# How light the lepton flavor changing gauge bosons can be

## Abstract

Spontaneous breaking of inter-family (horizontal) gauge symmetries can be at the origin of the mass hierarchy between the fermion families. The corresponding gauge bosons have flavor-nondiagonal couplings which generically induce the flavor changing phenomena, and this puts strong lower limits on the flavor symmetry breaking scales. However, for special choices of chiral horizontal symmetries the flavor changing effects can be naturally suppressed. For the sake of demonstration, we consider the case of leptonic gauge symmetry \(SU(3)_e\) acting between the right-handed leptons and show that the respective gauge bosons can have mass in the TeV range, without contradicting the existing experimental limits.

**1**. The replication of fermion families is one of the main puzzles of particle physics. Three fermion families are in identical representations of the Standard Model (SM) gauge symmetry \(SU(3)\times SU(2)\times U(1)\). Its electroweak (EW) part \(SU(2)\times U(1)\) is chiral with respect to fermion multiplets: the left-handed (LH) leptons and quarks, \(\ell _{Li}=(\nu _i,e_i)_L\) and \(Q_{Li}=(u_i,d_i)_L\), transform as weak isodoublets while the right-handed (RH) ones \(e_{Ri},u_{Ri},d_{Ri}\) as isosinglets, \(i=1,2,3\) being the family index. The chiral fermion content of SM has a remarkable feature that the fermion masses emerge only after spontaneous breaking of \(SU(2)\times U(1)\) by the vacuum expectation value (VEV) \(\langle \phi ^0\rangle = v_\mathrm{w} = 174\) GeV of the Higgs doublet \(\phi \), via the Yukawa couplings

*Z*boson and Higgs boson. In this way, the SM exhibits a remarkable feature of natural suppression of flavor-changing neutral currents (FCNC) [1, 2]: all FCNC phenomena are suppressed at tree level and emerge exclusively from radiative corrections. At present, the majority of experimental data on flavor changing and CP violating processes are in good agreement with the SM predictions. There are few anomalies, not definitely confirmed yet, which could point towards new physics beyond the Standard Model (BSM).

In a sense, the SM is technically natural since it can tolerate any Yukawa matrices \(Y_f^{ij}\), but it tells nothing about their structures which remain arbitrary. So the origin of the fermion mass hierarchy and their weak mixing pattern remains a mystery.

**2**. The key for understanding the fermion mass and mixing pattern may lie in symmetry principles. E.g. one can assign to fermion species different charges of abelian flavor symmetry *U*(1) [3]. Alternatively, one can introduce non-abelian horizontal gauge symmetries \(G_H\) as e.g. \(SU(3)_H\) [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] with the flavor gauge fields dynamically marking the family indices. Such a gauge theory of flavor can be considered as quantum flavordynamics, provided that it is built in a consistent way and sheds some more light on the origin of the fermion mass hierarchy.

Namely, one can envisage that the form of the Yukawa matrices in (1) is related to the VEV structures of horizontal scalar fields (known also as flavons) which spontaneously break \(G_H\), and the fermion mass hierarchy emerges from the hierarchy between the scales of this breaking. In Refs. [8, 9] this conjecture was coined as *hypothesis of horizontal hierarchies* (HHH). It implies that the fermion masses cannot be induced without breaking \(G_H\) so that it cannot be a vector-like symmetry, but it should have a chiral character transforming the LH and RH particle species in different representations. In such a picture the fermion Yukawa couplings should emerge from the higher order “projective” operators containing flavon scalars which transfer the VEV pattern of flavons to the structure of the Yukawa matrices \(Y_f\). In the UV-complete pictures such operators can be induced via renormalizable interactions after integrating out some extra heavy fields, scalars [4, 5, 6, 7] or verctor-like fermions [8, 9, 10, 11, 12, 13, 14]. In the context of supersymmetry, such horizontal symmetries can lead to interesting relations between the mass spectra of fermions and their superpartners and naturally realize the minimal flavor violation scenarios [15, 16, 17, 18, 19].

Discovery of the flavor gauge bosons and/or related FCNC effects would point towards new BSM physics of flavor. However, a direct discovery at future accelerators can be realistic only if the scale of \(G_H\) symmetry breaking is rather low, in the range of few TeV. Therefore, the following questions arise: (i) for which choice of symmetry group \(G_H\) one can realize the HHH paradigm, relating the fermion mass hierarchy to its breaking pattern, and (ii) which is the minimal scale of \(G_H\) symmetry allowed by present experimental limits – namely, can this scale be low enough to have \(G_H\) flavor bosons potentially within the experimental reach?

**3**. In the limit of vanishing Yukawa couplings, \(Y_f \rightarrow 0\), the SM acquires a maximal global chiral symmetry

*U*(3) groups respectively as \(\ell _{L}\sim 3_\ell \), \( e_{R} \sim 3_e\), etc. The Yukawa couplings (1) can be induced by the VEVs of flavons in mixed representations of these symmetry groups. One can consider the higher order operators e.g. for leptons

*SU*(5) grand unified theory (GUT) which unifies \(\ell _L\) and \(d^c_L\) fragments of each family in \(\bar{5}\)-plets and \(e^c_L\), \(u^c_L\) and \(Q_L\) fragments in 10-plets (\(\psi ^c_L= C\overline{\psi _R}^T\),

*C*is a charge conjugation matrix), the maximal symmetry reduces to two factors \(U(3)_\ell \times U(3)_e \):

*SO*(10) GUT all fermions of one family including the RH neutrino \(N_R\) reside in the spinor multiplet \(16_L = (\bar{5} + 10 + 1)_L\). Hence, there can be only one chiral symmetry

*U*(3) between three families of 16-plets, with all LH fermions \(\ell _L,Q_L\) transforming as triplets and the RH ones \(N_R,e_R,u_R,d_R\) as anti-triplets, in the spirit of chiral horizontal \(SU(3)_H\) of Refs. [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. For predictive models based on \(SO(10) \times SU(3)_H\) see e.g. in Refs. [20, 21, 22, 23].

It is tempting to consider some part of the maximal flavor symmetry (2), or its GUT-restricted versions, as a gauge symmetry \(G_H\). Gauging of chiral *U*(1) factors is difficult since they are anomalous with respect to the SM.^{1} Therefore, we consider a situation in which only some of non-abelian *SU*(3) parts in (2) are gauged. In particular, in this paper we concentrate on the lepton sector and discuss a simple model with a gauge symmetry \(G_H=SU(3)_e\) transforming the RH leptons as a triplet \(e_{R\alpha } = (e_1,e_2,e_3)_R\), while the LH leptons \(\ell _{Li} = \ell _{1,2,3}\) have no symmetry and \(i=1,2,3\) is just a family number.^{2}

We show that the lepton mass hierarchy \(m_\tau \gg m_\mu \gg m_e\) can be directly related to the hierarchy of \(U(3)_e\) symmetry breaking scales. As for the lepton flavor violating (LFV) phenomena induced by \(SU(3)_e\) gauge bosons, we show that they are strongly suppressed since the intermediate \(SU(2)_e\) subgroup acts as an approximate custodial symmetry.^{3} The respective scale is allowed to be as low as 2 TeV, without contradicting the present experimental limits on the LFV processes.^{4}

**4**. The LH and RH lepton fields of our model are in the following representations:

We assume that there is only one Higgs doublet \(\phi \) with the standard Higgs potential \(V(\phi ) = \lambda (\vert \phi \vert ^2 - v_\mathrm{w}^2)^2\). However, the Yukawa couplings of \(\phi \) with the fermions \(\ell _{Li}\) and \(e_{R\alpha }\) are forbidden by \(SU(3)_e\) symmetry. So, for generating the lepton masses this symmetry should be broken.

*M*(see Fig. 1, lower diagram).

For full breaking of gauge \(SU(3)_e\) symmetry, just two flavons with non-aligned VEVs are sufficient. An order of magnitude hierarchy between the scales \(v_2\) and \(v_3\), \(v_2/v_3 \sim m_\mu /m_\tau \), can emerge due to some moderate conspiracy of parameters admitting a natural “spread”, say within an order of magnitude, between the mass terms and coupling constants of \(\xi _2\) and \(\xi _3\) in (14). But large hierarchy \(v_1/v_3 \sim m_e/m_\tau \) at first sight requires a strong fine tuning. However, small \(v_1\) can be obtained naturally considering the case in which the VEV matrix \(\langle \xi _n^\alpha \rangle \) has rank 2 in the limit \(\mu \rightarrow 0\). This means that only two flavons, \(\xi _2\) and \(\xi _3\), get the VEVs oriented as in (9), because of their negative mass\(^2\) terms in (14), while the third flavon \(\xi _1\) has a positive mass\(^2\) term, i.e. \(\mu _1^2< 0\), and in the limit \(\mu =0\) it remains VEVless. The ratio of VEVs \(v_3/v_2 = (\mu _3/\mu _2) \sqrt{ \lambda _2/\lambda _3} \) can be of one order of magnitude due to natural fluctuation of the involved parameters. Then for \(\mu \ne 0 \) the last term in (14) explicitly breaks global \(U(1)_e\) symmetry and induces non-zero VEV \(\langle \xi _1 \rangle \), \(v_1 = \mu v_2 v_3/\mu _1^2\). Thus, taking \(\mu \) small enough, say \(\mu < v_2\), one can naturally get \(v_1\ll v_2\).

**5**. Gauge bosons \({{\mathcal {F}}}^\mu _a\) of \(SU(3)_e\), associated to the Gell-Mann matrices \(\lambda _a\), \(a=1,2,...\,8\), interact as \(g {{\mathcal {F}}}^\mu _a J_{a\mu }\) with the respective currents \(J_{a\mu } = \frac{1}{2} \overline{{{\mathbf {e}}}_R} \lambda _a \gamma _\mu {{\mathbf {e}}}_R\), where *g* is the \(SU(3)_e\) gauge coupling and \(\mathbf {e}_R = (e_1,e_2,e_3)_R^T\) denotes the triplet of the RH leptons. Clearly, these currents, in particular those related to non-diagonal generators \(\lambda _{1,2}\) etc., are generically FCNC. Nevertheless, as we shall see below, the processes mediated by flavor bosons exhibit no LFV in the initial eigenstates basis \(e_{R1},e_{R2},e_{R3}\) of the flavor diagonal generators \(\lambda _3\) and \(\lambda _8\).

*SU*(2) subalgebra of \(SU(3)_e\), and using the Fierz identities for these matrices one can rewrite (24) as

**6**. Let us question now how small the scale \(v_2\) can be without contradicting to existing experimental limits. The leading terms in (23) give rise to flavor conserving operators

*M*as

Experimental limits on the branching fractions for the LFV decays [37] vs. those predicted in our model. For the LFV decays of \(\tau \) lepton the branching ratio \(\mathrm{Br}(\tau \rightarrow \mu \nu _\tau {\bar{\nu }}_\mu ) = 0.174\) is taken into account. In last column we used Eq. (16) for the elements of mixing matrix \(V_R\) and set the scale \(v_2=2\) TeV from the compositeness limits. Both parameters \(\varepsilon \) and \(\tilde{\varepsilon }\) are normalised to \(1/20 \simeq m_\mu /m_\tau \); \(\varepsilon _{20} = 20\varepsilon \), \(\tilde{\varepsilon }_{20} = 20\tilde{\varepsilon }\). The latter choice is a clear overestimation and it would be more consistent to take \(\tilde{\varepsilon }\simeq m_e/m_\mu \simeq 1/200\). However, we see that all of the LFV limits are respected even with strongly overestimated \(\tilde{\varepsilon }\)

LFV mode | Experimental Br. | Predicted Br. | |
---|---|---|---|

\(\mu \rightarrow ee{\bar{e}}\) | \( < 1.0\times 10^{-12}\) | \(\frac{1}{8} \left( \frac{v_\mathrm{w}}{v_2}\right) ^4\left| V_{3e}^*V_{3\mu } + \varepsilon ^2 V_{2e}^*V_{2\mu } \right| ^2\) | \( \le 1.1 \times 10^{-13}\left( \big \vert \frac{g_{31}^*g_{32}}{g_{33}^2} \big \vert + \big \vert \frac{g^*_{21}}{g_{22}} \big \vert \right) ^2 \tilde{\varepsilon }_{20}^2 \varepsilon _{20}^4 \left( \frac{2\,\text {TeV}}{v_2}\right) ^4 \) |

\(\tau \rightarrow \mu e{\bar{e}}\) | \( < 1.8\times 10^{-8}\) | \(\frac{1}{4} \big (\frac{v_\mathrm{w}}{v_2}\big )^4 \left| V_{3\mu }^*V_{3\tau } \right| ^2 \, \mathrm{Br}(\tau \rightarrow \mu \nu _\tau {\bar{\nu }}_\mu )\) | \(= 6.2 \times 10^{-9 } \big \vert \frac{g_{32}}{g_{33}} \big \vert ^2 \varepsilon _{20}^2 \left( \frac{2\,\text {TeV}}{v_2}\right) ^4 \) |

\(\tau \rightarrow \mu \mu {\bar{\mu }}\) | \( < 2.1\times 10^{-8}\) | \( \frac{1}{8} \big (\frac{v_\mathrm{w}}{v_2}\big )^4 \big \vert V_{3\mu }^*V_{3\tau } \big \vert ^2 \, \mathrm{Br}(\tau \rightarrow \mu \nu _\tau {\bar{\nu }}_\mu )\) | \(= 3.1 \times 10^{-9 } \big \vert \frac{g_{32}}{g_{33}} \big \vert ^2 \varepsilon _{20}^2 \left( \frac{2\,\text {TeV}}{v_2}\right) ^4 \) |

\(\mu \rightarrow e \gamma \) | \(<4.2\times 10^{-13}\) | \(\frac{3\alpha }{2\pi } \big (\frac{v_\mathrm{w}}{v_2}\big )^4 \vert V_{3e}^*V_{3\mu } \vert ^2 \) | \(= 3.1\times 10^{-15} \big \vert \frac{g_{31}^*g_{32}}{g_{33}^2} \big \vert ^2 \tilde{\varepsilon }_{20}^2\varepsilon _{20}^4 \left( \frac{2\,\text {TeV}}{v_2}\right) ^4 \) |

\(\tau \rightarrow \mu \gamma \) | \(<4.4\times 10^{-8}\) | \(\frac{3\alpha }{2\pi } \big (\frac{v_\mathrm{w}}{v_2}\big )^4 \left| V_{3\mu }^*V_{3\tau } \right| ^2 \, \mathrm{Br}(\tau \rightarrow \mu \nu _\tau {\bar{\nu }}_\mu )\) | \( = 8.7 \times 10^{-11} \big \vert \frac{g_{32}}{g_{33}} \big \vert ^2 \varepsilon _{20}^2 \left( \frac{2\,\text {TeV}}{v_2}\right) ^4\) |

For \(\tau \) lepton decay modes as \(\tau \rightarrow \mu e \bar{e}\) and \(\tau \rightarrow 3\mu \) leading contributions arise from operator (23). From (27) we get the relevant constants as \(4G_{\tau \mu ee}\sqrt{2} =4G_{\tau \mu \mu \mu }/\sqrt{2} = V^*_{3\mu }V_{3\tau }/2v_2^{2}\). Hence, the widths of these decays are suppressed by a factor \(\sim \varepsilon ^2/v_2^4\), and are compatible with the experimental limits [37]. The summary of predicted branching ratios of relevant LFV processes compared with experimental limits is given in Table 1, with the parameters \(\varepsilon \), \(\tilde{\varepsilon }\) normalized to a benchmark value 1 / 20.

**7**. Let us remark that for promoting the chiral non-abelian factors in (2) as \(SU(3)_e\), etc. as gauge symmetries, one has to take care of anomaly cancellations. For more generality, let us consider

*SU*(3) horizontal groups respectively as

*ad hoc*fermions of the opposite chiralities which are singlets of the SM and are triplets under respective horizontal symmetry.

In particular, in our model “reduced” to leptons in which only \(SU(3)_e\) is considered as a gauge symmetry, triangle \(SU(3)_e^3\) anomaly is cancelled between the ordinary RH leptons \(e_{R\alpha }\sim (1,-2,3_e)\) and their LH mirror partners \(e'_{L\alpha }\sim (1,-2',3_e)\), where \(-2'\) denotes \(U(1)'\) hypercharge of mirror leptons.

*X*of additional gauge symmetry \(U(1)_X\) while ordinary leptons have no

*X*-charges. This additional charge is introduced in order to forbid the mixing of new fermions (44) with ordinary leptons (5) due to the mass term \(M \overline{{\mathcal E}_{L\alpha } } e_{R\alpha } \) and the Yukawa terms \(\overline{\ell _{Li}} {{\mathcal {E}}}_{Rj} \phi \) which would ruin the flavor structure induced by the operator (6). It is easy to check that by introducing extra fermions (44) and (45) the mixed triangle anomalies including \(U(1)\times SU(3)_e^2\), \(U(1)_X\times SU(3)_e^2\), \(U(1)\times U(1)_X^2\) and \(U(1)_X\times U(1)^2\) are all cancelled.

**8**. In this paper we discussed phenomenological implications of horizontal gauge symmetry \(SU(3)_e\) acting only in lepton sector, between three families of right-handed leptons. The lepton mass hierarchy \(m_\tau \gg m_\mu \gg m_e\) can be related to the hierarchy of the symmetry breaking scales \(v_3 \gg v_2 \gg v_1\). We have shown that the LFV effects induced by flavor changing gauge bosons are strongly suppressed due to custodial properties of \(SU(2)_e\subset SU(3)_e\) symmetry and respective scale can be as small as \(v_2=2\) TeV. This limit is in fact set from the compositeness limits on the flavor-conserving operators while the limits obtained from the LFV processes itself are weaker. Taken into account that the gauge coupling constant *g* of horizontal \(SU(3)_e\) can be less than 1, then masses of the \(SU(2)_e\) gauge bosons \(M_{1,2,3} \simeq (g/\sqrt{2}) v_2\) can be as small as 1 TeV or even smaller, and thus can be accessible at new electron-positron machines.

Analogously to \(SU(3)_e\), all *SU*(3) factors in (39) can be rendered anomaly free along the lines discussed in previous section, and thus they also can be gauge symmetries.^{5} The quark mass hierarchy can be related to hierarchies in breaking of \(SU(3)_Q\times SU(3)_d \times SU(3)_u\) gauge factors, i.e. with the ratios \(\varepsilon _d=v_2^d/v_3^d\) and \(\tilde{\varepsilon }_d=v_1^d/v_2^d\) between the VEVs of \(SU(3)_d\) triplet flavons, and the same for \(SU(3)_u\) and \(SU(3)_Q\). In this way, the hierarchy of down quark masses will go parametrically as \(1 : \varepsilon _d\varepsilon _Q : \varepsilon _d\tilde{\varepsilon }_d \varepsilon _Q \tilde{\varepsilon }_Q\) and of the up quarks as \(1 : \varepsilon _u\varepsilon _Q : \varepsilon _u\tilde{\varepsilon }_u \varepsilon _Q \tilde{\varepsilon }_Q\). The quark flavor violating processes mediated by gauge bosons of \(SU(3)_d\) and \(SU(3)_u\) will be suppressed due to custodial symmetry in the same way as the LFV processes mediated by \(SU(3)_e\) bosons. In particular, the operator \((\overline{s_R}\gamma ^\mu d_R)^2\) which induces \(K^0 -\bar{K}^0\) oscillation (analogously as leptonic operator (35) induces \(M-\overline{M}\) conversion) will be suppressed by a factor \(\sim \varepsilon _d^2 \tilde{\varepsilon }_d^2 \ll 1\). This can allow quark flavor changing gauge bosons to have masses in the range of few TeV, in fact limited only by the quark compositeness bounds. Interestingly, the flavor bosons of \(SU(3)_\ell \) and \(SU(3)_Q\) can give also anomalous contributions imitating the charged current \(\times \) current operators of the SM, and so they will have interference with the latter. E.g. \(SU(3)_\ell \) bosons induce operator \((\overline{e_L} \gamma _\rho \mu _L) (\overline{\nu _\mu } \gamma ^\rho \nu _e)\) which is nothing but the Fierz-transformed SM operator (33) responsible for the muon decay. Analogously, \(SU(3)_Q\) bosons should induce e.g. operator \((\overline{u_L} \gamma _\rho c_L) (\overline{s_L} \gamma ^\rho d_L)\) which also interferes with the charged current operators in the SM. Presence of such operators can affect the unitarity tests of the CKM mixing of quarks. Detailed analysis of these issues will be given elsewhere [30].

As far as the presence of mirror sector is concerned, mirror matters is a viable candidate for light dark matter dominantly consisting of mirror helium and hydrogen atoms [63, 64, 65, 66]. The flavor gauge bosons interacting with both ordinary and mirror fermions appear as messengers between two sectors and can give an interesting portal for mirror matter direct detection, complementary to the dark photon portal related to the photon-mirror photon kinetic mixing [67, 68]. However, they will also give rise to the mixing between the neutral ordinary and mirror mesons. Namely, the lighter flavor bosons induce mixings as \(\pi ^0-\pi ^{0\prime }\), \(K^0-K^{0\prime }\), etc. [48, 49], with implications for the invisible decay channels of neutral mesons (for a recent discussion, see also Ref. [69]). In the supersymmetric version the respective flavor gauginos, complemented by R-parity breaking, can induce the mixing between the ordinary and mirror neutral baryons. Interestingly, neutron-mirror neutron oscillation \(n-n'\) related to physics at the scale of few TeV [70, 71, 72] can be rather fast, in fact much faster than the neutron decay itself. Recent summary of experimental bounds on the \(n-n'\) oscillation time can be found in Ref. [73]). There are some anomalies in existing experiments on \(n-n'\) oscillation search [74] which can be tested in planned experiments on the neutron disappearance and regeneration [75, 76].

Oscillation phenomena between ordinary and mirror neutral particles are effective if they are degenerate in mass, i.e. mirror parity is unbroken and the weak scales \(\langle \phi \rangle = v_\mathrm{w}\) and \(\langle \phi ' \rangle = v'_\mathrm{w}\) are exactly equal in two sectors, \(v'_\mathrm{w} = v_\mathrm{w}\). However, the cancellation of horizontal anomalies between two sectors does not require that mirror parity is unbroken, and in fact one can consider models where it is spontaneously broken, e.g. \(v'_\mathrm{w} > v_\mathrm{w}\), with interesting implications for mirror dark matter properties and sterile mirror neutrinos [77] and axion physics [78]. In particular, in the context of the mirror twin Higgs mechanism for solving the little hierarchy problem, in supersymmetric [79, 80] or non-supersymmetric [81] versions, one expects \(v'_\mathrm{w}\) in the TeV range. Summarizing all, TeV scale still remains as a realistic scale for consistent supersymmetric models [82]. Supersymmetric extension of our model with TeV scale horizontal gauge symmetry can be a part of this picture as far as it is not problematic with respect to dangerous flavor-changing phenomena. It can be built in straightforward way, just by promoting all involved fields as superfields of corresponding chiralities and associating respectively the Yukawa terms (7) etc. with the superpotential terms. Moreover, this framework allows to realize MFV paradigm providing a theoretical motivation for the alignment of the fermion and sfermion masses along the lines discussed in Refs. [15, 16, 17, 18, 19]. In particular, in supersymmetrized version of our model the soft LFV breaking terms \(B_e^{i\alpha } \phi {\tilde{\ell }}_{Li}^\dagger \tilde{e}_{R\alpha }\), where tildes mark the slepton fields, in the leading approximation will be aligned to the Yukawa terms (10), \(B_e^{i\alpha } \propto Y_e^{i\alpha }\). Thus, for the slepton masses in the TeV range, the contributions of these soft trilinear terms to the LFV phenomena as \(\mu \rightarrow e\gamma \), \(\tau \rightarrow \mu \gamma \), etc. will be properly suppressed.

The following remark is in order. The viability of mirror sector is subject of strong cosmological restrictions. Namely, the Big Bang nucleosynthesis (BBN) constraints require that at the BBN epoch its temperature should be smaller than the temperature of the ordinary sector, \(T'/T < 0.6\) [63]. The constraints from the CMB and large scale structure are at least twice stronger, \(T'/T<0.2\div 0.3\) [65, 66]. On the other hand, the interactions (47) induce the process \(e \bar{e} \rightarrow e'\bar{e}'\) which in the early universe would bring two sectors into equilibrium. The freeze-out temperature \(T_d\) of this process can be easily estimated, just by rescaling by a factor \((2v_2/v_\mathrm{w})^{4/3}\) the neutrino decoupling temperature \(T_\nu \simeq 2\) MeV. Thus, for respecting the cosmological bounds, the reheating temperature of the universe should not exceed \(T_d \simeq (v_2/2\, \mathrm{TeV})^{4/3} \times 130\) MeV. (Analogous problem of low reheating temperature is typical also for the TeV scale gravity models with large extra dimensions, as discussed in Ref. [83], and for twin Higgs models [79, 80, 81].) Alternatively, one has to assume a possibility of additional entropy production in ordinary sector below the temperatures \(T_d\) which in turn implies the necessity of mirror symmetry breaking. In asymmetric mirror model, with \(v'_\mathrm{w} \gg v_\mathrm{w}\), \(T_d\) becomes significantly larger. In particular, for \(v'_\mathrm{w}\) larger than few PeV, as e.g. in heavy axion (axidragon) model [78], one can have \(T_d>1\) TeV or so.

Concluding, in this paper we demonstrated that in the TeV range there may exist a new physics related to the fermion flavor which can be revealed in future experiments at the energy and precision frontiers. In particular, the lepton-flavor changing gauge bosons can be as light as one TeV or even lighter, since the LFV processes are strongly suppressed by custodial symmetry. Nevertheless, some of these LFV processes, as e.g. \(\tau \rightarrow 3\mu \), can have widths close to present experimental limits and can be within the reach of future high precision experiments.

The work of Z.B. was partially supported by Shota Rustaveli National Science Foundation of Georgia, Grant DI-18-335/New Theoretical Models for Dark Matter Exploration. Preliminary version of this work was presented by B.B. at the European Physical Society Conference on High Energy Physics EPS-HEP 2017 [84].

## Footnotes

- 1.
- 2.
Alternatively, one could say that also \(SU(3)_\ell \) is a gauge symmetry but broken at some higher scales. More complete model with \(SU(3)_\ell \times SU(3)_e\) symmetry will be discussed elsewhere [30].

- 3.
Also the vector-like

*SU*(2) acting on both LH and RH fermion species has custodial properties [31]. However, it allows degenerate mass spectrum which makes problematic the naturalness of inter-family mass hierarchy. - 4.
For comparison, the naive lower limit on the scale of flavor changing bosons is over 100 TeV [32]. In the models [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] this scale was assumed to be close to the GUT scale, and in any case larger than a PeV, for avoiding the excessive FCNC. For an exception, see Ref. [33].

- 5.

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