Flavor changing in the flipped trinification

The flipped trinification, a framework for unifying the 3-3-1 and left-right symmetries, has recently been proposed in order to solve profound questions, the weak parity violation and the number of families, besides the implication for neutrino mass generation and dark matter stability. In this work, we argue that this gauge-completion naturally provides flavor-changing neutral currents in both quark and lepton sectors. The quark flavor changing happens at the tree-level due to the nonuniversal couplings of $Z'_{L,R}$, while the lepton flavor changing $l\rightarrow l'\gamma$ starts from the one loop level contributed significantly by the new charged currents of $Y_{L,R}$, which couple ordinary to exotic leptons. These effects disappear in the minimal left-right model, but are present in the framework characterizing a flipped trinification symmetry.


INTRODUCTION
The experiments of neutrino oscillations caused by nonzero small neutrino masses and flavor mixing have provided the most important evidences that prove the new physics beyond the standard model [1]. The compelling way to address the neutrino masses is to introduce right-handed neutrinos into the standard model, by which the neutrino mass generation is done by seesaw mechanisms [2]. The pioneering model that recognizes the seesaw mechanisms is the minimal left-right symmetric model [3], where the neutrino masses were predicted before the experimental confirmations.
The minimal left-right symmetric model offers a possibility to understand the origin of the parity violation of weak interactions, but as the standard model it neither shows why there are only three fermion generations nor addresses dark matter stability that accounts for more than 25% mass-energy density of the universe [4]. Indeed, the lightest right-handed neutrino may have a keV mass responsible for warm dark matter, but it would overpopulate the universe due to gauge interactions, which require nonstandard dilution mechanisms [5].
On the other hand, cold dark matter scenario that adds a new field as well as imposing a stabilizing symmetry remains to be arbitrary, ad hoc included [6].
It is well established that the 3-3-1 model [7] provides a potential solution to the generation number and addresses the issue of dark matter naturally [8]. Hence, we have recently proposed a theoretical model that unifies both the left-right and 3-3-1 symmetries, resulting in a SU (3) C ⊗ SU (3) L ⊗ SU (3) R ⊗ U (1) X gauge group, called flipped trinification [9] (for other interpretations, see [10]). This model inherits all nice features of both left-right and 3-3-1 models. Particularly, dark matter naturally exists which along with normal matter form gauge multiplets by the gauge symmetry, whereas the three generations emerge as a result of anomaly cancellation. Moreover, the origin of the matter parity and the dark matter stability are determined by a residual gauge symmetry. The new physics predicted occurs at TeV scale, giving rise to interesting signatures at current colliders.
An intriguing feature of the flipped trinification is that flavor violating interactions appear in both quark and lepton sectors. As a trinification symmetry is flipped, both left-and right-handed quark flavors transform differently under SU (3) L,R . Consequently, they lead to tree-level flavor-changing neutral currents (FCNCs) that coupe to Z L,R , where the relevant observables after integrating out Z L,R depend on both left-and right-handed quark mixing which matches a flipped trinification and preserves the SU (3) L and SU (3) R interchange.
The electric charge operator is given by where T iL,R (i = 1, 2, 3, ..., 8) and X are SU (3) L,R and U (1) X charges, respectively. The baryon minus lepton number is identified as which is non-commutative, in contrast to the usual (Abelian) extensions. We further define a basic electric charge as q = −(1 + √ 3β)/2.
Analogously, the fermion content is obtained from those of the 3-3-1 model by left-right symmetrization, which yields where a = 1, 2, 3 and α = 1, 2 are generation indices. The model predicts new fermions N a , J a , besides the right-handed neutrinos ν aR . The fermion sector is more economical than that of the well-known trinification [11]. In contrast to the trinification, the SU (3) L or SU (3) R anomaly cancellation requires the number of generations to match that of colors, and that the third quark generation transforms under SU (3) L,R differently from the first two quark generations, analogous to the 3-3-1 model [7].
To break the gauge symmetry and generate the masses appropriately, the scalar multiplets are supplied as which reflect the left-right symmetry. The corresponding VEVs are given by As shown in [9], the symmetry breaking is proceeded through several schemes, depending on the hierarchy arrangements of the VEVs. All the schemes lead to the existence of a residual discrete gauge symmetry that conserves every VEV, called matter parity gauge multiplets (see [9] for details of the dark sector and dark matter candidates). For consistency, we assume Λ R , w R , w u, u Λ L , w L , appropriate to the potential minimization.
[Indeed, the minimization conditions imply Λ L 0, w L 0, where the small nonzero values come from abnormal perturbative interactions, as seen in the next section]. This means that the flipped trinification is broken down to the standard model and matter parity, and then to the remnant SU (3) C ⊗ U (1) Q ⊗ W P , where the left-right asymmetry is explicitly recoginzed at the electroweak phase due to w = 0, w R = w L and Λ R = Λ L .

B. Fermion masses
First, we consider the physical states and masses of fermions. They arise from the Yukawa interactions as follows where M is a new physics scale that defines the effective interactions. The left-right symmetry demands that the couplings y, z, k, k are Hermitian, whereas x, t are generic.
After the symmetry breaking, the Yukawa Lagrangian yields fermion masses. The new leptons get a large mass at the new physical scale as follows The ordinary charged leptons obtain a mass at the weak scale, Note that the new leptons do not mix with the ordinary leptons due to the matter parity conservation. If neglecting the effective interactions, they have the same mixing matrices.
The Lagrangian (12) allows neutrinos having both kinds of mass terms: Dirac and Majorana. In the basis (ν L , ν c R ), the neutrino mass matrix is given by

C. Gauge boson masses
The presence of the scalar multiplets σ L , χ L does not significantly change the mass spectrum of the gauge bosons that was derived in [9]. Hereafter, we summarize the main results of the gauge sector. The gauge bosons W L , W R slightly mix, which yield eigenstates where the mixing angle ξ is defined by The W 1,2 masses are given by where g L , g R are SU (3) L,R couplings respectively, which match t R ≡ g R /g L = 1 at the flipped trinification scale due to the left-right symmetry. At the low energy, they may separate, t R = 1, due to the different contributions to the running couplings. W 1 is identical to the standard model W boson, implying u 2 + u 2 = (246 GeV) 2 , while W 2 is new.
Besides, the model predicts new non-Hermitian gauge bosons X ±q L,R and Y ±(q+1) L,R that couple to the charges T 4 ∓ iT 5 and T 6 ∓ iT 7 , respectively. The physical states are Here, the mixing angles ξ 1 , ξ 2 are obtained as The neutral gauge bosons A 3L,R , A 8L,R , B, that couple to the charges T 3L,R , T 8L,R , X respectively, mix via a 5 × 5 mass matrix, given in Appendix A. The photon field is which is massless, where t X ≡ g X /g L is U (1) X /SU (3) L coupling ratio. The sine of the Weinberg angle is s W = t X t R / t 2 X (1 + β 2 ) + t 2 R (1 + t 2 X (1 + β 2 )), obtained by matching the electromagnetic gauge coupling [12]. As usual, the standard model Z boson is given orthogonally to A by New neutral gauge bosons take the forms that are orthogonal to both A and Z L , i.e. to the U (1) Y gauge field in the parentheses, where ς = 1/ t 2 R + β 2 t 2 X and ς 1 = 1/ t 2 R + (1 + β 2 )t 2 X . In the new basis (A, Z L , Z L , Z R , Z R ), A is decoupled, while Z L infinitesimally mixes with (Z L , Z R , Z R ) where the relevant mixing angles are suppressed by (u, u ) 2 /(w, w R , Λ R ) 2 1.
Neglecting the mixing, Z L is a physical field and decoupled as the photon. We are left with diagonalizing the mass matrix of (Z L , Z R , Z R ), which yields the eigenstates Z L , Z R , Z R and corresponding masses as provided that Λ R w, w R , where the Z R -Z R mixing angle is finite, Analogously, we can diagonalize the mass matrix for the case Λ R w, w R , where Z R is decoupled, while Z L,R finitely mix. For the case Λ R ∼ w, w R , all the gauge bosons Z L,R , Z R finitely mix, which can be parameterized by the Euler angles. Note that w, w R are always taken in the same order, since they simultaneously break SU (3) L ⊗ SU (3) R → SU (2) L ⊗ SU (2) R and correspondingly reduce the left-right symmetry.

D. Higgs masses
Let us rewrite the scalar potential that includes the full scalar content. The full scalar potential takes the form, Here the interactions ζ 8,9 , f 1,2 are abnormal and subdominant since they can be removed by a global symmetry U (1) that nontrivially transforms any one of the fields.
Expanding the neutral scalar fields around their VEVs, we find minimization conditions, mass terms, and interactions. The mass terms are divided as where V S , V A include those of CP even and CP odd scalar fields, respectively, whereas V charged consists of those of the charged scalars. Considering q + 1-charged scalars, four and two massive Higgs fields, . These states are related to the gauge states by where where the mixing angle ξ 4 is defined by Concerning the singly-charged scalars, the model contains two massless Goldstone bosons (G ± W L , G ± W R ), which are eaten by W ± L , W ± R respectively, and two physical massive with the mixing angel ξ 5 defined by .
For the neutral scalars, they split into two parts: CP-odd and CP-even. The model contains only a light CP-even neutral scalar that is identified as the standard model Higgs boson, while the other CP-even states achieve large masses at the new physical scale. Additionally, the CP-odd part contains four massless Goldstone bosons, which are correspondingly eaten by the eleven massive gauge bosons Z, Z R , Z L , Z R , and three heavy scalar states.

III. FCNC
As mentioned, the tree-level FCNCs arise due to the discrimination of quark generations, i.e. the third generations of left-and right-handed quarks Q 3L,R transform differently from the first two Q αL,R under SU (3) L,R ⊗U (1) X gauge symmetry, respectively. Hence, the neutral currents will change ordinary quark flavors that nonuniversally couple to T 8L,R , since X is related to T 8L,R by the electric charge operator and that Q, T 3L,R conserve every flavor.
Indeed, with the aid of X = Q − (T 3L + T 3R ) − β (T 8L + T 8R ), the neutral currents of quarks take the form where Q L,R are summed over all the quark multiplets. All the terms coupled to T 3L,R , Q do not flavor change, because u L,R , d L,R are identical under such charges. Hence, the FCNCs exist only for the terms that couple to T 8L,R , In the basis (Z L , Z R , Z R ), the Lagrangian (50) is rewritten as where we denote q either u or d , Taking, for instance, the limit Λ R > w, w R and changing to the mass basis, we obtain It is noted that since g R A 8R − βg X B ∼ Z R and the large mixing Z R -Z R , both Z R and Z R contribute, whereas g L A 8L − βg X B composes Z L and these fields as it is not orthogonal to Z R . Consequently, the three fields Z L , Z R , Z R dominantly couple to the tree-level FCNCs, for i = j. The new observation is that Z R flavor changes due to the large mixing with Z R , in contrast to the minimal left-right symmetric model.
Integrating the heavy gauge bosons Z R , Z R , Z L out, we determine the effective Lagrangian that describes the meson mixings, where Generally, the fields Z L , Z R , Z R mix via a 3 × 3 mass matrix, as given in Appendix A. In this case, the mass eigenstates, Therefore, the couplings given in Eq. (56) are generalized by This effective Lagrangian contributes to mass splittings ∆m M between neutral mesons With the help of the mass matrix elements in [13], the mass differences computed from (57) and (58) are The total mass differences can be decomposed as where the first term comes from the standard model contribution given in [14] and the second term is the new physics contribution as derived in (59), (60), or (61). These predictions are compared to the experimental values [14]. Here, for the neutral kaon mixing we assume that the theory predicts the mass difference within 30% since the potential long-range uncertainties are large. In contrast, the intrinsic theoretical uncertainties for B s,d mass differences are small, assumed to be within 5%. In other words, the meson mass differences obey 16.8692/ps< (∆m Bs ) tot < 18.6449/ps.
For a numerical investigation, we take w = w R , g L = g R (i.e. t R = 1), V uL = V uR = I, and Λ R , w are beyond the weak scale and free to float,

IV. CHARGED LFV
One of the strongest bounds on the charged LFV is the decay µ → eγ. Hence, in this work we study that channel in detail and discuss other charged LFV processes which are potentially troublesome.
A. µ → eγ decay rate We are going to derive an expression for the branching decay ratio of µ → eγ in the flipped trinification, based upon SU (3) where U N L,R and U l L,R are basis-changing (mixing) matrices and unitary. If the left-right symmetry is not imposed, i.e. w L = 0 as in the previous study [9], U N L,R and U l L,R are not independent, because the mass matrices of N and l are solely generated by the same Yukawa coupling y ab [cf. Eqs. (13) and (14)]. It is easily realized that the one-loop diagrams with the mediation of charged gauge Y ±(q+1) or Higgs H ±(q+1) bosons that couple to l, N do not contribute to the decay µ → eγ, since the new leptons do not mix in the basis of charged lepton eigenstates due to the mentioned m l , m N ∼ y. Alternatively, when the left-right symmetry is included, the mass matrices of l and N generally differ due to the z ab coupling contribution, where note that 0 = w L w, w R ∼ M recognize a left-right asymmetry at the low energy. In this case, the new fields Y ±(q+1) /H ±(q+1) and N significantly contribute. That said, the two cases must be taken into account when we parametrize the mixing matrices for numerical investigation, in the following section.
The neutrino mixing matrix is denoted as U ν , which is a 6 × 6 unitary matrix, relating the gauge state X L ≡ (ν L , (ν R ) c ) T to the mass eigenstate X L , such as X L = U ν X L . We write U ν in terms of Hence, the Yukawa coupling x can be easily written in terms of diagonal light (called m L ) and heavy (called m R ) mass matrices and the mixing matrices U L,R,A,B , Similarly, the Yukawa coupling y can be expressed in terms of the diagonal matrices m l , m N that include respective charged and new lepton masses and the mixing matrices U l,N L,R , To derive the decay rate µ → eγ at one-loop approximation, we necessarily calculate the form factors of the relevant one-loop diagrams that contribute to the process. We list in doubly-charged Higgs scalars [16][17][18][19][20][21], as well as singly-charged Higgs scalars for the first time in [22]. In this paper, we present the results for the form factors of one-loop diagrams with the exchange of virtual general charged Higgs scalars and gauge bosons. To our best knowledge, this has not been done so far.
Vertex Couplinḡ The effective Lagrangian derived from calculations of the form factors of one-loop diagrams for µ → eγ with participation of virtual scalars and gauge bosons in the considering model can be simply expressed as Here A L,R are the form factors: where 2), and m k are the masses of associated fermions that along with either H Q or A Q µ form loops. The functions F (Q), F (r, s k , Q), G Q γ (x), and R Q γ (x) appearing in Eqs. (71) and (72) are defined as The branching ratio of µ → e + γ decay is obtained as [17,18] Br where α em = 1/128 is the fine-structure constant.
B. Numerical analysis/discussion: w L = 0 Before performing numerical calcualtions using the branching decay formula obtained in the previous section, let us estimate the magnitudes of relevant VEVs. Among the VEVs introduced, the smallest one could be Λ L , which is at eV scale responsible for the neutrino masses, much smaller than the weak scales u, u satisfying the constraint u 2 + u 2 = (246 GeV) 2 . Hence, we safely neglect the contributions of Λ L . The quark FCNC constraints imply w, w R , Λ R > ∼ O(50 − 100) TeV, appropriate to the collision bounds [9], where such VEVs break the flipped trinification to the standard model, significantly greater than the weak scales. Finally, Λ R can take a value, such that (i) Λ R w, w R , (ii) Λ R ∼ w, w R , or (iii) Λ R w, w R , depending on the symmetry breaking scheme. The viable dark mater scenarios [9,23] prefer the cases (i) and (ii), which will be taken into account.
Due to the condition Λ R , w, w R u, u w L , Λ L , the masses of the gauge bosons relevant to the process are approximated as m 2 where we have used g L = g R = g. Note that W 1 has the mass identical to the standard model, while m W 2 and m Y 1,2 are large, at TeV scale or higher. The masses of relevant new Higgs bosons H ± , H ±± , and H ±(q+1) depend on unknown parameters present in the scalar potential, which cannot be estimated precisely. However, their masses are all proportional to the new physics scales Λ R , w, w R , which should be large enough in order to escape from detections [15]. That said, it is reasonable to choose the new Higgs masses from hundreds of GeV to few TeV. Particularly, in hierarchical cases the largest masses can be chosen up to hundreds of TeV.
Let us parametrize the Yukawa couplings and mixing matrices, involved in the branching ratio µ → eγ in (77), in forms convenient to numerical investigation using the current data.
Without lost of generality, we work in the basis of charged lepton mass eigenstates, i.e. m l = yu / √ 2 is diagonal, implying U l L,R = I, in the same criteria used in the standard model. Besides, the new lepton masses are generated by the same Yukawa matrix y, with the relation between both kinds of masses has been given in (69) where note that N is a standard model singlet for β = −1/ √ 3, whereas it has an electric charge q = −1 for β = 1/ √ 3. The former is always viable in similarity to the case of a light sterile neutrino. However, the latter should be ruled out due to the electroweak precision test, unless the new physics scale is unexpectedly raised, w/u > ∼ 10 5 , so that the lightest new lepton is heavy enough to suppress the dangerous processes, e.g. Z → N N .
Note that at one-loop approximations, the diagrams with virtual neutral Higgs scalars do not contribute to LFV processes, including µ → eγ decay, because the interacting vertexes of two leptons with such a neutral scalar do not change flavor (i.e. conserving flavor). The vertex couplings are governed by the magnitudes of diagonal elements of the Yukawa matrix y as well as a mixing factor among neutral scalars. These vertexes are also constrained by the current experiments through the channels of the standard model like Higgs decay into two leptons h → ¯ . According to [15], h → ττ has been observed at a quite high precision, while h → µμ is likely observed, but at large uncertainty, and the branching decay h → eē can only be set by an upper limit, Br(h → eē) < 1.9 × 10 −3 . All these agree to the strengths of h interactions, set by the corresponding lepton masses. Due to the mixing, h can decay into light N 's, but the rate is highly suppressed by (u, u ) 2 /(w, w R , Λ R ) 2 1. The light N 's are undetectable due to weak interaction strengths. However, since no constraint has been made to their masses, they can take any values consistent to the scenario of interest.
Our study is interested in a model whose new physics scale is not too high, thus presenting has a diagonal form of the heavy neutrino masses.
In the following, we will present the results of numerical calculations for the case where the involved parameters are chosen as u = 0.1 GeV, w = 10 TeV, w R = 20 TeV, diag(M diag R ) = (10,20,30) TeV and U R (θ 12 , θ 13 , θ 23 , δ ) = U R (π/4, π/4, 0, 0). Note that the choice u = 0.1 GeV is in order to conserve the condition u 246 GeV, whose important implications have been discussed before. The other quantities such as Λ L , w L , and light neutrino masses are neglected due to the small effects for the process. Table I, it is realized that their vertexes do not contribute to the µ → eγ branching ratio at the one-loop level, as mentioned. The reason is similar to the case of vertexes of neutral Higgs scalars, which conserve lepton flavors. Indeed, all of the matrices relevant to them, such as U l R,L , U N R,L and y, are diagonal. In Figs (10,20,30) TeV and U L,R has canceled out the dependence of U mix on Λ R . Moreover, W 1 is the standard model W boson whose mass is fixed as M W 1 = 80 GeV. Therefore, the branching ratio lines shown in Fig. 2 are depicted as a function of the new boson mass M W 2 ∼ Λ R and mixing angle ξ w . For each value of ξ w , the branching ratio goes down due to the dominant contribution of W 2 to a constant value, as increasing Λ R . The constant line is preserved by a constant contribution of W 1 . Using the MEG current bound on the µ → eγ decay, one roughly estimates the lower bound Λ R ≥ 12, 10 TeV for ξ w = 10 −3 and ξ w ≤ 10 −4 , respectively. The strong dependence of the branching ratio on the mixing angle ξ w , which separates about two order between two successive lines for the range of large Λ R , suggests the domination of the interference terms in A L,R [cf. Eqs. (71) and (72)]. Indeed, the interference terms are proportional to m k mµ sin ξ w cos ξ w ≈ m k mµ ξ w . Thus, the branching ratio is proportional to w , which is consistent to the observation from the figure, whereas the other terms are either proportional to cos ξ 2 w 1 or suppressed by a factor ξ 2 w . It is figured out that the dominant interference terms are provided by the factor m k mµ ∼ 10 5 , for Whilst the sensitivity of the future MEG are possible to probe µ → eγ signal, provided that Λ R ≤ 1300, 120, 15.5, 13 TeV for ξ H = 10 −1 , 10 −2 , 10 −3 , 10 −4 , respectively.
In the next two figures, we demonstrate the dependence of the branching ratio as a single variable function of the new Higgs mass M H , where Λ R is fixed as 100 TeV. As we see from Figs. 4 and 5, the smaller the mixing angle ξ H is, the smaller the lower bound is set for the heavy Higgs masses. If the contributions to the diagrams include only the virtual singly charged scalar (Fig. 4, left panel), the lower bound for the scalar masses reduces from C. Numerical analysis/discussion: w L = 0 The flipped trinification discriminates from the minimal left-right symmetric model especially in the extended particle sectors, governed by the new gauge symmetry. Part of them produces the interesting quark FCNCs, as studied above. In this section, we argue that the presence of other part of them gives novel contributions to the charged LFV. It is stressed that such LFV processes e.g. µ → eγ can be altered in the case of non-vanishing w L . Although w L is constrained to be much smaller than M, Λ R , w R , w as well as not modifying the results discussed in the previous section related to H i and W i , the non-vanishing w L causes the mass matrices of ordinary charged and new leptons to be not simultaneously diagonalized. This provides the new sources of the LFV, which involve the (q + 1)-charged Higgs and gauge bosons (H 1,2 , Y 1,2 ) as well as the new leptons (N ) in the loops for µ → eγ, which is a new feature of the model.
In the basis of ordinary charged lepton mass eigenstates, the mentioned, new lepton mass matrix can be expressed as where Here ||y|| ∼ ||M l ||/u ∼ 10 −3 -10 −2 is constrained by the ordinary charged lepon masses and small. The coupling matrix z is generic and maybe sizable, but it generally obeys ||z|| ∼ 1 ||y||(w/w L ), provided that w/w L > ∼ 10 3 . For instance, if w = 10 TeV, one takes w L < ∼ 10 GeV. This leads to ||∆M N || ||M 0 N ||, which is also expected due to the contribution For brevity, in numerical calculation we assume z ab as a real symmetric matrix with z 12 = z 13 = z 23 = 1, which means that the new lepton mass matrix is invariant under the charge-conjugation and parity transformations. This choice leads to an approximation, We fix M = 100 TeV, w = 30 TeV, and w L appropriately ranging from an infinitesimal value to a few GeVs. All the remaining parameters take the same values as in the previous subsection.
In Fig. 6, we depict the dependence of the branching ratio Br ( Fig. 7 are smaller than the gauge ones by 14 orders of magnitude, which are about 13 orders of magnitude below the future MEG sensitivity. It is not hard to see that the branching ratios are strongly suppressed by the ordinary charged-lepton Yukawa couplings y 4 , where the biggest element is only y 3 ∼ 10 −2 , which are much smaller than the gauge contribution.

D. Other charged LFV processes
In this model, the charged LFV processes such as µ → 3e and τ → 3µ(3e) can exist at the tree-level, exchanged by the charged Higgs H ±± 1,2 . The µ → 3e branching ratio in the present scheme with a low scale of new physics of order 10-100 TeV is expected to be in the sensitive ranges of the current and near-future experiments. Present upper bound on branching ratios of τ → 3µ(3e) are in the order of 10 −8 [15], which are five orders less stringent than those of µ → 3e decay at 10 −12 [15]. Moreover, the µ → 3e experiment at Paul Scherrer Institute (PSI) is expected to determine signal of Br(µ → 3e) ≥ 10 −15 and its upgrade is sensitive to the µ → 3e branching ratio not smaller than 10 −16 [24]. Therefore, we need consider only the interested process of µ → 3e decay in this research.
It is easily verified that, in contrast to the previously mentioned processes the charged-LFV neutral-Higgs decays e.g. h → µτ receives only one-loop contributions. On the theoretical side, they are strictly suppressed by the heavy particle masses and the loop factor 1/16π 2 . It is easily proved that such processes satisfy all the current bounds with the chosen parameter regime, since such experimental bounds are less tight [15].
Using the relevant LFV vertexes given in Table I, while keeping in mind that the doubly charged Higgs bosons that dominantly contribute to the µ → 3e decay have the transferred momenta much smaller than their masses, one can write down the effective Lagrangian as Here, we denote M H i (i = 1, 2) to be the masses of doubly charged Higgs bosons and The branching ratio is straightforwardly obtained as [25] Br where As a specific case, taking θ 12 = θ 13 = θ 23 = π/4 and δ = 0, Fig. 8 describes the behavior of the µ → 3e branching ratio as a function of the doubly charged Higgs masses. The figure reveals a line of monotonically decreasing function as increasing of M H , which is consistent to the fact that the branching ratio is inversely proportional to M 4 H , mentioned above. The lower bounds of the doubly charged Higgs masses corresponding to the current limit, PSI experiment and its upgraded sensitivities are 14, 79, 143 TeV, respectively. Thus we apparently conclude that the future PSI experiment is more sensitive to the new physics of the considering model than the Mu to E Gamma experiment (MEG), which gives a lower bound M H = 53 TeV for the case ξ H = 0.1.

V. CONCLUSION
When a gauge symmetry is flipped, it leads to a deeper structure that defines a more fundamental theory. For instance, SU (2) L flipped yields electroweak unification; SU (5) flipped defines a seesaw mechanism; SO(10) flipped leads to E 6 and promising superstring theories. In this work, we have addressed such a nontrivial task, the flipped trinification and its novel consequences. First of all, a trinification flipped unifies both the 3-3-1 and left-right symmetries. Consequently, this flipped trinification resolves the generation number and the weak parity violation. Additionally, it generates neutrino masses and dark matter naturally via the gauge symmetry.
An important feature of the flipped trinification is that it presents the flavor changing currents in both quark and lepton sectors. We have probed that the quark FCNCs bound the new physics scale to be at or beyond several tens of TeV via the neutral meson mixings, B 0 d,s -B 0 d,s . The charged LFV via the decay µ → eγ yield mostly the same bound, whereas the other processes such as τ → 3µ(3e) and h → µτ are easily experimentally satisfied. The process µ → 3e receives tree level contributions by the doubly charged Higgs bosons and presents the same limit on the new physics as the meson mixing and µ → eγ do.
All the results indicate that the trinification is possibly flipped at tens of TeV. Additionally, the contributions of the new particles other than the left-right symmetric model are important to set the charged LFV and quark FCNC observables, which can be used to prove or rule out this proposal. order possess a mass matrix m 33 = u 2 1 + u 2 2 + 4(w 2 + w 2 L + Λ 2 L ) 3 , m 44 = t 2 R (u 2 1 + u 2 2 + 4(w 2 + w 2 R + Λ 2 R )) 3 where m ij are defined as where M 33 takes the form,