Flipped SU(6) Unification of the $SU(3)_c\times SU(3)_L\times U(1)_X$ Model

We propose to partially unify the sequential $SU(3)_c\times SU(3)_L\times U(1)_X$ model (with $\beta=1/\sqrt{3}$) into the flipped $SU(6)$ model with the gauge group $SU(6)\times U(1)_K$. Gauge anomaly cancellation can easily be satisfied. We discuss the relevant Higgs sector, the low energy $331$ model spectrum and the unification of $SU(3)_c$ and $SU(3)_L$ gauge couplings. Neutrino masses generation and successful gauge coupling unification can set lower/upper bounds on the $331$ breaking scale. The partial proton decay lifetime of various channels, for example, the $p\to e^+ \pi^0$ channel, in flipped SU(6) GUT are discussed. We find that certain parameter region with $M_{331}\sim 10^{15}$ GeV of case II (for case with $M_{331}$ scale ${\bf \tilde{H}_{3,8}}$ Higgs field) can predict a partial proton lifetime of order $10^{34}$ years for $p\to e^+ \pi^0$ mode, which can be tested soon by future DUNE and Hyper-Kamiokande experiments.


Introduction
The standard model (SM) of particle physics, based on the SU (3) c × SU (2) L × U (1) Y gauge group, has been extremely successful in describing phenomena below the weak scale.However, the SM still leaves some theoretical and aesthetical questions unanswered, for example, the origin of charge quantization, the values of the low energy parameters and the origin of the flavor structures.Such questions can be answered in the framework of Grand Unified Theory (GUT), such as SU (5) [1] and SO (10) [2] GUT.In the GUT framework, the matter contents of SM can be embedded into certain representations of the GUT group, indicating that the low energy Yukawa couplings can be obtained from a single Yukawa coupling (or few Yukawa couplings) at the GUT scale.The approximate unification of the SM couplings strongly indicate the existence of GUT.We know that the SU(5) GUT model unified the SM gauge group directly at the GUT scale without any intermediate partial unification step.If intermediate partial unification exists at a higher scale beyond M Z , for example, the Pati-Salam SU (4) c × SU (2) L × SU (2) R partial unification, genuine gauge couplings unification needs a larger GUT group, such as SO (10) in this case.So, it is interesting to seek other unification model with some intermediate partial unification steps, such as the (partial) GUT model with an intermediate SU (3) c × SU (3) L × U (1) X partial unification [3][4][5][6][7] step.
The measured value of the electroweak mixing angle sin 2 θ W (M Z ) = 0.23 ≲ 0.25 appears to obey an SU (3) symmetry in such a way that sin 2 θ W (µ) = 1/4 at some new fundamental energy scale µ upon TeV [8].By introducing an extra U (1) factor to accommodate quark sector, one can arrive at an SU (3) c × SU (3) L × U (1) X model (331 model).Depending on different choices of the β value (β = 1/ √ 3 [9,10] or β = √ 3 [9,11,12]) within the embedding of the electric charge, 331 models in the literatures need to introduce different electrically charged particles for the fitting of the SU (3) L representations.It is remarkable that the existence of three matter generations could be the consequence of gauge anomaly cancellation requirements.Besides, the heaviness of the top quark mass and the emergence of the Peccei-Quinn (PQ) symmetry can also possibly be explained in the 331 framework [13,14].
To understand the origin of charge quantization and the values of the low energy parameters, the intermediate 331 model needs to be unified into a true GUT theory.The unification of sequential 331 model into SU (6) model had been proposed in [15,16] and studied in [17][18][19].On the other hand, the genuine unification of 331 into SU (6) needs the introduction of additional adjoint fermions and scalars etc at some intermediate scale between the 331 scale (at O (TeV)) and GUT scale [15], reducing the predicability of the GUT theory.So, it is interesting to seek alternative (partial) unification steps for the 331 models.
We propose to partially unify the 331 gauge group into flipped SU (6) with the gauge group SU (6)×U (1) K .Similar to the flipped SU (5) model [20][21][22], the flipped SU (6) model can be well motivated from string theory models [23][24][25], which uses level-one Kac-Moody algebras and do not need adjoint Higgs fields for symmetry breaking [26].We know that the flipped SU (5) can adopt the economical missing-partner mechanism and possibly provide an unified cosmological scenario for inflation, dark matter and baryogenesis etc [27][28][29].We anticipate that such virtues can also be present for flipped SU (6).Flipped SU(6) model can also be unified into SO (12) or This paper is organized as follows.In Sec 2, we brief review the sequential 331 model and discuss the embedding of such 331 model into flipped SU(6) GUT model.In Sec 3, we discuss various sub-scenarios of 331 model and the corresponding gauge coupling unification.In Sec 4, we discuss the triggered proton decay modes and lifetimes in flipped SU (6).Sec 5 contains our conclusions.
1) X quantum numbers for the three generations.The filling of the matter contents are given as and ), with a = 1, 2, 3 and i = 1, 2 the family indices.Here (XD) iL , (XD) c iL and (XT ), (XT ) c L denotes some exotic vector-like quarks with the SM quantum numbers ), ). (2.3) We adopt the notation , where ψ c = C ψT and C the charge charge conjugate matrix.The relation of the hypercharge to the SU (3) L × U (1) X generators is given by with the choice of T 8 for fundamental representation 3 of SU (3) L as It can be checked that the gauge anomaly can cancel only if we take into account the contributions from all the generations.Ordinary 331 model contains a simple lepton sector and can be potentially tested in the TeV scale.There is an interesting variant 331 model called sequential 331 model [15], which, unlike ordinary 331 models, assign identically the matter quantum numbers for the three generations.Therefore, the gauge anomalies are canceled for each generation separately.The filling of the matter contents in sequential SU (3 and ). (2.7) with the SM quantum numbers for some exotic vector-like quarks and leptons (XL) L and (XL) c L given by ) , ) . (2.8) The relation of the hypercharge to the SU (3) L × U (1) X generators is the same as the non-sequential case.For later convenience, we show explicitly the U (1) Y charges for the three components within (1, 3, Q X ) representation of 331 model, which are given as (2.9)

The fitting of matter contents into flipped SU(6)
We propose to partially unify the 331 gauge group into SU (6) × U (1) K gauge group.The normalized U (1) P generator within SU (6), which is the (remaining) diagonal generator for SU (6) other than the diagonal ones in SU (3) c and SU (3) L , can be written as We can embed the matter contents of 331 model with β = 1/ √ 3 (denoted by their SU (3) c × SU (3) L × U (1) P × U (1) K quantum numbers) into flipped SU(6) representations for each . (2.11) The U (1) X quantum number is related to the corresponding U (1) K and U (1) P charges via after the breaking of SU (6 It is obvious to see that the SU (6)−SU (6)−SU (6) anomaly is canceled with two 6 representation and one antisymmetric 15 representation for each generation, as the anomaly coefficients for SU (6) representations are given by The U (1) K related anomalies are canceled because the U (1) K quantum numbers for fermions within each generation satisfy To avoid the gravitational violation of gauge symmetry, the anomaly related to gravity should vanish.It can be seen that the graviton-graviton-U (1) anomaly is canceled in our model, because only the abelian U (1) K generator is relevant and the U (1) K charges for the chiral fermions satisfy So, we have a possible anomaly free fitting of matter contents in flipped SU(6) ) (containing E c L ) are similar to that in flipped SU(5). 1 The case for the partial unification of 331 (with β = √ 3) into flipped SU( 6) is rather tricky, especially the relevant anomaly cancellation conditions.We will discuss it in our subsequent study.It seems not possible for such a case to unify in ordinary SU (6). 2 We note that the fitting of (XD) c L and D C L can be exchanged (also (LL, N s L ) and ((XL)L, N c L ) ).To ensure the VEV of 150;H is small, such a fitting ) 3 ) is not adopted here.We will discuss such alternative choices in our subsequent studies.
The fact that the gauge anomaly cancellation of the flipped SU(6) holds for each generation is the reminiscent of the gauge anomaly cancellation conditions of sequential 331 model.Such generation by generation gauge anomaly cancellation conditions will in general not hold for non-sequential 331 model fitting of flipped SU(6) model.
We should briefly comment on the anomaly cancellation conditions in non-flipped versus flipped SU(6) (partial) unification of (non-)sequential 331 models.In the ordinary SU(6) unification of the sequential 331 model, each generation will still be fitted into 15 ⊕ 6 ⊕ 6 representations so as that the gauge anomaly cancellation conditions are satisfied for each generation.In the ordinary SU(6) unification of the non-sequential 331 model, the RH-charged leptons E c aL ∼ (1, 1, 1) needs to be fitted into 20 representation of SU( 6).The quarks (including the exotic vector-like quarks) and LH leptons can be fitted into 15 ⊕ 6 ⊕ 6 for the first two generations, while those of the third generation needs to be fitted into 20 ⊕ 15 ⊕ 15 ⊕ 6.We can see that the gauge anomaly cancellation conditions for the first two generations in ordinary SU( 6) unification of non-sequential 331 model are satisfied for each generation while the third generation needs additional exotic fermions in 6 representation to cancel anomaly.Therefore, the anomaly cancellation for non-flipped SU(6) unification of both sequential and non-sequential 331 model always hold generation by generation, even though such conditions for the low energy non-sequential 331 model are satisfied non-trivially unless contributions from all the three generations are included.
In the flipped SU( 6) partial unification of non-sequential 331 model, which will discussed in detail in our future work [30], the RH-charged leptons E c aL ∼ (1, 1, 1) can still be kept as SU( 6) singlet.The gauge anomaly for the first two generations, which are given by 15 0 ⊕ 61/2 ⊕ 6−1/2 ⊕ 1 1 will no-longer cancel unless we include the third generation.We need to introduce only one 20 representation of SU(6) for the third generation, which needs fairly small additional exotic matter contents in contrast to ordinary non-flipped SU(6) unification of non-sequential 331 model.

The Higgs sector
The Higgs fields introduced in our model are responsible for the breaking of SU (6)×U (1) K gauge group, the breaking of the SU (3) c × SU (3) L × U (1) X gauge group and the mass generations for the matter contents, including the SM matter contents, the exotic vectorlike fermions and the sterile neutrinos.The total Higgs sector in our flipped SU (6)  with the gauge symmetry broken by the corresponding VEVs To break the flipped SU(6 with (2.21) The (1, 1, To break the residue SU (3) c × SU (3) L × U (1) X gauge symmetry into SM and generate masses for the SM matter contents, we need to introduce additional Higgs fields 6 1 2 ;H , 15 0,H and 6 − 1 2 ;H with their decompositions in terms of The Yukawa couplings for the matter contents can be written as 3 ) that contains doublet H d Higgs field in 2HDM), which are just needed to generate properly the masses for matter contents.As the 6 − 1 2 ;H Higgs is not responsible for the masses generations of SM matter contents, the simplest choice to break SU (3 3 ), which can be decomposed with the corresponding SM gauge quantum number along the (1, 1, 0) direction (in terms of SM gauge quantum number) as ⟨H 2 ⟩ = M 331 .The VEVs of the relevant Higgs fields can be written as with the VEVs of H 1 and H 3 trigger the breaking of the SM electrweak symmetry into U (1) Q .
It can be seen from the Yukawa couplings in eq.(2.23) that the 6 a 1 2 which will generate Dirac mass Y XD M 331 for vector-like heavy extra leptons (XL) L , (XL) c L and vector-like heavy quarks (XD) L , (XD) c L .Experimental measurements for the square of the mass differences [31] for neutrinos indicate that the heaviest neutrino mass should be of order 10 −2 eV.Such tiny neutrino masses can either be Dirac type or be Majorana type from dim-5 Weinberg operator.We should note that it is not possible to adopt only the Dirac type masses for neutrinos because the Yukawa terms involving Y U ;ab generate identical masses for both the up-type quark masses and Dirac-type neutrino masses at the flipped SU(6) breaking scale M X .Large hierarchy between the up-type quark masses and tiny Dirac type neutrino masses can not be generated by pure renormalization group equation (RGE) effects, that is, by RGE evolution from M X to M Z .Tiny Majorana neutrino masses from dim-5 Weinberg operator can be UV completed to various mechanisms, for example, the Type-I seesaw mechanism, which can be used to generate tiny neutrino masses after introducing additional Majorana mass terms for RH-neutrinos N c L .Bare Majorana mass terms are not allowed because the RH-neutrinos are fitted into non-singlet 15 0 representations of SU (6)×U (1) K .So, such Majorana mass terms for RH neutrinos can be only generated by a new term involving certain new Higgs field that couples to RH neutrinos.From the production of 15 0 representation we can see that the proper choice is 105 s 0 , which is decomposed as The term responsible for the generation of Majorana neutrino masses can be written as which will generate Majorana masses for RH neutrinos after the (1, 6, − 2 √ 3 ) 0;H component of 105 s develops a VEV along the (1, 1, 0) direction (in terms of SM quantum number).As ⟨105 s ⟩ = M S will also break the SU (3) c × SU (3) L × U (1) X gauge symmetry, we require that M S ≲ M 331 .The neutrino masses can be given by (2.27) The natural up-type quark masses, which are also the typical Dirac-type neutrino masses, are given by m U ≃ Y U v u ≃ M ν;D ∼ O( 102 ) GeV.To obtain tiny neutrino masses of order 10 −2 eV with the seesaw mechanism the 331 breaking scale M 331 is constrained to lie naturally at about 10 14 GeV for Y m ∼ O(1), otherwise the generated neutrino masses should be much larger than 10 −2 eV.The bounds on M 331 ∼ 10 14 GeV from neutrino masses can be relaxed to M 331 ≳ 10 14 GeV if the coupling Y m can be much smaller than identity.We should note that constraints on the scale of M 331 from neutrino masses can be relaxed if a mixed type I+ II seesaw mechanism is used for neutrino masses generations, within which a small VEV for an additional 21 1;H representation Higgs field along the SU (2) L triplet direction is needed.We will discuss such a possibility shortly after.On the other hand, it will be clear soon that successful gauge couplings unification for g 3c and g 3L requires the M 331 scale to be higher than 10 16  GeV.Such a bound can be relaxed unless certain additional colored Higgs field lies of order M 331 scale.Given the neutrino masses, the Yukawa coupling involved can be defined in terms of the physical neutrino parameters, up to an orthogonal complex matrix R [32], where mν , MR being the diagonal matrices for the light and heavy neutrino masses, and V P M N S being the PMNS lepton mixing matrix.In our case, the neutrino hierarchical spectrum can either be normally ordered (NO) or inversely ordered (IO), depending on the Yukawa parameters introduced in the theory.
If we adopt non-renormalizable Weinberg operator to generate tiny neutrino mass for flipped SU(6) without specifying its concrete UV completion model, the previous lower bound on M 331 from neutrino mass generation can be relaxed.Here M denotes the scale of the heavy modes, which are integrated out and responsible for the generation of Weinberg operator.As the non-renormalizability of the Weinberg operator requires M to be larger than the flipped SU(6) breaking scale M X , we thus obtain an upper bound for M X with M X < 10 14 GeV in this case.To be consistent, we need to ensure that such a constraint is satisfied for the choice of M 331 scale and the Higgs contents of the low energy 331 model.We leave the numerical discussions of this possibility in our future work.The new sterile neutrino component N s L within 6− 1 2 can also obtain masses after EWSB, which couples to the (XL) L and (XL) c L components via the Yukawa coupling terms involving Y U and Y

31)
So the mass matrix for the new sterile neutrinos can be given by in the basis of N s L , N (XL) , N (XL) c .Here N (XL) , N (XL) c denote the neutral components within (XL) L and (XL) c L , respectively.The M U , M E scales lie typically at the up-type quark mass scales O( 102 ) GeV and charged lepton mass scales O(1) GeV, respectively.After diagonalizing the mass matrix, we can obtain that the mass scale for the lightest new sterile neutrino is for Y XD ∼ O(1), which can contribute to additional light effective degrees of freedom ∆g * at the BBN era and cause cosmological difficulties.Therefore, we should try to push heavy such new sterile neutrinos, for example, by choosing unnaturally small Y XD .An interesting solution to such a problem without unnatural parameters is to introduce additional Majorana type masses for N s L .We can introduce new 21 1;H representation Higgs field, which has the following decomposition S all lie at the M 331 scale and will not cause cosmological difficulties.On the other hand, if M 331 ≫ ⟨21 1;H ⟩ ≳ 0.1 keV (or choosing Y XD ≲ 10 −6 for the first solution), the lightest sterile neutrinos with masses of order ⟨21 1;H ⟩ can act as a fermionic dark matter candidate, which also satisfy the Tremaine-Gunn (TG) bound [33].
Besides, if the (1, 3, 1) direction (in terms of SM quantum number) of (1, 6, 1 √ 3 ) 1;H component within 21 1;H Higgs field also develops a small triplet VEV (which also breaks the SM electroweak gauge symmetry), ordinary LH neutrinos of SM can also acquire Majorana masses so as that a mixed type I+II seesaw mechanism can be applied to the non-sterile neutrino sector.With a small non-vanishing (M ν ) 11 component for the the mass matrix (2.27), the 331 breaking scale M 331 can be much lower than 10 14 GeV with large fine-tuning among the type-I and type-II seesaw contributions for the neutrino masses.For example, the choice of M 331 ∼ 10 3 GeV requires O(10 −11 ) fine tuning for both contributions to get tiny neutrino masses of order 10 −2 eV.However, our numerical results indicate that successful gauge couplings unification for g 3c and g 3L still requires large M 331 scale (larger than 10 12 GeV) in the case with small SU (2) L triplet VEV, unless we keep some additional Higgs field as light as M 331 scale, for example, keeping the ( 3, 8, − 1 √ 3 ) 0 Higgs field within 105 s to lie at M 331 scale.
It is worth to note that the Higgs field in 6, 15, 20 representation of SU( 6) can be generated at the Kac-Moody level one [34].The large representation 105 s and 21 Higgs fields from higher Kac-Moody level can in fact be replaced by double 6 or 15 Higgs fields, similar to that appeared in the non-renormalizable Weinberg operators.

Gauge Coupling Unification
The GUT symmetry breaking chain is given by leads to the relations for the relevant gauge couplings holding at the SU (6) × U (1) K breaking scale M X (for 1/g 2 X ) and the M 331 scale (for 1/g 2 Y ), respectively.If we fit the flipped SU (6) gauge couplings within SO (12), the coupling g K should be normalized into canonical g K with g 2 K = g 2 K /3.It should be noted that the charge quantization can only be explained in the framework of SO( 12) or E 6 GUT instead of our partial unification scheme.The U (1) K charge assignments in our intermediate partial unification SU (6) × U (1) K model are still not quantized, which are constrained only by gauge anomaly cancellation conditions.
The one-loop beta function for the couplings are given by for Weyl fermions in r i f representation and complex scalars in r i s representations.In SUSY SU(5) GUT model, the doublet Higgs field that responsible for electroweak symmetry breaking should be much lighter than the colored triplet Higgs field so as that the dim-5 operator induced proton decay mode suppressed by the triplet Higgs mass can still be consistent with current proton decay bounds.There are many proposals to deal with such doublet-triplet (D-T) splitting problem, such as the missing partner mechanism [35], complicated version of sliding singlet mechanism in SU( 6) extension [36], missing VEV in SO (10) [37], pseudo Nambu-Goldstone bosons [38] etc.In missing partner mechanism, the color-triplet Higgs fields can couple with other colored fields to acquire large masses, whereas the doublet Higgs fields lack such partners so as that they can still be light.Missing partner mechanism can be elegantly realized in flipped SU(5) GUT model, which does not require adjoint or larger Higgs representations and can be seen as a virtue of flipped SU (5).We should note that the missing partner mechanism can also be realized in our flipped SU( 6) model (see the discussions in the appendix A.1).
In our following discussions, we assume that the splitting among the colored/uncolored Higgs fields can be successfully realized and we do not specify the origins of such splitting, for example, by missing partner mechanism or by orbifold projection (see appendix A.2). Therefore, rendering the colored Higgs fields to lie near the SU(6) breaking scale, the Higgs sector for the low energy 331 model contains We have the following coefficients for 331 model Upon the M X scale, where fields include the matter contents for three generations 6− 1 2 ;i , 6 1 2 ;i ,15 0;i and the Higgs fields 20 − 1 2 ;H ,21 1;H ,105 0;H ,6 1 2 ;H , 15 0,H and 6 − 1 2 ;H .The beta functions for SU (6) and U (1) K are given by which, after normalization into SO(12), gives After the breaking of 331 gauge group into SU (3) c × SU (2) L × U (1) Y at about the M 331 scale, the theory reduce to two Higgs doublet model (requiring M 331 ≳ 10 14 GeV) or two Higgs doublet Model plus an SU (2) L triplet (with relaxed M 331 scale, for example, at TeV scale, although large fine tuning is needed).The relevant beta functions are given by • Case II: 2HDM plus an SU (2 Note that we do not have the 3  5 factor for b Y because we do not normalize the g Y couplings within SU (5).We adopt the following inputs at M Z scale [39] α −1 em = 127.951± 0.009 , sin 2 θ W = 0.23129 ± 0.00033 , α s (M Z ) = 0.1185 ± 0.0016 , We can calculate the relevant SU (6)×U (1) K breaking scale M X for various low energy cases, after specifying the matter and Higgs contents of the low energy 331 models and their corresponding low energy models at the electroweak scale (in our case, the 2HDM or 2HDM plus an SU (2) L triplet).Given an M 331 scale, the SU (6) × U (1) K breaking scale and the corresponding SU (5) gauge coupling can be obtained numerically, using the corresponding beta functions for the gauge couplings given in previous discussions.In our partial unification model, the M X scale is defined as the intersection scale of the RGE evolution trajectories for SU (3) c and SU (3) L gauge couplings.The U (1) K coupling strength at M X can be obtained by the combinations of U (1) X coupling and the coupling of gauge field corresponding to the diagonal U (1) P generator within SU (6), which is just the SU(6) gauge coupling strength at M X .The RGE evolution trajectory of U (1) K gauge coupling upon M X will eventually intersect/unify with that of SU (6) gauge coupling at the SO( 12)/E 6 unification scale (or at the string scale M str with gravity).
We randomly scan the values of M 331 within the ranges that are compatible with the lower bound from neutrino masses generation and the upper Planck scale bound.Our numerical results indicate that the flipped SU (6) unification of 331 model can indeed be possible.In fig. 1  The relevant gauge interaction terms are given by with X β µ;A the heavy SU (6) gauge bosons, the index ′ β = 4, 5 ′ (corresponds to the SU (2) L index within SU (3) L ), A = 1, 2, 3 the SU (3) c color indices and V CKM , V l , V N being the mixing matrices for quarks, charged leptons and neutrinos, respectively.
Similar to flipped SU(5) [41], below the flipped SU (6) breaking scale, the dim-6 operator from integrating out the heavy gauge bosons of SU (6) can be written as with C 1 , C 2 the relevant coefficients.As the RH neutrinos are much heavier than the proton, only the first operator will contribute to proton decay.The effective dim-6 operator that trigger the decay of protons takes the form below the EW scale.The proton decay rate in our flipped SU(6) model can be calculated by generalizing the flipped SU(5) case [42] Γ(p for case II with light H3,8 Higgs.Expressions of A S1 for other scenarios can be straightforwardly obtained.The hadronic matrix elements can be obtained by lattice calculations [43] ⟨π within which the subscripts 'e' and 'µ' indicate that the matrix elements are evaluated at the corresponding lepton kinematic points.Using eq.( 4.4), we can calculate the partial lifetime of the p → e + π 0 mode in flipped SU(6) GUT models.The decay widths for other proton decay channels, such as p → π 0 µ + and p → K 0 e + can be similarly obtained after taking into account proper CKM and PMNS matrix elements.The leptonic PMNS mixing matrix is given by [44,45]: where c ij ≡ cos θ ij and s ij ≡ sin θ ij with the mixing angles θ ij = [0, π/2], the Dirac CP phase δ ∈ [0, 2π] and the Majorana phases α 2 and α 3 being set to vanish.For normally ordered (NO) or inversely ordered (IO) neutrino hierarchical spectrum, the matrix elements of U l = V * P M N S V ν can be given as We use the following best-fit value of the PMNS mixing and phase [46] θ 12 = 33.82 for IO and NO cases, respectively.Similar to ordinary flipped SU(5) case [47], we can calculate the relations for partial decay widths with Γ(p → π + ν) = i Γ(p → π + νi ).The p → K + νi channels can be proven to be forbidden by the unitary property of the CKM matrix.We also have the relation Γ(n → π − l + i ) = Γ(p → π 0 l + i )/2.By taking the ratio between the two partial decay widths, many of the factors in the expressions (such as the M X scale, the SU(6) gauge coupling g 6 and the A L;S factors) can be canceled, making the comparison of those ratios in various GUT models meaningful.
Table 2.The partial proton decay lifetimes of the (p → e + π 0 ) mode in flipped SU(6) GUT for NO neutrino masses hierarchy.The partial proton decay lifetimes of this channel for IO case is related to that of NO case by τ IO /τ N O ≈ 30.1.We calculate such partial life time for some benchmark points for various M 331 values in case I/case II with and without M 331 scale H3,8 Higgs, respectively.The definitions of ′ \ ′ symbol and the corresponding reasons R i are the same as that in Table 1 We show in Tab.2 the numerical results of p → e + π 0 partial life time for some benchmark points in case I/case II with and without M 331 scale H3,8 Higgs for NO neutrino mass hierarchy case.The partial proton decay lifetimes of p → e + π 0 for IO case is related to that of NO case by τ IO /τ N O ≈ 30.1.We can see that without light H3,8 Higgs, the partial proton decay lifetimes of the p → e + π 0 mode are rather large.Although the current proton decay bound τ (p → e + π 0 ) > 1.6 × 10 34 year [48] can easily be satisfied, such proton decay mode cannot be observed in the forthcoming experiments.On the other hand, with M 331 scale H3,8 Higgs, the partial proton decay lifetimes p → e + π 0 can be as low as 2.80 × 10 34 years for M 331 ≈ 5.0 × 10 14 GeV (of case II with NO neutrino hierarchy), although such a M 331 value lies near the upper bound on M 331 .It can be seen that such parameter regions for either cases will soon be tested by future DUNE [49] and Hyper-Kamiokande [51] experiments, which can reach as large as 1.3 × 10 35 years for p → e + π 0 channel in future Hyper-Kamiokande.
with ′ a, b ′ the family indices.The low energy Higgs fields in SU (3

1 √ 3 ) 1 ;
) in terms of SU (3) c × SU (3) L × U (1) P × U (1) K quantum numbers and the relevant Yukawa coupling is L ⊇ −y S;ab 6 H component of 21 1;H develops a VEV with ⟨21 1;H ⟩ = M S ′ ∼ M 331 along the (1, 1, 0) direction (in terms of SM quantum number), Majorana mass term can be generated for N S L .With new contribution (M ′ S ) 11 ∼ M 331 in eq.(2.32), the eigenvalues of M ′ , we show the RGE evolutions of the gauge couplings for scenarios with the low energy theory as 2HDM below M 331 scale (case I, left panel) and 2HDM plus SU (2) L triplet Higgs below M 331 scale (case II, right panel), respectively.The corresponding 331 symmetry breaking scale for the benchmark scenarios are M 331 = 10 16 GeV (left panel) and M 331 = 7.94 × 10 11 GeV (right panel), respectively.The corresponding flipped SU (6) unification scales are M X = 10 19.03 GeV with α −1 6 (M X ) ≈ 48.05 (left panel) and M X = 10 19.13GeV with α −1 6 (M X ) ≈ 45.12 (right panel).On the other hand, requiring the unification scales to lie below the Planck scale constrains M 331 ≳ 10 15.9 GeV for case I and M 331 ≳ 10 11.8 GeV for case II.Larger M 331 scale will lead to smaller flipped SU (6) GUT scale M X .Besides, the requirement that the SU (2) L coupling (within SU (3) L ) and SU (3) c coupling should not intersect below M 331 scale, that is, M X ≳ M 331 , set an upper bound for M 331 scale, which requires it to lie below 1017.6 GeV for case I and below 1015.3GeV for case II.

Figure 1 .
Figure 1.The RGE evolutions of the gauge couplings for the 331 models are shown for scenarios with 2HDM below M 331 (case I, left panels) and 2HDM plus SU (2) L triplet Higgs below M 331 (case II, right panels) without light H3,8 Higgs, respectively.With the 331 symmetry breaking scale M 331 = 10 16 GeV (left panels) and M 331 = 7.94 × 10 11 GeV (right panels), the SU (3) L and SU (3) c gauge couplings can be unified into the flipped SU (6) GUT model at the scale M X = 10 19.03 GeV (left panel) and M X = 10 19.13 GeV (right panel), respectively.The upper (and lower) panels correspond to the cases without (and with) the surviving M 331 scale H3,8 Higgs field, respectively.