Non-vanishing neutrino masses inferred from neutrino oscillation experiments [1–5], provided strong evidence for new physics beyond the standard model (BSM). The extensions of the SM to account for neutrino masses and mixing imply new sources of lepton flavor violation (LFV), which could explain the long-standing discrepancy between the SM prediction for the muon anomalous magnetic moment aμ=(gμ-2)/2 and its experimental measurement.

Recent experimental results indicate a possible 4.2σ difference between the measured value of the anomalous magnetic moments of muons aμ and the SM expectations [6–9], namely δaμ=aμexp-aμSM=(2.51±0.59)×10-9.(1)

We consider the explanation of the aμ anomaly in the left-right (LR) model with inverse seesaw mechanism (LRIS) to generate light neutrino masses and mixing at low energy scale. The salient feature of this class of models is the large neutrino Yukawa couplings, which allow for significant nonuniversal leptonic contributions to the aμ anomaly via diagrams mediated by charged Higgs bosons and right-handed neutrinos (RHNs). As constraints, we impose the experimental limits of the lepton flavor violation μ→eγ, μ-e conversion, and the electron anomalous magnetic moment [10–13].

The LR model is among the most natural extensions of the SM, which is motivated by grand unified theories (GUTs) and accounts for measured neutrino masses as well as providing an elegant explanation for the origin of parity violation in low-energy weak interactions. The LRIS has been analyzed in detail in Ref. [14]. We recall that it has a Higgs sector that consists of one scalar bidoublet and a scalar RH doublet only. In addition, the LRIS contains singlet fermions S1, S2 for adopting the IS mechanism of neutrino masses. Such a TeV scale LR model can be probed in current and future experiments as emphasized in Ref. [14]. Also, it was argued in Ref. [15] that the tension between the SM prediction and the experimental results of the RD and RD* ratios, defined by R({D*,D})=BR(B→{D*,D}τν)BR(B→{D*,D}ℓν), where ℓ=e, μ, can be resolved in this class of LRIS models. In fact there are several new physics scenarios that have been proposed to accommodate δaμ and δae results. See Refs. [16–30].

This paper is organized as follows. In Sec. II we highlight relevant interactions in the LRIS, as the details of the model are given in previous papers [14,15]. Section III is devoted for analyzing new LRIS contributions to aμ, specifically those due to the light and heavy Z,W,Z′,W′ gauge bosons and neutral and charged Higgs bosons with heavy neutrinos. Also, Sec. IV is devoted for the LFV constraints in LRIS. Finally, our conclusions are given in Sec. V.

As previously advocated, we consider the LRIS model [14], which is based on the gauge group GLR=SU(3)C×SU(2)L×SU(2)R×U(1)B-L. This model has the same fermion content as any other conventional left-right model [31–33], but with two extra singlet fermions per family S1 and S2 with opposite B-L charges=-2, =+2, respectively. The fermion singlet S2 is presumed to implement the IS mechanism for neutrino masses, while the other, S1, is added to cancel the U(1)B-L anomaly caused by S2. The LRIS has a simple Higgs sector consisting of one RH doublet χR that breaks down left-right symmetry to the SM gauge symmetry and one bidoublet ϕ that is broken down into two SM Higgs doublets. Furthermore, a Z2 discrete symmetry is assumed, with all particles having even charges except S1, which has an odd charge. This symmetry prevents the mixing mass term MS¯1cS2 from being used to allow for the IS mechanism.

The most general LRIS Yukawa Lagrangian is given by LY=∑i,j=13L¯Li(ϕyijL+ϕ˜y˜ijL)LRj+Q¯Li(ϕyijQ+ϕ˜y˜ijQ)QRj+L¯Riχ˜RyijsS2jc+H.c.,(2)where i, j are family indices, ϕ˜ is the dual bidoublet of the scalar bidoublet ϕ, defined as ϕ˜=τ2ϕ*τ2, and χ˜R is the dual doublet of the scalar doublet χR, given by χ˜R=iτ2χR*. A nonvanishing vacuum expectation value (VEV) of χR, ⟨χR⟩=vR/2 of order TeV breaks the RH electroweak (EW) sector together with B-L, namely SU(2)R×U(1)B-L down to the U(1)Y hepercharge symmetry. In addition, the VEVs of ϕ, ⟨ϕ⟩=diag(k1/2,k2/2), are of order O(100)  GeV, break the SM EW symmetry. The charged leptons acquire their masses via combinations of the lepton coupling Yukawa matrices yL and y˜L and tβ as defined below Eq. (3). Similarly, the quarks acquire their masses via combinations of the quark coupling Yukawa matrices yQ and y˜Q and tβ. The definition of the Yukawa couplings yL,Q and y˜L,Q in terms of physical fermion masses and mixing are recalled below from Ref. [14].

After B-L symmetry breaking and EW symmetry breaking, the following 9×9 neutrino mass matrix is obtained in the basis (νLc,νR,S2) Mν=(0MD0MDT0MR0MRTμs),(3)where the 3×3 matrix MD=v(yLsβ+y˜Lcβ)/2 is the Dirac neutrino mass matrix and the 3×3 matrix MR=ysvR/2. Here, we have assumed that k1=vsβ, k2=vcβ, as constrained from the W boson mass MW≃gLk12+k22/2 in LRIS, where v=246  GeV is the EW VEV, and sx=sinx, cx=cosx, and tx=tanx, henceforth. The neutrino mass matrix Mν can be diagonalized by 9×9 matrix U satisfying UMνUT=Mνdiag=diag(mνℓi,mνhj), yielding the physical light and heavy neutrino states νℓi, νhj, i=1,2,3, j=4,…,9, with the following light and heavy mass eigenvalues mνℓi=MDMR-1μs(MRT)-1MDT,i=1,2,3,(4)mνhj2=MR2±MD2,j=4,…,9.(5)where μs≲O(10-5)  GeV, MR∼O (a few TeV) and ys≲O(10-1). For these values, Eq. (4) shows that the light neutrino masses can be of order eV. The S1 fermions acquire radiative masses mS1∼μs∼O(KeV) and they do not mix with other neutrinos thanks to the Z2 discrete symmetry, so they are stable particles. As probable candidates of warm dark matter, one does not have to worry about the S1 fermions to overclose the Universe. It was demonstrated in [34] that S1 can account for the observed relic abundance and meanwhile it is not constrained by the constraints on sterile neutrino because its mixing with the active neutrinos vanishes identically in LRIS. On the other hand, the S2 fermions also acquire radiative mass terms ∼μs, but due to their large mixing with the RHN, which is ∼MR∼O (a few TeV), they acquire masses mS2∼MR as in Eq. (5).

The inverse relation of Eq. (4) is MD=UPMNSmνℓR(μs)-1MR,(6)where R is an orthogonal matrix and UPMNS is the 3×3 light neutrino mixing matrix [35–37].

In the following section, we will study the process aμ, which is dominated by the charged Higgs boson contributions at the loop level; thus, we provide a brief analysis for charged Higgs bosons masses and interactions based on the detailed previous work of Ref. [14]. In the flavor basis (ϕ1±,ϕ2±,χR±), the charged Higgs bosons symmetric mass matrix takes the form MH±2=α2(vR2sβ2c2βvR2s2β2c2β-vvRsβ.vR2cβ2c2β-vvRcβ..v2c2β),(7)where the scalar potential parameter α=α3-α2 as in [14]. This matrix can be diagonalized by the unitary matrix ZH±=(vc2βv2c2β2+vR2sβ20vRsβv2c2β2+vR2sβ2-12vR2s2β(v2c2β2+vR2sβ2)(v2c2β2+vR2)v2c2β2+vR2sβ2v2c2β2+vR2vvRcβc2β(v2c2β2+vR2sβ2)(v2c2β2+vR2)-vRsβv2c2β2+vR2-vRcβv2c2β2+vR2vc2βv2c2β2+vR2).(8)Thus, the mass eigenstates are given by (ϕ1±,ϕ2±,χR±)T=(ZH±)T(G1±,G2±,H±)T, where ZH±MH±2(ZH±)T=diag(0,0,mH±2). Here G1± and G2± represent the charged massless Goldstone bosons that are eaten by the charged gauge bosons Wμ and Wμ′ to acquire their masses and H± is the physical massive charged Higgs boson. The charged Higgs boson mass is given by mH±2=α2(vR2c2β+v2c2β).(9)We notice from Eq. (9) that α>0 as long as c2β>0 (i.e., tβ<1) and vice versa. Moreover, for vR∼O(10  TeV) and |α|∼O(10-2), the charged Higgs boson mass can be of order hundreds GeV. The physical charged Higgs boson is defined as a linear combination of the flavor basis fields ϕ1±,ϕ2±,χR±, i.e. (corrected from [14]), H±=Z31H±ϕ1±+Z32H±ϕ2±+Z33H±χR±.(10)It is worth noting that for vR≫v, vR∼O(TeV) is enough, the mixing Z33H±≪1 and the charged Higgs mass and combination reduce to the following approximations mH±≃vRα2c2β,(11)H±≃-(sβϕ1±+cβϕ2±).(12)Finally, the charged Higgs boson couplings with fermion families are given by Γu¯idjH±=-(∑a=13Vja*(yaiQ*Z32H±+y˜aiQ*Z31H±)) PL-(∑a=13Vja(yiaQZ31H±+y˜iaQZ32H±)) PR,(13)Γν¯kℓH±=-(∑i=13Uk,i+3*(yℓiL*Z31H±-y˜ℓiL*Z32H±)) PL+(∑i=13(Uki(y˜iℓLZ31H±-yiℓLZ32H±)-Uk,i+6yℓis*Z33H±)) PR,(14)where V is the 3×3 Cabibbo–Kobayashi–Maskawa quark mixing matrix and U is the 9×9 inverse seesaw neutrino mixing matrices defined after Eq. (3). The following parametrization will be used below Γu¯idjH±=CijPL+DijPR,(15)Γν¯kℓH±=ξkℓPL+ζkℓPR.(16)We fix vR∼O(10  TeV) for the extra gauge bosons W′,Z′ experimental limits on their masses and mixing with the corresponding electroweak gauge bosons [14]. Hence, as noted before Eq. (12), Z33H±≪1, and we can omit the third term ∑i=13Uk,i+6yℓis*Z33H± from numerical calculations of the charged Higgs boson couplings with leptons ζkℓ in Eqs. (14) and (16). Moreover, The nonunitairity limits of the 3×3 light neutrino mixing matrix UPMNS [36–41] ensures that for the charged Higgs boson and lepton couplings in Eq. (16) ξkℓ≪1 for light neutrinos (k=1, 2, 3) and ζkℓ≪1 for heavy neutrinos (k=4,…,9). Thus, and according to Eq. (12), the relevant charged Higgs boson couplings with fermions can be approximated to Cij≃∑a=13Vja*(yaiQ*cβ+y˜aiQ*sβ),(17)Dij≃∑a=13Vja(yiaQsβ+y˜iaQcβ),(18)ξkℓ≃∑i=13Uk,i+3*(yℓiL*sβ-y˜ℓiL*cβ),k=4,…,9,(19)ζkℓ≃∑i=13Uki(yiℓLcβ-y˜iℓLsβ),k=1,2,3.(20)It is clearly noticed that for tβ≪1 (tβ≫1) the couplings ξkℓ(ζkℓ) are y˜L(yL) dominant, and hence the couplings ξkℓ and ζkℓ are now uncorrelated. Moreover, if we closely investigate these couplings for ℓ=e, μ, we see that the family components yℓiL, y˜ℓiL can distinguish between the charged Higgs boson couplings to different lepton families. Successfully, this helps in explaining the aμ anomaly and satisfying the LFV results as clarified below. This can be achieved via controlling the entries of ys, μs and the orthogonal matrix R in Eq. (6), where the quark and lepton Yukawa couplings can be written in terms of the fermion masses as follows [14]: yQ=2vc2β(cβVMdV†-sβMu),(21)y˜Q=2vc2β(sβVMdV†-cβMu),(22)yL=2vc2β(cβMlp-sβMD),(23)y˜L=-2vc2β(sβMlp-cβMD),(24)where Mu, Md, Mlp are the quarks and charged leptons diagonal mass matrices and MD is the Dirac neutrino mass matrix defined after Eq. (3) and solved for it in Eq. (6). According to Eqs. (17) to (24), we can write the charged Higgs boson couplings to fermions in terms of the physical fermion masses as follows: Cij≃2vc2β∑a=13Vja*(VMdV†-s2βMu)ai*,(25)Dij≃2vc2β∑a=13Vja(s2βVMdV†-Mu)ia,(26)ξkℓ≃2vc2β∑i=13Uk,i+3*(s2βMlp-MD)ℓi,k=4,…,9,(27)ζkℓ≃2vc2β∑i=13Uki(Mlp-s2βMD)iℓ,k=1,2,3,(28)where the conjugate “*” is omitted from the matrices when they are (taken) real. As noted after Eq. (20), for tβ≪1 (tβ≫1), the couplings ξkℓ (ζkℓ) are MD (Mlp) dominant and uncorrelated, and the family components are discriminant. Similarly, the above discussion applies for the charged Higgs boson couplings with quarks as well.

Finally, we close this section by stating the scalar and pseudoscalar Higgs bosons sectors which were analyzed in detail with their couplings with charged leptons in [14] Γℓℓhi=v2mℓ(Zi1Hy˜ℓℓL+Zi2HyℓℓL),(29)ΓℓℓA=v2mℓ(Z31Ay˜ℓℓL-Z32AyℓℓL),(30)where ZH, ZA are the scalar and pseudoscalar Higgs mixing matrices, respectively [14,42]. More details about the LRIS Higgs and gauge sectors couplings and mixing and their parameters and spectra can be found in our previous work in Ref. [14].

In this section we analyze new contributions from the LRIS to the muon anomalous magnetic moment, aμ, induced by the light and heavy Z,W,Z′,W′ gauge bosons, as well as the neutral scalar and pseudoscalar and charged Higgs bosons h,A,H± shown in Fig. 1. We will also consider the constraints on these contributions imposed by the experimental limits of the electron anomalous magnetic moment, ae, and charged lepton flavor violations, particularly, the μ→eγ decay and the μ-e conversion. In this case, we can write δaμ=aμLRIS, where aμLRIS=aμW+aμW′+aμZ+aμZ′+aμh+aμA+aμH±.(31)The relevant amplitudes are, ignoring the W-W′ and Z-Z′ mixing, given by aℓW=GFℓ∑k=19|Uk,|ℓ||2(103+F2(xWνk)),(32)aℓW′=GFℓ∑k=19|Uk,3+|ℓ||2(103+F2(xW′νk))[1cwxW′W],(33)aℓZ=GFℓ(c4w-5)(13),(34)aℓZ′=GFℓ(c4w′-12c2w′-5)[tw248s2w′2xZ′W],(35)aℓh=GFℓ∑i=13xhiℓ(Γℓℓhi)2(76+log(xhiℓ)),(36)aℓA=GFℓ(-12)xAℓ(ΓℓℓA)2(76+log(xAℓ)),(37)where the lepton family order |ℓ|=1, 2 for ℓ=e, μ. The dimensionless coupling GFℓ=GFmℓ282π2 and the mass ratio parameters xba=ma2mb2,a=νk,W,ℓ,b=W,W′,Z′,hi,A,H±. The neutral gauge bosons mixing angles θw′ and the Weinberg angle θw are sw′=gYgR,sw=egL, where gY is the hypercharge coupling. The Z-Z′ mixing angle θw′ is constrained by tw′≲10-4 [43,44]. Also, the W′ mass is given by mW′=gLvR2+v2/2≳O(4  TeV) [43,44].

The analytical expressions of Eqs. (32)–(37) show that numerical values of the BSM contributions to aμ mediated by W′,Z′,hi,A are negligible due to the suppression of their masses ratios xW′,Z′W and xhi,Aℓ, and thus we exclude their minor contributions. Also, the second summation term ∑k=19|Uk,|ℓ||2F2(xWνk)≃∑k=49|Uk,|ℓ||2F2(xWνk) of Eq. (32), which represents the W-RHN loops contributions of Fig. 1 is typically ∼O(10-2); only ≲0.4% of the first term (103), and thus suppressed. This can be generally understood via the GIM cancellation mechanism [45] due to the unitarity of the full 9×9 neutrino mixing matrix U within the nonunitarity limits of the 3×3 UPMNS light neutrino mixing matrix [36–41]. In light of this, it can be generally concluded that any minimal BSM extension of the SM with RHN with any adopted seesaw mechanism can not account for the measured aμ anomaly and extra degrees of freedom are needed for this [36]. The LRIS with its extra degrees of freedom is a good candidate for such class of BSM models.

Finally, the charged Higgs boson H± contribution to aμ is given by aℓH±=GFℓΓγH±∑k=19(|ζ′kℓ|2F2(xH±νk)+2Re[ζ′kℓξ′kℓ*]F1(xH±νk)),(38)where the charged Higgs boson interaction couplings with leptons ξkℓ,ζkℓ appear in Eq. (16), and ζ′kℓ=vmνkζkℓ and ξ′kℓ=vmℓξkℓ. The charged Higgs boson interaction coupling with photons is ΓγH±=16e(gLU210+gRU310+(gBLU110-gLU210)(Z33H±)2)≃16e(gLU210+gRU310),(39)where the last approximation is for vR≫v where Z33H±≪1, as noted before Eq. (12). The matrix U0 is the neutral gauge bosons mixing matrix and gR is the SU(2)R coupling [14]. The loop functions Fk (k=1, 2) in Eqs. (32), (33), and (38) are given by Fk(y)=yPk(y)(y-1)k+1-6yk+1log(y)(y-1)k+2,k=1,2,(40)P1(y)=3y+3,(41)P2(y)=2y2+5y-1.(42)It is understood that, for y→1, the values of the loop functions Fk (k=1, 2) are given by their limits and F1(1)=1 and F2(1)=12. This happens when some heavy neutrinos are degenerate in mass with the charged Higgs boson as in Fig. 2. Asymptotically, the ratio F2(x)/F1(x) is increasing and bounded below and above, and F2(x),F1(x)→0 for x≪1 such that F2(x)/F1(x)→1/3 for x≪1, and F2(x)/F1(x)→2/3 for x≫1. Accordingly, the two loop functions F1 and F2 remain of the same order for all possible values of the argument x. Typically, the coupling ΓγH±∼0.076. Also, the first contribution term of Eq. (38) ∑k=19|ζ′kℓ|2F2(xH±νk)≃∑k=13|ζ′kℓ|2F2(xH±νk) is ∼O(10-13). For light neutrinos, this term is suppressed by the loop function F2(xH±νℓ), while, for heavy neutrinos, it is suppressed by their squared masses in the denominators of coefficients |ζ′kℓ|2(k=4,…,9). Indeed, the first term represents only 0.02% of the second term ∑k=192Re[ζ′kℓξ′kℓ*]F1(xH±νk)≃∑k=492Re[ζ′kℓξ′kℓ*]F1(xH±νk), which is O(10-9), where this time the second term is enhanced due to the charged lepton masses in the denominators of the coefficients ζ′kℓξ′kℓ*. Thus, the charged Higgs boson contribution to the aμ anomaly Eq. (38) can be approximated to aℓH±≃2GFℓΓγH±∑k=49Re[ζ′kℓξ′kℓ*]F1(xH±νk)≲3ΓγH±8π2mℓ∑k=49ζkℓξkℓmνk,(43)where 3ΓγH±8π2∼3×10-3 and the loop function F1 is increasing and bounded above such that F1(x)→3 for x≫1, and the complex notations “Re, *” are omitted as the couplings are (taken) real.

In the rest of this section, we analyze the parameter space of the LRIS for numerical scan for benchmark points (BPs). In the SM, all particles acquire their masses via the VEV of only one degree of freedom, the Higgs field, and each particle mass depends only on one parameter coupling, its coupling with the Higgs field. This feature almost fixes the SM parameters values, except maybe due to some measurements uncertainties. So, in the SM, couplings are fixed at the EW scale by particles masses. Conversely, in LRIS, there are many sources of VEVs and couplings for particles’ masses. So, in LRIS, VEVs, the Yukawa couplings and scalar potential parameters are in general free parameters (see Ref. [14]), and they can be varied while fermions and scalar masses are kept fixed. Also, in LRIS, the gauge couplings are constrained by the gauge bosons masses at the EW scale and by their renormalization group equations (RGEs) evolution up to GUT scale, especially when we fix vR∼O(10  TeV) for the extra W′,Z′ experimental mixings and masses limits as discussed after Eq. (37).

The neutrinos masses Eqs. (4) and (5) and their mixing matrix U after Eq. (3) are given in terms of MD, MR, and μs (or equivalently yL,y˜L,ys,μs,tβ, and vR). In our numerical analysis, we fix vR∼O(10  TeV). Also, we adopted the normal hierarchy of light neutrino masses mνℓ, as in Table I. The chosen light neutrino masses values satisfy Δm212=7.224×10-5  eV2 and Δm312=2.500×10-3  eV2. They agree with the 1σ ranges of the observed solar and atmospheric mass splittings values Δmsol2=7.420-0.200+0.210×10-5  eV2, and Δmatm2=2.517-0.028+0.026×10-3  eV2 [37]. But, to satisfy the aμ anomaly in the inverted hierarchy scenario of neutrino masses within the imposed LFV constraints, the nonunitarity limits of the UPMNS matrix should be violated. In this case, one has to set relatively large [∼O(10-10)] and thus almost degenerate light neutrino masses. Also, μs is enlarged for the aμ anomaly and other relevant LFV constraints, and it mixes with other entries in the neutrino mass matrix (3), thus violating the nonunitarity limits of UPMNS [37]. We chose to express MD (and hence, in correlation, yL,y˜L) in terms of ys, μs, and tβ as in Eqs. (6), (23), and (24). Accordingly, substituting MD and ys in the heavy neutrino masses Eq. (5) determine them. So, in our numerical analysis below, we fix the normal hierarchy light neutrino masses and the entries of the UPMNS matrix and scan over ys and μs as in Eq. (44) for the neutrino sector [37]. It is also worth mentioning here that the consistency of all numerical calculations of Table I is verified. For example, the MD matrix, when calculated from its main definition after Eq. (3) is found consistent with its values from Eq. (6). Also, the neutrino masses in Table I are consistent with Table II and Eqs. (4), (5), and (45). Finally, the orthogonal matrix R of Eq. (6) was fixed such that its nonvanishing components are R13=R21=R32=1 as given in Table II.

Also, the charged Higgs boson mass Eq. (9) is varied versus tβ and the scalar potential parameter α, and the charged Higgs mixing ZH± Eq. (8) is given in terms of tβ and vR. For the charged Higgs boson mass and mixing, we scan over tβ and α as in Eq. (44). As detailed above, after Eqs. (5), (9), and (20), in our numerical analysis of the neutrino and charged Higgs sectors, we scanned over the following independent parameters’ ranges [with vR∼O(10  TeV)] α∼[0.0050,0.0500],tβ∼[0.01,0.99],(ys)ij∼[0.01,0.50]δij,(μs)ij∼[10-9,10-5]δij  GeV.(44)We checked that all of our BPs are validated to satisfy the usual higgsbounds and higgssignals limits confronted with the latest LEP, Tevatron, and LHC data [46,47]. They provide important tests for compatibility of any BSM model. In our analysis, the LRIS model was first built in the sarah package, then it was passed to spheno for numerical spectrum calculations [48,49]. Specifically, we present one of our BPs in Table I with the corresponding observables in Table III and other parameters in Table II and Eq. (45).

The left (right) panel of Fig. 2 depicts the muon (electron) gμ(e)-2 anomalies δaμ(e) in LRIS, as given in Eqs. (32) and (38), resulting from the BSM contributions of the W-RHN loops and the charged Higgs boson contribution. We choose, without any loss of generality or independence, to show the distribution of δaμ(e) versus the mass ratio xH±ν5=mν52/mH±2 for its moderate and variable values, as it is clear from the masses values BP of Table I, but any other of the independent or dependent parameters or any one of the heavy neutrinos ratios xH±νj(j=4,…,9) would equally work for the same set of data. The green (red) borders indicate the 1σ (2σ) level of accuracy around the average δaμ as in Eq. (1). The electron anomaly δae is guaranteed to be within the allowed experimental uncertainty limits |δae|≲(10-15–10-13) [50,51]. So, all BPs used in Figs. 2 and 3 satisfy the electron ae anomaly limits. Furthermore, Fig. 3 shows that the LVF BR(μ→eγ) in Eq. (48) satisfies the experimental bounds for the same set of parameters values as in Fig. 2.

The 9×9 neutrino mixing matrix U of the BP of Table I, rounded to the O(10-4), with the UPMNS matrix [37], is U=(U3×3U3×6U6×3U6×6)T=(-0.82430.4535-0.338900000.0000-0.00010.54650.4812-0.68530000.00090.00020-0.1468-0.7453-0.64030000-0.11370-0.0004-0.00030.0004-0.7071000.707100-0.0120-0.0604-0.05170-0.7071000.702500.00010.00000.0003000.707100-0.7071-0.0004-0.00030.00040.7071000.7071000.01200.06040.05170-0.707100-0.70250-0.00010.00000.0000000.7071000.7071),(45)where each block matrix is parametrized such that [36,41] U3×6≃[03×3|F]3×6U6×6,(46)U3×3≃(13×3-12FFT)UPMNS,(47)where the nonunitarity limits of the UPMNS is encoded in F=MDMR-1 and the 3×6 extended matrix F=[03×3|F]3×6 is Fij=0 and Fi,j+3=Fij for i, j=1, 2, 3. Finally, U6×6 is the matrix which diagonalizes the {νR,S2} mass matrix.

Now, we turn to the constraints on the charged Higgs boson contributions to LFV rare processes. The LRIS W-RHN and the charged Higgs boson contributions to the BR(μ→eγ) and the μ-e conversion rates Rμ→e are in order. Experiments set upper bounds to these quantities, and the stringent experimental limits on these processes should be regarded as constraints on the charged Higgs boson contribution to aμ [11,13]. The LFV experiments set the upper limit BR(μ→eγ)≲4.2×10-13 with 90% confidence level [52]. In LRIS, the W-RHN and charged Higgs boson mediation for μ→eγ leads to BR(μ→eγ)LRIS=αw3sw2256π2mμΓμ(xWμ)2∑k=19|Uk,1Uk,2*F2(xWνk)+ζ′k,eζ′k,μ*F2(xH±νk)+(ζ′k,eξ′k,μ*+ξ′k,eζ′k,μ*)F1(xH±νk)|2.(48)As discussed after Eq. (37), the first term ∑k=19Uk,1⁢Uk,2*F2(xWνk)≃∑k=13(UPMNST)k,1(UPMNST)k,2F2(xWνk) of Eq. (48) of the W-ν contribution is ≲O(10-29), and thus negligible by the GIM cancellation mechanism [45]. The remaining W-RHN contribution in the first term ∑k=49Uk,1Uk,2*F2(xWνk) vanishes due to contributions from the first two rows of the 3×6 upper-right block matrix in Eq. (45) which gives Uk,1Uk,2*≊0, k=4,…,9. Accordingly, we only constrain the charged Higgs boson contribution. For this, as discussed in the paragraph before Eq. (43), the second term ∑k=19ζ′k,eζ′k,μ*F2(xH±νk)≃∑k=13ζ′k,eζ′k,μ*F2(xH±νk) is ∼O(10-9), and it is only about 0.004% of the third term ∑k=19(ζ′k,eξ′k,μ*+ξ′k,eζ′k,μ*)⁢F1(xH±νk)≃∑k=49(ζ′k,eξ′k,μ*+ξ′k,eζ′k,μ*)F1(xH±νk), which is ∼O(10-5) and need to be constrained. We can approximate Eq. (48) as BR(μ→eγ)LRIS≃αw3sw2256π2mμΓμ(xWμ)2∑k=49|(ζ′k,eξ′k,μ*+ξ′k,eζ′k,μ*)F1(xH±νk)|2,≲9αem256π4mμ5Γμ∑k=491mνk2(ζk,eξk,μmμ+ξk,eζk,μme)2,(49)where the factor 9αem256π4mμ5Γμ∼108 and the loop function F1(x)≲3 for x≫1, as noted after Eq. (43), and again the complex notations are omitted as the couplings are (taken) real.