered (given in Appendix B) to check the existence of pseudo-Goldstones which may be present when there is Higgs duplication. Spectra do not contain pseudo-Goldstone except the SM singlet G[1,1,0] sector which does not affect unification.
In all the cases studied acceptable Yukawa hierarchy and mixing is achieved but neutrino masses generated are too small. Type I contribution can be raised by suppressing f which provides unacceptable Yukawa structure. We have not considered the contribution of Type II seesaw generated by the VEV of symmetric multiplet O − .
Although we expect it to be small as compared to Type I as in MSGUTs, but some special points may yield significant contribution. Addition of 120-plet, which along with 10-plet is mainly responsible for generating charged fermion masses, is another way to get neutrino masses in experimentally measured range. The Rank[Ξ] = 5 case (considering 6 dim symmetric representation) phenomenology needs to be investigated because this provides sterile neutrino. With optimization, one can expect to find the flavour blind parameters of YUMGUT which can produce actual MSSM Yukawas. To completely demonstrate this idea we need to produce realistic SM mass mixing data respecting NMSGUT fitting features which will require a huge computational effort.
A number of experimental signals such as light moduli fields and singlet pseudo-Goldstones [146] which can also be DM candidates are associated with our proposal.
These fields also cause cosmological problems. Thus our work has laid the basis for an extensive program of future studies in unification and cosmology.
We have not yet found light smuon solution (which is very desirable for dark matter phenomenology and to resolve muon g-2 anomaly ) after including loop corrections.
We perhaps require either more searches or deeper RG analysis of the soft parameters. The refined NMSGUT fits have large A 0 , M H,H , µ and B parameters with heavy third s-generation like the tree level fits. These distinct predictions of the model will be tested at LHC with the discovery of Susy particles.
Branching ratio for different lepton flavour violating (l i → l j γ and l i → 3 l j γ) processes is calculated. These processes do not provide additional constraints on the soft mass matrices since the calculated BR is much smaller than the experimental upper bound because of the large sfermion masses and negative soft Higgs mass square parameters. We have calculated the ∆a µ for the loop corrected fits presented in Chapter 4 and 5. Since smuon is not light, this contribution is also very small.
We have not yet estimated charged lepton electric dipole moments but can and will do so in upcoming studies.
Another puzzle for the NMSGUT has been the necessity of using non universal Higgs doublets mass squared values that are negative : which is difficult when the soft terms come from supergravity. We calculated the NMSGUT RG equations and used them to study parameter evolution between M P lanck and M GU T . We found that the universal positive scalar mass squared parameters provided by SUGRY easily become negative due to the RG flow thus removing the problem in principle. Future fits will thus include this third stage of RG evolution.
Finally, we considered a new scenario based upon the SO(10) and O(N g )(family) gauge symmetry. In this framework, Higgs irreps of SO(10) also carry family indices whose VEVs break GUT and family symmetry thus generating matter Yukawas with enough structure to account for the observed hierarchy. So the number of SO(10) Yukawa couplings reduce dramatically. Therefore this scenario is called Yukawon ultra minimal grand unified theory (YUMGUT). Consistent SSB requires introduction of a special type ('Bajc-Melfo') of superpotential. Study of toy model (N g =2) and realistic three generation case show that the realistic MSSM data can be produced dynamically by the VEV of Yukawon field. Consideration of 126-126 VEV homogeneous equations of different rank provides new directions for model phenomenology such as existence of sterile neutrinos. YUMGUT also offers novel dark matter candidates from a SM singlet sector as well as from the hidden sector fields.
This study shows that NMSGUT is quite compatible with B and L violation experimental data. All these observations make NMSGUT a leading candidate for physics beyond SM and a mature theory of particle physics.