Tri-bimaximal-Cabibbo mixing: flavour violations in the charged lepton sector

The well understood structure of Upmns\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{pmns}$$\end{document} matrix mandates a Cabibbo mixing matrix in the first two generations of the charged lepton sector if we assume Tri-bimaximal mixing in the neutrino sector. This ansatz, called Tri-bimaximal-Cabibbo mixing, is ruled out immediately by the experiments searching for charged lepton flavour violating currents. In this article, we aim to show that the resurrection of the theoretically well motivated Tri-bimaximal mixing scenario comes naturally within Minimal Flavour Violation hypothesis in the lepton sector. We analyse the flavour violating currents μ→eee\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \rightarrow e e e$$\end{document}, μTi→eTi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu Ti \rightarrow e Ti$$\end{document}, μ→eγ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \rightarrow e \gamma $$\end{document}, π0→e+μ-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi ^0\rightarrow e^+ \mu ^{-}$$\end{document} and KL→μ+e-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_L \rightarrow \mu ^+ e^-$$\end{document} in this scenario and show that the New Physics that generates mixing among the charged lepton could lie within the reach of hadron colliders. In the minimal field content scenario, though the most stringent constrain on New Physics is ≳O(10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gtrsim {\mathcal {O}}(10$$\end{document} TeV) for maximal coupling, considering more natural couplings relaxes it to ≳O(4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gtrsim {\mathcal {O}}(4$$\end{document} TeV). On the other hand, New Physics with the extended field content is even more strongly constrained to ≳O(75\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gtrsim {\mathcal {O}}(75$$\end{document} TeV) for maximal coupling, while it gets relaxed to ≳O(31\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gtrsim {\mathcal {O}}(31$$\end{document} TeV) for natural scenario.


I. INTRODUCTION
The discovery of neutrino oscillation [1] has been of fundamental importance in understanding the flavour mixing in the neutral lepton sector.One such mixing matrix, motivated from discrete family symmetry like the A 4 , is the Tri-bimaximal mixing (U T B ) [2].But a non-zero measurement of the reactor angle by RENO [3] and Daya Bay [4] experiments rule out this hypothesis.Nevertheless, since it is theoretically well motivated and the rest of the angles predicted match the experimental results well enough, various variants based on the Tri-bimaximal mixing have been postulated [5].
On the other hand, we show that the Minimal Flavour Violation (MFV) hypothesis [11,12], with a choice of basis including the Cabibbo mixing among the charged leptons, can protect TBC mixing ansatz from these strong bounds.Under this hypothesis, all the operators in the lepton sector are assumed to transform under the flavour symmetry SU (3) L × SU (3) e .We compute the limit on the lepton flavour violating New Physics (NP) scale, using dim-6 operators, for observables like µ → eee, µT i → eT i, µ → eγ, π 0 → e + µ − and K L → µ + e − .We then extend the analysis to include right-handed neutrinos.Assuming modest values of couplings, we show that the NP, leading to the flavour violation in the former case, could lie within the LHC regime ∼ O(4 TeV).Whereas, in the extended field setup, the New Physics is strongly constrained to be much heavier ∼ O(31 TeV).
The article is organised as follows.In section:II we give a brief review of TBC mixing ansatz and compute the flavour violating leptonic decays of K L , π 0 and µ → eγ show that the current experimental bounds are violated, ruling out vanila scenario.We then discuss the TBC ansatz in MFV with minimal and extended field content in section:III.The basis choice with the ansatz is then analysed in the decays of µ → eee, π → eµ, µ → eγ, K L → µe and µN → eN .In section:IV we give our remarks and results.

II. TBC MIXING INDUCED LFV
In general the mixing in the lepton sector (U pmns ) can be written as, U pmns = U L e † U L ν .Where U L e is the mixing in the charged lepton sector and U L ν is the mixing in the neutrino sector.If we assume the TBC ansatz then, c e the TBC ansatz.Since TBC ansatz satisfies the experimentally observed angles, it motivates the study of mixing in the charged lepton sector.
The mixing in the first two generations of charged leptons are, in general, strongly constrained from charge flavour violation decays of mesons and muon.To illustrate this, consider a scalar operator in the mass basis, given by, where, Here, Λ is the energy scale of the NP.In the above currents, ℓ represents a leptonic field and q represents a quark field.C Q and C L are 3 × 3 matrix, with m, n representing quark generation and i, j representing lepton generation.
Rotating to the flavour basis, where ℓ ′ and q ′ represent the fermionic field in the flavour basis, the quark and lepton bilinears becomes, In this article, since we aim to study the effects of mixing among charged leptons, we now focus only on the lepton sector.Using Eq. ( 4) the Wilson Coefficient in the flavour basis C ′ L is related to the mass basis C L by, Scalar and dipole operators where the effects of this mixing are prominent are given in Table I.The relation between Wilson coefficients in mass basis and flavour basis is, are best studied using flavour violating decays like π 0 → e + µ − , K L → µ + e − , µ → eγ and muon conversion (µN → eN ).In fact, if we assume that U L e is the source of LFV, current experimental bounds from K L → µ + e − , π 0 → e + µ − and µ → eγ rules out the TBC mixing.
We consider only scalar and dipole operator here as they are more constraining than the vector and tensor operator.The reason is that, in lepton flavour violating decays of mesons, the Chiral PT matching of vector operators brings in momentum dependence (∝ p π 0 µ F 0 ), while the scalar operators are proportional to B 0 F 0 , where p π 0 µ , F 0 and B 0 are the pion momentum, pion decay constant and low energy constant given by B 0 = m 2 π 0 mu+m d = 2.667GeV respectively.A detailed discussion can be found in [13] where the authors show how relaxed the limits on vector operator are in comparison to the scalar operators for π 0 → µ + e − decay.
The Low Energy Effective Field theory operators in Table I need to be matched with operators in chiral perturbation theory to get meson decays.We use the matching and decay rate expressions used in Ref. [13][14][15].Operators that can contribute to K L → µ + µ − at tree-level are, The decay width of K L → µ + µ − in terms of the Wilson coefficient is then given by, Here, α where, ϵ is the CP violation parameter in the kaon oscillation with the value |ϵ| = 2.228 × 10 −3 [16].Values of other constants which are used is given in Table II.In Eq. ( 9) p µ is given by, Table II.Various input values in GeV.
By introducing LFV using U L e , the Wilson coefficients due to mixing in charged lepton sector become, Thus the lepton flavour violating decay of K L → µ + e − , generated by the off-diagonal component of the charged lepton mixing matrix, (U L e ) eµ , is given by, We take the ratio of flavour conserving Γ(K L → µ + µ − ) and the flavour violating decays Γ(K L → µ + e − ) to cancel out the hadronic factors and is given as, In the above equation |(U L e ) eµ | 2 ≈ 0.05.Using the experimental bounds on BR(K L → µ + µ − ) exp = (6.84±0.11)×10−9 [16] and BR(K L → µ + e − ) exp < 4.7 × 10 −12 [7] we get, This experimental result is in contradiction with the TBC mixing induced ratio given in Eq. ( 13), thus ruling out the TBC ansatz for operators containing down sector quarks.For completeness, we will also discuss the flavour violating pion decay.
The operators that contributes to the flavour diagonal pion decay π 0 → e + e − are, With these operators, the decay rate of π 0 → e + e − can be derived as, Introducing LFV through U L e , like in the case with kaons, here, the flavour off-diagonal Wilson coefficients become, Using this relation, the flavour violating decay rate of π 0 → e + µ − can be computed from Eq.( 16) as, Taking On the other hand, experimentally observed values for BR(π 0 → e + e − ) exp = (6.46 ± 0.33) × 10 −8 [16] and BR( Though this is only slightly below the ratio predicted by TBC ansatz, future experiments with higher sensitivities will probe it better.

C. µ → eγ
Before computing the µ → eγ, lets first look at the magnetic and electric dipole moments of muon.The relevant Lagrangian term with MDM a ℓ and EDM d ℓ is given by, Comparing this with the tensor operator given in Table I where τ µ = 2.19×10 −6 s is the mean life time of muon.Computing the µ → eγ using the Wilson Coefficient previously obtained, we get The above value is much higher than the current upper bound BR(µ → eγ) < 4.2 × 10 −13 [10].Thus, the mixing matrix U L e , as given in Eq. ( 1), being the only source of charged lepton flavour violation is strongly disfavoured by the decays K L → µ + e − , π 0 → e + µ − and µ → eγ and rules out the Tri-bimaximal-Cabibbo mixing scenario in its original form.Whereas, we here show that Minimal Flavour Violation hypothesis become a natural framework in which they can exist.

III. MINIMAL FLAVOUR VIOLATION WITH TBC MIXING
The Minimal Flavour Violation (MFV) hypothesis assumes that the Standard Model (SM) Yukawa couplings are the only source of flavour symmetry breaking [11,12].This means all the higher dimensional operators should be constructed out of the SM Yukawa couplings, satisfying the flavour symmetry The Yukawa couplings are considered as non-dynamical fields (spurions) which transform under the flavour symmetry Higher dimensional operators are constructed using these Yukawa couplings satisfying the flavour symmetry G F .For the purpose of this article, we will assume MFV in the lepton sector and not in the quark sector.
There are two different field contents possible for MFV hypothesis.First, with only SM fields called the minimal field content scenario and including the right-handed neutrinos called extended field content scenario [12,19].

A. Minimal Field Content
In the case of minimal field content, mass terms in the lepton sector are, where, Λ LN is the scale of the lepton number violation and v = 174 GeV is the vacuum expectation value of the Higgs field.The leptonic field transform under G LF as: In the above expression L L and e R represent SU (2) doublet and singlet leptonic field.In order to keep the Lagrangian invariant under G LF , Y e and g ν transform as: Assuming TBC mixing ansatz, the basis for MFV could be chosen as, where D e = 1 v diag(m e , m µ , m τ ) and m ν = diag(m ν1 , m ν2 , m ν3 ).Since this is different from the usual MFV where there is no mixing in the charged lepton sector, we call this scenario as Modified Minimal Flavour Violation (MMFV).A spurion that transforms as (8, 1) under the group G LF can be constructed as ∆

B. Extended Field Content
On the other hand, introducing right-handed neutrino appends flavour symmetry in the lepton sector (G LF ) to G ′ LF : G LF × SU (3) ν R .Now, the mass term in the lepton sector for extended field content scenario becomes, The right-handed neutrino mass term breaks SU (3) ν R symmetry to O(3) ν R and they are assumed to be in their mass basis, that is M ij ν = M ν δ ij .The Lagrangian remains invariant under the flavour symmetry G LF × O(3) ν R if the field and spurions transform as, Generating an effective left-handed Majorana mass matrix by integrating out the right-handed neutrinos, we get, If we take M ν = Λ LN then, g ν = Y T ν Y ν .Using G LF × O(3) ν R symmetry, we rotate the fields such that there is mixing in the charged lepton sector.In this basis, and one can construct ∆ = Y † ν Y ν such that it will transform as (8,1) under G LF .Since U L ν = U T B is real, the CP violation in U pmns arises from the Cabibbo mixing matrix.As a result Y † ν Y ν and Y T ν Y ν can be diagonalized by same unitary matrices.
. Note that ∆ is only proportional to m ν whereas in minimal field content it is proportional to m 2 ν .

C. Analysis
The operators that we consider are listed in Table III.Here we have kept only the dominant operators which are proportional to ∆ and Y e ∆, and have neglected the operators that go as Y e Y † e or higher orders of Y e .These operators in minimal field content scenario and extended field content scenario with fermions in their mass basis are shown in Table IV and V. Since we are interested in the protection offered by MFV hypothesis for charged lepton flavour violating currents induced by TBC mixing, we consider MFV only in the lepton sector and not in the quark sector 1 .
1 As the operators in Table IV and V shows, if we assume Minimal Flavour Violation flavour symmetry, our results do not depend directly on U L e,ν .Instead, they depend on Upmns.On the other hand, lepton number violating processes can be shown to be proportional to (U L e ) T U L ν .Since this is beyond the scope of this work we have not discussed it further in our work.Thus, new physics leading to mixing in the charged lepton sector finds a natural protection within MFV anzats.In addition to the lepton flavour violating currents discussed in [12], we also include µ → eee, K 0 → µe and π 0 → µe.Note that, on matching with new physics, the dipole operator is generated at 1-loop, while the scalar operator (in Table III) generating µ → eee is at tree-level.Thus, on translating the result to dynamical new physics parameters, µ → eee will become important.

Scalar
Dipole Table III.Operators satisfying flavour symmetry [12].QL and LL represents SU (2) doublet quark and lepton.uR, dR and eR represent SU (2) singlet up quark, down quark and charged lepton respectively.H is the SM Higgs field.
MMFV operators in minimal field content scenario Table IV.MMFV operators in minimal field content scenario.
MMFV operators in extended field content scenario.
MMFV operators in extended field content scenario..
The effective Lagrangian that we consider in our analysis is of the form, where Λ LF V is the scale of NP that generates LFV.In our analysis, we take the lepton number breaking scale as Λ LN = 10 13 GeV2 and m ν = diag 0, ∆m 2 12 , ∆m 2 12 + ∆m 2 23 for simplicity, with ∆m 2 12 = 7.53 × 10 −5 eV 2 and ∆m 2 23 = 2.43 × 10 −3 eV 2 [16].The mass of neutrinos is considered to be in normal hierarchy.There is no significant difference in the results if we consider inverted hierarchy.Below we discuss the limits on Λ LF V from various lepton flavour violating decays in both minimal field content scenario and extended field content scenario keeping the quark sector coupling (λ's) arbitrary.
µ − → e − e + e − : The operators that contribute to µ − → e − e + e − are: The BR(µ − → e − e + e − ) measured by SINDRUM collaboration gives an upper limit of < 1 × 10 −12 [9].Branching ratio of µ − → e − e + e − in case of a scalar operator [20] becomes, BR(µ − → e − e + e − ) = 1 12 Branching ratio in case of vector operator [20] is given by, BR(µ − → e − e + e − ) = 2 3 In the case of scalar operators, the constrain on the lepton flavour violation scale (Λ LF V ) is very weak due to an additional electron Yukawa coupling as given in Table .III. Whereas, vector operators are not suppressed and Fig. 1 shows the limits on Λ LF V from BR(µ − → e − e + e − ) exp for different values of λ V 2 11 in both minimal and extended field content scenarios.Λ LF V is more constrained in the case of extended field content scenario since ∆ is proportional to the mass of the neutrino m ν whereas, in minimal field scenario, it is proportional to the square of neutrino mass (m 2 ν ).
The branching ratio of π 0 → e + µ − in case of scalar operator, given in Eq.( 18), becomes, where τ π 0 = 8.43 × 10 −17 s is the mean life time of pion.The branching ratio in terms of the Wilson coefficients of vector operator [13] is given by, We find that Λ LF V is weakly constrained by BR(π 0 → e + µ − ) exp for both scalar operator and vector operator in minimal and extended field content scenarios.
The operators that contribute to K L → µ + e − are: We assume that λ The branching ratio in case of scalar operator, given in Eq.( 12), now becomes, where τ K L = 5.116 × 10 −8 s is the mean life time of K L .And, the branching ratio in case of vector operator [13] becomes, The limit on Λ LF V from BR(K L → µ + e − ) exp in vector operator scenario for different values of λ is shown in Fig. 2. In the case of scalar operator Λ LF V is weakly constrained by the BR(K L → µ + e − ) exp .µN → eN : The operators that contribute to µN → eN are:  In the case of a vector operator Here we have assumed λ V 1 11 = λ V 4 11 = λ V 3 11 for simplicity.The constants that have used to compute the above expressions are listed in Table .VI. Table VI.Various constants used in Eq. ( 45) and (46) [21].Fig. 3 shows the limit on Λ LF V from BR(µT i → eT i) for different values of λ S2 11 /λ V 1 11 in scalar and vector operator , we get a ℓ = 2m ℓ Re(C T LR ℓℓ ) and d ℓ = Im C T LR ℓℓ e.Now, using the experimental results for magnetic and electric dipole moments of muon, ∆a µ = a ℓ − a SM = (25.1 ± 5.9) × 10 −10 [17] and d µ < 1.9 × 10 −19 e cm [18], these Wilson coefficients gets constrained as, Re(C T LR µµ ) = 1.19 × 10 −7 GeV −1 and Im C T LR µµ < 0.96 × 10 −5 GeV −1 .Introducing the mixing matrix U L e , the flavour violating Wilson coefficient becomes C T LR eµ = C T LR µµ (U L e ) † eµ .Thus the contribution to µ → eγ, is given by,

( a )Figure 1 .
Figure 1.The plots show limits on ΛLF V from BR(µ − → e − e + e − )exp for different values of λ V 2 11 in both minimal and extended field content scenario.The region below the curve is allowed by the experiments

( a )
Scalar operator minimal field content scenario (b) Vector operator minimal field content scenario (c) Scalar operator extended field content scenario (d) Vector operator extended field content scenario

Figure 3 . 2 √ 2 g< 1 .
Figure 3.The plots show limits on ΛLF V from BR(µT i → eT i) for different values of λ S2 11 /λ V 1 11 in scalar and vector operators in minimal and extended field content scenarios.The region below the curve is allowed by the experiments