Charged Lepton Flavor Violation in a class of Radiative Neutrino Mass Generation Models

We investigate charged lepton flavor violating processes $\mu\rightarrow e \gamma$, $\mu\rightarrow e e \overline{e}$ and $\mu-e$ conversion in nuclei for a class of three-loop radiative neutrino mass generation models with electroweak multiplets of increasing order. We find that, because of certain cancellations among various one-loop diagrams which give the dipole and non-dipole contributions in effective $\mu e \gamma$ vertex and Z-penguin contribution in effective $\mu e Z$ vertex, the flavor violating processes $\mu\rightarrow e\gamma$ and $\mu-e$ conversion in nuclei become highly suppressed compared to $\mu\rightarrow e e \overline{e}$ process. Therefore, the observation of such pattern in LFV processes may reveal the radiative mechanism behind neutrino mass generation.


I. INTRODUCTION
Although we have observed lepton flavor violation (LFV) in the neutral fermion sector of the Standard Model (SM) in neutrino oscillation, the charged LFV in the SM has turned out to be highly suppressed. For example, by allowing massive neutrinos, m ν ∼ 1 eV and leptonic mixing matrix, Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix, U PMNS in the SM, the branching ratio of charged lepton violating process, µ → eγ, turns out to be about 10 −47 [1][2][3][4][5] which is beyond any experimental reach in the foreseeable future. But many physics beyond the standard model (BSM) scenario, specially new physics related to the generation and smallness of the neutrino mass, can lead to unsuppressed charged LFV processes [2,6,7] 1 which are within the reach of currently operating and future experiments. For theoretical and experimental status of charged LFV, please see [9][10][11][12][13][14].
A well motivated model of neutrino mass generation which addresses the origin of the neutrino mass and the particle nature of the Dark matter (DM) in our universe, is Krauss-Nasri-Trodden (KNT) model [15] where DM particle radiatively generate the mass of the neutrino at three loops and additional BSM particles having mass at the O(TeV) range, can be accessible to the LHC or future hadron colliders 2 . In [15], the additional BSM fields are two charged singlets S + 1 and S + 2 and three fermion singlets N R1,2,3 which are right handed (RH) neutrinos. A Z 2 symmetry with action {S + 2 , N Ri } → {−S + 2 , −N Ri } is also imposed to prevent the tree-level Dirac mass of the neutrino after electroweak symmetry breaking, and ensures stability of the lightest RH neutrino, N R1 , thereby giving a DM candidate.
Consequently, the three-loop topology of radiative neutrino mass diagram remains invariant [17], if one replaces S + 2 with larger scalar multiplet, Φ, which has integer isospin, j φ and hypercharge 3 , Y φ = 1 under SU (2) L × U (1) Y and N Ri are replaced with fermionic multiplet F i , i = 1, 2, 3, with integer isospin, j F and hypercharge, Y F = 0. In this scenario, the DM candidate is the lightest neutral component of F 1 , i.e. F 0 1 . Therefore the immediate generalization of KNT model is [18] where the particle content is taken as, Φ with (j φ , Y φ ) = (1, 1) and F i with (j F , Y F ) = (1, 0). Here, Z 2 symmetry is still needed to enforce the Dirac mass term of neutrinos to be zero at tree-level. * Electronic address: talal@du.ac.bd † Electronic address: snasri@uaeu.ac.ae 1 For general condition of tree-level and one-loop lepton flavor violating processes, please see [8]. 2 For a recent review on radiative generation of neutrino mass, please see [16]. 3 Here, the electric charge is In addition, no yukawa terms with SM fermion that give rise to the Dirac neutrino mass, are allowed in the Lagrangian if the KNT particle content is extended with, Φ that has (j φ , Y φ ) = (2, 1) and F i with (j F , Y F ) = (2, 0) to generate the neutrino mass at three-loop level [19]. Therefore, there is no need to use Z 2 symmetry for that purpose. But the viable dark matter candidate in the model, which is F 0 1 majorana fermion, has one-loop decay process which depends on λS − 1 Φ † .Φ.Φ term in the scalar potential. From the bound on dark matter mean life-time [20], which is of the order 10 25 − 10 27 sec, the λ coupling has to be λ ∼ 10 −26 − 10 −27 for TeV mass-ranged DM. Moreover, the neutrino sector of the model doesn't depend on this coupling in any way. Therefore in the limit, λ → 0, the softly broken accidental Z 2 symmetry becomes exact and ensures the stability of the DM.
Consequently, one can go to the next higher scalar and fermion representations in this class of generalized KNT models. In the case of Φ with (j φ , Y φ ) = (3, 1) and F i with (j F , Y F ) = (3, 0), the field content of the model not only prevents the appearance of Dirac mass term for neutrino but also the λ term in the scalar potential which would have prevented DM to be absolutely stable [21]. The direct product of two SU (2) scalar representations, Φ ⊗ Φ gives j φ ⊗ j φ = ⊕ J J ⊃ j φ where j φ has the same isospin value as j φ and therefore forms an invariant in the term λS − 1 Φ † ⊗ Φ ⊗ Φ but it is either symmetric or antisymmetric representation depending on the even-integer or odd-integer isospin value j φ respectively. As the antisymmetrized Φ ⊗ Φ representations are identically zero for odd-integer isospin, the λ term doesn't appear in the scalar potential at renormalizable level and the DM is stable.
In this article we present the generalized KNT model with larger electroweak multiplets in section II. In section III, we describe the relevant formulas of charged LFV processes µ → eγ, µ → eee and µ − e conversion rate in nuclei in generalized KNT model. Section IV contains the result of charged LFV processes in this model. We conclude in section V. Appendix A contains the loop functions used in calculations of charged LFV processes.

A. The field content
The three-loop radiative neutrino mass generation model contains a charged singlet S + 1 ∼ (0, 0, 1), a complex scalar multiplet, Φ ∼ (0, j φ , 1) and three real fermion multiplets, In this comparative study, we focus on four set of models in this class which we have referred as,

B. Mass splittings in the Multiplets
At the tree-level, the components of fermion multiplet, F i are mass degenerate. Moreover we work in the generation basis where M Fij = diag(M F1 , M F2 , M F3 ). We have also considered the non-degenerate mass for the three fermion multiplets, M F1 < M F2 < M F3 .
Consequently, after electroweak symmetry breaking (EWSB), the radiative corrections, for example, loops involving SM gauge bosons, lift the mass degeneracy in the component fields of the fermion multiplets. In the limit, M F M W , the mass splitting between the components of charge Q and Q is, [29].
On the other hand, the component fields of the scalar multiplet, after EWSB, can have splittings at the tree level due to the following term in V 1 (H, Φ), The maximum splitting among the masses of the component fields in the electroweak multiplet is bounded by the constraints on the Electroweak Precision observables (EWPO) [30,31]. Here we consider the constraint on the T parameter as it is the most sensitive EWPO on mass splitting in scalar multiplet or in other words, isospin breaking in the multiplet. Therefore, larger value of the coupling, λ Hφ2 leads to the larger splitting in the scalar component fields in the multiplet. On the other hand, if M 0 = 10 TeV and λ Hφ2 = 2π, the splitting between any two components of the scalar multiplet, allowed by the EWPO constraints, is very small as ∆m 2 ij /M 2 0 ∼ 10 −3 . Again, with M Fi ∼ 10 TeV, the radiative mass splittings between two components of fermionic multiplet leads to ∆m 2 Fij /M 2 0 ∼ 10 −4 . Therefore, for scalar and fermion multiplets' mass in the TeV range, the mass splittings are numerically negligible therefore we consider this scenario as 'near degenerate' case and make proper approximations in our subsequent analysis.

C. Three-loop radiative neutrino mass
The neutrino mass is generated radiatively at three loops. In the near degenerate case, we neglect the small mass splittings and have [15,18,19,21], where, c = 1, c = 3, c = 5 and c = 7 are for singlet, triplet, 5-plet and 7-plet cases respectively. Eq.(6) can be written in compact form, Here, M l is the diagonal charged lepton mass matrix and Λ = diag(Λ 1 , and the function I(r, β) is where The behavior of function F (α, β) with α and β is shown in Fig. 2.
The neutrino mass matrix, M ν of Eq.(6), can be diagonalized as where,m ν = diag(m ν1 , m ν2 , m ν3 ) and it contains 7 independent parameters which are two masses m ν2,3 that can be determined assuming either normal or inverted hierarchy by using experimentally measured [32] two mass squared differences ∆m 2 atm and ∆m 2 solar , three mixing angles θ 12 , θ 23 and θ 13 and still to be determined, one Dirac phase δ CP and one Majorana phase α M in U PMNS matrix. Here due to det F = 0, the lowest neutrino mass is m ν1 = 0 and it also implies one Majorana phase of U PMNS to be zero. On the other hand, the matrices F contains six and G contains 18 degrees of freedom. As there is no one-to-one correspondence between low energy neutrino parameters in M ν and the parameters of F , G and Λ, we numerically determine the set {f αβ , g iα , M F1,2,3 , M φ , m S , λ S } which satisfy the following relation, We have used this relation because there are no unique F and G which satisfy the low energy neutrino constraints, U PMNS . Therefore one can always find another set of F and G through orthogonal transformation, F → V F V T and bi-unitary transformation, G → W GY † where, V , W and Y are unitary matrices, to satisfy the low energy constraints.

III. CHARGED LEPTON FLAVOR VIOLATING PROCESSES
As the charged LFV processes, µ → eγ, µ → eee and µ − e conversion in Au and Ti nuclei have the most stringent experimental constraints, we focus our study on these three LFV processes in generalized KNT model with singlet, triplet, 5-plet and 7-plet respectively.
The branching ratio for µ → eγ, normalized by Br(µ → eν e ν µ ), is where where where m φ and q φ (q F ) are the corresponding mass and the electric charge respectively of the scalar (fermion) and φ (F i ) runs over all the charged components of the scalar (fermion) multiplet Φ (F i ). Note that A  In the generalized KNT model, the 3-body lepton flavor violating decay mode µ → eee receives the contributions from γ-penguin diagrams, Z-penguin diagrams, and Box diagrams. In this model, Higgs penguin diagram doesn't contribute to this process. Therefore, the branching ratio of µ → eee is given [33][34][35][36] as where A D and A N D are the dipole and non-dipole contributions from the photonic penguin diagrams respectively. F L Z and F R Z are given as Here, F Z is the Z-penguin contribution and g l L and g l R are the Z-boson coupling to the left-handed (LH) and right-handed (RH) charged leptons respectively. Also B represents the contribution from the box diagrams.

γ-penguin contribution
The γ penguin diagram can be obtained by attaching e − e fermion line to γ line in Fig. 3

(I), (II) and (III).
The dipole contribution of γ-penguin diagrams are same as in section III A. So we consider here the non-dipole contribution which is, Here, The non-dipole contributions A (1) N D and A

Z-Penguin Contribution
The Z-penguin diagram can be obtained from where F (1) Z is the contribution associated with Fig. 3 (I) and (II) with Z line. On the other hand, F Z is the contribution associated with Fig. 3 (III) (with Z line) and (IV). They are given as, where the sum over pairs (φ, F i ) implies the pairs of component fields from fermion and scalar multiplet entering into the one-loop process. gZF i F i , g Zφ , g Zνν and g ZS1 are the Z coupling of charged fermion components of F i , scalar components of Φ, tau neutrino and charged scalar S 1 respectively. Moreover, g l L and g l R are the Z coupling of the left handed and right handed charged leptons respectively.
In the near degenerate limit, for the triplet, the contributions from following pairs in Eq. (25) are, For the 5-plet, the contribution in Eq.(25) from the following pairs are whereas, for 7-plet, the contribution from the following pairs are, For singlet, there is only one contribution in F (1) Z which is coming from (φ − , F 0 i ) pair. Therefore in all cases, the only non-zero contribution in F And Z-couplings are g Zνν = g 2 cos θ W , g ZS1 = − g sin 2 θ W cos θ W and g l L = g cos θ W − 1 2 + sin 2 θ W . Therefore the total sum turns out to zero.

Box Contribution
The box contribution can be arranged into three parts, Here, B (1) is the contribution associated with neutral fermion in the loop and involves combination of one-loop box topologies Fig. 4 (left and center). It is given as, . On the contrary, the charged fermions running in the loop give contribution to B (2) and are associated to Fig. 4 (left). It is given as, The sum index F ranges over the charged components of fermion multiplets and φ 1,2 indices range over the corresponding scalar components set by the G yukawa term in Eq. (2). For the singlet, the only contribution is B (1) that comes box diagram that involves φ + and neutral fermion F 0 i . For larger scalar and fermion multiplets, apart from B (1) contribution also B (2) has contributions from charged fermions as follows. For triplet, box contribution B (2) involves φ 1,2 ∈ {φ ++ , φ 0 } for F = F + i . For 5-plet, φ 1,2 ∈ {φ +++ , φ − } for F = F ++ i and φ 1,2 ∈ {φ ++ , φ 0 } for F = F + i . On the other hand, for 7-plet, There is also box contribution coming from the charged scalar S + 1 (Fig. 4 (right)) which is given as C. µ to e Conversion Rate The conversion rate, normalized by the muon capture rate is [37][38][39] CR(µ − e, Nucleus) = Here, Z and N are the number of protons and neutrons in the nucleus, Z ef f is the effective atomic charge, F p is the nuclear matrix element and Γ capt represents the total muon capture rate. p e and E e are the momentum and energy of the electron which is taken as ∼ m µ . g (0) XK and g (1) XK (X = L, R and K = V, S) in the above expression are given as g XK(q) are the couplings in the effective Lagrangian describing µ − e conversion, G (q,p) , G (q,n) are the numerical factors that arise when quark matrix elements are replaced by the nucleon matrix elements, For the generalized KNT model, the µ − e conversion rate receives the γ and Z penguin contributions where the quark line is attached to photon and Z-boson lines in the respective penguin diagrams. It also doesn't receive any box contribution because there is no coupling between Φ and quarks. The relevant effective coupling for the conversion in this model is The relevant couplings are Here Q q is the electric charge of the quarks and Z boson couplings to the quarks are Also the relevant numerical factors for nucleon matrix elements are In the near degenerate limit, there will be cancellation in A D , A N D and F Z contributions for triplet, 5-plet and 7-plet cases as pointed out in sections III A, III B 1 and III B 2. Therefore, µ − e conversion rate will be also suppressed compared to the µ → eee process in the KNT model.

A. Viable Parameter Space
The parameter space of generalized KNT model for singlet, triplet, 5-plet and 7-plet in the near degenerate limit is taken as {f αβ , g iα , M F1,2,3 , M φ , m S , λ S } which enter into the neutrino mass generation in Eq. (6).
Here we briefly present the collider constraints and future reach on the masses of the fermion and scalar multiplets in this model. The sensitivity study [40] on the process e + e − → S + 1 S − 1 → l + α l − β + E miss in KNT model at future International Linear Collider (ILC) with √ s = 1 TeV showed that m S > ∼ 240 GeV. On the other hand, it was shown in [41] that for tri-lepton final states via pp → l ± l ± S − * 1 → l ± l ± l ± + E miss at LHC with √ s = 14 TeV and luminosity, L = 300 fb −1 , the discovery reach for S + 1 increases up to m S < ∼ 4 TeV. In addition, we have F 0 1 to be DM candidate and that sets M φ > M F1 . Based on searches of disappearing track signatures from long-lived charginos that is nearly mass-degenerate with a neutralino at LHC with √ s = 14 TeV and L = 36.1 fb −1 [42], we can re-interpret the exclusion limits for fermion components as m F ± 1 > ∼ 600 GeV for lifetime,τ F ± 1 = 1 ns. Moreover, for wino-like minimal DM models that resembles fermion multiplets of KNT model, future collider with √ s = 100 TeV and L = 3 ab −1 [43], will improve this limit to m F 0 1 > ∼ 3.2 TeV. Besides, the multi charged component of the scalar multiplet, for example φ ++ can be produced via qq → W + φ ++ φ − and consequently will have the cascade decay, φ ++ → φ + W + * etc, that will lead to multi-lepton final states and missing energy. The condition M φ > M F1 then also sets M φ > ∼ 3.2 TeV. We scan over M F1 ∈ (1, 50) TeV, M F2,3 ∈ M F1 + (1, 10) TeV, M φ ∈ (10, 100) TeV, m S ∈ (500 GeV, 50 TeV) and λ S ∈ (0.001, 0.1). The yukawa couplings, f αβ and g iα are chosen so that they satisfy the low energy neutrino constraints. Afterwards, the rate of charged LFV processes are determined for all cases in this near degenerate limit. Although the generalized KNT model can contain a viable DM candidate, here we have studied charged LFV aspects of the model. In the companion paper [44], we show that for standard freeze-out scenario, the DM relic density constraint leads to a very small window of mass at TeV range but if the DM content of the universe is set by non-thermal process, the constraint on the mass can be relaxed.

B. Charged LFV Processes
We can see from Fig. 5 that the rate of µ → eee is very large compared to the µ → eγ rate and µ−e conversion rate in Au and Ti nuclei. The main reason behind this suppressed rate in µ → eγ and µ−e conversion rate is the cancellations among several one-loop diagrams, as mentioned in section III A and III B 1, which have rendered dipole A In addition, the contribution to Z penguin, F Z also receives several cancellations in one-loop diagrams, as mentioned in section III B 2. On the other hand, such cancellations does not take place in the box contribution, B and in the near degenerate limit, all box diagrams coherently add up for each of the singlet, triplet, 5-plet and 7-plet cases. This can be seen from Fig. 6. Consequently, A D , A N D and F Z enter into µ → eγ, µ → eee and µ − e conversion rates whereas B also contributes into µ → eee rate. Finally we can see from Fig. 5 that for M F1 = 1 − 50 TeV range, part of the viable parameter space of generalized KNT model is already excluded by µ → eee rate set by SINDRUM and future Mu3e experiment will exclude almost all of the parameter space for all cases within this mass range. This implies that the masses of BSM fermion and scalar particles of KNT model had to be pushed beyond 50 TeV.
Also in Fig. 6 (right), for 7-plet case, the box contribution coming from diagrams with both neutral and charged fermions associated with G yukawa sector, |B (1) + B (2) | wins over |B (3) | associated with F yukawa because, due to larger scalar and fermion multiplets, more particles enter into the loop and therefore |B (1) +B (2) | becomes larger for M F1 than B (3) for m S . Similar pattern can be seen also in dipole contributions.  Here we can see that, box contributions are larger that dipole contributions. (Right) Similar comparison is made for the 7-plet case. As AND behaves similarly as AD and also FZ is comparatively smaller than AD and B, we have not included them in the figure.

V. CONCLUSION AND OUTLOOK
We have investigated charged lepton flavor violating processes µ → eγ, µ → eee and µ − e conversion in Au and Ti in the generalized KNT model with singlet, triplet, 5-plet and 7-plet. We have shown that due to the cancellation among several one-loop contributions to photonic dipole term, photonic non-dipole term and Z-penguin term A D , A N D and F Z respectively, the rates of µ → eγ and µ − e conversion in Au and Ti become highly suppressed compared to µ → eee. This is due to the coherent addition of one-loop box diagrams where no cancellations take place and leads to box contribution B which enters into µ → eee process. As a consequence, we have seen that for M F1 = 1 − 50 TeV mass range, the region of viable parameter space set by neutrino sector is already excluded by the limit from SINDRUM and future Mu3e will have enough sensitivity to exclude almost all of the parameter space in this mass range and thus push the mass of lightest fermionic component larger than 50 TeV in generalized KNT model.