# Exact one-loop results for \(l_i \rightarrow l_j\gamma \) in 3-3-1 models

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## Abstract

We investigate the decays \(l_i\rightarrow l_j \gamma \), with \(l_i=e,\mu ,\tau \) in a general class of 3-3-1 models with heavy exotic leptons with arbitrary electric charges. We present full and exact analytical results keeping external lepton masses. As a by product, we perform numerical comparisons between exact results and approximate ones where the external lepton masses are neglected. As expected, we found that branching fractions can reach the current experimental limits if mixings and mass differences of the exotic leptons are large enough. We also found unexpectedly that, depending on the parameter values, there can be huge destructive interference between the gauge and Higgs contributions when the gauge bosons connecting the Standard Model leptons to the exotic leptons are light enough. This mechanism should be taken into account when using experimental constraints on the branching fractions to exclude the parameter space of the model.

## 1 Introduction

We remark that 3-3-1 model is an active field of research and has a long history; see Ref. [2] and the references therein. In this work, we choose a general class of 3-3-1 models, which are similar to the models presented in Refs. [2, 3, 4] where new heavy leptons are introduced. However, there is an important difference: instead of fixing the electric charges of the new leptons to specific values being 0, \(+\,1\) or \(-\,1\), we let them be arbitrary. We will then study the dependence of the \(l_i \rightarrow l_j \gamma \) branching fractions on this arbitrary charge. That class of 3-3-1 models has been studied in many works; see e.g. [5, 6]. If we replace the new leptons with charge-conjugated partners of the SM leptons, we will have different 3-3-1 models with lepton-number violation; see e.g. Refs. [7, 8, 9, 10, 11]. The decays \(l_i \rightarrow l_j \gamma \) in these models have been discussed in Refs. [12, 13]; see also the recent review [14] and the references therein. We do not discuss these types of models in this work, but rather focus on the case with exotic leptons.

In the general class of 3-3-1 models here considered, there is one important parameter usually called \(\beta \), which together with *X*, the new charge corresponding to the group \(U(1)_X\), define the electric-charge operator. The electric charges of new particles therefore depend on \(\beta \). It has been known and widely accepted that \(\beta \) is one of the most important parameters to classify 3-3-1 models.

Recently, new efforts were made using 3-3-1 models to understand tensions between experimental measurements and the SM results in B physics; see e.g. [15, 16]. Motivated by this work, we want to use 3-3-1 models to understand the \(l_i \rightarrow l_j \gamma \) decays. Since the new leptons are assumed to be heavy, we expect large branching fractions. However, this is not totally obvious, because there are two contributions from gauge and Higgs sectors. Does a destructive interference effect occur?

The aim of this paper is manifold. First, we calculate the full and exact result for \(l_i \rightarrow l_j \gamma \) partial decay widths for a general class of 3-3-1 models with arbitrary \(\beta \). As a by product, we will perform numerical comparisons between the exact results (i.e. external lepton masses are kept) and approximate ones where external lepton masses are neglected. We note that approximate results have been almost exclusively used in the literature for the SM and many other models. We found this uncomfortable because the neutrino masses, which are much smaller than the lepton masses, are kept. We therefore want to know to what accuracy the approximate results valid, using the SM with arbitrary neutrino masses to answer this. As far as we know, this important point has never been addressed in the literature. We will also perform numerical studies for 3-3-1 model to see whether destructive interference effects occur and to see the dependence on \(\beta \), gauge boson and Higss masses. To the best of our knowledge, this is the first study of \(l_i \rightarrow l_j \gamma \) in 3-3-1 models with exotic leptons.

The paper is organized as follows. In the next section, we review the model and calculate the Feynman rules needed for \(l_i \rightarrow l_j \gamma \) decays. We then summarize the main calculation steps and present analytical results in Sect. 3. Numerical results are discussed in Sect. 4. In Sect. 4.1 we perform comparisons between the approximate and exact results for the neutrino contribution. In Sect. 4.2 we present results for the exotic-lepton contribution. Conclusions are in Sect. 5. Finally, we provide Appendices A and B to complete the results of Sect. 3.

## 2 3-3-1 model with arbitrary \(\beta \)

*SU*(3) generators \(T_3\), \(T_8\). Thus, the charge operator

*Q*depends on two parameters \(\beta \) and

*X*. With this information, we can write down the lepton representation as follows. Left-handed leptons are assigned to anti-triplets and right-handed leptons to singlets:

*g*and \(g_X\) are coupling constants corresponding to the two groups \(SU(3)_L\) and \(U(1)_X\), respectively. The matrix \(W^aT^a\), where \(T^a =\lambda _a/2\) corresponding to a triplet representation, can be written as

*B*is also the electric charge of the new leptons \(E_a\).

*A*,

*B*denote electric charges as defined in Eq. (7). These Higgses develop vacuum expectation values (VEVs) defined as

Vertices and couplings for \(l_i\rightarrow l_j\gamma \) decays in the 3-3-1 model with arbitrary \(\beta \) and new leptons. All momenta are defined as incoming. The photon field is denoted as \(A^\mu \), \(a,b = 1,2,3\) are family indices and \(\Gamma _{\lambda \mu \nu }(p_1,p_2,p_3)=(p_1-p_2)_\nu g_{ \lambda \mu } + (p_2-p_3)_\lambda g_{\mu \nu }+(p_3-p_1)_\mu g_{ \nu \lambda }\). Other notations are defined in the text

Vertex | Coupling | Vertex | Coupling |
---|---|---|---|

\(\overline{\nu }_{a}e_{b}H^+\) | \( \frac{ig}{\sqrt{2}m_W}U^{L*}_{ba} \left( \frac{m_{e_b}}{t_{v'v}} P_R+m_{\nu _a}t_{v'v} P_L\right) \) | \(\overline{e_a}\nu _bH^-\) | \( \frac{ig}{\sqrt{2}m_W}U^{L}_{ab}\left( \frac{m_{e_a}}{t_{v'v}} P_L+ m_{\nu _b}t_{v'v} P_R\right) \) |

\(\overline{E}_ae_b H^{+A}\) | \(\frac{-ig}{\sqrt{2}m_Y}V^{L*}_{ba}\left( \frac{m_{e_b}}{t_{v'u}} P_R+ m_{E_a} t_{v'u} P_L\right) \) | \(\overline{e}_aE_bH^{-A} \) | \(\frac{-ig}{\sqrt{2}m_Y}V^L_{ab}\left( \frac{m_{e_a}}{t_{v'u}} P_L+ m_{E_b} t_{v'u} P_R\right) \) |

\(\overline{\nu }_{a}e_{b}W^{+\mu }\) | \(\frac{ig}{\sqrt{2}}U^{L*}_{ba}\gamma _\mu P_L \) | \(\overline{e}_a\nu _b W^{-\mu }\) | \(\frac{ig}{\sqrt{2}}U^{L}_{ab}\gamma _\mu P_L \) |

\(\overline{\nu }_{a}e_{b}\phi _{W}^{+}\) | \(\frac{-ig}{\sqrt{2}m_W}U^{L*}_{ba}(m_{e_b}P_R - m_{\nu _a} P_L) \) | \(\overline{e}_a\nu _b \phi _{W}^{-}\) | \(\frac{-ig}{\sqrt{2}m_W}U^{L}_{ab}(m_{e_a} P_L - m_{\nu _b} P_R) \) |

\(\overline{E}_ae_b Y^{+A\mu }\) | \(\frac{-ig}{\sqrt{2}}V^{L*}_{ba}\gamma _\mu P_L\) | \(\overline{e}_aE_bY^{-A\mu } \) | \(\frac{-ig}{\sqrt{2}}V^L_{ab}\gamma _\mu P_L\) |

\(\overline{E}_ae_b \phi _{Y}^{+A}\) | \(\frac{-ig}{\sqrt{2} M_Y}V^{L*}_{ba}(m_{e_b}P_R - m_{E_a}P_L)\) | \(\overline{e}_aE_b\phi _{Y}^{-A} \) | \(\frac{-ig}{\sqrt{2}M_Y}V^L_{ab}(m_{e_a}P_L - m_{E_b}P_R)\) |

\(A^{\lambda }W^{+\mu }W^{-\nu }\) | \(-ie\Gamma _{\lambda \mu \nu }(p_A,p_{W^+},p_{W^-}) \) | \(A^{\lambda }Y^{+A\mu }Y^{-A\nu }\) | \( -ieA \Gamma _{\lambda \mu \nu }(p_A,p_{Y^{+A}},p_{Y^{-A}})\) |

\(A^{\lambda }W^{\pm \mu }\phi _W^{\mp }\) | \(iem_Wg_{\lambda \mu }\) | \(A^{\lambda }Y^{\pm A\mu }\phi _Y^{\mp A}\) | \(-ieAm_Y g_{\lambda \mu }\) |

\(A^{\mu }H^+H^-\) | \(ie(p_{H^+}-p_{H^-})_\mu \) | \(A^{\mu }H^{+A}H^{-A}\) | \(ieA(p_{H^{+A}}-p_{H^{-A}})_\mu \) |

\(A^{\mu }\phi _W^+ \phi _W^-\) | \(ie(p_{\phi _W^+}-p_{\phi _W^-})_\mu \) | \(A^{\mu }\phi _Y^{+A}\phi _Y^{-A}\) | \(ieA(p_{\phi _Y^{+A}}-p_{\phi _Y^{-A}})_\mu \) |

\(A^{\mu }\bar{l}_al_a\) | \(-ie\gamma _\mu \) | \(A^{\mu }\overline{E}_a E_a\) | \(ieB\gamma _\mu \) |

## 3 Analytical results

Equipped with the above Feynman rules, we can proceed to calculate the partial decay width of \(l_1 \rightarrow l_2 \gamma \) using standard techniques of one-loop calculation. We have done this in a careful way, with at least two independent calculations, and paid special attention to the relative sign between the gauge and Higgs contributions. This relative sign is very important because, as we will see in the numerical results, the interference term can be positive or negative.

In the literature, the calculation of \(l_1 \rightarrow l_2 \gamma \) is usually done by neglecting the external lepton masses. As stated in the introduction, we found this uneasy because the neutrino masses, which are much smaller than the lepton masses, are kept. We therefore want to check the validity of this approximation. To achieve this we have to keep the external lepton masses.

We have calculated the partial decay width of \(l_1 \rightarrow l_2 \gamma \) from scratch without approximation. In the following we summarize the key points and present exact analytical results. Results for the SM case are obtained as a special case and are discussed in Sect. 4.1.

The results can be further simplified if \(m_{E_a} \ll m_Y\) and \(m_{E_a} \ll m_{H^A}\) with \(a=1,2,3\) as presented in Appendix B.

Finally, we make an important remark on the dependence on coupling constants. From Eq. (30) we have \(D_{L,R}\propto eg^2\). Using Eq. (28) and noticing that \(G_F=g^2/(4\sqrt{2}m^2_W)\), we get \(\text {Br}(l_1 \rightarrow l_2 \gamma ) \propto e^2\), being independent of *g* or \(s_W\). Clearly, the coupling constant \(e=\sqrt{4\pi \alpha }\) should be calculated in the low-energy limit for the processes at hand. Therefore, we will use \(\alpha (0)\) as input parameter in our numerical analyses.

## 4 Numerical results

### 4.1 Neutrino contribution: approximate vs. exact

The approximate results calculated by neglecting the external lepton masses have been exclusively used in the literature. However, the justification is not totally obvious to us because the neutrino masses, which are much smaller than the lepton masses, are kept. We therefore here present compact formulas for the exact results (i.e. \(m_1\) and \(m_2\) kept) and perform a numerical comparison with the approximate ones.

*W*contribution and is given by \(D_{L,R}^{\nu W}\). Using the formulas in Appendix A we write the result in terms of scalar one-loop integrals \(A_0\), \(B_0\) and \(C_0\), which are also calculated in Appendix A. We obtain

Exact (i.e. \(m_1\) and \(m_2\) are kept in \(D_{L,R}\)) and approximate (i.e. \(m_1=m_2=0\)) branching fractions of \(l_1 \rightarrow l_2 \gamma \) at various hypothetical values of \(m_{\nu _1}\). Other two neutrino masses are fixed at tiny values calculated using \(m_{\nu _1}=0\) and the current known values of \(\Delta m^2_{21}\) and \(\Delta m^2_{32}\) specified in the text, namely\(m_{\nu _2}\approx 8.678\times 10^{-3}{\,\text {eV}}\) and\(m_{\nu _3}\approx 5.025\times 10^{-2}{\,\text {eV}}\). The neutrino mixing matrix is assumed being real and is calculated from three known mixing angles \(\theta _{12}\), \(\theta _{13}\) and \(\theta _{23}\) as given in the text. For the sake of comparison we set \(\text {Br}(l_1\rightarrow l_2\bar{\nu }_2\nu _1)=1\) for all three channels. The difference between exact and approximate results is defined as: \(\text {diff}=(\text {appr}-\text {exact})/\text {exact}\)

\(m_{\nu _1}\) [GeV] | Method | \(\mu \rightarrow e\gamma \) | \(\tau \rightarrow e\gamma \) | \(\tau \rightarrow \mu \gamma \) |
---|---|---|---|---|

0 | exact Br. | \(4.0969\times 10^{-55}\) | \(2.6800\times 10^{-55}\) | \(76.705\times 10^{-55}\) |

appr. Br. | \(4.0968\times 10^{-55}\) | \(2.6780\times 10^{-55}\) | \(76.377\times 10^{-55}\) | |

diff | \(-\,2.6\times 10^{-5}\) | \(-\,7.6\times 10^{-4}\) | \(-\,4.3\times 10^{-3}\) | |

\(10^{-13}\) | exact Br. | \(4.0968\times 10^{-55}\) | \(2.6801\times 10^{-55}\) | \(76.705\times 10^{-55}\) |

appr. Br. | \(4.0967\times 10^{-55}\) | \(2.6780\times 10^{-55}\) | \(76.377\times 10^{-55}\) | |

diff | \(-\,2.6\times 10^{-5}\) | \(-\,7.6\times 10^{-4}\) | \(-\,4.3\times 10^{-3}\) | |

\(10^{-1}\) | exact Br. | \(7.9502\times 10^{-17}\) | \(3.4303\times 10^{-17}\) | \(1.1400\times 10^{-17}\) |

appr. Br. | \(7.9500\times 10^{-17}\) | \(3.4277\times 10^{-17}\) | \(1.1351\times 10^{-17}\) | |

diff | \(-\,2.6\times 10^{-5}\) | \(-\,7.6\times 10^{-4}\) | \(-\,4.3\times 10^{-3}\) | |

\(10^{2}\) | exact Br. | \(1.3590\times 10^{-5}\) | \(0.58619\times 10^{-5}\) | \(0.19481\times 10^{-5}\) |

appr. Br. | \(1.3590\times 10^{-5}\) | \(0.58593\times 10^{-5}\) | \(0.19404\times 10^{-5}\) | |

diff | \(-\,2.5\times 10^{-5}\) | \(-\,4.4\times 10^{-4}\) | \(-\,4.0\times 10^{-3}\) | |

\(10^{16}\) | exact Br. | \(1.3278\times 10^{-4}\) | \(0.57261\times 10^{-4}\) | \(0.19030\times 10^{-4}\) |

appr. Br. | \(1.3278\times 10^{-4}\) | \(0.57249\times 10^{-4}\) | \(0.18959\times 10^{-4}\) | |

diff | \(-\,2.4\times 10^{-5}\) | \(-\,2.2\times 10^{-4}\) | \(-\,3.7\times 10^{-3}\) |

We now take into account the charged Higgs contribution. There are two additional parameters \(t_{v'v}\) and \(m_{H^\pm }\) (see the \(D_{L,R}^{\nu H^+}\) terms in Eq. (30)). We have calculated the difference between the exact and approximate results for four cases of \(t_{v'v}=1/50\) or 50 (we choose these exotic values so that the effect of \(t_{v'v}\) is large) and \(m_{H^\pm }=70\) or\(700{\,\text {GeV}}\). The result is very similar to the SM case: the difference is below permil level for \(\mu \rightarrow e \gamma \) and \(\tau \rightarrow e \gamma \) and is at the permil level for \(\tau \rightarrow \mu \gamma \). For the absolute value of \(\text {Br}(\mu \rightarrow e \gamma )\) the result is \(5\times 10^{-49}\) for \(t_{v'v}=50\) and\(m_{H^\pm }=70{\,\text {GeV}}\) and getting smaller for lower values of \(t_{v'v}\) and/or higher values of \(m_{H^\pm }\).

We make a technical remark here. Due to the huge hierarchy among the neutrino, charged leptons and *W* boson masses, the numerical calculation of the exact result is non-trivial because of numerical cancellation. To obtain the \(\mu \rightarrow e \gamma \) results in Table 2 we have used Mathematica 9 with at least 62 precision digits for\(m_{\nu _1}=10^{-13}{\,\text {GeV}}\) and about 180 precision digits for\(m_{\nu _1}=10^{16}{\,\text {GeV}}\).

### 4.2 Exotic-lepton contribution

In this numerical study we investigate the exotic-lepton contribution, to see how large the branching fractions can reach, what can be the dominant effects and dependence on the parameter \(\beta \), \(m_Y\) and \(m_{H^A}\). We will also show the gauge–Higgs interference effects.

In the previous section we have shown that the neutrino contribution is well below the current experimental limit. We will therefore neglect the neutrino contribution including interference effects with exotic leptons in the following. The external lepton masses will be neglected as justified in Sect. 4.1.

Summary of lower bounds on \(m_{Z'}\) for 3-3-1 models with exotic leptons obtained using ATLAS or CMS data at 7 and\(8{\,\text {TeV}}\). Exotic fermion contributions to the \(Z'\) total width are neglected, except for Ref. [24] where\(m_F = 1{\,\text {TeV}}\) is assumed for all exotic fermions. In the last column we have derived the bound on \(m_{Y^A}\) using the relation \(m_{Y^A} \approx m_{Z'}\sqrt{3[1-(1+\beta ^2)s^2_W]}/(2c_W)\) obtained using \(v,v'\ll u\) approximation [6] and \(s^2_W = 0.231\)

\(\beta \) | Data | Channel | Bound on \(m_{Z'}\) | Ref. | Bound on \(m_{Y^A}\) |
---|---|---|---|---|---|

\(-\,2/\sqrt{3}\) | CMS8 with\(20.6{\,\text {fb}}^{-1}\) | di-muon | \(\gtrapprox 3.2{\,\text {TeV}}\) | [25] | \(\gtrapprox 2.1{\,\text {TeV}}\) |

\(-\,1/\sqrt{3}\) | CMS7&8 | di-lepton | \(\gtrapprox 2.5{\,\text {TeV}}\) | [24] | \(\gtrapprox 2.1{\,\text {TeV}}\) |

\(-\,1/\sqrt{3}\) | ATLAS8 | di-lepton | \(\gtrapprox 2.89{\,\text {TeV}}\) | [26] | \(\gtrapprox 2.4{\,\text {TeV}}\) |

Branching fractions of \(l_1 \rightarrow l_2 \gamma \) at various values of \(\beta \), \(m_{Y^A}\). Other parameters are fixed as given in the text

\(\beta \) | \(\mu \rightarrow e\gamma \) | \(\tau \rightarrow e\gamma \) | \(\tau \rightarrow \mu \gamma \) | \(m_{Y^A}\)[TeV] | \(\mu \rightarrow e\gamma \) | \(\tau \rightarrow e\gamma \) | \(\tau \rightarrow \mu \gamma \) |
---|---|---|---|---|---|---|---|

0 | \(2.51\times 10^{-13}\) | \(1.94\times 10^{-14}\) | \(1.57\times 10^{-16}\) | 0.5 | \(1.79\times 10^{-9}\) | \(1.44\times 10^{-10}\) | \(1.69\times 10^{-12}\) |

\(1/\sqrt{3}\) | \(1.49\times 10^{-12}\) | \(1.33\times 10^{-13}\) | \(3.10\times 10^{-15}\) | 1 | \(6.62\times 10^{-11}\) | \(5.66\times 10^{-12}\) | \(1.03\times 10^{-13}\) |

\(-\,1/\sqrt{3}\) | \(4.95\times 10^{-12}\) | \(4.14\times 10^{-13}\) | \(6.52\times 10^{-15}\) | 1.5 | \(7.38\times 10^{-12}\) | \(6.49\times 10^{-13}\) | \(1.41\times 10^{-14}\) |

\(\sqrt{3}\) | \(2.18\times 10^{-11}\) | \(1.88\times 10^{-12}\) | \(3.69\times 10^{-14}\) | 2 | \(1.49\times 10^{-12}\) | \(1.33\times 10^{-13}\) | \(3.10\times 10^{-15}\) |

\(-\,\sqrt{3}\) | \(3.21\times 10^{-11}\) | \(2.73\times 10^{-12}\) | \(4.72\times 10^{-14}\) | 3 | \(1.64\times 10^{-13}\) | \(1.47\times 10^{-14}\) | \(3.61\times 10^{-16}\) |

A few remarks on the above default input-parameter choice are appropriate here. Concerning gauge bosons, the best ATLAS/CMS limits for 3-3-1 models with exotic leptons are summarized in Table 3. We note that, in almost all cases, the contributions from exotic leptons to the \(Z'\) total width are neglected, except for the case of Ref. [24] where\(m_F = 1{\,\text {TeV}}\) is assumed for all exotic fermions. When those contributions are properly taken into account, the bound on \(m_{Z'}\) will get weaker, because the branching fractions of \(Z'\rightarrow l^+ l^-\) with \(l=e,\mu \) will decrease. Therefore, the default choice in Eq. (35) may be acceptable. However, one should keep in mind that, strictly speaking, the ATLAS/CMS bound on \(m_{Z'}\) is unknown for our present numerical analysis, because it depends on the masses and electric charges of the exotic fermions (i.e. leptons and quarks) which have not been properly taken into account. We will therefore relax the constraint on \(m_{Y^A}\), varying it from 0.5 to\(3{\,\text {TeV}}\) for some plots. In this context, it is noted that, using LEP II data, the authors of Ref. [15] obtained\(m_{Z'}\gtrapprox 1{\,\text {TeV}}\) for \(\beta = \pm 1/\sqrt{3}\), \(2/\sqrt{3}\), leading to\(m_{Y^A}\gtrapprox 0.7{\,\text {TeV}}\). Phenomenological constraints on the masses of exotic Higgs bosons \(H^A\) and of the exotic leptons and their mixing angles are much more difficult to obtain and do not exist to the best of our knowledge.

With those difficulties in mind, we decided to choose the above default input parameters in a fairly random way following a few general principles: (i) \(u \gg v,v'\) (i.e. the \(SU(3)_L\) breaking scale is much larger than that of \(SU(2)_L\)), (ii) the exotic leptons are heavy and satisfy the unitary bound, (iii) and their mixing angles are large. We note that the choice of heavy masses are in agreement with the negative results of collider searches for physics beyond the SM. Large mixing angles are motivated by the PMNS matrix of the neutrino sector and the fact that we want to have large branching fractions close to the experimental limits.

In Table 4 we present the \(l_1 \rightarrow l_2 \gamma \) branching fractions for various values of \(\beta \), \(m_{Y^A}\). We observe the following features: the branching fractions are smallest at \(\beta = 0\) and increase with \(|\beta |\). The results exhibit a clear asymmetry under the transformation of \(\beta \rightarrow -\beta \), or in other words, they depend on the sign of \(\beta \). The right table shows a strong dependence on \(m_{Y^A}\). As expected, the branching fractions are large when \(m_{Y^A}\) is small. With the choice of exotic-lepton masses and mixing angles as given in Eq. (38), the branching fraction is largest for \(\mu \rightarrow e \gamma \) and smallest for \(\tau \rightarrow \mu \gamma \). With this setup, we see that the branching fractions of \(\tau \rightarrow e \gamma \) and \(\tau \rightarrow \mu \gamma \) all satisfy the current experimental constraints for all values of \(\beta \) and \(m_{Y^A}\) in Table 4. For the decay of \(\mu \rightarrow e \gamma \), only the cases of \(\beta = 0\) or\(m_{Y^A}=3{\,\text {TeV}}\) are below the experimental limit of \(4.2\times 10^{-13}\).

We now focus on the decay \(\mu \rightarrow e \gamma \) and discuss two density plots to see the dependence on \(\beta \), \(m_{Y}\) and \(m_{H^A}\). In Fig. 3 we show the density plot of \(\text {Br}(\mu \rightarrow e \gamma )\) as a function of \(\beta \) and \(m_Y\) (left) and of \(\beta \) and \(m_{H^A}\) (right). We observe from the left plot, consistently with Table 4, the branching fraction are smallest when \(\beta \) is around zero or when \(m_{Y}\) is large. From the right plot, we see a similar dependence on \(\beta \), but the dependence on \(m_{H^A}\) is much weaker than on \(m_{Y}\). From those two plots, we conclude that large branching fraction occurs at large \(|\beta |\), small \(m_Y\) and small \(m_{H^A}\).

In a series of six plots in Fig. 4 we would like to show again the dependence on \(\beta \), \(m_Y\) and \(m_{H^A}\), but with two-dimensional plots this time and for all three decays. We see clearly that the case of \(\beta = 0\) is special and different from the other cases of \(\beta = 1/\sqrt{3}, \sqrt{3}\). For \(\beta = 0\), the branching fractions of all three decays have a deep minimum when \(m_Y\) or \(m_{H^A}\) reach special values. The minimum positions are at low energies and are different for different decays, suggesting that they depend on the mixing angles. Together with Fig. 3 we conclude that a deep minimum occurs when \(|\beta |\) is small enough. This has a very important phenomenological consequence: for small values of \(|\beta |\), branching fraction can be very small even at small values of \(m_Y\) and \(m_{H^A}\). This means that, contrary to naive expectation, there can be small values of \(m_Y\) and \(m_{H^A}\) escaping the exclusion limit obtained using the experimental constraints on \(\text {Br}(l_i \rightarrow l_j \gamma )\), if \(|\beta |\) is small enough.

To understand the minimum occurring when \(\beta \) is around zero we have to study the dependence of the branching fraction on \(\beta \). This is shown in Fig. 5 (right). On the left plot we display again the dependence on \(m_Y\) for the special case of \(\beta = 0\). This time, differently from Fig. 4 (top-left), we focus on the low-energy region of\(m_Y \in [0.5,1]{\,\text {TeV}}\) and gauge, Higgs and interference contributions are also plotted. The left plot shows that the interference is strongly destructive and there is a spectacular cancellation between the sum of gauge and Higgs contributions and the interference term, leaving a very small branching ratio. The \(\beta \) dependence plot also shows a negative interference effect when \(\beta \in [0.035:0.26]\) for our default choice of input parameters. The insert in Fig. 5 (right) shows that the interference line crosses the zero branching fraction line when the gauge contribution (blue line) vanishes and when the Higgs term (brown line) vanishes. One should note that the gauge or Higgs contributions are non-negative. Overall, Fig. 5 shows that destructive interference effect tends to occur when \(|\beta |\) and \(m_Y\) are small.

## 5 Conclusions

In this paper, we have provided full and exact analytical results for the \(l_i \rightarrow l_j \gamma \) partial decay widths for a general class of 3-3-1 models with exotic leptons and with arbitrary \(\beta \). As a by product, we performed numerical comparisons between exact results (i.e. external lepton masses are kept) and approximate ones where \(m_i = m_j = 0\). We conclude that, for either extremely light neutrinos or very heavy leptons, the difference between exact and approximate results is less than permil level for \(\mu \rightarrow e \gamma \) and \(\tau \rightarrow e \gamma \) and is at the permil level for \(\tau \rightarrow \mu \gamma \). Therefore, unsurprisingly, approximation results widely used in the literature are excellently justified.

Concerning the exotic-lepton contribution, we found huge destructive interference between the gauge and Higgs contributions. This can happen when \(|\beta |\) and \(m_Y\) are small enough. This has an interesting consequence: the branching fractions can be small even for small \(m_Y\). Therefore, this destructive interference mechanism must be taken into account when using experimental constraints on \(\text {Br}(l_i \rightarrow l_j \gamma )\) to exclude parameter space. This in particular means that if one takes into account only the gauge contribution then the results can be completely off. It is likely that this destructive interference mechanism also occurs in \(b \rightarrow s\gamma \) and other similar processes.

Besides, we found that the gauge and Higgs contributions can be of similar size. Dependences on \(\beta \), \(m_Y\) and \(m_{H^A}\) have been shown. We observe that the branching fractions are very sensitive to \(\beta \) and \(m_Y\). They also depend on \(m_{H^A}\), but to a lesser extent. The dependence on \(\beta \) is interesting: the branching fractions are largest for \(|\beta |=\sqrt{3}\) and smallest around zero.

## Notes

### Acknowledgements

LTH would like to thank Theoretical Physics Group at IFIRSE for hospitality and supports during his stay at IFIRSE where part of this work was done. LDN would like to thank Jean Tran Thanh Van, Le Kim Ngoc and their team at ICISE for continuous supports and creating a beautiful environment for research. The work of LDN has been partly supported by the German Ministry of Education and Research (BMBF) under contract no. 05H15KHCAA. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number 103.01-2017.29.

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