$SU(3)_{F}$ Gauge Family Model and New Symmetry Breaking Scale From FCNC Processes

Based on the $SU(3)_{F}$ gauge family symmetry model which was proposed to explain the observed mass and mixing pattern of neutrinos, we investigate the symmetry breaking, the mixing pattern in quark and lepton sectors, and the contribution of the new gauge bosons to some flavour changing neutral currents (FCNC) processes at low energy. With the current data of the mass differences in the neutral pseudo-scalar $P^{0}-\bar{P}^{0}$ systems, we find that the $SU(3)_{F}$ symmetry breaking scale can be as low as 300TeV and the mass of the lightest gauge boson be about $100$TeV. Other FCNC processes, such as the lepton flavour number violation process $\mu^{-}\rightarrow e^{-}e^{+}e^{-}$ and the semi-leptonic rare decay $K\rightarrow \pi \bar{\nu} \nu$, contain contributions via the new gauge bosons exchanging. With the constrains got from $P^0-\bar{P}^0$ system, we estimate that the contribution of the new physics is around $10^{-16}$, far below the current experimental bounds.


Introduction
The last five decades have witnessed the great triumph of the standard model (SM). Especially the Higgs boson was finally discovered at the Large Hadron Collider (LHC) [1,2]. However, there are some solid experimental evidences hinting new physics beyond SM. These evidences include neutrino oscillations [3,4], dark matter (DM) [5,6] and baryon asymmetry of the universe (BAU) [7,8]. Neutrino oscillations can be explained by nonzero but tiny masses of neutrinos. And the observed nearly tri-bimaximal mixing pattern [9][10][11][12][13][14] strongly indicates new symmetries, discrete or continuous, in the neutrino flavour sector. In general, models [15][16][17][18][19][20][21][22][23] inhabited by these new flavour symmetries contain new heavy particles and new CP violation (CPV) phases. As a bonus, these models may provide candidates of the DM, and new CPV sources accounting for BAU. So the flavour symmetry can be a possible solution to the puzzles mentioned above.
In SM, before electroweak symmetry is spontaneously broken, quarks and leptons are all massless. Due to the universality of gauge interactions, no quantum number can distinguish the three families. Only the Yukawa interactions can tell them apart. Thus a simple extension to SM is to introduce a new flavour symmetry among the three families, which is then broken spontaneously. In this work we take the SU (3) as the flavour symmetry group, denoted as SU (3) F . The flavour structure of Minimal Flavour Violation in quark and lepton sectors based on family symmetries have been discussed in [24][25][26][27][28]. Models based on other family symmetry, such as SO(3) F symmetry, have been discussed in [16,17,[29][30][31][32].
In the SU (3) F gauged family symmetry model [18], there are new interactions among the three families. The extended gauge symmetry group becomes SU (3) F ⊗ SU (3) c ⊗ SU (2) L ⊗ U (1) Y . As the SM Higgs field being singlet under this new family symmetry transformation, new Higgs fields are needed to break the SU (3) F symmetry. A Hermitian field Φ = Φ † which is adjoint representation of the SU (3) F can do this job. Actually, to explain the mass and the mixing pattern both in quark and lepton sectors, we need two Hermitian fields Φ 1,2 = Φ † 1,2 . In the lepton sector, we also need right handed neutrinos N R and seesaw mechanism [33][34][35] to explain the tiny neutrino masses. So there should be a complex symmetric Higgs Φ ν = Φ T ν to generate Majorana mass terms for N R . The new Higgs fields transform under the SU (3) F gauge transformation as Φ 1,2 → gΦ 1,2 g † , Φ ν → gΦ ν g T , g(x) ∈ SU (3) F .
For the representation of SU (3), one has 3⊗3 = 6⊕3 where the 6 representation denoted as (2, 0) in p − q notation is symmetric while3 is anti-symmetric. Here the Φ ν is the symmetric 6 representation of SU (3) F . Seesaw mechanism can also be used to explain the mass hierarchy structures in quark and charged lepton sectors. There could also be new heavy charged fermion fields as cousins of N R , and a new SU (3) F singlet Higgs φ s to couple these new heavy fields with SM fields together. We can write down the general particle contents based on SU (3) F gauge family symmetry with features mentioned above, as listed in Table.1. For the new gauge transformation acting in the same way on the left handed and right handed parts of all fermions, no chiral anomaly occurs here. The general form of the Lagrangian is where L G contains the kinetic and self-interaction terms of gauge bosons, including the new gauge bosons. L k is the covariant kinetic term of the SM fermions, and contains the new gauge interactions among the three families's fermions mediated by the eight new gauge bosons. And  With the eight new gauge bosons, there are tree level flavour changing neutral currents (FCNC), as well as processes that violate CP or lepton flavour numbers. These processes are suppressed in SM. In this work we use the experimental data of these processes, to get constraints on the breaking scale of this new SU (3) F gauge symmetry.
We show the breaking pattern of the new family symmetry in Sec.2, and then give out the new effective Hamiltonian mediated by the new gauge bosons in Sec.3. After that the current experimental results of the neutral pseudo-scalar meson systems are used to constrain the broken scale of this family symmetry in Sec.4. Then we use these constraints to estimate new contributions to the semi-leptonic rare Kaon decay in Sec.5 and the lepton flavour number violating (LFNV) processes in Sec.6. A short conclusion is given in Sec.7.

Spontaneous Breaking of the SU (3) F family symmetry
Masses of the SU (3) F family gauge bosons come from their interactions with the Higgs fields Φ 1 , Φ 2 and Φ ν , as described by the covariant derivative terms of Φ 1,2 = Φ † 1,2 and Φ ν = Φ T ν in L Higgs , The covariant kinetic terms are We use Φ 1,2 to generate masses for quarks and charged leptons, for only one Hermitian Φ cannot produce the observed mixing in quark sector. And Φ ν generates neutrino masses through seesaw mechanism [35]. We assume that the vacuum expectation values (VEV) of Φ ν are higher than that of Φ 1,2 and dominate the contribution to the new gauge bosons masses, since neutrinos are much lighter than the charged fermions. To show that, we /ξ e , to generate charge leptons masses. The corresponding Yukawa interactions are The nearly tri-bimaximal mixing pattern of neutrinos can be explained by a residual Z 2 symmetry after SSB of SU (3) F . The VEVs of the Higgs fields are assumed as the following forms [18] where is the tri-bimaximal neutrino mixing matrix among three families, as a result of the residual Z 2 symmetry. V j (j = 0, 1, 2) is the VEV of component field of Φ ν , which possesses a residual Z 2 symmetry. After diagonalising Φ ν , we get To get the mass eigenstates, diagonalising the mass matrices of neutrino and charged leptons as follows One has U ν = U T B due to the Z 2 symmetry and U e ∼ 1 due to the approximate global U (1) symmetries after spontaneous symmetry breaking [18]. U e is expected to has similar hierarchy structure to Cabbibo-Kobayashi-Maskawa (CKM) mixing matrix [36], which gives Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [37][38][39] U P MN S = U † e U T B some deviation from U T B with non-zero θ 13 . One can get the mass spectrum of SM charged leptons and neutrinos are where the index i = 1, 2, 3 stands for charged leptons mass eigenstates e, µ, τ . And j = 1, 2, 3 stand for neutrinos mass eigenstates ν 1 , ν 2 , ν 3 . The observed neutrinos' mass hierarchy suggests Taking all the Yukawa couplings to be nature and of order 1, we get their masses are Assuming M j ν ∼ 0.1eV and using m e ∼ 0.5MeV we can get V 0 ∼ 10 14 GeV, v 1 ∼ 10 5 v s . The Yukawa couplings can be tuned to reduce all the scales. With ξ e , ξ ν ∼ 1, tuning y e L , y e R ∼ 10 −2 and y ν L , y ν R ∼ 10 −4 , we get v 1 ∼ 10v s , V 0 ∼ 10 3 TeV. With the assumption that v s ∼TeV, there is |V 0 | ≫ v 1 . So we can safely neglect contribution from Φ 1,2 in Eq.(4) and only consider that from Φ ν . There is another benefit for this interval of v s 's value. The Higgs field φ s can mixing with the SM Higgs field and be a cold dark matter candidate.
In the following parts of this paper, we denote A a F,µ , A a F,µ T a as F a µ , F µ for short. They can be parameterised by the Gell-Mann matrices with T a = λ a /2, The gauge family symmetry breaks down to residual Z 2 symmetry with nonzero V 0,1,2 . If V 0 = 0 and V 1 = V 2 = 0, the SU (3) F symmetry is broken down to SO(3) F symmetry. Then there are 5 gauge family fields, F 1 , F 3 , F 4 , F 6 and F 8 , gaining degenerate masses m = 2g F V 0 . The other 3 fields F 2 , F 5 , F 7 , which corresponding to the unbroken SO(3) F symmetry, remain massless. The SO(3) F is besides broken with non-zero V 1,2 and a Z 2 symmetry is left. The masses of F 2 , F 5 , F 7 are smaller comparing with the other five since V 1,2 < V 0 . We denote that and assume r 1 and r 2 are of same order of the Wolfenstein parameter λ ∼ 0.22. A detailed analysis of neutrinos mass spectrum [18] shows r 1 ∼ λ, r 2 ∼ ∓2λ can be used to explain the normal and inverted mass hierarchy spectrum of left handed neutrinos. We can use V 0 , V 1 , V 2 , or equally V 0 , r 1 , r 2 to get the mass spectrum of the new family gauge bosons. With the abbreviations the mass terms can be expressed as where the matrices are with r 0 ≡ 2 + 4r 1 + 2r 2 , and The matrix elements of δM 2 5×5 and M 2 3×3 are the of same order. The M 2

3×3
and M 2 5×5 can be diagnosed, where u T B and U 5 are the mixing matrices The analytical form of mixing matrix U 5 is too complex to list here. If we take the assumption r 1 ∼ λ, and r 2 ∼ −2λ(r 2 ∼ 2λ) for normal hierarchy (inverted hierarchy), the numerical results are It's notable that although the mass eigenvalues depend on r 1 , r 2 , the mixing matrix u T B do not, which is guaranteed by the residual Z 2 symmetry. With δM 2 5×5 treated as perturbation, we get the mass eigenstates of the family gauge bosons The masses of the five heavy gauge bosons are And the masses of the three light gauge bosons, which are related to the SO(3) F symmetry, are

Low Energy Effective Hamiltonian
In general the family eigenstates of the fermions are different from weak eigenstates. After the SSB of SU (3) F family symmetry, the interactions between the new family gauge bosons and SM fermions are where all the fermion triplets are weak eigenstates, and the corresponding mixing matrices are the clashes between weak eigenstates and family eigenstates. All the mass matrices of quarks and charged leptons are gained through SM Higgs H and Φ 1,2 , which are hermitian. Assuming all the Yukawa couplings to be real, as the situation in models with spontaneous CP violation, we get hermitian mass matrices, and the SSB of the new gauge symmetry and seesaw mechanism give out where U e , U ν are the mixing matrices in Eq.(9) and U u , U d are similar to U e . The mixing matrices satisfy that Experimental measurement shows that the deviation between U MN SP and U T B is small. So we can take U e ∼ 1 as the leading-order approximation. Hence the charged lepton mass eigenstates are coincident with the family eigenstates.
All the mixing matrices are physical and can be measured via the interactions among SM fermions and SU (3) F gauge bosons. It's quite different from that in SM, where U u , U d and U e , U T B are not all observable, only their clashes U CKM and U P MN S hold physical meanings.
We can also assume that U u , U d and U e have the same hierarchy structures as U CKM and can be parameterised via Wolfenstein method [40] For U e , we replace A, λ, ρ, δ by A e , λ e , ρ e , δ e . A detailed analysis of the allowed values of these parameters and the CP violation phases can be find in [41]. For the mixing matrix in up(down) quark sectors, we have mixing matrix U u (U d ) with the parameters A, λ, ρ, δ replaced by A u , λ u , ρ u , δ u (A d , λ d , ρ d , δ d ). Eq. (27) gives out the relations of the Wolfenstein parameters in U CKM , U u and U d as follows, It's known that the SM Dirac CP phase δ is not enough to generate the observed BAU [42][43][44]. And the new Dirac CP phases δ e , δ u , δ d may help to solve the baryogenesis problem.
The low energy effective Hamiltonian mediated by these new family gauge bosons can be written down easily, And the coefficients are Mixing matrix among SU (3) F gauge bosons is a block diagonal matrix made up by U 5×5 and u T B , Quite a lot of effective operators occur. To suppress these new operators' contribution, we expect that the new energy scale V 0 , V 1 , V 2 ≫ v ∼ 173GeV . There are also some FCNC operators which are absent in SM at tree level. Such operators can contribute to the processes including the P 0 -P 0 mixing in neutral meson systems, as well as some LFNV processes and some CPV processes. These processes appear in SM at loop level through penguin diagrams and box diagrams, and are suppressed comparing with the tree level processes. The new gauge bosons can contribute to these processes at tree level directly. So we may find hints of these new gauge bosons in these interesting processes. In the following parts we will find the constraints given by these processes respectively.

Mass difference of
In neutral meson systems, P 0 can mix withP 0 , where P 0 refers to either K 0 , D 0 , B 0 d or B 0 s . Such mixing violates CP symmetry and has been studied widely [46][47][48][49][50][51]. We take K 0 -K 0 as an example. In SM, K 0 andK 0 are mixed by ∆S = 2 interactions through box diagrams [52]. The measured tiny mass difference between K 0 L and K 0 S [53] puts stringent constraints on tree level FCNC beyond SM. The SU (3) F family gauge bosons and their mixing can contribute to this process at tree level. So the measured mass difference can give hint of the new gauge bosons' masses.
All the eight new gauge bosons can contribute to this mass difference. Noticed that Z 6 , Z 7 , Z 8 are lighter than the other 5 gauge bosons, we may ignore the heavy ones and focus on these lighter ones. This approximation makes V ∼ u T B . The form of U 5 is not concerned here.
The mass difference between K 0 andK 0 can be calculated using methods in [32,54,55]. The Hamiltonian can be written as H = H 0 + H 2 , with H 0 refers to the strong and electromagnetic interaction parts, which conserves the strange number. And H 2 is the weak interaction term and induces ∆S = 2 processes. The real parts of eigenvalues of H are denoted as m L , m S . Their mass difference is The new low energy effective Hamiltonian responsible for K −K mixing is Here we treat λ d as a small parameter and get the coefficient in Eq. (35) to the order of λ 2 d . At higher order the heavy family gauge bosons' effects should be take into consideration. The coefficient C K is where The contribution of G K (V 1 , V 2 ) are at order of λ 2 d . If we assume λ and λ d are of the same order, then the contribution of G K (V 1 , V 2 ) can be omitted as the contributions of the heavy gauge bosons. This approximation is equivalent to setting the mixing matrix U d ∼ 1.
To get the matrix element K 0 |O|K 0 , we use the vacuum insertion approximation (VIA). The result is where [56] N 1 ≡ K 0 |sγ 5 d|0 0|sγ 5 d|K 0 , With the definition of Kaon decay constant f K , we get To the lowest order of λ d , The hadronic matrix uncertainties will modify the relation above [45,57]. From Eq.(34), the new family interaction contributes to the mass difference via a new term in addition to that in SM as If the new contribution saturate the mass difference, then With Eq. (13), it's easy to get Using the experimental data [53,58] listed in Table.2, and taking the assumption that r 1 ∼ λ and r 2 ∼ 2λ, we can get the bounds of the symmetry broken scales which are about The lower bounds of V 0 , V 1 and V 2 as functions of r 1 , r 2 are shown in Fig.1 where and    To the lowest order, we neglect the mixing matrices U u , U d , and the same mixture of Z i in F 2 and F 5 lead to the result F D = F Bs = F K . Using data from [53,[58][59][60][61] we can get other lower bounds, which are list in the Table.2. It's obvious from Table.2 that the K 0 -K 0 system and D 0 -D 0 system give the most stringent constraints on V 1 . The lower bounds turn out to be about 70 ∼ 84 TeV. V 0 can be got through V 1 with Eq. (13), which turns out to be about 300TeV. To apply seesaw mechanism at this scale, we need tuning the Yukawa coupling to 10 −4 . Although not very nature, it's much better than the situation in SM. It is notable that the constrains on the scales are not depend on the gauge coupling strength g F . If we take it on the same order as the weak interaction, the mass of the new lightest gauge family boson can be about 100TeV. This energy scale is at the reach of the next generation 100TeV colliders.

Semi-leptonic decay of Kaon
In SM FCNC processes occur at loop level through box diagrams and penguin diagrams [45,62]. These processes are suppressed by high order coupling, loop factor 1/16π 2 , and CKM factors in power of λ ∼ 0.22. With the new gauge bosons, FCNC process can happen at tree level. The new gauge bosons may manifest themselves and play a crucial roles in such processes. On the other hand, due to their heavy masses, there is almost no significant effect on the SM tree level allowed channels. For example, the rare kaon decay process K → πνν, and LFNV processes µ → eee.
In SM, the rare Kaon decay processes are induced by Z-penguin diagram and box diagram. And the channel K L → π 0 νν violates CP directly [63], providing same flavour contents of the final neutrino pair.
The couplings between SM fermions and the new gauge bosons provide several new |∆S| = 1 low energy effective Hamiltonian terms, for the final neutrinos with arbitrary flavour contents, the effective Hamiltonian terms are: where l, m = e, µ, τ , and the numerical values of the coefficient matrix elements for r 1 ∼ λ, r 2 ∼ 2λ are The diagonal matrix elements correspond to same flavour neutrino final states. We can sum these channels incoherently and get the coefficient being l |ζ ll | 2 ∼ 0.16. We only focus on the left-handed neutrinos, thus the leptonic current takes a V -A form. As for the hadronic current, since π|A µ |K = 0, the final result only depends on π|V − A|K . We have where the neutrino pairs belong to weak eigenstates and have the same flavour.
We find the contributions from new gauge bosons are far below the SM prediction in Eq. (56). The CP violation in K L → πνν is still dominated by SM contribution.
The process µ → eγ are not influenced by the new gauge bosons at tree level. However, for µ − → e − e + e − , there are tree level contributions mediated by the new gauge bosons. Here with the assumption that U e ∼ 1, we get the effective Hamiltonian for this process is where G(r 1 , r 2 ) = 216r 1 3 − 72r 1 2 r 2 2 + 432r 1 2 r 2 + 198r 1 2 − 96r 1 r 2 3 + 216r 1 r 2 2 +264r 1 r 2 + 60r 1 − 24r 2 4 + 16r 2 3 + 74r 2 2 + 40r 2 + 6, Taking r 1 ∼ λ, r 2 ∼ 2λ, we get F (r 1 , r 2 )/G(r 1 , r 2 ) ∼ −0.12. The branching ratio for this channel is Assuming V 0 ≥ 3 × 10 2 T eV , we get This result is much larger than the SM prediction in Eq.(57) but still below the experimental bound [53]. The contribution of new physics in this process is of same order as that in K L → πνν. Both of their initial flavours are changed. And they are induced by the mixing among the heavy family gauge bosons F 1 , F 4 , F 6 and F 3 , F 8 . There are many similar processes, such as the rare B decays through B → X s µ − µ + , rare Kaon decay through K L → π 0 e + e − , K L → µ + µ − . Their branching ratios are of the same order, i.e. 10 −16 from the new gauge bosons' contributions. And they are all below the various experimental bounds. These results make the lower bound V 0 ∼ 300T eV safe.

conclusion
We have investigated the structure of SU (3) F gauge family symmetry model and its low energy phenomenal results in flavour physics. This family symmetry undergoes spontaneous breaking to SO(3) F and then to a residual Z 2 symmetry. Seesaw mechanism is widely used both in leptonic sector and quark sector to explain the observed mass hierarchy and mixing structure, especially the neutrinos' mass spectrum. The equality of seesaw scale and flavour symmetry breaking scale needs a tuning of the Yukawa couplings, about 10 −4 , which are much softer than SM. New scalar field is introduced and may be a dark matter candidate. Also new CP violation phases appear and may provide a solution to the baryon asymmetry in the universe. The symmetry breaking mode makes the new gauge bosons can be divided into two groups. Their mass scales can be constrained through the mass differences of P 0 -P 0 meson systems. We get the broken scale of the new gauge family symmetry is about V 0 ≥ 300 TeV, and mass of the lightest new gauge boson can be low as 100 TeV. These new gauge bosons can induce FCNC processes at tree level, and their contributions are suppressed by their heavy masses and the resulting branching ratios are about 10 −16 , which is 4 ∼ 5 order below the current experimental bounds. We expect the improvement of the rare FCNC processes' measurements, as well as some exotic processes' discovery, which may be found in the next running of LHC and the next generation colliders of 100 TeV, can throw some light upon this new flavour symmetry. The field strengths of all gauge fields, including the SU (3) family symmetry, are defined as

Acknowledgments
(A.1) We define the covariant derivative as The full Lagrangian is with each term defined as follows L k = u L,R iγ µ D µ u L,R + d L,R iγ µ D µ d L,R + e L,R iγ µ D µ e L,R + ν L iγ µ D µ ν L ,