Unification with Vector-like fermions and signals at LHC

We look for minimal extensions of Standard Model with vector like fermions leading to precision unification of gauge couplings. Constraints from proton decay, Higgs stability and perturbativity are considered. The simplest models contain several copies of vector fermions in two different (incomplete) representations. Some of these models encompass Type III seesaw mechanism for neutrino masses whereas some others have a dark matter candidate. In all the models, at least one of the candidates has non-trivial representation under $SU(3)_{color}$. In the limit of vanishing Yukawa couplings, new QCD bound states are formed, which can be probed at LHC. The present limits based on results from 13 TeV already probe these particles for masses around a TeV. Similar models can be constructed with three or four vector representations, examples of which are presented.


Introduction
For the past few decades, the path to Beyond Standard Model (BSM) physics has been dictated mostly by solutions to the hierarchy problem [1]. However, with no experimental evidence to support this endeavor, from either LEP, Tevatron or the LHC so far, one might wish to explore alternate paths which do not contain a solution to the hierarchy problem. Furthermore, there could solutions to the hierarchy problem which do not introduce any new particles all the way up to GUT scales. The relaxion idea and its variant for example, propose a cosmological solution to the hierarchy problem without introducing any new physics at the weak scale [2][3][4][5].
One of the guiding principles for these alternate paths is the unification of gauge coupling constants. Popular models like split supersymmetry [6][7][8] have been proposed which have part of the MSSM particle spectrum at the weak scale and rest (scalar spectrum) at an intermediate scale.
The current limits on the stable, long lived R-hadrons which are a prediction of these models are about 1.5-1.61 TeV [9,10]. However, this framework depends crucially on the underlying MSSM framework. Generalization without supersymmetry are important to explore.
With this view point, we revisit extensions of the Standard Model with vector-like fermions which lead to precision gauge coupling unification (for earlier works in this direction, please see ). There are several virtues of these models: (i) They have minimal constraints from electroweak precision parameters, especially from S and T parameters [33][34][35][36][37][38][39], as long as the mixing between vector-like fermions and SM fermions is small.
(ii) They do not lead to any anomalies as they are vector in nature.
(iii) They can be tested directly at the collider experiments like LHC. The kind of signals depend whether on the amount of mixing they have with the Standard Model fields.
(iv) If they have mixing with SM quarks, it is possible that they can be probed indirectly in flavour physics.
To our knowledge, there has not been a recent survey of models containing vector fermions leading to gauge coupling unification. An earlier analysis was done in Ref. [13] with the available LEP data at that time. We have updated where those models stand in the Appendix, F. In addition to improvements in the gauge coupling measurements and theoretical threshold calculations which are now available at NNLO, an important role is played by the experimental discovery and the (almost) precise determination of the Higgs mass. It has been shown that the Higgs potential becomes unstable from scales close to 10 11 GeV [40], depending on the exact values of the top mass and alpha s . Thus a Grand Unified Theory should not only lead to gauge coupling unification but also keep Higgs potential stable all the way up to the GUT scale.
In the models presented here as we will see the Higgs potential naturally remains stable all the way up to the GUT scale. In the view that the primary existence of these vector particles is unification of gauge couplings, we dub them "unificons". However, as we will see later, these models do not restrict themselves only to unification. In some models, we find solutions with a provision for Type III seesaw mechanism for neutrino masses, and in some others there is a WIMP (Weakly Interacting Massive Particle) dark matter candidate. Thus "unificon" models can indeed have wide phenomenological reach solving other problems in Standard Model like neutrino masses and dark matter.
As a search for all possible models with extra vector-like fermions would be a herculean task, we resort to minimality. We assume unification of gauge couplingsá la SU (5). Additional vector-like particles appear as incomplete representations of SU (5). We have looked at all possible incomplete decompositions emanating from SU (5) representations up to dimension 75. The number of copies in each representation is taken to be n which is an integer between 1 and 6. The mass range of these additional vector-like fermions is chosen to be m ∼ k TeV, where k is a O(1) number taken to be approximately between 1/4 to 5. There are no solutions with successful gauge coupling unification as long as the vector-like fermions come in one single representation. This holds true even if increase the number of copies all the way to six, the maximum we have allowed per representation 1 . The minimal set of successful models with two different representations each with varied number of copies is listed in Table 2. All these models satisfy constraints from proton decay and the stability of the Higgs potential. Both representations come in several copies. Some solution allows for degeneracy between the fermions of the different representations, where as in some cases require non-degeneracy of the fermions in representation 1 and fermions in representation 2.
Interestingly all models have at least one representation with non trivial colour quantum numbers which makes them attractive from LHC point of view. In the limit of negligible Yukawa couplings, these colour states in SU(3) representations of the type 3, 6, 8 form bound states and are produced at LHC. The present limits on these bound states from 13 TeV run of LHC are already touching the 1 TeV mark, depending on the decay mode and the final states. We provide in detail limits on the relevant SU(3) representation bound states.
We also looked for solutions with three and four different representations. Unlike the two representation case, we considered degenerate spectrum for all the vector-like fermions in these two cases. Several solutions are found which are listed in Appendix B and Appendix C. The rest of the paper is organised as follows: In the next Section 2 we recap the essential RG required for gauge coupling unification and stability of the Higgs potential. In Section 3 we present the results for two fermion different representation case. In Section 4 we present the properties of each successful model. In Section 5, we discuss the bound state formalism of the colour vector-like fermions and limit from LHC. We close with a conclusion and outlook. In Appendix D we have tabulated all forty representations of SU (3) × SU (2) × U (1) coming from SU (5) representations upto dimension 75 [41] with their Dynkin index. In Appendix E constraints on mixing between SM fermions with vector-like quark is summarized. Appendix G summaries the two-loop RG equation of Standard Model.

One loop gauge unification
It is well known that gauge couplings do not unify precisely in the Standard Model. If one insists on unification of the guage couplings at the GUT scale, the required sin 2 θ W (M 2 Z ) is 0.204 (for one loop beta functions) instead of the current experimental value of sin 2 θ W (M 2 Z ) = 0.23129 ± 0.00005 [42]. As argued in the introduction, in the present work, we look for additional vector-like matter fermions, close to the weak scale, which can compensate the deviation and lead to successful gauge coupling unification. At the 1-loop level, the beta functions for the three gauge couplings are given as where is l = {U (1), SU (2), SU (3)} runs over all the three gauge groups. The b l functions have the general form: Here C(R) is quadratic Casimir and T (R) is Dynkin index of representation R. V, F and S represents vector, Weyl fermion and complex scalar field respectively. For U(1) group T (R 1 ) and C(R 1 ) are For SU(N) group T (R) is defined as follows: The following are the list of Dynkin Indices for lower dimensional Representations: Second Rank symmetric tensor n+2 2 More complete list on quadratic casimirs can be found in [43]. Within SM, the beta functions take the value In the presence of a vector-like fermion V 1 at the scale M 1 greater than weak scale, given the gauge coupling unification at M GU T , the (Eq. (2.1)) take the form: .
Where the second line can be derived from Eq. (2.6) assuming unification at M GU T . The parameters ∆b lk are defined as b l − b k . In the absence of new vector-like particles,b is independent of the running scale µ. In their presence however, there is a µ dependence but it is typically mild. For the case where the new particles are close to weak scale ∼ TeV, and when µ in = M Z , the log factor, ln(M 1 /M GU T )/ ln(µ/M GU T ) is close to one. In this case, the expression for unified theories is given byb Note that the Eq. (2.7) can purely be determined from experiments at M Z . Its value is given bȳ b(M Z ) = 0.718, (2.10) In the SM, if we insist on unified gauge couplings at M GU T , at the weak scale,b takes the value 0.5 clearly in conflict with experiments. In MSSM,b turns out to be 5/7. Of course, these arguments are valid only at one loop. There is deviation in Eq. (2.9) when higher loops are considered. In our analysis, most of the successful models have ab of 0.67 to 0.833. The above discussion can be easily generalised for more than one Vector field V i at scales M i . It has the following general form at the 1-loop level.b .
where we assumed the hierarchy of the scales as M 1 < M 2 < M 3 etc.

Two loop RG evolution of gauge couplings
To improve the precision in unification of gauge couplings, we consider two loop beta functions. At the two loop level, the beta functions involve Yukawa couplings which makes them model dependent.
Vector-like fermions which typically have "bare" mass terms in the Lagrangian, can also mix with the Standard Model fermions through Yukawa interactions if allowed by the gauge symmetry. However, this mixing is subject to strong phenomenological constraints [33][34][35][36][37]. A detailed Discussion on the mixing constraints can be found in Appendix E.
In the present analysis, we restrict ourselves to models with minimal or zero vector-like fermion and SM mixing through the Higgs mechanism. With this assumption, we can safely neglect the Yukawa contribution from the new sector to the gauge coupling unification. The RG equations at the two loop level are given by [44][45][46]: where the first term in the right hand side is due to one-loop which was discussed in the previous subsection. The second term is purely from gauge interactions whereas the third terms involves the Yukawa terms Y u,d,e where the suffixes mean the up-type,down-type and lepton-type couplings. The expression for the coefficients appearing in the second term of the above equation are as follows [44,47]: In the third term of Eq. (2.12), we have the coefficients C lf and for the standard model particles it has the following form: (2.16) As we are considering the Yukawa couplings between the vector-like fermions with Higgs boson to be negligible 2 , the contribution of vector-like particles to C lf coefficient can be taken as zero. On the other hand δm ij = 0, where δ is used to indicate contribution from additional vector-like fermions. We'll give explicit values of δm ij for each of the viable models in Section 4. Two-loop RG running for the Yukawa couplings is given as The SM RG for these Yukawa couplings are shown in Appendix G. Here we will address the effect of new fermion fields in RG of Yukawa couplings [47][48][49][50]. The one loop beta functions of these couplings are not be affected by new matter(fermion) fields because we considered the Yukawa couplings between the vector-like fermions with Higgs boson to be negligible. Two loop beta functions get contributions from the diagrams shown in Fig. (1), which results in the following terms: in the safe region. While this is true, the evolution of the SM Yukawa couplings is itself modified in these models as seen in the previous sub-section. It is thus worthwhile to check explicitly the stability of Higgs self coupling along with gauge coupling unification.

Evolution of Higgs Self coupling
To check the Higgs stability we follow [40,51,52] who have checked for the stability using three loop beta functions and NNLO matching conditions. We use the beta function of the λ at the two loop and put a condition that λ is always positive at all scales of evolution. Two-loop RG running for the Higgs quartic coupling are shown below where beta functions for SM Higgs quartic couplings are defined in Appendix G. The effect of new fermion fields in RG of Higgs quartic couplings are: To solve the RG equations we need boundary values of the coupling constants and masses at the top mass (M t ) scale. The quantities of interest are Higgs quartic coupling (λ), Yukawa couplings and gauge coupling, which can be calculated in terms of physical observables W-boson mass (M W ), Z-boson mass (M Z ), Higgs mass (M h ) and α 3 (M Z ) at the two loop level. The input parameters are calculated in the M S-scheme. More detailed can be found in [52]. For the RG running we use the central value of Top mass. The input values of SM parameters and couplings are listed in Table 1.
Values of the relevant couplings at scale M t are as follows: where all the parameters with mass dimension has written in GeV. Central values of the above couplings are calculated upto NNLO ( [40] for λ) order for all of them except the y t (M t ) for which we considered NNNLO [57][58][59]. The value of α 3 (M Z ), is extracted from the global fit of Ref. [56] in the effective SM with 5 flavours. Including RG running from M Z to M t at 4 loops in QCD and at 2 loops in the electroweak gauge interactions, and 3 loop QCD matching at M t to the full SM with 6 flavours, the strong gauge couplig is calculated. The contribution of the bottom and tau Yukawa couplings, are computed from the M S b-quark mass (M b (M t ) = 2.75 GeV) and Tau mass (M τ (M t ) = 1.742 GeV) [60].
Threshold Corrections at GUT Scale One of the main concerns which remains now is the possible effect of threshold corrections at the GUT scale, which can be quite significant. These corrections are highly model dependent. In some GUT models, with no extra matter at the weak scale (other than the Standard Model particle content), it is possible to achieve gauge coupling unification through large threshold corrections at the GUT scale [61]. While such extreme situations are no longer valid due to the constraint on the stability of the Standard Model Higgs potential, it is still possible that GUT scale threshold corrections could play an important role. To study the impact of threshold corrections on gauge coupling unification, we define the following parameters: α ave. (µ) = (α 1 (µ) + α 2 (µ) + α 3 (µ))/3 and¯ i (µ) = (α i (µ) − α avg (µ))/α ave (µ). Note that α ave coincides with α GU T when all¯ i → 0, at the scale M GU T . In the presence of threshold corrections, one could allow for deviations in α GU T in terms of¯ i at the GUT scale 4 Defining = max(¯ i ), we see that is as large as 6% in the Standard Model. In our survey of models below, we have allowed for variations in up to 1.2%. A more conservative set of models is tabulated in Appendix A which have of 3%.

Gauge coupling unification with vector-like fermions
As mentioned in Introduction, in our search for successful models with gauge coupling unification, we focus on vector-like matter in incomplete representations of SU(5). We have considered (incomplete) representations [41] up-to dimension 75, which contains a 15 of SU(3) of QCD as the largest component. The full list of incomplete representations is presented in Appendix D. As can be seen from the Table 8, there are 40 representations which we have considered. Note that representations 4, 5 in Table 8 do not come as incomplete representations of SU(5) instead they are singlet representations of SU (5). Our search strategy is start with n i copies of representation i , with all the n i copies degenerate in mass, m i and look for unification of the gauge couplings. The maximum number of copies is taken to be 10. The number of representation types i considered simultaneously is restricted up to four. An important constraint comes from proton decay, which restricts the scale of unification to lie above (at least) 10 15 GeV. As mentioned above, in addition to unification, we also consider that the Higgs potential should be stable all the way up to the GUT scale. In the computations, we have also varied the input parameters to lie within their two sigma regions. The masses of new vector-like are assumed to lie between 250 GeV -5 TeV.
For i = 1 we searched for the mass of the vector-like fermion from 250 GeV -5TeV, considering number of vector-like fermions n 1 = 6. These masses for n 1 copies have been considered degenerate for simplicity and no successful model was observed. The simplest solutions we found contain at least two different representation content each with a different number of copies. We call these solutions "minimal unificon models". These are listed in Table 2. We now explain the notation used in the Table. The two representations considered are called Rep1 and Rep2. The representation is described as n i (R SU (3) , R SU (2) , R U (1) ), where n i introduced earlier is the number of copies of the representation, R G is the representation of the field under the gauge group G of the SM.

MRep2
One Two Vaccum  Furthermore, in the above, we mentioned only one part of the representation instead of the complete vector multiplet for brevity. For example, 1, 2, 1 2 actually means 1, 2, 1 2 ⊕ 1, 2, − 1 2 . Colored representations like 3, 1, 2 3 may mean two possibilities: (a) 3, 1, 2 On the other hand, the real representations like (1, 3, 0) and (8, 1, 0) are not short-hand notations. In the second last column, the entries are written in units of 10 16 GeV. Thus except the first model, all the models have unification scale larger than 10 16 GeV. All models appeared as the solution of one loop RG equation. In the third and fifth columns, we show the mass range of the vector-like fields. One can see that if we increase the mass of one representation, the mass of the other field also increases (as shown in Fig Solutions with three types of representations are also possible. These are listed in Table 6 of Appendix B. Here we made a restricted choice that all the representations and their copies are degenerate in mass of about 1 TeV. As can been seen from the Table, the minimum number of extra vector-like fermions required is seven over the three representations, where as the maximum number is eighteen. All of them have unification scale less than 10 16 GeV, which puts them at risk with Proton decay. The life time of the proton in these models is of order ∼ 10 32 years which in contrast with the experimental value > 10 32 − 10 34 years [63]. The maximum number of representations we have chosen simultaneously is four. Searching for models with different masses for each copy and each representation is computationally very intensive. Thus, we have considered all the four representations and their copies to be degenerate in mass at 1 TeV. The list of successful models is given in Table 7 of Appendix C. The minimum number of vector-like particles required over all representations is five and the maximum is twenty. As with the three representation case, we find that the Unification scale is smaller than 10 16 GeV with the exception of one model (Model No 17 of Table 7). As before from the arguments of Proton decay, these models can have potentially small proton life times in conflict with experiment. We do not address this issue here.

Minimal unificon models
In this section we concentrate on the Minimal vector-like fermion Unification Models. The list of such of models is given in Table 2. Several interesting features are evident from the Table 2.
(a) Except for the first and eigth model, all the models have unification scale above 10 16 and thus are safe with proton decay. (b) The minimalist model is model 7, with only two vector-like fermions one with a mass range of 0.250-5 TeV and another within a mass range of 250-5000 GeV. This model might have constraints from direct searches of vector-like quarks at LHC and elsewhere if there is significant mixing with SM particles. In its absence, as we assumed here, the bound will be different. We will discuss it in the next section. (c) The maximum number of vector-like fermions needed is nine in Model 9.
We now discuss each of these models in detail.

Model 1
In this model 5 , we have six copies of 1, 2, 1 2 , which we called Rep1, with mass range between 250 GeV to 5000 GeV and one copy of 6, 1, 1 3 , called Rep2, with mass range from 250 GeV to 5000 GeV. Rep1 field is lepton doublet like field and thus it can interact with right handed electron and the Higgs field through Yukawa interactions. This field mainly decays to gauge bosons like Z boson and W ± . For the sake of simplicity of the two loop gauge coupling RG running, we impose appropriate Z N symmetries to these fermion doublets. This symmetry cut-down all the Yukawa terms involving these fields at the renormalisable level and only gauge couplings are allowed. Lightest neutral component of these fermions can be a dark matter candidate. This type of dark matter is called inert fermion doublet dark matter [65,66]. Rep2 is more exotic and at the renormalisation level, it can interact with the gauge bosons only. It cannot decay to any standard model particles. Thus they form bound states. Phenomenology of this is studied in detail in the next Section 5. (4.1) 5 We have cross-checked our Two Loop RG equation of this model with the publicly code SARAH [64] for consistency.

Model 2
We got six copies of Rep1 = 1, 2, 1 2 in mass range between 250 GeV to 2000 GeV and two copies of Rep2 = (8, 1, 0) with mass range from 500 GeV to 5 TeV. Similar to the previous model, Rep1 field is lepton like field and thus all the comments are applicable here. Rep2 is gluino like and at the renormalisation level, it can interact with the gluons only and does not have any decay chain. Possibility of any higher dimension decaying operators and its collider phenomenology are studied in the next Section 5.
In the model, M Rep1 is always less than M Rep2 . The change in the beta functions in the two thresholds are as follows:    (4.14)

Model 3
In this model, we got two copies of Rep1 = (1, 3, 0) and four copies of Rep2 = 3, 1, 1 3 . The mass ranges of Rep1 and Rep2 are (250 GeV,5 TeV) and (250 GeV,5 TeV) respectively. Rep1 can be a viable candidate of type III [67,68] seesaw model with fermion mass of M. The neutrino masses are generically given by a factor v 2 /M , where v is the vacuum expectation value of the Higgs field. For large M (of the order of 10 14 GeV), small neutrino masses are generated even for Yukawa couplings of ∼ 1. On the other hand, either smaller Yukawa couplings ∼ 10 −11 (which would not effect the RG running) or extended seesaw mechanisms, such as those of the inverse seesaw models [69], are  required to obtain small neutrino masses while keeping M close to a few hundreds of GeV. However, we can also impose appropriate Z N symmetries. This symmetry removes all the Yukawa terms involving these fields at the renormalisable level and only gauge couplings are allowed 6 . Neutral component of these fermions is a viable dark matter candidate. This type of dark matter are referred as wino like dark matter and have been discussed in [70][71][72][73].
Rep2 has same representation like the down quark. This colour vector-like fermion can form a bound state and annihilate to diphoton, dijet etc. event, which we studied in Section 5.
Seesaw requires Yukawa couplings, our model does not have a seesaw mechanism for neutrino masses.
A sample unification point is shown in Fig. (6)(a), three copies of weak isospin triplet vector-like fermions with degenerate mass of 4.16 TeV and one copy of color sextet vector-like fermion with a mass of 280 GeV is considered. The Fig. (6)(a) shows unification clearly. Fig. (6)(b) shows the mass distribution in Rep1-Rep2 mass plane.

Model 6
This model consist of one copy of Rep1=(1, 4, 1 2 ) and two copies of Rep2=(6, 1, Gauge coupling unification and running of y t and λ are also shown in Fig. (7)(a), with one copy of weak isospin fourplet vector-like fermions with degenerate mass of 1.51 TeV and two copies of color sextet vector-like fermion with a mass of 4.81 TeV. The Fig. (7)(b) has the mass distribution in Rep1-Rep2 mass plane.

Model 7
This model consist of one copy of Rep1=(3, 1,

Model 8
This model consist of four copies of Rep1=(1, 2, 1 2 ) and one copy of Rep2=(8, 1, 0). The mass range for Rep1 and Rep2 are (300 GeV to 5 TeV) and (300 GeV to 5 TeV) respectively. This representation has been discussed in Model 2 with different number of particles for each represenatation. The difference can been studied with their bound state decay to diphoton channel and dijet, as shown in Section 5.
For most points in this model vector-like fermions in Rep1 can be degenerate with vector-like fermions in Rep2 (M Rep1 ∼ M Rep2 ) as shown in Fig. (9)

Model 9
This model consist of three copies of Rep1=(1, 3, 0) and six copies of Rep2= (3, 1, 1 3 ). The mass range for Rep1 and Rep2 are (1.1 TeV to 5 TeV) and (250 GeV to 1.8 TeV) respectively. This representation has been discussed in Model 3 with different number of particle for each represenatation. The difference can been studied with their bound state decay to diphoton channel and dijet, as shown in Section 5. In the model, M Rep1 is greater than M Rep2 . The change in the beta functions in the two thresholds are as follows:

Collider Signature of Minimal vector-like fermion models
The models listed in Table 2 have several exotic states lying close to electroweak scale, which can be probed at LHC. Models have exotic lepton like states (uncoloured) mostly in doublet, triplet and fourplet representation of SU (2). These states are produced at the LHC through Drell-Yan process and typically have cross-section of the order 10 fb [74](roughly slepton production or exotic lepton production). These particles decay through Yukawa interaction to lighter SM leptons. In the limit of vanishing Yukawa couplings, these particles can manifest as missing energy and disappearing charge track at LHC and limits from monojets and disappearing tracks could apply to our model. The LHC at 14 TeV with integrated luminosity 3000 fb −1 is only sensitive to mass of order 400 GeV [75]. In the following we will concentrate on the strongly interacting exotic sector; which appears in all the successful models.

Decay Operators
The models tabulated in the above has exotic fields and some of these fields don't have renormalisation level decay operators. These fields are (i) 6, 1, 1 3 , (ii) 6, 1, 2 3 and (iii) (8, 1, 0). Now question is whether we can have higher dimensional operators or not. Note that if there exists any higher dimensional operator then there must be some new fields which got integrated out in some higher scales. Now this scale has to be high (close to the GUT scale) as otherwise unification will be disturbed. These higher dimensional operators are suppressed as where dim(O) is the dimension of the operator O. Six-dimensional operators are suppressed by square of the GUT scale and thus life-time of the particle is expected to be High (∼ 10 33 years).
Thus we are focusing only on the five dimensional operators. Any five dimensional operator for decay of such particle must have the forms: (1) Exotic field × a standard model fermion × Higgs × Higgs (5.2) (2) Exotic field × a standard model fermion × Gauge boson × Gauge boson, (5.3) where in the place of the Higgs and SM fermions fields one can use their conjugate fields. Thus colour charge of the exotic field has to be neutralized by SM fermion to form a five dimensional operator involving the Higgs. In the SM, there is no such field and hence possibility (1) is not possible. For the second case, colour representation of the exotic times that of the SM fermion field must transform as any one of 1, 8, 10 and 27 dimensional representation. However we don't have SM field with above representation hence, this second possibility is also ruled out. These exotic fields can form a bound state and in the next subsection we'll discuss this in details.

Formalism for Bound state
In this section we investigate the possibility of producing bound states of the colour vector-like fermions. The idea of bound state has been studied, in understanding bottom and charm quark through their bound states. For the formation of bound state, we assume the new vector-like fermion (ψ) is long lived so that it has time to form a bound state prior to decaying. This condition is easily satisfied in our case, as the Yukawa coupling between the new vector-like fermions and SM particle is assumed to be negligible. The bound state formalism has been studied in [76,77], where they focus on pair-produced colour particle Beyond the Standard Model by the observation of diphoton, dijet etc. resonances arising from QCD bound state. We assume that the only interaction that contribute to the production of bound state is the Standard Model SU(3) colour gauge interaction. We estimate the annihilation rates and parton-level cross-section at leading order, along with NLO MSTW parton distribution functions [78], to compute the LHC signals for √ s= 8 TeV, 13 TeV and 14 TeV evaluated at scale m ψ . The production cross section of colour singlet spin zero bound state from constituent vector-like fermion with colour representation 3, 6, 8 are shown in Fig. (11) and Fig. (14). As pointed out in Ref. [79], NLO corrections to cross-section can increase the diphoton resonance arising from stoponium by 25%. Therefore, large uncertainties are expected in our result of factor of two or so. This still can allow us to constraints minimal vector-like fermion model. Further uncertainty in our results arises because of limits extracted from ATLAS and CMS result, which is obtained for a fixed spin and production channel. Signal shape have some dependence on the acceptance, intrinsic width and whether a jet is due to parton-level gluon or quark, this adds to some uncertainties. A pair of ψψ near threshold can form a QCD bound state, which we defined as O. If the decay width of O is smaller than its respective binding energy, it can be observed as a resonance which annihilates to SM particles. For particles (ψ) of mass m ψ Λ QCD , the Bohr radius of relevant bound state is much smaller than the QCD scale and the velocity of its constituents is non relativistic, we can estimate bound state as modified hydrogenic approximation. For a particle ψ in the colour representation R, the potential between ψ andψ depends on the colour representation R of the ψψ pair through the casimirs of R and R as whereᾱ s is defined as the running coupling at the scale of the average distance between the two particle in the corresponding hydrogenic state, which is order of the Bohr radius a 0 = 2/(Cᾱ s m ψ ) (for which we used Ref. [80]). The binding energy of the wave functions at the origin for the ground state are given by The quantum number of ψ determines the production as well as the decay modes of bound state particle O. The cross-section for the bound state O to be produced by initial-state partons x and y is given asσ where M = 2m ψ + E b is the mass of the bound state, β(ŝ) is the velocity of ψ orψ in CM frame. The production cross-section of any narrow resonance O of mass M and spin J from parton x and y, and the decay rate of bound state to x and y, are related bŷ where D O denotes the colour representation of particle O.
In the next subsection we will strict ourself to study the colour singlet and spin zero (J=0) bound state system. Assuming the production cross-section of ψψ is dominated by gluon fusion. The gluon fusion partonic production cross-section of bound state is given bŷ Depending on the quantum number of ψ, bound state O can decay to diphoton, dijet, Zγ, ZZ and W + W − channels. The production of preceding pair events produced in proton-proton collisions in LHC can be predicted as σ(pp → O) × BR(O → X 1 X 2 ).
Here we will identify the channels in which the bound state resonance would be most easily measurable and compute the corresponding cross-section as a function of the mass, colour representation and charge of the constituent particles. The promising final states that we analyzed are diphoton and dijet channels. In the case of SU(2) multiplet the large mass splitting is constrained by Electroweak precision test, which modifies the oblique parameter T and S [38], hence we have analysed our results in degenerate mass scenario.

5.
3.1 γγ, ZZ, Zγ, W + W − channel Any spin half particle can be produced in pairs (in gg collisions ) in an S-wave J = 0 colour singlet bound state, which can decay as typically narrow γγ, ZZ, Zγ resonance. The decay width of the γγ, ZZ, Zγ signal due to spin J = 0 bound state is given as [81] Γ  (2) doublet. This can also decay to W + W − channel, which is comparable to γγ channel. The decay width for W + W − is given as [81], where The branching fraction of the isoweak singlet fermions which satisfied the gauge coupling unification and vacuum stability are tabulated in Table 3.
Both ATLAS and CMS have performed a search of resonant production of photon pairs for scalar particle (J=0). ATLAS [82] analysis is based on data corresponding to an integrated luminosity of 15.4 fb −1 at √ s=13 recorded in 2015 and 2016. CMS [83] data sample correspond to luminosity 12.9 fb −1 at √ s=13 in 2016, combined statistically with the previous data of 2012 and 2015 at √ s=8 and √ s=13 respectively, with luminosity of 19.7 and 3.3 fb −1 .

Dijet channel
S-wave bound state with spin J = 0 can be produced via gg → O and annihilating mostly to gg. For j=1/2 there is also a comparable contribution from S-wave J = 1 colour octet bound states produced via qq → O and annihilating to qq, which we will not discuss here. The decay width of gg signal due to spin J = 0 colour singlet bound state is,   (×2 for Complex Representation of constituent fermion) Search for narrow resonances decaying to dijet final states in proton-proton collision has been performed by the ATLAS and CMS collaborations using the LHC run data at √ s 8 TeV as well as 13 TeV. CMS [84] study has been performed with integrated luminosity 18.8 fb −1 at √ s = 8 TeV using a novel technique called data scouting. ATLAS [85] has studied with √ s= 8 TeV using full integrated luminosity of 20.3 fb −1 masses upto 4.5 TeV. In run-II, ATLAS [86] with centre-of-mass energy √ s= 13 has studied the dijet search using the data collected in 2015 and 2016 with luminosity 3.5 fb −1 and 33.5 fb −1 respectively and CMS [87] has presented a data with luminosity 36 fb −1 considering masses above 600 GeV.

Limits on Signals from CMS and ATLAS
In next section we examine the constraints on masses of bound state from dijet and diphoton bounds considering one copy of constituent vector-like fermions. We have used the recent limits of ATLAS and CMS for diphoton resonance at centre of energy √ s=13 TeV from 2015 and as well as 2016 data. Dijet bounds has been considered for centre of energy √ s=8 and 13 TeV from both ATLAS and CMS.
As we have n number of copies of vector-like fermions described in the in Section 4 for two fermions representation, we will give the exclusion limits of vector-like fermion particle occurring in

Model
Representation different models with n number of copies in the Table 4.

Dijet Bounds
In Fig. (12 Fig. (12)(b) shows the dijet limits from ATLAS(black) [88] 13 TeV and CMS(blue) [89] 13 TeV data. We can clearly say that the dijet limits are not strong enough to rule any of the models, if they have only one copy of constituent fermions. In the Fig. (15), we have plotted (black solid line) the projected limit for 14 TeV LHC at 3000 fb −1 for the dijet cross Section [90] . Assuming Z B model, 14 TeV limits on mass of Z B and coupling between Z B gauge field with quark has been calculated in Ref. [90]. Using this limit, we have calculated 14 TeV projected limit on dijet cross-section. We have found that mass of vector-like   fermion with colour representation six can be excluded up to 800-900 GeV at the HL-LHC.

Diphoton Bounds
The diphoton channel has played a very important role in discovering the Higgs Boson. It can be a very important channel to look at BSM physics. We present the production of diphoton channel as a function of the resonance mass considering one copy of constituent vector-like fermions in Fig. (13). Black line is the upper limit on this cross-section from ATLAS [82] 13 TeV and blue line is from CMS [83] 13 TeV data. It can be observed that the upper limits on cross-section can give stringent bound on the masses of vector-like fermions (m ψ = M/2).
There has been searches in Zγ, ZZ and W W resonances from these bound states. ATLAS [91] has performed a combination of individual searches in all-leptonic, and all hadronic final states to search for heavy bosons decaying to ZZ and W W with integrated luminosity of 20.3 fb −1 at 8 TeV. The sensitivity is weaker than γγ channel for ATLAS [92] at 8 TeV by around 1000. Both CMS [93] and ATLAS [94] have performed a resonance decaying to Zγ at centre-of-mass energy of 8TeV at integrated luminosity 20.3 and 19.7 fb −1 respectively. Where sensitivity is weaker than diphoton channel is weaker by order 10.
CMS [95] has performed a searches in Zγ resonance in leptonic channel final decay state at centre of mass energies of 8 and 13 TeV. The bounds are weaker than diphoton bounds by factor of 200. ATLAS [96] has searched for heavy resonance decaying to ZZ and ZW pair decaying to leptonic and hadronic channels at a centre of mass energy 13 TeV with total integrated luminosity 13.2 fb −1 . The sensitivity is still weaker by factor 1000 with respect to diphoton channel.

Summary and Outlook
Unification of gauge couplings is one of the most important signatures of a successful Grand Unified Theory beyond the electroweak scale. We look for models with extra vector-like fermions at the weak scale which can lead to successful unification of gauge couplings. With two representation, we find a class of nine models leading to successful unification of gauge couplings. An interesting aspect of these is that all of them contain coloured vector-like fermion in the spectrum. The coloured set of the vector-like fermions can be probed at LHC by looking for bound states formed by them and their probable decays. We have already listed the present bounds from LHC for each successful model. The future runs of LHC are sensitive to further mass ranges of these particles. Finally, it would be interesting to look for complete GUT models with this particle spectrum.

B Three Representation Case
Here we enlist the models which satisfy gauge coupling unification and positivity of higgs potenatial for three fermion representation model. Unlike Two Representation case, we made a restricted choice that all the representations and their copies are degenerate in mass of about 1 TeV, with up to ten copies in each representation. All of the models have unification scale less than 10 16 GeV, which does not satisfy with Proton decay constraint. The models are listed below in Table 6 ModelNo. Rep Table 6. Models satisfying three fermion representation of gauge coupling unification and stable higgs potenatial with degenerate mass of 1TeV. The representation is described as ni(R SU (3) , R SU (2) , R U (1) ), where ni introduced earlier is the number of copies of the representation, RG is the representation of the field under the gauge group G of the SM.

C Four Representation Case
Here we enlist the models which satisfy gauge coupling unification and stable higgs potenatial upto grand unified scale for four fermion representation model. Here also we restricted representations and their copies are degenerate in mass of about 1 TeV. We have allowed for up to ten copies in each model. Except one model, all of the models have unification scale less than 10 16 GeV, which does not satisfy with Proton decay constraint. The models are listed below in Table 7 ModelNo. Rep Table 7. Models satisfying four fermion representation of gauge coupling unification and stable higgs potenatial with degenerate mass of 1 TeV. The representation is described as ni(R SU (3) , R SU (2) , R U (1) ), where ni introduced earlier is the number of copies of the representation, RG is the representation of the field under the gauge group G of the SM.

D Representations and Dynkin indices
We considered all the SU (3) × SU (2) × U (1) representations coming from SU (5) representations upto dimension 75. In Table 8, we listed those forty representations [41] with their contribution to beta function (i.e. Dynkin index) considering them as scalar fields. One can straight-forwardly derive corresponding Dynkin indices if the fileds are vector-like fermion just by multiplying the tabulated value with 2 if the representation is real and by multiplying with 4 if the considered representation is complex.  Table 8. Representation of fields considered in this paper. In the column entitled with "SM Rep" we put incomplete multiplets of SU (5) and the entries inside the brackets are SU (3), SU (2) and U (1) representations respectively. In the column with title we'd written the SU (5) representations from which those representations are coming. Dynkin indices are calculated assuming the fields are scalar fields. Note that we had considered up the SU(5) representation of dimension 75. There are some extra representations as well.
Here we will briefly overview the leading order constraints coming from EW precision tests, direct searches at colliders and Higgs physics. It is reasonable to assume that, only third family of SM fermions have sizable contribution from new vector-like fermions.

E.1 Vector like quarks
Due to mixing of the SM top and bottom quark with vector-like fermions partners, the resulting physical up and down type quark mass eigenstates u 0 , c 0 , t 0 , T 0 and d 0 , s 0 , b 0 , B 0 may contain non-zero T and B components, leading to a deviation in their couplings to Z and W± bosons. In this case, the relation between weak and mass eigenstates for up quark can be parameterized as two  Table 9. Vector-like fermions, that provide a consistent extension of the SM and modify the Higgs boson couplings [37]. where m q and m Q are the mass of SM fermion and vector-like fermion respectively. This mixing gives new contributions to the oblique parameters S and T [104], which is precisely measured at LEP and SLC. The contributions to S, T in models with T, B singlets and (T B) doublets are studied in [34,37,38,105], which would give a constraints in mixing parameters between SM and their vector-like fermions partners. For singlet B quark, the constraints from R b is strong, which gives upper bound on mixing sin θ d L to be 0.04. For singlet T quark upper bound of sin θ u L is 0.15 to 0.10 for mass range 600 GeV to 2 TeV respectively, from S and T parameter. For (T B) doublet, the constraints from EW precision gives upper bound on sin θ d R to be 0.06 and, sin θ u R between 0.13 to 0.09 for mass range 600 GeV to 2 TeV respectively, considering the splitting between M B and M T of 2 GeV.
Direct Searches A full model of vector-like Quark decaying to SM particles and search strategies to discover at LHC has been studied in Ref. [34,106,107]  T B doublet assuming that they couple to the third generation, are the same as for singlets, We would summaries the mass constraints coming from direct searches of VLQ at the LHC. For Integrated luminosity of 19.5 fb −1 at √ s = 8 TeV CMS [108] experiment at the Large Hadron Collider searched for the T quark decaying into three different final states, bW, tZ, and tH. The search is carried out using events with at least one isolated lepton. The lower limits are set on the T quark mass at 95% confidence level between 687 and 782 GeV for all possible values of the branching fractions into the three different final states assuming strong production.
A search in CMS [109] is performed in five exclusive channels: a single-lepton channel, a multilepton channel, two all-hadronic channels optimized either for the bW or the tH decay, and one channel in which the Higgs boson decays into two photons. A statistical combination of these results is performed and lower limits on the T quark mass are set. Depending on the branch-ing fractions, lower mass limits between 720 and 920 GeV at 95 % confidence level are found. A search similar to Top like vector quark, heavy B quark vec-tor couplings to W, Z, and H bosons, is carried out by CMS experiment [110]. The B quark is assumed to be pair produced and to decay in one of three ways: to tW, bZ, or bH. The search is carried out in final states with one, two, and more than two charged leptons, as well as in fully hadronic final states.Each of the channels in the exclusive final-state topologies is designed to be sensitive to specific combinations of the B quark-antiquark pair decays. A statistical combination of these results gives lower limits on the B quark mass between 740 GeV and 900 GeV with 95 % confidence level, depending on the values of the branching fractions of the B quark to tW, bZ, and bH.
ATLAS has also searched for exotic quark, heavy X quark with Q = 5/3 decaying to tW gives a lower bound of mass 840 GeV [111] with 95% C.L. . Quark Y with chagre Q = −4/3 decaying to Wb gives lower bound of mass 770 GeV [112] with 95% C.L.. The experimental searches assume pair production via strong interactions and prominent decays in the indicated channels.

E.2 Vector like leptons
In this section we discuss new colourless fermions. Weak iso-triplet with zero hyper-charge vector-like fermion can couple to left l handed SM fermions and higgs as: Tr ΣM Σ Σ c + h.c. , (E.6) where the matrix notation of Σ is as follows The contribution of Σ to the EW precision parameters is vanishingly small [33], since the mixing angle are suppressed by ∼ m ν /M Σ and the loop induced mass splitting between the M Σ ± − M Σ 0 = 164 − 165 GeV [113]. In the limit Y Σ M Σ /v we can realize it as a type III seesaw model [67] with neutrino mass m ν = Y 2 Σ v 2 /M Σ . In the limit Y Σ → 0, this can be realized as a wino like dark matter [70]. SM fermions can also couple to four different possible vector-like leptons, a weak singlet E, a weak doublet L or Λ, a weak triplet ∆. The effect of these vector-like leptons on modification on the Higgs decays, anomalous magnetic moment to the muon and lepton flavour violation decays are studied in Refs. [35,100,[114][115][116][117].
Direct search The limits on M strongly depend on the SM generation that couples to the heavy leptons. The limits on doublet L, couplings only to the third generation is M L > 270 GeV and coupling with e and µ gives bound of M L > 450 GeV, Ref. [114], while the LEP limit remains more constraining in the case of the singlet E, M E > 100 GeV. For the exotic doublet Λ with a doubly-charged component, Ref. [115] reports M Λ > 320 GeV.