Renormalizable SU(5) Completions of a Zee-type Neutrino Mass Model

We explore the potential of a selected model of radiative neutrino masses to be implemented in a renormalizable SU(5) unification framework. The Zee-type model under consideration uncovers the SU(5) representations in which the new fields are embedded and which may contain also other light states leading to the unification of gauge couplings. We perform an exhaustive search which reveals specific patterns of new states and demonstrate that such patterns are consistent with a general choice of relevant scalar potential. It turns out that all of the specific scenarios which lead to successful unification include the colored scalars testable at the LHC.


Introduction
Despite all the phenomenological success of the Standard Model (SM), certain theoretical and experimental issues like neutrino masses, dark matter, charge quantization, hierarchy problem etc. seem to indicate the need to go beyond its well-established framework. The ultraviolet completions motivated by neutrino mass models may address the open questions and pave the road beyond the Standard Model (BSM). For example, the neutrino masses in canonical type-I [1,2,3,4,5], type-II [6,7,8,9,10,11], and type-III [12] tree-level seesaw models percolate down from a single scale that may be linked to the unification point of SM gauge couplings, hinted first within the SU(5) Grand Unified Theory (GUT) of Georgi and Glashow [13]. After realizing that there is no single gauge coupling crossing in this simplest GUT, it was noticed that augmenting the SM by the second Higgs doublet and the corresponding supersymmetric (SUSY) partners enables a successful minimal SUSY SM (MSSM) unification [14]. A decisive role [15] played by incomplete (or split) irreducible representations (irreps) 5 H in the MSSM unification success, motivated the corresponding non-SUSY attempts to cure the crossing problem [16,17] with just six copies of the SM Higgs doublet field and nothing more. Still, the scale of such unifications would be too low.
Further studies of unification in the context of non-SUSY SU(5) GUTs employed incomplete SU (5) irreps which contain new states introduced by tree-level seesaw models. The studies in [18,19,20] employed adjoint SU (5) representation 24 F which contains both the fermion singlet and the TeVscale fermion triplet fields providing a low scale hybrid of type-I and type-III seesaw models. Similarly, Refs. [21,22,23] employed 15 S SU(5) representation with the TeV-scale complex scalar triplet, employed in the type-II seesaw mechanism.
When considering possible GUT-embeddings of a radiative neutrino mass generating mechanism, we opt for genuine radiative Zee-type models, genuine in the sense that no additional symmetries are required to make them the dominant contribution to neutrino mass. At the same time, by avoiding fermion singlets we are choosing the SU(5) embedding and discard the SO (10) one. The first one-loop model proposed by Zee [24] has introduced only new scalar fields, the charged singlet and the second complex doublet, which do not lead to competing tree-level seesaw mechanisms. The embedding of original Zee model in renormalizable non-SUSY SU(5) setup has been studied in [25].
Our focus here will be on the variant of the Zee model presented in [26], which in the following we call the BPR model. It keeps the Zee's charged scalar singlet, but a real scalar triplet replaces Zee's second Higgs doublet. Finally, BPR model introduces three copies of vector-like lepton doublet fields which, if embedded in split 5 F , may influence the gauge running as twelve Higgs doublets. Let us note that besides the genuine one-loop model [26] there exist three-loop radiative neutrino models [27,28], where an automatic protection from the tree-level or lower loop-contributions has been achieved by introducing appropriate larger weak multiplets. However, the appearance of ∼ 10 6 GeV Landau pole (LP) for the SU(2) L gauge coupling g 2 [29] eliminates these models from a unification framework. In contrast, as demonstrated in [30], the BPR one-loop model with the scalar triplet as the largest weak representation exhibits, in addition to the absence of LP, perturbativity and stability up to the Planck scale. Therefore, we proceed here with the study of the gauge coupling unification in the context of the BPR loop model [26] for which the above requirements with respect to Yukawa and quartic couplings may remain valid when including extra color octet or color sextet scalar fields [31]. As it will turn out, adding these fields may be crucial to achieve the proper gauge unification.
In Sec. 2 we first present the set of BSM particles from the neutrino model [26], dubbed BPR particles, and then present the gauge-unification conditions which the newly introduced states have to satisfy. In Section 3, we will study the conditions under which the gauge couplings unify, and the particle spectra which make a proper unification possible. Then in Section 4 we will show that the appropriate particle spectra are consistent with the scalar potential of our SU(5) GUT scenarios. We conclude in Sec. 5. The details of the algorithm for search are given in App. A and the details of SU(5) representations in App. B.

BPR model from GUT perspective 2.1 BPR-model states
We adopt a simple and predictive TeV-scale radiative model [26] in which the loop contribution is genuine, i.e. self-protected like in the original Zee model. In its present variant the color singlet, weak triplet, hypercharge zero scalar field ∆ ∼ (1, 3, 0), is supplemented by a charged scalar singlet Figure 1: The one-loop BPR [26] neutrino mass mechanism. and by additional three generations of vector-like lepton doublets which are needed to close the neutrino mass loop diagram displayed in Fig. 1. The corresponding vertices in the loop diagram are provided by Yukawa and quartic couplings in The vacuum expectation value v SM of the SM Higgs doublet H leads to the neutrino mass matrix where f (m 1 , m 2 , m 3 ) is a loop function specified in [26]. Assuming like in [26] the mass values M E ∼ m ∆ + ∼ m h + ∼ 200−500 GeV, Eq. (5) leads to m ν ∼ 0.1 eV for the couplings y 1,2 and λ 7 of the order of 10 −4 . For definiteness, we will in the most of this work keep masses of these new states fixed at 500 GeV.
In principle, even much larger masses would lead to a viable neutrino mass model, with larger but still perturbative values of y 1,2 and λ 7 . Still, as we shall discuss later, such scenarios would not bring much additional insight from the GUT perspective. We display in Table 1 the SU(5) embedding of the SM extended by states in the neutrino mass model at hand. We note that additional potentially light BPR particles are described by the same SM group representations as those already populated by the SM states: new vector-like fermionic doublets E R,L belong to the same representation as the Higgs H c , and similarly for the charged scalar singlet h + and the SM lepton singlet e c R , or the scalar adjoint triplet ∆ and the spin one triplet W i µ . Understanding the quantum numbers of SM particles was one of the main motivations which led to the development of GUTs. The fact that BPR states populate already established SM representations could be viewed as an additional motive to study them in the GUT setup.

Matching BPR states with SU(5) irreps
One of the strongest arguments in favor of original Georgi-Glashow GUT scenario is a neat embedding of all SM fermion representations, with apparently arbitrary quantum numbers, into sum 5 F ⊕ 10 F of just two complete lowest SU (5) representations. Since the gauge bosons have to belong to adjoint multiplet of the SU(5) group, essentially the only remaining unknown has been the structure of the scalar sector. In the present study, this generalizes to question of incorporating the well motivated BPR set of particles into the same GUT context.
Following the pattern of SM states, and general principles of economy and elegance, BPR states at hand may be expected to belong to the lowest possible representations of the SU(5) group. This would put scalar ∆ in an adjoint 24, scalar h + in 10, and vector-like leptons E R,L in appropriate number of 5+5, which is a choice displayed in the right column of Table 1. These new SU(5) irreps contain additional states displayed in Table 1, that are not needed for BPR neutrino mass mechanism. Some among these additional states will prove crucial in obtaining the desired gauge coupling unification.
To completely specify the structure of our model Lagrangian, we need to choose the SU(5) irreps that will contain the Standard Model Higgs H.
Here the most economical choice would be 5, but, as we shall see, this would not lead to a viable GUT model. Thus, we will consider also the options where the Higgs state belongs to 45 or 70 irreps 1 , and we complete SU(5) ⊃ SU(3) c × SU(2) L × U(1) Y branching rules from Table 1 with: As is known, Higgs in a mixture of 5 and 45 can improve the GUT fermion mass relations, like in the Georgi-Jarlskog mechanism [32]. However, in the present work we will not study the pattern of SM fermion masses.
On the other hand, a pattern of scalar masses and of new vector-like lepton masses is important for our considerations, since these particles decisively affect the running of gauge couplings. We need then to check for any of these possibilities whether unification can be achieved for large enough M GUT , whether scalar Lagrangian at renormalizable level allows for required masses of particles and, finally, whether such scenario is in compliance with phenomenological constraints and general theoretical requirement of perturbativity [31,33].

Gauge coupling unification criteria
The unification of gauge couplings is controlled by the renormalization group equations (RGE) which govern the running of gauge couplings with the oneloop β coefficients given by Here, the Dynkin indices for generatorsT a in gauge, fermion and scalar representations G i , F i and S i , respectively, and are conventionally normalized to 1 2 for fundamental representations of SU(N ) groups (and thus to 3 with coefficients and threshold weight factor of BSM state k, defined as with value between 1 (for m k = m Z ) and 0 (for m k = M GUT ), depending on the mass of the BSM particle m k . The sum in (10) goes over all BSM states, with ∆b being the increase in the β coefficients at the threshold of a given BSM state, and b As first, the unification condition α 1 (M GUT ) = α 2 (M GUT ) = α 3 (M GUT ) ≡ α GUT can be expressed in the form of the so-called B-test [34,35]: where we used the average numerical values for the constants at m Z scale, as given in [36]. The comparison to the corresponding SM value 0.528 indicates that the couplings do not unify in the SM.
Second, the associated GUT scale yields for the SM the value M GUT = 10 13 GeV. Therefore, additional BSM states should improve unification and increase its scale up to at least 5 × 10 15 GeV which is in agreement with proton lifetime bounds [37]. Such additional BSM states must therefore provide a negative net contribution to B 12 and positive to B 23 .

Effect of BPR states on gauge unification
Before embedding in the SU(5) GUT framework, we first investigate how the new states, needed for neutrino mass mechanism, influence the RGE running and to what extent they alone could satisfy the unification criteria from Sect. 2.3.
In Table 2 we list extra BPR states together with their contribution to pertinent combinations of β-function coefficients. As already stressed, the states with positive B 23 and negative B 12 are promising for unification. It can be readily seen that only h + is not of this kind.
If we consider the default configuration with all BPR states close to the electroweak scale (i.e. weight factors from Eq. (11) being r k ∼ 1), one immediately observes that the B-test combination increases to B 23 /B 12 = 0.974, from the SM value 0.528, considerably overshooting the required value of 0.718 from Eq. (12). This is mostly due to the strong effect of three copies of BPR vector-like lepton doublets. Since they actually double the RGE effect of previously mentioned six Higgs doublets, one can similarly achieve correct unification if they are set at the intermediate scale with the factor r k ∼ 0.5. However, again like in the six Higgs doublet case, the unification scale would be too low.
Indeed, one observes that there is no way to obtain high enough unification scale by using only BPR states. Namely, even if negative effects of scalar h + state are avoided by putting it at some very high scale, the total contribution of ∆ and E L,R states to ∆B 12 is at most only −17/15 (for r k = 1) resulting in M GUT = 1.1 × 10 15 GeV as the maximal possible GUT scale, even if one would completely disregard the condition of the gauge coupling crossing.
In conclusion, BPR states alone cannot lead to a successful unification that would at the same time respect proton decay bounds. To achieve that, some other states below the GUT scale should be invoked. Such states are naturally provided by embedding BPR model in a SU(5) unification framework, as we shall show in detail.

Higgs doublet in 5 H
As explained in Sect. 2.2, we will first try the simplest possible SU(5) embedding of the BPR mechanism, where SM Higgs doublet becomes a member of the 5 H . The scalar sector of this model contains 5 H , 10 S and 24 S multiplets (we use subscript H on those scalar irreps of SU(5) which contain SM Higgs field), and there are N vec generations of vector-like matter in 5 F ⊕ 5 F , on top of the Standard Model quarks and leptons in n g = 3 generations of 10 F and 5 F , and gauge bosons in the adjoint 24 g representation, as displayed in Table 1. Their contributions to the RGE running are listed in Table 3. Table 3: BSM contributions to RGE running in the simplest SU(5) embedding of the BPR mechanism. H stands for SM Higgs doublet whose contribution has already been accounted for by b (SM ) i . The massless scalar leptoquarks X and Y get absorbed into longitudinal components of massive gauge bosons, as dictated by the Nambu-Goldstone mechanism -the β coefficients of these scalars thus enter at the same scale as heavy vectors (i. e. r k ≈ 0).
It is known that the simplest Georgi-Glashow SU(5) GUT suffers from the doublet-triplet splitting problem, where leptoquark S 1 = (3, 1, −1/3), which completes 5 H together with SM Higgs, has to be much heavier than the Higgs so that it doesn't induce the fast proton decay. There is nothing preventing the other scalar SU(5) multiplets to be split, and we will check in Sect. 4 that our splitting patterns are consistent with the structure of the general scalar potential. Still, whatever the actual mechanism responsible for the multiplet splitting, there is no reason to assume that this mechanism is somehow aligned with the neutrino mass mechanism. Thus, we will be quite general in allowing the splitting of masses within SU(5) multiplets.
With this freedom, and having at our disposal variety of states from Table 3 with different RGE behaviour, the unification prospects look promising. Indeed, we have found several scenarios where coupling constants correctly unify (cross at the single point). However, we also find that, whatever the masses of BSM states between M Z and M GUT , the highest possible unification scale in this model is M GUT < 10 15 GeV, in violation of experimental bounds on proton decay widths. Thus, this simplest embedding of the proposed neutrino mass model with the Higgs doublet restricted to 5 H irrep is ruled out.

Higgs doublet in 45 H
Next we consider the scenario where the SM Higgs doublet is embedded into 45 H instead of 5 H , or in some mixture of both. The larger particle content can help raising the unification scale and, as a bonus, this setup can serve to correct the wrong mass relations between charged leptons and down-type quarks at the renormalizable level which are usually obtained in the simplest SU(5) models. The β coefficients of the extra states from scalar 45 H can be found in Table 4, which should be added to states in Table 3 to obtain a complete embedding of the SM Higgs and the BPR states into SU(5) multiplets.
In this more realistic model one finds many ways in which one can achieve a correct unification, so we need to specify some criteria that will lead to a set of models covering all interesting scenarios; let us list those implemented in our study.
• First, note that if all states of a given SU (5) irrep appear at the same mass scale, their effect on RGE cancels (contributions to either B 23 or B 12 from all states of a given SU(5) irrep in, e. g., Table 3, taking the same r k , add up to zero). Thus, we will fix the BPR states close to the electroweak scale (for definiteness, we put them at 500 GeV), and by doing so we don't loose much generality, from the standpoint of RGE, because the effect on unfication of making e. g. BPR vector-like  • Next, since, as discussed before, we allow general splitting of SU(5) multiplets, with any experimentally allowed mass for the rest of BSM states (see Sect. 4), we have enough freedom to achieve the exact gauge coupling unification i.e. a fulfilment of the B-test, see Eq. (12). Then, we require the GUT scale M GUT larger than 2 × 10 15 GeV. The lowest experimental bound, coming from proton decay searches, is actually about 5 × 10 15 ; however, it turns out that for most of the scenarios presented here, a simplified analysis (ignoring the Yukawa contributions to 2-loop RGE) shows that improving RGE to 2-loop accuracy increases M GUT beyond 5 × 10 15 GeV.
• We will also exclude scenarios with very heavy new BSM particles, with masses between ∼ 10 11 GeV and M GUT . Otherwise, one can always take any successful model, add some particles slightly below M GUT that will have only small influence on running, and thus obtain many more models which will be qualitatively the same as the ones presented in this paper, only more complicated. This requirement at the same time excludes from consideration leptoquarks S 1 = (3, 1, −1/3),S 1 = (3, 1, 4/3) and S 3 = (3, 3, −1/3) which, if lighter than ∼ 10 11 GeV would naturally lead to proton decay in violation of experimental limits.
• For all BSM particles we take 500 GeV as a lower bound on their masses. Direct LHC searches sometimes put higher bounds on such  [39,40], where couplings to loop fermions in production and decay are taken to be of order one (see also Ref. [41]).
• We include only particles from single copies of scalar SU(5) irreps 5 H , 10 S , 24 S and 45 H (or, in the next Sect. 70 H ) and N vec = n g = 3 copies of vector-like 5 F , which are all already needed for embedding the BPR neutrino mass mechanism.
Under these conditions, we performed the exhaustive search of the parameter space, using the algorithm specified in Appendix A, and resulting in successful scenarios listed in Table 5. As explained in the Appendix A, when a given set of new BSM states offers a continuum of possible GUT scenarios (with different spectra), we represent this continuum by a specific choice of spectrum with minimal average mass of particles 2 . Such a choice is motivated, besides the need for definiteness, by the desire to focus on models which have maximal discovery potential at LHC and future colliders.
Note that a light (8, 2, 1/2) is the only other allowed representation in 45 H with negative B 12 contribution needed to increase M GUT and thus suppressing the proton decay. Of course by itself it doesn't help the unification due to negative B 23 (acting alone it can decrease B 23 /B 12 to 0.470), but its strong effect on unification scale is important for all models displayed in Table 5.

Higgs doublet in 70 H
If we opt for SM Higgs belonging to 70 H instead of 45 H (in addition to 5 H ), the search proceeds under the conditions explicated in the previous subsection, and the β coefficients of the extra states can be found in Table 6.
In this setup, we find three unification scenarios displayed in Table 7 to which scenarios A1, A3, A4 and A6 from Table 5 should be added, since they employ only states from 45 H that are also present in 70 H . From this search we have explicitly excluded representation (15, 1, −1/3) which, if light, leads to Landau poles below M GUT . To avoid this, it should be heavier than at least 10 7 GeV [31] so that its effect on the RG running would be diminished. Including also this representation leads to 16 additional scenarios beyond those in Tables 5 and 7, which we have chosen not to list.
Interestingly, all viable scenarios in this setup, as displayed in Table 7, involve the color triplet (3, 1, 1/3) at the same scale (500 GeV) as BPR vectorlike leptons, making the irrep 5 F complete and nullifying its influence on the RGE running. Thus, these states can all be at any other scale as well, without changing the gauge unification property of the model. We note that there are no viable scenarios with such unsplit fermion 5 F and with Higgs in 45 H , i.e. in the framework of Sect. 3.3. • 24 S : an adjoint 24-dimensional real traceless representation χ i j ; • 5 H : a fundamental 5-dimensional complex representation H i ; • 10 S : an antisymmetric 10-dimensional complex representation φ ij ; • 45 H : a 45-dimensional complex 2-index antisymmetric traceless representation Σ ij k , which get decomposed under the SM gauge group as displayed in Table 1 and Eq. (6). For the scenario from Sec. 3.4, Σ ij k is replaced with with the SM decomposition displayed in Eq. (7). The details of individual representations can be found in the Appendix B. The following fields from this set can develop potentially non-vanishing VEVs: • the SM singlet field from χ i j whose GUT scale VEV (1, 1, 0) • the neutral components of the weak doublets from H i and Σ ij k whose • the neutral components of (1, 2, + 1 2 ) Ω and (1, 4, + 1 2 ) Ω from Ω ij k can also develop VEVs of the order of electroweak scale; • the neutral component of the weak triplet from χ i j can develop a tiny (few GeV) VEV (1, 3, 0) χ severely constrained by the measured electroweak precision ρ parameter.

Scalar potential with 5 H ⊕ 10 S ⊕ 24 S ⊕ 45 H
We are only interested in the part of the renormalizable scalar potential that provides the SU Here and the summation over one upper and one lower repeating index is assumed. The potential contains nine real parameters {m χ , µ χ , m H , µ H , m φ , µ φ , m Σ , µ Σ , µ Σ } and one complex parameter {τ } with positive dimension of mass.
The spectrum presented in Tables 8 and 9 is computed in the minimum of the scalar potential (the vacuum state) obtained for where and v GUT is kept as a free parameter. The six massless complex scalar states (3, 2, − 5 6 ) χ are absorbed into longitudinal components of twelve heavy gauge bosons. Table 8: The scalar spectrum for the simplest SU(5) embedding of BPR model with only 5 H , 10 S and 24 S multiplets, which corresponds to setting to zero the parameters in V 45 and V mix . Their masses remain unchanged even after adding 45 H or 70 H to the particle content. Note that the parameters µ H , µ φ and µ χ should be understood as multiplied by v GUT and α 1 , α 2 , β 1 , β 2 , β 3 , λ 1 and λ 2 by v 2 GUT , while each of the masses is a sum of the pertinent contributions. For example, Table 9: Additional contribution to the scalar spectrum in the vacuum after adding 45 H to the 5 H ⊕ 10 S ⊕ 24 S model. The last two rows represent the mixing between 5 H and 45 H . Again, the parameters µ Σ , µ Σ and τ should be multiplied by v GUT and η 1 , η 2 , η 3 , η 4 , η 5 , η 6 , κ 1 and The two (1, 2, + 1 2 ) representations from 5 H and 45 H mix to form a SM Higgs doublet responsible for electroweak symmetry breaking. To compute their physical masses one needs to diagonalize the matrix Similar diagonalization proceeds for the states (3, 1, − 1 3 ) from H and Σ. One of the masses needs to be around the weak scale to correspond to the SM Higgs. It can as well be fine-tuned to zero, since in our case we have neglected all the v SM contributions to spectrum.
When both of the Higgs doublets develop a non-vanishing VEVs the Georgi-Jarlskog mechanism can be implemented to account for the observed masses of light fermions. It is also interesting to note that by excluding the mixing terms (V mix with coefficients τ , κ 1 and κ 2 ), as for example in the scenario without 5 H where the Higgs doublet belongs entirely to 45 H , the masses of fields from Σ are not linearly independent, and the following relation among them holds However, in the most general case the above expressions for scalar masses are all linearly independent. One can simplify the spectrum even further by imposing an additional Z 2 symmetry under which in Eq. (15) µ χ → 0, thus imposing a strong correlation between the weak triplet and the color octet masses 4.2 Scalar potential with 5 H ⊕ 10 S ⊕ 24 S ⊕ 70 H As long as we are interested only in the v GUT -proportional spectrum, the form of the scalar potential remains unaltered upon replacing Σ ij k with Ω ij k in Eqs. (14)- (19). The corresponding scalar spectrum is shown in Tables 8  and 10.
There are two major differences from the previous case with 45 H . As before, disabling the mixing terms in the scalar potential introduces the linear dependence among the masses of but now this dependence is preserved even after the states (1, 2, + 1 2 ) Ω and (3, 1, − 1 3 ) Ω effectively decouple through the mixing with 5 H . As can be seen from Eq. (27) the rest of the states remain linearly dependent, and since some of them are heavy (e.g. the leptoquarks) a certain fine-tuning is needed to make a particular state light as required by unification.
The second difference comes from the fact that, when considering the full scalar potential for 45 H and 70 H , they are not of the same form any more due to different symmetry properties of Σ ij k and Ω ij k . Namely, since φ ij is antisymmetric and Ω ij k is symmetric, all terms contracting Ω ij k with φ * ij vanish. Consequently, the Georgi-Jarlskog mechanism cannot be used in this case and we have to rely on either non-renormalizable Yukawa terms or some other mechanism to explain the pattern of SM fermion masses.

Conclusions
Although the SM particle set has been completed with the discovery of the Higgs boson in 2012, it is far from being established as unique, isolated set [42]. Our search for possible additional particles proceeds with an aim to both explain the neutrino masses and to achieve the unification of gauge couplings. With this in mind, we rely on the BSM states employed in the selected Zee-type BPR neutrino model [26]. This set of states allows us to introduce incomplete SU (5) representations, that have a potential to improve gauge coupling crossing. Still, this set alone leads to too low unification scale M GUT < 10 15 GeV, if the Higgs doublet is restricted to belong to 5 H irrep.
In contrast, there are immense possibilities to achieve the successful unification if the SM Higgs doublet is embedded into 45 H . Therefore we specify a plausible set of criteria under which our search algorithm shrinks the number of possibilities to seven successful scenarios listed in Table 5. In all of them, a light colored scalar (8, 2, 1/2) provided by 45 H plays a decisive role. If we choose the SM Higgs belonging to 70 H instead of 45 H , our search algorithm selects four scenarios (A1, A3, A4 and A6) from Table 5, and allows for three additional scenarios displayed in Table 7. Notably, in these new scenarios the BPR vector-like leptons are assigned to complete irrep 5 F , that do not affect the RGE running. Since in these latter scenarios only the scalar SU (5) irreps are incomplete, an eventual verification of them would be in support of a conjecture [43] that only scalar irreps may be split.
To conclude, in our procedure of renormalizable SU(5) embedding, the colorless BPR particles employed in the neutrino mass model get accompanied by the colored partners to enable a successful unification. We decide to keep sufficiently heavy those among the colored leptoquark scalars which present a threat to proton stability, and the other colored states may play a model-monitoring role both through the LHC phenomenology [34,35] and through tests at Super(and future Hyper)-Kamiokande [37] experiments.
We also point out that in most of the allowed parameter space the color octet scalar (8, 2, 1/2) is the most promising BSM state for the LHC searches, and as such is studied already in [25]. Additional colored states in the specific gauge unification scenarios in Tables 5 and 7 call for a study of characteristic exotic signals at the LHC, which may make some among these specific models falsifiable.

A Algorithm for optimal unification search
When studying GUT models with several new BSM states, one needs a welldefined procedure for identifying the viable unification scenarios. To this end, it is of some advantage to "linearize" the B-test (12) by first separating the fixed contribution of SM states from a contribution of the variable mass BSM states In the next step we write the gauge coupling crossing condition in the form Finally, we separate the β-function coefficients ∆b where c k = (∆b 2 ). For example, the E L,R , h + and ∆ states responsible for the neutrino masses in BPR model, if they are at electroweak scale (r k ≈ 1), contribute to this sum with c k r k ≈ 3, and significantly overshoot the required value (30). Thus, we need to add extra states with total negative contribution c k r k ≈ −1.6. By choosing some particular set of states we solve for r k .
In principle, there is an experimental uncertainty of constant c (related to the uncertainty of b in (28)), but we don't need to discuss it because most of the time we will be able to require the exact gauge crossing, regardless of possible small variations in the value of c.
Along the same lines, expression for the GUT scale (13) with s k = ∆b 2 . Obviously, if we have just one new variable BSM state (N=1) at our disposal, we can just solve the crossing condition (31), obeying any existing experimental lower bounds m k ≥ m k,min −→ r k ≤ r k,max (34) and check that GUT scale is high enough. For two states, N = 2, M GUT can also be chosen at will, and we can either require it equal to experimental lowest bound, or look for the range of possible M GUT for which solution of (31) and (33) exists. Regardless of this choice, for more than two particles, N ≥ 3, the problem becomes under-determined and we need another criteria.
To obtain definite scenarios we choose to maximize the norm of the vector which means that we choose scenarios with roughly minimal masses of new particles, or, in other words, we choose scenarios which have maximal discovery potential. Following further this principle of maximal discovery potential, one could also try to minimize at the same time the distance of M GUT to the existing experimental lower bound. We have tried this, but a necessary choice of relative weight of two optimization objectives brings a complication which we deem unnecessary at this point. Thus, we performed, for each choice for a set of BSM states, a one-dimensional scan with ever increasing fixed M GUT , to find the range of M GUT for which the unification scenario works. The problem can be organized as a standard linear algebra matrix equation and if we are temporary not concerned with bounds on r k , it can be solved using Lagrange multiplier method to obtain a solution with extremal r , which is r = A T (AA T ) −1 a. One can also choose to make a variable change r k → x k ≡ r k,max − r k and minimize x instead of maximizing r . To take the bounds on r k properly into account, more sophisticated optimization algorithm is needed and we used the sequential least squares programming algorithm SLSQP [44].

B Details of SU(5) representations
In this Appendix we present the structure and normalizations of used SU (5) representations.

Adjoint representation
is the adjoint 24-dimensional real traceless representation which decomposes under the SM group as The symbols σ, ∆, O, X, Y (and their complex conjugates X and Y ) denote the SM singlet field, the weak triplet, the color octet and the lower and upper components of SU(2) L doublet (3, 2, − 5 6 ), respectively. Note that the singlet σ, the electrically neutral triplet component ∆ 0 and colorless octet components O 1 and O 2 are real fields so that their mass terms come with a prefactor 1 2 .
Fundamental representation H i and H * i are the 5-dimensional fundamental and antifundamental complex representations with SM decomposition whose fields have the same quantum numbers as and where the weak doublet can play the role of Standard Model Higgs and potentially mix with its counterpart from Σ ij k or Ω ij k .
2-index antisymmetric representation φ ij and φ * ij are 10-dimensional complex antisymmetric representations with the SM decomposition whose fields have the same quantum numbers as The normalization factor 1 √ 2 is there only for convenience to avoid the double counting of fields in the mass term.
45-dimensional representation Σ ij k and Σ * k ij are 45-dimensional complex representations satisfying the antisymmetry and tracelessness conditions Under the SM it is decomposed as