Constraining $A_4$ Leptonic Flavour Model Parameters at Colliders and Beyond

The observed pattern of mixing in the neutrino sector may be explained by the presence of a non-Abelian, discrete flavour symmetry broken into residual subgroups at low energies. Many flavour models require the presence of Standard Model singlet scalars which can promptly decay to charged leptons in a flavour-violating manner. We constrain the model parameters of a generic $A_4$ leptonic flavour model using a synergy of experimental data including limits from charged lepton flavour conversion, an 8 TeV collider analysis and constraints from the anomalous magnetic moment of the muon. The most powerful constraints derive from the MEG collaborations' limit on Br$\left(\mu\to e\gamma\right)$ and the reinterpretation of an 8 TeV ATLAS search for anomalous productions of multi-leptonic final states. We quantify the exclusionary power of each of these experiments and identify regions where the constraints from collider and MEG experimental data are complementary.


Introduction
Since the discovery of neutrino oscillations by Super-Kamiokande [1], two puzzling aspects of neutrino physics have emerged. First, neutrinos have very small but non-zero masses and second the leptonic mixing or Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix, U , has a strikingly different structure from the quark mixing matrix. One of the most fruitful beyond the Standard Model (SM) ideas applied to the neutrino sector is the introduction of a non-Abelian flavour symmetry to explain the observed structure of the PMNS matrix. These models generally propose a discrete flavour symmetry which is broken spontaneously, leaving the leptonic mass terms invariant under residual symmetries. Through symmetry considerations alone, without the specification of a detailed flavour model, it is possible to reduce the number of degrees of freedom between mixing parameters and thereby predict sum-rules which will be testable at upcoming long (T2HK and DUNE) [2,3] and medium (JUNO) [4] baseline neutrino oscillation experiments.
Such a task can be completed in a model-independent or dependent manner. In the latter case, the breaking of the flavour symmetry is realised by SM singlet scalar fields, also known as flavons, which have non-trivial quantum numbers associated to the non-Abelian flavour group. These flavons acquire vacuum expectation values (VEVs) which spontaneously break the flavour symmetry to its Abelian residual symmetries in the charged lepton and neutrino sector. In general, two flavons are sufficient; however for larger symmetry groups and supersymmetric setups additional flavon multiplets are necessary for model construction [21,22,31,32,38,[43][44][45][46][47][48][49][50][51][52]. Typically, the corrections to the leading order mixing pattern are provided by higher dimensional operators formed between the flavons and charged leptons [23][24][25]. An alternative possibility was proposed in the work of [53], where the cross-coupling between the flavons of the neutrino and charged lepton sector may slightly break the Abelian residual symmetries and thereby provide the needed deviation from exact TBM mixing in the context of an A 4 flavour model.
The rich phenomenology of flavour models has been explored in the quark and lepton sector using both Abelian and non-Abelian flavour symmetries. In the case of Abelian family symmetries, which manifests from the Froggatt-Nielsen (FN) mechanism [5], the collider and flavour violating phenomenology of a single flavon was explored [54]. Although our model and theirs markedly differ, we reach a similar conclusion to their work: limits from MEG can largely exclude the flavour breaking scale of less than ∼ 1 TeV. In addition, there has been work completed on constraining quark flavour model parameter space using collider constraints including Higgs-flavon mixing, electroweak oblique parameters and direct production of the flavon at current and future colliders [55][56][57]. Moreover, the observed flavour violating decay of the Higgs (h → µτ ) was investigated in the context of a FN mechanism [58].
Using non-Abelian discrete symmetries the CLFV processes in A 4 was first discussed in [59], where channels allowed by the residual symmetry Z 3 were emphasised. CFLV processes mediated by flavons were studied in [60,61]. Specifically, correlations between Z 3 -breaking channels and the correction to TBM were discovered in [61]. Constraints on the flavon mass in supersymmetric A 4 leptonic flavour models have been studied [62]. Moreover, the observed flavour violating decay of the Higgs (h → µτ ) was investigated in the context of A 4 [63]. It was found that the flavon could be produced at colliders if it is sufficiently light.
There is also prospect for direct production, without reliance upon the flavon-Higgs mixing, at lepton colliders [64].
The primary aim of this work is to exclude regions of parameter space of a nonsupersymmetric A 4 leptonic flavour model. To do so we apply a synergy of experimental data ranging from recasting an 8 TeV collider analysis to applying limits from charged lepton flavour violating (CLFV) processes determined by the MEG collaboration. To our best knowledge we are the first to undertake such a rigorous investigation of a relatively generic leptonic flavour model [61]. We begin with a discussion of the motivation for and the basic principles underlying leptonic flavour models. We further elucidate on the specific model in Section 2.2 with a particular emphasis on the relevant interactions for the 8 TeV ATLAS analysis and charged lepton flavour violation limits we recast. In Section 3 we discuss the model parameter space and sampling strategy. We first confront the model with experimental data from g − 2 and MEG as detailed in Section 3.1 and Section 3.2 respectively. The implementation of the Higgs-scalar mixing and Higgs width constraints are presented in Section 3.3 and Section 3.4 respectively. The aforementioned constraints can be calculated analytically; however, excluding regions of the parameter space using a collider data reinterpretation is a more involved process and the tool-chain, ATLAS analysis and CLs method are discussed at length in Section 3.5. Finally, we present our results and make concluding remarks in Section 4 and Section 5.

Basic Mechanism
The threefold repetition of fermion generations and their subsequent masses and mixing structure, is arguably one of the most puzzling features of the SM. One plausible explanation of the pattern of fermionic mixing is an underlying flavour symmetry. In regards to the lepton sector, non-Abelian, discrete groups are a popular choice of family symmetry. This derives from the observation that leptonic mixing is large and generically, before the reactor mixing angle was measured, the entries of the PMNS matrix resembled Clebsch-Gordan coefficients of discrete groups.
The basic premise of leptonic flavour models is that at sufficiently high-energies there exists an underlying family symmetry, typically non-Abelian and discrete, which unifies the three generations of leptonic doublets into a single mathematical structure, such as a triplet of the flavour group. From the observation of neutrino oscillations, it is clear that leptonic masses are non-degenerate and therefore the non-Abelian flavour group cannot be a symmetry of the low-energy effective Lagrangian. As a consequence, it is assumed that the full flavour symmetry must be broken at low energies into two Abelian residual symmetry groups which are unbroken in the charged-lepton and neutrino sectors. The realisation of this breaking manifests through the introduction of new scalars, known as flavons which are usually assumed to be SM gauge singlets. The scalar potential of these flavons is invariant under transformations of the non-Abelian flavour symmetry at highenergies. However, at the flavour breaking scale, the non-trivial alignment of the VEVs of the flavons spontaneously break the non-Abelian flavour symmetry to Abelian residual symmetries in the neutrino and charged lepton sectors. The forms of the residual symmetries derive from consideration of the largest possible symmetry of each sector and the structure inherited from the larger non-Abelian flavour group. The most general discrete residual symmetry of the charged lepton sector is a direct product of cyclic groups, Z n . In contrast, the largest symmetry of the complex, symmetric Majorana neutrino mass matrix is Z 2 ×Z 2 . However, it is possible a subgroup thereof, namely Z 2 , could be a residual symmetry of the neutrino mass matrix. These remnant symmetries constrain the structure of the charged lepton and neutrino mass matrices and thereby the structure of the leptonic mixing matrix.

A 4 Leptonic Flavour Models
S T Figure 1: The tetrahedral group A 4 as the full flavour symmetry and its subgroups T = Z 3 and S = Z 2 as residual symmetries in the charged lepton and neutrino sectors, respectively.
In this work, we begin with the general A 4 model setup. The tetrahedral group or the rotational symmetries of a tetrahedron (A 4 ), as geometrically represented in Fig. 1, is the smallest discrete group (order 12) containing three-dimensional irreducible representations. The generators of the group, T and S, satisfy the relations: T 3 = S 2 = (ST ) 3 = 1. This group contains four irreducible representations: three singlets 1, 1 1 , and one triplet, 3. In the Altarelli-Feruglio basis, the triplet representation matrices of generators S and T are given by where ω = e 2iπ/3 . T and S are respectively the generator of the residual symmetries Z 3 = {1, T, T 2 } and Z 2 = {1, S} after A 4 symmetry breaking. The only physically inequivalent Abelian subgroups of A 4 are Z 3 and Z 2 where all other Abelian subgroups are conjugate to either Z 3 or Z 2 . In flavour model building, at least two triplet flavons are required: one is needed for charged lepton and the other for neutrino mass generation. We denote these flavons as ϕ and χ, respectively. These flavons could be a pseudo-real or a complex triplet of A 4 . In this present work, we focus on the former scenario where the three components of ϕ satisfy ϕ 1 = ϕ * 1 and ϕ 2 = ϕ * 3 . Such an assumption allows for a minimal number of model parameters and degrees of freedom.
In most A 4 models, the electroweak leptonic doublets (denoted as L = (L e , L µ , L τ ) T with L e = (ν eL , e L ) T , L µ = (ν µL , µ L ) T and L τ = (ν τ L , τ L ) T ) are often arranged to belong to a 3 of A 4 . And the right-handed charged leptons e R , µ R and τ R belong to singlets 1, 1 and 1 , respectively. The Higgs, H, is assigned as a trivial singlet 1 of A 4 . At leading order, the general Lagrangian terms responsible for lepton masses have the following form whereH = iσ 2 H * and the subscript r (r = 1, 1 , 1 , 3) denotes the irreducible 1 r-plet product of the fields in the bracket. The scale Λ is a new scale higher than v ϕ , v χ and may arise as a consequence of the decoupling of some heavy A 4 multiplet particles. In order to generate tiny Majorana neutrino masses, we apply the traditional dimension-five Weinberg operator (LHH T L c ) where Λ W is the related scale, which may be different from Λ 2 . The most widely studied mixing pattern is TBM mixing which predicts sin θ 12 = 1/ √ 3, sin θ 23 = 1/ √ 2 and sin θ 13 = 0. Naturally, corrections are required to render TBM mixing consistent with neutrino oscillation data, in particular with the non-zero valued reactor angle, θ 13 ≈ 8 • . One great success of A 4 models is that they naturally predict TBM mixing (at leading order) based on the following symmetry argument.
In order to ensure Z 3 and Z 2 as residual symmetries in charged lepton and neutrino sectors respectively, the vacuum alignment of these flavons is preserved under transformation of the residual symmetries As a consequence, VEVs of ϕ and χ have to take the following forms . We note that the mass matrices of the charged lepton (m l ) and neutrino (m ν ) satisfy the aforementioned residual symmetries in the following manner: in which T and S are the generators of Z 3 and Z 2 , respectively, as mentioned before. In order to generate a leptonic mixing matrix consistent with current global fit data [65], there must be a slight breaking of either the residual symmetry of the neutrino or charged lepton sector or possibly both. Although radiative corrections from the SM break the exact TBM mixing, such contributions are too small to induce |U e3 | ≈ 0.1. It is possible that the necessary deviations from TBM result from higher order operators in the flavon potential or couplings between flavon and charged lepton sector (for reviews see e.g. [23][24][25]). Such cross-couplings result in a VEV shift, i.e., ϕ 2 = 0, which thereby breaks the residual Z 3 symmetry. Without loss of generality, one can always perform the following parameterisations Using this parametrisation, the shifted VEV ϕ can be always represented as where ϕ is a complex parameter. One requirement in this bottom-up approach is that, to be consistent with oscillation data, the Z 3 breaking effect should be small. In particular, the shift | ϕ | 1. This shift could be one of the main sources of the deviations. In fact, as stated in [53], if we assume all corrections to the mixing are obtained from the ϕ VEV shift, θ 13 and δ are predicted to be sin θ 13 = √ 2Im( ϕ ) and δ = 270 • − 2Arg( ϕ ) for Im( ϕ ) > 0. Furthermore, to be consistent with all oscillation data, ϕ has to satisfy 0.10 | ϕ | 0.17 and 38 • < Arg( ϕ ) < 142 • . However, in this work, we do not limit our discussion to a specific model. Instead, we will vary ϕ in a relatively wide range, | ϕ | ∈ [10 −3 , 1] and Arg( ϕ ) ∈ [0, 2π) as shown in Table 1. Such an approach allows us to be agnostic about the origin of the corrections to the mixing; the needed correction could derive from a number of sources including the shift in the VEV of χ or higher dimensional operators responsible for the lepton masses.

Interactions Relevant for Phenomenology
We study the observable phenomenology of this well motivated flavour model and therefore concentrate on the interactions of the flavon associated with the charged lepton sector (ϕ). The flavon χ in the neutrino sector has a lesser experimental visibility 3 , which is why we do not consider its particle excitation and fix its VEV. The ϕ flavon communicates with the SM in two ways. The first is via the modification of leptonic mass terms. The second is through the portal coupling of the flavons with the Higgs. In order to illustrate the effective interactions involving flavons, we expand the flavons and Higgs about their VEVs: For the charged-lepton-portal interaction, we can straightforwardly write the couplings between flavons and charged leptons from the Lagrangian terms of Eq. (2.2) in the Altarelli-Feruglio basis The Higgs-portal interaction is obtained from the scalar potential. In principle, the full scalar potential should include self-couplings of H, ϕ and χ, as well as their crosscouplings. However, as we ignore the excitation of χ, the scalar potential can be simplified and effectively represented as a potential involving only the self-and cross-couplings of H and ϕ, and the VEV of χ contributes as a correction to the potential.
The self-couplings of Higgs are identical to the SM Higgs potential, given by where the minimum of this potential is achieved by µ 2 H < 0 and λ > 0. In the unitary gauge, the Higgs doublet takes the form, H = (0, v H / √ 2) T . The cross-coupling between χ and H, H † H(χχ) 1 , only corrects the quadratic coupling of the Higgs after χ acquires a VEV. Such a term can be absorbed by the redefinition of the parameter µ H and need not be written out explicitly.
The flavon can communicate with the visible sector via the Higgs portal coupling which cannot be forbidden by imposing any symmetries. The only renormalisable A 4 -invariant operator is (H † H)(ϕϕ) 1 . We write it out explicitly in the potential as where is a real parameter. As the Higgs is A 4 -invariant, the cross-coupling does not alter the ϕ VEV direction. Consequently, this cross-coupling term does not contribute to leptonic flavour mixing. As we shall see later, this term will lead to mixing between the Higgs and flavon and therefore plays an important role for the flavon production at the Large Hadron Collider (LHC). The self-couplings of ϕ is the origin of the breaking of A 4 to Z 3 . To simplify the couplings, an additional Z 2 symmetry (or a larger Abelian symmetry including the transformation ϕ → −ϕ) is usually imposed. With these considerations in mind, the most general A 4 -and Z 2 -invariant self-couplings of ϕ is given by where (2.14) In order to achieve a nontrivial and stable vacuum, the conditions µ 2 ϕ < 0 and g 1 + g 2 /3 > 0 are required and applied throughout this work.
With the present terms of the Higgs and flavon potential (c.f. Eq. (2.11) and Eq. (2.13)), after spontaneous flavour breaking the leptonic mixing matrix would have exact TBM structure. In order to achieve the necessary deviation needed, the cross-coupling terms between charged lepton and neutrino flavons must be present. The cross-couplings between the Higgs and χ can be absorbed by the redefinition of µ 2 H and therefore the only crosscoupling term left is the Z 3 -breaking one, (ϕϕ) 1 (χχ) 1 . This term is effectively represented as where A is a complex parameter with mass dimension two. The other cross-couplings are trivial. For example, (ϕϕ) 1 (χχ) 1 with χ = χ can be absorbed by the redefinition of µ ϕ , and (ϕϕ) 3 (χχ) 3 = 0 for the Z 2 -preserving VEV χ . Hence, the effective potential is constructed from Equations (2.11), (2.12), (2.13), and (2.15).
After minimisation of the Higgs and flavon potential, these parameters satisfy the following condition We note that the shifted VEV, ϕ = (1, ϕ , * ϕ ) T v ϕ , gives rise to non-zero θ 13 and CP violation. The parameter A may be determined from Eq. (2.17) in the following manner We now consider the masses of the Higgs and flavons modified by the Z 3 -breaking coupling. After the shifted VEV ϕ = (1, ϕ , * ϕ ) T v ϕ is included, the mixing between ϕ 1 and ϕ 2 , as well as the Higgs with ϕ 2 , is predicted. We obtain the mass term for all scalars in the basisΦ where the mass matrix M 2 Φ is a real 4 × 4 symmetric matrix with the following entries andφ 1 (φ 2 ) denotes the particle excitation around the VEV of ϕ 1 (ϕ 2 ), i.e., Eq. (2.9).
The SM Higgs is denoted as h (m h = 125 GeV) 4 with the three other scalar mass eigenstates denoted as s 1 , s 2 , s 3 .
We relate the gauge to the mass basis in the following way Before we proceed we will summarise the model parameter space relevant for limit setting. From Eqs. (2.18) and (2.20), we observe this model contains the following parameters: , ϕ , v ϕ , g 1 , g 2 . We note that all the parameters are real apart from ϕ and therefore there are six free parameters. Some salient features of this model include: • The parameter controls the cross-coupling of the flavons with SM Higgs. This parameter is of crucial importance for the Higgs-flavon mixing and therefore the production of flavons at collider and will be further discussed in Section 3.5.
• ϕ parametrises the breaking of Z 3 in the charged lepton sector. As ϕ → 0, the Z 3 -preserving limit is reached and TBM mixing is recovered. Therefore, ϕ 2 does not acquire a VEV and the only mixing that occurs is between h and ϕ 1 . Subsequently, only CLFC interactions are present.
• The flavour breaking scale is parametrised by the VEV of ϕ: v ϕ . Moreover, the masses of the flavons are functions of v φ and therefore the presence of these flavons, at colliders or otherwise, will be increasingly suppressed as the flavour breaking scale becomes larger 5 .
In the limit M 2 jj ≈ M 2 ii , the mixing between the scalars becomes ill defined. In order to avoid this regime, we explore regions of the parameter space where the flavon masses are non-degenerate 6 and the three flavon masses lie outside a 10 GeV window of the Higgs mass as will be explained further in Section 3.5.
• In the majority of the parameter space the flavons can promptly decay to two charged leptons both in a manner which is charged lepton flavour conserving and violating. Moreover, as the coupling of the flavons to the charged leptons is proportional to the charged lepton mass, the dominant decay channel of the flavons is tau-dominated.
• g 1 and g 2 are dimensionless couplings present in the A 4 ×Z 2 invariant flavon potential as shown in Eq. (2.13). Their role is most easily understood in the limit ϕ → 0 where an exact Z 3 residual symmetry in the charged lepton sector is recovered. In such a framework, ϕ 2 does not acquire a VEV, and masses of s 2 and s 3 are obtained from the quadratic terms of the scalar potential, m s 2 = m s 3 = − 1 3 g 2 v 2 ϕ . However, ϕ 1 , does acquire a VEV and therefore mixes with the Higgs. Subsequently, the mass eigenstate of the Higgs and ϕ 1 are Unfortunately for a 4 × 4 mass matrix there are no closed analytic forms for the masses of s1, s2 and s3; however the masses are complicated functions of all the parameters in p and are approximately linear in vϕ. 6 We ensure the difference between the flavon masses is ≥ 10 MeV. (2.23) In the realistic regime we are interested in, namely ϕ = 0, the masses of m s 1 and m s 2 (m s 3 ) are proportional to (g 1 + g 2 /3) and g 2 respectively 7 , however the relation no longer has a closed analytic as the 4×4 mass matrix must be diagonalised numerically.

Confronting the Model with Experimental Data
In the previous section, we reviewed the pertinent features of the model and presented its free parameters. The objective of this section is to evaluate the extent to which existing measurements are able to constrain the allowed parameter space. In order to do so, we compare predictions of the model with dedicated data from the g − 2, MEG and Atlas experiments. The former two experimental limits can be directly compared with analytic calculations and are discussed in Section 3.1 and Section 3.2 respectively. The comparison of the model prediction with collider data requires a rather involved tool-chain based on Monte-Carlo event generators and analysis software. We discuss the signatures of this flavour model at the LHC, Monte-Carlo event generation, the Atlas measurement and statistical methodology used in Section 3.5.

Parameter Space and Sampling
To simplify the numerical treatment, we apply the polar form of the complex parameter ϕ = | ϕ | · e iθϕ and define our parameter space in terms of its magnitude and phase. In those cases where a model parameter span several orders of magnitude, the sampling is performed logarithmically. The parameter sampling boundaries are given in Table 1.
Parameter p min(p) max(p)

Sampling Strategy
The sampling is undertaken in a random uniform fashion. By doing so, we ensure that samples from sub-spaces exhibit the same uniform structure as the global sampling space. A sampled point is then used to construct the mass matrix, M 2 Φ (Eq. (2.20)), which we diagonalise numerically. We reject a sampled point if any of the following conditions on the resulting scalar masses m s 1 , m s 2 , m s 3 is fulfilled: 1. Any flavon mass is too light, i.e. m s i < 10 GeV, i = 1, 2, 3.

Any flavon mass which is not the Higgs is close to being
The conditions 1 and 2 ensure that the theory is well behaved and that the production cross-section of the new scalars is not too small while the requirements 3 and 4 protects against being in the regime of resonant mixing. The final two constraints 5, 6 guarantee vacuum stability [67] and that the scalar masses are positive.
All points sampled from the parameter space specified in Table 1 that pass all six prerequisites are further tested in terms of compatibility with experimental data as detailed in the following.

g − 2
The most recent measurements of the anomalous magnetic moment of the muon at Brookhaven National Laboratory (BNL) [68,69] indicate a deviation from the Standard Model precision calculation. The Muon g − 2 experiment E989 based at Fermilab aims to make a factor of four improvement upon the current measurement [70].
New physics models, with additional scalars coupling to charged leptons, may explain this deviation. The muon anomalous magnetic moment is defined to be a µ = (g − 2) µ /2 and its deviation from the SM is given by In the flavour model we investigate, the leading contribution to g − 2 has a τ -lepton running in the loop as shown in Fig. 2. Completing a standard calculation (see for example Refs. [71,72]) we find the magnetic moment to be We note that there is a one-loop level contribution to ∆a µ fromφ 1 ; however, as the couplings of each vertex ∝ m µ /v ϕ , there is a O 10 −3 suppression relative to that of the process shown in Fig. 2 and therefore its contribution is negligible. In order to test whether a parameter point p is excluded by the result in (3.1), we interpret the latter as an upper boundary on a µ and demand that ∆a µ (p) ≤ 3.68 × 10 −9 .

MEG Result on Br (µ → eγ)
SM processes which violate charged lepton flavour, induced by massive neutrinos, occur at unobservable rates ∼ O 10 −48 . However, new physics models which modify the charged lepton sector could enhance such processes to detectable rates and provide crucial information in complement to direct searches. The Z 3 -breaking flavour model discussed in Section 2.2 has both a rich flavour and chiral structure. Moreover, as the flavons couple to the charged leptons, such a model will alter CLFV rates. There has been a systematic improvement in the sensitivity to a wide range of CLFV processes. The current bounds on the branching ratio of τ -CLFV radiative decay processes are ∼ O 10 −8 [73][74][75]. The upper limit on the branching ratio of µ → eγ flavour conversion processes are currently ∼ O 10 −12 [75,76] with the most stringent constraint from the MEG collaboration, with Br (µ → eγ) ≤ 4.2 × 10 −13 [77] at 90% C.L. A MEG upgrade (MEG II) is envisaged to further constrain the upper limit on this CLFV process to ∼ 4 × 10 −14 in the near term [78]. The Mu2e experiment at Fermilab [79] and COMET [80] based in JPARC aims to even further increase the sensitivity to this rare decay, Br (µ → eγ) ≤ 10 −16 − 10 −17 .
Consequently, as experimental constraints from µ → eγ flavour conversion provide the most severe limits on CLFV processes, we shall use this particular CLFV process to constrain the parameters of the Z 3 -breaking scenario 8 . This process is loop-induced and is mediated byφ 2 as shown in Fig. 2. The contribution of the flavon in the loop was studied in great detail in [61] where the assumption of small ϕ was applied. We do not apply this assumption and present the analytic expression for the relevant bound on the branching ratio below (3.5) The functional form of G is and finally A is given by the sum Through the perturbative expansion of W ij , using ϕ as the expansion parameter, this result is consistent with that found in [61]. The calculation is straight forward and we can compare the flavon model prediction for Br (µ → eγ) at any test point p with the experimentally found upper limit. We find that the MEG data exhibits the strongest exclusionary power in this work.

Higgs-Scalar Mixing Contraint
Extending the scalar sector of the SM has been a popular option to address various beyond SM phenomena such as providing a dark matter candidate [81][82][83]. Therefore, implications on the Higgs sector in the context of a single pure scalar singlet have been explored in a number of works [84][85][86][87][88]. These works have constrained the mixing of the new scalar with the Higgs. In general, the mixing is small and the 125 GeV Higgs boson we observed at the LHC appears to be mostly comprised of the SM Higgs mass eigenstate. In our case, there are three additional scalars which acquire VEVs and therefore all three flavons mix with the Higgs. The constraint on p enforced from mixing is imposed via the following requirement:

Higgs-Width Constraint
In certain regions of the model parameter space, p, it is possible that the coupling of s 1 , s 2 and s 3 to h will modify the Higgs total width. Theoretical calculations, assuming purely SM interactions, predict the Higgs total width (Γ SM ) to be ≈ 4 MeV. However, the constrained upper limits of the total width, using measurements of on-and off-shell decay rates to Zbosons, indicate the upper limit to be 22 MeV at a 95% C.L. [89]. In such regions of the model parameter space, we assume that the deviation between the theoretically predicted and measured Higgs width derives entirely from new physics associated to our model as outlined in the Appendix 9 . We consider a point p excluded by the Higgs-width results if the calculated width exceeds 22 MeV.

Reinterpretation of Atlas Search for High Multiplicity Leptonic Final States
Reinterpreting a collider analysis is a more involved procedure compared to the experimental constraints discussed in the previous sections. We shall therefore initially give a brief explanation of the general workflow followed by more specific descriptions of the Atlas analysis, event simulation and statistical methods applied.

General Workflow
We need to simulate fully differential collider events and analyse them as faithful to the original data analysis as possible in order to be able to compare predictions to the measured observation and background events. The procedure requires writing the physics model in question in a language a Monte-Carlo (MC) event generator is able to understand. We use FeynRules [90] to code the Lagrangian and derive a model file in the Universal Feyn-Rules Output (UFO) format. This UFO format is understood by the MC event generator Sherpa [91,92] which in turn is then able to simulate proton-proton collisions according to the flavon model including QCD and QED radiation effects as well as hadronisation and hadron decays.
The simulated events are analysed by the dedicated tool Rivet [93]. For the Atlas analysis we have chosen to reinterpret, we are greatly helped by the fact that the analysis team provided a validated Rivet routine of their measurement. The latter contains exactly the same selection criteria and analysis logic as applied in the original data analysis. One caveat, namely that the presented data and background distributions have not been corrected for detector effects (not "unfolded") is overcome by the fact that the analysis team included a machinery that applies all resolutions and efficiencies to the simulated particles such that a fair comparison between our signal MC and the data is possible.
In order to make a statement on whether a sampled point p yields a prediction that is compatible with the data we apply a hypothesis test known as the CL s method. This method allows to distinguish, on a certain confidence level, whether the observed data is more likely to be explained by the SM background only or by the signal plus background hypothesis.  [94] used for reinterpretation in this work. The definition of the signal regions S1 . . . S4 is given in Table 2. The data and background counts are explicitly listed in Table 4. Copyright 2018 CERN for the benefit of the ATLAS Collaboration. CC-BY-4.0 license

The Atlas Search Analysis
There are many beyond the SM (BSM) scenarios which have anomalous production of leptonic final states and therefore there have been a number of dedicated analyses which have searched for three or more charged, prompt and isolated leptons. These analyses have shown little deviation from the SM expectation and therefore may be effective in excluding regions of parameter space for many models. The ATLAS collaboration has conducted a number of supersymmetry searches which have multi-lepton final states [95][96][97] and indeed a model-independent analysis was performed, providing limits using 7 TeV data [98]. The CMS Collaboration has also performed a similar analysis using both 7 TeV [99] and 8 TeV data [100]. The analysis we choose to constrain our model parameter space with uses a data sample collected in 2012 by the ATLAS experiment with a centre of mass energy of 8 TeV and corresponding integrated luminosity of 20.3 fb −1 [94]. This analysis searches for the anomalous production of at least three charged leptons in the final state. Moreover, as this analysis searches for events which have at least one tau final state, it is particularly well suited to our model as the flavons dominantly decay to taus and muons. The analysis logic and data have been preserved, validated and made publicly available by the Atlas collaboration within the analysis tool Rivet [93] as ATLAS_2014_I1327229. We found this measurement, of all publicly available and validated analysis, to be the most suitable for constraining our model parameter space. We re-use the observed data and total background estimates published by Atlas as presented in Table 4 to perform our statistical analysis.
We shall not reiterate the full details of the analysis but rather present the most relevant features for our work. The analysis first applies a veto on Z-bosons and then divides the events (which contain at least three leptons) into four disjoint signal regions based on charged lepton flavour pairs and leptonic content.
• OSSF: events which contain an opposite sign same flavour (OSSF) charged lepton pair.
• no OSSF: events which do not contain an OSSF pair.
• 2e/µ ≥ 1τ had : events containing two electrons or muons with and at least one hadronically decaying tau lepton.
Depending on the signal region, different kinematic variables are used in the measurement: : the scalar sum of p T of the leptons used to categorise the event.
• Minimum lepton p T : minimum lepton transverse momentum.
• m eff m eff m eff : the effective mass of the event which combines the scalar sum of missing energy, scalar sum of the jets and the total p T of the leptons in the event.
An overview of the signal regions and observables applied is provided in Table 2 and Table 3.
The selection of histograms from [94] we reinterpret in this work are shown in Fig. 3 10 . As can be seen, the dominant SM processes which contribute to multi-leptonic final states are diboson production and production of a top quark pair in association with a weak gauge boson. The statistics associated to the m eff kinematic variable are relatively low in comparison with the minimum lepton p T and H leptons T . Although, there is generally good agreement between the SM predictions and the data; there are regions where the observed event yield is lower than the expected yield by more than three times the uncertainty on the expectation, and this occurs for the low statistics histograms.   Table 4: Observed (N obs ) and background (N BG ) counts reported by Atlas for each signal region (SR) used in this analysis .

Event Simulation and Analysis
The Universal FeynRules Output (UFO) for the model is generated using FeynRules [90] The model information in the Universal FeynRules Output format is imported into the Sherpa event generator [91] to provide a full simulation of BSM processes at the particle level.
As the Higgs portal coupling, shown in Eq. (2.12), is the only way the flavons can be produced at a hadron collider, gluon fusion will be the most relevant production channel for the flavons. Although, BSM is available at next-to-leading order (NLO) accuracy in NLOCT prepackaged in FeynRules [101], this feature is currently not implemented into Sherpa. Therefore, for the BSM processes in this model we simply correct the tree-level cross sections with a K-factor of 2.2 [102]. This K-factor is computed at next-to-next-to leading order [103][104][105], for gluon fusion which is the dominant production mechanism of the Higgs [106]. As the flavons decay leptonically, additional radiations from the final states should not affect our results significantly. There are several kinematic regimes which are important for flavon production: In this kinematic regime, the flavons may be pair produced by the Higgs.
The three-body decay h i → sf f is kinematically allowed, however this will be suppressed by phase space.
• 2m h < m s i The pair production of the Higgs becomes kinematically accessible.
From Eq. (2.10), we observe that three-body decays of the scalars to scalar and two leptons are possible but they are expected to be subdominant due to phase space suppression. However, for certain points in the model parameter space, p, it is possible that the three-body decays are non-negligible [88] and therefore are included in our Monte-Carlo simulations. The dominant configuration is two scalars in the final state where all possible combinations are generated. The subdominant contribution is one scalar in the final state with two leptons (both in charged lepton flavour conserving and violating combinations) as shown in Fig. 4. Sherpa uses the matrix element generator Comix [107] to find all contributing diagrams.
To summarise, for each test point p we simulate 10 6 events 11 with Sherpa in a setup that includes the following processes: • pp → s i s j where the flavons are denoted as s i and s j for i, j = {1, 2, 3}.
• pp → s i ¯ where s i for i = {1, 2, 3} and denotes the charged leptons.
The generated scalars are decayed according to internally calculated branching ratios. Furthermore, Sherpa takes care of QED and QCD radiation as well as hadronisation and effects such as hadron decays and multiple parton interactions. These events are then passed through the ATLAS analysis using Rivet and the output is a signal histogram of expected event yields for each p that can be compared with data and background estimates since Rivet applies all relevant detector effects. Figure 4: Flavon production channels.

The CLs Method
In order to infer information on the viability of a given point p, the expected event yield within the signal regions of the analysis is compared to the observed data and estimated backgrounds as reported by the collaboration on HepData [108]. As the experiment has not reported a discrepancy between data and the estimated backgrounds, we set an upper limit on the signal cross section normalised to the nominal cross section as calculated by SHERPA through frequentist interval estimation based on the profile likelihood [109]. In this model, the probability of observing data D is a function of parameters that are grouped into two sets: the parameters of interest (POIs), in this case the normalised cross section µ, as well as nuisance parameters θ. The log-likelihood ratio is given by whereμ andθ are the best-fit value for an unconstrained fit of the model against the data andθ are the best-fit values for the nuisance parameters for a constrained fit with a constant signal strength µ. We choose to use the reported per-bin uncertainty as a shape-systematic on the reported estimated background, such that the model has one nuisance parameter γ i for each bin entering the fit. In order to set an upper limit, we use the test statistic q µ q µ = −2 log λ(µ) forμ ≤ µ 0 forμ > µ, (3.9) where the choice of test statistic is only dependent on the parameter of interest and avoids counting upward fluctuations in which the best fit valueμ exceeds the tested signal strength µ as evidence against signal hypothesis. For the hypothesis test, we evaluate the modified p-value CL s which is commonly used by collider experimentalists and is defined to be in which p(q µ |µ ) is the distribution of the test statistic q µ for data, which is populated according to a signal strengh µ which we compute using the asymptotic formulae derived in [109]. To compute the CL s values, the Python-based implementation of HistFactory [110] pyhf was applied [111].

Results
The total number of points passing the cuts in Section 3 is N tot = 8865, which we found to be a sufficient number of points to cover the model parameter space. For each type of data, d, we count the number of points not excluded by it, N The main results of this study are summarised in Table 5 and Table 6. As can be seen from Table 5, MEG excludes 65.6% of the total model parameter space, while the 8 TeV collider analysis excludes 40.0%. The constraints from Higgs measurements provide a total In addition to quantifying the absolute exclusionary power of each measurement, we have demonstrated the complementarity between the collider analysis and that of MEG. As detailed in Table 6, of the 3045 (34.3%) points which cannot be excluded by MEG, we found 378 of those points (12.4%) can be conservatively excluded by a combination of Higgs width, Higgs-scalar mixing and the ATLAS analysis. Moreover, of the 5317 (60.0%) points which are not excluded by the collider analysis, 2386 (44.87%) of those may be excluded which cannot be excluded by the relevant experiment while the dark blue denotes the regions which are excluded at 95% C.L. In the two-dimensional projection plots the colours have the same meaning (yellow to dark blue represents lesser to greater exclusionary power in those parameter space) but there is also additional information encoded in the size of the coloured dots: the larger the dot size the higher the density of sampled points in that region 12 . Fig. 5 shows the results using Atlas data exclusively. We observe that for higher values of v ϕ (the VEV of the flavons) the exclusionary power decreases. This is to be expected as the flavon masses increase with v ϕ and therefore their production cross section decreases. However, for the majority of the model parameter space, v ϕ < 10 2.0 GeV is excluded at 95% C.L. In the case of the cross-coupling between the flavons and the Higgs, , the smaller values ( < 10 −2 ) become increasingly more difficult to exclude. This is simply due to the fact that the production cross section of the flavons decreases for smaller values of . Moreover, we note that there is periodic behaviour in the polar coordinate, θ ϕ . We observe that the exclusionary power in regards to g 1 and g 2 is not particularly strong.
The exclusion from MEG alone is shown in Fig. 6. It can be observed that the exclusionary power on v ϕ , which is the parameter that encapsulates the flavour breaking scale, is greater than that of Atlas alone and that v ϕ < 10 2.1 GeV is excluded at 95% C.L. Moreover, the cross-coupling is particularly constrained to a corner of the parameter space, < 10 −2.5 . We note that the constraints on g 1 and g 2 are much more aggressive compared to the constraint from Atlas.
The plot in Fig. 7 shows the combined MEG and Atlas constraints. Finally, Fig. 8 shows the constraints from all included experimental data. We find that there is not a significant qualitative difference between the two plots (as the Higgs and g − 2 constraints are very weak). We observe that the flavour breaking scale, parametrised by v ϕ , must be greater than ∼ 10 2.5 GeV. Moreover, the cross-coupling between this flavour sector and the SM must be < 10 −2 . The absolute value of the parameter which controls how much the residual Z 3 symmetry of the charged lepton sector is broken, is particularly constrained to | ϕ | < 10 −2.75 13 . However, the polar coordinate of the Z 3 breaking parameter is constrained to θ ϕ < 1.0 radians. In summary, the majority of the chosen model parameter space can be excluded through the combination of the measurements from the Atlas analysis, MEG, g − 2 experiments and Higgs measurement data.

Summary
Explaining the origin of the flavour structure, in both quark and lepton sectors, has been a recurring theme in many proposed extensions of the SM. One such approach to explain the pattern of leptonic mixing is the application of discrete, non-Abelian flavour symmetries. This flavour symmetry must be broken in the low-energy effective theory but its residual symmetries survive and play an important role in predicting the structure of the leptonic mixing matrix.
In this work, we did not presuppose the flavour breaking scale was high (close to the GUT scale) and therefore it is an interesting endeavour to constrain this flavour model's parameter space using a synergy of experimental data. In order to exclude regions of the model parameter space, we applied constraints from g − 2, MEG, Higgs-scalar mixing and Higgs width measurements. In addition, we recasted an 8 TeV Atlas analysis which searched for events with high-multiplicities of leptonic final states. We believe we are the first to combine, in both a conservative and rigorous manner, such experimental data to constrain a leptonic flavour model.
One of our primary aims was to be as generic as possible in constraining the parameter space of a well-motivated flavour model, such as A 4 . Therefore, we chose to investigate a simplified description of an A 4 model. At leading order this model has the general features of most A 4 models, where the residual symmetry Z 3 (Z 2 ) is preserved in the charged lepton (neutrino) sector and consequently TBM mixing is predicted after A 4 symmetry breaking. However, at sub-leading order, the residual symmetries are slightly broken due to the shift in one of the flavon VEVs. As a consequence, a pattern of mixing consistent with neutrino oscillation data is achieved. We mainly focus on the phenomenology of the flavon, ϕ which couples to charged leptons as it has greater experimental visibility than its neutrino flavon counterpart, χ.
We conducted an exploration of the six-dimensional model parameter space and found most of the constraints could be calculated analytically. The collider reinterpretation of an 8 TeV Atlas measurement was a more involved process. We benefited greatly from the Atlas collaboration both preserving and validating their analysis; moreover, the analysis was publicly available via Rivet which we used as our analysis tool. We believe we are the first to utilise the Monte-Carlo event generator, Sherpa, for BSM purposes, which was particularly amenable as the fully-showered and hadronised Monte-Carlo events could directly fed into the analysis tool. Although we focused on an economical model, most basic features of leptonic flavour models have been included, e.g., interactions related to lepton mass generation, interactions leading to the breaking of flavour symmetry and residual symmetries, as well as the Higgs-portal interactions. An alternative model may increase the number of free parameters but preserve most of these features and this investigation remain relevant.
We found the most aggressive constraint is derived from the CLFV limit set by the MEG experiment; approximately ∼ 60% of the parameter space could be excluded. The exclusionary power exhibited by the Atlas analysis came second only to MEG; excluding ∼ 40%. The remaining experimental data had marginal but non-negligible exclusionary power. Interestingly, the exclusionary power of MEG and and the collider experimental data complement each other: the collider analysis, combined with Higgs width and Higgsscalar mixing constraints, can exclude regions MEG simply cannot and vice versa. This is because the collider search is directly sensitive to the interactions of the flavons with the SM Higgs doublet, while the constraints from MEG is not. We hope the collider experimentalists view this optimistically, that searches for highmultiplicity leptonic final states exclude sizeable regions of the leptonic flavour model parameter space and complement limits set by experiments dedicated to searching for CLFV. Moreover, at higher centre of mass energies, the exclusionary power of collider searches for anomalous production of leptonic final states may seriously compete with those of MEG. Work to precisely quantify this statement is of interest but relegated for future studies. In addition, the construction and optimisation of an analysis for final states such as τ τ µe, which would be a useful step in constraining the flavour breaking scale, if indeed such a scale exists.

A Group Theory for A 4
In this paper, as in [61], we work in the Altarelli-Feruglio basis [13], where T is diagonal. T and S are respectively given by The products of two 3-dimensional irreducible representations a = (a 1 , a 2 , a 3 ) T and b = (b 1 , b 2 , b 3 ) T can be expressed as (A.2)

B Higgs Width Constraint
As introduced in Section 2.2, the gauge and mass eigenstates of the scalar sector in the Z 3 breaking scenario, may be related by the unitary matrix Using the above notation, the Z 3 breaking triplet couplings are given below