We consider an effective theory (ET) approach to flavour-violating processes beyond the Standard Model (SM), where the breaking of flavour symmetry is described by spurion fields whose low-energy vacuum expectation values are identified with the SM Yukawa couplings. Insisting on canonical mass dimensions for the spurion fields, the large top-quark Yukawa coupling also implies a large expectation value for the associated spurion, which breaks part of the flavour symmetry already at the UV scale Lambda of the ET. Below that scale, flavour symmetry in the ET is represented in a non-linear way by introducing Goldstone modes for the partly broken flavour symmetry and spurion fields transforming under the residual symmetry. As a result, the dominance of certain flavour structures in rare quark decays can be understood in terms of the 1/Lambda expansion in the ET. We also discuss the generalization to 2-Higgs-doublet models with large tan(beta).
We study the charged lepton flavour violation in a popular neutrino mass model with $A_4$ discrete symmetry. This symmetry requires the presence of multiple Higgs doublets in the model and it also dictates the flavour violating Yukawa couplings of the additional neutral scalars of the model. Such couplings lead to the decays of the neutral mesons, the top quark and the $τ$ lepton into charged leptons of different flavours at tree level. The $A_4$ symmetry of the model leads to certain characteristic signatures in these decays. We discuss these signatures and predict the rates for the most favo…
The formalism of combined finite modular and generalised CP (gCP) symmetries for theories of flavour is developed. The corresponding consistency conditions for the two symmetry transformations acting on the modulus $τ$ and on the matter fields are derived. The implications of gCP symmetry in theories of flavour based on modular invariance described by finite modular groups are illustrated with the example of a modular $S_4$ model of lepton flavour. Due to the addition of the gCP symmetry, viable modular models turn out to be more constrained, with the modulus $τ$ being the only source of CP vi…
We investigate the implications for fermion mass models in heterotic orbifolds of the modular symmetry mixing twisted states localized at different fixed points. We show that, unlike in the case of continuous gauge symmetries, the mass eigenstates do not mix under the symmetry; thus coupling constants in the low-energy theory are invariant functions of the moduli.
We show how a mass mixing matrix can be generated dynamically, for two massless fermion flavours coupled to a Lorentz invariance violating (LIV) gauge field. The LIV features play the role of a regulator for the gap equations, and the non-analytic dependence of the dynamical masses, as functions of the gauge coupling, allows to consider the limit where the LIV gauge field eventually decouples from the fermions. Lorentz invariance is then recovered, to describe the oscillation between two free fermion flavours, and we check that the finite dynamical masses are the only effects of the original L…
We show how an accidental $U(1)$ Peccei-Quinn (PQ) symmetry can arise from a discrete $A_4$ family symmetry combined with a discrete flavour symmetry $ \mathbb{Z}_3 \times \mathbb{Z}_5^2 $, in a realistic Pati-Salam unified theory of flavour. Imposing only these discrete flavour symmetries, the axion solution to the strong $ CP $ problem is protected from PQ-breaking operators to the required degree. A QCD axion arises from a linear combination of $ A_4 $ triplet flavons, which are also responsible for fermion flavour structures due to their vacuum alignments. We find that the requirement of a…
We discuss the role played by the horizontal flavour symmetry in supersymmetric theories. In particular, we consider the horizontal symmetry $SU(3)_H$ between the three fermion families and show how this concept can help in explaining the fermion mass spectrum and mixing pattern in the context of SUSY GUTs.
We show that a flavour symmetry a la Froggatt-Nielsen can be naturally incorporated in models with gauge-Higgs unification, by exploiting the heavy fermions that are anyhow needed to realize realistic Yukawa couplings. The case of the minimal five-dimensional model, in which the SU(2)_L x U(1)_Y electroweak group is enlarged to an SU(3)_W group, and then broken to U(1)_em by the combination of an orbifold projection and a Scherk-Schwarz twist, is studied in detail. We show that the minimal way of incorporating a U(1)_F flavour symmetry is to enlarge it to an SU(2)_F group, which is then comple…
We present flavour-symmetric couplings for the OZI-allowed three-meson vertices of effective meson theories, which for the case of the two-meson channels to which the flavour-singlet scalar meson couples, are endowed with a correction factor with respect to the standard formula.
We study the properties of coloured three particle s-wave quark clusters when flavour symmetry is broken. The relevance of such clusters for models of pentaquarks is shortly mentioned.
After we study the 6-dimensional ${\cal N} = (1, 1)$ supersymmetry breaking and $R$ symmetry breaking on $M^4\times T^2/Z_n$, we construct two ${\cal N} = (1, 1)$ supersymmetric $E_6$ models on $M^4\times T^2/Z_3$ where $E_6$ is broken down to $SO(10)\times U(1)_X$ by orbifold projection. In Model I, three families of the Standard Model fermions arise from the zero modes of bulk vector multiplet, and the $R$ symmetry $U(1)_F^{I} \times SU(2)_{{\bf 4}_-}$ can be considered as flavour symmetry. This may explain why there are three families of fermions in the nature. In Model II, the first two fa…
In order to explain the fermions masses and mixing parameters appearing in the lepton sector of the Standard Model, one proposes the extension of its symmetry. A discrete, non-abelian subgroup of $U(3)$ is added to the gauge group $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ . Apart from that, one assumes the existence of one extra Higgs doublet. This article focuses mainly on the mathematical theorems and computational techniques which brought us to the results.
Combining Pati-Salam (PS) and flavour symmetries in a renormalisable setup, we devise a scenario which produces realistic masses for the charged leptons. Flavour-symmetry breaking scalar fields in the adjoint representations of the PS gauge group are responsible for generating different flavour structures for up- and down-type quarks as well as for leptons. The model is characterised by new heavy fermions which mix with the Standard Model quarks and leptons. In particular, the partners for the third fermion generation induce sizeable sources of flavour violation. Focusing on the charged-lepton…
We show how a specific sequential breaking pattern of a U(2) flavour symmetry occurs automatically in a broad framework. The relative orientation in U(2) space of the spurion fields that breaks the U(2) symmetry is uniquely fixed, thus determining the form of the fermion mass matrices in a predictive way.
The Flavour Symmetry of the Standard Model (SM) gauge sector is broken by the fermion Yukawa couplings. Promoting the Yukawa matrices to scalar spurion fields, one can break the flavour symmetry spontaneously by giving appropriate vacuum expectation values (VEVs) to the spurion fields, and one encounters Goldstone modes for every broken flavour symmetry generator. In this paper, we point out various aspects related to the possible dynamical interpretation of the Goldstone bosons: (i) In an effective-theory framework with local flavour symmetry, the Goldstone fields represent the longitudinal m…
A new mechanism for spontaneous breaking of flavour symmetry is demonstrated. An exactly flavour symmetric model with degenerate bare nonets and with sufficiently strong tri-linear meson couplings is shown to lead to self-consistency equations which are unstable. Instead there exists a stable solution, which break flavour symmetry spontaneously in the mass spectrum. For a C-degenerate meson spectrum the stable mass spectrum obeys the Okubo-Zweig-Iizuka (OZI) rule and the approximate equal spacing rule.
While solving the supersymmetric flavour problem, a U(2) flavour symmetry might be at the origin of the pattern of fermion masses and mixings. The consequences of this hypothesis are spelled out concerning the parameters of the CKM matrix and their observability in B- and K-physics. The relevance of searching for lepton flavour violation and for the electric dipole moments of the electron and the neutron is also emphasized.
We show how two principles - strong coupling and discrete symmetry - can work together to generate the flavour structure of the Standard Model. We propose that in the UV the full theory has a discrete flavour symmetry, typically only associated with tribimaximal mixing in the neutrino sector. Hierarchies in the particle masses and mixing matrices then emerge from multiple strongly coupled sectors that break this symmetry. This allows for a realistic flavour structure, even in models built around an underlying grand unified theory. We use two different techniques to understand the strongly coup…
We study the problem of modulus stabilisation in the framework of the modular symmetry approach to the flavour problem. By analysing simple UV-motivated CP-invariant potentials for the modulus $τ$ we find that a class of these potentials has (non-fine-tuned) CP-breaking minima in the vicinity of the point of $\mathbb{Z}_3^{ST}$ residual symmetry, $τ\simeq e^{2πi/3}$. Stabilising the modulus at these novel minima breaks spontaneously the CP symmetry and can naturally explain the mass hierarchies of charged leptons and possibly of quarks.
We demonstrate that an $SU(2)_L\times U(1)_Y$ model with extended Higgs sector gives rise to bi-maximal neutrino mixing through the incorporation of SO(3) flavour symmetry and discrete symmetry. The neutrino and the charged lepton masses are generated due to higher dimensional terms. The hierarchical structures of neutrinos and charged leptons are obtained due to inclusion of SO(3) flavour symmetry and discrete symmetry.The model can accommodate the vacuum oscillation solution of solar neutrino problem, through reasonable choice of model parameters along with the atmospheric neutrino experimen…
The collection of a few anomalies in semileptonic $B$-decays invites to speculate about the emergence of some strikingly new phenomena. Here we offer a possible interpretation of these anomalies in the context of a weakly broken $U(2)^5$ flavour symmetry and lepto-quark mediators.
We study UV-complete Froggatt-Nielsen-like models for the generation of mass and mixing hierarchies, assuming that the integrated heavy fields are chiral with respect to an abelian Froggatt-Nielsen symmetry. It modifies the mixed anomalies with respect to the Standard Model gauge group, which opens up the possibility to gauge the Froggatt-Nielsen symmetry without the need to introduce additional spectator fermions, while keeping mass matrices usually associated to anomalous flavour symmetries. We give specific examples where this happens, and we study the flavourful axion which arises from an …
In a generic supersymmetric extension of the Standard Model, whether unified or not, a simple and well motivated U(2) symmetry, acting on the lightest two generations, completely solves the flavour changing problem and necessarily leads to a predictive texture for the Yukawa couplings.
Flavour models may display a relation between the CP-violating asymmetry for leptogenesis and low-energy parameters. If the flavour symmetry produces an exact mass independent lepton mixing scheme at leading order (with type I see-saw) the CP-violating asymmetry would vanish in the absence of corrections. We present a model displaying the link between deviations from the mixing scheme and leptogenesis.
After a brief review on flavour symmetry breaking (FSB) in the unpolarized nucleon sea, we discuss theoretical predications for FSB in the polarized nucleon sea from meson cloud and `Pauli blocking'.
The gauge sector of the Standard Model (SM) exhibits a flavour symmetry which allows for independent unitary transformations of the fermion multiplets. In the SM the flavour symmetry is broken by the Yukawa couplings to the Higgs boson, and the resulting fermion masses and mixing angles show a pronounced hierarchy. In this work we connect the observed hierarchy to a sequence of intermediate effective theories, where the flavour symmetries are broken in a step-wise fashion by vacuum expectation values of suitably constructed spurion fields. We identify the possible scenarios in the quark sector…
I estimate the SU(3) flavour symmetry breaking in the ratio of penguin-to-tree ratios of the decays B_s\to K^+ K^- and B_d\toπ^+π^-, given an assumption on the flavour symmetry breaking of the hadronic input parameters. The decay amplitudes are calculated in QCD factorization. Implications for the determination of gamma are discussed.
In the framework of the modular symmetry approach to lepton flavour, we consider a class of theories where matter superfields transform in representations of the finite modular group $Γ_5 \simeq A_5$. We explicitly construct a basis for the 11 modular forms of weight 2 and level 5. We show how these forms arrange themselves into two triplets and a quintet of $A_5$. We also present multiplets of modular forms of higher weight. Finally, we provide an example of application of our results, constructing two models of neutrino masses and mixing based on the supersymmetric Weinberg operator.
We present flavour-symmetric results for the couplings of quark-antiquark systems to meson-meson channels in the harmonic-oscillator expansion. We tabulate their values for all possible open and closed decay channels of pseudo-scalar, vector and scalar mesons. We compare the predictions of a model that employs these flavour-symmetric couplings, both with the results of a model which uses explicitly flavour-dependent couplings, and with experiment.
A classification scheme of hadrons is proposed on the basis of the division algebra H of quaternions and an appropriate geometry. This scheme suggests strongly to understand flavour symmetry in another manner than from standard symmetry schemes. In our approach, we do not start from `exact' symmetry groups like SU(2) \times SU(2) chiral symmetry and impose various symmetry breaking mechanisms which collide with theorems wellknown from quantum field theory. On the contrary, the approximate symmetry properties of the hadron spectrum at low energies, usually classified by `appropriately' broken c…
Hadronic spectroscopy can be introduced to students and developed rather far without requiring SU(N) flavour symmetry. In such a "minimalist" presentation, we are naturally led to comment and clarify the concept of the "generalized" Pauli principle.
We discuss our recently proposed S3(down)xS3(up) flavour-permutation-symmetric mixing observables, giving expressions for them in terms of (moduli-squared) of the mixing matrix elements. We outline their successful use in providing flavour-symmetric descriptions of (non-flavour-symmetric) lepton mixing schemes. We develop our partially unified flavour-symmetric description of both quark and lepton mixings, providing testable predictions for CP-violating phases in both B decays and neutrino oscillations.
In models with flavour symmetries added to the gauge group of the Standard Model the CP-violating asymmetry necessary for leptogenesis may be related with low-energy parameters. A particular case of interest is when the flavour symmetry produces an exact mass independent lepton mixing scheme, leading to a vanishing CP-violating asymmetry. We present a model-independent discussion that confirms this always occurs for unflavoured leptogenesis in type I see-saw scenarios.
We study classes of models which are based on some discrete family symmetry which is completely broken such that the observed neutrino flavour symmetry emerges indirectly as an accidental symmetry. For such "indirect" models we discuss the D-term flavon vacuum alignments which are required for such an accidental flavour symmetry consistent with tri-bimaximal lepton mixing to emerge. We identify large classes of suitable discrete family symmetries, namely the $Δ(3n^2)$ and $Δ(6n^2)$ groups, together with other examples such as $Z_7\rtimes Z_3$. In such indirect models the implementation of the …
The chromomagnetic interaction, with full account for flavour-symmetry breaking, is applied to S-wave configurations containing two quarks and two antiquarks. Phenomenological implications are discussed for light, charmed, charmed and strange, hidden-charm and double-charm mesons, and extended to their analogues with beauty.
We quantitatively analyze a quark-lepton flavour model derived from a six-dimensional supersymmetric theory with $SO(10)\times U(1)$ gauge symmetry, compactified on an orbifold with magnetic flux. Two bulk $\mathbf{16}$-plets charged under the $U(1)$ provide the three quark-lepton generations whereas two uncharged $\mathbf{10}$-plets yield two Higgs doublets. At the orbifold fixed points mass matrices are generated with rank one or two. Moreover, the zero modes mix with heavy vectorlike split multiplets. The model possesses no flavour symmetries. Nevertheless, there exist a number of relations…
Can a theory of flavour capable of describing the spectrum of fermion (including neutrino) masses and mixings also contain within it the seeds for a solution of the SUSY flavour and CP problems? We argue that supergravity together with a non-Abelian family symmetry can completely resolve the SUSY flavour and CP problems in a broad class of theories in which family symmetry and CP is spontaneously broken in the flavon sector. We show that a simple superpotential structure can suppresses the F-terms of the flavons and GUT scale Higgs fields and that, if this mechanism is implemented, the resulti…
A new mechanism for breaking an internal symmetry spontaneously is discussed, which is intermediate between the Nambu-Goldstone and Wigner modes of symmetry breaking. Here the quark-antiquark sea takes the role of the vacuum of the Nambu-Goldstone case. Flavour symmetry becomes a discrete permutation symmetry of the valence quarks with respect to the sea quarks, which can be spontaneously broken without generation of massless Goldstone bosons.
A variety of lepton flavour violating effects related to the recent discovery of neutrino oscillations and mixings is here systematically discussed in terms of an S_3-flavour permutational symmetry. After a brief review of some relevant results on lepton masses and mixings, that had been derived in the framework of a Minimal S_3-Invariant Extension of the Standard Model, we derive explicit analytical expressions for the matrices of the Yukawa couplings and compute the branching ratios of some selected flavour changing neutral current (FCNC) processes, as well as, the contribution of the exchan…
We study the phenomenology of a model where an SU(2)^3 flavour symmetry acting on the first two generation quarks is gauged and Yukawa couplings for the light generations are generated by a see-saw mechanism involving heavy fermions needed to cancel flavour-gauge anomalies. We find that, in constrast to the SU(3)^3 case studied in the literature, most of the constraints related to the third generation, like electroweak precision bounds or B physics observables, can be evaded, while characteristic collider signatures are predicted.
We construct a three-Higgs doublet model with a flavour non-universal ${\rm U}(1)\times \mathbb{Z}_2$ symmetry. That symmetry induces suppressed flavour-changing interactions mediated by neutral scalars. New scalars with masses below the TeV scale can still successfully negotiate the constraints arising from flavour data. Such a model can thus encourage direct searches for extra Higgs bosons in the future collider experiments, and includes a non-trivial flavour structure.
A brief overview of flavour-symmetry strategies to extract the angle gamma of the unitarity triangle is given, focusing on B --> pi K modes and the B_d --> pi^+ pi^-, B_s --> K^+ K^- system. We discuss also a variant of the latter approach for the e^+ e^- B-factories, where B_s --> K^+ K^- is replaced by B_d --> pi^{+/-} K^{+/-}.
An important class of flavour groups, that are subgroups of $U(3)$ and that predict experimentally viable lepton mixing parameters including Majorana phases, is the $Δ(6n^2)$ series. The most well-known member is $Δ(24)=S_4$. I present results of several extensive studies of lepton mixing predictions obtained in models with a $Δ(6n^2)$ flavour group that preserve either the full residual $Z_2\times Z_2$ or a $Z_2$ subgroup for neutrinos and can include a generalised CP symmetry. Predictions include mixing angles and Dirac CP phase generally; and if invariance under a generalised CP symmetry is…
Many studies have been made on extensions of the standard model with $B-L$ gauge symmetry. The addition of three singlet (right-handed) neutrinos renders it anomaly-free. It has always been assumed that the spontaneous breaking of $B-L$ is accomplished by a singlet scalar field carrying two units of $B-L$ charge. This results in a very natural implementation of the Majorana seesaw mechanism for neutrinos. However, there exists in fact another simple anomaly-free solution which allows Dirac or inverse seesaw neutrino masses. We show for the first time these new possibilities and discuss an appl…
We describe in detail the method we have used to determine the CKM angles gamma, alpha and beta using flavour symmetries between non-leptonic B decays. This method is valid in the context of the SM but also in presence of New Physics not affecting the amplitudes.
If R-parity violation turns out to be a true aspect of Nature, a speculation about its possible origin could add a new dimension to the supersymmetric flavour problem. It has been shown in the past by Barbieri, Hall and their collaborators that the small breaking parameters of an approximate non-abelian flavour symmetry could govern the light quark and lepton masses and at the same time could account for the near degeneracies of squarks and sleptons. A possible connection of the above feature to the natural suppressions of R-parity-violating couplings has been investigated here. With some modi…
We study the realisation of supersymmetric discrete flavour symmetry models to the thermal history of our universe. We focus on the evolution of the pseudo moduli field among the flavons by taking into account finite temperature corrections. We show that the pseudo moduli flavon dominates the energy density of our universe and this domination makes crucially difficult to realise the flavour symmetry models in our universe. We also discuss possible extensions of the supersymmetric discrete flavour symmetry models which can ensure the consistency of the models with the thermal history of our uni…
The current state of knowledge of the order parameters of the spontaneously broken chiral symmetry, the quark condensate and the pseudoscalar decay constant in the chiral limit, is briefly reviewed, based on available phenomenological fits and lattice QCD calculations. We argue that while the theory is pretty well understood in the two-flavour case, there is still a gap in the knowledge of the characteristics of the QCD vacuum in the three-flavour one. Our constraints for the three-flavour parameters obtained by a Bayesian statistical analysis of the decays of eta to three pions are presented.…
We show that there is a minimal discrete symmetry which leads to the extended flavour democracy scenario constraining the Dirac neutrino, the charged lepton and the Majorana neutrino mass term ($M_R$) to be all proportional to the democratic matrix, with all elements equal. In particular, this discreet symmetry forbids other large contributions to $M_R$, such as a term proportional to the unit matrix, which would normally be allowed by a $S_{3L}\times S_{3R}$ permutation symmetry. This feature is crucial in order to obtain large leptonic mixing, without violating 't Hooft's, naturalness princi…
One of the key elements in the testing of the Standard-Model description of CP violation through $B$-meson decays is the direct determination of the angle $γ$ of the unitarity triangle in a variety of ways. We give a brief overview of the implications of the current $B$-factory data for flavour-symmetry strategies, and discuss new, theoretically clean methods employing pure tree decays of neutral $B_{d,s}$ mesons.
Lepton-flavour symmetry in the Standard Model is broken by small masses for charged leptons and neutrinos. Introducing neutrino masses via dimension-5 operators associated to lepton-number violation at a very high scale, the corresponding coupling matrix may still have entries of order 1, resembling the situation in the quark sector with large top Yukawa coupling. As we have shown recently, in such a situation one may introduce the coupling matrices between lepton and Higgs fields as non-linear representations of lepton-flavour symmetry within an effective-theory framework. This allows us to s…
In a supersymmetric scenario, we study masses and mixings for leptons through the ${\bf Q}_{6}$ flavour symmetry. In the simplest case, the ${\bf M}_ν$ effective neutrino mass matrix, that comes from the type I see-saw mechanism, breaks the $μ\leftrightarrow τ$ interchange symmetry. As consequence, the reactor and atmospheric angles deviate from $0^{\circ}$ and $45^{\circ}$, respectively. At first glance, the model might accommodate very well the reactor and atmospheric angles in good agreement with the experimental data.
Discrete non-Abelian Symmetries have been extensively used to reproduce the lepton mixings. In particular, the S4 group turned out to be suitable to describe predictive mixing patterns, such as the well-known Tri-Bimaximal and the Bimaximal schemes, which all represent possible first approximations of the experimental lepton mixing matrix. We review the main application of the S4 discrete group as a flavour symmetry, first dealing with the formalism and later with the phenomenological implications. In particular, we summarize the main features of flavour models based on S4, commenting on their…
In split-family SUSY, one can use a U(2)^3 symmetry to protect flavour observables in the quark sector from SUSY contributions. However, attempts to extend this procedure to the lepton sector by using an analogous U(2)^5 symmetry fail to reproduce the neutrino data without introducing some form of fine-tuning. In this work, we solve this problem by shifting the U(2)^2 symmetry acting on leptons towards the second and third generations. This allows neutrino data to be reproduced without much difficulties, as well as protecting the leptonic flavour observables from SUSY. Key signatures are a $μ\…
We construct an extension of the Standard Model with a scalar leptoquark $ϕ\sim (3,1,-\tfrac13)$ and the discrete flavour symmetry $G_f=D_{17}\times Z_{17}$ to explain anomalies observed in charged-current semi-leptonic $B$ meson decays and in the muon anomalous magnetic moment, together with the charged fermion masses and quark mixing. The symmetry $Z_{17}^{\rm diag}$, contained in $G_f$, remains preserved by the leptoquark couplings, at leading order, and efficiently suppresses couplings of the leptoquark to the first generation of quarks and/or electrons, thus avoiding many stringent experi…
This study focuses on hard diffractive events produced in proton-proton collision at LHC exhibiting one intact proton in the final state which can be tagged by forward detectors. We report prospective results on the W boson charge asymmetry measured for such events, which allow to constrain the quark diffractive density functions in the Pomeron.
It is shown that a simple breaking of the subnuclear democracy leads to a successful description of the mixing between the second and third family. In the lepton channel the $ν_{μ} - ν_{τ}$ oscillations are expected to be described by a mixing angle of $2.65^ {\circ }$ which might be observed soon in neutrino experiments.
Theories with non-Abelian flavour symmetries lead at zeroth order to neutrino degeneracy and massless charged fermions. The flavour symmetry is spontaneously broken sequentially to give hierarchies $Δm_{atm}^2 \gg Δm_{\odot}^2$ and $m_τ\gg m_μ\gg m_e$, and a misalignment of the vacuum between neutrino and charged sectors gives large $θ_{atm}$. Explicit models are given which determine $θ_{atm} = 45^\circ$. A similar construction gives vacuum misalignment for the lighter generations and gives a vanishing $ββ_{0 ν}$ rate, so that there is no laboratory constraint on the amount of neutrino hot da…
Low scale seesaw models, like low scale type II, inverse (ISS), and linear seesaw (LSS) models provide an interesting mechanism to obtain light neutrino masses and mixings, as they can be tested in future TeV scale experiments. Discrete flavour symmetry groups like $A_4$ can be incorporated to explain the flavour structure of particles. But, so far, nothing is known about dynamics of flavour symmetry - scale of its breaking, or VEV alignment of the flavon fields. In a recent study [1], we have investigated and shed light on how to pinpoint the favoured VEV alignment of the flavon field using l…
