Lepton-flavor changing decays and non-unitarity in the inverse seesaw mechanism

The key role played by gauge symmetry in the description of fundamental physics and the origin of mass through the occurrence of spontaneous symmetry breaking have been greatly supported by the measurement, by the ATLAS and CMS Collaborations at the CERN, of the long-awaited Higgs boson HiggsATLAS ; HiggsCMS . In the minimal theoretical scheme, provided by the scalar sector of the Standard Model SMGlashow ; SMSalam ; SMWeinberg , the Brout-Englert-Higgs mechanism breaks the electroweak gauge symmetry group ${\rm SU}(2)_{L}\otimes{\rm U}(1)_{Y}$ into the electromagnetic group ${\rm U}(1)_{e}$ EnBr ; Higgs , which comes along with the definition of the masses of all the currently measured particles, except for the neutrinos, assumed to be massless in this framework. Neutrino masses have not ever been measured PDG , and yet the widely accepted interpretation that neutrino oscillations Pontecorvo imply massiveness of these particles calls for an explanation emerging from some high-energy formulation of physics beyond the Standard Model. Though the addition of three sterile right-handed neutrino fields to the Standard-Model particle spectra, together with a set of Dirac-type Yukawa terms, does the job, the resulting neutrino masses are determined by ad hoc “unnaturally” small values of Yukawa constants. In view of the massiveness and electromagnetic neutrality of neutrinos, the proper description of these particles could rather correspond to Majorana spinor fields Majorana , in which case global ${\rm U}(1)$ invariance is lost and, therefore, lepton number is not conserved. If lepton-number non-preservation is assumed, the effective Lagrangian for the electroweak Standard Model LLR ; BuWy ; Wudka gets extended, then allowing for the emergence of Majorana mass-terms Weinbergoperator ; BaLe ; CSV , presumably originating in some description of fundamental physics characterized by a high-energy scale, $\Lambda$. In particular, the Weinberg operator Weinbergoperator yields neutrino masses matching the mass profile that characterizes the seesaw mechanism MoSe1 ; MoSe2 , in which, besides the known neutrinos, a set of heavy-neutral-lepton partners, dubbed “heavy neutrinos”, comes about. The heavy-neutrino masses, $m_{N_{j}}$, determined by the high-energy scale as $m_{N_{j}}\sim\Lambda$, are linked to the masses, $m_{\nu_{j}}$, of the light neutrinos, in this scheme given by $m_{\nu_{j}}\sim\frac{v^{2}}{\Lambda}$, with $v$ the vacuum expectation value of the Standard-Model Higgs potential. In this context, current upper bounds on light-neutrino masses Planck ; cosmonumass ; KATRIN push the masses of heavy neutrinos towards enormous values, thus leaving direct production of heavy neutrinos off the table and also severely attenuating their contributions, as virtual particles, to Standard-Model observables.

So even though the seesaw mechanism provides a nice explanation for the tininess of neutrino mass, in connection with fundamental physics beyond the Standard Model, it is quite difficult to probe. This inconvenience has motivated the realization of seesaw variants, aimed at a relaxation of the restriction on the energy scale $\Lambda$, in order to bring it closer to current experimental sensitivity. Ref. CHLR provides a review on the seesaw mechanism and its variants. The framework for the present paper is defined by the neutrino-mass generation approach known as the “inverse seesaw mechanism” MoVa ; GoVa ; DeVa . More concretely, we consider a realization of the inverse seesaw in which the neutrino sector is enriched, as compared to the case of the Standard Model, by the introduction of three right-handed neutrino fields, together with a set of three further left-handed lepton fields, all of them singlets with respect to the ${\rm SU}(2)_{L}\otimes{\rm U}(1)_{Y}$ gauge-symmetry group. These new fermion fields introduce a slight violation of lepton number through Majorana-like mass terms characterized by two small-valued matrices, here denoted by $\mu_{S}$ and $\mu_{R}$, both proportional to an energy scale $v_{\sigma}$, of spontaneous symmetry breaking, which abide by the hierarchy condition $v_{\sigma}\ll v\ll\Lambda$. In the mass-eigenspinor basis, this extended neutrino sector yields three light neutrinos and a total of six heavy neutral leptons. The mass matrix of light neutrinos, $m_{\nu}$, is proportional to $\mu_{S}$, which therefore attenuates the pressure on $\Lambda$, thus allowing for smaller and more reasonable values of the heavy-neutrino masses, as compared to what happens in the original version of the seesaw mechanism.

In the electroweak Standard Model, the absence of right-handed neutrinos and the assumption of lepton-number conservation prevents neutrino-mass terms from being generated. Moreover, lepton-flavor-violating processes, also forbidden in such a context, are well-motivated means to search for traces pointing towards the presence of new physics. Despite the large amount of experimental work dedicated to the search for charged-lepton-flavor violation, no process of this kind has ever been observed PDG , though notice that neutrino mixing, required for the occurrence of neutrino oscillations Pontecorvo , implies that such sort of processes are actually allowed, in which case their measurement could be eventually achieved in experimental facilities. The present investigation readdresses the one-loop contributions, generated by the whole set of massive neutral leptons defined in the context of the inverse seesaw, to the charged-lepton-flavor-changing decays $\mu\to e\,\gamma$, $\tau\to e\,\gamma$, and $\tau\to\mu\,\gamma$. These decay processes have been formerly examined in Refs. SFLYC ; SoRu ; GPH . Currently, the most stringent experimental constraints on these decay processes are the ones reported by the MEG, Belle, and BaBar collaborations MEGlfvbound ; Bellelfvbound ; BaBarlfvbound . The MEG II and Belle II upgrades, expected to improve experimental sensitivities by $\sim 1$ order of magnitude MEG2lfvestimation ; Belle2lfvestimation , are also to be borne in mind. Our estimations show that, in accordance with such experimental works, the inverse-seesaw contribution to $\mu\to e\,\gamma$ yields the most promising result among the charged-lepton-flavor-violating decay processes under consideration. A worthwhile aspect of the inverse seesaw regards the matrix that characterizes the mixing of light neutrinos with charged leptons. In the simplest scenarios, as the Standard Model endowed with three singlet right-handed Dirac neutrino fields GiKi or the one given by the introduction of the Weinberg operator with the assumption that lepton number is not preserved Weinbergoperator , light-neutrino mixing is characterized by the Pontecorvo-Maki-Nakgawa-Sakata (PMNS) matrix MNSmatrix ; Pontecorvomatrix , which is unitary. However, the presence of a light-neutrino mixing matrix which is not unitary occurs in a large class of models aimed at generating neutrino masses FGLY . Constraints imposed by the charged-lepton-flavor-violating decays $\mu\to e\,\gamma$, $\tau\to e\,\gamma$, and $\tau\to\mu\,\gamma$ on these non-unitary effects have been estimated in Refs. FHL ; BFHLMN . In the present work, we analyze the aforementioned lepton-flavor-changing decays in terms of their relation with such non-unitary effects. The most restrictive constraints proceed from the contributions to $\mu\to\,e\gamma$ and its comparison with the MEG limit. Furthermore, we find that the sensitivity expectation estimated for MEG II should yield a potential improvement, given by a factor $\sim\frac{1}{3}$, of restrictions for the non-unitary effects.

The reminder of the paper has been organized as follows: in Section II, the inverse seesaw mechanism is reviewed, which includes the main expressions for the phenomenological calculation to be carried out later; we describe the execution of our analytical calculation of the charged-lepton-flavor-violating decays $\mu\to e\,\gamma$, $\tau\to e\,\gamma$, and $\tau\to\mu\,\gamma$ at one loop in Section III; our estimations and analyses of the contributions are developed throughout Section IV; we conclude the paper by providing, in Section V, a summary.

In this section, we discuss the inverse seesaw mechanism MoVa ; GoVa ; DeVa , which is the framework behind our phenomenological investigation. Aiming at a more definite presentation, we address this task by considering a completion in which lepton number is spontaneously broken CMP . Think of an extension of the Standard Model, characterized by a Lagrangian invariant under the electroweak gauge group ${\rm SU}(2)_{L}\otimes{\rm U}(1)_{Y}$. Assume that the Standard-Model field content has been increased by the introduction of a set of three chiral fermion fields $\nu_{1,R}$, $\nu_{2,R}$, and $\nu_{3,R}$, which are right handed with lepton number $L(\nu_{R})=+1$, and by three more fermion fields $S_{1,L}$, $S_{2,L}$, and $S_{3,L}$, with definite left chirality and lepton number $L(S_{L})=+1$. All these fields are assumed to be singlets with respect to electroweak gauge transformations. Then, a complex scalar field, $\sigma$, assumed to be an ${\rm SU}(2)_{L}\otimes{\rm U}(1)_{Y}$ singlet and to have lepton number $L(\sigma)=+2$, is introduced as well. The addition of these new-physics fields translates into extensions of both the lepton-Yukawa sector and the scalar sector of the Standard Model.

The Standard-Model Higgs doublet $\Phi$ and the scalar singlet $\sigma$ define a scalar potential, nested within the scalar sector, which reads

	$\displaystyle V(\Phi,\sigma)=-\mu_{\Phi}^{2}\Phi^{\dagger}\Phi-\mu_{\sigma}^{2}\sigma^{\dagger}\sigma+\lambda_{\Phi}\big{(}\Phi^{\dagger}\Phi\big{)}^{2}$
	$\displaystyle\hskip 36.98866pt+\lambda_{\sigma}\big{(}\sigma^{\dagger}\sigma\big{)}^{2}+\lambda_{\Phi\sigma}\big{(}\Phi^{\dagger}\Phi\big{)}\big{(}\Phi^{\dagger}\Phi\big{)},$		(1)

where $\mu_{\Phi}^{2}$, $\mu_{\sigma}^{2}$, $\lambda_{\Phi}$, $\lambda_{\sigma}$, and $\lambda_{\Phi\sigma}$, are positive quantities¹¹1A discussion on the vacuum stability of $V\big{(}\Phi,\sigma\big{)}$ can be found in Ref. MRSV .. Two stages of spontaneous symmetry breaking come about, the first of which we assume to occur at the energy scale $v=\sqrt{\frac{2(2\lambda_{\sigma}\mu_{\Phi}^{2}-\lambda_{\Phi\sigma}\mu_{\sigma}^{2})}{4\lambda_{\Phi}\lambda_{\sigma}-\lambda_{\Phi\sigma}^{2}}}$, identified as the electroweak vacuum expectation value. By this symmetry breaking, the gauge group ${\rm SU}(2)_{L}\otimes{\rm U}(1)_{Y}$ is broken down to the electromagnetic group ${\rm U}(1)_{e}$. Then, at $v_{\sigma}=\sqrt{\frac{2(2\lambda_{\Phi}\mu_{\sigma}^{2}-\lambda_{\Phi\sigma}\mu_{\Phi}^{2})}{4\lambda_{\Phi}\lambda_{\sigma}-\lambda_{\Phi\sigma}^{2}}}$, the global symmetry ${\rm U}(1)$ associated to the preservation of lepton number is broken. After this, two massive scalar fields are defined, namely, the Higgs boson field, $h$, and a novel scalar field, $s_{0}$, with masses given by

$$m_{h}^{2}=\lambda_{\Phi}v^{2}+\lambda_{\sigma}v_{\sigma}^{2}+\sqrt{\big{(}\lambda_{\Phi}v^{2}-\lambda_{\sigma}v_{\sigma}^{2}\big{)}^{2}+v^{2}v_{\sigma}^{2}\lambda_{\Phi\sigma}^{2}},$$

(2)

$$m_{s_{0}}^{2}=\lambda_{\Phi}v^{2}+\lambda_{\sigma}v_{\sigma}^{2}-\sqrt{\big{(}\lambda_{\Phi}v^{2}-\lambda_{\sigma}v_{\sigma}^{2}\big{)}^{2}+v^{2}v_{\sigma}^{2}\lambda_{\Phi\sigma}^{2}}.$$

(3)

Moreover, as a consequence of the rupture of the ${\rm U}(1)$ global symmetry, a physical Goldstone boson $J$, widely known as the Majoron, emerges CMP . Originated from the symmetry breaking of a global group, the Majoron is born massless. Yet, authors have evoked mechanisms to give it a mass, thus turning this scalar into a viable dark matter candidate ABMS ; BeVa ; RBS ; BLRV ; GMS ; EJJRTV ; GaHe ; BrPa .

Prior to electroweak symmetry breaking, the Yukawa sector, ${\cal L}_{\rm Y}$, is given by the Lagrangian

	$\displaystyle{\cal L}_{\rm Y}=\sum_{j=1}^{3}\sum_{k=1}^{3}\Big{(}-Y^{\nu}_{jk}\overline{L_{j,L}}\tilde{\Phi}\nu_{k,R}$
	$\displaystyle\hskip 22.76228pt-M_{jk}\overline{S_{j,L}}\nu_{k,R}-\frac{1}{2}\lambda_{jk}^{(S)}\,\sigma\,\overline{S_{j,L}}S_{k,L}^{\rm c}$
	$\displaystyle\hskip 22.76228pt-\frac{1}{2}\lambda_{jk}^{(N)}\,\sigma^{*}\overline{\nu_{j,R}^{\rm c}}\nu_{k,R}+{\rm H.c.}\Big{)}+\cdots,$
			(4)

where

$$L_{j,L}=\left(\begin{array}[]{c}\nu^{\prime}_{j,L}\vspace{0.15cm}\\ l^{\prime}_{j,L}\end{array}\right)$$

(5)

is the $j\textrm{-th}$ ${\rm SU}(2)_{L}$ lepton doublet, with $j=1,2,3$, and where the definition $\tilde{\Phi}=i\sigma_{2}\Phi^{*}$, with $\sigma_{2}$ the imaginary Pauli matrix, has been used. This expression for ${\cal L}_{\rm Y}$ shows explicitly only those Yukawa couplings involving the fermion fields that extend the field content of the Standard Model, around which we develop our discussion, whereas the ellipsis denote further Yukawa terms, including those made exclusively of Standard-Model fields. The lagrangian terms explicitly displayed in Eq. (4) incorporate four sorts of Yukawa constants, each of which defines a $3\times 3$ matrix: $Y^{\nu}$, $M$, $\lambda^{(S)}$, and $\lambda^{(N)}$. In particular, we assume the matrix $M$ to originate from a stage of spontaneous symmetry breaking taking place at a high-energy scale $\Lambda$, of some gauge group that characterizes a high-energy fundamental description beyond the Standard Model, in which case we work under the premise that $v\ll\Lambda$. Further, $M\sim\Lambda$ is reasonably assumed. The lepton-number assignments for the fields $S_{j,L}$, $\nu_{j,R}$, and $\sigma$ guarantee invariance of ${\cal L}_{\rm Y}$ with respect to the global group ${\rm U}(1)$. Next, electroweak gauge symmetry is broken at $v$, which allows us to get to the charged-lepton mass-eigenspinor basis by implementing the unitary transformations $l_{L}=V^{\ell{\dagger}}_{L}l^{\prime}_{L}$ and $l_{R}=V^{\ell{\dagger}}_{R}l^{\prime}_{R}$, with $l^{\prime}_{j,R}$ the $j$-th right-handed ${\rm SU}(2)_{L}$ lepton singlet, then defining the $\alpha$-flavor charged-lepton field as $l_{\alpha}=l_{\alpha,L}+l_{\alpha,R}$. This results in the charged-lepton mass terms $-m_{l_{\alpha}}\overline{l_{\alpha}}\,l_{\alpha}$. After breaking the ${\rm SU}(2)_{L}\otimes{\rm U}(1)_{Y}$ gauge symmetry and then using $\sigma=\frac{v_{\sigma}}{\sqrt{2}}+\cdots$, to break the global ${\rm U}(1)$ group, the Yukawa Lagrangian ${\cal L}_{\rm Y}$, Eq. (4), acquires the form

	$\displaystyle{\cal L}_{\rm Y}=\sum_{\alpha=e,\mu,\tau}\sum_{k=1}^{3}\Big{(}-(m_{\rm D})_{\alpha k}\,\overline{\nu_{\alpha,L}}\,\nu_{k,R}+{\rm H.c.}\Big{)}$
	$\displaystyle\hskip 22.76228pt+\sum_{j=1}^{3}\sum_{k=1}^{3}\Big{(}-M_{jk}\overline{S_{j,L}}\nu_{k,R}-\frac{(\mu_{S})_{jk}}{2}\overline{S_{j,L}}S_{k,L}^{\rm c}$
	$\displaystyle\hskip 22.76228pt-\frac{(\mu_{R})_{jk}}{2}\overline{\nu_{j,R}^{\rm c}}\nu_{k,R}+{\rm H.c.}\Big{)}+\cdots.$
			(6)

By considering the left-handed unitary-transformation matrix $V^{\ell}_{L}$, we found it convenient to define $\nu_{L}=V^{\ell{\dagger}}_{L}\nu^{\prime}_{L}$. Then the first term in the right-hand side of this equation features the $3\times 3$ complex matrix $m_{\rm D}=\frac{v}{\sqrt{2}}V^{\ell{\dagger}}_{L}Y^{\nu}$. Moreover, the $3\times 3$ matrices $\mu_{S}=\frac{v_{\sigma}}{\sqrt{2}}\lambda^{(S)}$ and $\mu_{R}=\frac{v_{\sigma}}{\sqrt{2}}\lambda^{(N)}$ emerge as well. Note that both these matrices are symmetric. The $\mu_{S}$ and $\mu_{R}$ terms are no longer invariant under global ${\rm U}(1)$ and therefore spoil lepton-number conservation. Assuming, on the grounds of naturalness tHooftnaturalness , that these matrices are small, which can be achieved if the condition $v_{\sigma}\ll v$ is fulfilled, ${\rm U}(1)$ global symmetry is only slightly violated. We follow such an assumption from here on. Notice that, from the assumed origin for the $M$ matrix, we have the inverse-seesaw hierarchy condition

$$v_{\sigma}\ll v\ll\Lambda,$$

(7)

among the three involved energy scales.

The Yukawa sector can be rearranged as

$${\cal L}_{\rm Y}=\frac{1}{2}\left(\begin{array}[]{ccc}\overline{\nu_{L}}&\overline{\nu_{R}^{\rm c}}&\overline{S_{L}}\end{array}\right){\cal M}_{\nu}\left(\begin{array}[]{c}\nu_{L}^{\rm c}\vspace{0.1cm}\\ \nu_{R}\vspace{0.1cm}\\ S_{L}^{\rm c}\end{array}\right)+{\rm h.c.}+\cdots,$$

(8)

where ${\cal M}_{\nu}$ is a $9\times 9$ symmetric matrix given, in terms of the $3\times 3$ block matrices $m_{\rm D}$, $M$, $\mu_{R}$, and $\mu_{S}$, as

$${\cal M}_{\nu}=\left(\begin{array}[]{ccc}0&m_{\rm D}&0\vspace{0.1cm}\\ m_{\rm D}^{\rm T}&\mu_{R}&M^{\rm T}\vspace{0.1cm}\\ 0&M&\mu_{S}\end{array}\right).$$

(9)

Moreover, the definitions

$$M_{\rm M}=\left(\begin{array}[]{cc}\mu_{R}&M^{\rm T}\vspace{0.1cm}\\ M&\mu_{S}\end{array}\right),$$

(10)

$$M_{\rm D}=\left(\begin{array}[]{cc}m_{\rm D}&0\end{array}\right),$$

(11)

where the sizes of $M_{\rm M}$ and $M_{\rm D}$ respectively are $6\times 6$ and $3\times 6$, allow one to give ${\cal M}_{\nu}$ the form of the neutral-lepton mass matrix that characterizes the type-1 seesaw mechanism. Since ${\cal M}_{\nu}$ is symmetric, the existence of a unitary-diagonalization matrix $\Omega$ is ensured, with the diagonalization proceeding as Takagi

$$\Omega^{\rm T}{\cal M}_{\nu}\,\Omega=\left(\begin{array}[]{ccc}M_{n}&0&0\vspace{0.1cm}\\ 0&M_{N}&0\\ 0&0&M_{X}\end{array}\right).$$

(12)

In this equation, $M_{n}$ is a $3\times 3$ diagonal matrix, whose in-diagonal elements are the light-neutrino masses, which we denote as $m_{n_{j}}$, with $j=1,2,3$. Furthermore, the matrix $M_{N}$ is $3\times 3$ sized and diagonal, with its diagonal entries corresponding to the masses $m_{N_{j}}$ of three heavy neutrinos , so $j=1,2,3$ as well. Finally, the $3\times 3$ diagonal matrix $M_{X}$ comprises, nested within its diagonal, the masses $m_{X_{j}}$ of three further heavy neutral leptons. This diagonalization procedure entails the following change of basis:

$$\left(\begin{array}[]{c}n_{L}\vspace{0.1cm}\\ N_{L}\vspace{0.1cm}\\ X_{L}\end{array}\right)=\Omega^{\rm T}\left(\begin{array}[]{c}\nu_{L}\vspace{0.1cm}\\ \nu_{R}^{\rm c}\vspace{0.1cm}\\ S_{L}\end{array}\right),\hskip 8.5359pt\left(\begin{array}[]{c}n_{R}\vspace{0.1cm}\\ N_{R}\vspace{0.1cm}\\ X_{R}\end{array}\right)=\Omega^{\dagger}\left(\begin{array}[]{c}\nu_{L}^{\rm c}\vspace{0.1cm}\\ \nu_{R}\vspace{0.1cm}\\ S_{L}^{\rm c}\end{array}\right).$$

(13)

Here, $n_{L}$ and $n_{R}$ are $3\times 1$ matrices for chiral left- and right-handed neutrino fields, respectively. Meanwhile, $N_{L}$, $N_{R}$, $X_{L}$, and $X_{R}$ are also $3\times 1$ sized, comprised by left-handed and right-handed neutral spinor fields. Putting all the pieces together, the Yukawa-sector Lagrangian is written as

	$\displaystyle{\cal L}_{\rm Y}=\sum_{j=1}^{3}\sum_{k=1}^{3}\Big{(}-\frac{1}{2}m_{n_{j}}\overline{n_{j}}n_{j}$
	$\displaystyle\hskip 17.07182pt-\frac{1}{2}m_{N_{j}}\overline{N_{j}}N_{j}-\frac{1}{2}m_{X_{j}}\overline{X_{j}}X_{j}\Big{)}+\cdots$		(14)

for which the non-chiral fermion fields $n_{j}=n_{j,L}+n_{j,R}$, $N_{j}=N_{j,L}+N_{j,R}$, and $X_{j}=X_{j,L}+X_{j,R}$ have been defined.

The diagonalization matrix $\Omega$ is conveniently expressed as

$$\Omega=U\,V,$$

(15)

with $U$ and $V$ a couple of $9\times 9$ unitary matrices. The matrix $U$ is intended to block-diagonalize the mass matrix ${\cal M}_{\nu}$. This unitary matrix can be expressed in block-matrix form as

$$U=\left(\begin{array}[]{cc}U_{11}&U_{12}\vspace{0.1cm}\\ U_{21}&U_{22}\end{array}\right),$$

(16)

where $U_{11}$ and $U_{22}$ are square matrices, the former $3\times 3$ sized and the latter $6\times 6$ sized. Meanwhile, $U_{12}$ is a $3\times 6$ matrix and $U_{21}$ is $6\times 3$. While the last equation is largely generic, the unitary character of $U$ allows for the following matrix-block parametrization KPS ; DePi :

$$U=\left(\begin{array}[]{cc}\big{(}{\bf 1}_{3}+\xi^{*}\xi^{\rm T}\big{)}^{-\frac{1}{2}}&\xi^{*}\big{(}{\bf 1}_{6}+\xi^{\rm T}\xi^{*}\big{)}^{-\frac{1}{2}}\vspace{0.1cm}\\ -\xi^{\rm T}\big{(}{\bf 1}_{3}+\xi^{*}\xi^{\rm T}\big{)}^{-\frac{1}{2}}&\big{(}{\bf 1}_{6}+\xi^{\rm T}\xi^{*}\big{)}^{-\frac{1}{2}}\end{array}\right),$$

(17)

where ${\bf 1}_{3}$ and ${\bf 1}_{6}$ stand for the $3\times 3$ and the $6\times 6$ identity matrices, respectively. Further, $\xi$ is a $3\times 6$ matrix fulfilling $M_{\rm D}-\xi M_{\rm D}^{\rm T}\xi^{*}-\xi M_{\rm,M}=0$. If the entries of $\xi$ are small, this condition yields the expression

$$\xi=M_{\rm D}M_{\rm M}^{-1},$$

(18)

corresponding to a large suppression provided by $M_{\rm M}^{-1}$. From here on, we assume Eq. (18) to hold. Then, the block parametrization for $U$, Eq. (17), can be approximated as

$$U\approx\left(\begin{array}[]{cc}{\bf 1}_{3}-\frac{1}{2}M_{\rm D}^{*}\big{(}M_{\rm M}^{-1}\big{)}^{*}M_{\rm M}^{-1}M_{\rm D}^{\rm T}&M_{\rm D}^{*}\big{(}M_{\rm M}^{-1}\big{)}^{*}\vspace{0.1cm}\\ -M_{\rm M}^{-1}M_{\rm D}^{\rm T}&{\bf 1}_{6}-\frac{1}{2}M_{\rm M}^{-1}M_{\rm D}^{\rm T}M_{\rm D}^{*}\big{(}M_{\rm M}^{-1}\big{)}^{*}\end{array}\right).$$

(19)

The afore-announced block-matrix diagonalization, driven by $U$, goes as follows:

$$U^{\rm T}{\cal M}_{\nu}U=\left(\begin{array}[]{cc}M_{\rm light}&0\vspace{0.1cm}\\ 0&M_{\rm heavy}\end{array}\right),$$

(20)

where, the matrix $M_{\rm light}=-M_{\rm D}M_{\rm M}^{-1}M_{\rm D}^{\rm T}+{\cal O}\big{(}(M_{\rm M}^{-1})^{3}\big{)}$ is $3\times 3$ sized, whereas $M_{\rm heavy}=M_{\rm M}+{\cal O}\big{(}M_{\rm M}^{-1}\big{)}$ is a $6\times 6$ matrix. The matrix $V$, introduced in Eq. (15), is written as

$$V=\left(\begin{array}[]{cc}\hat{V}&0\vspace{0.1cm}\\ 0&\tilde{V}\end{array}\right),$$

(21)

with the $3\times 3$ matrix $\hat{V}$ and the $6\times 6$ matrix $\tilde{V}$ respectively diagonalizing $M_{\rm light}$ and $M_{\rm heavy}$ as

$$\hat{V}^{\rm T}M_{\rm light}\,\hat{V}=M_{n},$$

(22)

$$\tilde{V}^{\rm T}M_{\rm heavy}\tilde{V}=\left(\begin{array}[]{cc}M_{N}&0\vspace{0.1cm}\\ 0&M_{X}\end{array}\right),$$

(23)

in accordance with Eq. (12). The inverse of $M_{\rm M}$ turns out to be DePi

$$M_{\rm M}^{-1}=\left(\begin{array}[]{cc}\big{(}\mu_{R}-M^{\rm T}\mu_{S}^{-1}M\big{)}^{-1}&-\big{(}\mu_{R}-M^{\rm T}\mu_{S}^{-1}M\big{)}^{-1}M^{\rm T}\mu_{S}^{-1}\vspace{0.1cm}\\ -\big{(}\mu_{S}-M\mu_{R}^{-1}M^{\rm T}\big{)}^{-1}M\,\mu_{R}^{-1}&\big{(}\mu_{S}-M\mu_{R}^{-1}M^{\rm T}\big{)}^{-1}\end{array}\right),$$

(24)

from which the inverse-seesaw light-neutrino mass formula,

$$M_{n}\approx\hat{V}^{\rm T}m_{\rm D}M^{-1}\mu_{S}\,(M^{\rm T})^{-1}m_{\rm D}\hat{V},$$

(25)

follows. According to Eq. (25), the tinniness characterizing the masses of light neutrinos does not rely only on the suppression introduced by $M^{-1}$, as in the canonical seesaw mechanism, because the smallness of $\mu_{S}$ establishes a further suppression, thus reducing the stress on the high-energy scale $\Lambda$, associated to $M$, which in this manner evades huge values.

As shown in Eq. (23), the unitary matrix $\tilde{V}$, which diagonalizes the heavy-neutral-lepton mass matrix $M_{\rm heavy}$, yields the diagonal mass matrices $M_{N}$ and $M_{X}$, which correspond to the sets of heavy-neutral leptons $\{N_{1},N_{2},N_{3}\}$ and $\{X_{1},X_{2},X_{3}\}$, respectively. In pursuit of some insight regarding the mass spectra for these fields, let us assume that the matrices $M$, $\mu_{R}$, and $\mu_{S}$ are diagonal, in which case $\tilde{V}$ can be written as

$$\tilde{V}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{rcr}i({\bf 1}_{3}+\frac{1}{4}\chi)&&{\bf 1}_{3}-\frac{1}{4}\chi\vspace{0.2cm}\\ -i({\bf 1}_{3}-\frac{1}{4}\chi)&&{\bf 1}_{3}+\frac{1}{4}\chi\end{array}\right),$$

(26)

where $\chi=M^{-1}\big{(}\mu_{S}-\mu_{R}\big{)}$ and $i$ is de imaginary unity. Thus, from Eqs. (23) and (26), the diagonal matrices

$$M_{N}=M-\frac{\mu_{S}}{2}-\frac{\mu_{R}}{2},$$

(27)

$$M_{X}=M+\frac{\mu_{S}}{2}+\frac{\mu_{R}}{2},$$

(28)

emerge, which then lead us to the neutral-lepton mass terms displayed in Eq. (14). Now consider the difference $M_{X}-M_{N}=\mu_{S}+\mu_{R}$, according to which masses $m_{N_{j}}$ and $m_{X_{j}}$, all of them large as dictated by the hierarchy condition shown in Eq. (7), are very similar to each other. Thus the complete heavy-neutral-lepton mass spectrum is quasi-degenerate for each pair of fields $N_{j}$ and $X_{j}$, with $j=1,2,3$.

To recapitulate, after the spontaneous symmetry breaking of the Standard-Model electroweak gauge group ${\rm SU}(2)_{L}\otimes{\rm U}(1)_{Y}$ into the electromagnetic group ${\rm U}(1)_{e}$, at $v$, and the ulterior breaking of the global ${\rm U}(1)$ symmetry at $v_{\sigma}$, the neutral-lepton field content comprehends three light-neutrino fields, denoted by $n_{j}$, also three heavy-neutrino fields, which we have referred to as $N_{j}$, and three further heavy-neutral-lepton fields, represented by $X_{j}$, with $m_{N_{j}}\approx m_{X_{j}}$, for each pair labeled by $j=1,2,3$. Let us jointly denote all the heavy-neutral-lepton fields as

$$f_{j}=\left\{\begin{array}[]{l}N_{k},\textrm{ if }j=1,2,3,\vspace{0.1cm}\\ X_{k},\textrm{ if }j=4,5,6,\end{array}\right.$$

(29)

where $N_{k}=N_{1},N_{2},N_{3}$ and $X_{k}=X_{1},X_{2},X_{3}$. With this in mind, the charged-currents lagrangian term can be written as

	$\displaystyle{\cal L}_{W\nu l}=\sum_{\alpha=e,\mu,\tau}\Big{(}\sum_{j=1}^{3}\big{(}{\cal B}_{\alpha n_{j}}W^{-}_{\mu}\overline{l_{\alpha}}\gamma^{\mu}P_{L}n_{j}+{\rm h.c.}\big{)}\vspace{0.1cm}$
	$\displaystyle\hskip 34.14322pt+\sum_{j=1}^{6}\big{(}{\cal B}_{\alpha f_{j}}W^{-}_{\mu}\overline{l_{\alpha}}\gamma^{\mu}P_{L}f_{j}+{\rm h.c.}\big{)}\Big{)},$		(30)

for which the definitions

$${\cal B}_{\alpha n_{j}}=\big{(}U_{11}^{*}\hat{V}^{*}\big{)}_{\alpha j},$$

(31)

$${\cal B}_{\alpha f_{j}}=\big{(}U_{12}^{*}\tilde{V}^{*}\big{)}_{\alpha j},$$

(32)

have been used. The factors $\big{(}U_{11}\big{)}_{\alpha\beta}$ and $\big{(}U_{12}\big{)}_{\alpha k}$, in Eqs. (31) and (32), are components of the matrices $U_{11}$ and $U_{12}$, which are part of the generic block-matrix form of $U$, given in Eq. (16). Moreover, $\hat{V}_{\beta j}$ and $\tilde{V}_{kj}$ are entries of the unitary matrices introduced in Eq. (21). The whole set of quantities ${\cal B}_{\alpha n_{j}}$ and ${\cal B}_{\alpha f_{j}}$ constitute the $3\times 3$ matrix ${\cal B}_{n}$ and the $3\times 6$ matrix ${\cal B}_{f}$, respectively, which can be accommodated into the $3\times 9$ matrix ${\cal B}=\big{(}{\cal B}_{n}\,,\,{\cal B}_{f}\big{)}$, with entries

$${\cal B}_{\alpha j}=\left\{\begin{array}[]{l}{\cal B}_{\alpha n_{k}},\textrm{ if }j=1,2,3,\vspace{0.2cm}\\ {\cal B}_{\alpha f_{k}},\textrm{ if }j=4,5,6,7,8,9,\end{array}\right.$$

(33)

where $n_{k}=n_{1},n_{2},n_{3}$ and $f_{k}=f_{1},f_{2},f_{3},f_{4},f_{5},f_{6}$, for any fixed $\alpha$. The matrix ${\cal B}$ fulfills a sort of semi-unitarity property, that is

	$\displaystyle\displaystyle\sum_{j=1}^{9}{\cal B}_{\alpha j}{\cal B}_{\beta j}^{*}=\delta_{\alpha\beta},\vspace{0.1cm}$		(34)
	$\displaystyle\displaystyle\sum_{\alpha=e,\mu,\tau}{\cal B}_{\alpha j}^{*}{\cal B}_{\alpha k}={\cal C}_{jk}.$		(35)

The quantities ${\cal C}_{jk}$, involved in Eq. (35), form a $9\times 9$ matrix, ${\cal C}$, which can be written in block-matrix form as

$${\cal C}=\left(\begin{array}[]{cc}{\cal C}_{nn}&{\cal C}_{nf}\vspace{0.1cm}\\ {\cal C}_{fn}&{\cal C}_{ff}\end{array}\right),$$

(36)

where ${\cal C}_{nn}$ is $3\times 3$, the size of ${\cal C}_{ff}$ is $6\times 6$, ${\cal C}_{nf}$ is a $3\times 6$ matrix, and ${\cal C}_{fn}$ is $6\times 3$ sized. Furthermore, the entries of ${\cal C}$ relate to those of these block matrices as

$${\cal C}_{jk}=\left\{\begin{array}[]{l}C_{n_{l}n_{i}},\textrm{ if }j=1,2,3\textrm{ and }k=1,2,3,\vspace{0.2cm}\\ C_{n_{l}f_{i}},\textrm{ if }j=1,2,3\textrm{ and }k=4,5,6,7,8,9,\vspace{0.2cm}\\ C_{f_{l}n_{i}},\textrm{ if }j=4,5,6,7,8,9\textrm{ and }k=1,2,3,\vspace{0.2cm}\\ C_{f_{l}f_{i}},\textrm{ if }j=4,5,6,7,8,9\textrm{ and }k=4,5,6,7,8,9,\end{array}\right.$$

(37)

where $n_{l},n_{i}=n_{1},n_{2},n_{3}$ and $f_{l},f_{i}=f_{1},f_{2},f_{3},f_{4},f_{5},f_{6}$. These block matrices are defined as

$${\cal C}_{n_{j}n_{k}}=\big{(}\Omega_{11}^{\rm T}\Omega_{11}^{*}\big{)}_{jk},$$

(38)

$${\cal C}_{n_{j}f_{k}}=\big{(}\Omega_{11}^{\rm T}\Omega_{12}^{*}\big{)}_{jk},$$

(39)

$${\cal C}_{f_{j}n_{k}}=\big{(}\Omega_{12}^{\rm T}\Omega_{11}^{*}\big{)}_{jk},$$

(40)

$${\cal C}_{f_{j}f_{k}}=\big{(}\Omega_{12}^{\rm T}\Omega_{12}^{*}\big{)}_{jk}.$$

(41)

From the $U$ block parametrization displayed in Eq. (19), the expression for $M^{-1}_{\rm M}$ as given in Eq. (24), and the explicit form shown in Eq. (26) for $\tilde{V}$, the matrices ${\cal B}_{n}$ and ${\cal B}_{f}$ can be written as

$${\cal B}_{n}=\Big{(}{\bf 1}_{3}-\frac{1}{2}m_{\rm D}M^{-2}m_{\rm D}^{\dagger}\Big{)}\hat{V}^{*},$$

(42)

$${\cal B}_{f}=\Big{(}\begin{array}[]{lcr}\frac{i}{\sqrt{2}}m_{\rm D}M^{-1}&&\frac{1}{\sqrt{2}}m_{\rm D}M^{-1}\end{array}\Big{)}.$$

(43)

In the so-called “minimally extended Standard Model” GiKi , characterized by the addition of three right-handed Dirac-neutrino fields, Yukawa terms for right-handed neutrinos give rise to neutrino mass terms. In the neutrino mass-eigenspinor basis, the lepton charged currents, $j^{\mu}_{{\rm SM},W}=2\,\overline{l_{L}}\,U_{\rm PMNS}\gamma^{\mu}\,n_{L}$, involve lepton-flavor change, driven by the PMNS matrix, $U_{\rm PMNS}$, which is the lepton-sector analogue of the Kobayashi-Maskawa matrix, lying in the quark sector. The neutrino-flavor basis is then defined by the transformation $\nu_{L}=U_{\rm PMNS}\,n_{L}$, by means of which the charged currents are simply expressed as $j^{\mu}_{{\rm SM},W}=2\,\overline{l_{L}}\gamma^{\mu}\,\nu_{L}$. An important feature of $U_{\rm PMNS}$ is its unitary property. A similar discussion can be developed if neutrino masses are rather defined through the inclusion, in the framework of a lepton-number-violating effective Lagrangian for Standard Model, of the Weinberg operator. Getting back to the theoretical framework of the present paper, the charged currents in which light neutrinos participate, displayed in the first line of Eq. (30), carry neutrino mixing characterized not by the $U_{\rm PMNS}$ unitary matrix, but by the $3\times 3$ matrix ${\cal B}_{n}$, which is not unitary. Expressing this matrix as ${\cal B}_{n}=\big{(}1-\eta\big{)}\hat{V}^{*}$, the $3\times 3$ matrix

$$\eta=\frac{1}{2}m_{\rm D}M^{-2}m_{\rm D}^{\dagger}$$

(44)

is understood as an object characterizing non-unitary effects in light-neutrino mixing.

In this section, we present and discuss our analytical calculation of contributions from virtual light neutrinos neutrinos $n_{j}$, heavy neutrinos $N_{j}$, and heavy neutral leptons $X_{j}$, to the charged-lepton-flavor-violating decay $l_{\alpha}\to l_{\beta}\,\gamma$, where the indices $\alpha$ and $\beta$ label charged-lepton flavors, so $\alpha=\mu,\tau$ and $\beta=e,\mu$. In the presence of massive neutrinos, and as long as neutrino mixing takes place, such a process receives contributions from Feynman diagrams since the one-loop order, whatever Standard-Model extension is considered. Note that, in contraposition, this decay process is forbidden in the Standard Model. In general, physical processes which are either not allowed by the Standard Model or suppressed in such a framework bear great relevance, because manifestations from the high-energy description might be more easily detected.

In the context of the inverse seesaw mechanism, the contributing diagrams emerge at one loop as a result of lepton mixing in the charged currents displayed in Eq. (30), which are characterized by the matrix ${\cal B}=\big{(}{\cal B}_{n},{\cal B}_{f}\big{)}$. In general, the set of Feynman rules by which Majorana fermions abide differ from those for fermions described by Dirac fields. Refs. DEHK ; GlZr provide detailed discussions on the matter. For instance, the assumption of Majorana fermions opens the possibility of having a larger number of contributing diagrams, as compared with the Dirac case. However, by following the Wick’ theorem Wick , we found no extra diagrams. Moreover, all the determined contributing diagrams were found not to distinguish among the Majorana and Dirac cases.

While gauge invariance is a key element in the construction of field theories, as it provides a criterion to build Lagrangian terms, the quantization of gauge theories imperiously requires the elimination of this symmetry, which is achieved by fixing the gauge. Then, gauge freedom ensures that any physical quantity must be gauge independent. In particular, the gauge choice must be innocuous to the decay amplitude we are calculating. The linear and the non-linear gauge-fixing approaches FLS ; Shore ; EiWu ; MeTo , where the choice of the gauge is parametrized by some gauge-fixing parameter, are well-known gauge-fixing schemes. Another, often considered, gauge choice is the “unitary gauge”, distinguished by the absence of pseudo-Goldstone bosons, which are born in the process of spontaneous symmetry breaking but which lack physical degrees of freedom. We carry out the calculation of the amplitude for $l_{\alpha}\to\,l_{\beta}\gamma$ in the unitary gauge, so no contributing Feynman diagrams featuring pseudo-Goldstone-boson lines are to be taken into account, in contraposition to what happens in other gauges. Therefore, the amplitude for $l_{\alpha}\to\,l_{\beta}\gamma$ is given by $i{\cal M}=\overline{u_{\beta}}\,\Gamma_{\mu}^{\beta\alpha}u_{\alpha}\,\varepsilon^{\mu*}$, where $u_{\alpha}$ and $u_{\beta}$ are momentum-space spinors, whereas $\varepsilon^{\mu}$ is the polarization vector associated to the electromagnetic field. The vertex function $\Gamma_{\mu}^{\beta\alpha}$ is expressed as

$$\Gamma_{\mu}^{\beta\alpha}=\sum_{j=1}^{3}\sum_{\psi=n,N,X}\Bigg{(}\begin{gathered}\vspace{-0.2cm}\includegraphics[width=65.44142pt]{diag1}\end{gathered}+\begin{gathered}\vspace{-0.2cm}\includegraphics[width=65.44142pt]{diag2}\end{gathered}+\begin{gathered}\vspace{-0.2cm}\includegraphics[width=65.44142pt]{diag3}\end{gathered}\Bigg{)}.$$

(45)

The presence of loop-momentum integrals opens the possibility of having ultraviolet-divergent contributions. About this, note that the superficial degrees of divergence of the diagrams shown in Eq. (45) are 0, 1, and 1, respectively, so these pieces might bear ultraviolet divergences growing as fast as linearly. To deal with this, a regularization method must be implemented. Among the different options, we followed the dimensional-regularization approach BoGi ; tHVe . This regularization method has the advantage of preserving gauge symmetry, ensured by fulfillment of Ward identities Ward , and is also well suited for its implementation through software tools. In the dimensional-regularization method, the dimension of spacetime is assumed to be $D$, with 1 time-like dimension and $D-1$ space-like dimensions. Then, loop integrals are modified as $\int\frac{d^{4}k}{(2\pi)^{4}}F(k)\to\mu_{\rm R}^{4-D}\int\frac{d^{D}k}{(2\pi)^{D}}F(k)$, where $\mu_{\rm R}$ is the renormalization scale, which has units of mass and whose task is to preserve the units of loop integrals, whereas $F(k)$ represents some function of the loop 4-momentum $k$. An analytic continuation of the $D$-dimensional loop integrals is defined by assuming the dimension $D$ to be a complex quantity, with $D\to 4$.

To perform the analytic calculation of the amplitudes corresponding to the contributing diagrams of Eq. (45), use has been made of the software packages FeynCalc SMO1 ; SMO2 ; MBD and Package-X Patel , implemented through Mathematica, by Wolfram. With these software tools, calculations have been carried out by following the tensor-reduction method PaVe ; DeSt . Furthermore, the momenta conventions displayed in Fig. 1

have been used to perform the calculation, where $q=p_{\alpha}-p_{\beta}$, due to 4-momentum conservation. The expression we found for the vertex function $\Gamma_{\mu}^{\beta\alpha}$ has the gauge-invariant and Lorentz-covariant structure

$$\Gamma_{\mu}^{\beta\alpha}=\mu^{\beta\alpha}\,\sigma_{\mu\nu}q^{\nu}+d^{\beta\alpha}\,\sigma_{\mu\nu}q^{\nu}\gamma_{5},$$

(46)

where $\mu^{\beta\alpha}$ and $d^{\beta\alpha}$ are the transition magnetic form factor and the transition electric form factor, respectively NPR ; BGS . The corresponding branching ratio is then given by

$${\rm Br}(l_{\alpha}\to\,l_{\beta}\gamma)=\frac{\big{(}m_{\alpha}^{2}-m_{\beta}^{2}\big{)}^{3}}{8\pi m_{\alpha}^{3}\Gamma_{\rm tot.}}\Big{(}|\mu^{\beta\alpha}|^{2}+|d^{\beta\alpha}|^{2}\Big{)},$$

(47)

where $m_{\alpha}$ and $m_{\beta}$ respectively denote the masses of the charged-leptons $l_{\alpha}$ and $l_{\beta}$, whereas $\Gamma_{\rm tot.}$ is the total decay rate for $l_{\alpha}$.

The transition magnetic and electric form factors, $\mu^{\beta\alpha}$ and $d^{\beta\alpha}$, can be written as

$$\mu^{\beta\alpha}=\sum_{j=1}^{3}{\cal B}_{\beta n_{j}}{\cal B}_{\alpha n_{j}}^{*}\mu^{\beta\alpha}_{n_{j}}+\sum_{j=1}^{6}{\cal B}_{\beta f_{j}}{\cal B}_{\alpha f_{j}}^{*}\mu^{\beta\alpha}_{f_{j}},$$

(48)

$$d^{\beta\alpha}=\sum_{j=1}^{3}{\cal B}_{\beta n_{j}}{\cal B}_{\alpha n_{j}}^{*}d^{\beta\alpha}_{n_{j}}+\sum_{j=1}^{6}{\cal B}_{\beta f_{j}}{\cal B}_{\alpha f_{j}}^{*}d^{\beta\alpha}_{f_{j}}.$$

(49)

The $\mu^{\beta\alpha}_{n_{j}}$ and $d^{\beta\alpha}_{n_{j}}$ factors, corresponding to the sum of contributing Feynman diagrams which exclusively involve the $j$-th virtual light neutrino $n_{j}$, are functions of the mass $m_{n_{j}}$, whereas $\mu^{\beta\alpha}_{f_{j}}$ and $d^{\beta\alpha}_{f_{j}}$, associated to the sum of diagrams in which only the $j$-th virtual heavy-neutral lepton $f_{j}$ participates, is either $m_{N_{j}}$ or $m_{X_{j}}$ dependent. All the aforementioned form-factor contributions depend on the masses $m_{\alpha}$ and $m_{\beta}$, of the external leptons, while they also depend on the $W$-boson mass, $m_{W}$. Moreover, since use has been made of the tensor-reduction method, the resulting expressions for all these contributing form factors are given in terms of Passarino-Veltman scalar functions PaVe . More precisely, 2-point and 3-point Passarino-Veltman scalar functions, respectively defined as

$$B_{0}(p_{1}^{2},m_{0}^{2},m_{1}^{2})=\frac{(2\pi\mu_{\rm R})^{4-D}}{i\pi^{2}}\int d^{D}k\frac{1}{\big{(}k^{2}-m_{0}^{2}\big{)}\big{(}(k+p_{1})^{2}-m_{1}^{2}\big{)}},$$

(50)

$$C_{0}(p_{1}^{2},(p_{1}-p_{2})^{2},p_{2}^{2},m_{0}^{2},m_{1}^{2},m_{2}^{2})=\frac{(2\pi\mu_{\rm R})^{4-D}}{i\pi^{2}}\int d^{D}k\frac{1}{\big{(}k^{2}-m_{0}^{2}\big{)}\big{(}(k+p_{1})^{2}-m_{1}^{2}\big{)}\big{(}(k+p_{2})^{2}-m_{2}^{2}\big{)}},$$

(51)

are involved. The scalar functions $B_{0}$ are the sources of ultraviolet divergences, whereas 3-point functions $C_{0}$ are finite in this sense. A latent drawback of our gauge choice, regarding the ultraviolet behavior of the amplitude $\Gamma_{\mu}^{\beta\alpha}$, is that the growth of ultraviolet divergences might be worsened by the unitary gauge because, by this election, the gauge-boson propagators increase the superficial degree of divergence of the diagrams and, thus, has the potential of complicating the elimination of ultraviolet divergences. However, let us point out that any factor $\mu^{\beta\alpha}_{n_{j}}$, $\mu^{\beta\alpha}_{f_{j}}$, $d^{\beta\alpha}_{n_{j}}$, or $d^{\beta\alpha}_{f_{j}}$ has the following structure: $\sum_{k}B_{0}^{(k)}f^{(k)}+\sum_{n}C_{0}^{(n)}g^{(n)}+h$, where $f^{(k)}$ is a mass-dependent function accompanied by some 2-point scalar function, $B_{0}^{(k)}$, whereas $g^{(n)}$ is another function depending on masses, which appears multiplied by a 3-point scalar function, here referred to as $C_{0}^{(n)}$. The sums $\sum_{k}$ and $\sum_{n}$ run over all the 2- and 3-point Passarino-Veltman scalar functions featured by the partial contribution under consideration. Finally, $h$ is another mass-dependent function, which is not multiplied by any loop scalar function. We have been able to verify that the first term, $\sum_{k}B_{0}^{(k)}f^{(k)}$, is written as a sum made exclusively of terms of the form $\big{(}B_{0}^{(r)}-B_{0}^{(s)}\big{)}w^{(r,s)}$, with $w^{(r,s)}$ some combination of the aforementioned mass-dependent functions $f^{(k)}$. In other words, any partial form-factor contribution $\mu^{\beta\alpha}_{n_{j}}$, $\mu^{\beta\alpha}_{f_{j}}$, $d^{\beta\alpha}_{n_{j}}$, or $d^{\beta\alpha}_{f_{j}}$ can be written in such a way that the presence of 2-point functions exclusively occurs as differences among pairs of them. Keep in mind that all the $B_{0}$ functions, no matter what their arguments are, share the same divergent part, that is, $B_{0}^{(k)}=\Delta_{\rm div.}+\eta^{(k)}_{\rm fin.}$ for all $k$, where $\Delta_{\rm div.}$ gathers all the divergent contribution, whereas $\eta^{(k)}_{\rm fin.}$ represents the non-divergent part, which is determined by the specific arguments of the 2-point function under consideration. Then, differences $B_{0}^{(r)}-B_{0}^{(s)}$ are ultraviolet finite, thus implying that $\mu^{\beta\alpha}_{n_{j}}$, $\mu^{\beta\alpha}_{f_{j}}$, $d^{\beta\alpha}_{n_{j}}$, or $d^{\beta\alpha}_{f_{j}}$ are free of ultraviolet divergences.

While the matrix ${\cal B}$ is not unitary, Eq. (34) allows for a sort of Glashow-Iliopoulos-Maiani mechanism to operate GIM . An adequate implementation of this mechanism, during numerical evaluation, is imperative in order to avoid inaccurately large contributions. By usage of Eq. (34), with the flavor-change assumption $\alpha\neq\beta$, the transition electromagnetic moments $\mu^{\beta\alpha}$ and $d^{\beta\alpha}$ can be conveniently written as

	$\displaystyle\mu^{\beta\alpha}=\sum_{j=1}^{3}{\cal B}_{\beta n_{j}}{\cal B}_{\alpha n_{j}}^{*}\big{(}\mu_{n_{j}}^{\beta\alpha}-\mu_{f_{6}}^{\beta\alpha}\big{)}$
	$\displaystyle\hskip 22.76228pt+\sum_{j=1}^{5}{\cal B}_{\beta f_{j}}{\cal B}_{\alpha f_{j}}^{*}\big{(}\mu_{f_{j}}^{\beta\alpha}-\mu_{f_{6}}^{\beta\alpha}\big{)},$		(52)

	$\displaystyle d^{\beta\alpha}=\sum_{j=1}^{3}{\cal B}_{\beta n_{j}}{\cal B}_{\alpha n_{j}}^{*}\big{(}d_{n_{j}}^{\beta\alpha}-d_{f_{6}}^{\beta\alpha}\big{)}$
	$\displaystyle\hskip 22.76228pt+\sum_{j=1}^{5}{\cal B}_{\beta f_{j}}{\cal B}_{\alpha f_{j}}^{*}\big{(}d_{f_{j}}^{\beta\alpha}-d_{f_{6}}^{\beta\alpha}\big{)}.$		(53)

Now notice that all contributions $\mu^{\beta\alpha}_{n_{1}}$, $\mu^{\beta\alpha}_{n_{2}}$, $\mu^{\beta\alpha}_{n_{3}}$, $\mu^{\beta\alpha}_{f_{1}}$, $\mu^{\beta\alpha}_{f_{2}}$, $\mu^{\beta\alpha}_{f_{3}}$, $\mu^{\beta\alpha}_{f_{4}}$, $\mu^{\beta\alpha}_{f_{5}}$, $\mu^{\beta\alpha}_{f_{6}}$ come from Feynman diagrams sharing the very same structure, only distinguished among each other by the virtual neutral lepton flowing through the loop, which can be appreciated from the diagrammatic expression displayed in Eq. (45). For this reason, these partial contributions to $\mu^{\beta\alpha}$ differ of each other only by their neutral-lepton-mass dependence. For instance, if the change $m_{n_{3}}\to m_{f_{5}}$ is implemented in $\mu_{n_{3}}^{\beta\alpha}$, the resulting expression is the one for $\mu_{f_{5}}^{\beta\alpha}$. The same argumentation applies for the electric-dipole partial contributions $d^{\beta\alpha}_{n_{1}}$, $d^{\beta\alpha}_{n_{2}}$, $d^{\beta\alpha}_{n_{3}}$, $d^{\beta\alpha}_{f_{1}}$, $d^{\beta\alpha}_{f_{2}}$, $d^{\beta\alpha}_{f_{3}}$, $d^{\beta\alpha}_{f_{4}}$, $d^{\beta\alpha}_{f_{5}}$, $d^{\beta\alpha}_{f_{6}}$. Then note that terms which do not depend on heavy-neutral-lepton mass vanish from the differences $\big{(}\mu_{n_{j}}^{\beta\alpha}-\mu_{f_{6}}^{\beta\alpha}\big{)}$, $\big{(}\mu_{f_{j}}^{\beta\alpha}-\mu_{f_{6}}^{\beta\alpha}\big{)}$, $\big{(}d_{n_{j}}^{\beta\alpha}-d_{f_{6}}^{\beta\alpha}\big{)}$, and $\big{(}d_{f_{j}}^{\beta\alpha}-d_{f_{6}}^{\beta\alpha}\big{)}$, in Eqs. (52) and (53). Moreover, notice that further cancellations from these differences take place, thus yielding a delicate balance in which a fine suppression of contributions happens.