Roadmap of left-right models based on GUTs

Roadmap of left-right models based on GUTs

Joydeep Chakrabortty joydeep@iitk.ac.in Department of Physics, Indian Institute of Technology, Kanpur-208016, India    Rinku Maji mrinku@iitk.ac.in Department of Physics, Indian Institute of Technology, Kanpur-208016, India    Subhendra Mohanty mohanty@prl.res.in Theoretical Physics, Physical Research Laboratory, Ahmedabad-380009, India    Sunando Kumar Patra sunando.patra@gmail.com Department of Physics, Indian Institute of Technology, Kanpur-208016, India    Tripurari Srivastava tripurar@iitk.ac.in Department of Physics, Indian Institute of Technology, Kanpur-208016, India
Abstract

We perform a detailed study of the grand unified theories and with left-right intermediate gauge symmetries of the form . Proton decay lifetime constrains the unification scale to be GeV and, as discussed in this paper, unwanted cosmological relics can be evaded if the intermediate symmetry scale is GeV. With these conditions, we study the renormalisation group evolution of the gauge couplings and do a comparative analysis of all possible left-right models where unification can occur. Both the D-parity conserved and broken scenarios as well as the supersymmetric (SUSY) and Non-supersymmetric (Non-SUSY) versions are considered. In addition to the fermion and scalar representations at each stage of the symmetry breaking, contributing to the -functions, we list the intermediate left-right groups which successfully meet these requirements. We make use of the dimension-5 kinetic mixing effective operators for achieving unification and large intermediate scale. A significant result in the supersymmetric case is that to achieve successful unification for some breaking patterns, the scale of SUSY breaking needs to be at least a few TeV. In some of these cases, intermediate scale can be as low as GeV, for SUSY scale to be TeV. This has important consequences in the collider searches for SUSY particles and phenomenology of the lightest neutralino as dark matter.

I Introduction

Grand Unified Theories (GUTs) are the theories which attempt to discover a single gauge group for the unification of the strong, weak, and electromagnetic interactions and where the three couplings of the standard model (SM) are unified at a high scale (called the GUT scale, ) to a single coupling of the GUT gauge group. The grand unified gauge group must be in form of either or , as it must posseses an unified coupling . The SM is expected to emerge from the unified symmetry group, thus the minimum rank of the GUT group must be 4. Some of the successful candidates for such theory are , , and . In this paper we focus on and , as we are interested in those unified groups which contain left-right gauge symmetries as the subgroup. The motivation behind left-right models as intermediate symmetry group is to raise P and CP violation to the same status as gauge symmetry breaking which take place via vacuum expectation values of specific scalar representations. As the rank of these groups are 5 and 6 respectively, it is indeed possible to accommodate multiple intermediate symmetries in the desert between and .

In this paper we focus on the economical scenario of one intermediate symmetry at scale () below the GUT scale which further breaks to the SM directly. Among the numerous possibilities for the intermediate symmetry groups, we concentrate only on the left-right models, which are of the form , where is any group or product of groups. These specific breaking patterns can be achieved by the suitable choice of representations and orientations of the vacuum expectation values of the GUT breaking scalarsFritzsch and Minkowski (1975); Lazarides et al. (1981); Clark et al. (1982); Aulakh and Mohapatra (1983); Hewett et al. (1986); Deshpande et al. (1993); Amaldi et al. (1991); Hung and Mosconi (2006); Howl and King (2008); Chakrabortty and Raychaudhuri (2009); Bertolini et al. (2010); Chakrabortty et al. (2011); De Romeri et al. (2011); Arbeláez et al. (2014); Chakrabortty and Raychaudhuri (2010a). Many phenomenological studies have been performed both in presence and absence of supersymmetry (SUSY) for Lazarides et al. (1981); Chakrabortty et al. (2011); De Romeri et al. (2011); Patra et al. (2016); Bandyopadhyay et al. (2016); Babu et al. (2016); Bandyopadhyay and Raychaudhuri (2017) and Gursey et al. (1976); Achiman and Stech (1978); Hewett et al. (1986); Hung and Mosconi (2006); Howl and King (2008); Chakrabortty et al. (2014); Miller and Morais (2014); Chakrabortty et al. (2015); Gogoladze et al. (2014); Younkin and Martin (2012); Calmet and Yang (2011); Wang (2011); Biswas et al. (2011); Atkins and Calmet (2010). Successful generation of neutrino and fermion masses is one of finest achievements of GUT models Mohapatra and Senjanovic (1980); Bajc et al. (2003); Goh et al. (2003a); King and Ross (2003); Goh et al. (2003b); Mohapatra et al. (2004); Bertolini et al. (2004); Dev and Mohapatra (2010); Joshipura et al. (2009); Chakrabortty et al. (2011); Patel (2011); Blanchet et al. (2010); Bhupal Dev et al. (2011); Joshipura and Patel (2011); Bhupal Dev et al. (2012); Meloni et al. (2014); Babu et al. (2017); Meloni et al. (2017). Recently, different aspects of unification have been discussed in the context of dark matter Babu and Khan (2015); Nagata et al. (2015); Garcia-Cely and Heeck (2015); Brennan (2017); Boucenna et al. (2016); Mambrini et al. (2015); Parida et al. (2017); Nagata et al. (2017); Sahoo et al. (2017); Arbeláez et al. (2017). The implication of domain walls in presence of left-right symmetry in SUSY framework has been studied in Mishra and Yajnik (2010); Borah and Mishra (2011); Borah (2012).

Our aim here is to check all possible intermediate groups that arise from and for both SUSY and Non-SUSY varieties, both in presence and absence of gravitational smearing at the unification scale. Moreover, we also want to check the viability of such scenarios which pass the scrutiny of proton decay limits and cosmological constraints, namely topological defects and baryon asymmetry of the universe.

To be consistent with the observed limit Abe et al. (2017) on proton lifetime111One can find the recent development in lattice computation for proton decay in Ref. Aoki et al. (2017). (), the unification scale222 In principle this bound needs to computed for individual model. But we have considered the conservative limit without loss of generality. has to be raised above GeV. One way to achieve this is to include the contribution from the Planck mass-suppressed effective dimension-5 operators. These are expected to arise by integrating out the full quantum gravity theory or string compactification leading to an effective GUT theory at . We study the effects of the Planck-suppressed effective dimension-5 operators along with the RG evolution of the couplings and limit the Wilson coefficients of these operators from the requirement that GeV.

There are some critical constraints on the intermediate left-right symmetry models from cosmology, which are related to the existing D-parity in such models. It was shown by Shaposhnikov and Kuzmin Kuzmin and Shaposhnikov (1980), that the net baryon asymmetry must be zero in models with unbroken D-parity. Another cosmological problem that arises, is the formation of string-bounded stable domain walls, when D-parity is broken Kuzmin and Shaposhnikov (1980). A way out of both of these problems is if the inflation takes place after GUT symmetry breaking when one of the GUT scalars acts as the inflaton. Viable inflation from scalars as the inflaton, has been constructed Garg and Mohanty (2015); Ellis et al. (2016) and it is seen that the reheat temperature at the end of inflation is GeV. If the scale of D-parity breaking is above the reheat temperature ( GeV), there is no problem of stable domain walls and baryogenesis can be achieved via leptogenesis by heavy neutrino decay, in GUT models with left-right intermediate symmetries Fukugita and Yanagida (1986). In this paper, we impose the criterion that the D-parity breaking of the intermediate scale must be above GeV and study the parameter space and gauge groups of the intermediate scale which satisfy this criterion. This ensures that after reheat the universe is in the SM phase and the harmful cosmological defects are not created. We do a detailed study of the role of the abelian mixing operators (which arise when there is a product of groups in the intermediate scale) in raising the intermediate scale symmetry to above GeV and limit the range of couplings of the abelian mixing operators using this criterion.

The rest of the paper is organized as follows: In section II we lay down some aspects of grand unified theories which are used for selecting the intermediate scale symmetries consistent with proton decay and cosmology. Here we have briefly discussed extended survival hypothesis, D-parity, renormalisation group evolutions (RGEs) of gauge couplings. We have also noted the modifications in the boundary conditions of RGEs at different scales due to threshold correction and Planck scale physics. We have concluded this section by introducing the homotopy structure of the vacuum manifolds and their respective topological defects. In section III, we have discussed all possible one-intermediate scale breaking patterns that carry explicit left-right gauge symmetry. We have computed the two loop beta functions for SUSY and Non-SUSY scenarios for each breaking chain. Then in section IV we have determined the values of the intermediate and unification scales and the unified gauge coupling, in accordance with the present experimental bounds of the low scale parameters, by simultaneously solving the RGEs and performing a goodness of fit test with a constructed statistic. This enables us to obtain the bounds with correlation among these high scale parameters including abelian mixing. The constraints due to topological defects and proton lifetime are implemented. We conclude by discussing their impacts on symmetry breaking scales and other free parameters of the theory.

Ii Some aspects of unification

In this section we study some aspects of GUTs which have a bearing on fixing the unification and the intermediate symmetry scales.

ii.1 Extended Survival Hypothesis

The direct breaking of GUT group to SM is not favoured as it does not predict the correct Weinberg angle () at low energy333This more specifically applicable for Non-supersymmetric scenario and also with minimal particle content. One can explain this by adding more particles and including their threshold corrections.. One or more intermediate scales are therefore necessary. As the SM has rank 4, the GUT groups need to have large ranks () to posses one or more intermediate symmetry groups. We need extra scalars to break these intermediate gauge groups. These scalars are usually embedded in large representations under the GUT group. But unlike the GUT breaking scalars, they contribute in the RG between intermediate and unification scales. Due to their large dimensionality, their contribution to the beta coefficients may be large enough to spoil the unification picture. Also, the presence of such representations may require a significant fine tuning in the scalar potential to achieve correct vacuum structure. Thus to avoid the catastrophe due to the unnecessary sub-multiplets, a prescription named Extended Survival Hypothesis (ESH) has been proposed del Aguila and Ibanez (1981). According to this, at every stage of the symmetry breaking chain, only those scalars are light and relevant that develop a vacuum expectation value at that or the subsequent levels of the symmetry breaking. These sub-multiplets play a crucial role in generating the fermion masses, specifically neutrino masses and without much fine tuning of the parameters of the scalar potential. We will use ESH to understand the symmetry breaking within a minimal fine tuned scenario.

ii.2 D-parity

D-parity is an important ingredient in the context of grand unified theories. Historically, D-parity was first introduced in Mohapatra and Pati (1975); Senjanovic and Mohapatra (1975); Senjanovic (1979); Chang et al. (1984a, b) in case of , which contains as a maximal subgroup. D-parity, which plays a role analogous to charge conjugation, is defined as the product where ’s are the antisymmetric generators of . As an example, a multiplet under is related to its conjugate representation by D-parity. D-parity is not realised in all possible intermediate symmetries. The characteristics of the vacuum orientation, in the wake of the breaking of GUT symmetry, decides whether the D-parity is broken or not. Though it is possible for the intermediate symmetry to have the form in both cases, it is the D-parity which decides whether and , the respective gauge couplings, will be the same or not at the intermediate scale.

The minimum rank of the GUT group must be 5 to obtain the preferred form, mentioned in last paragraph, of the intermediate symmetry groups; thus is the minimal choice. As is of rank 6 and it contains as a subgroup, we can realize D-parity through a few of its sub-groups. All these possibilities will be discussed in a later part of this paper.

D-parity and the scale at which it is broken has some significant implications for cosmology. If the intermediate symmetry is , then the coupling constants of the two groups must be same () in the unbroken D-parity phase. In such a case, as pointed out by Kuzmin and Shaposhnikov Kuzmin and Shaposhnikov (1980), baryon asymmetry cannot be generated by the decay of leptoquarks in the D-symmetric phase. To generate baryon asymmetry through leptoquarks Ignatiev et al. (1979), the masses of these leptoquarks must be close to the unification scale. This implies that the D-parity breaking must take place close to the unification scale in the left-right models, in the conventional GUT-baryogenesis scenario Ignatiev et al. (1979).

A different cosmological problem associated with D-parity breaking is the formation of string-bounded domain walls which do not decay and would dominate the density of the late universe Kuzmin and Shaposhnikov (1980). The formation of domain walls and monopoles is undesirable in the phase transition associated with the symmetry breaking, as it would dominate the energy density of the universe. On the other hand, textures harmlessly decay in the early universe while string networks are sub-dominant, can be accommodated in the energy density of the universe and may have observable signature in small angle anisotropy of the CMB Fraisse et al. (2008).

One scenario, that provides a way out of these cosmological problems associated with the string bounded domain walls (induced by D-parity breaking) and other harmful cosmological relics, is inflation Starobinsky (1980); Guth (1981); Albrecht and Steinhardt (1982). Inflation can take place with the GUT scalars as inflaton and viable inflation models with scalars playing the role of the inflation have been constructed Aulakh and Garg (2012); Garg and Mohanty (2015); Ellis et al. (2016); Gonzalo et al. (2017); Leontaris et al. (2017). Any domain walls or other topological defects will be inflated away in these models, where inflation takes place following the GUT symmetry breaking scale. Following inflation, the reheat temperature from the decay of the inflation is around . If the intermediate symmetry and D-parity is broken at a scale above the reheat temperature of , then the dangerous walls bounded by strings Kibble et al. (1982a) will not form in the radiation era after inflation. The problem of baryogenesis can be solved through leptogenesis Fukugita and Yanagida (1986) in these models. This is possible with the decay of heavy right handed neutrinos with masses lower than the reheat temperature and the subsequent conversion of the lepton asymmetry to baryon asymmetry through sphalerons Kuzmin et al. (1985) in the electroweak era. We will follow this cosmological scenario in this paper and will impose the criterion that only those GUT models are phenomenologically acceptable where the D-parity breaking scale is above the reheat temperature of .

ii.3 RGEs of gauge couplings

The RGEs of the gauge couplings (upto two loop) can be written in terms of group theoretic invariants which encapsulate the contributions from the respective scalars and fermions of the theory Caswell (1974); Jones (1974, 1982); Machacek and Vaughn (1983, 1984, 1985). These invariants depend solely on the representations of those scalars and fermions, under the gauge symmetries we are considering. Following Ref. Jones (1982), the beta functions upto two-loop for gauge couplings for a product group can be written as444Here, we have not included the contributions of the Yukawa couplings.:

(1)

Here are the scalar and fermion representations transforming under group . for the chiral fermions, otherwise it is 1. The are the quadratic Casimir for scalar, fermion and adjoint representation for respectively. is the dimensionality of the representation and , the normalisation of the generators in dimensional representation. These group theoretic factors are related to each other by , where is the number of generators of the group. These quantities have special values for abelian groups, e.g. where are the normalised abelian charges..

In case of supersymmetry, the beta functions upto two-loop can be given as in Ref. Jones (1982):

(2)

Here the dimensions of the representations are assigned for the super-multiplets.

ii.4 Abelian mixing

In a theory when we have more than one abelian gauge group, the Lagrangian posses extra gauge invariant term in the gauge kinetic sector. Let us consider there are two abelian groups and are their respective gauge invariant field strength tensors. Then, apart from their individual gauge kinetic terms there will be a term which leads to the abelian mixing. As a result abelian gauge couplings start mixing with each other even at the one-loop level Holdom (1986); del Aguila et al. (1988); Lavoura (1993); del Aguila et al. (1995); Luo and Xiao (2003); Fonseca et al. (2013) and one needs to modify the structures of functions accordingly. In presence of multiple abelian gauge groups, e.g. , the RGEs can be written as:

(3)

where

(4)

and is the gauge coupling matrix, represented as

(5)

where runs over number of groups. For example, for two gauge symmetries, and the above matrix will be of order 2.

The ’s are defined as Bertolini et al. (2009); Chakrabortty and Raychaudhuri (2010b):

(6)

The beta coefficients can be written as Bertolini et al. (2009); Chakrabortty and Raychaudhuri (2010b):

(7)

where is the ’s normalized charge and is the dimensionality of the non-singlet representations (fermion/scalar) that carry this charge. We would like to mention that for , we get the abelian mixing terms.

This mixing may lead to more complicated structures at two-loop level and the functions are given as Bertolini et al. (2009); Chakrabortty and Raychaudhuri (2010b):

(8)
(9)
(10)

with -coefficients given as

(11)

At two-loop level, this abelian mixing gets entangled with non-abelian gauge couplings too. This affects their mutual running as follows Bertolini et al. (2009); Chakrabortty and Raychaudhuri (2010b):

(12)

where . Here, is the non-abelian gauge coupling and stands for abelian mixing with non-abelian gauge couplings, with as the non-abelian index. The abelian mixing has been discussed in detail in Refs. Bertolini et al. (2009); Chakrabortty and Raychaudhuri (2010b); Fonseca et al. (2013), in the context of and GUT groups. In the context of supersymmetric GUT, the effects of abelian mixing in SUSY spectrum, more precisely for gaugino masses, has been discussed in Fonseca et al. (2012); Braam and Reuter (2012); Hirsch et al. (2012); Rizzo (2012); O’Leary et al. (2012).

ii.5 Matching conditions

In the instance of the breaking of a simple or a product gauge group into its subgroups, the gauge couplings of the broken groups are redistributed in terms of the unbroken symmetries. Thus the parent and the daughter gauge couplings need to be matched at the symmetry breaking scale which has been discussed in detail in Hall (1981); Weinberg (1980); Chang et al. (1985); Binger and Brodsky (2004). If we neglect the heavy-mass-dependent logarithmic effects, we can write the matching condition of two gauge couplings as:

(13)

where is the quadratic Casimir of group in adjoint representation. This matching condition will get modified in presence of abelian gauge couplings. As an example, let us consider an abelian daughter group and let the respective gauge coupling be . The generator of this unbroken group () is an outcome of the spontaneous breaking of generators , i.e. . Here indicates the number of broken generators and are the suitable weight factors leading to normalised charge and satisfy the following relation: . Now the the matching condition is given as Bertolini et al. (2009); Chakrabortty and Raychaudhuri (2010b):

(14)

for the abelian group. In the presence of more than one abelian groups, spontaneously broken at same scale and contributing to the charge, this matching condition is further modified. As we have discussed in the last section, the gauge couplings get mixed in the presence of two or more abelian gauge groups and we need to treat the full gauge coupling matrix together, in place of a single coupling (see Eqn. 5). In this case, the matching condition reads as Bertolini et al. (2009); Chakrabortty and Raychaudhuri (2010b):

(15)

where the matrix is given in Eqn. 5. is a row vector in the above equation and satisfies the relation . In the absence of non-abelian groups in the parent sector, the above equation reduces to with .

ii.6 dimension-5 operators and unification boundary conditions

At the GUT scale, the unified renormalisable gauge kinetic term is written as:

(16)

where the unified gauge field strength tensor , and ’s are the generators of unified group and they are normalized as . This contains an unified gauge coupling .

In a typical unified theory, all the fundamental forces are included apart from gravity. Still, as the unification scale is fairly close to the Planck scale(), it is possible for string compactification or quantum gravity to have some impact on the unification boundary condition Hall (1981); Hill (1984); Shafi and Wetterich (1984); Hall and Sarid (1993). These effects are expected to be through the higher dimensional operators suppressed by Planck scale and can be written as:

(17)

where is a dimensionless parameter.

Group Scalar Representation
650
Table 1: The group theoretic factors (’s) arise from dimension-5 operators for the following breaking .

Here transforms as the adjoint representation of the GUT group, and thus restricts the choice of which can belong to only the symmetric product of two adjoint representations. The GUT symmetry is spontaneously broken once the acquires vacuum expectation value (VEV), , and the gauge couplings get additional contributions from the effective operator Eq. 17. These contributions are unequal due to the non-singlet nature of and modify the unification boundary conditions as: , where . It is worthy to mention that these effects could be important to evade the proton decay constraints. The extra free parameter allows a range of solutions for the unification scale, and may help to revive certain breaking patterns which will be discussed in a later part of this paper.

Group Scalar Representation
54
210
770
Table 2: The group theoretic factors (’s) arise from dimension-5 operators for the following breaking .

The relevant and necessary dimension-5 contributions are tabulated in Tables 1, 2, and 3 (see Chakrabortty and Raychaudhuri (2009); Martin (2009); Chakrabortty and Raychaudhuri (2010a) for more). A set of new results has been provided in Table 4 for breaking pattern for both D-parity conserved and broken cases. We would like to mention here that these dimension-5 operators may affect the unification scenario for and GUT groups Patra and Parida (1991); Vayonakis (1993); Chakrabortty and Raychaudhuri (2010b); Calmet et al. (2010) and these corrections lead to the non-universality of gaugino masses in the SUSY case Drees (1985); Ellis et al. (1985); Anderson et al. (1996); Chakrabortty and Raychaudhuri (2009); Martin (2009); Chakrabortty and Raychaudhuri (2010a); Bhattacharya and Chakrabortty (2010) leading to different phenomenology Bhattacharya and Chakrabortty (2010); Atkins and Calmet (2010); Biswas et al. (2011); Calmet and Yang (2011); Wang (2011); Younkin and Martin (2012); Chakrabortty et al. (2014); Gogoladze et al. (2014); Miller and Morais (2014); Chakrabortty et al. (2015) compared to the usual minimal supersymmetric standard model.

Group Scalar Representation
54 + 210
210 + 45
770 + 210
Table 3: The group theoretic factors (’s) arise from dimension-5 operators for the following breaking .
Group Scalar Representation
Table 4: The group theoretic factors (’s) arise from dimension-5 operators for the following breaking . D-parity is consereved and broken when and (under ) components of 650-dimensional scalars acquire VEVs respectively.

ii.7 Topological defects associated with spontaneous symmetry breaking

It is worthwhile to properly analyse the topological structures of vacuum manifolds in spontaneously broken gauge field theories. Ref.s Kibble et al. (1982b); Vachaspati (1997) note that various types of topological defects, namely domain walls, cosmic strings, monopoles, and textures may appear. Investigating the homotopy groups of the respective vacuum manifolds can shed light on these structures. In this paper, we concentrate on those defects, which may appear from the subsequent breaking of GUT gauge groups to the SM Lazarides et al. (1982); Kibble et al. (1982a); Bhattacharjee et al. (1992); Davis and Jeannerot (1995a, b); Jeannerot et al. (2003). During the breaking of a group down to its subgroup , we can study the homotopy groups of the vacuum manifold to see whether topological defects form during the phase transition associated with the said breaking. Topological defects are formed if . Various types of topological defects which can form are : domain walls (), cosmic strings (), monopoles (), and textures (). Out of these, monopoles and domain walls are undesirable, as they dominate the energy density and would surpass that of the universe. Textures decay rapidly and leave no trace in the present universe. The energy density budget of the universe can accommodate cosmic strings and those may have observable signatures in the small angle anisotropy of the CMB Fraisse et al. (2008). We will later discuss whether these defects are isolated or hybrid ones.

We list the homotopy of different groups below, which will appear in different stages of symmetry breaking using Bott periodicity theorem Bott (1959):
  (I)

(18)
(19)

  with and .
   where .
  (II)

(20)
(21)
(22)

  with .

We would like to mention a few useful special cases Bott (1959); Harris (1961); Mimura (1967); Lundell and Tosa (1990):

Lie zeroth Homotopy Fundamental 2nd homotopy 3rd homotopy
Group group group group
Table 5: Homotopy classification of Lie Groups.

We can define the homotopy as for a product group. The vacuum manifold is defined as for a given symmetry breaking chain . To investigate the topological structure, i.e., homotopy of the vacuum manifold we can write when .

It becomes easy to classify the possible emergence of different topological defects, once we identify the homotopy of the vacuum manifold at every stages of symmetry breaking. We can assure the appearance of domain walls, cosmic strings, monopoles and textures for respectively, when the homotopy of vacuum manifold is non-trivial. Here we demonstrate the generation of topological defects using two examples where we assume that :

  • Consider a symmetry breaking of the form Davis and Jeannerot (1995b): . Analysing the vacuum manifold of the first stage of symmetry breaking, we note the following:

    (23)

    In the second stage of symmetry breaking, we find

    (24)
  • Now consider another symmetry breaking of the form Davis and Jeannerot (1995b): . If we analyse the vacuum manifold of first stage of symmetry breaking, we note the following:

    (25)

    In the second stage of symmetry breaking, we find

    (26)

One can have hybrid topological defects Lazarides et al. (1982); Kibble et al. (1982b); Vilenkin (1982) in case of a sequential symmetry breaking. For example, monopoles are produced in the first stage of symmetry breaking in case-I (see Eq. II.7) and we will have strings due the second stage of symmetry breaking (see Eq. 24). The monopole-antimonopole pair is connected by the strings in this scenario. Unlike case-I, the strings, which are the outcomes of the first stage of symmetry breaking, are topologically unstable in the next type of symmetry breaking (see Eq. 25) but the domain walls, which are produced in the latter step, are stable (see Eq. 26). All these discussions and conclusions regarding the topological defects are equally applicable for supersymmetry and Non-supersymmetric scenarios. The topological structures are based on the homotopy of the vacuum manifold corresponding to the spontaneous breaking of some Lie groups. Interestingly enough, the supersymmetry algebra is validated by the Lie algebra, and we can find the Lie algebra for SUSY by exponentiating the infinitesimal super-transformation. This has been discussed in detail in Albert (1948); Srivastava (1975); Santilli (1978); Davis and Jeannerot (1995b).

Throughout this paper, we will impose the constraint that the scale of symmetry breaking (producing the harmful monopoles and domain walls) should be above the post-inflation reheat temperature of , so that these defects do not form after inflation in the inflationary cosmology. This will restrict the symmetry breaking pattern which are acceptable vis-à-vis cosmology.

Iii RGEs of gauge couplings: coefficients

iii.1 Breaking of to SM:

, whose rank is five and dimensionality of the adjoint representation is 45, is considered to be one of the favourite candidates for unification. Here we have considered all possible breaking patterns of to the SM through a single intermediate gauge group that includes structure. These breaking patterns are all rank-conserving (see Fig. 1). We stick to minimal field configurations, especially in scalar sectors. Using the novelty of extended survival hypothesis (ESH), we only make those sub-multiplets lighter, which participate in the process of symmetry breaking, including the electro-weak ones. Only these sub-multiplets participate in the evolution of the -function. We have illustrated the situation both in the presence and absence of D-parity.

Figure 1: Adopted one intermediate step breaking of to the SM.

iii.1.1

spontaneously breaks to through the VEVs of possible scalars , and which contain the sub-multiplet under . This ensures the presence of the desired intermediate symmetry.

Scalars
-
-
- -
- -
Fermions
Table 6:

In Table 6 we have listed the fermion and scalar representations which contribute to the RGEs of the gauge couplings from to . The VEVs of and conserve D-parity, while that of does not. We have explicitly discussed both D-parity conserved and broken cases.

At the intermediate scale , is spontaneously broken through the VEV of . Here remains unbroken, ensured by the singlet structure of . The SM hyper-charge generator () is formed out of and ; pops out of the itself. This leads to the following matching conditions of the gauge couplings at the intermediate scale

(27)
(28)

where .

We have computed the coefficients for the RGEs of the gauge couplings from to scale upto two loop level for Non-SUSY and SUSY cases respectively:

D-parity not conserved

D-parity conserved

iii.1.2

can be spontaneously broken to through the VEVs of the possible scalars . These two fields contain sub-multiplets under . One can also think of this possible breaking via , using the combined VEVs of these fields and the fields mentioned in earlier section.

-
 Scalars - -
- -
 Fermions
Table 7:

All representations of fermions and scalars, which take part in the RG evolution of the gauge couplings from to scale and contribute to the respective coefficient computation, are tabulated in Table. 7. The VEVs of and conserve and break D-parity respectively.

is broken spontaneously through the VEV of at the intermediate scale and we find as a remnant symmetry. Here, remains unbroken, ensured by the singlet structure of . The generator of is a linear combination of the generators of and at the intermediate scale ; this helps us write the matching condition at this scale:

(29)

We have computed the coefficients which are relevant for the running between and scales upto two loop level for both Non-SUSY and SUSY cases. These are listed below:

D-parity not conserved