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Froggatt-Nielsen models with a residual Z_4^R symmetry

The Froggatt-Nielsen mechanism provides an elegant explanation for the hierarchies of fermion masses and mixings in terms of a U(1) symmetry. Promoting such a family symmetry to an R-symmetry, we explicitly construct supersymmetric Froggatt-Nielsen models which are gauged, family dependent U(1)_R completions of the Z_4^R symmetry proposed by Lee, Raby, Ratz, Ross, Schieren, Schmidt-Hoberg and Vaudrevange in 2010. Forbidden by Z_4^R, the mu-term is generated around the supersymmetry breaking scale m_3/2 from either the Kahler potential or the superpotential. Neutrinos acquire their mass via the type I seesaw mechanism with three right-handed neutrino superfields. Taking into account the Green-Schwarz anomaly cancellation conditions, we arrive at a total of 3 x 34 distinct phenomenologically viable charge assignments for the standard model fields, most of which feature highly fractional charges.


I. INTRODUCTION
The standard model of particle physics postulates three families of fermions. The masses and mixings of the quarks and leptons are dictated by Yukawa couplings, that is, interactions of two fermionic fields ψ i and ψ j with the recently discovered Higgs boson H [1,2]. In order to correctly describe the hierarchies observed in the fermionic masses and mixings without relying on fine-tuned correlations among the entries of a Yukawa matrix Y , it is necessary to assume that Y itself features a hierarchical pattern. Such a special structure of a priori random coupling constants calls for an explanation. In 1979 one was proposed by Froggatt and Nielsen in the form of an underlying family dependent U(1) gauge symmetry extension [3]. At the effective level, the Froggatt-Nielsen (FN) mechanism requires the introduction of a standard model neutral but U(1) charged field which, like the Higgs, develops a vacuum expectation value (VEV). Appropriate powers of this so-called flavon field φ compensate non-zero charges of the original trilinear Yukawa terms, thereby giving rise to a hierarchy in the entries of the Yukawa matrices. Despite motivating a hierarchical structure, the exact value of each Yukawa interaction depends on unknown order-one coefficients. The number of free parameters can only be reduced compared to a theory which does not invoke the FN mechanism, if the extra U(1) family symmetry is spontaneously broken down to a residual discrete Z N symmetry. Such a possibility has been proposed and studied carefully in supersymmetric FN models with residual matter (or equivalently R-) parity [4], baryon triality [5], and proton hexality [6], and it has been demonstrated how these residual discrete symmetries are obtained exactly from certain simple linear conditions on the U(1) charge assignments. In the present paper, we consider a similar situation where a discrete R-symmetry (Z R N ) arises from a family dependent gauged U(1) R symmetry. In particular, we focus on the Z R 4 symmetry proposed in [7,8], see also [9,10]. This family independent symmetry is compatible with grand unification and anomaly-free by means of the (discrete) Green-Schwarz (GS) mechanism [11]. Furthermore, it provides an attractive solution to the µ-problem [12] by forbidding the bilinear Higgs superpotential term. The µ-term is generated dynamically at the electroweak scale when the Z R 4 symmetry is broken to standard R-parity, Z R 2 , 1 by the breakdown of supersymmetry, either in the Kähler potential [13] or in the superpotential [14]. Starting with a U(1) R family symmetry, the class of Froggatt-Nielsen models constructed in this paper is characterized by the following symmetry breaking chain, where SM refers to the standard model gauge group SU(3) C × SU(2) W × U(1) Y . Our main concern shall be the first step, where we identify constraints on the U(1) R charge assignments originating from the following requirements: • phenomenologically acceptable quark and lepton masses and mixings, including the neutrinos, • absence of the µ-term, • anomaly freedom of the gauged U(1) R family symmetry via the Green-Schwarz mechanism, • a flavon VEV φ ∼ ǫM grav induced by the Dine-Seiberg-Wen-Witten mechanism [15][16][17][18] (here ǫ ∼ 0.2 is an expansion parameter similar in size to the Wolfenstein parameter λ c of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, and M grav ∼ 2.4 × 10 18 GeV denotes the gravitational scale), • existence of a residual Z R 4 symmetry after U(1) R breaking. The paper is organized as follows. In Section II we discuss how the hierarchies of the fermion masses and mixings can arise from a family dependent U(1) R symmetry à la Froggatt and Nielsen. In Section III we formulate the anomaly constraints on possible U(1) R charge assignments, and show how non-vanishing anomaly coefficients fix the VEV of the flavon field. Demanding an exact residual Z R 4 symmetry, we derive the corresponding linear conditions in Section IV. The neutrino constraints are studied in Section V. Collecting all results, we arrive at a set of 3 × 34 possible charge assignments which are listed in Appendix A. We then conclude in Section VI.

II. FROGGATT-NIELSEN MECHANISM USING A U(1)R SYMMETRY
In this section we focus on the first step of the breaking chain depicted in Eq. (2). The Froggatt-Nielsen mechanism [3] provides a symmetry argument to explain the hierarchies of the fermion masses and mixings based on a family dependent U(1) extension of the SM gauge group. We examine the situation where the FN U(1) family symmetry is altered to a gauged U(1) R-symmetry [19]. The VEV of the SM singlet flavon breaks the gauged U(1) R symmetry down to Z R 4 . We defer the discussion of anomalies [20,21] as well as residual discrete symmetries to later sections, and consider first the changes to the FN mechanism due to a non-trivially R-charged superspace coordinate θ.
It is convenient to fix the Froggatt-Nielsen U(1) R charge normalization such that the flavon field φ carries R-charge R φ = −1. Then the charge of the superspace variable θ takes the general value R θ , and a local U(1) R transformation, parametrized by α(x), maps As a consequence, a term allowed in the U(1) R symmetric superpotential has to carry a charge of +2R θ . The total charge R total of a term with the flavon field removed has to equal 2R θ + n, where n is a non-negative integer, in order to be generated effectively after U(1) R breaking. In the unbroken phase, non-zero n can be compensated by n powers of the flavon field which, in our normalization has U(1) R charge R φ = −1.
To illustrate the origin of hierarchical Yukawa matrices Y , let us consider the example of the operator Y ij u Q i H uŪj . Denoting the total U(1) R charge of this operator by R ij total = R Qi + R Hu + RŪ j , it originates from an underlying U(1) R symmetric term of the form where 1 √ 10 ≤ y ij u ≤ √ 10 are undetermined order-one coefficients, M grav denotes the gravitational scale, and ensures that the superpotential remains holomorphic and satisfies the cluster decomposition principle [22,23]. When U(1) R is broken by the flavon VEV φ , the hierarchies of the Yukawa couplings Y ij u are generated from Eq. (5) due to family dependent and non-negative integer powers of the expansion parameter ǫ ≡ φ Mgrav < 1. In general, the FN suppression of an operator with a total U(1) R charge of R total is given by ǫ n , with n = R total − 2R θ . As we argue in Section III, it is possible to identify the expansion parameter ǫ with the Wolfenstein parameter λ c ∼ 0.22 [24].
The first goal is to formulate constraints on the U(1) R charges such that the desired hierarchies of the GUT scale fermion masses and mixings are achieved [4,[25][26][27][28][29][30][31][32]. The singular values of the mass matrices [33] must be related as where z = 0, 1, x = 0, 1, 2, 3 and tan β = vu v d is the ratio of the up-type and the down-type Higgs VEVs. Furthermore, m t ∼ v u , and the CKM mixing matrix is parametrised by, where y = −1, 0, 1, with y = 0 being the preferred value. We note that the parametrisation in terms of x, y, z allows for a high degree of flexibility in which to interpret our final results for the U(1) R charge assignments. In order to derive the physical masses and mixings in Froggatt-Nielsen models, it is necessary to bring the kinetic terms into their canonical form [4,5,29,32,34]. Luckily, the required non-unitary transformations do not change the FN structure of the Yukawa couplings, but only affect the order-one coefficients which are unknown anyway. Therefore, the left-chiral unitary transformations V UL and V DL which are subsequently required to diagonalise the up-type and the down-type quark mass matrices take the form leading to a CKM matrix of the same structure [35], Enforcing the desired structure of the CKM matrix in Eq. (8) and making use of the fact that successive generations of quarks have smaller U(1) R charges, i.e. R Q3 ≤ R Q2 ≤ R Q1 , we can re-express R Q3 and R Q2 in terms of R Q1 and the parameter y, With these relations, dictated by the structure of the CKM matrix, the mass ratios of Eqs. (7a) to (7c) are realised if the U(1) R charge assignments satisfy [6] RD 2 = RD 1 − 1 + y , Finally we can make use of Eqs. (7d) and (7e) as well as the relation m t ∼ v u to re-express the charges of all right-chiral superfields in terms of the charges of the left-chiral superfields. Inserting the relations of Eqs. (11) and (12a) to (12c), we readily find the conditions Through these manipulations we have now re-expressed all charges in terms of the integer parameters x, y, z, ∆ L 21 and ∆ L 31 as well as the remaining charges R Q1 , R L1 , R Hu , R H d and, since we are discussing U(1) R symmetries, the charge R θ of the superspace variable.
In order to further reduce the number of free parameters we impose anomaly freedom of the theory in the next section. However, we first comment on the possible origin of the µ-term. In principle, a term of the form can arise (effectively) in the superpotential either before or after supersymmetry breaking, with the exponent n given by The former case is analogous to the generation of the Yukawa operators discussed above, with M = M grav , and requires the exponent n = n b to take a non-negative integer value. To arrive at a µ-parameter around the order of the electroweak scale requires n = n b > 20 [4]. Here we shall not pursue this direction but rather consider the second option where the term in Eq. (14) is generated when supersymmetry is broken. This can happen in two different ways [7,8,36], both of which require n = n a to take an integer value. Adopting the Giudice-Masiero mechanism [13], the µ-term arises from a non-renormalisable, non-holomorphic operator in the Kähler potential, involving the complex conjugate of a SM × U(1) R neutral hidden sector chiral superfield Z whose F -term acquires the VEV F Z ∼ m 3/2 M grav . This VEV breaks both supersymmetry and the residual Z R 4 symmetry, thereby effectively generating the superpotential term in Eq. (14) with M being of the order of the gravitino mass m 3/2 . We emphasise that the Giudice-Masiero mechanism is compatible with the parameter n = n a in Eq. (15) being either positive or negative, where the latter case requires one to use the complex conjugate of the flavon field in Eq. (14). 3 Provided that the integer n = n a in Eq. (15) is non-negative, an effective µ-term can additionally be obtained from a holomorphic non-renormalisable superpotential operator à la Kim and Nilles [14]. In this case, the product where W hidden denotes the superpotential of a hidden sector [36,37]. When supersymmetry is broken, the hidden sector superpotential develops a VEV W hidden ∼ m 3/2 M 2 grav , which in turn induces the effective superpotential term of Eq. (14) with M ∼ m 3/2 . A possible realisation of this mechanism is through a gaugino condensate [38][39][40].
For the purpose of this paper, it is irrelevant whether we rely on the Giudice-Masiero or the Kim-Nilles mechanism to generate the µ-term as both mechanisms result in M ∼ m 3/2 , see Eq. (14). Moreover, both options require n = n a = R Hu + R H d to be an integer, which must be non-negative for the Kim-Nilles mechanism. It is interesting to compare the above situation with the conventional, i.e. non-R FN scenario. For the conventional case, in order to generate an effective µ-term, the U(1) charges of the two Higgs fields must add up to an integer n R θ =0 . For nonnegative n R θ =0 , the regular holomorphic contribution to the µ-term dominates over the non-holomorphic Giudice-Masiero contribution. In order to fall in the regime of an electroweak-scale µ-parameter, n R θ =0 > 20 is required. On the other hand, with negative n R θ =0 , the holomorphic contribution is removed, and the Giudice-Masiero contribution leads to an attractive solution to the µ-problem.
In contrast, adopting a U(1) R Froggatt-Nielsen symmetry allows to solve the µ-problem using either nonholomorphic terms à la Giudice and Masiero or holomorphic terms à la Kim and Nilles. In both cases, the U(1) R charges of the two Higgs fields must add up to an integer n = n a = R Hu + R H d . The regular superpotential term (including powers of the flavon field φ) which would typically dominate the µ-term, is forbidden as long as

III. ANOMALY CONSTRAINTS
To ensure a consistent theory after extending the SM gauge group by a local FN family symmetry, one must enforce that all anomaly coefficients are zero or that anomaly cancellation proceeds via the Green-Schwarz mechanism [11].
Here we make use of the latter option, or, more precisely its four-dimensional analogue [41], as applied for instance in [4,[26][27][28][29][30]32]. We adopt a gauged U(1) R symmetry, which was first discussed in [19]. The anomalies were first studied in [20,21]. The components of a general left-chiral superfield S = ϕ + θψ + θ 2 F transform as where R S is the U(1) R charge of S. The shifted charge R S − R θ of the spin one-half particle ψ contributes to the anomaly coefficients. The gaugino component λ of a gauge vector superfield V also contributes to the anomaly coefficients. With V being necessarily U(1) R neutral, the gaugino carries charge R θ and transform as With these remarks, the mixed SU(3) C − SU(3) C − U(1) R anomaly can be calculated easily, yielding [42] where ℓ(r) denotes the Dynkin index of an SU(3) representation r, normalised such that ℓ(r fund ) = 1 2 for the fundamental representation. Similarly, we obtain the anomaly coefficients for the remaining mixed anomalies The three mixed anomaly coefficients of Eqs. (18) to (20) must satisfy the Green-Schwarz anomaly cancellation condition where k ... denotes the Kac-Moody level of the respective gauge symmetry. Assuming gauge coupling unification in the context of string theory [43], these are related as [4] Combining Eqs. (21) and (22), this yields two conditions which allow us to derive new constraints on the U(1) R charge assignments. Using the phenomenological constraints obtained in Section II, we can re-express R Q1 and R Hu as In addition to the GS conditions arising from the mixed anomalies, there exists another anomaly coefficient, quadratic in U(1) R , which severely constrains the allowed charge assignments. 4 As a potential anomaly cannot be absorbed by the GS mechanism, this anomaly coefficient must vanish identically, This allows us to re-express the charge R H d as Since we are constructing the explicit high-energy model the comments on non-linear anomalies in [44] do not apply.
In summary, anomaly considerations give rise to three new constraints as stated in Eqs. (23), (24) and (26). The allowed charge assignments are thus determined by the integer parameters x, y, z, ∆ L 21 and ∆ L 31 as well as the charges R L1 and R θ .
In the following section we will further reduce the number of free parameters by demanding the discrete Z R 4 symmetry to arise from the breaking of the continuous U(1) R symmetry. Before proceeding with this central part of our paper let us however, first discuss the scale at which the flavon acquires its VEV. Thanks to the anomalous nature of the U(1) R symmetry, the Dine-Seiberg-Wen-Witten mechanism [15][16][17][18] radiatively generates a non-vanishing Fayet-Iliopoulos term proportional to the gravitational anomaly. The latter can in turn be related to the A SU(3)C −SU(3)C −U(1)R anomaly using the GS mechanism. The non-vanishing Fayet-Iliopoulos term then induces a flavon VEV such that [4] where we have calculated the colour anomaly using the constraints on the U(1) R charges derived above.
As previously stated, in order for the FN mechanism to explain the fermion mass hierarchies we must be able to identify ǫ with the Wolfenstein parameter, that is ǫ ∼ λ c ∼ 0.22. However one should bear in mind that the FN mechanism does not fix order-one coupling constants. This implies that one can allow for values of ǫ which vary by small amounts from λ c . Mindful of this and the possible choices for x and z, we can limit the allowed region for R θ within a small interval. Using the GUT scale value of the strong coupling, g C ∼ 0.72, we find conservatively As will be shown in Section IV, the requirement of a residual Z R 4 symmetry constrains R θ to odd-integer multiples of 1/4. Together with Eq. (28), this entails six possible values for R θ , namely R θ = −7/4, −5/4, −3/4, −1/4, 1/4, 3/4. We shall, however, not include all possibilities in our subsequent analyses since many of these solutions generate a highly FN suppressed µ-term. The exponent |n| = |n a | = |R Hu + R H d | in Eq. (14) can be determined from the anomaly condition of Eq. (24) as a function of R θ and z. With z = 0, 1, the four cases R θ = −7/4, −5/4, −3/4, 3/4 lead to a value of |n| ≥ 5. We will therefore only consider the two cases In this section we explicitly construct the U(1) R gauge completion of the discrete Z R 4 model. In order to identify FN models where the U(1) R symmetry is broken to a residual Z R 4 symmetry we must place additional constraints on the U(1) R charge assignments. To do so we consider the charge of the most general operator, with possible factors of the flavon field removed, Here i = 1, 2, 3 label the generations of fermions and n S denotes the number of a specific superfield S occurring in a given operator. For example the operator Q i H dDj has n Qi = 1, n H d = 1 and nD j = 1. Note that we have included three right-handed neutrinosN i in Eq. (30) in order to generate neutrino masses via the type I seesaw mechanism [45][46][47][48]. We point out that the presence of the standard model neutralN i does not alter the anomaly constraints derived in Section III, i.e. Eqs. (23), (24) and (26). Neither are the results of Section II affected by adding right-handed neutrinos. Beyond the relations of Eqs. (11), (12a) to (12c), (13a) to (13c) and (15) we get from demanding the Dirac neutrino mass terms L i H uNj to be generated in the superpotential (the second relation follows from the first). 5 Without loss of generality we can take RN 3 ≤ RN 2 ≤ RN 1 . Mgrav , and an extraordinarily large soft breaking scale m 3/2 of at least 500 TeV, see also [6]. Therefore, such a situation is highly unnatural, and we do not consider it any further.  Inserting all the constraints on the U(1) R charge assignments, we can re-express R total in terms of x, y, z, ∆ L 21 , ∆ L 31 , R L1 , R θ , a few irrelevant integers from Eq. (31) and the numbers n Si which specify the number of times S i occurs in a given operator. Having specified an operator by the sets of integers n Si , it is clear that not every term is allowed by the standard model gauge symmetry. Enforcing standard model gauge invariance translates into the following three constraints [4] SU (3) : SU (2) : where C, W ∈ Z. Together with the constraints on the charge assignments obtained in Section II, these gauge group constraints allow us to eliminate n Qi , n Li , nD i and nĒ i from Eq. (30). Then the total U(1) R charge of the most general SM neutral operator can be written as where Z indicates an integer which can be compensated by appropriate powers of the flavon field. So far we have not yet imposed the condition that the U(1) R symmetry gets broken down to the residual discrete Z R 4 symmetry of Table I. To this end, in the next step we demand that each operator allowed (forbidden) by Z R 4 be allowed (forbidden) at the level of the broken U(1) R symmetry. The explicit constraint for the latter is given below in Eq. (38). Considering R-symmetries, it is necessary to discuss operators arising in both the superpotential as well as the Kähler potential. With the charges given in Table I, the Z R 4 transformation property of any given operator can be parametrised as for superpotential operators , for Kähler potential operators , where N d denotes an integer, and S = 1 for a term in the superpotential while S = 0 for a term in the Kähler potential. The integer C 4 can then be chosen to take values between 0 and 3 and indicates whether a term is allowed (C 4 = 0) or forbidden (C 4 = 1, 2, 3) by the Z R 4 symmetry. Combining Eq. (36) with Eqs. (32) to (34), it is possible to eliminate n H d from Eq. (35). As a result we can re-express the total charge of an operator as Eq. (37) is our master equation which we scrutinise in the rest of this section. The integers C, W, nŪ i , nN i , n Hu , N d as well as C 4 and S parametrise a particular operator of the superpotential (S = 1) or Kähler potential (S = 0), which for C 4 = 0 ( C 4 = 1, 2, 3) is allowed (forbidden) under Z R 4 . Choosing special sets of these parameters, we discuss classes of operators and compare their compatibility with being allowed (for C 4 = 0) or forbidden (for C 4 = 1, 2, 3) at the level of the broken U(1) R symmetry, where the condition for being allowed is R total = N + 2R θ , for superpotential operators , Z , for Kähler potential operators . (38) This comparison provides constraints on the U(1) R charges [3R Q1 + R L1 ], [R H d − R L1 ] and R θ which will allow us to formulate simple conditions for Z R 4 to arise as a residual symmetry of the Froggatt-Nielsen U(1) R . In the following we focus on the class of operators which satisfy Their total U(1) R charge is given by We first study cases with C = 0 to derive constraints on R θ and [R H d − R L1 ]. Equipped with these results, we then consider C = 1 in order to constrain [3R Q1 + R L1 ].
(i) C = 0: A Z R 4 invariant operator has C 4 = 0 and a total U(1) R charge of Since we want complete agreement between the Z R 4 symmetry of Table I and the remnant symmetry after U(1) R breaking, this equation must hold for all N d . Comparing the above for arbitrary N d with the condition for a U(1) R allowed operator, see Eq. (38), it is clear that must be satisfied. This requirement removes the third term in Eq. (40), that is the one proportional to ( N d − C).
With this simplification, we consider Z R 4 violating operators with C 4 = 1 and a total U(1) R charge of Demanding that such a term be forbidden at the U(1) R level, tells us that [R H d − R L1 ] / ∈ Z. In order to extract further or more specific information on the charge assignments, it is necessary to consider various powers of the Z R 4 violating operators with C 4 = 1. From Eq. (36), one finds that the square of a (forbidden) superpotential term with C 4 = 1 will be allowed in the superpotential due to its resulting Z R 4 charge of 2. In contrast, only the fourth power of a (forbidden) Kähler potential term with C 4 = 1 will be allowed in the Kähler potential, while the square and the cube will still be forbidden by the Z R 4 symmetry. Translating these observations to constraints on the U(1) R charges we get from superpotential operators , from Kähler potential operators .
Consistency of these conditions requires 2R θ / ∈ Z. Together with Eq. (42), the most general choice for R θ consistent with a residual Z R 4 symmetry is an odd-integer multiple of 1 4 , and can be written as This in turn entails that [R H d − R L1 ] must also be an odd-integer multiple of 1 4 , hence (ii) C = 1: We proceed with determining the allowed values of 3R Q1 + R L1 . We reconsider Eq. (40), this time setting C = 1. The U(1) R charge of a Z R 4 invariant operator is given by Comparison with Eq. (38) immediately reveals that  We have constrained all three parameters in Eq. (40) (38) is fulfilled for all Z R 4 allowed (C 4 = 0) operators while it is not satisfied for all Z R 4 forbidden (C 4 = 1, 2, 3) operators, regardless of the values of the free integer parameters C, W, nŪ i , nN i , n Hu and N d .
Using notation similar to Refs. [4][5][6] we re-express the constrained parameters as, where both ∆ H and ζ are independent integers, and Eq. (48) is obtained through the use of Eqs. (23) and (24), namely, With x, y, z and 4R θ being integers, the combination ∆ L 21 + ∆ L 31 − z + 4R θ must be an integer multiple of 3. We collect all the information on the constrained charge assignments derived in this and the previous sections in Table II. The U(1) R charges of the standard model charged fields depend on the parameters x, y, z, ∆ L 31 , ∆ H , ζ and R θ , where the latter has to be an odd-integer multiple of 1 4 . The charges of the right-handed neutrinos must be consistent with Eq. (31), but remain otherwise unconstrained at this point. We discuss this in more detail in the next section.

V. NEUTRINO CONSTRAINTS
In this section, we discuss the possibilities of introducing neutrino masses to the model which are compatible with the residual Z R 4 symmetry. As Z R 4 includes R-parity, the only way to generate neutrino masses in the absence of right-handed neutrinos is through the effective Weinberg operator L i H u L j H u [49]. Being generated gravitationally, this term gives rise to a neutrino mass matrix of the form where S = 1 if it originates in the superpotential, and S = 0 if it arises from the Kähler potential. As the FN expansion parameter ǫ can only lead to further suppression, the absolute neutrino mass scale is bounded from above by the ratio between the square of the Higgs VEV and the gravitational scale. With v u ∼ m t ∼ 175 GeV and M grav ∼ 2.4 × 10 18 GeV, we obtain the maximum mass scale of the neutrinos to be of order 10 −5 eV if the term originates in the superpotential and even smaller if the Weinberg operator is generated in the Kähler potential. This result is in conflict with atmospheric neutrino oscillations observations which require the absolute neutrino mass scale to be bigger than 10 −2 eV. Hence, we do not consider this option any further.

A. Seesaw mechanism
To account for realistic neutrino masses we introduce three right-handed neutrinosN i and impose the type I seesaw mechanism [45][46][47][48]. This mechanism relies on a Majorana mass term for the right-handed neutrinos as well as a Dirac mass term which couples left-and right-handed neutrinos. Each of these terms can originate in either the superpotential (requiring R total = 2R θ +N) or the Kähler potential (requiring R total = Z). However, we argued in Section IV that the Dirac term should arise in the superpotential, and this requirement is in fact built into the charge assignments listed in Table II. As a consequence, we now show, the Majorana mass term cannot originate from the Kähler potential. Consider the square of the Dirac operator in Eq. (52). Since R θ is an (odd-)integer multiple of 1 4 , see Eq. (44), the squared Dirac term must have integer total U(1) R charge, i.e. R Li + R Lj + 2R Hu + RN i ′ + RN j ′ ∈ Z. However, using the U(1) R charges of Table II we have R Li + R Lj + 2R Hu = 1 2 + Z, so that Hence, the Majorana mass term of Eq. (51) can only arise from the superpotential, cf. Eq. (38). Thus both the Majorana and the Dirac mass terms are generated in the superpotential. The corresponding mass matrices are then of the form where the exponents must be non-negative integers for any choice of i, j = 1, 2, 3. Due to the orderings Assuming only moderate ǫ suppressions, the Majorana and Dirac masses take values around their natural scale, and the seesaw formula is applicable as long as max Inserting the matrices of Eq. (54) into the seesaw formula M ν = M D M −1 R M T D yields an effective light neutrino mass matrix of the form where, in contrast to Eq. (50), the exponent R Li + R Lj + 2R Hu − 2R θ can now take negative integer values. Hence, it is possible to end up with a phenomenologically acceptable neutrino mass scale above 10 −2 eV. In the next subsection we study Eq. (57) with the aim of extracting further constraints on the possible U(1) R charge assignments.

B. Neutrino masses and mixings
The neutrino mass matrix of Eq. (57) can be diagonalised by a unitary matrix V ij νL ∼ ǫ |RL i −RL j | which has the same ǫ structure as the left-chiral unitary transformation V ij eL required to diagonalise the charged lepton mass matrix. The singular values m i resulting from diagonalising Eq. (57) by V ij νL are related as As the ordering of the neutrino masses is still unknown, the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix can be written as where T indicates a matrix which permutes the singular values m i appropriately to make it consistent with either a normal or an inverted mass ordering. In accordance with Refs. [50][51][52], we can now parametrise the experimentally observed PMNS matrix in the following manner, where the 0, 1 indicates the allowed exponents. 6 Due to the democratic pattern of Eq. (60), U PMNS T −1 and U PMNS have identical ǫ structure, and Eq. (59) entails ǫ 0,1 ∼ ǫ |RL i −RL j | . Thus, the PMNS mixing constrains the U(1) R charges R Li such that only three different combinations of the parameters Disregarding the ordering of the masses in Eq. (58), which can be corrected by the permutation matrix T , the three possible choices for (∆ L 21 , ∆ L 31 ) are compatible with different neutrino mass spectra as follows: • (0, 0) allows for normal hierarchy, inverted hierarchy and degenerate neutrinos, • (0, −1) allows for normal hierarchy, • (−1, −1) allows for normal hierarchy and inverted hierarchy.
Turning to the absolute neutrino mass scale, it is given by M 33 ν of Eq. (57). With the charges listed in Table II, we find Solving for the exponent yields Thanks to R θ being an odd-integer multiple of 1 4 , the parameter Z is an integer. Its value is further constrained by ǫ as well as the absolute neutrino mass scale m ν abs . With three possible neutrino mass spectra [59] Table III. Notice that larger values of ǫ cannot be achieved once we restrict ourselves to R θ = ± 1 4 , see the end of Section III.

C. Collection of all constraints and resulting models
The charges of Table II depend on the parameters x, y, z, ∆ L 31 , ∆ H , ζ and R θ . As argued at the end of Section III, we constrain ourselves to cases where All remaining six parameters are integers which are restricted to the values where Z and ∆ L 21 must comply with Table III and Eq. (61), respectively. In particular, parameter sets for which the right-hand side of Eq. (68f) does not yield an integer are excluded. Moreover, as discussed below Eq. (61), only certain mass spectra are allowed for a given choice of (∆ L 21 , ∆ L 31 ). Taking into account all these constraints, we end up with a limited number of viable models which are defined by the sets of parameters listed in Table IV. In addition to the free parameters defined above the table shows the allowed neutrino mass spectra as well as the ranges for m ν abs and ǫ. Ignoring the right-handed neutrinos, we thus have identified 3 × 34 different phenomenologically acceptable FN U(1) R models which give rise to the residual Z R 4 symmetry of Table I. The explicit U(1) R charge assignments can be found in Appendix A.
We remark that the origin of the µ-term is model dependent. As all the parameters are fixed for a given model,   67) and (68a) to (68f) and Table III together with the corresponding mass spectra allowed by the given choice of (∆ L 21 , ∆ L 31 ). The last two columns show the possible ranges for m ν abs and ǫ. These two values are calculated from Eqs. (27) and (62) respectively, using the freedom in the parameter x but disregarding potential contributions from order-one factors.
all their subcases) yield a negative value of n, so that the µ-term necessarily has to arise via the Giudice-Masiero mechanism. All the other cases of Table IV (including all their subcases) have n > 0, so that both a Kim-Nilles as well as a Giudice-Masiero origin are possible.

D. Right-handed neutrino charges
Before concluding this section, we wish to comment on the charges of the right-handed neutrinosN i . As Eq. (57) does not depend on RN i , there exists no constraint from low-energy neutrino physics. However, as discussed above, consistency requires the charges to satisfy the conditions of Eqs. (55) and (56). 7 A further constraint can be imposed if one demands successful baryogenesis via leptogenesis [60]. In the case of "vanilla" leptogenesis, the mass of the lightest right-handed neutrino has to be larger than M lept ∼ 10 9 GeV [61][62][63], which translates into Since the mass scale of the Dirac neutrino term is necessarily smaller or equal to the Higgs VEV v u ∼ 175 GeV, the constraint of Eq. (56) is always satisfied. We are then left with the conditions of Eq. (55) yielding a lower bound on RN 3 , as well as the leptogenesis constraint and hence an upper bound on RN 1 . With the ordering RN 3 ≤ RN 2 ≤ RN 1 it is now straightforward to count the number of all distinct allowed right-handed neutrinos charge assignments for each model in Table IV.

VI. CONCLUSION
In this paper we have successfully constructed a viable set of flavour models with a gauged U(1) R family symmetry. We have determined these models such that they act as a U(1) R gauge completion of the Z R 4 symmetry. Through the use of the Green-Schwarz mechanism for anomaly cancellation and the insistence of the residual Z R 4 symmetry after U(1) R breaking we are able to constrain the free parameters of the theory. Furthermore, if one insists on phenomenologically acceptable quark and lepton masses and mixings at the GUT scale, as well as the inclusion of neutrino masses using the type I seesaw mechanism, we obtain unique sets of charge assignments under the original U(1) R symmetry. Although the right-handed neutrino charges remain undetermined, they are severely constrained by the required assumptions for a successful type I seesaw mechanism, as well as by the minimum right-handed neutrino mass required for successful leptogenesis.
Additionally we have discussed the origin of the µ-term in these models. We insist that it remains forbidden under the U(1) R symmetry and is dynamically generated during the breaking of Z R 4 to Z R 2 through either the Kim-Nilles or the Giudice-Masiero mechanism. While both mechanism have different origins the results in the context of the models presented are indistinguishable.
The result of this work is a set of 3 × 34 models presented in Appendix A. Adopting a U(1) R charge normalisation where the flavon field has charge R φ = −1, most models suffer from highly fractional charges. The most aesthetically pleasing ones are • case 3a of One of the key outcomes of this paper results from the insistence of the residual discrete Z R 4 symmetry. By promoting the FN U(1) to an R-symmetry and by fixing the charge of the flavon field to R φ = −1 we are left with the charge of the superspace coordinate θ as a free parameter. However this parameter is fixed by the residual symmetry, which in the case of a Z R 4 symmetry results in R θ = 2Z+1 4 , that is an odd-integer multiple of a quarter.
In this appendix we present the resulting possible charge assignments for each of the parameter values in Table IV, where y = −1, 0, 1 is displayed in Tables V to VII respectively. In the case where multiple x values are possible for a given case we use a, b, c and d after the case number, which is given in the first column of Table IV, such that the case is distinguishable. It should be noted once again that y = 0 corresponds to the scenario where the CKM matrix is in the best agreement with experiment without relying on tuning through the order-one coupling coefficients.  Table IV for the scenario where y = −1.    Table IV for the scenario where y = 1.