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Proton Hexality from an Anomalous Flavor U(1) and Neutrino Masses - Linking to the String Scale

We devise minimalistic gauged U(1)_X Froggatt-Nielsen models which at low-energy give rise to the recently suggested discrete gauge Z_6 symmetry, proton hexality, thus stabilizing the proton. Assuming three generations of right-handed neutrinos, with the proper choice of X-charges, we obtain viable neutrino masses. Furthermore, we find scenarios such that no X-charged hidden sector superfields are needed, which from a bottom-up perspective allows the calculation of g_string, g_X and G_SM's Kac-Moody levels. The only mass scale apart from M_grav is m_soft.


Introduction
In this paper, we consider low-energy discrete symmetries, N , as extensions of the SU(3) × SU(2) × U(1) gauge symmetry of the Minimal Supersymmetric Standard Model (MSSM). We focus on the case, where the N is the remnant of a spontaneously broken local gauge symmetry, in order to avoid potentially harmful gravity effects [1]. Such discrete symmetries originating in a gauge theory are called discrete gauge symmetries (DGSs) [2]. In Refs. [3,4], a systematic study was performed of all the DGSs resulting from Abelian, anomaly-free gauge symmetries, U(1) X , which leave the MSSM invariant. Specifically, the following assumptions were made in these studies 1 • The only light, low-energy fields are those of the MSSM. All beyond-the-MSSM fields are heavy.
• At least the following superpotential terms are N -invariant: where we have made use of the standard notation for the MSSM chiral superfields, see for example [6]. The invariance of the first three terms implies that the Nsymmetry, but not necessarily the original U(1) X , is family-universal.
Given these assumptions, the only possible DGS resulting from an anomaly-free U(1) X are the 2 -symmetry matter parity (M p ), the 3 -symmetry baryon triality (B 3 ) and the 6 -symmetry proton hexality (P 6 = M p × B 3 ) [3,4]. In Refs. [7,8], the U(1) X gauge charges were determined, which lead to a low-energy M p , B 3 , or P 6 , respectively. See also Refs. [9,10] for related work on the conditions for DGSs in GUTs.
It is now of great interest to see whether realistic flavor models for the Standard Model (SM) fermion masses and mixings can be constructed in each case. Employing the original U(1) X in a minimal Froggatt-Nielsen (FN) scenario [11] and using the Green-Schwarz (GS) mechanism [12] to cancel the U(1) X anomalies, a successful M p -model was constructed in Ref. [7] and its implications for suppressed proton decay were discussed in Refs. [13,14]. Later, a corresponding B 3 -model was constructed in Ref. [8], with a detailed discussion of the neutrino masses.
It is the purpose of this note to construct a P 6 -FN flavor model, in order to complete this program. Furthermore, from the phenomenological point of view, proton hexality is a very attractive symmetry. It combines the advantages of the M p and the B 3 models [4]: the lightest supersymmetric particle (LSP) is stable and the dangerous dimension-four and dimension-five proton decay operators are forbidden. We shall proceed analogously to Refs. [7,8] and refer the reader to these publications for an explanation of our notation and an introduction to for example the Giudice-Masiero/Kim-Nilles (GM/KN) mechanism [15,16].
There has been extensive previous work on anomalous flavor models employing the Green-Schwarz mechanism and with breaking slightly below the Planck scale, see for example Refs. [17,18,19,20,21,22]. However, we believe this is the first work on such a model aiming for a remnant "gauged" P 6 . There are also some non-anomalous flavor models with U(1) X breaking at the TeV scale [23,24,25,26,27,28,29].
This note is structured as follows: In Sect. 2, we discuss the constraints on the Xcharges which are not related to neutrino phenomenology. In Sect. 3, we then focus on the neutrino sector and how it fixes the X-charges; corresponding tables are given in Appendix B. In Sect. 4, we discuss the possibility and the implications of excluding Xcharged hidden sector superfields, enabling us to calculate the string coupling constant. We conclude in Sect. 5.

Non-Neutrino Constraints on the X-Charges
In the following we proceed as in Refs. [7,8] and consider only one flavon chiral superfield A, with U(1) X -charge X A = −1. In order to obtain a viable flavor model, the U(1) X charges of the P 6 -FN models must satisfy several phenomenological and consistency constraints. They must (a) reproduce phenomenologically acceptable charged SM fermion masses and mixings, see Ref. [30], (b) reproduce phenomenologically acceptable neutrino masses and mixings, (c) satisfy the Green-Schwarz mixed linear anomaly cancellation conditions (with gauge coupling unification), as well as guarantee that the mixed quadratic anomaly vanishes on its own, e.g. Ref. [7], (d) imply the desired low-energy DGS P 6 , i.e. give rise to the following discrete familyindependent 6 -charges for the MSSM chiral superfields [4,31]: and (as will be argued later) z N = 3 for the additional right-handed neutrino (SM singlet) chiral superfields.
Excluding the conditions (b) and (d) for a moment, it was shown in Table 1 of Ref. [7] that all 20 X-charges of the MSSM+N i superfields can be expressed in terms of nine real numbers. Note that for simplicity, we assume three generations of right-handed neutrinos, unlike in Ref. [7] where only two generations were introduced.
x = 0, 1, 2, 3, Here X F denotes the U(1) X -charge of the field F . A few comments are in order: • ∆ L 31 and ∆ L 21 can only take integer values.
• x is related to the ratio of the vacuum expectation values (VEVs) of the two Higgs as well as (see the argument in Item 3 in Sect. 3.1) the three X-charges of the righthanded neutrinos being half-odd-integer. Inserting the expression for X Q 1 of Table 1 in Ref. [7], we can rewrite this as where ∆ H , ζ ∈ . We thus impose proton hexality by trading the parameters X L 1 and ∆ L 21 of Eq. (2.1) for the integer parameters ∆ H and 3ζ + p. The resulting constrained X-charges are shown in Table 1. 3 Neutrino Constraints on the X-Charges

The Origin of P 6 Neutrino Masses
Next we take the remaining constraints (b) into account, i.e. the experimental data from the neutrino sector. To do so, let us first consider the possible sources of neutrino masses in a P 6 invariant FN scenario.
. From the observed atmospheric neutrino oscillations, we however know that the absolute mass scale must be at least 5·10 −2 eV. Thus the neutrino mass matrix cannot (solely) originate from the non-renormalizable operator L i H U L j H U . This is not the case if we allow for the mass scale which suppresses L i H U L j H U to be lower than M grav , see e.g. the model in Ref. [4]. Note that in the case where X L i + X L j + 2X H U < 0, the operator L i H U L j H U is generated from the Kähler potential via the GM/KN-mechanism in combination with the FN-mechanism, leading to an even stronger suppression by a factor of m soft 3. When enlarging the particle spectrum by three generations of right-handed neutrinos N i , i.e. particles which couple trilinearly to L i H U , a new possibility for the neutrino mass term arises. Since L i H U L j H U is P 6 -allowed and the term L i H U N j by definition as well, but L i H U is P 6 -forbidden, the right-handed neutrinos must carry a half-odd-integer X-charge. Thus the Majorana mass term N i N j is necessarily also P 6 -allowed. In Ref. [5], the possibility of DGSs which allow for L i H U N j but forbid N i N j and L i H U L j H U will be discussed.
Throughout this article, we consider the third possibility above as the only viable source of neutrino masses in our scenario. The flavon field A and the right-handed neutrinos N i have a lot in common. Apart from their U(1) X -charges, both are uncharged. But there are also certain important differences: 1.) After U(1) X breaking A will not carry any -charge, whereas the N i will. 2.) The flavon field A acquires a VEV, whereas the N i are assumed not to. This is just like the MSSM non-Higgs scalar fields, which are not supposed to acquire a VEV, in order to e.g. preserve color and/or electromagnetism. Note that A = ǫM grav , but N i = 0 is consistent with the requirement of SUSY being unbroken at ǫM grav , i.e.
In the discussion of the constraints on the X-charges coming from the neutrino sector, we have to distinguish between four cases. These differ in the origin of the superpotential terms L i H U N j and N i N j . Depending on the overall X-charge, the terms are either of pure FN origin or effectively generated via the GM/KN-mechanism in combination with the FN-mechanism. For the Majorana mass terms, the low-energy effective superpotential terms are 2 while for the Dirac mass terms we have The labeling of the four different cases is shown in the following table.
(This can be compared also to Table 5 of Ref. [7]: Case I contains their 1.+2., Case II 6., Case III 3. and Case IV 4.+5.) When determining the masses of the light neutrino degrees of freedom we have to diagonalize the 6 × 6 neutrino mass matrix (3.6) We have approximated the (1, 1) entry of the matrix above to be the 3 × 3 zero matrix, because we already concluded earlier [see below Eq. (3.1)] that M (ν) LH U LH U does not contribute substantially enough to the absolute neutrino masses.
Under the assumption that the ǫ-suppression is not able to compensate the gravitational scale M grav such that one arrives at m soft or H U (which would be ∼ 24 powers of ǫ), we see from Eqs. For Case (II.iii), the ǫ-suppression must lower H U ∼ 200 GeV down to the neutrino mass scale, in order to be phenomenologically viable. This corresponds to about 20 powers of ǫ and we do not consider it any further. In Case (II.ii) one would naturally, i.e. without finetuning among the submatrices M (D) and M (M) , expect the neutrino mass matrix to have six singular values (masses) of the same order; as for (II.iii), extreme ǫ-suppression is required to obtain three sub-eV neutrinos. Hence, we also discard Case (II.ii). For the rest of this article, we refer to Case (II.i) as Case II.
Regardless of the Case (I -IV), in the following the light neutrino mass matrix is derived from the see-saw mechanism [32,33,34,35] and is given as (discarding the contributions from L i H U L j H U ) For later convenience we change the basis of the right-handed neutrinos so that M (M) is diagonal. Such a basis transformation is unproblematic after U(1) X is broken. As discussed in Ref. [8], this basis transformation does not alter the ǫ-structure of M (D) in Eqs. (3.4) and (3.5). It is now straightforward to determine M (ν) for the upper four cases: Here we have made use of Table 1 and the definition ∆ L i1 ≡ X L i − X L 1 . Note that the dependence on the X-charges of the right-handed neutrinos drops out in Cases I and IV, as has been shown analytically in Ref. [7]. Thus the masses of the light neutrinos do not depend on the charges X N a . For Cases II and III one might naïvely expect that although the overall mass scale of the light neutrinos depends on the X N a , their mass ratios m 3 : m 2 : m 1 do not. The latter however is not true, as is shown explicitly for Case II in Appendix A. Making use of the orderings 3 Assuming X N 2 − X N 3 ≥ 1, the second largest neutrino mass would be suppressed by a factor of at least ǫ 4 compared to the heaviest neutrino. Even when including the effects of unknown O(1) coefficients, this suppression is too large to be consistent with the data (see Sect. 3.3). For Case II, we must therefore constrain the X-charges of the right-handed neutrinos by (3.13) for (normal and inverted) hierarchy and degeneracy, respectively (see Sect. 3.3). Similarly for Case III: Here one obtains the condition X N 1 = X N 2 for (normal and inverted) hierarchical light neutrinos, and X N 1 = X N 2 = X N 3 for degenerate scenarios.

Constraints from Neutrino Mixing
The ǫ-structure of the light neutrino mass matrix is determined by ∆ L 21 and ∆ L 31 . We have M Both types of matrices are diagonalized by a unitary transformation U with Case I : Case II : As mentioned above and discussed in greater detail in Appendix A, the ratios of the light neutrino masses depend on the X-charges of the right-handed neutrinos in Cases II and III [see Eqs. (A.16) and (A. 17)]. Recalling the orderings X L 3 ≤ X L 2 ≤ X L 1 and respectively. In order to compare the theoretically derived mixing matrices U (ν) with neutrino phenomenology, it is convenient to define the matrix U (ν) ≡ U MNS † , so that Here m 1 ≤ m 2 ≤ m 3 for normal and m 3 ≤ m 1 ≤ m 2 for inverted ordering of the neutrino masses, see e.g. Ref. [36]. U MNS is the Maki-Nakagawa-Sakata matrix [37] for mixing in the lepton sector. Working in a basis with diagonal charged leptons, cf. Ref. [8], this mixing is solely due to the neutrino sector. Comparing Eqs. (3.15, 3.21), we can easily determine the relation between U (ν) and U (ν) and thus the theoretically predicted structure of the MNS matrix for the various scenarios: • Considering the Cases I & II and a normal neutrino mass ordering, we simply have • while an inverted mass ordering leads to • For Cases III & IV, we similarly find that for a normal neutrino mass ordering

24)
• and for an inverted mass ordering we obtain severe constraints on the possible values for ∆ L i1 from the experimentally allowed ǫ-structure of the MNS matrix [8] Here, multiple possibilities for the exponents of ǫ are separated by commas. Depending on T ... we have four different equations for The resulting ǫ-structures of U (ν) are shown in Table 2 together with the compatible values for the pairs (∆ L 21 , ∆ L 31 ). Notice that due to the ordering Having derived the constraints on the parameters ∆ L i1 from neutrino mixing, we must also satisfy the second condition of Eq. (2.4), which states that ∆ L 21 + ∆ L 31 − z must not be a multiple of three. As mentioned earlier, we choose to work with z = 1 in order to have the µ-term generated by the GM/KN+FN-mechanism. Therefore the choice (∆ L 21 , ∆ L 31 ) = (−1, −1) is incompatible with the requirement of P 6 conservation, and in the remainder of this article we -of course -do not consider this P 6 violating solution.
We conclude the discussion of the neutrino mixing with some observations regarding the CHOOZ [38] mixing angle, θ 13 . In our notation this angle is parameterized by the entry ǫ 0,1,2,... in the mixing matrices U (ν) of Table 2. As the CHOOZ angle is small, one should try to find solutions in terms of (∆ L 21 , ∆ L 31 ) where this entry is ǫ 1 or ǫ 2 . Comparing with the four matrices in Table 2, we see that a normal mass ordering with (0, −1) or (−1, There exist of course other possible explanations for the smallness of θ 13 . For example, in Ref. [39] this is achieved by separating the effective neutrino mass matrix as a sum of two parts; each contains only a 2 × 2 block and is of rank one. Alternatively, there is a plethora of models adopting non-Abelian discrete symmetries like e.g. A 4 [40,41,42,43], ∆ (27) [44,45], S 3 [46,47], S 4 [48,49], 7 ⋊ 3 [50], P SL 2 (7) [51] to give rise to the tribimaximal mixing pattern [52], in which θ 13 is exactly zero.

Constraints from Neutrino Masses
Before discussing the Cases I -IV individually, some general remarks concerning the magnitude of the three light neutrino masses are in order. We shall combine the results of the solar [53,54], atmospheric [55], reactor [56], and accelerator [57] neutrino oscillation experiments, 4 as well as the upper bound on the absolute neutrino mass scale originating from the kinematic mass measurements [60]. This leads to three possible scenarios, see e.g. Refs. [36,61]: Assuming a (normal or inverted) hierarchical scenario, the absolute upper neutrino mass scale m ν abs ≡ max (m 1 , m 2 , m 3 ) is about 0.05 eV, a value which is consistent with the cosmological upper bound on the sum of the neutrino masses, i m i ≤ 0.7 eV [62,63]. For an inverted hierarchy, two neutrinos must have a mass around this scale, while the third neutrino is much lighter. As the suppression between the masses of the two heavier neutrinos is given by [cf. Eqs. (3.16-3.19), respectively] Case I : Case III : For the degenerate case, m ν abs can take values within the range [0.2 eV, 2.2 eV], where the lower end of the interval is estimated such that it satisfies the condition 0.05 eV ≪ m ν abs . Concerning the cosmological bound, high values for the neutrino masses are more or less disfavored, depending on which cosmological observations are included in the derivation of the bound [62,63]. We return to this issue in the discussion of our results. Within our P 6 FN-framework, the degenerate scenario is only possible if we have ∆ L 21 = ∆ L 31 = 0. This in turn requires a certain amount of finetuning among the O(1) coefficients in order to get correct neutrino masses and mixing.
We now turn to the discussion of each of the individual Cases I -IV. In our calculations we take M grav = 2.4 · 10 18 GeV, 100 GeV ≤ m soft ≤ 1000 GeV, (3.32) and H U ∼ m t = 175 GeV. 5 In addition we assume z = 1, as well as Eq. (2.2).
(I) From Eq. (3.8) and the ordering ∆ L 31 ≤ ∆ L 21 ≤ ∆ L 11 = 0, we get the absolute neutrino mass scale as Solving for the exponent yields -For a normal or inverted hierarchical scenario m ν abs ≈ 0.05 eV. Inserting this and the limiting values for ǫ, we arrive at the following allowed range where the lower value corresponds to both small x and small m ν abs . Thus we have for the degenerate neutrino scenario a value which is compatible with all x = 0, 1, 2, 3. x = 0 leads to a neutrino mass scale of m ν abs ≈ 1.7 eV and x = 3 to m ν abs ≈ 0.5 eV. Taken at face value, both are in conflict with the cosmological upper bound on the sum of the neutrino masses. However, O(1) coefficients can alleviate this tension. In the comment column of Table 6 we give the naïve sum of the neutrino masses assuming all O(1) coefficients are exactly one.
All possible sets of parameters (∆ L 21 , ∆ L 31 , 3ζ + p, ∆ H , x) are summarized in Table 3. The compatibility with the various neutrino mass scenarios is denoted by the symbol . Note that by virtue of Eq. (2.4), the first three parameters are not independent of each other. As pointed out earlier, we assume z = 1. The allowed values for y = −1, 0, 1 remain unconstrained by the neutrino sector. Altogether we can find 4 × 4 × 3 = 48 distinct sets of X-charge assignments (including also less favored possibilities), which fulfill the constraints of Tables 1+3. They are given in Appendix B, Table 5. Table 3: The sets of parameters which are compatible with neutrino phenomenology in Case I, where the terms L i H U N j and N i N j have pure FN origin. We assume z = 1. The hierarchical scenarios slightly prefer small x and disfavor large (denoted by the parentheses). The parameter y = −1, 0, 1 remains unconstrained.
For Case I, the X-charges of the right-handed neutrinos are not directly constrained by neutrino phenomenology. Recall however that this case requires by definition Table 1 and z = 1 this translates into Depending on ǫ, the first term on the right-hand side is numerically between 22.1 and 24.7. With the latter, i.e. for the case where ǫ = 0.222, we arrive at the upper bounds of 2∆ N 1 − ∆ N 3 ≤ 20 for the degenerate and 2∆ N 1 − ∆ N 3 ≤ 21 for the hierarchical case, respectively. In Sect. 4, we will constrain the ∆ N i by requiring the absence of X-charged hidden sector superfields.
It is worth noting that thermal leptogenesis requires the lightest right-handed neutrino to be not too light: M (M) 11 4 × 10 8 GeV if the spectrum is hierarchical (no close states) but otherwise with rather conservative assumptions [64]. Even though the considerations here do not determine the X-charges of N i , and hence their masses, we do obtain quite restrictive constraints once we require that all anomalies are canceled without introducing additional (hidden) fields charged only under U(1) X but not the standard model. See Appendix B for more details.
(II) Proceeding with Case II, we obtain from Eq. (3.9) and the orderings ∆ The hierarchical scenarios require   Table 1, we can define an integer n as Since X N i +X N j < 0, the X-charges of the right-handed neutrinos must be negative, hence n ≥ 0. Another condition is that Inserting respectively Eqs. (3.43,3.45) shows that this is automatically satisfied. However, there is yet another relation to be met. Recall that for the see-saw mechanism we require M (D) ≪ M (M) . This provides us with a lower bound on X N i , as can be seen in the following. From Eqs. (3.3) and (3.4), the lightest righthanded neutrino has a Majorana mass of the order m soft ǫ − 2X N 3 and the heaviest Dirac mass is of order As a conservative estimate, we take m soft = 1000 GeV, yielding m soft H U ≈ ǫ −1 for the left-hand side. Therefore In Table 4, we give all sets of parameters (∆ L 21 , ∆ L 31 , 3ζ + p, ∆ H , x, n), which comply with the phenomenology of neutrino masses and mixings for Case II. We assume z = 1, and the parameter y = −1, 0, 1 remains unaffected by the neutrino sector.
Compared to the analogous table for Case I, we have added the parameter n ∈ AE, which is defined by the X-charge of the right-handed neutrino N 3 [cf. Eq. (3.46)] and determines the parameter ∆ H . Limiting ourselves to Case II restricts the allowed values for n. Altogether we can thus find [(4 × 8 + 3 × 9) + (3 × 9 + 2 × 10) + (3 × 9 + 2 × 10) + (3 × 9 + 2 × 10)] × 3 = 600 sets of X-charge assignments, including also less favored possibilities. Some of these charge assignments, however, are identical due to the first two rows of Table 4. There are 504 distinct sets of Xcharges. A selected subset resulting from Table 4 is given in Appendix B, Table 7. For the relevant criteria see the next Section.
It is worth noting that there is a constraint from neutrino oscillation due to the presence of right-handed states. The most stringent limit comes from appearance experiment searches: ν µ → ν ℓ , for ℓ = e, τ . Because we required the right-handed neutrino masses to be much higher than the light neutrino masses, Eq. (3.48), the oscillation probability is averaged out, and hence we obtain an upper limit on the mixing angle. The effective "sin 2 2θ" must be less than about 3 × 10 −4 [65, 66, 67, 68, 59]. In terms of the mixing matrices, the limit is therefore |U µi U * ℓi | ≈ and thus the soft scale has to be extraordinarily large, at least 500 TeV. This renders Case III highly unattractive. We will therefore not elaborate on the possibility of the Dirac mass matrix being generated by GM/KN+FN any further.
(IV) As for Case III.
4 An X-charged Hidden Sector?
The GS cancellation of chiral anomalies often requires the introduction of further Xcharged matter fields, which are singlets under the Standard Model gauge group, i.e. hidden sector superfields, for examples see Refs. [7,8,69]. 6 But as we now explain, in our P 6 conserving FN study, it is possible to have GS anomaly cancellation without exotic, hidden sector, matter. In such a case, anomaly considerations open up a window on the underlying string theory. It should be stressed that the condition of no further X-charged matter is an option which does not affect any of the previous considerations. Two of the GS conditions are given as 7 [7] A where the positive real parameters k ... are the affine or Kač-Moody levels, which take integer values for non-Abelian gauge groups. A ... denote the anomaly coefficients, with G standing for "gravity", C for SU(3) C , and X for U(1) X . The k ... are related to the corresponding gauge coupling constants at the unification scale These 2 + 2 equations give Assuming, as in deriving Table 1, that all non-MSSM superfields are color singlets, we have

4)
Here and in Eq. (4.2), we have used the standard GUT-normalization of non-Abelian groups with generators t a such that tr[t a t b ] = 1 2 δ ab . With Table 1, we get e.g. A CCX = 3 2 (6 + x + z), (4.7) So despite the 17 MSSM X-charges being known, cf. Tables 3 and 4, we cannot give numerical values for {g string , g X , k X , k C }, since the ∆ N i , A hidden GGX and A hidden XXX are still 6 However, in Ref. [7], with three instead of two generations of right-handed neutrinos and k C = 3 the GS anomaly cancellation conditions could also have been satisfied without exotic matter. 7 We differ from Ref. [7] by a factor of 3 in the denominator of the third ratio.
unknown. But now let us suppose that the left-chiral MSSM superfields, as well as the N i and the flavon A are the only X-charged superfields. Hence A hidden GGX and A hidden XXX vanish. 8 We can then scan all 48+504 X-charge assignments, defined by the parameters {x, z, ∆ L 21 , ∆ L 31 , ∆ H }, for solutions to the fourth equality of Eq. (4.3) with the requirement of k C being an integer: As pointed out above, the integers ∆ N i are already constrained. Besides the required • For Case I, see below Eqs. (3.39) and (3.40),

11)
• and for Case II, see Eqs. (3.13,3.14,3.46), with n given in Table 4, We then find that the 48 sets of Case I are all in accord with k C = 3. The required values for i ∆ N i are given in Table 6. The conditions on ∆ N i however do not determine the X-charges of the right-handed neutrinos uniquely; see Appendix B for a complete list of the remaining possibilities in each case. On the other hand, there exist six cases (# 25, 26, 27, 37, 38, 39) which are also compatible with k C = 2. In these models, the constraints on ∆ N i fix their individual values uniquely, cf. Table 6. Turning to Case II, the X-charges of the right-handed neutrinos have to satisfy stronger constraints due to Eqs. (4.12,4.13). Demanding Eq. (4.9), only 24 of the 504 models in Table 4 survive; they are displayed in Table 7. In all 24 cases we have k C = 2, and ∆ N i is fixed uniquely as given in Table 8. A brief comment about the number of possible models before and after imposing Eq. (4.9) is in order. Excluding the right-handed neutrinos, we start with 48 distinct sets of X-charge assignments in Case I and 504 in Case II. This huge difference is due to the fact that in Case II the dependence of the effective neutrino mass matrix M (ν) on the right-handed neutrinos N i , see Eq. (3.9), allows for a variation of ∆ H parameterized by n. In Case I, such a dependence and thus a similar parameter is absent. Taking the right-handed neutrinos into account, the dependence of M (ν) on N i strongly limits the possible X-charges for N i in Case II [cf. Eqs. (4.12,4.13)], whereas for Case I, X N i can be chosen from an interval [cf. Eqs. (4.10,4.11)]. When it comes to finding solutions to Eq. (4.9), this freedom of assigning X N i in Case I allows each of the 48 sets of X-charges to be consistent without an X-charged hidden sector. In Case II, the situation is much more constrained, reducing 504 models to only 24 viable ones.
Having determined the Kač-Moody levels k C which are consistent with the assumption of no exotic X-charged matter, we can calculate the string coupling constant g string from Eq. (4.2). Inserting g C [M string ] ≈ g C [M GUT ] = 0.72 we get g string ∼ 0.88 , for k C = 3 , (4.14) g string ∼ 0.72 , for k C = 2 . From k C we can obtain the other Kač-Moody levels of G SM from the gauge coupling unification relation 9 which adopts the Y -normalization with Y L = 1/2, and has already been implemented when deriving Table 1, cf. Ref. [7]. Thus, the models of Case I with k C = 3 have k W = 3 and k Y = 5, while those with k C = 2 (i.e. six models of Case I and all models of Case II) demand k W = 2 and k Y = 10/3. The question arises whether Kač-Moody levels k C and k W higher than 1 can be obtained from string model building. Actually, such models have been considered, e.g. [73,74,75], but a systematic investigation of this issue is lacking. Nevertheless, there are indications that higher Kač-Moody levels might occur rather generically, see e.g. Ref. [76]. Also, from the phenomenological point of view, models with higher levels have already been discussed, e.g. in Ref. [77]. This is important regarding the possible representations for the Higgs fields in the theory [78,79,80].
In addition to the Kač-Moody levels of G SM , we can, from a bottom-up perspective, calculate the U(1) X gauge coupling constant g X in those cases, where the ∆ N i are uniquely fixed, i.e. for all models with k C = 2. Evaluating the second equality of Eq. The results for each of the 6+24 models with uniquely fixed X-charge assignments are listed in Tables 6+8. We point out that the k X corresponding to the above determined g X are quite high integers, e.g. 8839 for # 6 of Case II. This underlines that the scenarios without X-charged exotic matter are to be taken more as an existence proof rather than concrete models.

Discussion and Conclusion
In this note, we have devised FN models in which the anomalous U(1) X gauge symmetry is broken down to the discrete 6 -symmetry, proton hexality. The masses of the light neutrino states are generated by introducing right-handed neutrinos N i and applying the see-saw mechanism. For Case I, the Majorana mass terms of N i originate only from the FN-mechanism, while for Case II they result effectively from a combination of the FN-and the GM/KN-mechanism. Requiring phenomenologically acceptable fermion masses and mixings, the GS mixed anomaly cancellation conditions with gauge coupling unification, as well as the low-energy remnant discrete symmetry P 6 , we are led to 48 X-charge assignments for Case I (cf. Table 3) and 504 X-charge assignments for Case II (cf. Table 4). Under the assumption of no exotic X-charged particles, all 48 sets of Case I, but only 24 of the 504 sets of Case II are compatible with the GS anomaly cancellation conditions. The X-charges of the resulting 48+24 sets are shown in Tables 5 and 7. Furthermore, we can determine the Kač-Moody levels of G SM in these models. For k C = 2, the X-charges of the right-handed neutrinos are fixed uniquely. This enables us to calculate the gauge coupling constant g X of U(1) X in these cases.
All results are listed in Tables 6 and 8 together with the obtained light neutrino mass spectrum, the maximal denominator of the X-charges, as well as some additional comments on each of the models. We emphasize here that all are phenomenologically acceptable because the unknown O(1) coefficients allow a certain flexibility. However, if asked to select "preferred" models, one can consider the following three criteria: (1) "nice" CKM matrix, (2) naturally small CHOOZ mixing angle, (3) small maximal denominator for the X-charges.
Sets with y = 0 lead to our preferred ǫ-structure of the CKM matrix, see Sect. 2. These amount to one third of all the models. The CHOOZ mixing angle corresponds to the (1, 3) entry of the MNS matrix. This is naturally suppressed in our models if ∆ L 31 = −2 (U MNS 13 ∼ ǫ 2 ) or ∆ L 31 = −1 (U MNS 13 ∼ ǫ), see the end of Sect. 3.2. Altogether 24+21 sets lead to a naturally small CHOOZ angle by virtue of the U(1) X charge assignments. Finally, we have labeled the 10+3 models with a maximal denominator ≤ 54 by "denom." in the comments. From the aesthetical viewpoint, the most appealing set is # 6 of Case II (Table 8) where all X-charges are multiples of 1/6. This model features a small CHOOZ angle but, unfortunately, a not so nice CKM matrix. With regard to criterion (3), we however emphasize that models with highly-fractional X-charges are very common, especially when fulfilling phenomenological constraints, see Ref. [81].
Looking for models which satisfy all of the above three criteria, we find that -remarkably enough -only one remains: namely # 32 of Case I (Table 6). This model has a normal hierarchical neutrino mass spectrum with U MNS 13 ∼ ǫ, the maximal denominator of the X-charges is 30. Without X-charged hidden sector matter, k C = 3 and i ∆ N i = 18, leading to 16 distinct X-charge assignments for the right-handed neutrinos, cf. Appendix B.

A X N a -Dependence of the Neutrino Masses
For Case II [cf. Eq. (3.9)], the Dirac and the Majorana mass matrices can be written as with A ≡ H U ǫ X H U and B ≡ m 3/2 . The dimensionless coefficients α ij and β ij are of order one. In our basis, M (M) ij and thus β ij is diagonal. With this notation the effective light neutrino mass matrix reads In the last step we have defined a ik ≡ α ik √ β kk ǫ X L i +2X N k . The light neutrino masses m = A 2 B λ can now be obtained from the characteristic polynomial 10 of M (ν,II) ,

22
C 2 = p 1 p 2 (a 2 33 + a 2 32 + a 2 31 ) + p 1 p 3 (a 2 23 + a 2 22 + a 2 21 ) + p 2 p 3 (a 2 13 + a 2 12 + a 2 11 ), (A.5) As a ik ∼ ǫ X L i +2X N k , the order of the coefficients C ... can be readily determined. With we get C 3 λ + C 2 = 0, which yields where "equal" applies only if X L 2 = X L 3 and X N 2 = X N 3 . Similarly, the order of the second singular value is derived from the quadratic and the linear term of Eq. (A.3) where "equal" applies only if either X L 2 = X L 3 and X N 2 = X N 3 or X L 1 = X L 2 and X N 1 = X N 2 . Finally, the order of λ 1 is obtained from the linear and the constant term where "equal" applies only if X L 1 = X L 2 and X N 1 = X N 2 . This yields the following ratios for the light neutrino masses Analogously, we obtain for Case III that
For k C = 2, the ∆ N i are uniquely fixed and given in Table 6. It is interesting to note that the lower limit from thermal leptogenesis, M 7. We observe that there is only a small number of combinations allowed within this limit [e.g. for i ∆ N i = 20 only (7, 7, 6) is okay]. On the other hand, some of the solutions above predict no hierarchy between N 1 and N 2 , and the bound may be less severe, e.g. 2 × 10 7 GeV in Ref. [82]. In the extreme case of resonant enhancement, one can allow for even TeV scale right-handed neutrinos [83].
Case II is treated similarly. However, displaying explicitly the 504 sets of X-charges which are hinted at in Table 4 would fill more than 12 pages. We content ourselves with presenting those 24 models which are consistent without X-charged exotic matter. They are given in Tables 7 and 8. Small maximal denominators of the X-charges are obtained for cases # 6 (6), # 7 (30), # 9 (42).      Table 6: The features of the X-charge assignments in Table 5 (Case I). In the comments we state the reason for preferring individual cases: "CKM" means that this model naturally exhibits a nice CKM matrix, i.e. y = 0. "CHOOZ" refers to a naturally small CHOOZ angle: sin θ 13 ≈ ǫ |∆ L 31 | , with |∆ L 31 | = 1, 2. We write "denom." to label cases where the X-charges have a maximal denominator ≤ 54. For the degenerate scenarios we show the naïve sum of the neutrino masses, i m i , without O(1) coefficients. Assuming no exotic matter, the three ∆ N i are uniquely fixed for k C = 2, unlike for k C = 3. 26 2243 210  Table 8: The features of the X-charge assignments in Table 7 (Case II). In the comments we state the reason for preferring individual cases: "CKM" refers to a nice CKM matrix, "CHOOZ" to a naturally small CHOOZ angle (|∆ L 31 | = 1, 2), and "denom." labels models where the X-charges have a maximal denominator ≤ 42.