Fermion Mass Hierarchies, Large Lepton Mixing and Residual Modular Symmetries
SSISSA 09/2021/FISIIPMU21-0009CFTP/21-003
Fermion Mass Hierarchies, Large Lepton Mixingand Residual Modular Symmetries
P. P. Novichkov a, , J. T. Penedo b, , S. T. Petcov a,c, a SISSA/INFN, Via Bonomea 265, 34136 Trieste, Italy b CFTP, Departamento de F´ısica, Instituto Superior T´ecnico, Universidade de Lisboa,Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal c Kavli IPMU (WPI), University of Tokyo, 5-1-5 Kashiwanoha, 277-8583 Kashiwa,Japan
Abstract
In modular-invariant models of flavour, hierarchical fermion mass matrices mayarise solely due to the proximity of the modulus τ to a point of residual symme-try. This mechanism does not require flavon fields, and modular weights are notanalogous to Froggatt-Nielsen charges. Instead, we show that hierarchies depend onthe decomposition of field representations under the residual symmetry group. Wesystematically go through the possible fermion field representation choices whichmay yield hierarchical structures in the vicinity of symmetric points, for the foursmallest finite modular groups, isomorphic to S , A , S , and A , as well as fortheir double covers. We find a restricted set of pairs of representations for whichthe discussed mechanism may produce viable fermion (charged-lepton and quark)mass hierarchies. We present two lepton flavour models in which the charged-leptonmass hierarchies are naturally obtained, while lepton mixing is somewhat fine-tuned.After formulating the conditions for obtaining a viable lepton mixing matrix in thesymmetric limit, we construct a model in which both the charged-lepton and neu-trino sectors are free from fine-tuning. E-mail: [email protected] E-mail: [email protected] Also at Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784Sofia, Bulgaria. a r X i v : . [ h e p - ph ] F e b ontents τ sym = i ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.2 τ sym = i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1.3 τ sym = ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Decomposition under residual symmetries . . . . . . . . . . . . . . . . . . 113.3 Hierarchical structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3.1 From entries to masses . . . . . . . . . . . . . . . . . . . . . . . . . 133.3.2 Example and results . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 Promising models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4.1 A (cid:48) models with L ∼ , E c ∼ (cid:48) . . . . . . . . . . . . . . . . . . . . 183.4.2 S (cid:48) models with L ∼ ˆ2 ⊕ ˆ1 , E c ∼ ˆ3 (cid:48) . . . . . . . . . . . . . . . . . . 20 S (cid:48) models with τ (cid:39) ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Introduction
Understanding the origins of flavour in both the quark and lepton sectors, i.e., theorigins of the patterns of quark masses and mixing, of the charged-lepton and neutrinomasses, of neutrino mixing and of the CP violation in the two sectors is one of the mostchallenging unresolved fundamental problems in particle physics [1]. Within the reference three neutrino mixing scheme, the lepton flavour problem con-sists of three basic elements or sub-problems, namely, understanding:i) the origin of the hierarchical pattern of charged-lepton masses: m e (cid:28) m µ (cid:28) m τ , m e /m µ (cid:39) / m µ /m τ (cid:39) / m ν j are much smaller than the masses of charged leptons andquarks, m ν j ≪ m (cid:96),q , q = u, c, t, d, s, b and (cid:96) = e, µ, τ , with m ν j ∼ < . m (cid:96) ≥ . m q ∼ > m (cid:28) | ∆ m | with∆ m / | ∆ m | (cid:39) /
30, where ∆ m ij ≡ m i − m j .Each of these three sub-problems is by itself a formidable problem. As a consequence,individual solutions to each of them have been proposed. The hierarchical pattern ofcharged-lepton masses can most naturally be understood within the Froggatt-Nielsenmechanism based on the U (1) FN flavour symmetry [2] and its extensions. The enormousdisparity between the neutrino masses and the masses of the charged leptons and quarkscan be understood within the seesaw or radiative models of neutrino mass generation orelse employing the Weinberg effective operator idea [3] (for a concise review see, e.g., [4]).All these approaches lead naturally to massive Majorana neutrinos. Arguably the mostelegant and natural explanation of the observed pattern of neutrino (or lepton) mixingof two large and one small mixing angles is obtained within the non-Abelian discretesymmetry approach to the problem (see, e.g., [5–9]).In the case of the quark sector, the flavour problem similarly has two basic sub-problems, namely, understanding:i) the origins of the hierarchies of the masses of the charge 2 / − /
3) quarks;ii) the origins of the relatively small values of the three quark mixing angles.The most natural qualitative solution of these two problems is arguably provided bythe Froggatt-Nielsen approach [2], although the approach based on non-Abelian discretesymmetries has been applied to the quark flavour problem as well. Solutions to the twoflavour problems within the theories with extra dimensions have also been proposed. “Asked what single mystery, if he could choose, he would like to see solved in his lifetime, Weinbergdoesn’t have to think for long: he wants to be able to explain the observed pattern of quark and leptonmasses.” From Model Physicist , CERN Courier, 13 October 2017. A rather comprehensive discussion of the past proposed approaches to the lepton and quark flavourproblems can be found in the review article [1]. ν µ and ¯ ν µ by the Super-Kamiokandeexperiment [11] and lead, in particular, to the determination of the pattern of neutrinomixing, stimulated renewed attempts to seek alternative viable approaches to the leptonas well as to the quark flavour problems. A step in this direction was made in [12] wherethe idea of using modular invariance as a flavour symmetry was put forward. This neworiginal approach based on modular invariance opened up a new promising direction inthe studies of the flavour problem and correspondingly in flavour model building.The main feature of the approach proposed in [12] is that the elements of the Yukawacoupling and fermion mass matrices in the Lagrangian of the theory are modular formsof a certain level N which are functions of a single complex scalar field τ – the modulus– and have specific transformation properties under the action of the modular group.In addition, both the couplings and the matter fields (supermultiplets) are assumed totransform in representations of an inhomogeneous (homogeneous) finite modular groupΓ ( (cid:48) ) N . For N ≤
5, the finite modular groups Γ N are isomorphic to the permutation groups S , A , S and A (see, e.g., [13]), while the groups Γ (cid:48) N are isomorphic to the doublecovers of the indicated permutation groups, S (cid:48) ≡ S , A (cid:48) ≡ T (cid:48) , S (cid:48) and A (cid:48) . These discretegroups are widely used in flavour model building. The theory is assumed to possess themodular symmetry described by the finite modular group Γ ( (cid:48) ) N , which plays the role ofa flavour symmetry. In the simplest class of such models, the VEV of the modulus τ isthe only source of flavour symmetry breaking, such that no flavons are needed. Anotherappealing feature of the proposed framework is that the VEV of τ can also be the only3ource of breaking of the CP symmetry [14]. When the flavour symmetry is broken,the elements of the Yukawa coupling and fermion mass matrices get fixed, and a certainflavour structure arises. As a consequence of the modular symmetry, in the lepton sector,for example, the charged-lepton and neutrino masses, neutrino mixing and the leptonicCPV phases are simultaneously determined in terms of a limited number of couplingconstant parameters. This together with the fact that they are also functions of a singlecomplex VEV – that of the modulus τ – leads to experimentally testable correlationsbetween, e.g., the neutrino mass and mixing observables. Models of flavour based onmodular invariance have then an increased predictive power.The modular symmetry approach to the flavour problem has been widely imple-mented so far primarily in theories with global (rigid) supersymmetry. Within the SUSYframework, modular invariance is assumed to be a feature of the K¨ahler potential and thesuperpotential of the theory. Bottom-up modular invariance approaches to the leptonflavour problem have been exploited first using the groups Γ (cid:39) A [12,16], Γ (cid:39) S [17],Γ (cid:39) S [18]. After the first studies, the interest in the approach grew significantly andmodels based on the groups Γ (cid:39) S [19–26], Γ (cid:39) A [26–28], Γ (cid:39) A [23, 29–50],Γ (cid:39) S [51, 52] and Γ (cid:39) P SL (2 , Z ) [53] have been constructed and extensively stud-ied. Similarly, attempts have been made to construct viable models of quark flavour [54]and of quark-lepton unification [55–62]. The formalism of the interplay of modular andgCP symmetries has been developed and first applications made in [14]. It was exploredfurther in [63–65], as was the possibility of coexistence of multiple moduli [66–69], con-sidered first phenomenologically in [19, 30]. Such bottom-up analyses are expected toeventually connect with top-down results [70–91] based on ultraviolet-complete theories.While the aforementioned finite quotients Γ N of the modular group have been widelyused in the literature to construct modular-invariant models of flavour from the bottom-up perspective, top-down constructions typically lead to their double covers Γ (cid:48) N (see,e.g., [73, 75, 76, 92]). The formalism of such double covers has been developed and viableflavour models constructed in Refs. [93], [94, 95] and [96, 97] for the cases of Γ (cid:48) (cid:39) T (cid:48) ,Γ (cid:48) (cid:39) S (cid:48) and Γ (cid:48) (cid:39) A (cid:48) , respectively.In almost all phenomenologically viable flavour models based on modular invarianceconstructed so far the hierarchy of the charged-lepton and quark masses is obtained byfine-tuning some of the constant parameters present in the models. Perhaps, the onlynotable exceptions are Refs. [98, 99], in which modular weights are used as Froggatt-Nielsen charges, and additional scalar fields of non-zero modular weights play the roleof flavons.In the present article we develop the formalism that allows to construct models inwhich the fermion (e.g. charged-lepton and quark) mass hierarchies follow solely from theproperties of the modular forms present in the fermion mass matrices, thus avoiding thefine-tuning without the need to introduce extra fields. We consider theories described by Possible non-minimal additions to the K¨ahler potential, compatible with the modular symmetry, mayjeopardise the predictive power of the approach [15]. This problem is the subject of ongoing research. By fine-tuning we refer to either i) high sensitivity of observables to model parameters or ii) unjus-tified hierarchies between parameters which are introduced in the model on an equal footing. (cid:48) N with N ≤ N ). It was noticedin [19] and further exploited in [27, 30, 64] that for the three fixed points of the VEVof τ in the modular group fundamental domain, τ sym = i , τ sym = ω ≡ exp( i π/
3) = − / i √ / τ sym = i ∞ , the theories based on the Γ N invariancehave respectively Z S , Z ST , and Z TN residual symmetries. In the case of the double covergroups Γ (cid:48) N , the Z S residual symmetry is replaced by the Z S and there is an additional Z R symmetry that is unbroken for any value of τ (see [94] for further details). Theindicated residual symmetries play a crucial role in our analysis.The fermion mass matrices are strongly constrained in the points of residual symme-tries. This suggests that fine-tuning could be avoided in the vicinity of these points ifthe charged-lepton and quark mass hierarchies follow from the properties of the modularforms present in the corresponding fermion mass matrices rather than being determinedby the values of the accompanying constants also present in the matrices. Relativelysmall deviations of the modulus VEV from the symmetric point might also be neededto ensure the breaking of the CP symmetry [14].We note that in [100] flavour models in the vicinity of the residual symmetry fixedpoints, τ sym = i, ω, i ∞ , have been investigated within the modular invariant A frame-work ( N = 3). The authors find viable lepton (quark) flavour models in the vicinityof each of three residual symmetry values of τ sym (of τ sym = i ), in which the mixingarises seemingly without fine-tuning. At the same time, the charged-lepton and quarkmass hierarchies are obtained by fine-tuning the values of the constants present in therespective mass matrices.The aim of this study is to investigate the possibility of obtaining fermion masshierarchies – and, in models of lepton flavour, large mixing – without fine-tuning. Thearticle is structured as follows. After introducing the necessary tools in section 2, wedescribe how one can naturally generate hierarchical mass patterns in the vicinity ofsymmetric points in section 3.1. In section 3.2, the role of decompositions under theresidual symmetry groups is highlighted. We perform a systematic scan of attainablehierarchical patterns for N ≤
5, the results of which are reported in section 3.3. Theanalysis of two promising lepton flavour models in section 3.4 motivates the discussion,in section 4.1, of necessary conditions to avoid fine-tuned leptonic mixing. We are thendriven to a subset of viable models, the most promising of which is explored in section 4.2.We summarise our results and conclude in section 5.5
Framework
We start by briefly reviewing the modular invariance approach to flavour. In this su-persymmetric (SUSY) framework, one introduces a chiral superfield, the modulus τ ,transforming non-trivially under the modular group Γ ≡ SL (2 , Z ). The group Γ isgenerated by the matrices S = (cid:18) − (cid:19) , T = (cid:18) (cid:19) , R = (cid:18) − − (cid:19) , (2.1)obeying S = R , ( ST ) = R = , and RT = T R . Elements γ of the modular group acton τ via fractional linear transformations, γ = (cid:18) a bc d (cid:19) ∈ Γ : τ → γτ = aτ + bcτ + d , (2.2)while matter superfields transform as ‘weighted’ multiplets [12, 92, 101], ψ i → ( cτ + d ) − k ρ ij ( γ ) ψ j , (2.3)where k ∈ Z is the so-called modular weight and ρ is a unitary representation of Γ.In using modular symmetry as a flavour symmetry, an integer level N ≥ ρ ( γ ) = for elements γ of the principal congruence subgroupΓ( N ) ≡ (cid:26)(cid:18) a bc d (cid:19) ∈ SL (2 , Z ) , (cid:18) a bc d (cid:19) ≡ (cid:18) (cid:19) (mod N ) (cid:27) . (2.4)Hence, ρ is effectively a representation of the (homogeneous) finite modular group Γ (cid:48) N ≡ Γ (cid:14) Γ( N ) (cid:39) SL (2 , Z N ). For N ≤
5, this group admits the presentationΓ (cid:48) N = (cid:10) S, T, R | S = R, ( ST ) = , R = , RT = T R, T N = (cid:11) . (2.5)The (lowest component of the) modulus τ acquires a VEV which is restricted to theupper half-plane and plays the role of a spurion, parameterising the breaking of modularinvariance. Additional flavon fields are not required, and we do not consider them here.Since τ does not transform under the R generator, a Z R symmetry is preserved insuch scenarios. If also matter fields transform trivially under R , one may identify thematrices γ and − γ , thereby restricting oneself to the inhomogeneous modular group Γ ≡ P SL (2 , Z ) ≡ SL (2 , Z ) / Z R . In such a case, ρ is effectively a representation of a smaller(inhomogeneous) finite modular group Γ N ≡ Γ (cid:14) (cid:10) Γ( N ) ∪ Z R (cid:11) . For N ≤
5, this groupadmits the presentationΓ N = (cid:10) S, T | S = , ( ST ) = , T N = (cid:11) . (2.6)In general, however, R -odd fields may be present in the theory and Γ and Γ (cid:48) N are thenthe relevant symmetry groups. As shown in Table 1, the finite modular groups Γ N N S A S A Γ (cid:48) N S A (cid:48) ≡ T (cid:48) S (cid:48) ≡ SL (2 , Z ) A (cid:48) ≡ SL (2 , Z )dim M k (Γ( N )) k/ k + 1 2 k + 1 5 k + 1 Table 1:
Finite modular groups and dimensionality of the corresponding spaces ofmodular forms, for N ≤
5. Note that for N = 2 only even-weighted modular forms exist. and Γ (cid:48) N are isomorphic to permutation groups and to their double covers for small N .Group-theoretical results for the Γ N groups are collected in appendix B of [14], whilefor the double cover groups Γ (cid:48) N they can be found in Refs. [93, 94, 96].Finally, to understand how modular symmetry may constrain the Yukawa couplingsand mass structures of a model in a predictive way, we turn to the Lagrangian – whichfor an N = 1 global supersymmetric theory is given by L = (cid:90) d θ d ¯ θ K ( τ, ¯ τ, ψ I , ¯ ψ I ) + (cid:20) (cid:90) d θ W ( τ, ψ I ) + h.c. (cid:21) . (2.7)Here K is the K¨ahler potential, while the superpotential W can be expanded in powersof matter superfields ψ I , W ( τ, ψ I ) = (cid:88) (cid:16) Y I ...I n ( τ ) ψ I . . . ψ I n (cid:17) , (2.8)where one has summed over all possible field combinations and independent singlets ofthe finite modular group. By requiring the invariance of the superpotential under mod-ular transformations, one finds that the field couplings Y I ...I n ( τ ) have to be modularforms of level N . These are severely constrained holomorphic functions of τ , which undermodular transformations obey Y I ...I n ( τ ) γ −→ Y I ...I n ( γτ ) = ( cτ + d ) k ρ Y ( γ ) Y I ...I n ( τ ) . (2.9)Modular forms carry weights k = k I + . . . + k I n and furnish unitary representations ρ Y ofthe finite modular group such that ρ Y ⊗ ρ I ⊗ . . . ⊗ ρ I n ⊃ . Non-trivial modular formsof a given level exist only for k ∈ N , span finite-dimensional linear spaces M k (Γ( N )),and can be arranged into multiplets of Γ ( (cid:48) ) N . The fact that these spaces have low di-mensionalities for small values of k and N (as shown in Table 1) is at the root of thepredictive power of the described setup, since only a restricted number of τ -dependentYukawa textures are allowed in the superpotential.Note that modular forms are functions of τ and are thus invariant under R . Inorder to compensate the ( − k factor in eq. (2.9), odd-weighted forms must furnishrepresentations with ρ Y ( R ) = − (we use hats to denote such representations). Foreven-weighted modular forms, one has instead ρ Y ( R ) = . While we restrict ourselves to integer k , it is also possible for weights to be fractional [76, 102–104]. In theories of supergravity W transforms under the modular symmetry with a certain weight [92,101],shifting the required weights k of the modular forms. .2 Residual symmetries The breakdown of modular symmetry is parameterised by the VEV of the modulus andthere is no value of τ which preserves the full symmetry. Nevertheless, at certain so-called symmetric points τ = τ sym the modular group is only partially broken, with theunbroken generators giving rise to residual symmetries. Recall that the R generator isunbroken for any value of τ , so that a Z R symmetry is always preserved. − − / / Re τ / / p / I m τ ie πi/ i ∞D Figure 1:
The fundamental domain D of the modular group Γ and its three symmetricpoints τ sym = i ∞ , i, ω . The value of τ can always be restricted to D by a suitablemodular transformation. Figure from Ref. [94]. The fundamental domain D of the modular group is shown in Figure 1, along withits symmetric points. There are only three inequivalent symmetric points, namely [19]: • τ sym = i ∞ , invariant under T , preserving Z TN × Z R ; • τ sym = i , invariant under S , preserving Z S (recall that S = R ); and • τ sym = ω ≡ exp(2 πi/ ST , preserving Z ST × Z R .Finally, it is worth noting that these symmetric values preserve the CP symmetry ofa CP- and modular-invariant theory (e.g. a modular theory where the couplings satisfya reality condition) [14, 94]. A Z CP2 symmetry is preserved for Re τ = 0 or for τ lying onthe border of D , but is broken at generic values of τ . In theories where modular invariance is broken only by the modulus, the flavour structureof mass matrices in the limit of unbroken supersymmetry is determined by the value of8 and by the couplings in the superpotential. At a symmetric point τ = τ sym , flavourtextures can be severely constrained by the residual symmetry group, which may enforcethe presence of multiple zero entries in the mass matrices. As τ moves away from itssymmetric value, these entries will generically become non-zero. The magnitudes ofsuch (residual-)symmetry-breaking entries will be controlled by the size of the departure (cid:15) from τ sym and by the field transformation properties under the residual symmetrygroup (which may depend on the modular weights). This is shown in what follows.Consider a modular-invariant bilinear ψ ci M ( τ ) ij ψ j , (3.1)where the superfields ψ and ψ c transform under the modular group as ψ γ −→ ( cτ + d ) − k ρ ( γ ) ψ ,ψ c γ −→ ( cτ + d ) − k c ρ c ( γ ) ψ c , (3.2)so that each M ( τ ) ij is a modular form of level N and weight K ≡ k + k c . Modularinvariance requires M ( τ ) to transform as M ( τ ) γ −→ M ( γτ ) = ( cτ + d ) K ρ c ( γ ) ∗ M ( τ ) ρ ( γ ) † . (3.3)Taking τ to be close to the symmetric point, and setting γ to the residual symmetrygenerator, one can use this transformation rule to constrain the form of the mass matrix M ( τ ). We consider each of the three symmetric points in turn. τ sym = i ∞ The representation basis for the group generators S and T typically found in the liter-ature is the T -diagonal basis, in which ρ ( c ) ( T ) = diag( ρ ( c ) i ). This basis is particularlyuseful for the analysis of models where τ is ‘close’ to τ sym = i ∞ , i.e. models with largeIm τ . By setting γ = T in eq. (3.3), one finds M ij ( T τ ) = ( ρ ci ρ j ) ∗ M ij ( τ ) . (3.4)It is convenient to treat the M ij as a function of q ≡ exp (2 πiτ /N ), so that (cid:15) ≡ | q | = e − π Im τ/N (3.5)parameterises the deviation of τ from the symmetric point. Note that the entries M ij ( q )depend analytically on q and that q T −→ ζq , with ζ ≡ exp (2 πi/N ). Thus, in terms of q ,eq. (3.4) reads M ij ( ζq ) = ( ρ ci ρ j ) ∗ M ij ( q ) . (3.6) Note that in the case of a Dirac bilinear ψ and ψ c are independent fields, so in general k c (cid:54) = k and ρ c (cid:54) = ρ, ρ ∗ . q , one finds ζ n M ( n ) ij (0) = ( ρ ci ρ j ) ∗ M ( n ) ij (0) , (3.7)where M ( n ) ij denotes the n -th derivative of M ij with respect to q . This means that M ( n ) ij (0) can only be non-zero for values of n such that ( ρ ci ρ j ) ∗ = ζ n .It is clear that in the symmetric limit q → M ij = M (0) ij is only allowedto be non-zero if ρ ci ρ j = 1. More generally, if ( ρ ci ρ j ) ∗ = ζ l with 0 ≤ l < N , M ij ( q ) = a q l + a q N + l + a q N + l + . . . (3.8)in the vicinity of the symmetric point. It crucially follows that the entry M ij is expectedto be O ( (cid:15) l ) whenever Im τ is large. The power l only depends on how the representationsof ψ and ψ c decompose under the residual symmetry group Z TN . This point will be madeexplicit in section 3.2. τ sym = i For the analysis of models where τ is in the vicinity of τ sym = i , it is convenient toswitch to the basis where the S generator is represented by a diagonal matrix. In this S -diagonal basis, one has ρ ( c ) ( S ) = diag( ρ ( c ) i ). It is useful to define and work with˜ ρ ( c ) i ≡ i k ( c ) ρ ( c ) i , (3.9)which not only simplify the algebra, but also correspond to representations of the residualsymmetry group, see eq. (3.23). By setting γ = S in eq. (3.3), one finds M ij ( Sτ ) = ( − iτ ) K ( ˜ ρ ci ˜ ρ j ) ∗ M ij ( τ ) . (3.10)We now treat the M ij as functions of s ≡ τ − iτ + i , (3.11)so that, in this context, (cid:15) ≡ | s | parameterises the deviation of τ from the symmetricpoint. Note that the entries M ij ( s ) depend analytically on s and that s S −→ − s . Thus,in terms of s , eq. (3.10) reads M ij ( − s ) = (cid:18) s − s (cid:19) K ( ˜ ρ ci ˜ ρ j ) ∗ M ij ( s ) ⇒ ˜ M ij ( − s ) = ( ˜ ρ ci ˜ ρ j ) ∗ ˜ M ij ( s ) , (3.12)where we have introduced ˜ M ij ( s ) ≡ (1 − s ) − K M ij ( s ). Expanding both sides in powersof s , one obtains ( − n ˜ M ( n ) ij (0) = ( ˜ ρ ci ˜ ρ j ) ∗ ˜ M ( n ) ij (0) , (3.13) Although we make use of the same notation, the ρ ( c ) i depend on the basis under consideration. M ( n ) ij denotes the n -th derivative of ˜ M ij with respect to s .It should be clear from eq. (3.13) that for τ (cid:39) i the mass matrix entry M ij ∼ ˜ M ij isonly allowed to be O (1) when ˜ ρ ci ˜ ρ j = 1. If instead ˜ ρ ci ˜ ρ j = −
1, the entry M ij ∼ ˜ M ij isexpected to be O ( (cid:15) ), with (cid:15) = | s | . Note that, unlike in the previous section, the relevantfactors ˜ ρ ( c ) i depend on the weights k ( c ) via eq. (3.9). τ sym = ω Finally, for the analysis of models where τ is in the vicinity of τ sym = ω , we consider thebasis where the product ST is represented by a diagonal matrix. In this ST -diagonalbasis where ρ ( c ) ( ST ) = diag( ρ ( c ) i ), it is useful to define˜ ρ ( c ) i ≡ ω k ( c ) ρ ( c ) i , (3.14)which are representations under the residual symmetry group, see eq. (3.24). By setting γ = ST in eq. (3.3), one finds M ij ( ST τ ) = [ − ω ( τ + 1)] K ( ˜ ρ ci ˜ ρ j ) ∗ M ij ( τ ) . (3.15)It is now convenient to treat the M ij as functions of u ≡ τ − ωτ − ω , (3.16)so that, in this context, (cid:15) ≡ | u | parameterises the deviation of τ from the symmetricpoint. Note that the entries M ij ( u ) depend analytically on u and that u ST −−→ ω u . Thus,in terms of u , eq. (3.15) reads M ij ( ω u ) = (cid:18) − ω u − u (cid:19) K ( ˜ ρ ci ˜ ρ j ) ∗ M ij ( u ) ⇒ ˜ M ij ( ω u ) = (˜ ρ ci ˜ ρ j ) ∗ ˜ M ij ( u ) , (3.17)where ˜ M ij ( u ) ≡ (1 − u ) − K M ij ( u ). Expanding both sides in powers of u , one obtains ω n ˜ M ( n ) ij (0) = ( ˜ ρ ci ˜ ρ j ) ∗ ˜ M ( n ) ij (0) , (3.18)where ˜ M ( n ) ij denotes the n -th derivative of ˜ M ij with respect to u .It follows that for τ (cid:39) ω the mass matrix entry M ij ∼ ˜ M ij is only allowed to be O (1)when ˜ ρ ci ˜ ρ j = 1. More generally, if ˜ ρ ci ˜ ρ j = ω l with l = 0 , ,
2, then the entry M ij ∼ ˜ M ij isexpected to be O ( (cid:15) l ) in the vicinity of τ = ω , with (cid:15) = | u | . Like in the previous section,the factors ˜ ρ ( c ) i depend on the weights k ( c ) , see eq. (3.14). We have just shown that, as τ departs from a symmetric value τ sym , entries of the massmatrices are subject to corrections of O ( (cid:15) l ), where (cid:15) parameterises the deviation of τ τ sym . The powers l are extracted from products of factors which, in this section,are shown to correspond to representations of the residual symmetry group. One cansystematically identify these residual symmetry representations for the different possiblechoices of Γ (cid:48) N representations of matter fields. This knowledge will later be exploited toconstruct hierarchical mass matrices via controlled corrections to entries which are zeroin the symmetric limit.We start by noting that matter fields ψ furnish ‘weighted’ representations ( r , k ) ofthe finite modular group Γ (cid:48) N . Whenever a residual symmetry is preserved by the valueof τ , matter fields decompose into unitary representations of the residual symmetrygroup. Modulo a possible Z R factor, these groups are the cyclic groups Z TN , Z S , and Z ST (cf. section 2.2). A cyclic group Z n ≡ (cid:104) a | a n = 1 (cid:105) has n inequivalent 1-dimensionalirreducible representations (irreps) k , where k = 0 , . . . , n − a is represented by one of the n -th roots of unity, k : ρ ( a ) = exp (cid:18) πi kn (cid:19) . (3.19)For odd n , the only real irrep of Z n is the trivial one, 1 (the reality of an irrep isindicated by removing the boldface). For even n , there is one more real irrep, 1 n/ . Allother irreps are complex, and split into pairs of conjugated irreps: ( k ) ∗ = n − k .To illustrate the aforementioned decomposition of representations at symmetric points,we take as an example a ( , k ) triplet ψ of S (cid:48) . It transforms under the unbroken γ = ST at τ = ω as ψ i ST −−→ ( − ω − − k ρ ( ST ) ij ψ j = ω k ρ ( ST ) ij ψ j . (3.20)One can check that the eigenvalues of ρ ( ST ) are 1, ω and ω , and so in a suitable( ST -diagonal) basis the transformation rule explicitly reads ψ ST −−→ ω k ω
00 0 ω ψ = ω k ω k +1
00 0 ω k +2 ψ , (3.21)which means that ψ decomposes as ψ (cid:32) k ⊕ k +1 ⊕ k +2 under the residual Z ST .One can find the residual symmetry representations for any other ‘weighted’ multipletof a finite modular group in a similar fashion. For a given level N , the decompositionsof fields under a certain residual symmetry group only depend on the pair ( r , k ). Ingeneral: • At τ = i ∞ , ψ ∼ ( r , k ) transforms under the unbroken γ = T as ψ i T −→ ρ r ( T ) ij ψ j = ρ i ψ i , (3.22)where for the last equality we have assumed to be in a T -diagonal basis (no sumover i ). The phase factors ρ i correspond to the Z TN irreps into which ψ decomposes.It follows that each ρ i is a power of ζ = exp(2 πi/N ), depending on r but not on k . See the discussion in appendix A. At τ = i , ψ ∼ ( r , k ) transforms under the unbroken γ = S as ψ i S −→ ( − i ) − k ρ r ( S ) ij ψ j = i k ρ i ψ i , (3.23)where for the last equality we have assumed to be in an S -diagonal basis (no sumover i ). The phase factors ˜ ρ i = i k ρ i correspond to the Z S irreps into which ψ decomposes. It follows that each ˜ ρ i is a power of i which depends both on r andon k (mod 4). • At τ = ω , ψ ∼ ( r , k ) transforms under the unbroken γ = ST as ψ i ST −−→ ( − ω − − k ρ r ( ST ) ij ψ j = ω k ρ i ψ i , (3.24)where for the last equality we have assumed to be in an ST -diagonal basis (nosum over i ), as in the explicit example of eq. (3.21). The phase factors ˜ ρ i = ω k ρ i correspond to the Z ST irreps into which ψ decomposes. It follows that each ˜ ρ i isa power of ω which depends both on r and on k (mod 3).After identifying the ( ~ ) ρ i and ( ~ ) ρ ci factors for the fields ψ and ψ c entering a bilinear(equivalently, their irrep decompositions), one can apply the results of the previoussection to determine the structure of a mass matrix in the vicinity of a symmetric pointin terms of powers of (cid:15) , in the appropriate basis. It follows from the above that, in theanalysis with large Im τ , the product ( ρ ci ρ j ) ∗ matches some power ζ l with 0 ≤ l < N ,while in the analysis corresponding to τ (cid:39) ω one necessarily has ˜ ρ ci ˜ ρ j = ω l with l = 0 , , τ (cid:39) i context, ˜ ρ ci ˜ ρ j is some integer power i l , with l = 0 , , ,
3. It turns out that only two out of the four possibilities are viable, namely l = 0 , ρ ci ˜ ρ j = ±
1, as considered in section 3.1.2. This is due to the fact that M ( τ ) ij is R -even and thus the fields ψ ci and ψ j need to carry the same R -parity (seealso appendix A).We list in Tables 6 – 9 of appendix A the decompositions of the weighted repre-sentations of Γ (cid:48) N ( N ≤
5) under the three residual symmetry groups, i.e. the residualdecompositions of the irreps of Γ (cid:48) (cid:39) S , Γ (cid:48) (cid:39) A (cid:48) = T (cid:48) , Γ (cid:48) (cid:39) S (cid:48) = SL (2 , Z ), andΓ (cid:48) (cid:39) A (cid:48) = SL (2 , Z ). We are in a position to use the results found so far and construct hierarchical massmatrices in the vicinity of a symmetric point. We have seen that in an appropriate basis M ( τ ( (cid:15) )) ij ∼ O ( (cid:15) l ). For each ( i, j ) pair, the power l can be obtained from the residualsymmetry group decompositions of Tables 6 – 9.Note that a modular-symmetric mass matrix M ( τ ( (cid:15) )) depends analytically on thesmall real parameter (cid:15) , defined in section 3.1 for each symmetric point. Physical masses13re the singular values of M ( τ ) and are also analytic functions of (cid:15) . After SUSY-breaking, the leading superpotential contribution to each fermion mass is thus expectedto be proportional to a power of (cid:15) which depends on the hierarchical structure of theentries of M . To find out which, one can make use of the following set of relations, validfor any n × n complex matrix M [106]: (cid:88) i <...
0. Then, m ∼ (cid:88) i,j | M ij | = Tr M † M ,m m ∼ (cid:88) | det M × | ⇒ m ∼ (cid:80) | det M × | Tr M † M ,m m m = | det M | ⇒ m ∼ | det M | (cid:80) | det M × | , (3.26)where ∼ refers to power counting in (cid:15) and not necessarily to a reliable approximation.Note that so far the considered mass spectrum is generic. This is to be contrasted withthe special case of a hierarchical × d > d > d ≥ m (cid:28) m (cid:28) m . In this case, eqs. (3.26) turn into useful approximations, m (cid:39) (cid:88) i m i = Tr M † M ,m m (cid:39) (cid:88) i 5) and a common normalisation N . The power structure matches thatof eq. (3.28) and na¨ıvely this corresponds to the desired (1 , (cid:15), (cid:15) ) spectrum. Upon closerinspection, however, one realises that the determinant of M vanishes identically for any value of τ ,det M ∝ √ Y Y Y − Y Y + Y (cid:0) Y − √ Y Y (cid:1) + Y (cid:0) Y − Y Y (cid:1) = 0 , (3.30)meaning that at least one fermion is massless. In the vicinity of τ sym = i ∞ , we have( m , m , m ) ∼ (1 , (cid:15), K = 4, for which themodular multiplets Y (5 , , Y (5 , , , and Y (5 , , are available. In this case the spectrumfollows a (1 , (cid:15), (cid:15) ) pattern, without a massless fermion (see section 3.4.1).Let us pause and describe our philosophy going forward. We are interested in iden-tifying 3 × ψ and ψ c . While the representations r and r c are in general reducible, we focus on the casewhere the same weight is shared between the irreps into which they decompose. Thus,in our search, we take r ( c ) to be either irreducible or a direct sum of irreps sharing the The freedoms in choosing i) the normalisations of modular multiplets and ii) the normalisations ofClebsch-Gordan coefficients introduce ambiguities in the identification of hierarchies. In the interest ofminimizing their impact, when possible we make use of modular form multiplets obtained from tensorproducts of a single k = 1 multiplet with itself, via canonically normalised Clebsch-Gordan coefficients,as in [94]. Using this procedure, one expects that relative normalisations of modular multiplets cannotbe responsible for hierarchies, at least within the same weight k . ρ ( R ). While it is possible for r ( c ) to be a direct sum of hatted and unhatted rep-resentations, the requirement of a common weight k ( c ) would result in the co-existenceof R -odd and R -even fields within ψ ( c ) . The fact that M ( τ ) is R -even would then implythe isolation of these sectors by the Z R symmetry and the vanishing of some mixingangles.Finally, it is straightforward to recognise that if all mass matrix entries are either O (1) or O ( (cid:15) ), then leading contributions to the masses themselves are not expected tobe smaller than O ( (cid:15) ), unless one resorts to cancellations. Therefore, for τ (cid:39) i one cannotproduce the desired hierarchical patterns solely as a consequence of the smallness of (cid:15) .The result of our analysis is given in Tables 10 – 13 of appendix B. These tablessummarise, for each of the levels N ≤ 5, the patterns which may arise in the vicinityof the two potentially viable symmetric points, τ sym = ω and i ∞ , for all ( r , r c ) pairs of3-dimensional representations and all weights k ( c ) . One finds that it is only possible toobtain hierarchical spectra for a small list of representation pairs, the most promising ofwhich are collected here, in Table 2. N Γ (cid:48) N Pattern Sym. point Viable r ⊗ r c S (1 , (cid:15), (cid:15) ) τ (cid:39) ω [ ⊕ ( (cid:48) ) ] ⊗ [ ⊕ ( (cid:48) ) ⊕ (cid:48) ] τ (cid:39) ω [ a ⊕ a ⊕ (cid:48) a ] ⊗ [ b ⊕ b ⊕ (cid:48)(cid:48) b ]3 A (cid:48) (1 , (cid:15), (cid:15) ) τ (cid:39) i ∞ [ a ⊕ a ⊕ (cid:48) a ] ⊗ [ b ⊕ b ⊕ (cid:48)(cid:48) b ] with a (cid:54) = ( b ) ∗ S (cid:48) (1 , (cid:15), (cid:15) ) τ (cid:39) ω [ a , or ⊕ ( (cid:48) ) , or ˆ2 ⊕ ˆ1 ( (cid:48) ) ] ⊗ [ b ⊕ b ⊕ (cid:48) b ](1 , (cid:15), (cid:15) ) τ (cid:39) i ∞ ⊗ [ ⊕ , or ⊕ ⊕ (cid:48) ], (cid:48) ⊗ [ ⊕ (cid:48) , or ⊕ (cid:48) ⊕ (cid:48) ], ˆ3 (cid:48) ⊗ [ ˆ2 ⊕ ˆ1 , or ˆ1 ⊕ ˆ1 ⊕ ˆ1 (cid:48) ], ˆ3 ⊗ [ ˆ2 ⊕ ˆ1 (cid:48) , or ˆ1 ⊕ ˆ1 (cid:48) ⊕ ˆ1 (cid:48) ]5 A (cid:48) (1 , (cid:15), (cid:15) ) τ (cid:39) i ∞ ⊗ (cid:48) Table 2: Hierarchical mass patterns which can be realised in the vicinity of symmetricpoints. These patterns are unaffected by the exchange r ↔ r c and may only be viable forcertain weights (see appendix B). Subscripts run over irreps of a certain dimension, and (cid:48)(cid:48)(cid:48) a = a for N = 3, while (cid:48)(cid:48) a = a for N = 4. Primes in parenthesis are uncorrelated. We have excluded from this summary table reducible representations made up of threecopies of the same singlet, since in those cases at least three independent modular mul-tiplets of the same type must contribute to the mass matrix to avoid a massless fermion,and the number of superpotential parameters is unappealingly high. Still, such casesmay result in other interesting hierarchical patterns such as (1 , (cid:15) , (cid:15) ) and ( (cid:15), (cid:15) , (cid:15) ) andcan be found in the tables of appendix B.Inspired by these results, we now turn to the construction of realistic models of leptonflavour where mass hierarchies are a consequence of the described mechanism.16bservable Best-fit value and 1 σ range m e /m µ . ± . m µ /m τ . ± . δm / (10 − eV ) 7 . +0 . − . | ∆ m | / (10 − eV ) 2 . +0 . − . . +0 . − . r ≡ δm / | ∆ m | . ± . . ± . θ . +0 . − . . +0 . − . sin θ . +0 . − . . +0 . − . sin θ . +0 . − . . +0 . − . δ/π . +0 . − . . +0 . − . Table 3: Best-fit values and 1 σ ranges for neutrino oscillation parameters obtained fromthe global analysis [107], and for charged-lepton mass ratios, given at the scale 2 × GeV with the tan β averaging described in [12], obtained from Ref. [108]. The parametersentering the definition of r are δm ≡ m − m and ∆ m ≡ m − ( m + m ) / To ascertain the viability of modular-invariant models of lepton flavour one must confronttheir predictions with experimental data on ratios of charged-lepton masses, neutrinomass-squared differences and leptonic mixing angles, see Table 3 (the constraints on theDirac CPV phase δ are ignored in our fit). The reader is referred to [19] for details onour numerical procedure.The minimal-form K¨ahler potential is here considered, K ( τ, τ , ψ I , ψ I ) = − Λ log( − iτ + iτ ) + (cid:88) I | ψ I | ( − iτ + iτ ) k I , (3.31)with Λ of mass dimension one. We further take Higgs doublets H u and H d to be sin-glets under the modular group. Charged-lepton masses are obtained from their Yukawainteractions, W ⊃ (cid:88) s α s (cid:16) Y ( N,k Y ) r s ( τ ) E c L (cid:17) ,s H d , (3.32)where L and E c denote lepton doublet and charged-lepton singlet superfields withweights k L and k E , respectively. Neutrino masses are generated either by the Wein-berg operator, W ⊃ (cid:88) s g s (cid:16) Y ( N,k W ) r s ( τ ) L (cid:17) ,s H u , (3.33)17r within a type-I seesaw UV completion, W ⊃ (cid:88) s g s (cid:16) Y ( N,k Y ) r s ( τ ) N c L (cid:17) ,s H u + (cid:88) s Λ s (cid:16) Y ( N,k M ) r s ( τ ) ( N c ) (cid:17) ,s , (3.34)where at least 2 neutrino gauge-singlet superfields N c of weight k N are present in themodel. To compensate the modular weights of field monomials, the modular formsentering the Weinberg and Majorana terms need to have weights k W = 2 k L and k M =2 k N , while those in Yukawa terms need instead k Y = k L + k E and k Y = k L + k N .The relevant superpotentials can be cast in the form W = λ ij E ci L j H d + c ij L i L j H u (Weinberg) Y ij N ci L j + 12 ( M N ) ij N ci N cj (Seesaw) . (3.35)After electroweak symmetry breaking, with (cid:104) H u (cid:105) = (0 , v u ) T and (cid:104) H d (cid:105) = ( v d , T , theseresult in the Lagrangian mass terms for leptons L ⊃ − (cid:0) M e (cid:1) ij e iL e jR − (cid:0) M ν (cid:1) ij ν ciR ν jL + h.c. , (3.36)which have been written in terms of four-spinors. Here, M e = v d λ † , while M ν = v u c (Weinberg) − v u Y T M − N Y (Seesaw) . (3.37)Finally, aiming at minimal and predictive examples, we impose a generalised CP sym-metry on the models, enforcing the reality of coupling constants [14]. A (cid:48) models with L ∼ E c ∼ (cid:48) We start by considering the most ‘structured’ series of hierarchical models, i.e. the casewith both fields L , E c furnishing complete irreducible representations of the finite mod-ular group. According to Table 2, the only such possibility arises at level N = 5 in thevicinity of τ = i ∞ when L and E c are different triplets of the finite modular group A (cid:48) .For definiteness, we choose L ∼ ( , k L ), E c ∼ ( (cid:48) , k E ). The predicted charged-leptonmass pattern is ( m τ , m µ , m e ) ∼ (1 , (cid:15), (cid:15) ).We have performed a systematic scan restricting ourselves to models requiring mod-ular forms of weight not higher than k = 6, and involving at most 8 effective parameters(including Re τ and Im τ ). Models producing a massless electron are rejected. For neu-trino masses generated via a type I seesaw, we have considered gauge-singlet superfields N c furnishing a complete irrep of dimension 2 or 3. Out of the 36 models thus identified,we have selected the only one which i) is viable in the regime of interest, ii) produces18 charged-lepton spectrum which is not fine-tuned, and iii) is consistent with the ex-perimental bound on the Dirac CPV phase δ . For this model, k L = 3, k E = 1 and N c ∼ ( ˆ2 (cid:48) , W = (cid:104) α (cid:16) Y (5 , E c L (cid:17) + α (cid:16) Y (5 , , E c L (cid:17) + α (cid:16) Y (5 , , E c L (cid:17) (cid:105) H d + (cid:104) g (cid:16) Y (5 , ˆ6 , N c L (cid:17) + g (cid:16) Y (5 , ˆ6 , N c L (cid:17) + g (cid:16) Y (5 , ˆ6 , N c L (cid:17) (cid:105) H u + Λ (cid:16) Y (5 , (cid:48) ( N c ) (cid:17) . (3.38)The modular forms entering W are obtained from the lowest weight ( k = 1) sextet [97], Y (5 , ˆ6 = (cid:16) ε + θ , θ − ε , ε θ , √ ε θ , − √ ε θ , ε θ (cid:17) T , (3.39)where θ and ε are functions of the modulus, θ ( τ ) = 1 + 3 q / q / − q / 125 + . . . and ε ( τ ) = q (cid:0) − q / q / 25 + 37 q / 125 + . . . (cid:1) , with q = exp (2 πiτ / | q | is small and the sextet modular multiplet approximately reads Y (5 , ˆ6 (cid:39) (1 , , q, , , | (cid:15) | with (cid:15) ≡ ε/θ (cid:39) q , used only in the context of this section. The charged-leptonmass matrix is then approximated by M † e (cid:39) √ √ v d α θ (cid:0) ˜ α − ˜ α (cid:1) √ (cid:0) − + ˜ α − ˜ α (cid:1) (cid:15) √ (4 + ˜ α ) (cid:15) √ (cid:0) − + ˜ α − ˜ α (cid:1) (cid:15) − (cid:0) + ˜ α − α (cid:1) (cid:15) − (cid:0) + ˜ α − ˜ α (cid:1) (cid:15) √ (cid:0) − + ˜ α − α (cid:1) (cid:15) (2 − ˜ α ) (cid:15) − (cid:0) + ˜ α − ˜ α (cid:1) (cid:15) , (3.40) with ˜ α ≡ √ α /α , ˜ α ≡ √ √ α /α , matching the pattern in eq. (3.28). It follows thatthe charged-lepton mass ratios are given by m e m µ (cid:39) (cid:12)(cid:12)(cid:12)(cid:12) (4 + ˜ α − α ) (10 + 7 ˜ α − α ) (2 − α + 5 ˜ α )(2 − ˜ α ) (9 ˜ α − 10 ˜ α ) (cid:12)(cid:12)(cid:12)(cid:12) | (cid:15) | ,m µ m τ (cid:39) (cid:12)(cid:12)(cid:12)(cid:12) − ˜ α α − 10 ˜ α (cid:12)(cid:12)(cid:12)(cid:12) | (cid:15) | , (3.41)at leading order in | (cid:15) | . These expressions alone isolate viable ( (cid:15) -independent) regionsin the plane of ˜ α and ˜ α . Taking the 1 σ ranges for charged-lepton mass ratios fromTable 3, we plot these regions in Figure 2. The superimposed contours refer to theBarbieri-Giudice measure of fine-tuning [109] in the charged-lepton sector, max(BG),corresponding to the largest of four quantities | ∂ ln(mass ratio) /∂ ln ˜ α , | .Expansions similar to (3.40) for the neutrino Yukawa and mass matrices are notuseful since | (cid:15) | is not the only small parameter in the neutrino sector. In particular,some entries of M ν are proportional to 1 + (5 / g /g − √ g /g which is forced by the Note that fitting simultaneously mass ratio and mixing angle data may drive the model parametersto tuned values, even if no tuning is expected at the level of charged-lepton masses. igure 2: Values of the charged-lepton Yukawa couplings of the A (cid:48) model with largeIm τ which, when eq. (3.41) is applicable, allow to reproduce charged-lepton mass ratiosat 1 σ (green). The red regions are not accessible due to an upper limit on | (cid:15) | within thefundamental domain. Contours refer to a Barbieri-Giudice measure of fine-tuning (seetext). The yellow star shows the location of the best-fit point for this model. fit to be O (10 − ) in the viable region. Fine-tuning in the neutrino sector is expected (seesection 4.1) and is related to the fact that the residual symmetry constrains not only thecharged-lepton masses, but also the lepton mixing pattern. By sending Im τ → ∞ whilekeeping the couplings fixed to their best-fit values, one can check that the (13), (23),(31) and (32) entries of the PMNS matrix become zero. In particular, sin θ → 0, andtherefore the symmetric limit does not provide an approximate description of leptonicmixing.The result of the fit of this A (cid:48) model is summarised in the first column of Table 5,which includes best-fit values and the corresponding 3 σ ranges. The viable region in the τ plane is shown on the left side of Figure 5 and corresponds to a neutrino spectrumwith inverted ordering (IO). S (cid:48) models with L ∼ ˆ2 ⊕ ˆ1, E c ∼ ˆ3 (cid:48) In the second most ‘structured’ case, one of the fields L , E c is an irreducible triplet,while the other decomposes into a doublet and a singlet of the finite modular group.This possibility is realised at level N = 4 in the vicinity of τ = i ∞ , see Table 2. For defi-20iteness, we take L = L ⊕ L with L ∼ ( ˆ2 , k L ), L ∼ ( ˆ1 , k L ), and E c ∼ ( ˆ3 (cid:48) , k E ). Thecharged-lepton mass pattern in this regime is predicted to be ( m τ , m µ , m e ) ∼ (1 , (cid:15), (cid:15) ).We have performed a systematic scan restricting ourselves to models involving atmost 8 effective parameters (including Re τ and Im τ ; no limit on modular form weights).Once again, models predicting a massless electron are rejected, while the N c (whenpresent) furnish a complete irrep of dimension 2 or 3. Out of the 60 models thus iden-tified, we have selected the only one which i) is viable in the regime of interest and ii)produces a charged-lepton spectrum which is not fine-tuned. This model turns out tobe consistent with the experimental bound on the Dirac CPV phase. It corresponds to k L = k E = 2 and N c ∼ ( , 1) and the superpotential reads: W = (cid:104) α (cid:16) Y (4 , E c L (cid:17) + α (cid:16) Y (4 , (cid:48) E c L (cid:17) + α (cid:16) Y (4 , E c L (cid:17) (cid:105) H d + (cid:104) g (cid:16) Y (4 , ˆ3 N c L (cid:17) + g (cid:16) Y (4 , ˆ3 (cid:48) N c L (cid:17) + g (cid:16) Y (4 , ˆ3 (cid:48) N c L (cid:17) (cid:105) H u + Λ (cid:16) Y (4 , ( N c ) (cid:17) . (3.42)The modular forms entering W are obtained from the lowest weight ( k = 1) triplet [94], Y (4 , ˆ3 = (cid:16) √ εθ, ε , − θ (cid:17) T , (3.43)where θ ( τ ) = 1 + 2 q + 2 q + . . . and ε ( τ ) = 2 q + 2 q + 2 q + . . . , with q = exp ( iπτ / | (cid:15) | with (cid:15) ≡ ε/θ (cid:39) q , the charged-lepton mass matrix is approximatelygiven by M † e (cid:39) √ v d α θ (cid:15) ( ˜ α + √ ) √ (cid:15) ( α −√ ) √ (cid:15) − ˜ α ( √ α +9 ) √ (cid:15) ( √ α − ) √ (cid:15) ˜ α (cid:15) − ˜ α √ (cid:15) ˜ α √ (cid:15) , (3.44)with ˜ α ≡ α /α and ˜ α ≡ α /α . It matches the expected power structure in | q | , asone can check. One can also find approximate expressions for the charged-lepton massratios, which read m e m µ (cid:39) √ (cid:12)(cid:12) ˜ α ( ˜ α − (cid:12)(cid:12) | ˜ α | (cid:0) ( ˜ α + √ + 12 ˜ α (cid:1) | (cid:15) | ,m µ m τ (cid:39) (cid:114) (cid:113) ( ˜ α + √ + 12 ˜ α | ˜ α | | (cid:15) | . (3.45)The expressions (3.45) isolate viable ( (cid:15) -independent) regions in the plane of couplingconstants, say ˜ α − = α /α and ˜ α / ˜ α = α /α . We plot these regions in Figure 3, Aside from consulting the third column of Table 8, one must keep in mind the ordering of the ρ ( c ) i ,which depends on the representation basis for the group generators (we are using that of [94]). igure 3: Values of the charged-lepton Yukawa couplings of the S (cid:48) model with largeIm τ which, when eq. (3.45) is applicable, allow to reproduce charged-lepton mass ratiosat 1 σ (green). The red regions are not accessible due to an upper limit on | (cid:15) | within thefundamental domain. Contours refer to the Barbieri-Giudice measure of fine-tuning (seetext). The yellow star shows the location of the best-fit point for this model. including contours quantifying the degree of fine-tuning involved in the relation betweencharged-lepton mass ratios and superpotential parameters (as described in the previoussection). Note that the model best-fit point in particular corresponds to a small valueof max(BG) (cid:39) . a | (cid:15) | , b − b | (cid:15) | , b + b | (cid:15) | ), which naturally leadsto IO. However, we were unable to find viable regions with IO and, instead, the modelpredicts a neutrino spectrum with normal ordering (NO). Within the viable region, theapproximate pattern ( a | (cid:15) | , b − b | (cid:15) | , b + b | (cid:15) | ) is not accurate since | (cid:15) | is not theonly small parameter in the neutrino sector. In particular, some of the entries of M ν areproportional to (1+ √ g /g ) which is forced to be O (10 − ) in our fit. As in the previoussection, the fine-tuning in the neutrino sector is explained by the necessity to introducelarge corrections to the symmetric-limit PMNS matrix, which has a zero either in the(31) or (33) entry in the case of NO or IO, respectively.The result of the fit of this S (cid:48) model is summarised in the second column of Table 5.The viable region in the τ plane is located near the point τ = 2 . i and is shown on theleft side of Figure 5. 22 Large mixing angles without fine-tuning We have seen in the previous sections that a slightly broken residual modular symmetryallows to accommodate hierarchical charged-lepton masses without fine-tuning of thecorresponding couplings. However, the resulting models are still subject to fine-tuningin the neutrino sector, since residual symmetries typically constrain not only the charged-lepton masses, but also the form of the PMNS matrix by forcing some of its entries to bezeros. This raises the question of whether it is possible to have a PMNS matrix whichis close to the observed one even in the symmetric limit, i.e. such that either none of itsentries vanish, or only the (13) entry vanishes as (cid:15) → L and E c give rise to aPMNS matrix which is viable in the symmetric limit (as defined above). Most notably,there are only two such cases consistent with hierarchical charged-lepton masses:1. L (cid:32) ⊕ ⊕ E c (cid:32) ⊕ r , where 1 is some real singlet of the flavour symmetry,and r is some (possibly reducible) representation such that r (cid:54)⊃ L (cid:32) ⊕ ⊕ ∗ , E c (cid:32) ∗ ⊕ r , where is some complex singlet of the flavoursymmetry, ∗ is its conjugate, and r is some (possibly reducible) representationsuch that r (cid:54)⊃ , ∗ .The above original result makes use of the assumption that one charged-lepton massand at least one neutrino mass does not vanish in the symmetric limit. However, one canalso deduce from the analysis performed in [110] that the PMNS matrix is genericallyunconstrained in the symmetric limit when the opposite is true. Therefore, we extendthe list of viable cases with the following two:3. all charged-lepton masses vanish in the symmetric limit, i.e. the correspondinghierarchical pattern involves only positive powers of (cid:15) , e.g. ( (cid:15), (cid:15) , (cid:15) );4. all light neutrino masses vanish in the symmetric limit, i.e. L decomposes intothree (possibly identical) complex singlets none of which are conjugated to eachother.It follows that a modular-symmetric model of lepton flavour with hierarchical charged-lepton masses may be free of fine-tuning if it satisfies any of the properties 1-4. Applyingthis filter to the promising hierarchical cases of Table 2, one is left with the representa-tion pairs listed here, in Table 4. In this summary table, we have once again disregardedreducible representations made up of three copies of the same singlet.We now proceed by constructing such a model in the following section (clearly, themodels described in section 3.4 do not satisfy any of the properties 1-4). As a final23 Γ (cid:48) N Pattern Sym. point Viable r E c ⊗ r L Case2 S (1 , (cid:15), (cid:15) ) τ (cid:39) ω [ ⊕ ( (cid:48) ) ] ⊗ [ ⊕ ( (cid:48) ) ⊕ (cid:48) ] 1 or 4 τ (cid:39) ω [ a ⊕ a ⊕ (cid:48) a ] ⊗ [ b ⊕ b ⊕ (cid:48)(cid:48) b ] 2[ ⊕ ⊕ (cid:48) ] ⊗ [ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48) ],3 A (cid:48) (1 , (cid:15), (cid:15) ) τ (cid:39) i ∞ [ ⊕ ⊕ (cid:48)(cid:48) ] ⊗ [ (cid:48) ⊕ (cid:48) ⊕ (cid:48)(cid:48) ] 24 S (cid:48) (1 , (cid:15), (cid:15) ) τ (cid:39) ω [ a , or ⊕ ( (cid:48) ) , or ˆ2 ⊕ ˆ1 ( (cid:48) ) ] ⊗ [ b ⊕ b ⊕ (cid:48) b ] 1 or 45 A (cid:48) − − − − Table 4: Hierarchical charged-lepton mass patterns which may be realised in the vicinityof symmetric points without fine-tuned mixing (PMNS close to the observed one in thesymmetric limit). The property which is satisfied (from 1-4, see text) is given in the lastcolumn and may depend on the weights k and k c . The case N = 3 with τ (cid:39) ω is theonly one in the table for which r E c ↔ r L may be required, and for which not all k ( c ) choices are viable. For other notation, see the caption of Table 2. remark, we note that the argument of Ref. [110] is only valid in the case when the flavoursymmetry analysis can be applied directly to the light neutrino mass matrix. In oursetup, this corresponds to the situation when light neutrino masses arise either directlyfrom a modular-invariant Weinberg operator, or via a type-I seesaw UV completion suchthat none of the gauge-singlet neutrinos N c becomes massless in the symmetric limit (sothat they can be integrated out). This is the case in the two models considered so farand in the model described in the following section. S (cid:48) models with τ (cid:39) ω We finally turn to the most ‘structured’ cases within the surviving lepton flavour modelsof Table 4. These arise at level N = 4 in the vicinity of τ = ω and correspond to E c and L being a triplet and the direct sum of three singlets of the finite modular group S (cid:48) , respectively. The expected charged-lepton mass pattern is ( m τ , m µ , m e ) ∼ (1 , (cid:15), (cid:15) ).We have performed a systematic scan restricting ourselves to promising models in-volving the minimal number of effective parameters (9, including Re τ and Im τ ). Onceagain, models predicting a massless electron are rejected, while the N c furnish a com-plete irrep of dimension 2 or 3 ( N c are present since Weinberg models require moreparameters). Out of 48 models, we have identified the only one which i) is viable in theregime of interest, ii) is not fine-tuned in this regime, and iii) is consistent with the 2 σ range for the Dirac CPV phase, predicting δ (cid:39) π while other models lead to δ (cid:39) 0. Forthis model, L = L ⊕ L ⊕ L with L , L ∼ ( ˆ1 , L ∼ ( ˆ1 (cid:48) , E c ∼ ( ˆ3 , 4) and24 c ∼ ( (cid:48) , W = (cid:104) α (cid:16) Y (4 , (cid:48) , E c L (cid:17) + α (cid:16) Y (4 , (cid:48) , E c L (cid:17) + α (cid:16) Y (4 , (cid:48) , E c L (cid:17) + α (cid:16) Y (4 , (cid:48) , E c L (cid:17) + α (cid:16) Y (4 , E c L (cid:17) (cid:105) H d + (cid:104) g (cid:16) Y (4 , ˆ3 N c L (cid:17) + g (cid:16) Y (4 , ˆ3 N c L (cid:17) + g (cid:16) Y (4 , ˆ3 (cid:48) N c L (cid:17) (cid:105) H u + Λ (cid:16) Y (4 , ( N c ) (cid:17) . (4.1)Since L and L are indistinguishable, one of the constants α i , with i = 1 , . . . , 4, iseffectively not an independent parameter and can be set to zero by a suitable rotationwithout loss of generality. We choose to set α = 0.At leading order in the small parameter | (cid:15) | , with (cid:15) ≡ − √ − i εθ and | (cid:15) | (cid:39) . (cid:12)(cid:12)(cid:12) τ − ωτ − ω (cid:12)(cid:12)(cid:12) in the context of this section, the charged-lepton mass matrix reads M † e (cid:39) − √ − √ v d α θ α + √ ˜ α i √ ˜ α √ (cid:15) √ (cid:16) ˜ α − √ ˜ α (cid:17) (cid:15) i √ ˜ α (cid:15) (cid:15) (cid:0) 10 ˜ α + √ 13 ˜ α (cid:1) (cid:15) − i √ √ ˜ α (cid:15) , (4.2)while the charged-lepton mass ratios are given by m e m µ (cid:39) | ˜ α ˜ α | (cid:113) (cid:0) α + √ 13 ˜ α (cid:1) + 39 ˜ α α + (cid:104) (cid:0) ˜ α − √ 13 ˜ α (cid:1) (cid:105) ˜ α | (cid:15) | ,m µ m τ (cid:39) √ (cid:114) α + (cid:104) (cid:0) ˜ α − √ 13 ˜ α (cid:1) (cid:105) ˜ α (cid:0) α + √ 13 ˜ α (cid:1) + 39 ˜ α | (cid:15) | , (4.3)with ˜ α i ≡ α i /α , i = 3 , , 5. With respect to charged-lepton mass ratios, the modelbest-fit point is found to correspond to max(BG) (cid:39) . K , the light neutrino mass matrix is instead given by: M ν (cid:39) K (cid:15) g g ˜ g ˜ g ˜ g ˜ g i (cid:113) ˜ g (4.4)at leading order in | (cid:15) | , where ˜ g i ≡ g i /g , i = 2 , 3. Note that the smallness of | (cid:15) | doesnot constrain the M ν contribution to the mixing matrix, which depends only on thecouplings g i , and large mixing angles are allowed.From the form of M ν it is clear that, in the limit of unbroken SUSY, there is amassless neutrino, even though N c is a triplet. This follows from the modular-symmetric25uperpotential, which implies the proportionality of the first two columns of Y , reducingits rank and therefore the rank of M ν . The neutrino masses thus read m = 0 , m , (cid:39) (cid:114) K ˜ g (cid:32)(cid:115) g )2˜ g ∓ (cid:33) | (cid:15) | , (4.5)and imply the (cid:15) -independent prediction r = m − m m − ( m + m ) / (cid:39) g + 8˜ g − (cid:113) (cid:0) g (cid:1) + 4˜ g | ˜ g | g + 4˜ g + 6 (cid:113) (cid:0) g (cid:1) + 4˜ g | ˜ g | , (4.6)which, by taking into account the 1 σ range for r in Table 3, isolates a viable region inthe plane of coupling constants. We show this region in the plane of g /g and g /g in Figure 4. Contours refer to a Barbieri-Giudice measure of fine-tuning for the ratio ofneutrino mass-squared differences, given by max {| ∂ ln r/∂ ln ˜ g | , | ∂ ln r/∂ ln ˜ g |} . At themodel best-fit point, it has an acceptable value of 2 . 9. Additionally, the 3 σ ranges for˜ g , are not especially narrow.The result of the fit of this S (cid:48) model is summarised in the last column of Table 5.The viable region in the τ plane corresponds to a neutrino spectrum with NO and islocated very close to τ sym = ω , as can be seen from the magnified plot on the right sideof Figure 5.In summary, in the vicinity of the symmetric point, i.e. for small | (cid:15) | , this modelcan naturally lead to the observed charged-lepton mass hierarchies, see eq. (4.3). Theneutrino mass-squared difference ratio r is, in this region, insensitive to (cid:15) and dependsonly on the two ratios ˜ g , of neutrino couplings, see eq. (4.6). Furthermore, it is notespecially sensitive to these couplings. Finally, since light neutrino masses vanish inthe symmetric point, the symmetric limit allows for a generic mixing matrix (case 4 ofsection 4.1). Therefore, the fit is not expected to be tuned in a way that compensatessome ‘wrong PMNS’ symmetric prediction. In fact, we have numerically verified thatsending τ → ω ( (cid:15) → 0) has almost no effect on the values of mixing angles. This canbe understood by considering, in turn, each of the contributions to the mixing matrix.The rotation to the mass basis in the neutrino sector, on the one hand, is independentof (cid:15) in the region of interest, see eq. (4.4), and thus has a well-defined limit as (cid:15) → g , of neutrino couplings. On the other hand, one can check thatthe charged-lepton rotation arising from the diagonalisation of M e M † e , with M † e givenin eq. (4.2), also has a well-defined limit as (cid:15) → (cid:15) , and depends only on the ratios ˜ α , , of charged-lepton couplings.26 igure 4: Values of the neutrino Yukawa couplings of the S (cid:48) model with τ (cid:39) ω which,when eq. (4.6) is applicable, allow to reproduce the ratio r at 1 σ (green). Contoursrefer to a Barbieri-Giudice measure of fine-tuning (see text). The yellow star shows thelocation of the best-fit point for this model. Figure 5: Allowed regions in the τ plane for the models discussed in sections 3.4 and 4.2(left). The region corresponding to the model of section 4.2 is magnified (right). Thegreen, yellow and red colours correspond to different confidence levels (see legend). Pointsoutside the fundamental domain, while redundant, are kept for illustrative purposes. odel Section 3.4.1 ( A (cid:48) ) Section 3.4.2 ( S (cid:48) ) Section 4.2 ( S (cid:48) )Re τ − . +0 . − . . +0 . − . − . +0 . − . Im τ . +0 . − . . +0 . − . . +0 . − . α /α . +0 . − . − . +2 . − . — α /α . +0 . − . . +5 . − . . +0 . − . α /α — — − . +0 . − . α /α — — 1 . +0 . − . g /g − . +0 . − . − . +0 . − . . +0 . − . g /g . +0 . − . . +0 . − . . +0 . − . v d α , GeV 0 . +0 . − . . +1 . − . . +1 . − . v u g / Λ, eV 0 . +1 . − . . +9 . − . . +0 . − . (cid:15) ( τ ) 0 . +0 . − . . +0 . − . . +0 . − . CL mass pattern (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) )max(BG) 5 . 579 0 . 738 0 . m e /m µ . +0 . − . . +0 . − . . +0 . − . m µ /m τ . +0 . − . . +0 . − . . +0 . − . r . +0 . − . . +0 . − . . +0 . − . δm , 10 − eV . +0 . − . . +0 . − . . +0 . − . | ∆ m | , 10 − eV . +0 . − . . +0 . − . . +0 . − . sin θ . +0 . − . . +0 . − . . +0 . − . sin θ . +0 . − . . +0 . − . . +0 . − . sin θ . +0 . − . . +0 . − . . +0 . − . m , eV 0 . +0 . − . . +0 . − . m , eV 0 . +0 . − . . +0 . − . . +0 . − . m , eV 0 0 . +0 . − . . +0 . − . Σ i m i , eV 0 . +0 . − . . +0 . − . . +0 . − . |(cid:104) m (cid:105)| , eV 0 . +0 . − . . +0 . − . . +0 . − . δ/π . +0 . − . . +0 . − . ± O (10 − ) α /π . +0 . − . . +0 . − . α /π . +0 . − . ± O (10 − ) N σ . 431 0 . 649 0 . Table 5: Best-fit values and 3 σ ranges of the parameters and observables for the modelsdiscussed in sections 3.4 and 4.2 Summary and Conclusions We have investigated the possibility to obtain fermion mass hierarchies without fine-tuning in modular-invariant theories of flavour, which do not include flavons. In thesetheories, hierarchical fermion mass matrices may arise solely due to the proximity ofthe VEV of the modulus τ to a point of residual symmetry. In our analysis we haveconsidered theories with flavour symmetry described by a finite inhomogeneous or homo-geneous modular group, Γ N or Γ (cid:48) N , with N ≤ 5. For N ≤ 5, the finite modular groupsΓ N are isomorphic to the permutation groups Γ (cid:39) S , Γ (cid:39) A , Γ (cid:39) S and Γ (cid:39) A ,while the groups Γ (cid:48) N are isomorphic to their double covers S (cid:48) ≡ S , A (cid:48) ≡ T (cid:48) , S (cid:48) and A (cid:48) .In the simplest class of such models considered by us, the VEV of the modulus τ isthe only source of flavour symmetry breaking, such that no flavons are needed. Anotherappealing feature of the proposed framework is that the VEV of τ can also be theonly source of CP symmetry breaking in the theory [14]. There is no value of τ whichpreserves the full modular symmetry. Nevertheless, at certain so-called symmetric points τ = τ sym the modular group is only partially broken, with the unbroken generators givingrise to residual symmetries. There are only three inequivalent symmetric points in thefundamental domain of the modular group [19]: τ sym = i , τ sym = ω ≡ exp( i π/ 3) = − / i √ / τ sym = i ∞ . In these three points, the theories basedon Γ N invariance have respectively Z S , Z ST and Z TN residual symmetries. In the case ofthe double cover groups Γ (cid:48) N , there is an additional Z R symmetry that is unbroken forany value of τ [94], thus enlarging the residual symmetries Z ST and Z TN by the factor Z R , while the Z S symmetry is enlarged to a Z S one. In each of the three symmetricpoints the standard Z CP2 symmetry may also be conserved [14].The indicated residual symmetries play a crucial role in our analysis. In Γ ( (cid:48) ) N modularinvariant theories of flavour the fermion mass matrices are modular forms of a given level N . As we show, the mass matrices can be strongly constrained in the vicinity of pointsof residual symmetries by the properties of the respective modular forms. For each of thethree symmetric points, we have developed the formalism which allows to determine thedegree of suppression of the elements of the fermion mass matrices, and correspondingly,of their singular values – the fermion masses – in the vicinity of a given symmetric point.More specifically, our analysis showed that, if (cid:15) parameterises the deviation of τ from agiven symmetric point, | (cid:15) | (cid:28) 1, the degree of suppression is given by | (cid:15) | l , where l is aninteger and can take valuesi) l = 0 , , ..., N − τ sym = i ∞ ,ii) l = 0 , , τ sym = ω , andiii) l = 0 , τ sym = i .These results show, in particular, that it is impossible to obtain the charged-lepton andquark mass hierarchies in the vicinity of the symmetric point τ sym = i as a consequenceonly of the smallness of | (cid:15) | . As we have proven, the specific value of the power l dependsonly on how the representations of the fermion fields in the mass term bilinear, denoted29or brevity as ψ i and ψ cj , decompose under the considered residual symmetry group.We have derived the decompositions of the weighted irreducible representations of Γ (cid:48) N ( N ≤ 5) under the three residual symmetry groups, i.e., the residual decompositions ofthe irreducible representations (irreps) of Γ (cid:48) (cid:39) S , Γ (cid:48) (cid:39) A (cid:48) = T (cid:48) , Γ (cid:48) (cid:39) S (cid:48) = SL (2 , Z ),and Γ (cid:48) (cid:39) A (cid:48) = SL (2 , Z ) (they are listed in Tables 6 – 9 in Appendix A). The resultsinclude also the case of irreps of Γ N , since they represent a subset of the irreps of Γ (cid:48) N .Having these results we proceeded to identify 3 × ψ ci M ( τ ) ij ψ j , M being the mass matrix, and considered all possible3-dimensional representations for the fields ψ i and ψ cj , i, j = 1 , , 3. While the represen-tations of these fields r and r c are in general reducible, we focused on the case wherethe same weight is shared between the irreps into which they decompose. The resultsof this analysis are given in Tables 10 – 13 of Appendix B. These tables summarise, foreach of the levels N ≤ 5, the patterns of the three fermion masses which may arise inthe vicinity of the two potentially viable symmetric points, τ sym = ω and i ∞ , for all( r , r c ) pairs of 3-dimensional representations and all weights k ( c ) . We have found thatit is only possible to obtain hierarchical spectra for a small list of representation pairs,the most promising of which are collected in Table 2 and correspond to the patterns(1 , (cid:15), (cid:15) ), (1 , (cid:15), (cid:15) ) and (1 , (cid:15), (cid:15) ).Using the developed formalism and the aforementioned results we have performed ascan searching for phenomenologically viable models of lepton flavour where the charged-lepton mass hierarchies are a consequence of the described mechanism. The charged-lepton masses were obtained from their Yukawa interactions, while neutrino masses aregenerated either by the Weinberg operator or within a type-I seesaw UV completion.Aiming at minimal and predictive models, we have imposed a generalised CP symmetryon the models, enforcing the reality of coupling constants [14] and restricted the scanto models involving at most 8 constant parameters (including Re τ and Im τ ). Modelsproducing a massless electron were rejected. Out of the many models thus identified, wehave selected only those which i) are phenomenologically viable in the regime of interest,and ii) produce a charged-lepton spectrum which is not fine-tuned.We have found two viable models and both are in the vicinity of τ = i ∞ . The firstis an A (cid:48) model where the lepton doublets L and charged-lepton singlets E c are differenttriplets of A (cid:48) . For this model, the charged-lepton, Dirac-neutrino and N c Majorana massterms involve modular forms of weights 4, 5 and 4, respectively. The neutrino masses aregenerated by the seesaw mechanism with the gauge-singlet neutrino fields N c furnishinga doublet of A (cid:48) . The predicted charged-lepton mass pattern is ( m τ , m µ , m e ) ∼ (1 , (cid:15), (cid:15) ).The best description of the input data on the charged-lepton and neutrino masses andmixing ( N σ = 0 . 43) was found to be obtained for Re τ = − . 47, Im τ = 3 . S (cid:48) modular symmetry. In this modelthe charged-lepton, Dirac-neutrino and N c Majorana mass terms involve modular formsof weights 4, 3 and 2, respectively. The charged-lepton mass pattern is predicted to be( m τ , m µ , m e ) ∼ (1 , (cid:15), (cid:15) ). The viable region in the τ plane is centred around τ = 2 . i .30oth the A (cid:48) and S (cid:48) viable models were found to require a certain amount of fine-tuning when describing the neutrino masses and mixing. The presence of fine-tuning inthe neutrino sector is explained by the necessity to introduce large corrections to thesymmetric-limit PMNS matrix. Addressing the problem of fine-tuning in the neutrinosector, we have found that a modular-symmetric model of lepton flavour with hierarchicalcharged-lepton masses is expected to be free of fine-tuning if it satisfies at least oneof four conditions (see section 4.1). Two of the conditions were formulated earlier inRef. [110] for arbitrary flavour symmetry groups.We have constructed a viable model based on S (cid:48) modular symmetry in the vicinityof τ = ω , which is free of fine-tuning in both the charged-lepton and neutrino sectors.It has altogether nine parameters. The neutrino masses are generated via the seesawmechanism. The charged-lepton mass pattern is predicted to be ( m τ , m µ , m e ) ∼ (1 , (cid:15), (cid:15) ).The model predicts, in particular, δ (cid:39) π and m (cid:39) 0. We have found also other viablenon–fine-tuned S (cid:48) models which, however, predict δ (cid:39) σ allowed ranges of i) Re τ , Im τ and the superpotential parameters, of ii)the charged-lepton masses and neutrino mass and mixing observables used as input inthe statistical analysis of the models, and of iii) the predicted lightest neutrino mass,the Dirac and Majorana CPV phases, the sum of the neutrino masses and effectiveneutrinoless double beta decay Majorana mass.The results obtained in the present article show, in particular, that the requirementof absence of fine-tuning in both the charged lepton and neutrino sectors in lepton flavourmodels based on modular invariance is remarkably restrictive. It is hoped that usingthis requirement it might be possible to identify not more than a few, if not just one,modular-invariant models providing a simultaneous, viable and appealing solution toboth the lepton and quark flavour problems. Note Added. While this work was in its conclusion, Ref. [111] appeared on the arXivin which the authors investigated the possibility to generate the charged-lepton masshierarchy in the vicinity of the symmetric point τ sym = i . The two models presented inRef. [111] are restricted to level N = 3. In these scenarios, the electron is massless byconstruction, m e = 0, and m e (cid:54) = 0 is expected to arise either through SUSY breaking orfrom a dim-6 operator. The ratios m e /m τ and m µ /m τ are then associated to independentparameters and not to different powers of the same expansion parameter, as is the caseof our work. Acknowledgements This project has received funding/support from the European Union’s Horizon 2020 re-search and innovation programme under the Marie Sk(cid:32)lodowska-Curie grant agreement In other words, it is possible to have a PMNS matrix which is close to the observed one even in thesymmetric limit, i.e., such that either none of its entries vanish, or only the (13) entry vanishes as (cid:15) → Residual group decompositions The multiplets of Γ (cid:48) N are ‘weighted’, i.e. are described by a pair ( r , k ). At a symmetricpoint these multiplets decompose into 1-dimensional representations of the correspond-ing residual symmetry group. In this appendix we present the decompositions of Γ (cid:48) N multiplets ( N ≤ 5) under the three residual groups of interest (Tables 6 – 9). As seen insection 2.2, these are Z S , Z ST × Z R and Z TN × Z R .Before proceeding, let us comment on the Z R factors in Z ST × Z R and in Z TN × Z R .While kept as part of the residual symmetry group definition in this appendix, they havebeen omitted in the main text of section 3.2. To understand why they can be ignoredwithout loss of generality, note that a direct product Z n × Z ≡ (cid:10) a, b | a n = b = 1 , ab = ba (cid:11) has 2 n irreps ± k , k = 0 , . . . , n − 1, which are simply given as products of the Z n and Z irreps: ± k : ρ ( a ) = exp (cid:18) πi kn (cid:19) , ρ ( b ) = ± . (A.1)In this notation, 1 +0 is the trivial irrep. The representation under Z is just a sign anddoes not affect the reality/complexity of a representation. Hence real irreps are 1 +0 , 1 − and, for even n , 1 + n/ , 1 − n/ (one also has ( ± k ) ∗ = ± n − k ). Since M ( τ ) in the bilinear ofeq. (3.1) is a function of τ alone, it is R -even. The fields ψ and ψ c are then constrained tocarry the same R -parity, i.e. transform with the same sign under Z R . Fields in unhattedrepresentations r – for which ρ r ( R ) = – are even (odd) under Z R if k is even (odd),while the opposite happens for hatted representations. Keeping this in mind, one canomit the Z R factor and ignore the superscript signs in the following tables.Finally, notice that a Z R factor is hidden in the residual Z S , as S = R . Fieldstransforming under Z S as 1 or 1 are R -even while fields transforming as or are R -odd. Requiring that ψ and ψ c carry the same R -parity implies that one effectivelyworks with Z S / Z R (cid:39) Z , which is why it is generic to consider ˜ ρ ci ˜ ρ j = ± r Z S ( τ = i ) Z ST × Z R ( τ = ω ) Z T × Z R ( τ = i ∞ ) k ± k ± (cid:48) k +2 ± k ± k ⊕ k +2 ± k − ⊕ ± k +1 ± ⊕ ± Table 6: Decompositions of ‘weighted’ ( r , k ) multiplets of Γ (cid:48) (cid:39) S under the residualsymmetry groups. Irrep subscripts should be understood modulo n , where n = 4 , k . As in [94], we denote with a hat representations r for which ρ r ( R ) = − . Z S ( τ = i ) Z ST × Z R ( τ = ω ) Z T × Z R ( τ = i ∞ ) k ± k ± (cid:48) k ± k +1 ± (cid:48)(cid:48) k ± k +2 ± ˆ2 1 k +1 ⊕ k +3 ∓ k ⊕ ∓ k +1 ∓ ⊕ ∓ ˆ2 (cid:48) k +1 ⊕ k +3 ∓ k +1 ⊕ ∓ k +2 ∓ ⊕ ∓ ˆ2 (cid:48)(cid:48) k +1 ⊕ k +3 ∓ k ⊕ ∓ k +2 ∓ ⊕ ∓ k ⊕ k +2 ⊕ k +2 ± k ⊕ ± k +1 ⊕ ± k +2 ± ⊕ ± ⊕ ± Table 7: Decompositions of ‘weighted’ ( r , k ) multiplets of Γ (cid:48) (cid:39) A (cid:48) = T (cid:48) under theresidual symmetry groups. Irrep subscripts should be understood modulo n , where n =4 , k . r Z S ( τ = i ) Z ST × Z R ( τ = ω ) Z T × Z R ( τ = i ∞ ) k ± k ± ˆ1 1 k +1 ∓ k ∓ (cid:48) k +2 ± k ± ˆ1 (cid:48) k +3 ∓ k ∓ k +2 ⊕ k ± k +1 ⊕ ± k +2 ± ⊕ ± ˆ2 1 k +1 ⊕ k +3 ∓ k +1 ⊕ ∓ k +2 ∓ ⊕ ∓ k +2 ⊕ k ⊕ k ± k ⊕ ± k +1 ⊕ ± k +2 ± ⊕ ± ⊕ ± ˆ3 1 k +1 ⊕ k +1 ⊕ k +3 ∓ k ⊕ ∓ k +1 ⊕ ∓ k +2 ∓ ⊕ ∓ ⊕ ∓ (cid:48) k +2 ⊕ k +2 ⊕ k ± k ⊕ ± k +1 ⊕ ± k +2 ± ⊕ ± ⊕ ± ˆ3 (cid:48) k +1 ⊕ k +3 ⊕ k +3 ∓ k ⊕ ∓ k +1 ⊕ ∓ k +2 ∓ ⊕ ∓ ⊕ ∓ Table 8: Decompositions of ‘weighted’ ( r , k ) multiplets of Γ (cid:48) (cid:39) S (cid:48) = SL (2 , Z ) underthe residual symmetry groups. Irrep subscripts should be understood modulo n , where n = 4 , k . Z S ( τ = i ) Z ST × Z R ( τ = ω ) Z T × Z R ( τ = i ∞ ) k ± k ± ˆ2 1 k +1 ⊕ k +3 ∓ k +1 ⊕ ∓ k +2 ∓ ⊕ ∓ ˆ2 (cid:48) k +1 ⊕ k +3 ∓ k +1 ⊕ ∓ k +2 ∓ ⊕ ∓ k ⊕ k +2 ⊕ k +2 ± k ⊕ ± k +1 ⊕ ± k +2 ± ⊕ ± ⊕ ± (cid:48) k ⊕ k +2 ⊕ k +2 ± k ⊕ ± k +1 ⊕ ± k +2 ± ⊕ ± ⊕ ± k ⊕ k ⊕ k +2 ⊕ k +2 ± k ⊕ ± k ⊕ ± k +1 ⊕ ± k +2 ± ⊕ ± ⊕ ± ⊕ ± ˆ4 1 k +1 ⊕ k +1 ⊕ k +3 ⊕ k +3 ∓ k ⊕ ∓ k ⊕ ∓ k +1 ⊕ ∓ k +2 ∓ ⊕ ∓ ⊕ ∓ ⊕ ∓ k ⊕ k ⊕ k ⊕ k +2 ⊕ k +2 ± k ⊕ ± k +1 ⊕ ± k +1 ⊕ ± k +2 ⊕ ± k +2 ± ⊕ ± ⊕ ± ⊕ ± ⊕ ± ˆ6 1 k +1 ⊕ k +1 ⊕ k +1 ⊕ k +3 ⊕ k +3 ⊕ k +3 ∓ k ⊕ ∓ k ⊕ ∓ k +1 ⊕ ∓ k +1 ⊕ ∓ k +2 ⊕ ∓ k +2 ∓ ⊕ ∓ ⊕ ∓ ⊕ ∓ ⊕ ∓ ⊕ ∓ Table 9: Decompositions of ‘weighted’ ( r , k ) multiplets of Γ (cid:48) (cid:39) A (cid:48) = SL (2 , Z ) under the residual symmetry groups. Irrep subscripts should beunderstood modulo n , where n = 4 , k . Possible hierarchical patterns In this appendix we list the hierarchical patterns which may arise in the vicinities ofthe two symmetric points of interest (see main text). We consider in turn the finitemodular groups Γ (cid:48) (cid:39) S , Γ (cid:48) (cid:39) A (cid:48) = T (cid:48) , Γ (cid:48) (cid:39) S (cid:48) = SL (2 , Z ), and Γ (cid:48) (cid:39) A (cid:48) = SL (2 , Z )(Tables 10 – 13). We have focused on 3-dimensional (possibly reducible) representations( r , r c ) entering the bilinear (3.1). Dependence on the weights k ( c ) may only arise for τ ∼ ω and through the combination K = k + k c , modulo 3.Note that for N = 2 the residual symmetry group at τ sym = i ∞ is Z T . Mass matrixentries are then expected to be either O (1) or O ( (cid:15) ) and, as was the case for τ (cid:39) i , onecannot obtain the sought-after hierarchical patterns from the smallness of (cid:15) alone. Assuch, only τ (cid:39) ω is considered in Table 10. Table 10: Leading-order mass spectra patterns of bilinears ψ c ψ in the vicinity of thesymmetric point ω , for 3d multiplets ψ ∼ ( r , k ) and ψ c ∼ ( r c , k c ) of the finite modulargroup Γ (cid:48) (cid:39) S . Spectra are insensitive to transposition, i.e. to the exchange ψ ↔ ψ c .Congruence relations for k + k c are modulo 3. r r c τ (cid:39) ωk + k c ≡ k + k c ≡ k + k c ≡ ⊕ ⊕ (1 , , 1) (1 , , 1) (1 , , ⊕ ⊕ (cid:48) (1 , , 1) (1 , , 1) (1 , , ⊕ (cid:48) ⊕ (cid:48) (1 , , 1) (1 , , 1) (1 , , ⊕ (cid:48) ⊕ ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ⊕ (cid:48) ⊕ (cid:48) ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ⊕ (cid:48) (cid:48) ⊕ ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ⊕ (cid:48) (cid:48) ⊕ (cid:48) ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ⊕ ⊕ ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ⊕ (cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ⊕ (cid:48) ⊕ ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ⊕ (cid:48) (cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (cid:48) ⊕ ⊕ (cid:48) ⊕ ⊕ (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (cid:48) ⊕ ⊕ (cid:48) ⊕ (cid:48) ⊕ (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (cid:48) ⊕ (cid:48) ⊕ (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ⊕ ⊕ (cid:48) ⊕ ⊕ (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ⊕ ⊕ (cid:48) ⊕ (cid:48) ⊕ (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (cid:48) ⊕ ⊕ (cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ⊕ ⊕ ⊕ ⊕ (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) )36 able 10: (cont.) r r c τ (cid:39) ωk + k c ≡ k + k c ≡ k + k c ≡ ⊕ ⊕ (cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (cid:48) (cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) Table 11: Leading-order mass spectra patterns of bilinears ψ c ψ in the vicinity ofthe symmetric points ω and i ∞ , for 3d multiplets ψ ∼ ( r , k ) and ψ c ∼ ( r c , k c ) of thefinite modular group Γ (cid:48) (cid:39) A (cid:48) = T (cid:48) . Spectra are insensitive to transposition, i.e. to theexchange ψ ↔ ψ c . Congruence relations for k + k c are modulo 3. r r c τ (cid:39) ω τ (cid:39) i ∞ k + k c ≡ k + k c ≡ k + k c ≡ (1 , , 1) (1 , , 1) (1 , , 1) (1 , , (cid:48)(cid:48) ⊕ (cid:48) ⊕ (1 , , 1) (1 , , 1) (1 , , 1) (1 , , (cid:48) ⊕ ⊕ (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (cid:48)(cid:48) ⊕ ⊕ (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (cid:48)(cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48) (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) ⊕ ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (cid:48)(cid:48) ⊕ (cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48) ⊕ (1 , , 1) (1 , , 1) (1 , , 1) (1 , , (cid:48) ⊕ ⊕ (cid:48)(cid:48) ⊕ (cid:48) ⊕ (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (cid:48)(cid:48) ⊕ ⊕ (cid:48)(cid:48) ⊕ (cid:48) ⊕ (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48) ⊕ (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (cid:48)(cid:48) ⊕ (cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (cid:48)(cid:48) ⊕ (cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (cid:48)(cid:48) ⊕ (cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48) (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) ⊕ ⊕ (cid:48)(cid:48) ⊕ (cid:48) ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (cid:48) ⊕ ⊕ (cid:48) ⊕ ⊕ (1 , , (cid:15) ) (1 , (cid:15) , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (cid:48) ⊕ ⊕ (cid:48)(cid:48) ⊕ ⊕ (1 , , 1) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:48) ⊕ ⊕ (cid:48) ⊕ (cid:48) ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , 1) (1 , (cid:15), (cid:15) )37 able 11: (cont.) r r c τ (cid:39) ω τ (cid:39) i ∞ k + k c ≡ k + k c ≡ k + k c ≡ (cid:48) ⊕ ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , (cid:15) , (cid:15) ) (1 , , (cid:15) ) (cid:48) ⊕ ⊕ (cid:48)(cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , (cid:15) , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , (cid:15) , (cid:15) ) (cid:48) ⊕ ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48) (1 , (cid:15), (cid:15) ) (1 , , 1) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (cid:48)(cid:48) ⊕ ⊕ (cid:48)(cid:48) ⊕ ⊕ (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (cid:48)(cid:48) ⊕ ⊕ (1 , , (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (cid:48)(cid:48) ⊕ ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (1 , (cid:15), (cid:15) ) (1 , , 1) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (cid:48)(cid:48) ⊕ ⊕ (cid:48)(cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , 1) (1 , (cid:15), (cid:15) ) (cid:48)(cid:48) ⊕ ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , (cid:15), (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (cid:48) ⊕ (cid:48) ⊕ (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , (cid:15), (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (1 , , 1) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:48) ⊕ (cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , (cid:15), (cid:15) ) (1 , , 1) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48) (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (cid:48) (cid:48)(cid:48) ⊕ (cid:48) ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (cid:48)(cid:48) ⊕ (cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (1 , (cid:15) , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , (cid:15) , (cid:15) ) (cid:48)(cid:48) ⊕ (cid:48) ⊕ (cid:48) (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (1 , , (cid:15) ) (1 , (cid:15) , (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , 1) (1 , (cid:15), (cid:15) ) (cid:48)(cid:48) ⊕ (cid:48) ⊕ (cid:48) (cid:48)(cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , , (cid:15) ) (1 , , (cid:15) ) (1 , (cid:15) , (cid:15) ) (1 , , (cid:15) ) (cid:48)(cid:48) ⊕ (cid:48) ⊕ (cid:48) (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48) (1 , , 1) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48) (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48) (1 , , (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) (1 , , (cid:15) ) ⊕ ⊕ (cid:48) ⊕ ⊕ (1 , , (cid:15) ) ( (cid:15), (cid:15) , (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) ⊕ ⊕ (cid:48)(cid:48) ⊕ ⊕ (1 , , (cid:15) ) (1 , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , , (cid:15) ) ⊕ ⊕ (cid:48) ⊕ (cid:48) ⊕ (1 , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , , (cid:15) ) (1 , (cid:15) , (cid:15) ) ⊕ ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) ( (cid:15), (cid:15) , (cid:15) ) (1 , (cid:15), (cid:15) ) ⊕ ⊕ (cid:48)(cid:48) ⊕ (cid:48) ⊕ (cid:48) ( (cid:15), (cid:15) , (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) ( (cid:15), (cid:15) , (cid:15) ) ⊕ ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48) ( (cid:15), (cid:15), (cid:15) ) (1 , , (cid:15) ) (1 , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (cid:48) ⊕ ⊕ (cid:48) ⊕ (cid:48) ⊕ (cid:48) ( (cid:15), (cid:15) , (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) ( (cid:15), (cid:15) , (cid:15) ) (cid:48) ⊕ ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) ( (cid:15), (cid:15) , (cid:15) ) (1 , (cid:15), (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (cid:48) (cid:48)(cid:48) ⊕ ⊕ (1 , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , , (cid:15) ) (1 , (cid:15) , (cid:15) ) (cid:48)(cid:48) ⊕ ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ( (cid:15), (cid:15), (cid:15) ) (1 , , (cid:15) ) (1 , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (cid:48) ⊕ (cid:48) ⊕ (cid:48) ( (cid:15), (cid:15), (cid:15) ) (1 , , (cid:15) ) (1 , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) (1 , , (cid:15) ) (1 , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , , (cid:15) )38 able 11: (cont.) r r c τ (cid:39) ω τ (cid:39) i ∞ k + k c ≡ k + k c ≡ k + k c ≡ (cid:48) ⊕ (cid:48) ⊕ (cid:48) (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (1 , , (cid:15) ) ( (cid:15), (cid:15) , (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ( (cid:15), (cid:15) , (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) ( (cid:15), (cid:15) , (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (cid:48) (cid:48)(cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) ( (cid:15), (cid:15) , (cid:15) ) (1 , (cid:15), (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (cid:48) (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48) (1 , , (cid:15) ) (1 , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , , (cid:15) ) (cid:48)(cid:48) ⊕ (cid:48) ⊕ (cid:48) (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) (1 , , (cid:15) ) ( (cid:15), (cid:15) , (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48) (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) (1 , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , , (cid:15) ) (1 , (cid:15) , (cid:15) ) ⊕ ⊕ ⊕ ⊕ (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , , ⊕ ⊕ (cid:48) ⊕ (cid:48) ⊕ (cid:48) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ⊕ ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ( (cid:15), (cid:15), (cid:15) ) (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (cid:48) (cid:48) ⊕ (cid:48) ⊕ (cid:48) ( (cid:15), (cid:15), (cid:15) ) (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (cid:48) (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , , (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ⊕ (cid:48)(cid:48) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) Table 12: Leading-order mass spectra patterns of bilinears ψ c ψ in the vicinity of thesymmetric points ω and i ∞ , for 3d multiplets ψ ∼ ( r , k ) and ψ c ∼ ( r c , k c ) of the finitemodular group Γ (cid:48) (cid:39) S (cid:48) = SL (2 , Z ). Spectra are insensitive to transposition, i.e. to theexchange ψ ↔ ψ c . Congruence relations for k + k c are modulo 3. r r c τ (cid:39) ω τ (cid:39) i ∞ k + k c ≡ k + k c ≡ k + k c ≡ (1 , , 1) (1 , , 1) (1 , , 1) (1 , , (cid:48) (1 , , 1) (1 , , 1) (1 , , 1) (1 , , (cid:15) ) (1 , , 1) (1 , , 1) (1 , , 1) (1 , , (cid:15) ) (cid:48) (1 , , 1) (1 , , 1) (1 , , 1) (1 , , (cid:15) ) (cid:48) (cid:48) (1 , , 1) (1 , , 1) (1 , , 1) (1 , , (cid:48) ˆ3 (cid:48) (1 , , 1) (1 , , 1) (1 , , 1) (1 , , (cid:15) ) ˆ3 3 (cid:48) (1 , , 1) (1 , , 1) (1 , , 1) (1 , , (cid:15) ) ˆ3 ˆ3 (1 , , 1) (1 , , 1) (1 , , 1) (1 , , (cid:15) ) ˆ3 ˆ3 (cid:48) (1 , , 1) (1 , , 1) (1 , , 1) (1 , , ˆ3 (cid:48) ˆ3 (cid:48) (1 , , 1) (1 , , 1) (1 , , 1) (1 , , (cid:15) ) ⊕ (1 , , 1) (1 , , 1) (1 , , 1) (1 , (cid:15), (cid:15) ) ⊕ (cid:48) (1 , , 1) (1 , , 1) (1 , , 1) (1 , (cid:15), (cid:15) )39 able 12: (cont.) r r c τ (cid:39) ω τ (cid:39) i ∞ k + k c ≡ k + k c ≡ k + k c ≡ ⊕ ˆ1 (1 , , 1) (1 , , 1) (1 , , 1) (1 , , (cid:15) ) ⊕ ˆ1 (cid:48) (1 , , 1) (1 , , 1) (1 , , 1) (1 , , (cid:15) ) (cid:48) ⊕ (1 , , 1) (1 , , 1) (1 , , 1) (1 , (cid:15), (cid:15) ) (cid:48) ⊕ (cid:48) (1 , , 1) (1 , , 1) (1 , , 1) (1 , (cid:15), (cid:15) ) (cid:48) ˆ2 ⊕ ˆ1 (1 , , 1) (1 , , 1) (1 , , 1) (1 , , (cid:15) ) (cid:48) ˆ2 ⊕ ˆ1 (cid:48) (1 , , 1) (1 , , 1) (1 , , 1) (1 , , (cid:15) ) ˆ3 2 ⊕ (1 , , 1) (1 , , 1) (1 , , 1) (1 , , (cid:15) ) ˆ3 2 ⊕ (cid:48) (1 , , 1) (1 , , 1) (1 , , 1) (1 , , (cid:15) ) ˆ3 ˆ2 ⊕ ˆ1 (1 , , 1) (1 , , 1) (1 , , 1) (1 , (cid:15), (cid:15) ) ˆ3 ˆ2 ⊕ ˆ1 (cid:48) (1 , , 1) (1 , , 1) (1 , , 1) (1 , (cid:15), (cid:15) ) ˆ3 (cid:48) ⊕ (1 , , 1) (1 , , 1) (1 , , 1) (1 , , (cid:15) ) ˆ3 (cid:48) ⊕ (cid:48) (1 , , 1) (1 , , 1) (1 , , 1) (1 , , (cid:15) ) ˆ3 (cid:48) ˆ2 ⊕ ˆ1 (1 , , 1) (1 , , 1) (1 , , 1) (1 , (cid:15), (cid:15) ) ˆ3 (cid:48) ˆ2 ⊕ ˆ1 (cid:48) (1 , , 1) (1 , , 1) (1 , , 1) (1 , (cid:15), (cid:15) ) ⊕ ⊕ (1 , , 1) (1 , , 1) (1 , , 1) (1 , , ⊕ ⊕ (cid:48) (1 , , 1) (1 , , 1) (1 , , 1) (1 , , (cid:15) ) ⊕ ⊕ ˆ1 (1 , , 1) (1 , , 1) (1 , , 1) ( (cid:15), (cid:15), (cid:15) ) ⊕ ⊕ ˆ1 (cid:48) (1 , , 1) (1 , , 1) (1 , , 1) ( (cid:15), (cid:15), (cid:15) ) ⊕ (cid:48) ⊕ (cid:48) (1 , , 1) (1 , , 1) (1 , , 1) (1 , , ⊕ (cid:48) ˆ2 ⊕ ˆ1 (1 , , 1) (1 , , 1) (1 , , 1) ( (cid:15), (cid:15), (cid:15) ) ⊕ (cid:48) ˆ2 ⊕ ˆ1 (cid:48) (1 , , 1) (1 , , 1) (1 , , 1) ( (cid:15), (cid:15), (cid:15) ) (cid:48) ⊕ ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (cid:48) ⊕ ˆ1 ⊕ ˆ1 (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) (cid:48) (cid:48) ⊕ ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (cid:48) (cid:48) ⊕ (cid:48) ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (cid:48) ˆ1 (cid:48) ⊕ ˆ1 ⊕ ˆ1 (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) (cid:48) ˆ1 (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) ⊕ ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ( (cid:15), (cid:15) , (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ⊕ ˆ1 ⊕ ˆ1 (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15) , (cid:15) )40 able 12: (cont.) r r c τ (cid:39) ω τ (cid:39) i ∞ k + k c ≡ k + k c ≡ k + k c ≡ (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (cid:48) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (cid:48) ⊕ ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (cid:48) (cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ( (cid:15), (cid:15) , (cid:15) ) (cid:48) ˆ1 ⊕ ˆ1 ⊕ ˆ1 (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (cid:48) ˆ1 (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (cid:48) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15) , (cid:15) ) ˆ2 ⊕ ˆ1 ˆ2 ⊕ ˆ1 (1 , , 1) (1 , , 1) (1 , , 1) (1 , , (cid:15) ) ˆ2 ⊕ ˆ1 ˆ2 ⊕ ˆ1 (cid:48) (1 , , 1) (1 , , 1) (1 , , 1) (1 , , ˆ2 ⊕ ˆ1 (cid:48) ˆ2 ⊕ ˆ1 (cid:48) (1 , , 1) (1 , , 1) (1 , , 1) (1 , , (cid:15) ) ˆ3 1 (cid:48) ⊕ ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) ˆ3 1 (cid:48) ⊕ (cid:48) ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) ˆ3 ˆ1 (cid:48) ⊕ ˆ1 ⊕ ˆ1 (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ˆ3 ˆ1 (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ˆ3 (cid:48) (cid:48) ⊕ ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) ˆ3 (cid:48) (cid:48) ⊕ (cid:48) ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) ˆ3 (cid:48) ˆ1 (cid:48) ⊕ ˆ1 ⊕ ˆ1 (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ˆ3 (cid:48) ˆ1 (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ˆ3 1 ⊕ ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15) , (cid:15) ) ˆ3 1 (cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ˆ3 ˆ1 ⊕ ˆ1 ⊕ ˆ1 (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ˆ3 ˆ1 (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (cid:48) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ( (cid:15), (cid:15) , (cid:15) ) ˆ3 (cid:48) ⊕ ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ˆ3 (cid:48) (cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15) , (cid:15) ) ˆ3 (cid:48) ˆ1 ⊕ ˆ1 ⊕ ˆ1 (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ( (cid:15), (cid:15) , (cid:15) ) ˆ3 (cid:48) ˆ1 (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (cid:48) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ⊕ (cid:48) ⊕ ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , ⊕ (cid:48) ⊕ (cid:48) ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) ⊕ (cid:48) ⊕ ˆ1 ⊕ ˆ1 (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ⊕ (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ⊕ (cid:48) (cid:48) ⊕ ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) ⊕ (cid:48) (cid:48) ⊕ (cid:48) ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , ⊕ (cid:48) ˆ1 (cid:48) ⊕ ˆ1 ⊕ ˆ1 (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ⊕ (cid:48) ˆ1 (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ( (cid:15), (cid:15), (cid:15) )41 able 12: (cont.) r r c τ (cid:39) ω τ (cid:39) i ∞ k + k c ≡ k + k c ≡ k + k c ≡ ⊕ ⊕ ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) ⊕ (cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15) , (cid:15) ) ⊕ ⊕ ˆ1 ⊕ ˆ1 (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ⊕ (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (cid:48) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ( (cid:15), (cid:15) , (cid:15) ) ⊕ (cid:48) ⊕ ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15) , (cid:15) ) ⊕ (cid:48) (cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) ⊕ (cid:48) ˆ1 ⊕ ˆ1 ⊕ ˆ1 (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ( (cid:15), (cid:15) , (cid:15) ) ⊕ (cid:48) ˆ1 (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (cid:48) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ˆ2 ⊕ ˆ1 1 (cid:48) ⊕ ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ˆ2 ⊕ ˆ1 1 (cid:48) ⊕ (cid:48) ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ˆ2 ⊕ ˆ1 ˆ1 (cid:48) ⊕ ˆ1 ⊕ ˆ1 (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) ˆ2 ⊕ ˆ1 ˆ1 (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , ˆ2 ⊕ ˆ1 (cid:48) (cid:48) ⊕ ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ˆ2 ⊕ ˆ1 (cid:48) (cid:48) ⊕ (cid:48) ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ˆ2 ⊕ ˆ1 (cid:48) ˆ1 (cid:48) ⊕ ˆ1 ⊕ ˆ1 (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , ˆ2 ⊕ ˆ1 (cid:48) ˆ1 (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) ˆ2 ⊕ ˆ1 1 ⊕ ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ˆ2 ⊕ ˆ1 1 (cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ( (cid:15), (cid:15) , (cid:15) ) ˆ2 ⊕ ˆ1 ˆ1 ⊕ ˆ1 ⊕ ˆ1 (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15) , (cid:15) ) ˆ2 ⊕ ˆ1 ˆ1 (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (cid:48) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) ˆ2 ⊕ ˆ1 (cid:48) ⊕ ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ( (cid:15), (cid:15) , (cid:15) ) ˆ2 ⊕ ˆ1 (cid:48) (cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ˆ2 ⊕ ˆ1 (cid:48) ˆ1 ⊕ ˆ1 ⊕ ˆ1 (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , , (cid:15) ) ˆ2 ⊕ ˆ1 (cid:48) ˆ1 (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (cid:48) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15) , (cid:15) ) (cid:48) ⊕ ⊕ (cid:48) ⊕ ⊕ (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , , (cid:48) ⊕ ⊕ (cid:48) ⊕ (cid:48) ⊕ (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , , (cid:15) ) (cid:48) ⊕ ⊕ (cid:48) ⊕ ˆ1 ⊕ ˆ1 (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (cid:48) ⊕ ⊕ (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (cid:48) ⊕ (cid:48) ⊕ (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , , (cid:48) ⊕ (cid:48) ⊕ (cid:48) ⊕ ˆ1 ⊕ ˆ1 (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ⊕ ⊕ (cid:48) ⊕ ⊕ (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , , (cid:15) )42 able 12: (cont.) r r c τ (cid:39) ω τ (cid:39) i ∞ k + k c ≡ k + k c ≡ k + k c ≡ ⊕ ⊕ (cid:48) ⊕ (cid:48) ⊕ (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , (cid:15) , (cid:15) ) ⊕ ⊕ (cid:48) ⊕ ˆ1 ⊕ ˆ1 (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ⊕ ⊕ (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ( (cid:15), (cid:15) , (cid:15) ) (cid:48) ⊕ ⊕ (cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , (cid:15) , (cid:15) ) (cid:48) ⊕ ⊕ (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (cid:48) (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ( (cid:15), (cid:15) , (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , , (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (cid:48) (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (cid:48) ˆ1 (cid:48) ⊕ ˆ1 ⊕ ˆ1 (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ( (cid:15), (cid:15) , (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (cid:48) ˆ1 (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ⊕ ⊕ ⊕ ⊕ (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , , ⊕ ⊕ (cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ( (cid:15) , (cid:15) , (cid:15) ) ⊕ ⊕ ⊕ ˆ1 ⊕ ˆ1 (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ⊕ ⊕ (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (cid:48) (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ( (cid:15) , (cid:15) , (cid:15) ) (cid:48) ⊕ (cid:48) ⊕ (cid:48) (cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , , (cid:48) ⊕ (cid:48) ⊕ (cid:48) ˆ1 (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (cid:48) (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ˆ1 (cid:48) ⊕ ˆ1 ⊕ ˆ1 ˆ1 (cid:48) ⊕ ˆ1 ⊕ ˆ1 (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , , (cid:15) ) ˆ1 (cid:48) ⊕ ˆ1 ⊕ ˆ1 ˆ1 (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , , ˆ1 (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 ˆ1 (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , , (cid:15) ) ˆ1 ⊕ ˆ1 ⊕ ˆ1 1 (cid:48) ⊕ ⊕ (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ˆ1 ⊕ ˆ1 ⊕ ˆ1 1 (cid:48) ⊕ (cid:48) ⊕ (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ( (cid:15), (cid:15) , (cid:15) ) ˆ1 ⊕ ˆ1 ⊕ ˆ1 ˆ1 (cid:48) ⊕ ˆ1 ⊕ ˆ1 (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , (cid:15) , (cid:15) ) ˆ1 ⊕ ˆ1 ⊕ ˆ1 ˆ1 (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , , (cid:15) ) ˆ1 (cid:48) ⊕ ˆ1 ⊕ ˆ1 ˆ1 (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (cid:48) (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , , (cid:15) ) ˆ1 (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 ˆ1 (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (cid:48) (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , (cid:15) , (cid:15) ) ˆ1 ⊕ ˆ1 ⊕ ˆ1 1 (cid:48) ⊕ (cid:48) ⊕ (cid:48) (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ( (cid:15) , (cid:15) , (cid:15) ) ˆ1 ⊕ ˆ1 ⊕ ˆ1 ˆ1 ⊕ ˆ1 ⊕ ˆ1 (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ( (cid:15) , (cid:15) , (cid:15) ) ˆ1 ⊕ ˆ1 ⊕ ˆ1 ˆ1 (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (cid:48) (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , , ˆ1 (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (cid:48) ˆ1 (cid:48) ⊕ ˆ1 (cid:48) ⊕ ˆ1 (cid:48) (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) ( (cid:15) , (cid:15) , (cid:15) )43 able 13: Leading-order mass spectra patterns of bilinears ψ c ψ in the vicinity of thesymmetric points ω and i ∞ , for 3d multiplets ψ ∼ ( r , k ) and ψ c ∼ ( r c , k c ) of the finitemodular group Γ (cid:48) (cid:39) A (cid:48) = SL (2 , Z ). Spectra are insensitive to transposition, i.e. to theexchange ψ ↔ ψ c . Congruence relations for k + k c are modulo 3. r r c τ (cid:39) ω τ (cid:39) i ∞ k + k c ≡ k + k c ≡ k + k c ≡ (1 , , 1) (1 , , 1) (1 , , 1) (1 , , (cid:48) (1 , , 1) (1 , , 1) (1 , , 1) (1 , (cid:15), (cid:15) ) (cid:48) (cid:48) (1 , , 1) (1 , , 1) (1 , , 1) (1 , , ⊕ ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (cid:48) ⊕ ⊕ (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15), (cid:15) ) (1 , (cid:15) , (cid:15) ) ⊕ ⊕ ⊕ ⊕ (1 , , 1) ( (cid:15) , (cid:15) , (cid:15) ) ( (cid:15), (cid:15), (cid:15) ) (1 , , eferences [1] F. Feruglio, Pieces of the Flavour Puzzle , Eur. Phys. J. C (2015) 373[ ].[2] C. D. Froggatt and H. B. Nielsen, Hierarchy of Quark Masses, Cabibbo Anglesand CP Violation , Nucl. Phys. B (1979) 277.[3] S. Weinberg, Baryon and Lepton Nonconserving Processes , Phys. Rev. Lett. (1979) 1566.[4] S. T. 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