Lepton flavor violation

4.5 Non-unitary effect

4.5.2 Lepton flavor violation

In view of the presence of this non-unitarity effect, the neutrino states (ν_αL with α = e, µ, τ) appearing in the SM charged current interaction Lagrangian now can be written as,

ν_αL= [(1−C₁)U_ν]αi ν_i+ [K]αj N_j, (4.54) where the matrix W3×6 (see Eq. (4.48)) is conventionally denoted by K. ν_i=1,2,3 and Nj=4,5,...9 are the light and heavy neutrino mass eigenstates respectively. Then in a basis where charged leptons are diagonal (as in our case), the charged current interactions have contributions involving three light neutrinosνi and six heavy neutrinos Nj as

−LCC = g

√2

¯lαγ^µ{[(1−C1)Uν]αi νi+ [K]αj Nj}W_µ⁻+ h.c.. (4.55) These nine neutrino states can therefore mediate lepton flavor violating decays likelα → l_βγ in one loop (e.g µ−→eγ). Resulting branching ratio for such processes ( in the limit m_β →0) now can be written as [217, 218, 220–226],

BR(L_α→L_βγ)' α³_Wsin²θ_Wm⁵_l

α

256π²m⁴_WΓlα

3

X

j=1

[(1−C₁)U_ν]^∗_αj[(1−C₁)U_ν]_βkI_γ m²_νj m²_W

!

+

9

X

l=4

K^∗αlKβlI_γ m²_N

l

m²_W

!

2

, (4.56)

where

I_γ(x) = 10−43x+ 78x²−49x³+ 18x³lnx+ 4x⁴

12(1−x)⁴ , with x= m²_ν,N

m²_W . (4.57) Here α_W = g²/4π, with g as the weak coupling, θ_W is electroweak mixing angle, m_W is W^± boson mass and Γ_lα is the total decay width of the decaying charged lepton lα.

120 non-unitarity Current upper bound for the branching ratio of the LFV decays are [4] (at 90% CL)

BR(µ−→eγ) < 5.7×10⁻¹³, (4.58)

BR(τ −→eγ) < 3.3×10⁻⁸, (4.59)

BR(τ −→µγ) < 4.4×10⁻⁸. (4.60)

Another important lepton flavor violating decay µ→ eee is also worthy to mention and details of computation of branching ratio calculation can be found in [217, 218]. Current upper limit for this decay is BR(µ−→eee)<1.0×10⁻¹² (90% CL) [4].

Since the flavor structure of the neutrino mass matrix is already fixed in our present scenario (from the A₄ and additional symmetry consideration), it would provide some concrete understanding for the LFV processes in this inverse seesaw model. Both the W3×3 = (1−C₁)U_ν and W3×6=K matrices play the instrumental role here. Remember that, Uν is the diagonalizing matrix for the light neutrinos, defined by Uν =UT BU1Um

as discussed in section 4.3. Hence this can be obtained in terms of α, β, k, φ_ba and φ_da. The non-unitary parmeterC1 is required to satisfy,C1=λv²/2v_f² <8×10⁻⁴ as discussed earlier. Therefore we can completely evaluate W3×3 = (1−C1)Uν, once a specific value of C₁ is chosen.

On the other hand the rectangular matrixK is approximately given by [215, 216]

K '(−F µM⁻¹, F)Uh, (4.61)

where Uh is the diagonalizing matrix of mνheavy given in Eq. (4.46). As previously mentioned,F =m_DM⁻¹ in our scenario is proportional to identity matrix of order 3×3, i.e. F = _y^y1v

2vfI3×3=f₁I3×3. Hence, the matrix K turns out to be

K = (−f₁µM⁻¹, f₁I3×3)U_h. (4.62) Using Eqs. (4.8) and (4.9), we find

µM⁻¹= a y2v_f









1−²₃αe^{iφba 1}₃αe^iφba+βe^{iφda 1}₃αe^iφba

1

3αe^iφba 1 +¹₃αe^iφba −²₃αe^iφba+βe^iφda

1

3αe^iφba+βe^iφda −²₃αe^iφba 1 +¹₃αe^iφba.









, (4.63)

where we have used the explicit flavor structure ofµ and M.

We now proceed to find out the form ofU_h, the diagonalizing matrix ofmνheavy. Note that the m_νheavy matrix can first be block diagonalized by V₀ as

m⁰_νheavy = (V0)^Tmν_heavyV0' −M+µ/2 0

0 M +µ/2

!

, (4.64)

where V₀ is given by (in our scenario bothµand M are symmetric matrices)

V0 ' 1

√2

I+^µM₄⁻¹ I−^µM₄⁻¹

−I+^µM₄⁻¹ I+^µM₄⁻¹

!

. (4.65)

Here we have neglected the terms involving higher orders inµM⁻¹ as expected in inverse seesaw scenario in general. Now the upper (−M+µ/2) and lower (M+µ/2) block matrices of m⁰_ν

heavy carry the form of µ matrix itself (or m_ν). The presence of M just redefines the previous parameter a by a1,2 = a/2∓y2v_f (see Eq. (4.8) and (4.9)). Therefore we can follow the similar prescription for diagonalizing these blocks as we did in case of m_ν diagonalization. Hence m⁰_νheavy can further be diagonalized by V^Tm⁰_νheavyV with

V = U_{T B}.V₁(θ₁, ψ₁) 0 0 U_{T B}.V₂(θ2, ψ₂)

!

, (4.66)

where Vi has the form similar toU1,i.e.

V_i =









cosθ_i 0 sinθ_ie^−iψi

0 1 0

−sinθie^iψi 0 cosθi









. (4.67)

Therefore the diagonalizing matrix of m_νheavy can be written as

U_h ' 1

√2

I+^µM₄⁻¹ I−^µM₄⁻¹

−I+^µM₄⁻¹ I+^µM₄⁻¹

! UT B.V1(θ1, ψ1) 0 0 U_{T B}.V₂(θ₂, ψ₁)

!

. (4.68)

In order to find Uh, we use µM⁻¹ as obtained in Eq. (4.63). Furthermore, we get θ1,2

and ψ_1,2 appearing in V_1,2 as discussed earlier. Hence following the same way as in Eq.

122 non-unitarity (5.17) and Eq. (5.18) we find

tan 2θ_i =

√3β_icosφ_da

(β_icosφ_da−2) cosψ_i+ 2α_isinφ_basinψ_i, (4.69) tanψi = sinφ_da

α_icos(φ_ba−φ_da), (4.70)

with i= 1,2 and we use the definition ofα_i and β_i as, α1,2= |b|

|a| ∓2|y₂|v_f and β1,2 = |d|

|a| ∓2|y₂|v_f. (4.71) For simplicity we discard phase difference between y₂ anda, and setφ_y2a= 0.

Note that from our understanding in Sections 4.3-4.4, we can have estimates over the parameters α, β, k along with the phases φ_ba, φ_da in order to satisfy sinθ₁₃, other mixing angles, r, individual solar and atmospheric splittings, also to be consistent with the upper bound on sum of the light neutrino masses. Specific choice of C₁ enables us to compute magnitude of the flavon vev v_f and hence |a| from Eq. (4.53). With all these values in hand we can finally evaluate parameters θi, ψi, αi andβi appearing inU_h. Here we consider ³ |y₂|= 1. Now following the analytic expressions in Eqs. (4.61-4.63), (4.66-4.71), we can estimateW3×3andKand hence the corresponding contribution to the branching ratio (see Eq. (4.56)). Due to particular flavor structures of the matrices µas well as mD and M, we find W3×3 and K are such that this scenario predicts vanishingly small branching ratio (∼10⁻³⁵) for LFV decays.

In addition, we have performed the evaluation numerically also. In order to evaluate it, we need to diagonalize the entire 9×9 neutrino mass matrix Mν. Since the neutrino mixings are entirely dictated by the flavor structure of µ matrix, we could have find the entireMν numerically with the choices ofα, β, kalong with the phasesφba, φdaas done in cases A, B, C, D and the general case. However to computem_D andM, we need consider to |y1|and |y2|separately (for example, to haveλ= 1, we assume |y1|=|y2|= 1). Then following Eq. (4.7), we can entirely construct the M_ν matrix numerically. Then with the help of Mathematica⁴, we are able to find the diagonalizing matrix W (and hence K matrix also) and have estimate over the LFV decays. It turns out that the numerical

3A common phase φ0 as described in the discussion above Eq. (4.26) in Section 4.3, is irrelevant for neutrino phenomenology and hence we put it at zero. We also set phases off1 anda/y2 to zero.

4We also use Takagi factorization [219] to findW.

estimate coinsides with our analytical evaluation of vanishingly small branching ratios for LFV decays to a good extent.

4.5.3 Neutrinoless double beta decay and contribution of heavy neutri-